STEVEN VOGEL
Life in Moving Fluids
THE PHYSICAL BIOLOGY
OF FLOW
Second Edition
Revised and Expanded
Illustrated by
Susan Tanner Beety
and the Author
PRINCETON UNIVERSITY PRESS
PRINCETON, N.J.
Copyright © 1994 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
Chichester, West Sussex
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Vogel, Steven, 1940-
Life in moving fluids : the philosophical biology of flow /
Steven Vogel.—2nd ed., rev. and expanded.
p. cm.
Includes bibliographical references and index.
ISBN 0-691-03485-0
ISBN 0-691-02616-5 (pbk.)
1. Fluid mechanics. 2. Biophysics. I. Title
QH505.V63 1994
574.19'1—dc20 93-46149
This book has been composed in Baskerville
Princeton University Press books are printed on acid-free
paper and meet the guidelines for permanence and
durability of the Committee on Production Guidelines for
Book Longevity of the Council on Library Resources
Second printing, and first paperback printing, 1996
Printed in the United States of America
by Princeton Academic Press
10 98765432
To the Department of Zoology of Duke University
as it existed from the mid-sixties to the mid-nineties—
a cohesive oasis of intellectual stimulation, academic excellence,
and interpersonal support and sensitivity
Contents
Preface to the New Edition
Acknowledgments
Chapter k
Chapter 2.
Chapter 3.
Chapter 4.
Chapter 5.
Chapter 6.
Chapter 7.
Chapter 8.
Chapter 9.
Chapter 10,
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Remarks at the Start
What Is a Fluid and How Much So
Neither Hiding nor Crossing Streamlines
Pressure and Momentum
Drag, Scale, and the Reynolds Number
The Drag of Simple Shapes and Sessile Systems
Shape and Drag: Motile Animals
Velocity Gradients and Boundary Layers
Life in Velocity Gradients
. Making and Using Vortices
. Lift, Airfoils, Gliding, and Soaring
. The Thrust of Flying and Swimming
. Flows within Pipes and Other Structures
. Internal Flows in Organisms
. Flow at Very Low Reynolds Numbers
. Unsteady Flows
. Flow at Fluid-Fluid Interfaces
. Do It Yourself
List of Symbols
Bibliography
and Index of Citations
Subject Index
ix
xiii
3
16
32
50
81
106
132
156
174
204
230
262
290
308
331
362
378
398
403
407
441
Preface to the New Edition
About a dozen years ago, calling up a degree of hubris that I now find
quite inexplicable, I wrote a book about the interface between biology and
fluid dynamics. I had never deliberately written a book, and I had never
taken a proper course in fluids. But I had learned through teaching—both
something about the subject and something about the dearth of material
that might provide a useful avenue of approach for biologist and engineer.
Each seemed dazzled and dismayed by the complexity of the other's
domain. The book happened in a hurry, in a kind of race against the
impending end of a sabbatical semester, and in a kind of mad fit of passion driven
by the simple realization (and astonishment) that it was actually happening.
The reception of Life in Moving Fluids turned out to surpass my most self-
indulgent fantasies—it reached the people I had hoped to reach, from
ecologists and marine biologists to physical and applied scientists of various
persuasions, and it seems to have played a catalytic or instigational role in
quite a few instances. Quite clearly the book has been the most important
thing of a professional sort that I've ever done; certainly that's true if
measured by the frequency with which the first punning sentence of its
preface is flung back at me. (That my writing has been more important
than my research in furthering my area of science suggests that doing
hands-on science, which I enjoy, is really just a personal indulgence—quite
a curious state of affairs!)
But the book was done quickly, and I was so concerned about keeping my
feet and mouth decently distant when talking about physical matters that I
barely realized how thin was my coverage of the biology. Some omissions
got apologetic mention, others were quietly given the blind eye, and a lot
represented simple personal innocence. Mistakes were made, most of
which were not accidental, and people wrote to point out (always kindly)
the errors of my ways. Mistakes and ambiguities became particularly
distressing when I found them contaminating the primary literature—that's
not the ideal measure of a book's influence!
Doing a new version is an enormous luxury, one afforded only a small
fraction of authors of instructional material. In the present case,
correcting errors has turned out to be the smallest part of the task. What I've been
able to do is to rewrite the book with what was almost entirely lacking
before—a sense of who would use it and what role it would serve. In effect,
I now have a criterion by which to judge appropriate content, level, and so
forth. I'm still trying to make something that serves a variety of roles. The
primary one, as before, is as guide for the biologist who needs to know
IX
PREFACE TO THE NEW EDITION
something about fluids in motion. But I've given more attention to the
problem of the biologist or engineer who wants to know a little about what
people interested in biological fluid mechanics have been up to. That's what
most of the additional material, the near doubling of the number of words,
is about. In the earlier version, the three hundred or so references
represented the full depth of my plumbing of the relevant literature. In the
present one, the seven hundred-odd references are a culling from several
thousand sources given some degree of attention. Even so, I'm
uncomfortably aware that the job has been a bit superficial and spotty. Part of that
reflects limitations of time and energy; part represents the positive
conviction that compendia and textbooks are different creatures and that this
intends to be one of the latter. But I must f orthrightly admit that omissions
don't always represent informed or even specific judgments—the book at
best gives the flavor of a very heterogeneous area of inquiry and has a story
line driven by the physical rather than biological content.
Certain topics have been deliberately omitted to maintain the intended
level of presentation, an entry-level work for people with backgrounds
typical of biologists. Vorticity, stokeslets, potential flow, the Navier-Stokes
equations—none lacks biological relevance, but usefully lacing them into
the present discussion didn't seem practical. At least I would have had to
violate my first rule for writing—explain, don't just mention. And some
perfectly biological material just got too complicated and specialized, in
particular some of the fancier aspects of swimming and flying. For the
latter, I've tried to direct the reader toward other sources.
Other topics were omitted as judgments of scope and for simple reasons
of space. My focus as a biologist is on organisms, not cells, molecules,
habitats, or ecosystems. Fluid mechanics is quite as relevant at levels of
organization other than organismal, but the present book simply doesn't
worry much about them. Thus for eddy viscosity, for much on gravity
waves, for atmospheric, oceanic, lacustrine, and other large-scale motion,
for diffusion with drift and flows where the mean free path of molecules is
significant, for non-Newtonian and intracellular flows, for convective heat
transfer—for these the reader will have to turn elsewhere. Again, I've tried
to suggest some appropriate sources. Finally, the appendices on techniques
of the earlier edition have disappeared. I just couldn't figure out how to do
them justice in a reasonable space, and I think we're now at a stage at which
the proper vehicle is some network-accessible bulletin board that permits
interaction and continuous alteration.
On the other hand, the new version has gained whole new topics, notjust
a lot more biological detail and citations. Swimming has surfaced, pumps
are now of prime concern, blood flow is no longer dismissed with some
sanguine phrases, unsteady flows and the acceleration reaction get a
proper start, events at the air-water interface get more than vaporous
X
PREFACE TO THE NEW EDITION
mention, jet propulsion isn'tjust recoiled from, Peclet number is present if
perhaps peculiarly done, and Froude propulsion efficiency is pushed with
dispatch if not great efficiency.
In the preface to the earlier version I offered to send my accumulated
teaching material to anyone who wrote to me. Quite a few people took
advantage of the offer, and I make it again. The set of problems proved
more useful than the other items, so problems are what you'll get if you
write (answers, too, unless I get suspicious about your motives).
In going through the bibliography I find citations of no fewer than forty-
two people who were part of one or another class in front of which I said my
piece. I happily admit the grossest bias in choosing cases and sources,
mainly because of my joy in discovering that such large-scale favoritism has
been possible. I hope these written words have some similar effect.
XI
Acknowledgments
A large number of people have given suggestions and advice for this
rewriting; that so many took so much time is a most pleasant reminder that
the book has been useful. The ones I can specifically recall, and to whom I
express my gratitude, are Sarah Armstrong, Douglas Craig, Hugh
Crenshaw, Mark Denny, Robert Dudley, Olaf Ellers, Charles Ellington,
Shelley Etnier, Matthew Healy, Carl Heine, Mimi Koehl, Anne Moore,
Charles Pell, Roy Plotnick, John J. Socha, Lloyd Trefethen, Vance Tucker,
Jane Vogel, Stephen Wainwright, and Paul Webb. In addition, I appreciate
the continuous flow of advice and support that has come from the
members of the comparative biomechanics and functional morphology group
("BLIMP") of the Duke Zoology Department. All suggestions were actually
taken seriously, and a few suggestions were actually taken. At Princeton
University Press, Emily Wilkinson stoked the fires of the project, keeping it
off the back burner, and Alice Calaprice steered me around turbulence
and stagnation points and dealt deftly with both my allusions and illusions.
The following figures have been redrawn from existing material. The
author gratefully acknowledges permission for reuse from the copyright
holders (for previously published material) and the original creator (for
other material). Figure 3.8: David M. Fields and Jeannette Yen; 4.6, 14.5b:
Marine Biological Laboratory (Biological Bulletin)', 4.11a: Lisa S. Orton;
6.1: Cornell University Press (Aerodynamics by T Von Karman); 6.5a:
Barbara A. Best; 6.5b: Douglas A. Craig; 6.6, 11.12: The Company of
Biologists, Ltd. (Journal of Experimental Biology); 6.11b, 10.9a, 10.9b: American
Society of Limnology and Oceanography (Limnology and Oceanography);
7.5: Springer-Verlag (Journal of Comparative Physiology); 8.6; American
Society of Zoologists (American Zoologist); 9. la, 9.1b: Birkhauser Verlag (Swiss
Journal of Hydrology); 9.1e, 15.3: John Wiley and Sons, Inc. (Fresh Water
Invertebrates of the United States by R. W. Pannak); 9.2: Olaf Ellers (Ph.D.
dissertation); 9.5a: Cambridge University Press (The Invertebrata by L. A.
Borradaile et al.); 10.8: National Research Board of Canada (Canadian
Journal of Zoology); 11.7, 11.2: The Royal Society (Philosophical Transactions
of the Royal Society); 11.8: Springer-Verlag (Oecologia); 12.5b: Cambridge
University Press (Biological Reviews); 13.6: United Engineering Trustees,
Inc. (Fluid Mechanics for Hydraulic Engineers by H. Rouse); 14.2b: Holt,
Rinehart, and Winston, and Mrs. Alfred S. Romer (The Vertebrate Body, by
A. S. Romer); 14.5a: Cambridge University Press (Journal of the Marine
Biological Association of the United Kingdom); 15.2: Springer-Verlag (Marine
Biology); 16.1, 16.2, 16.3: Thomas L. Daniel (Ph.D. dissertation).
X 1 11
LIFE IN MOVING FLUIDS
CHAPTER 1
Remarks at the Start
With the easy confidence of long tradition, the biologist measures
the effects of temperature on every parameter of life. Lack of
sophistication poses no barrier; heat storage and exchange may be ignored
or Arrhenius abused; but temperature is, after time, our favorite abscissa.
One doesn't have to be a card-carrying thermodynamicist to wield a
thermometer.
By contrast, only a few of us measure the rates at which fluids flow,
however potent the possible effects of winds and currents on our particular
systems. Fluid mechanics is intimidating, with courses and texts designed
for practicing engineers and other masters of vector calculus and similar
arcane arts. Besides, we've developed no comfortably familiar and
appropriate technology with which to produce and measure the flow of fluids
under biologically interesting circumstances. So the effects of flow are too
commonly either ignored or else relegated to parentheses, speculation, or
anecdote.
A life immersed in a fluid—air or water—is, of course, nothing unusual
for an organism. Almost as commonly, the organism and fluid move with
respect to each other, either through locomotion, as winds or currents
across some sedentary creature, or as fluid passes through internal
conduits. Clearly, then, fluid motion is something with which many organisms
must contend; as clearly, it ought to be a factor to which the design of
organisms reflects adaptation.
It is this particular set of phenomena—the adaptations of organisms to
moving fluids—that this book is mainly about. Its intended messages are
that such adaptations are of considerable interest and that fluid flow need
not be viewed with fear or alarm. Flow may indeed be one of the messier
aspects of the physical world, but most of the messiness can be explained in
words, simple formulas, and graphs. Quantitative rules are available to
bring respectable organization to a wide range of phenomena. With a little
experience, one's intuition can develop into a reasonably reliable guide to
flows and the forces they generate. Even the technology for experimental
work on flows proves less formidable than one might anticipate. Indeed,
the underlying complexity of fluid mechanics can be something of a boon,
since it greatly restricts the possibilities of exact mathematical solutions or
trustworthy simulations. Thus the investigator must often resort to the
world of direct experimentation and simple physical models, a world in
which the biologist can feel quite at home.
3
CHAPTER 1
Supplying a comprehensive review of what's known about the
interrelationships of the movements of fluids and the design of organisms isn't my
intention. I will cite no small number of phenomena and investigations;
but they're mainly meant to illustrate the diversity of situations to which
flow is relevant and the ways in which such situations can be analyzed.
Instead, the main objective is an attempt to imbue the reader with some
intuitive feeling for the behavior of fluids under biologically interesting
circumstances, to supply some of the comfortable familiarity with fluid
motion that most of us have for solids. I take the view that with that
familiarity the biologist is likely to notice relationships and phenomena and to
put hydrodynamic hypotheses to proper experimental scrutiny. And I feel
strongly that such investigations should be unhesitatingly pursued by any
biologist and should not constitute the peculiar province of some au courant
priesthood.
One might organize a book such as this with either biology or physics as
framework. The physical phenomena, though, flow more easily in an
orderly and useful sequence; my attempts to make good order of the
biological topics inevitably have a more severe air of artificiality. So the physics of
flow will provide the skeleton, fleshed out, in turn, by consideration of the
bioportentousness of each item. Where relevance to a particularly large
segment of biology wants examination, as when considering velocity
gradients or drag, wholly biological chapters will be interjected. While I hope
that the arrangement will be effective for both the biologist seeking an easy
entry into fluids and the fluid mechanist dazzled by biological diversity, I've
opted for the biologist where hard choices had to be made. The reader may
be a little vague on the distinctions between work and power or stress and
strain but is assumed to be quite sound on vertebrates and invertebrates as
well as cucumbers and sea cucumbers. For one thing, I'm very much a
biologist myself; for another, the relevant biological details are easy to
obtain from textbooks or other references. Since the framework is physical
rather than biological, topics such as seed dispersal and suspension feeding
will wander in and out; since specific biological topics sometimes involve
several applications of fluid mechanics, a little redundancy has been
inevitable.
Biological examples will vary from well-established through half-baked
to wildly speculative; I'll try to indicate the degree of confidence with which
each should be vested. Speculation is the crucial raw material of science,
and it seems especially important at this stage of this subject. If some
assertion generates either enough antipathy or enthusiasm to provoke a
decent investigation, then we'll all be a bit farther ahead. In any case, I
claim no proprietary rights to any hypothesis here, whether it's explicit or
implied. (But neither does any idea come warranted against post facto
silliness.) Incidentally, it's an entirely practical procedure to fix on some
4
REMARKS AT THE START
physical phenomenon and then go looking to see how organisms have
responded to it. While this may sound like shooting at a wall and then
drawing targets around the bullet holes, science is, after all, an
opportunistic affair rather than a sporting proposition. But whatever the
respectability of the approach, it does at least take some advantage of the way things
are organized here.
It's probably not the best idea to use this book solely as a reference, with
the index for intromission. The reader coming from biology ought to move
in sequence through at least the basic material on viscosity, the principle of
continuity, Bernoulli's equation, the Reynolds number, and the
characteristics of velocity gradients. The book certainly can be used out of order
or with parts omitted, but certain pitfalls lurk. Biologists have been all too
willing to use equations with no more than a guileless glance to check that
the right variables were represented. As a result, much mischief has been
perpetrated. Bernoulli's equation doesn't work well in boundary layers and
has little direct applicability to circulatory systems. The Hagen-Poiseuille
equation presumes that flow is laminar and that one is far from the
entrance to a pipe. Stokes' law applies (in general) to small spheres, not large
ones, and it usually needs a correction for spheres of gas moving in a liquid
medium. Equations certainly abound in these pages, but they can be found
as easily elsewhere. More important are the discussions of how and where
they apply—when to rush in and when to fear to tread. Especially in this
latter matter, the specific sequential treatment of topics is of significance.
Inclusions, Exclusions, General Sources
At least for physical phenomena, some idea of what's covered in this book
is apparent from the table of contents. Naturally, the omissions are less
evident, but it's important that they be mentioned. Physical fluid mechanics
ramifies in many directions; much of it carries the unmistakable odor of
our technological concerns and has little relevance to biology. Thus we can
safely ignore such things as high speed (compressible) flows and the flows
of rarefied gases. For reasons that flatter neither author nor most readers,
the mathematical niceties of fluid mechanics will not loom large here. More
attention will be given to phenomena that I judged simple or widespread
than to ones that I regarded as more complex or specialized. The focus
here is on flow in and around organisms, decent-sized chunks of
organisms, and small assemblages of organisms, and this focus has necessitated
some omissions. Only a little will be said about the statics of fluids, bio-
meteorology, and the flow of substances that are incompletely fluid, such as
the contents of cells and life's various slurries and slimes. Thus I'll largely
eschew the treacherous quagmires of mitosis and cyclosis, of surface waves
and ocean currents, of the hydraulics of streams and rivers, of atmospheric
5
CHAPTER 1
winds and wind-driven circulation in lakes, and of flow through porous
media. In addition, I've made some essentially arbitrary exclusions and will
have little to say about the sensory side of responses to flows and the effects
of flow on chemical communication in either air or water. Flows either
driven by or involving temperature differences—convective heat
transfer—are a major omission rationalized only by a lack of space and
steam. Most of these topics are treated well elsewhere, and sources of
enlightenment will be mentioned at appropriate points.
More extensive and detailed information on fluid dynamics may be
obtained from conventional engineering textbooks, such as those by Streeter
and Wylie (1985) or Massey (1989). The basic processes that will be of
concern here were about as well understood fifty or sixty years ago as they
are today, so the age of the source is usually immaterial. Indeed, the earlier
generations of fluid dynamicists may have worried more than their
successors about low-speed phenomena and other items that turn out to have
biological relevance; and inexpensive reprints of several quite useful old
texts are currently available. Prandtl and Tietjens' (1934) Applied Hydro-
and Aeromechanics is particularly good on boundary layers and drag. Fluid
Mechanics for Hydraulic Engineers (1938) contains a fine treatment of
dimensional analysis and the origin of the common dimensionless indices, while
Elementary Mechanics of Fluids (1946) covers an unusually diverse range of
topics; both are by Hunter Rouse. Goldstein's (1938) two volumes, Modern
Developments in Fluid Dynamics, have nice verbal descriptions of phenomena
in between the equations. Mises (1945) gives excellent explanations of how
wings and propellers work in Theory of Flight. With several of these books at
hand one can usually find an understandable and intuitively satisfying
explanation of a puzzling aspect of flow. Finally, both aesthetically and
technically pleasing, there's An Album of Fluid Motion (1982), by Van Dyke.
Trustworthy popular accounts are surprisingly scarce, with no treatment
of fluid mechanics coming anywhere near the breadth and elegance that
Gordon's (1978) Structures brings to solid mechanics. My favorite three are
Von Karman's (1954) Aerodynamics, Sutton's (1955) The Science of Flight, and
Shapiro's (1961) Shape and Flow. The first is particularly witty, the second is
especially clear and honest, and the third, mirabile dictu, focuses on fluid
phenomena that biologists should encounter.
Several other general sources deserve mention. Fluid Behavior in
Biological Systems, by Leyton (1975), is nearest to this book in intent. It gives less
attention to drag, boundary layers, and propulsion but more to flow in
porous media, heat transfer, non-Newtonian fluids, thermodynamics, and
micrometeorology. Ward-Smith's Biophysical Aerodynamics (1984) focuses
on seed dispersal and animal flight. Waves and Beaches (1980) by Bascom,
Biology and the Mechanics of the Wave-Swept Environment by Denny (1988),
and Air and Water, also by Denny (1993), are invaluable for anyone working
6
REMARKS A I I H E S I A R T
along the edge of the ocean; in effect the Denny picks up in both elegance
and focus where the Bascom leaves off. And the privately published
compendium of Hoerner (1965), Fluid-Dynamic Drag, is rilled with information
about the behavior of simple objects to which analogous data for organisms
can be compared.
Technology
The how-to-do-it aspects of biological fluid mechanics present special
snares. High technology is certainly no stranger to the microscopist,
molecular biologist, or neurophysiologist. But few of us are facile at making
and measuring moving fluids, and there's rarely a lab-down-the-hall where
folks are already, so to speak, well immersed in flow. Faced with some
upcoming investigation, one's first impulse is to seek out a friendly
engineer, who then prescribes a hot-wire or laser anemometer; the problem is
thereby reduced to a search for kilobucks. There is, though, a second and
less ordinary problem. The technology used by the engineers is a product
of, by, and for engineers. The range of magnitude of the flows we usually
need to produce and of the forces we typically want to measure is rather
different, and engineering technology is often as inappropriate as it is
expensive.
In over thirty years of facing problems of flow, I've found that the devices
I've had to use were, compared to those of my colleagues in more
established fields of biology, rather cheap. On the other hand, they have rarely
been available prepackaged and precalibrated, with a factory to phone
when all else fails. Flow tanks, wind tunnels, flow meters, anemometers,
and force meters have simply been built in my laboratory as needed. I've
developed a fair contempt for fancy commercial gear except for items of
very general applicability—digital voltmeters, power supplies, poten-
tiometric chart recorders, variable speed motors, gears and pulleys,
electronic stroboscopes, analog-to-digital converters, video cameras and
recorders, and so forth. The consistently most valuable tools have been lathe,
milling machine, and drill press—but I'm a quite unreconstructed primiti-
vist with the perverse passion of the impecunious and impatient. The first
edition of this book included appendices on making and measuring flows.
I've not retained them, since the earlier edition is still available as a
reference and since I'm now convinced that it would take a whole book to do the
job properly.
Dimensions and Units
Fluid mechanics makes use of a wide array of different variables, some
(density, viscosity, lift, drag) familiar and others (circulation, friction fac-
7
CHAPTER 1
tor, pressure coefficient) out of the biologist's normal menagerie of terms.
A list of symbols and quantities used in this book precedes the index; it
might be worth putting a protruding label on that page. Matters will be
somewhat simplified if the reader pays a little attention to the dimensions
that attach to each quantity. By dimensions I don't mean units. Thus
velocity always has dimensions of length per unit time, whether data are given in
units of meters per second, miles per hour, or furlongs per fortnight.
Dimensions take on rather special significance in fluid mechanics notjust as
a result of the general messiness of the subject but because of a condition
that may sound trivial but proves surprisingly potent. For an equation to have
any applicability to the real world, not only must the two sides be numerically equal,
but they must be dimensionally equal as well. The general statement asserts that
proper equations must be dimensionally homogeneous—each term must
have the same dimensions. When theory, memory, or intuition fails, this
injunction can go a long way toward indicating the form of an appropriate
equation.
An example should clarify the matter. Assume you want an equation
relating the tension in the wall of a sphere or a cylinder to the pressure
inside—perhaps you know the surface tension of water and want to know
what this does to the internal pressure of a gas bubble (in connection with
Chapter 15 or 17). Tension has dimensions of force per unit length (as you
can tell, if need be, from the units in which surface tension is quoted).
Pressure has dimensions of force per unit area (as in pounds of force per
square inch of area). To relate pressure to tension, clearly pressure must be
multiplied by some linear dimension of the sphere or cylinder. Thus the
equation will be of the form
tension = constant x radius x pressure,
where we know nothing about the constant except that it's dimensionless.
Evidently a given tension generates a larger pressure when the radius is
small than when the radius is large. Even without further information
about the constant we're no longer surprised that (surface tension being
constant) tiny bubbles have high internal pressures. And it ought to take a
higher pressure to generate a given tension when the radius is small, so
we're much less mystified about why it's relatively hard to start blowing up a
balloon despite the obvious flaccidity of the rubber. We're no longer
surprised that a small plant cell can withstand pressure differences of many
atmospheres across its thin walls, nor are we puzzled at how arterioles can
get by with much thinner walls than that of the aorta when both are
subjected to similar internal pressures.
The easiest way to compare the dimensions of different variables is to
reduce them to combinations of a few so-called fundamental dimensions.
We will require only three such dimensions here: length, mass, and time
8
REMARKS AT THE START
(temperature is a frequent fourth). This reduction is accomplished by use of
definitions or previously memorized functions. For instance, force is mass
times acceleration, and therefore force has fundamental dimensions of
MLT-2. Pressure or stress is force per area, so both have dimensions of
ML_1T~2. Incidentally, this latter example emphasizes the fact that just
because two quantities have the same dimensions doesn't mean that they
are synonymous or equivalent. With such simple manipulations, the
fundamental dimensions of each term in almost any equation can be obtained.
Considerable use will be made of this sort of dimensional reasoning in
forthcoming chapters. More extensive and formal introductions to the
subject of dimensional analysis can be found in books by Bridgman (1931)
and Langhaar (1951); biological contexts are supplied by McMahon and
Bonner (1983) and Pennycuick (1992).
Not only constants but also variables may be dimensionless and still be
rich in relevance to the real world. Any number that is the ratio of two
quantities measured in the same dimensions will be dimensionless. A fairly
simple example is strain, as used in solid mechanics. Strain is the ratio of the
extension in length of a stressed object to its original, unstressed length.
Unstressed length is simple and commonly fixed; things get even more
interesting when several of the quantities in a dimensionless index decide
to vary. Such indices turn out to be scales that achieve quite useful
simplifications of otherwise complex situations and that can lead to
remarkable insights into what really matters beneath a confusion of varying
quantities. Thus surface-to-volume ratio depends on size as well as shape; it has
a dimension of inverse length (L~!). By contrast, surface cubed over
volume squared is dimensionless and quite indifferent to the size of an object
per se; so, for instance, a cube has the same value whatever its size. It's
therefore much more useful as an index of shape. Dimensionless numbers,
usually named after their first promulgators, find wide use in fluid
mechanics; while initially they seem odd, one rapidly achieves proper
contemptuous familiarity even with graphs of one dimensionless number
plotted against another. I hardly need mention that dimensionless numbers
are automatically unitless and therefore quite indifferent to which system
of units is in use.
Units, though, can't be completely ignored. All variables in a dimension-
ally proper equation ought to be given in a consistent set of units. An
earlier generation of biologists, when they meant metric units, usually used
something approximating the physicist's centimeter-gram-second (COS)
system, along with a few oddities such as heat as calories and pressure as
millimeters of mercury in a column. We're now enjoined to adopt a specific
version of the metric system common to all of science, the SI or "Systeme
Internationale," which will be used here. Fundamental units (for the
fundamental dimensions) are kilograms (mass), meters (length), and seconds
9
CHAPTER 1
(time). SI allows the common prefixes going up or down by factors of 1000
(mega-, milli-, micro-, and so forth) to be attached to any fundamental or
derived unit. Only a single prefix, though, may be used with a single unit,
and the prefix must attach to the numerator. Thus meganewtons per
square meter is legitimate but newtons per square millimeter is not. I'll only
stoop to such f rowned-upon units as centimeters, liters, and kilometers as a
kind of vernacular where no calculations are contemplated. SI units often
get mildly ludicrous in the context of organisms. Thus the drag of a
cruising fruit fly (Vogel 1966) is about three micronewtons. And Wainwright et
al. (1976) give the strength of spider silk as about 109 newtons per square
meter of cross section; it would take one hundred billion (U.S.) strands to
get that combined area. But consistency is really a sufficiently
compensatory blessing.
Table 1.1 gives a list of quantities with their fundamental dimensions and
SI units. For further introduction to SI units and conventions, see
Quantities, Units, and Symbols by the Symbols Committee of the Royal Society
(1975) or The International System of Units by Goldman and Bell (1986). For
the inevitable nuisance of dealing with different systems of units, Pen-
nycuick's (1988) booklet, Conversion Factors, is an absolute godsend. It's
cheap enough so one might scatter a few copies around home, office, and
laboratory.
Table 1.1 Common quantities fundamental dimensions, and SI
UNITS.
Quantities
Dimensions
L
L2
L*
T
LT"1
LT~2
M
MLT-2
ML-*
ML2T~2
ML2T"3
ML-'T"2
ML-'T"1
L*t- •
MT"2
SI units
meter (m)
square meter (m2)
cubic meter (m*)
second (s)
meter per second (m s_ •)
meter per second squared (m s-2)
kilogram per cubic meter (kg m_H)
newton (N or kg m s-2)
kilogram per cubic meter (kg m-3)
joule (J or N m)
watt (W or J s_l)
pascal (Pa or N m-2)
pascal second (Pa s)
square meter per second (m2 s_')
newton per meter (N m_ ')
Length, distance (1)
Area, surface (S)
Volume (V)
Time (t)
Velocity, speed (U)
Acceleration (a)
Mass (m)
Force, weight (F, W)
Density (p)
Work, energy (W)
Power (P)
Pressure, stress (p, t)
Dynamic viscosity (|x)
Kinematic viscosity (v)
Tension, surface t. (7)
Notes* For the dimensions, M = mass, L = length, and T = time Notice that some
symbols are shared between quantities and units; since units are never indicated in
formulas and text can be made explicit, no ambiguity need result.
10
REMARKS AT THE START
If you're decently scrupulous about consistency of dimensions and use of
units then you never have to specify units when giving equations, a
considerable convenience. The main place where the system falters at all is when
equations with variable exponents are fitted to empirical data, as in any
statement such as "metabolic rate is proportional to body mass to the 0.75
power." The exponent of proportionality comes out the same in any set of
units, but the constant with which it forms an equation does not. Moreover,
the rule about dimensional homogeneity is violated unless one tacitly heaps
all the unpleasantness on the constant of proportionality. If the expression
is written as an equation rather than a proportionality, the usual practice is
merely to specify the set of units that must be used. Here the practice
among fluid mechanists and biologists is about the same, both being
practical people who have scattered data, imperfect theories, and the like.
Measurements and Accuracy
First, a few words about what's meant by accuracy. As Eisenhart (1968)
has pointed out, lack of accuracy reflects two distinct disabilities in data.
First, there's imprecision, or lack of repeatability of determinations. And
second, there's systematic error, or bias, the gap between the measured
value and some actual, "true" value—the tendency to measure something
other than what was really intended. In the kinds of problems we'll discuss,
unavoidable imprecision is usually pretty horrid, at least by the standards
of physical rather than biological sciences. Thus it's rarely worth great
attention to fine standards for calibration where these just drive
systematic errors well below what proves to be the more intractable problem of
imprecision.
Quantities such as density can be measured very precisely. But the
inevitable irregularities in flows, the nature of vortices and turbulence, and
quite a few other phenomena severely limit the precision with which the
behavior and effects of moving fluids can be usefully measured. The drag
of an object measured in one wind tunnel will often differ considerably
from that measured in another, while a third datum will result from towing
the object through otherwise still air. If a figure of, say, 1 m s~l is cited as
the transition point from laminar to turbulent flow in some pipe, that
figure should not be interpreted as 1 ± 0.01 or often even 1 ± 0.1 m s_1.
With extreme care it may be possible to postpone the transition to 10 ms-1
or more. Some empirical formulas given in standard works, especially
those for convective flows, use constants with three significant figures. My
own experience suggests that such numbers should be viewed with
enormous skepticism for anything except, perhaps, the very specific
experimental conditions under which they were determined. And citing
calculations for Reynolds numbers (Chapter 5), for instance, without minimally
1 1
CHAPTER 1
rounding off to the nearest part in a hundred is at the least a bit self-
deceptive.
The development of electronic calculators has had a pernicious effect on
our notions of precision. The art estimating just how much precision is
truly necessary to resolve the question at hand has suffered from the
demise of the slide rule. As a practical matter, the flow of fluids, even
without organisms, is a subject that enjoys barely a slide rule level of
precision—rarely better than one percent and often much worse.
Relative Motion
Another matter ought to be set straight at the start, an item that arises
with some frequency among biologists taking their first look at fluid flow.
Frames of reference can be chosen for convenience, and the surface of the
earth upon which we walk has no absolute claim as a "correct" reference
frame. One might imagine that a seed carried beneath a parachute of fluff y
fibers will trail behind the fibers as the unit is blown across the landscape by
a steady wind. In fact the image is quite wrong for anything beyond the
initial events of detachment from the plant—when the surface of the earth
is still an active participant—as can be seen in Figure 1.1. Farther along (if
the wind is steady) neither any longer "knows" anything about what the
ground's doing so the seed hangs below the fluff—the surface of the earth
now constitutes a reference frame that's both misleading and unnecessarily
complex. People who've traveled in. balloons commonly comment on the
silence and windlessness they experience and its incongruence with clear
visual evidence that the ground is moving beneath.
A more drastic if less commonly observed case is that of a "ballooning"
spider. (For a general account of the phenomenon, see Bishop 1990.) A
young spider spins a long silk strand that extends downwind from it.
Eventually it lets go and drifts along. One might expect that the spider is
thereafter pulled behind the silken line until the whole system settles to the
earth. What will happen in the absence of gusts, vortices, and gradients is
that the spider will fall downward but at a speed that's much reduced by the
drag of the line. The line will gradually shift from running horizontally to
running vertically, forming a relatively high drag, low weight element that
extends downstream (here, of course, upward) from the falling spider. The
line lags behind, slowing descent as a result of its high drag relative to its
weight. (Humphrey 1987 estimates that the line has over 75% of the drag
of the system while contributing less than a tenth of a percent of the
weight.)
Failing to keep in mind a proper reference frame can generate odd
misconceptions as well as obscure some real problems. From time to time
statements appear in the avian literature about a problem facing (or, per-
12
REMARKS AT THE START
Figure 1.1. A dandelion seed carried by a steady wind orients vertically,
as if falling through still air. Only during detachment does the fibrous
end tilt downwind.
haps we should say chasing) a bird flying downwind—it must somehow
keep from getting its tail feathers ruffled. But consider: once launched, a
bird simply does not (and, indeed, cannot) fly downwind with respect to the
local flow around it. If the wind with respect to the ground goes in the same
direction as the bird, then the bird just flies that much faster with respect to
the ground. The problem is a really nasty one for slow fliers such as
migrating monarch butterflies that can't make progress (once again with respect
to the ground) against even modest winds (Gibo and Pallett 1979).
Worse, you might really get stung by confusion about relative motion
and frames of reference. Imagine being chased by a swarm of mayhem-
minded killer bees, as in Figure 1.2. A decent breeze is blowing, so to get a
little more speed you run downwind. Bad move—you're quickly bee-set.
Honeybees can fly (despite a lot of lore to the contrary) only about 7.5 m s~l
(Nachtigall and Hanauer-Thieser 1992), but that's equivalent to about a
four-minute mile. With a 4.5 m s~l breeze from behind, they'll go all of 12
m s~l with respect to the ground. You may gain a little from the tail wind, but
the bees will automatically get full value. What if you run upwind? You may
be slowed down slightly, but the bees will be dramatically retarded—down
to 3 m s~l with respect to the ground. Only about a nine-minute mile is needed
for you to stay ahead.
The ability to shift reference frames for our convenience can effect more
than conceptual simplifications. No specialjustification is needed for using
an experimental system in which the organism is stationary and the fluid
environment moves as a substitute for a reality in which the organism
swims, flies, or falls through a fluid stationary with respect to the surface of
the earth. That's the main reason for the popularity of such devices as wind
tunnels and flow tanks for working on flying and swimming. I've used a
13
CHAPTER 1
Figure 1.2. Going downwind, bees are faster than any running human;
a head wind slows the human only a little, but the bees are badly
hindered—unless they've looked ahead to Chapter 8.
vertical wind tunnel to provide upward breezes just equal to the falling
rates of seeds. Shrewd choice of reference frame is an old tradition—
assuming (not proving!) a sun-centered system permitted Copernicus to
simplify enormously the complex cosmology of Ptolemy.
Balancing Forces
When considering solid bodies moving through fluids, Newton's first law
has to be put into a somewhat more distant perspective than might have
been the practice in one's first physics course. Bodies may continue in
steady, rectilinear motion unless imposed upon by external forces; but
imposition of external forces is just what fluids do to solid bodies that have
the temerity to force passage. Chief among the external forces is, of course,
drag.1 In practice, then, a body traveling steadily with respect to a fluid
(ah—frames of reference) must be exerting a force on the fluid exactly
sufficient to balance the fluid's force on the body. That force might be
provided by the action of wings or tail acting as thrust generators. Or it
might be supplied by the action of gravity on the body, as when a body sinks
steadily, when the downward force of gravity balances the combined
upward forces of buoyancy and drag. Or it might be provided by some other
solid structure that transmits forces—a mounting strut supporting an
object in a flow tank or wind tunnel or the branches and trunk supporting the
leaves of a tree.
In combination, force balance and reference frame can make some
subtle but substantial trouble, and the utility of quite a lot of literature is
compromised by insufficient attention to them. Say you persuade an insect
1 Perhaps someone (me, for instance) ought to say a word against the common usage
of that needlessly redundant and pretentious term, "drag force." Drag is a force, is
always a force, and is nothing but a force.
14
REMARKS AT THE START
to beat its wings as hard as it can while attached to a fine wire, and you even
manage to attach the fine wire to a device that will measure the force the
insect produces. What will this tell you about how fast the insect can fly or
about how much force it can generate at top speed? Almost nothing! Even
if you (separately) measure how fast the insect can fly, you are on very
unsafe ground if you multiply that speed and the force you've measured to
get the insect's power output. Similarly, if you want to know the drag of the
insect at its maximum speed, you can't just put the thing in a wind tunnel
set to that speed and measure the force it exerts on a mount. If the insect is
doing exactly what it ought to, then the mount will feel no force at all
because the drag of the insect's body will exactly balance the thrust
produced by the wings. If it's not beating its wings then you get a force where
normally no net force would be exerted. Fortunately, there are ways
around such difficulties; what's important at this point is that they be
recognized.
I do regret a little the admonitory tone of much of this initial chapter. A
lot of it reflects some scar tissue induced by abrasions in the form of written
material that I've been expected to evaluate. Even though I quite obviously
mean what I've said here, its general character contrasts a bit with my first
rule of fluid mechanics—you have to be breezy if you expect to make waves.
15
CHAPTER 2
What is a Fluid and How Much So?
Perhaps you were long ago told of three common states of matter—
gas, liquid, and solid. Perhaps you were also told of a handy rule for
distinguishing them—the rule that solids have size and shape; that liquids
have size but no shape; and that gases have neither size nor shape. Perhaps
you've even heard that while intermediates between solids and liquids are
common, intermediates between liquids and gases are certainly not.
Where, in all this, do fluids enter? In common parlance, "fluid" is just a
synonym for "liquid", a definition the reader must immediately expunge
from memory. For present purposes, indeed for all of the science of fluid
mechanics, a fluid is either a gas or a liquid but positively not a solid.
Definitions are, of course, in some measure arbitrary, but a bald one such
as the present assertion about fluids raises some distinctly nonarbitrary
questions. Why do we find it convenient to lump the gaseous and liquid
states even when we know of no intermediates? Why do we persist in
making a general distinction between liquids and solids when, in practice,
we're made up of substances of intermediate character? And, of most
immediate concern, how can we reliably distinguish fluids from solids?
A simple experiment can be used (conceptually, anyway) to tell a fluid
from a solid, an experiment that makes no qualitative distinction between
gases and liquids. Consider an apparatus consisting of two concentric
cylinders of ordinary size, such as that shown in Figure 2.1a. Although the outer
one is fixed, the inner cylinder may be rotated on its long axis. Between the
cylinders is a gap that we can fill with any material of any state. We further
assume that the material adheres to or "wets" the walls of the cylinders and
that rotation of the inner cylinder is resisted only by forces caused by the
presence of the outer wall and transmitted through the material between
them.
One naturally expects that the inner cylinder can be turned more or less
easily, depending on the material in the gap. Indeed, that's just what
happens; not surprisingly, the force required varies over an enormous range.
But a less obvious distinction emerges. For some materials, the force
necessary to turn the inner cylinder depends on how far we've already turned it
from "rest." For others, the force is quite independent of how far we've
distorted the material and depends only on how fast we turn the cylinder. In
each case we've imposed a "shearing" load on the material; that is, we've
attempted to deform it by sliding one surface relative to another concentric
16
WHAT IS A FLUID?
(a) (b)
Figure 2.1. (a) A pair of concentric cylinders. If the space between
them is filled with a fluid, then rotation of the inner one will exert a force
tending to rotate the outer one. (b) The practical version, an ice cream
freezer. The operator monitors the viscosity increase that denotes
progress by feeling the increasing difficulty of turning the dasher relative to
the container.
(essentially parallel) one. As it turns out, we've just distinguished solids
from fluids.
Solids resist shear deformation—they care how far they're deformed.
Generally the further one wishes to deform them, the more force is
required. To put the matter quantitatively, consider a rectangular block of
material as in Figure 2.2 in which the top surface is pushed sideways but the
bottom surface is fixed. The pushing force tends to deform the rectangular
solid into one in which two opposite sides are parallelograms. If 6 (theta) is
the angle by which the material is deformed and S the surface area over
which the force, F, is applied, then
^ = G9. (2.1)
G is called the shear modulus (similar to the elastic or Young's modulus)
and corresponds to our common notion of shearing distortions such as
happen when we twist things. The equation implies that shear stress (F/S) is
proportional to shear strain (6); in short, it's a statement of Hooke's law. If G
does indeed stay constant as stress and strain vary, then the material is
spoken of as "Hookean."
By no stretch do proper fluids have a shear modulus, high or low—
they're infinitely distortable, magnificently oblivious to their shapes, past
or present. But deformation still matters, although in quite a different way.
What fluids care about is how rapidly they are deformed. Thus a much
different expression is needed to describe what happens to a fluid when
17
CHAPTER 2
Figure 2.2. A rectangular block of material is deformed by a shear
force, F, or a shear stress, F/S, so two faces become parallelograms
of tiltO.
(assuming that by some preposterous prestidigitation nothing leaks out)
the same shear stress is applied to the system in Figure 2.2:
F
S
|x9
(2.2)
6// is the rate of shear and |x (mu) is a property called the dynamic viscosity.
The latter, to be tediously explicit, indicates how much a fluid resists not
shear but rate of shear. If |x remains constant as shear stress (what you do to
the fluid) and shear rate (how the fluid responds) vary, the fluid is said to be
"Newtonian"—the analog of Hookean behavior in solids. For different
fluids, though, the actual value of the dynamic viscosity varies enormously,
from low values for gases to exceedingly high values for substances such as
tar or glass that, over long periods of time, behave as fluids.1 Values of
viscosity can be usefully determined or assigned to systems ranging from
the insides of cells to glaciers or even the mantle of the earth.
This distinction between solids and fluids is precisely the same as that
between springs and shock absorbers. That's a very real difference for
anyone who installs one or the other on a car—a shock absorber can easily
be set to whatever length is needed to slip onto its mounts, but a spring has a
forceful preference for a particular length.
The No-Slip Condition
The properly skeptical reader may have detected a peculiar assumption
in our demonstration of viscosity: the fluid had to stick to the walls of the
1 Reiner (1964) invented a dimensionless index that provides a guide to whether a
system behaves like a solid or a fluid; with biblical allusion, he called it the Deborah
number" (De). It's defined as the ratio of the time a process takes to the time required for
significant plastic deformation of the system H De « 1, you assume you're dealing with
a solid; if De >> 1, then for practical purposes your system is fluid. De rarely appears in
the biological literature (but see, for instance, Jenkinson and Wyatt 1992), but I think it
deserves to be better known.
18
WHAT IS A FLUID'
inner and outer cylinders in order to shear rather than simply slide along
the walls. Now fluid certainly does stick to itself. If one tiny portion of a
fluid moves, it tends to carry other bits of fluid with it—the magnitude of
that tendency is precisely what viscosity is about. Less obviously, fluids stick
to solids quite as well as they stick to themselves. As nearly as we can tell
from the very best measurements, the velocity of a fluid at the interface with a
solid is always just the same as that of the solid. This last statement
expresses something called the "no-slip condition"—fluids do not slip with
respect to adjacent solids. It is the first of quite a few counterintuitive
concepts we'll encounter in this world of fluid mechanics; indeed, the
dubious may be comforted to know that the reality and universality of the
no-slip condition was heatedly debated through most of the nineteenth
century. Goldstein (1938) devotes a special section at the end of his book to
the controversy. The only significant exception to the condition seems to
occur in very rarefied gases, where molecules encounter one another too
rarely for viscosity to mean much.
Yet another peculiarity of this no-slip condition is that the nature of the
solid surface makes very little difference. If water is flowing over a solid
without an air-water interface to complicate matters, the no-slip condition
holds whether the solid is hydrophilic or hydrophobic, rough or smooth,
greasy or clean. The nature of the solid surface matters only when we have
a liquid-gas interface present as well—in short, where surface tension
becomes a factor.
The no-slip condition has a number of important ramifications. In
particular, it means that any time a fluid flows across a solid, a velocity gradient
is present. That is, velocity varies with distance above the surface; or, put
more formally, dUldz isn't zero. That's what "boundary layers" (Chapter 8)
are all about. Again, these velocity gradients are developed entirely within
the fluid, not (as with two sliding solids) between one material and the
other. If the fluid is an ordinary homogeneous one, its local velocity must
smoothly approach that of the surface as the surface is approached; there
can be no discontinuity within the fluid. While it may seem a little strange at
first, even though the velocity may be zero at the surface, dUldz, the velocity
gradient, isn't. Furthermore the fluid velocity cannot asymptotically
approach the speed of the solid, for that would require a variable viscosity—it
would have to get lower in fluid closer to the solid surface to account for the
increasing steepness of the velocity gradient. In practice, the no-slip
condition explains (in part) why dust and grime accumulate on fan blades, why
pipes (including blood vessels) encounter trouble from accumulation of
deposits rather than from wearing thin, and why a bit of suspended rock is
needed in water for the latter to become effectively erosive.
The no-slip condition is as easy to demonstrate as its universality is hard
to prove. Simply fill a circular basin with water, stir the water into a
19
CHAPTER 2
smoothly circuitous motion, and inject a small bit of dye on the bottom or
side wall. The last layer of dye will remain there despite quite a lot more
stirring well above. Eventually, of course, diffusion and dilution will get rid
of any adhering colorant, but the time needed is notable. Alternatively, just
consider why dishcloths and mops are so much more effective for cleaning
surfaces than any mere rinse.
Most often the region near a solid surface in which a velocity gradient is
appreciable is a fairly thin one, measured in micrometers or, at most,
millimeters. Still, its existence requires the convention that when we speak
of velocity we mean velocity far enough from a surface so the combined
effect of the no-slip condition and viscosity, this velocity gradient, doesn't
confuse matters. Where ambiguity is possible, we'll use the term "free
stream velocity" to be properly explicit.
Assumptions and Conventions
Before proceeding further, I'd like to establish a series of assumptions
that will underlie the forthcoming chapters unless one or another of them
is explicitly relaxed. They are at least mildly preposterous; but, like pan-
mictic populations and point masses, they are convenient fictions.
1. Fluids are"Newtonian." As mentioned, many substances combine
properties of both fluids and solids. But we will disallow any trace of solidity as
previously defined—the fluids here will have no memory of previous shape
nor any elasticity. In effect, we draw a jurisdictional border, putting
molasses or treacle and most syrups on the inside and the various jams,
yoghurts, and gelatins on the outside. As Shakespeare says in King Lear, "Out,
vile jelly."
Many biological materials are in the complex, multidimensional
continuum of non-Newtonian fluids and viscoelastic solids—whole blood,
synovial fluid, mucuses of various consistencies—but we'll largely ignore them
and, in doing so, avoid perhaps the messiest and least understood branch
of fluid mechanics, rheology. As noted already, a Newtonian fluid is one
which shows a direct proportionality, a linear relationship, between the
applied shear stress and the resulting rate of deformation. That being the
case, a value of viscosity may be found that's independent of the specific test
conditions. Air and water, the fluids of main concern here, are virtually
perfect Newtonian fluids.
2. Fluids are continua. We'll assume that fluids are nonparticulate and
infinitely distortable and divisible. We're going to turn the picture of De-
mocritus to the wall and deny molecules. But consider: Is there anything in
your everyday world that requires you to make the distinctly odd contrary
assumption, that matter is particulate or molecular? Why should cheese,
sliced thinly enough, ultimately stop being cheese? Molecules are appar-
20
WHAT IS A FLUID'
ently necessary to explain the physical basis of viscosity; but once viscosity is
accepted we have little further need for them. A "fluid particle" will bejust
a linguistic convenience for specifying an arbitrarily small element of a
moving fluid that enjoys our particular scrutiny.
3. Fluids are incompressible. This contradicts the experience of anyone who
has wielded a bicycle pump. If air were really incompressible, then when
you put a thumb over the orifice you wouldn't be able to push the handle
down at all. Gases, surely, compress easily, even if liquids are fairly
recalcitrant. Put aside these subversive thoughts! While compressing gases with
static devices such as piston pumps is easy, getting compression through
flow is no small matter. Given the choice, fluids behave as if they would
rather flow than squeeze—at least up to speeds within a decent fraction of
the speed of sound, about 340 m s-1 in air and 1500 m s-1 in water.
If you direct a stream of air at a wall, at some point at least the air comes
to a halt. Anticipating the discussion of Bernoulli's principle (Chapter 4),
we can estimate how much compression is maximally involved in that
deceleration. At an ambient pressure of one atmosphere, a flow of 10 ms-1
gives a compression of 0.06%; a flow of 20 m s~l gives 0.24%; and a flow of
30 ms-1 (67 mph or 108 km hr~l) gives only 0.53%. The latter is about as
fast a flow of air as is experienced (or, say, survived) by any ordinary
biological system—a report that a deer fly achieved 350 m s~l was carefully
demolished by Langmuir (1938), who showed that the fly would either be
crushed by its drag or would consume its own weight in fuel each second. So
the assumption of incompressibility is quite a safe and conservative one; it
affords a huge simplification relative to the world with which airplane
designers must contend. Density becomes a constant for most present
purposes, with the same value for a fluid in any sort of motion as for the fluid at
rest, and with its value varying only with hydro- and aerostatic changes as in
deep dives and ascents to high altitudes.
4. Flows are always "steady." As we will use the term, so-called steady flow
doesn't deny that fluid speeds up or slows down as it flows along. Instead it
means that at a given point in space (with respect to some specified frame of
reference) the speed of flow doesn't vary with time in any regular manner.
The frame of reference may be moving with respect to the earth—one can
speak of the steady flow of air above a point on the wing of a gliding bird.
And the definition is limited to "regular" variation—the ubiquitous
random fluctuations of turbulent flow are for this purpose ignored. Chapter
16 will be given to unsteady, time-dependent flows; otherwise the
assumption will be operative.
5. Fluids make no interfaces with other fluids. Gases, of course, will not
discretely interface with each other, but liquid-liquid and liquid-gas
interfaces (especially the latter) are household events. We'll get around to these
interfaces late in the book, giving them a whole chapter (17) of their own.
21
CHAPTER 2
Elsewhere we'll arbitrarily limit our view to situations where they don't
occur and thus effect another great simplification at little explanatory cost.
Not only won't we have to worry about the chemical characteristics of
surfaces (mentioned already), but we can defer phenomena such as surface
waves and capillarity. So—bodies of fluid will be considered unbounded
except by solids.
One might note that with assumptions (3) and (5) in place the differences
between liquids and gases become quantitative rather than qualitative, and
no easy and absolute test tells whether a fluid is a liquid or a gas. While that
may seem odd or unreal, it proves a useful way to view the world.
In addition to these four assumptions, we'll assume that bodies of fluid
are at uniform temperature and otherwise homogeneous, and (usually)
that the solids bounding or immersed in them are perfectly rigid.
Properties of Fluids
Table 2.1 gives representative values of the main properties that will
matter for the phenomena we'll consider.
Density
For fluids, using mass is a bit of a mess since they may be unbounded and
often move continuously through our frame of reference. In practice,
mass is replaced by density, or mass per unit volume. The symbol used is p
(rho), the fundamental dimensions are ML~'*, and the SI unit is the kg
m~3. The COS unit,2 in this unusual case 10s times larger, is the g cm-3.
Some contrary sources (even outside the southern hemisphere) use specific
volume instead of density; one is the reciprocal of the other.
For both air and water at ordinary temperatures, density drops as
temperature increases. Only in air, though, is the effect really substantial; air
density is inversely proportional to absolute temperature. But the
biological consequences of water's minor density differences are profound and
are discussed in almost every book on limnology. In fresh water, density is
maximal at about 4° C; it drops slightly when the freezing point is
approached, and it drops further with the formation of ice (so ice floats and
only a pond's surface freezes). In seawater, —3.5° C is the temperature of
maximum density, a temperature at which ice crystals have ordinarily
begun to form.
2 While the COS or "centimeter-gram-second" system is passe, we still have to contend
with a considerable body ofliterature that uses it.
22
WHAT IS A FLUID?
Table 2.1 SI values of some physical variables at atmospheric
PRESSURE.
Air
Fresh water
Seawater
Acetone
Glycerin
" 90% aq.
Mercury
0°
20°
40°
0°
20°
40°
0°
20°
30°
20°
20°
20°
20°
Dynamic
Viscosity
(Pas)
17.09 x 10"6
18.08 x lO-6
19.04 x 10"6
1.787 x l0-»
1.002 x lO-3
0.653 x lO-3
1.890 x 10-3
1.072 x 10-3
0.870 x 10-3
0.326 x 10-3
1.490
0.219
1.554 x 10-3
Density
(kg m~3)
1.293
1.205
1.128
1.000
0.998
0.992
1.028
1.024
1.022
0.792
1.261
1.235
13.546
x lO3
X 103
X lO3
X lO3
X lO3
X lO3
X lO3
X 103
X lO3
X lO3
Kinematic
Viscosity
(m2s~1)
13.22 x 10-6
15.00 x 10-6
16.88 x 10-6
1.787 x 10-6
1.004 x lO"6
0.658 x lO"6
1.838 x lO"6
1.047 x lO"6
0.851 x 10-6
0.412 x 10"6
1.182 x lO"3
0.177 x lO"3
0.115 x 10-6
Notes: Values
for seawater
for seawater presume a salinity of 35%o (parts per thousand). Note that
the highest temperature is 30° rather than the 40° for air and fresh water.
Dynamic Viscosity
We met this property when defining a fluid and will now flesh out its
definition. One way of viewing it is to envision a large stack of very thin
sheets of paper. If shearing is the sliding of each sheet with respect to the
one beneath, then dynamic viscosity is the interlamellar stickiness or
friction between the sheets. To examine the relevant variables, a slightly
fancier version (Figure 2.3) of the last figure is useful. Imagine two negligibly
thick parallel flat plates of the same shape and area, separated by a fluid-
filled space. The lower plate is fixed, and we push on the upper one with a
steady force, F, causing it to move in its plane of orientation at a uniform
velocity, U. How is the relationship between F and U affected by the
viscosity of the fluid and the geometry of the system? The force required for the
upper plate to achieve a given velocity is the product of that velocity, the
area of a plate (S), and the property that we're calling the dynamic viscosity
(|x), divided by the distance (z) between the plates:
IxcAS
or |x
US
(2.3)
Dynamic viscosity thus has fundamental dimensions of ML~ lT~l or, less
fundamentally, (force)(time)/(area). Since SI uses the pascal for force per
23
CHAPTER 2
Figure 2.3. A pair of thin, flat plates of area S, z units apart, moving at
speed U under the impetus of force F. If the lower plate is fixed, a force
is needed to keep the upper one moving. The magnitude of that force is
proportional to the dynamic viscosity of the fluid between them. The
length of each horizontal arrow between the plates is proportional
to the local flow speed relative to the bottom plate.
area, the SI unit of viscosity is the "pascal second," Pa s; the unit has no
specific name. Much tabulated data are given in "poises," the comparable
COS unit, equivalent to dyne seconds per square centimeter. A poise is ten
times smaller than a pascal second.
Minor definitional matters. Very commonly dynamic viscosity is simply
called "viscosity" (as already done here), although the adjective in the initial
term usefully distinguishes it from "kinematic viscosity," about which more
just ahead. Ambiguity about which one is meant can usually be resolved by
noting just what units are used. Dynamic viscosity is occasionally referred
to as "molecular viscosity" to distinguish it from "eddy viscosity," a larger-
scale measure of turbulent intensity used mainly by oceanographers
(Sverdrup et al. 1942). Physical chemists traditionally use the symbol nq (eta)
instead of |x for dynamic viscosity.
Note that the dynamic viscosity of air increases slightly with
temperature. In water, by contrast, viscosity drops dramatically as temperature
rises. Few physical properties have as extreme a temperature coefficient as
does the viscosity of ordinary liquids at ordinary temperatures. Seawater
and fresh water behave in a similar manner, with the small differences in
viscosity following in close proportion to the concentration of salts.
On the other hand, the dynamic viscosity of either air or water is
substantially independent of the value of density—counterintuitive, perhaps, but
a great convenience. For water density varies only slightly with pressure: at
20° C the increase in the density of seawater descending from the surface to
the bottom of the deepest ocean trench is only about 5%. And the latter is
associated with an increase in viscosity an order of magnitude less—only
about 0.5% (Stanley and Batten 1968).
Put more formally, dynamic viscosity is the coefficient that relates shear
24
WHAT IS A FLUID5
stress (t or tau) to the local velocity gradient or shear rate (dU/dz); the view
given in Figure 2.3 still applies:
T-n-3- (2-4)
Kinematic Viscosity
Upon first encountering the symbol v (nu) for viscosity you'd naturally
ask, "What's v?" As it happens, it's the so-called kinematic viscosity, nothing
more than the ratio of dynamic viscosity to density:
v = ^ . (2.5)
P
Kinematic viscosity has fundamental dimensions of L2T~l; the SI unit is
the m2 s~l. The COS unit, the cm2 s~l (104 times smaller, which we won't
use here) is called the "stokes"3 or St.
Why bother with such a nearly redundant unit? In many situations the
character of a flow happens to depend very much on this particular ratio. It
determines the practical "gooiness" of a fluid—how easily it flows, how
likely it is to break out into a rash of vortices, how steep its velocity gradients
will be. Dynamic viscosity determines the interlamellar stickiness of the
fluid, or how much a fluid particle is likely to be affected by any non-
synchrony in the movement of adjacent particles. Density determines what
might loosely be regarded as the inertia of a fluid particle, or its tendency to
continue as it has been going regardless of the activity of its neighbors.
Their ratio, the kinematic viscosity, as Batchelor (1967) put it, "measures
the ability of molecular transport to eliminate the nonuniformities of fluid
velocity."
Note in the table that the kinematic viscosities of air and water are not
especially different—only about 15-fold at 20° C. Moreover, air is the more
kinematically viscous fluid, demonstrating, if nothing else, that this
property does not take kindly to raw intuition. In air, kinematic viscosity
increases with temperature slightly more than does dynamic viscosity. In
water, kinematic viscosity shows the same dramatic decrease with
temperature as does dynamic viscosity. So that's what's v.
Additional values of density and of dynamic and kinematic viscosities of
gases, liquids, and aqueous solutions (including seawater) are most easily
obtained from recent editions of the Handbook of Chemistry and Physics.
Other such data are given in the appendices of most textbooks of fluid
mechanics, hydraulics, aerodynamics, and physical oceanography.
3 Many otherwise respectable sources call the unit the "stoke." The name honors Sir
George G Stokes (1819-1903), known for the Navier-Stokes equations and Stokes' law.
That his name happens to have a final s is no excuse for unauthorized truncation.
25
CHAPTER 2
Measuring Viscosity
One way to take an initial look at the consequences of viscosity is to ask
how to measure the quantity—really how to make it do something that can
be measured. Admittedly, if pure water, seawater, or some simple solution
is in use, looking up a value for either dynamic or kinematic viscosity is
simpler than making such measurements. Sometimes, though, a solution is
used for which tabulated values are unavailable; fortunately, measuring
the viscosity of most liquids is not especially difficult. Onejust needs (who'd
ever guess) a viscometer. (The word "viscosimeter" is synonymous but a
linguistic barbarism.) Viscosity is, of course, a measure of resistance to rate
of shearing; so all viscometers involve some scheme for shearing fluid. The
simplest and cheapest commercially available device is an Ostwald
viscometer (Figure 2.4). It forces a liquid through a fine tube and counts on the no-
slip condition to generate sufficient shear in the flow. Improvised
alternatives can be made of capillary tubing, fine pipettes, catheter tubes, and so
forth.
For use, the viscometer is mounted in a constant temperature bath, and a
known (usually 5 cm3) quantity of liquid is introduced into the arm without
the capillary. The liquid is then sucked up through the capillary tube until
the top meniscus is above the upper reference line. The liquid is allowed to
fall, and the time is recorded for how long it takes the top meniscus to drop
from the upper to the lower reference line. Not surprisingly, the rate of
flow is proportional to the density of the liquid—that's just gravity in
action. It's also inversely proportional to the dynamic viscosity—that's the
situation first described by Hagen and Poiseuille around 1840 (Chapter
13). As a result, the time one measures is directly proportional to the kinematic
viscosity. The proportionality constant is obtained by timing the descent of
a liquid of known kinematic viscosity. If what one wants is dynamic viscosity,
one needs nothing more than a separate measurement of density.
If you purchase an Ostwald viscometer from your favorite purveyor of
scientific glassware, make sure it's appropriate for the expected range of
kinematic viscosities—these aren't all-purpose devices, and aqueous
solutions run with dismaying rapidity through meters designed for motor oils.
The cheapest viscometers lack a glass connection across the top of the arms
of the "U" and as a result are quite fragile. One should provide some
appropriate brace before use. Also, these glass devices should be kept
scrupulously clean and between uses should be rinsed with acetone or some
other solvent that leaves no residue.
Lots of other sorts of viscometers are commercially available, but one
should hesitate before getting anything much more complex—unless, say,
non-Newtonian fluids are involved. For very viscous fluids such as glycerin
or sugar syrups one can get fairly good viscosity values by just timing the
26
WHAT IS A FLUID?
upper
mark
lower
mark
capillary
Figure 2.4. An Ostwald viscometer, slightly fatter than life. The time
needed for a meniscus to drop from upper to lower marks as the liquid
passes through the capillary is proportional to the kinematic viscosity of
the liquid.
fall of a sphere of known size and weight and applying Stokes' law (Chapter
15); the main precaution is that the walls of the container should be
sufficiently far from the falling sphere.
The biologist will rarely have occasion to measure the viscosity of a gas,
and the Ostwald device won't work for gases. But an analogous
arrangement can easily be contrived; I'll leave the design as an exercise for the
reader.
Consequences of the Inverse Viscosity-Temperature
Relationship
At 5° C water is about twice as viscous (dynamically or kinematically) as at
35° C; organisms live at both temperatures and, indeed, at ones still higher
and lower. Some experience an extreme range within their lifetimes—
seasonally, diurnally, or even in different parts of the body simultaneously.
Does the consequent variation in viscosity ever have biological
implications? Quite a number of cases are at least possible although very few can be
considered well established.
Consider the body temperatures of animals. At elevated temperatures
27
CHAPTER 2
less power ought to be required to keep blood circulating if the viscosity of
blood follows the normal behavior of liquids. And, in our case, it does
behave in the ordinary way—human blood viscosity (ignoring blood's
minor non-Newtonianism) is 50% higher at 20° than at 37° C (Altman and
Dittmer 1971). Is this a fringe benefit of having a high body temperature?
Probably the saving in power is not especially significant—circulation costs
only about 6% of basal metabolic rate. More interesting is the possibility of
compensatory adjustments in the bloods and circulatory systems of animals
that tolerate a wide range of internal temperatures. The red blood cells of
cold-blooded vertebrates, and therefore presumably their capillary
diameters, are typically larger than either the nucleated cells of birds or the
nonnucleated ones of mammals (Chien et al. 1971). The shear rate of blood
is greatest in the capillaries; must these be larger in order to permit
circulation at adequate rates without excessive cost in a cold body?
Have some animals arranged for their bloods to behave like "multi-
viscosity" motor oils, which resist excessive thickening when cold and
thinning when hot? That might be helpful if an animal has to be fully
functional with the same circulatory machinery at different temperatures.
Fletcher and Haedrich (1987) found that the viscosity of rainbow trout
blood has an unusually low temperature dependence, but trout seem
unlikely animals to encounter highly variable ambient temperatures.
Is the severe temperature dependence of viscosity perhaps a
serendipitous advantage on occasion? A marine iguana of the Galapagos basks
on warm rocks, heating rapidly, and then jumps into the cold Humboldt
current to graze on algae, cooling only slowly. Circulatory adjustments as
the animal takes the plunge have been postulated (Bartholomew and Las-
iewski 1965), but no one seems to have looked at whether part of the
circulatory reduction in cold water is just a passive consequence of an
increase in viscosity. A variety of large, rapid, pelagic fish have circulatory
arrangements that permit locomotory muscles to get quite hot when
they're in use (Carey et al. 1971); blood flow ought to increase
automatically at just the appropriate time.
A less speculative case is that of antarctic mammals and birds; Guard and
Murrish (1975) found that the apparent viscosity of their bloods changed
with temperature even more drastically than in more ordinary animals
such as humans and ducks. Antarctic animals must commonly contend
with cold appendages, since full insulation of feet and flippers would be
quite incompatible with their normal functions. The circulation of such an
appendage often includes a heat exchanger at the base of the limb so that,
in effect, a cold-blooded appendage and a warm-blooded body can be run
on the same circulatory system without huge losses of heat (Scholander
1957). Changes in blood viscosity will reduce flow to appendages when they
get cold quite without active adjustments within the circulatory system.
28
WHAT IS A FLUID?
More generally, they have shown that the temperature coefficient of
viscosity is certainly a variable that can be to some extent controlled, either in an
evolutionary or an immediate sense.
While we're considering the Antarctic, I should say a word about some
very peculiar fish that live there, the so-called ice fish (Ruud 1965). They
are the most transparent of adult vertebrates, but the transparency
appears to have come at the price of loss of red blood cells and a consequent
reduction in oxygen-carrying capacity, making them fairly sluggish
creatures. Still, there's a compensatory benefit—ice fish blood flows
substantially more readily than the bloods of more ordinary fish from the same
habitat (Wells et al. 1990).
Might warm, swimming animals release heat through the skin in such a
way that kinematic viscosity is locally lowered and drag is reduced? Aleyev
(1977) cites Parry (1949) to support a claim that the scheme is well
established for cetaceans, but I find no support in Parry's paper beyond some
calculations showing that cetaceans produce a whale of a lot of heat, and
some evidence that a complex circulation in blubber can actively control
the outward passage of heat. The latter is more likely just a scheme to
augment heat dissipation during and after high-speed swimming (Palmer
and Weddell 1964). Aleyev also cites Walters (1962), who speculated on the
possibility that tuna locally release heat behind a structure called the
"corselet" more or less amidships to reduce their drag. Walters is dubious, Webb
(1975) is dubious, and so am I.
Lower viscosity at higher temperature implies steeper velocity gradients
and thus thinner gradient regions on surfaces. In effect, a solid surface is
less shielded from free stream flow. That ought to help the exchange of
dissolved material between an organism and moving water around it. Does
the phenomenon aid gas exchange in fish gills and compensate, in part, for
the decreased solubility of oxygen in water at high temperatures?
Similarly, changes in viscosity ought to change the performance of filter-
feeding devices. Are the dimensions of such devices adjusted intra- or
interspecifically to reflect changes in ambient temperature? In at least one
case what changes is the food caught. An antarctic echinoderm larva
catches bacteria; a very similar larva from California catches microalgae
but not bacteria (Pearse et al. 1991). If California larvae feed in cultures to
which a viscosity-increasing polymer (methyl cellulose or
polyvinylpyrrolidone) has been added, they then take up bacteriophagy (Pearse, pers.
comm.).
Planktonic organisms are quite commonly more dense than the water
around them; according to Stokes' law (Chapter 15) they should sink more
rapidly at higher temperatures, although, as Hutchinson (1967) pointed
out, passive changes in their own densities might offset theef feet of altered
viscosity. Reportedly (by Sverdrup et al. 1942, for example) plankton from
29
CHAPTER 2
tropical waters are smaller and more angular and ramose than plankton
from polar water. That's what one would expect if keeping sinking rates
relatively constant were what mattered and if shape and size were the only
variables at work. But, as we'll see with a more specific case, this particular
world is a little hard to second-guess.
Individuals of many species of the microcrustacean, Daphnia, have larger
heads, much longer and more curved crests or helmets, or extra spines
when in warmer water (Figure 2.5). An old suggestion is that this
phenomenon, termed "cyclomorphosis," is an adaptation to viscosity differences
between warm and cool water. The difference is evident between
generations raised at different temperatures, seasonally within a single species, as
seasonal replacement of one group of species by another, and in
comparison with the fauna of climatically different ponds and lakes. Similar
seasonal polymorphisms are known in dinoflagellates and in rotifers. Certainly a
larger head and more flattened body should reduce sinking rates in less
viscous water, but that explanation has (as Hutchinson 1967 originally
noted) substantial difficulties. Hebert (1978) presented an alternative hy-
drodynamic explanation involving changes in the muscles used for
locomotion to offset sinking (Daphnia use their second set of antennae to swim,
so muscles in the head are entirely germane). But exposure to chemicals
released by certain predators will also induce the development of crests or
spines in Daphnia. While predation is not as effective on crested or spinose
individuals, the latter are less successful in terms of other aspects of their
life histories (Grant and Bagley 1981; Havel and Dodson 1987). To
complicate things further, the induction of crests and spines involves both genetic
as well as environmental factors.
Even the more general observation, mentioned earlier, that cold-water
plankton are larger and less surface-rich in shape is hard to tie
unequivocally to viscosity differences. The main problem is that sinking rate
depends on the difference between the density of the organism and that of
the medium, and assuming constant density is clearly unsafe. For instance,
embryos otEuphausia (another microcrustacean) sink at different rates at
different developmental stages (Quetin and Ross 1984); while large
specimens ofDaphnia pulex sink faster than small ones, the differences are much
less than considerations of size would predict (Dodson and Ramcharan
1991); and cyanobacteria (which are photosynthetic) change their density
with changes in light intensity (Kromkamp and Walsby 1990). To make
matters still worse, the density of active planktonic organisms is a bit tricky
to measure and isn't commonly done. (Walsby and Xypolyta 1977 describe
a technique using a specific gravity bottle and a nonpenetrating tracer to
determine the volume not occupied by organisms; alternatively sinking
distance might be watched in density gradients of some osmotically inactive
material such as "Percoll.")
30
WHAT IS A FLUID?
Figure 2.5. The microcrustacean water flea, Daphnia: (a) the warm-
water form; (b) the cold-water form.
At the very least, one ought not attribute some decrease in activity of an
aquatic organism at low temperature to reduction in metabolic rate
without checking to see if an increase in viscosity is contributory; the results of
Podolsky and Emlet (1993) on swimming speeds of sand dollar larvae
certainly make that point. But we're getting ahead of ourselves—less ex
cathedra treatment of each of these cases requires that additional physical
material be developed, which will happen for pipes and blood flow in Chapters
13 and 14, for filter-feeding and sinking plankton in Chapter 15.
31
CHAPTER 3
Neither Hiding nor Crossing Streamlines
We're now ready to examine what happens when fluids flow,
beginning with the most universally applicable notions and then moving
on to increasingly specific phenomena. Especially in this and the next few
chapters, the reader should bear in mind the ultimate artificiality of a
linear narrative. In particular, the biological examples may be nothing but
the truth, but—for want of material to be developed further along—
they're hardly the whole truth.
The Principle of Continuity
Consider a pipe that's open at both ends. If the pipe has rigid walls, it
must have a constant internal volume. From our assumption that fluids are
incompressible, it follows that, if a given volume of fluid enters one end,
then the same volume has to come out the other. Not only must the volumes
be equal, but entry and exit must take precisely the same time, and so the
volumes-per-time must be equal. This trivial idea, termed the "principle of
continuity," turns out to be surprisingly rich with biological applications
and implications. On occasion, though, it has been less than obvious to
practicing biologists.
To view the matter more formally, consider a pipe whose cross-sectional
area varies from one part to another, as in Figure 3. la. We'll call the cross
section near the entry S{ and that near the exit S2. If a small volume of
fluid, S{dl{, enters the pipe in an interval of time, dt, then an equal volume,
S2dl2, must leave the pipe in the same period. Thus
vj i at i o oflio
dt dt
But any dlldt is, of course, a velocity, so (using Ul and U2 for entry and exit
velocities)
SlUl = S2U2. (3.1)
The equation says that the product of the cross-sectional area and the
average velocity normal to the plane of that area is the same in both places
in the pipe. And the rule should hold for any cross section whatsoever—no
matter how the pipe expands, contracts, or changes shape, the product of
cross-sectional area and velocity will remain constant. Put another way, the
32
STREAMLINES
(a) (b) IS2
3»
u2
3*
i
i
i ~ -^
Figurl 3.1. (a) An increase in the overall cross-sectional area of a pipe
must be concomitant with a decrease in the average speed of flow,
(b) Dividing the pipe into a parallel array makes no difference. Local
velocities are proportional to the lengths of the arrows.
volume flow rate, or volume flux (we'll call it 0, does not change within a
conduit.
Now consider a pipe that branches, as in Figure 3.1b. We now have to
look not just at entry and exit but at any cross section, adding up the
contributions of the parallel conduits. But again fluid inside has no hiding
place, so for every volume that enters, an equal volume must leave. So the
rule is still a useful one since the sums of the area-velocity products must be
the same everywhere—at least if every bit of fluid passes our monitors once
and only once:
2 SlUl = £ S2U2. (3.2)
Note that no assumptions were made about energy or about friction; the
argument is purely geometric. That gives it enormously wide applicability.
In fact, were incompressibility not assumed, we would merely have had to
substitute pS for S and mass flux for volume flux for the principle still to
work. The principle of continuity has the same role and the same
generality for fluid mechanics that conservation of mass has for solids; it's really
just a specific application of the idea of conservation of mass; it's our
principal principle, continuously useful.
One garden-variety device based on continuity is the nozzle attached to a
garden hose. By constricting an aperture the water is persuaded to speed
up, and its increased momentum carries it a far greater distance than it
would go without the nozzle. Volume and thus mass flux is unchanged,
except for a little reduction due to the extra friction of the nozzle. But
momentum flux, loosely mass flux times velocity, may be increased several-
fold. The trick involves no power input by the nozzle; indeed, additional
power from a pump upstream might be needed to offset the extra fric-
tional loss. Analogously, with an appropriate input of power one can make
Ui
Si
33
CHAPTER 3
Figure 3.2. The column of liquid coming from a spigot contracts as it
accelerates downward, as it must according to the principle of continuity.
a stream of fluid constrict without a nozzle. That's what an ordinary axial
fan or a propeller does—since flow is more rapid downstream the effective
cross section of the moving stream must be less. Which is why, despite the
same volume flow rate in both places, you feel a fan's wind more strongly
downstream than upstream.
To emphasize the wide applicability of the principle of continuity we
might look at a relatively unusual use. If liquid falls freely from a
downwardly directed orifice (Figure 3.2), the column of liquid contracts. The
effect will be most noticeable with a highly viscous liquid, whose initial
speed can be kept low. The column contracts as it accelerates because its
cross-sectional area must always be inversely proportional to its velocity.
Knowing the acceleration of gravity, it's possible to obtain the rate of
discharge of the pipe from nothing more than two measurements of the width
of the column and the vertical distance between them. Alternatively, given
these latter measurements, a stopwatch, and a container of known volume,
one can make a fair estimate of the acceleration of gravity—one just
combines equation (3.1) with the ordinary equation for distance covered at
constant acceleration beyond an initial speed.
Continuity continually makes large-scale mischief. The old London
bridge, the one that lasted from 1209 to 1832, rested on a set of boat-
shaped piers so wide that almost half the Thames was blocked. As a result,
the flow between the piers should have almost doubled the already rapid
tidal currents. In fact, the cutwaters that were necessary to protect the piers
from scour made the situation even worse. Only small boats could get
through, and these had to take careful aim and considerable risk in doing
what was known as "shooting the bridge" (Gies 1963).
34
STREAMLINES
Applying Continuity to Bounded Flows
Largish creatures, whether plants or animals, devote considerable
anatomy to internal fluid transport systems. The very existence of macroscopic
organisms is predicated on such systems, devices to circumvent the nasty
difficiency of diffusion for all but very short distance transport. LaBarbera
and I (1982) repeated Krogh's (1941) argument on the point rather
elaborately, and I've pushed it more recently as well (Vogel 1988a, 1992a). We
give these systems that move fluid in bulk various names—circulatory,
respiratory, translocational—but they all do much the same job, moving
fluid from one site of material exchange to another. Each must contend
with several partly conflicting physical imperatives. For decent power
economy, long-distance transport is best done in pipes of large cross-
sectional area. Large pipes will have greater volume relative to their wall
area and thus relatively less surface at which (recall the no-slip condition)
viscosity can work its malicious mischief. For effective exchange of material
between the fluid and the surrounding tissue, pipes of small cross-sectional
area are much better. Exchange is ultimately dependent on diffusion;
again, that's an agency effective over only short distances within either fluid
or tissue.
Consider an implication of the principle of continuity for such a system.
Simply narrowing the large pipes at sites of diffusive exchange would,
according to continuity, result in very high speeds. Power losses would be
great, and little time would be available for diffusion to occur between
tissue and any element of fluid. All internal bulk fluid transport systems,
whatever their function, do use both large and small pipes; but in practice
fluid always moves fastest in the largest pipes and slowest in the smallest
ones. While this may sound like a violation of the principle of continuity, it's
not. Rather, organisms just apply equation (3.2) and make the total cross-
sectional area of the small pipes very much larger than that of the large
ones, as in Figure 3.3 and Table 3.1.
People
We have perfectly ordinary plumbing. The output of each side of the
heart of a resting human is about 6 liters per minute (10~4 m3s-1). The
ascending aorta and the main pulmonary artery have internal diameters of
about 2.5 cm and thus cross sections of about 5 cm2 (5 x 10~4 m2). Thus the
resting rate of flow in either is around 0.2 m s~1. A single capillary is about
6 |xm in diameter or about 30 |xm2 (30 x 10~12 m2) in cross section. Blood
flows through it at roughly 1 mm s_1 (10~3m s-1). Thus the product of
velocity and cross section for a capillary is some 3 x 109 (3 U.S. billion)
3 5
CHAPTER 3
left heart
pulmonary
capillaries
systemic
capillaries
right heart
Figure 3.3. An extremely diagrammatic view of an avian or a
mammalian circulation. Lower speed in the capillaries than in the heart is
possible only if the total cross-sectional area of capillaries exceeds
that of each half-heart.
Table 3.1. The sizes and numbers op ihe pipes versus ihe speeds
of flow in several fluid transport systems.
Element
Area
(mm2)
Number
Total Area Flow Speed
(mm2) (mms-1)
Oak tree, 2.5 cm trunk (Kramer and Koslowski 1960; Lundegardh 1966)
Xylem vessels, trunk 7.9 x 10"3 380 3.0 10
Leaves 5000 350 1.8 x 10fi 1.7 x 10"'
Dog circulatory system (Caro et al. 1978)
Aorta
Large arteries
Arterioles
Capillaries
200
20
2.1 x 10-^
3.0 x 10"r>
Sponge, 2.4 cm^ volume (Reiswig 1975a)
Ostia (input holes)
Incurrent canal apertures
Flagellated chambers
Excurrent canal apertures
Osculum
3.3 x 10"1
0.031
7.1 x 10- <
0.11
3.4
Idealized human lung (Weibel 1963)
Trachea
5th generation bronchi
10th generation bronchi
15th generation bronchi
20th generation bronchi
250
10
1.3
0.34
0.17
1
20
6 x 10«
1.9 x 10(»
9.4 x l(K>
3,400
2.9 x 107
280
1
1
32
1000
3.3 x 1()»
1.0 x l()«
200
400
1.2 x i0-»
5.7 x 104
310
100
2 x l()i
31
3.4
250
310
1300
1.1 x l()»
1.7 x K)r>
200
100
3.2
0.7
0.57
1.7
0.0087
0.57
0.07
200
160
38
4.4
0.3
36
STREAMLINES
times less than the equivalent product for the ascending aorta or the main
pulmonary artery. So there must be very roughly 3 billion parallel
capillaries in the systemic circulation receiving the output of the aorta. These
should have an aggregate cross-sectional area of about a tenth of a square
meter—approximately a square foot. That total cross section is some 200
times greater than that of the aorta, accounting (by continuity) for the 200-
fold drop in the speed of flow of the blood.
Admittedly these figures (from Caro et al. 1978 and Milnor 1990) are
very rough ones. At rest not all capillaries are open and operational, and
lots of other features of the microcirculation, such as arterio-venous
shunts, have been ignored. But however approximate, that estimate of a
couple of billion parallel capillaries was bought at very low cost! In our
mammalian kind of circulatory system, the volume flow rate through the
lungs must be exactly equal to that through the entire systemic set of pipes.
Thus if lung capillaries were exactly the same size and conveyed blood at
the same speed as capillaries elsewhere, their aggregate cross section would
have to be the same tenth of a square meter. As it happens, lung capillaries
aren't quite the same; nonetheless, lungs obviously have a lot of capillaries
and are exceedingly bloody organs.
Trees
The ascent of sap in the xylem of trees (Figure 3.4a) provides another
case where one of the variables in equation (3.2) cannot easily be measured.
It's no minor matter to invade a system in which the largest elements are
conduits that are only a fraction of a millimeter in diameter, with stiff,
strong walls where internal pressures are negative by many atmospheres.
To give some idea of what transpires in trees, I've taken some data from
Lundegardh (1966) for the leaf area of an inch-diameter (2.5 cm) oak tree
(Quercus robur) with 300-400 leaves. I've combined these with sap velocities
for the trunk and with transpiration rates for the leaves of Q. rubra, from
Kramer and Koslowski (I960), to address some questions raised by P. J.
Kramer (1959), Zimmermann (1983), and others.
If the tree has two square meters of leaf area (S{) and transpires water at
1.5 x 10~8 nvss~l (a liquid volume of about two ounces per hour) for each
square meter of area, then the volumetric water loss (Sjf/j) is 3 x 10~8
mHs-1. The 100 |xm vessels of the trunk make up about 7% of its total cross
section; thus their aggregate cross-sectional area is 1.5 x 10~4 m2 (S2).
Dividing SlUl by S2, we calculate a rate of sap ascent in the vessels of 2 x
10~4 m s-1, a fifth of a millimeter per second. But the rate of ascent of sap
is not hard to measure, at least roughly. A pulse of heat is applied to the
trunk, and the time is noted for the arrival of heated sap beneath a sensor a
few centimeters higher. Such measurements give ascent rates of about a
37
CHAPTER 3
Figure 3.4. Changing cross-sectional area and thus speed of flow in
(a) a tree, (b) a sponge, and (c) a clam. The numbers in the text and in
Table 3.1 should emphasize the impossibility of drawing such hydraulic
systems with consistent scales!
centimeter per second, fully fifty times higher than the calculated rate.
Apparently the majority of the vessels are filled with air—that is, they have
embolized—and are nonfunctional. Air embolisms, mainly resulting from
gas release when sap freezes, are a major hazard for systems in which an
aqueous liquid is under great negative pressure (see, for instance, Tyree
and Sperry 1988). While some embolisms can be repaired, it's not
uncommon for embolized vessels simply to be left unused as conduits for sap
(Zimmermann 1983).
One doesn't associate rapid flows with plants, but continuity demands
that transpiration from the great surface areas of broad-leafed plants be
reflected in rapid sap movement in the vessels of stems and trunks. A
centimeter per second, while relatively rapid and characteristic only of
trees with fairly wide vessels, is far from any record. In vessels of the
taproot of wheat plants, Passioura (1972) measured speeds up to 25 cm s~l,
and even these rates could be raised 3-fold by experimental manipulations.
Sponges
Sponges are little more than highly elaborate manifolds of pipes, with
lots of small pores and one or a few large (commonly apical) openings on
their surfaces (Figure 3.4b). Grant (1825) established that flow was
unidirectional—into the small openings and out through the larger
ones—and he was much impressed by the rate at which a sponge could
move water. But what was doing all the pumping? Sponges were known to
38
SI REAMLINES
have flagella, but spongologists persisted in invoking muscles that just as
persistently remained undetectable. It just wasn't credible that tiny, slow,
flagella could make a sufficient pump. The pumping is certainly
impressive—a sponge ordinarily pumps a volume of water equal to its own
volume every five seconds (Reiswig 1974), a rate roughly a hundred times
that of a human heart relative to body volume. In the fine words of Bidder
(1923), a sponge is "a moment of active metabolism between the unknown
future and the exhausted past." It was Bowerbank (1864), almost forty
years after Grant, who recognized that flagella could do the job. The
solution to the problem entailed nothing more than application of the
principle of continuity.
Again, let's view the situation quantitatively. (I'm using mainly the
morphological data given by Reiswig 1975a.) A small sponge of 100,000 mm3
will ordinarily have an output opening 100 mm2 in cross-sectional area and
an output velocity of 0.2 m s~l. It's difficult to envision a flagellum only 25
|xm long pumping water at more than 50 |xm s_1, a velocity 4000 times
lower than that of the sponge's outflow. But the total cross-sectional area of
the flagellated chambers proves to be nearly 6000 times greater than the
area of the output opening. Thus one needn't make any heroic
assumptions about the pumping speeds of flagella, much less make muscles
mandatory. Interestingly, small sponges put a final constriction on their output
apertures. Bidder (1923) recognized these as nozzles that increased the
speed and coherence of the output jet and thus minimized the chance that
the animal might uselessly refilter its own output.
Similar arrangements are evident in other filter-feeding animals such as
bivalve mollusks (Figure 3.4c). In these latter, though, both intake and
output apertures have low cross sections and consequently high velocities;
input and output conduits are connected by distributing and collecting
manifolds to large, slow-speed, ciliary filter pumps. For that matter, one
can imagine a circulatory system much like our own but driven by ciliated
capillaries. Such a system might achieve the normal speeds in the large
vessels in the complete absence of a pumping heart. One drawback of such
a scheme is its operating cost—muscles manage much better than cilia or
flagella by that criterion. For sponges and bivalves, the functions of pump
and filter are combined, so the economic argument must not be so directly
applicable (LaBarbera and Vogel 1982). Also, as we'll see in Chapter 14,
ciliary and flagellar pumps do better for low-pressure, high-volume
applications than for systems with pressures as high as those of mammalian
circulations.
Less Completely Bounded Systems
At times the rate at which water is carried by a stream or river (the
"discharge") increases substantially. A placid stream may metamorphose
39
CHAPTER 3
into a raging torrent in which the depth and width obviously increase and
the velocity seems far greater than normal. How much does the velocity
really increase? If it were to increase in proportion to the discharge, then,
by the principle of continuity, the river shouldn't rise! Clearly, average flow
speed doesn't increase as fast as discharge; in fact, the increase in velocity at
a given place that is submerged under both normal and flow conditions
turns out to be quite modest. The graphs given by Dury (1969) and by
Leopold and Maddock (1953) indicate that mean current speed at a
particular station along a river only doubles for every 10-fold increase in
discharge. Since peak velocity in a larger cross section will occur farther from
bottom and sides of a channel, the actual speeds near the interfaces are
likely to increase even less. Being swept downstream by the high water
speeds of floods may just not be as worrisome to small attached or bottom-
living organisms as we might otherwise think.
As the discussion in Hynes (1970, pp. 224-229) implies, abrasion and
alternations of the form of the bottom are more important than velocity
increases per se. Thus during a long-term study of an ordinarily well-
behaved stream in Denmark, a one-day, 70 mm rainfall caused a brief and
rare flood. The effect on the fauna of the gravel and sand on the bottom
was slight; the main casualties were due to movement of boulders—some
snails (Ancylus) and some caddisfly pupae (Wormaldia) that lived on those
rocks got crushed as they rolled (Thorup 1970).
An analogous situation occurs near an irregular bottom beneath moving
water. If a portion of the bottom is elevated above the general terrain, then
the speed of flow will be greater across it—it can be viewed as half of a
longitudinally sliced nozzle. The constriction effect isn't quite as severe as
in a system with a solid boundary all around or even one with an air-water
interface just above, but the speeding up of flow and its biological
consequences are far from negligible. To give just one example, Genin et al.
(1986) looked at seamounts beneath several thousand meters of ocean.
Regions of 2-fold increases in flow on the seamounts were associated with
3-5-fold increases in densities of a black coral (Stichopathes) and some
gorgonian corals. The density differences were attributed to better
recruitment and growth of these passive (current-dependent) suspension feeders,
since the concentration of edible material on the seamounts was quite
clearly lower than on the nearby floor.
Streamlines
How might this principle of continuity be applied to situations other
than flow through pipes or flow in discrete, homogeneous streams? Can we
make full use of the principle for open fields of flow, in particular for cases
in which the velocity of flow varies across the field of flow as well as in the
4 0
STREAMLINES
direction of flow? The principle, if anything, proves even more potent in
such situations—in part because we are less likely to use it intuitively, and so
the results of its application are less self-evident. The conceptual device
that enables us to apply the principle of continuity to open flow fields is
something called a "streamline."
A streamline, it should be stressed at the start, has a very special meaning
in fluid mechanics, a meaning that bears only an indirect connection to its
vernacular use. A streamline is a line to which the local direction of fluid movement
is everywhere tangent. Loosely then, a streamline traces a path through a field
of flow along which some particles of fluid travel. Let's assume we know the
direction of flow at all points in a flow field at some instant in time. To create
a streamline, we start at some upstream point and draw a very short line in
the direction of flow at that point; we then consider the direction of flow at
the far end of the line and extend the line a short distance in the new
direction. We continue the process until the line wends its way across the
entire field, as in Figure 3.5. What have we done by this apparently trivial
and obviously awkward procedure? The line follows the stream, and the
direction of flow is always along the line. Therefore any component of
velocity normal to the line must be zero; in short, fluid does not cross the line.
Viscosity may move momentum, conductivity may move heat, diffusion
may move molecules, but fluid in bulk doesn't ordinarily move across the
line.
It's no more difficult to create a second streamline, running in tandem
with the first; and with two, we can get to the real point. Streamlines
provide a conceptual device for dividing a complex field of flow into an
array of pipes with nonmaterial walls. For a simple, two-dimensional flow
(no convergence or divergence outside the plane of the paper or the screen
of the machine), the principle of continuity must apply between the pair of
streamlines. If we want to deal with properly three-dimensional flow, we just need
a set of streamlines that surround some fluid, a so-called stream tube, within
which continuity is equally applicable. So what all of these lines and tubes
lead to is really quite wonderful. Where a pair of streamlines diverge or a
stream tube becomes wider, we know immediately that the fluid is traveling
more slowly. If streamlines converge or tubes become narrower as one
moves downstream, that's a definitive indication that velocity is increasing.
Pathlines and Streaklines
But how can we draw streamlines in the real world? The direction of flow
at all points is rarely a matter of public record; indeed, making streamlines
is more commonly used to determine flow direction than the other way
around. In practice, two fundamentally different schemes can be used. In
the first, a visible marker or particle of some sort is released near the
41
CHAPTER 3
Figure 3.5. A streamline (thick line) can be drawn from information
about the local direction of flow (arrows) since it's always tangent to the
local flow.
upstream end of the flow in question. Ideally, the marker is neutrally
buoyant and very small, so it always travels in the local flow direction. A
time exposure or repetitive photograph of its travel gives a solid or dotted
line recording the history of the marker; the record is called a "pathline" or
particle path, as shown in Figure 3.6a. In the second, a continuous stream
of particles, dye, tiny bubbles, or smoke is introduced at a fixed point; and
some time during the process an instantaneous photograph is taken. It
gives the present position of fluid that has, over a period of time, passed by
the injection point; it is called a "streakline" or "filament line." Most often,
streams are introduced at an array of points normal to the flow direction,
so a whole set of simultaneously produced streaklines are recorded in a
single image, as in Figure 3.6b.
If the flow is steady, that is, if velocity at all points is constant over time,
then pathlines and streaklines coincide, and both mark the streamlines. If
flow is unsteady, neither, strictly speaking, marks streamlines, and
everything becomes much more complicated. But, as mentioned earlier, we'll
deal mainly with steady flows. Photographs showing streamlines both
encapsulate a vast amount of information about a field of flow and provide an
overall view of complex flows that we visual creatures find intuitively useful
and satisfying. Besides that, they are commonly objects of great esthetic
appeal—one has only to look at Van Dyke's (1982) collection to be forever
convinced of that.
Biologists don't make as much use of this valuable tool as I think they
ought to. As qualitative views, streamline patterns are very useful for
understanding the complex flows both around organisms of irregular shapes
and in common environments. Exploration with a syringe of dye of the
flow in any stream with irregularities in its bed leads to a whole new
appreciation of the diversity of habitats with respect to flow. Upwellings, down-
wellings, local upstream flows, places with unsuspected periodic flows,
places in which flow either enters or leaves the substratum—all emerge
from even a casual and unsystematic survey.
Moreover, quantitative information can be extracted; where organisms
resist embellishment with instruments, photographs of streak- or pathlines
42
SI REAMLINES
.
• • •
. •
# • • .
*
(a) Pathlines
(b) Streaklines
Figure 3.6. Two kinds of streamlines, (a) "Pathlines" show the paths
taken by particles as they flow through the field of view—the view
extends over a period of time, as in a multiple- or long-exposure
photograph, (b) "Streaklines" give an instantaneous view of the position of
markers (dye streams, for instance) steadily released upstream.
may be the most informative and least abusive way to get information on
velocities in their full, three-dimensional vectorial splendor—and from
these a picture of drag, thrust, locomotory mechanisms, and so forth. Thus
Kokshaysky (1979) induced a bird to fly through a cloud of wood particles
while these were photographed with a repetitive stroboscope. The views he
and others obtained of the vortex rings behind flying birds have
fundamentally changed our view of the aerodynamics of avian flight; we'll have a
lot to say about these vortices in Chapter 10. Similarly, a view of wind
pollination as a very much more sophisticated business than anyone
guessed earlier has come from a series of papers by Niklas (see, especially,
1985 and 1992) on airflow patterns around flowers and conifer cones.
Apparently plants contrive shapes that interact with local flows as pollen
traps and that may even achieve some degree of specificity for the
appropriate kind of pollen.
Some instruments yield data for direction of flow—for instance, tufts of
string attached to test objects or probing needles; some instruments give
data for speed of flow without specifying direction—hot wire and
thermistor anemometers are examples. Streamlines can be used to obtain the
complementary data from the other. If one has the pattern of streamlines
determined from information on direction together with a single datum
43
CHAPTER 3
for speed, one can use the principle of continuity and the definitional
unlawfulness of crossing a streamline to get speed at all other points in the
flow field. If one knows a single line of flow across a flow field (such as the
location of the substratum), one can use a map of speeds to derive
the pattern of streamlines and thus the direction of flow at all other points.
Small size doesn't preclude marking streamlines to get information on
speed and direction of flow at specific places. If anything, flow visualization
really comes into its true glory on the size scales relevant to individual
organisms a millimeter or centimeter long. On the one hand, little else may
be available for mapping flows (or measuring forces—see Chapter 4); on
the other, both the low speeds and the absence of turbulence greatly
simplify matters. Streaklines can be made down nearly to a tenth of a
millimeter in diameter by injecting dye from a micrometer-driven syringe through
a piece of polyethylene catheter tubing with the tip drawn out. With the
injector on a manipulator, the investigator can look at how flow is drawn
into and passes through biological filters. Lidgard (1981) used the
technique to map colonywide surface currents and functional excurrent
chimneys in encrusting bryozoa while watching through a stereomicroscope.
And LaBarbera (1981) mapped flow patterns in, through, and out of
several kinds of brachiopods only a centimeter or two long, showing how they
avoided either internal mixing or any recirculation of previously filtered
water. At this scale a stream of fluorescein dye passes right through a
lophophore of tentacles and emerges intact as a beautifully laser-straight,
very slow (10 mm s_1) green jet.
Pathlines are at least as useful. Figure 3.7 (from Vogel and Feder 1966)
gives a view of pathlines around a model of a fruit-fly wing immersed in a
moving liquid. This shift from air to water will be described in Chapter 5;
for present purposes it brings up the important point that flow
visualization, clearly the most direct way to obtain either streaklines and pathlines,
is considerably easier in liquids than in gases. Incidentally, the biologist
should not overlook organisms as sources of particles for marking
pathlines—such things as freshly hatched brine shrimp (Artemia) and
cultured algal cells are of handy size and density.
Timelines and Isotachs
Two other kinds of flow maps are of considerable use; neither is a form of
streamline although streamlines can be derived from either. The first is
something usually called a "timeline." At a given instant a marker in the
form of a continuous or periodically interrupted line of dye, smoke, or
electrolytic bubbles is introduced, usually normal to the direction of flow.
That line then moves downstream and is periodically illuminated. A time-
exposure photograph thus gives a set of lines, with the distance between
44
STREAM LI NES
Figure 3.7. Pathlines around an inclined flat plate immersed in a
rotating bowl of water in which particles have been suspended. The plate was
8.5 mm across, and the rate of flashing of the stroboscope was 12 s_i.
equivalent loci on adjacent ones inversely proportional to the speed of flow.
Instead of moving around immersed objects as do streamlines, timelines
enwrap them in tangles. They're particularly useful for showing local
reversals of the direction of flow, as often occurs behind points of flow
separation (Chapter 5).
The second nonstreamline map is one of speeds without consideration
of direction, one obtained directly from traverses with a flowmeter or
anemometer. Lines are drawn to follow equal values of flow speed, like the
isotherms and isobars for temperature and barometric pressure on a
weather map. Unfortunately no name for these lines is universally
accepted; I like the term "isotach," but I've seen "isovel" and other synonyms
at least as commonly.l We'll encounter a specific use of isotachometric maps
in Chapter 4; Figure 3.8 gives an example of one. Such diagrams may be
1 "Isovel" is a miscegenation of Greek and Latin as offensive as "automobile," which
I've been told should properly be "autokineton."
4 5
CHAPTER 3
Figure 3.8 Isotachs for flow around a swimming copepod (a micro-
crustacean), Pleuromamma xiphias, made by David Fields andjeannette
Yen. The animal is about a millimeter long and is swimming upward.
uncommon, but they ought to be of special value to biologists dealing with
sessile organisms that protrude into spatially irregular flows or with the
habitats of organisms that prefer exposure to certain velocity gradients.
Laminar and Turbulent Flows
Turbulence was mentioned as a nuisance for visualizing flow; since it has
surfaced, we should no longer postpone the business of distinguishing
between laminar and turbulent flow. The existence of these two radically
different regimes of flow is another of the strange complications in the
behavior of fluids and another phenomenon that gets pretty poor
treatment in common practice or parlance. Perhaps the way to begin is to admit
that when introducing viscosity in the last chapter, another assumption was
tacitly made. We assumed that "layers" of fluid slipped smoothly across one
another with all particles of fluid moving in an orderly, unidirectional
fashion. And when defining streamlines, I slipped in the word "ordinarily"
when forbidding fluid to cross streamlines. These presumptions that all
fluid particles move very nearly parallel to each other in smooth paths are
strictly valid only for what we term "laminar flow." In it the large- and
46
STREAMLINES
small-scale movements of the fluid are the same, at least down to the level at
which molecular diffusion becomes an appreciable mode of transport.
In "turbulent flow," by contrast, tiny individual fluid particles (a
particularly useful polite fiction at this point) move in a highly irregular manner
even if the fluid as a whole appears to be traveling smoothly in a single
direction. Intense small-scale motion in all directions is superimposed on
the main large-scale flow. Turbulence is essentially a statistical
phenomenon, and descriptions of overall motion in turbulent flows should not be
presumed to describe the paths of individual particles. The easiest analogy
is to diffusion, in which Fick's law works admirably for the overall
phenomenon but says almost nothing about what any molecule will be doing at any
particular instant. The difference between turbulence and diffusion is
mainly one of scale—diffusion is a molecular phenomenon while turbulent
motions happen on much larger (if still sometimes quite small) scales. By
convention—which is to say for convenience—turbulent flows aren't
considered automatically unsteady.
In turbulent flow, then, not only is momentum transferred across the
flow—which is what viscosity accomplishes on a more limited scale—but
actual mass similarly shifts around in directions other than that of the
overall flow. The transfer of mass is formally analogous to that of
momentum, and something analogous to dynamic or molecular viscosity can be
used as a measure of the intensity of turbulence. That's the "eddy viscosity"
mentioned in the last chapter; its value is zero for laminar flow. (See
Sverdrup et al. 1942, Hutchinson 1957, Massey 1989, or books on
meteorology or physical oceanography for further information.) It's the eddy
viscosity to which reference is made in some often-quoted doggerel of the
meteorologist L. F. Richardson (itself a parody of the more ecological
original):
Big whirls have little whirls
Which feed on their velocity;
And little whirls have lesser whirls,
And so on to viscosity.
The distinction between these two regimes of flow has been recognized
for a long time, as has the abrupt character of the transition between them.
One can describe it no better than did Osborne Reynolds (1883), who
introduced a filament of dye into a tube of flowing water and watched how
it behaved as the current was altered (Figure 3.9):
When the velocities were sufficiently low, the streak of colour
extended in a beautiful straight line across the tube. If the water in the
tank had not quite settled to rest, at sufficiently low velocities, the
streak would shift about the tube, but there was no appearance of
47
CHAPTER 3
laminar flow
-^
turbulent flow
Figure 3.9. A diagrammatic version of what Reynolds did—the
behavior of a stream of dye in laminar and turbulent pipe flow. The apparatus
isn't entirely impractical. Flow in the upper pipe will be slower than that
in the lower pipe, and the rates can be adjusted by changing the height of
water in the tank so that the upper stream is laminar and the bottom one
turbulent.
sinuosity. As the velocity was increased by small stages, at some point in
the tube, always at a considerable distance from the trumpet or intake,
the colour band would all at once mix up with the surrounding water.
Any increase in the velocity caused the point of break-down to
approach the trumpet, but with no velocities that were tried did it reach
this. On viewing the tube by the light of an electric spark, the mass of
colour resolved itself into a mass of more or less distinct curls showing
eddies.
What both quotations should emphasize for the reader is that, just as one
swallow never makes a summer, one vortex or even a few discrete vortices
does not mean the flow is turbulent. Turbulence, again, consists of
temporally and spatially irregular motion superimposed on the larger pattern of
flow.
While engineers are almost exclusively concerned with turbulent flows,
both kinds of flows are of biological interest, as are cases in which part but
not all of a flow field is turbulent. In general, small, slow organisms and tiny
pipes experience laminar flows, and large, fast organisms and large pipes
experience turbulent flows. Reynolds is best known for deducing the basic
rule governing the transition point for flow in pipes (see Chapter 5). At this
point, I'm mainly concerned that the reader bear in mind that (1) two such
48
STREAMLINES
regimes exist, (2) the transition is often abrupt, (3) the practical formulas
for dealing with the two are quite different, and (4) much of biological
consequence occurs near the transition point, where our a priori
expectations are least reliable and where, as a result, nature has unusually rich
opportunities to surprise us.
49
CHAPTER 4
Pressure and Momentum
Force is the basic currency of Newtonian mechanics—all three laws of
motion are stated in terms of force—and force will play an important
role here. For many purposes, though, force provesjust a little too abstract.
While the force of a blunt knife might be the same as that of a sharp one,
the latter may penetrate where the former does not. What most often
matters is the force divided by the area over which it's applied. That,
though, can be expressed by either of two variables with the same
dimensions and the same units, but which are most emphatically not the same.
The simpler variable, conceptually, is stress, something we met in the form
of shear stress when defining viscosity. It's especially useful in solid
mechanics—push on an object, and a force per unit area, a stress, is
exerted in the direction of the push. In other directions both force and stress
are reduced by the usual rules for resolution of forces. Push lengthwise on
a rod, and you exert a stress on whatever is in contact with the end of the
rod—a stress equal to the force with which you push divided by the area of
the rod's end. Any outward or radial force can ordinarily be ignored.
Fluids, of course, have no preferred shape—if you push on a (suitably
contained) fluid, it tries to squidge out in every direction with equal
urgency. Push on a piston in a cylinder (such as a hypodermic syringe or
automotive shock absorber), and you not only exert a stress on the end of
the cylinder but on the side walls as well. For such an omnidirectional
response we drop the word "stress" and substitute pressure. You exert a
pressure on the inner walls of the cylinder as well as on the inner walls of
anything with which the fluid in it is continuous. Moreover, that pressure is
the same everywhere (ignoring gravity)—the force you exert times the
area of the face of the piston1 on which you push, as in Figure 4. la. Make a
tiny hole anywhere in the system, and the fluid will squirt out with equal
impetus.
It sounds like something for nothing—force ends up exerted,
undiminished, over a very much greater area than that over which it was
applied. But no conservation law has been violated since nothing has
moved and thus no work has been done; the gain is no different from that
brought with a lever or system of pulleys. You just have to get used to this
1 Or, if the face isn't normal to the force, then the normal area of the face of the piston.
5 0
PRESSURE AND MOMENTUM
Figure 4.1. (a) Stationary fluids exert their pressures equally in all
directions, which means that (b) the pressure on the bottom of a column of
liquid depends on the height of the column but not on its shape. A wider
and thus heavier column has more nonvertical surface somewhere that
takes the extra force; pressure, force per area, is unimpressed.
odd omnidirectional character. It's at its queerest when one considers
what's sometimes called the "hydrostatic paradox." A column of liquid
(Figure 4. lb) of a specific height but of any shape at all will exert a specific
pressure on its bottom or on any side near the bottom. That pressure will be
determined by three variables only—the height, the density of the liquid,
and the acceleration of gravity. Neither the shape of the column nor the
total weight of the liquid matters. Odd—but that's the great advantage of
using pressure as a variable. Were we to consider the downward force or
stress of the column of liquid, the downward component of the force it
exerts on all surfaces exposed to it, then we'd have to do some pretty
complicated bookkeeping to account for all the orientations of all the bits of
surface.
Thus if one dives beneath the surface of a body of water, the pressure
rises in proportion to one's depth at a rate of about a tenth of an
atmosphere per meter (more specifically, 9800 Pam-1 in fresh water and 10,000
Pa m_1 in seawater). As with the enclosed column, only height or depth
matters. And the pressure is exerted equally in all directions, even upward;
so you aren't awkwardly accelerated in any direction as a consequence of
that pressure. Similarly, the weight of the atmosphere above you,
approximately equal to that of two elephants, presses on you equally in all
directions and thus normally makes no trouble. These pressure are termed
"static pressures" in general or "hydrostatic pressures" for the special case
of diving.
51
CHAPTER 4
Bernoulli's Principle
But what about pressures caused by fluid motion, the sorts of pressures
that blow trees over and keep birds aloft? These get a little more
complicated; indeed in a sense they're what the rest of this book is about. The most
convenient place to start is with a most peculiar notion, the principle (or
equation) of Bernoulli, in particular of Daniel Bernoulli (1700-1782), one
of several mathematical eminences of a single disputatious family. Recall
that the principle of continuity was obtained solely from geometry and is a
simple and general concept with no exceptions lurking about to trap the
unwary. Bernoulli's principle may be the only bit of fluid mechanics that
most of us were taught; it involves such disquieting assumptions that one
can reasonably wonder if it is ever trustworthy.
The worst of these assumptions is that of an "ideal fluid," a really oxy-
moronic notion. An ideal fluid is the name given to a fluid of zero viscosity,
and viscosity was the property used to recognize a fluid in the first place.
The point is that, lacking viscosity, the fluid doesn't lose momentum to
internal friction or to the walls of any container through which it flows.
To get a feeling for the others, it's perhaps best to go through a
derivation of the principle.2 Assume that an element of fluid moves through a
length of pipe varying in internal diameter (S) and in height above the
earth (z) , a pipe of the form shown in Figure 4.2. The volume of the
element of fluid will be Sdl; its mass will be pSdl. Reverting to differentials,
the height difference between the ends of the region of pipe will be dz, and
the pressure difference between the ends will be dp.
We apply Newton's second law, the rule that force equals mass times
acceleration, to the element of fluid. In this case the overall force is the sum
of any pressure force and any gravitational force; since increases in either
pressure or height will slow the fluid, both forces must have minus signs:
— pressure force — gravitational force = mass x acceleration
dU
Sdp — pgS dz = pSdl
dt
If the terms are now divided by pSdl, they then have dimensions of
force/mass; dividing and rearranging, we get
dP + g dz + du _ 0
pdl dl dt
2 In the earlier edition I followed the practice of most physics textbooks and worked
from conservation of energy. It's easier than what's done here, but I was informed by
Stanley Corrsin that it wasn't quite legitimate, it didn't really expose the assumptions, and
it jumbled history.
52
PRESSURE AND MOMENTUM
Figure 4.2. Bernoulli's principle. The decrease in velocity associated with
an increase in the cross-sectional area of a pipe causes an increase in
pressure (center). An increase in elevation of the pipe (right) causes
a decrease in pressure.
This dt is awkward for a continuing process. To get rid of it we assume
steady motion so U = dlldt and that dt = dllU. Consequently,
dp gdz U dU _
pdl
dl
dl
Now we want to integrate with respect to dl; to do that we have to assume
that the density, p, is constant. We get
- + gz + —— = constant,
p 2
(4.1)
The result is Bernoulli's equation. It states that the sum of a pressure
term, a gravitational (or height) term, and a velocity term remains constant
as the fluid flows through the pipe. The dimensions, though, are unhandy,
so we multiply by density to get forces over areas, or pressure terms:
pU2
P + — + PS*
constant.
(4.2)
Not only do these terms have dimensions of pressure, but they really do
determine the magnitude of the explicit and ordinary pressure in the first
term—the gauges in Figure 4.2 can be quite real items in lab or classroom.
Decrease the second term by expanding the pipe (using of the principle of
continuity) and the first will locally increase. Increase the third by elevating
part of the pipe and the first will decrease there.
Bernoulli's equation in pressure terms is used to define some terms (in
the verbal sense of "term.") The sum of the "static pressure" (/?), the
"dynamic pressure" (pc/2/2), and the "manometric height" (pgz) remains con-
53
CHAPTER 4
stant as the fluid flows. Static pressure is the ordinary sort that we
encountered when talking about hydrostatic pressures, the kind one measures on
pressure gauges. Dynamic pressure is the pressure invested in movement of
the fluid—if the fluid were suddenly brought to a halt without fractional,
thermal, or other such shifts in who's-got-the-energy, then that pressure
would appear as an increase in static pressure. Manometnc height is height
expressed in terms of pressure—the pressure exerted by a column of
liquid of a specified density and height.
The constants in the previous equations are a minor nuisance. One way
to circumvent them is to consider the changes in the magnitude of each
term as the fluid moves from an upstream point (subscript 1) to a
downstream point (subscript 2). We're rarely interested in absolute pressures
anyway—how often do you have to use/? rather than A/?? Bernoulli's
equation then becomes
(Pi ~ Pi) + 9{U2' 2 U,2) + pg(z* ~ z'> = °- (0)
And frequently the gravitational term can be omitted; even if it's ultimately
relevant, one can often arrange a simplified situation in which the fluid has
no net upward or downward motion, so
<,,-,,, + «£Ll££=o. (4.4)
Incidentally, the figures that were used in Chapter 2 to argue for the
practical incompressibility of air were obtained by using this last equation.
One just takes the most extreme case of air with an initial velocity of U2
brought to a halt so Ul = 0 and calculates a A/?. That pressure difference is
then compared to the atmospheric pressure, 101,000 Pa. An initial speed
of 20 m s_1 would, for example, give a maximum local compression of
about half of one percent.
Manometry
To begin exploring how Bernoulli's principle can do useful service in a
real world, let's consider some devices based on it, especially machines that
can be used to measure speeds of flow. But as a start, assume a fluid at rest;
clearly the second term in equation (4.3) will be zero, and
A/? = pgAz. (4.5)
That equation might look familiar—it's the basic formula for column
manometry, in which a difference in height (Az) of ends of a manometric fluid
of density p is used to measure differences in pressure, as in Figure 4.3a.
54
(a)
PRESSURE AND MOMENTUM
air bubble
Az
r,
Figure 4.3. (a) A U-tube manometer for use with a liquid in air—the
difference in height of the columns multiplied by liquid density and
gravitational acceleration gives the port-to-port pressure difference,
(b) A multiplier manometer, in which a small change in the heights of
liquid in the two jars causes (by the principle of continuity) a very large
movement of the air bubble in the top pipe.
This is worth remembering on another account as well, since this is the rule
you use to get "real" units of pressure, proper forces per areas, from quaint
quantities such as earthbound inches of water or mundane millimeters of
mercury.
Even a bit of biology emerges here. Systolic pressure at one's heart has to
be high enough to supply blood at some minimal pressure to the capillaries
in an elevated head and brain, so the A/? at the aorta must be at least this
minimal pressure plus the pgAz for the height difference between heart
and head. (You might imagine that a syphon arrangement could
circumvent any difficulty; but it turns out, perhaps because the system can't
handle subambient pressures safely, that little if any syphoning takes place—
see Hicks and Badeer 1989.) Most mammals have about the same average
aortic blood pressure, about 13,000 Pa (100 mm Hg). Large ones, though,
have higher pressures, with horses commonly about twice and giraffes
almost three times that figure (Warren 1974). Because of the difficulty of
supplying an elevated brain, one both expects and finds a constraint on
circulatory design—the manometric height of an organism cannot exceed
its systolic blood pressure. Manometric height is, of course, a pressure and
uses the density of blood, which is about the same as water.
55
CHAPTER 4
The rule is a somewhat crude one, with the fact that the heart is about
halfway up from the ground roughly offsetting the minimum pressure
needed at the brain. But it's an interesting constraint, a rule that the dimen-
sionless ratio of manometric height to systolic blood pressure cannot
ordinarily exceed unity. One wonders about whether terrestrial dinosaurs
could possibly have had typical reptilian blood pressures of around 5,000
Pa. At the same time one understands why arboreal snakes have higher
aortic pressures than other snakes and why their hearts are in an unusually
anterior position (Lillywhite 1987).
As we saw when calculating the compression they cause in air, the
pressure differences involved in low-speed flows are minuscule compared to
very ordinary hydrostatic pressures. Both the common mercury
sphygmomanometer that warns us of incipient hypertension and the aneroid
barometer that warns us of incipient hurricanes prove hopelessly inadequate
for the pressure differences resulting from small velocities. Equation (4.5),
though, suggests two ways of achieving higher sensitivity. A liquid of low
density may be used in the manometer—water, isooctane, or acetone
instead of mercury. Or some way of reading very small height differences
may be devised. Both schemes have been used in fluid mechanics, and the
technology behind inclined-tube manometers, Chattock gauges, and their
ilk is discussed in older sources such as Prandtl and Tietjens (1934) and
Pankhurst and Holder (1952). The most direct solution is the use of a high-
sensitivity electronic manometer, but such internally calibrating,
computer-compatible instruments are not for the impecunious.
Still, direct manometry shouldn't be dismissed as anachronistic or un-
af fordable. For an investment of ten dollars or so, the device shown in
Figure 4.3b provides sensitivity to pressure differences as low as a millionth
of an atmosphere (0.1 Pa) even under field conditions. Using acetone (to
minimize surface tension) in half-pint vacuum bottles (to minimize thermal
volume changes of the air inside) and glass tubing of about 3 mm internal
diameter, rise or fall of the air-liquid interfaces moves a bubble about 200
times as far. Thus a centimeter of travel of the bubble marks a 50 |xm height
change in the primary acetone manometer; by equation (4.5) that's a
pressure difference of 0.386 Pa. In practice the instrument still runs into a little
trouble from surface tension in the bottles and is best used either following
specific calibration or in a null-balancing mode in which the apparatus is
tilted to restore the bubble's position. With some practice, I got quite useful
results from it (Vogel 1985).
For measuring pressures in water, one can take advantage of the fact that
water-immiscible liquids can be made up with densities arbitrarily close to
that of water. In manometry it isn't really the absolute density that matters
(equation 4.8 is a bit oversimplified) but rather the difference between the
densities of ambient fluid and manometer fluid. Only when the ambient
56
PRESSURE AND MOMENTUM
fluid is air (or any gas) and the manometer is filled with liquid can we safely
ignore the density of the former. The main difficulty in making simple
liquid manometers for working with pressures in water is that rather wide-
bore glass tubes must be used to keep interfacial effects within tolerable
limits. Here again sensitivity to tiny pressure change is bought with a
requirement for large volume changes—work must be done; there's no free
lunch. By the way, these two-liquid manometers may be inverted to form
what we might call O-tube instruments if it's handier to use a manometer
fluid of lower density than that of the ambient water. Oh, yes—be sure to
keep air out of such systems or you'll have accidental manometers turning
up wherever least convenient.
Measuring Flows with Bernoulli-based Devices
Just as measuring a change in height can give a measure of change in
pressure, measuring pressure can provide a measure of velocity. One just
uses a different pair of terms in the Bernoulli equation. And velocity is
something we very much need to measure if we're to work with flowing
fluids. If the difference between the heights of upstream and downstream
points is negligible (or is accounted for elsewhere), then equation (4.4) can
be applied:
Ap = | (t/,2 - t/.,2).
Two practical problems become immediately evident. The equation
doesn't use velocities per se but the difference between velocities. And the
pressure change is proportional to the difference in the squares of the
velocities. The former requires that we worry about two locations, not just
one; the latter generates great problems with sensitivity at low speeds, the
reason I pressed the issue of high-sensitivity pressure-measuring devices.
Venturi Meters
One very useful instrument makes use of the principle of continuity to
circumvent (partly) the differential measurement implied in equation
(4.4). It's called a "Venturi meter" and consists of a contraction of known
size in a pipe, which locally speeds the flow to U2 (Figure 4.4). Since S { and
S2 are known, equations (3.1) and (4.4) can be combined to eliminate U2'-
A Venturi meter is worth remembering as a cheap substitute for an
expensive in-line volume flow meter. I've made considerable use of one
5 7
S2 _-
•> U2
T Az
Figure 4.4. A Venturi meter, in which a contraction to a predetermined
cross-sectional area speeds the flow, lowers the pressure, and thus
permits determination of velocity or volume flow in the uncontracted pipe.
built originally just as a demonstration device. At modest speeds even a
crudely machined annulus glued into an ordinary pipe seems to give
decent results. What's especially important with a crude one is that the man-
ometric port from the uncontracted pipe be upstream from the
contraction. Note that no arbitrary constants in equation (4.6) mandate
calibration. As we'll see, several organisms use arrangements quite similar
to Venturi meters. Laboratory aspirators and automobile carburetors work
on the same principle as well; the main difference from a meter in all of
these applications is that gas or liquid flows through the bypass that would
be occupied by stationary manometer fluid.
Pitot Tubes
Most often the biologist is interested in measuring flow at a single point
in an open field of flow rather than across some closed conduit inserted into
a system of plumbing; for such point measurements the Venturi meter is
nearly useless. How can we contrive an adequate reference velocity in order
to apply Bernoulli's equation? The trick is one already mentioned—the
dynamic pressure appears as a manometrically measurable pressure if the
fluid is suddenly brought to a halt. It then need only be compared with the
static pressure that characterizes the local unobstructed flow. And
"compare," as with the Venturi meter, implies no more than connection to
opposite ends of a manometer.
The device for suddenly bringing a moving fluid to a halt is called a "Pitot
tube" or "Pitot-static tube" (Figure 4.5); to it applies a minor variant of the
Bernoulli equation,
CHAPTER 4
Sl_
*Ui
58
PRESSURE AND MOMENTUM
static holes
£.
^
coaxial pipes
^
V
/
to
pressure-
measuring
device
Figure 4.5 A Pitot-static tube, in longitudinal (sagittal) section and
much fatter than life. One aperture faces upstream, while a ring of small
"static holes" are parallel to the flow. If a static hole on a remote surface
is used instead of these, the device is simply called a "Pitot tube."
*£+>-«.
(4.7)
where H has the amusing name of "total head" and represents the sum of
dynamic and static pressures, the two terms on the left side of the equation.
The aperture facing upstream is designed so that it samples fluid locally
brought to rest, so it's exposed to both dynamic and static pressures and
thus to total head. The "static hole" (or holes), whether adjacent or remote,
is exposed only to the local static pressure. The Ap of a manometer located
between the upstream-facing and static holes therefore indicates (H — p),
or the dynamic pressure. The applicable equation is simplicity itself:
A/?
pi/2
2
(4.8)
To use a Pitot tube together with an ordinary liquid-filled manometer, you
need only combine equations (4.5) and (4.8). But it's worth remembering
that you're dealing with two different densities, that of the ambient fluid
(here p„) in equation (4.8) and that of the manometer fluid (here p7„) less the
ambient fluid in equation (4.5):
jj2 = 2ffAz(p„, ~ 9a)
9a
(4.9)
The portable version, the Pitot-static tube, is inexpensive and rugged.
Much effort has gone into the design of the upstream end to reduce its
sensitivity to minor misalignments with respect to the current direction
and to determine the best size and location for the static holes. Neverthe-
59
CHAPTER 4
less, for the biologist interested in low-speed flows it has a serious problem
of insensitivity; again that square of velocity in the Bernoulli equation is
trouble. To read a Pitot tube to 0.1 m s~l instead of 1ms-1 requires a 100-
fold greater sensitivity of the manometer. Using the acetone-multiplier
manometer described earlier, a one-centimeter deflection corresponds to
an airflow of about a meter per second, or two miles per hour. For water
flow, even using a two-liquid manometer with a density difference of a
tenth the density of water and reading it to tenth of a millimeter, the
detectable current is still no less than 14 mm s_1. In short, the scheme is
workable, but only with some manometric audacity. The advantages gained
are freedom from any external source of power and from the necessity of a
known flow for calibration.
The Pitot Tubes of Life
Quite a few organisms make use of devices that approximate Pitot tubes,
although (as with the Venturi-meter analogs) the pressure difference is
used to drive some secondary flow rather than to press on a static column of
liquid. Wallace and Merritt (1980) note with evident amusement that the
larva of Macronema, a lotic hydropsychid caddisfly, constructs a Pitot tube in
the middle of which it spins a catch net (Figure 4.6a). One opening faces
upstream and is exposed to almost full static plus dynamic pressure (a little
of the pressure is relieved as a result of flow through the structure or, to put
it another way, flow isn't quite halted in front). The other opening is normal
to the current and must experience very nearly ambient static pressure. An
ascidian, Styela montereyensis, has to be a bit fancier, since it lives in shallow
seawater where the direction of flow changes periodically. As shown in
Figure 4.6b, it attaches itself to the substratum by a flexible stalk and can
passively reorient, as does a weathervane, to keep the incurrent aperture
facing the flow (Young and Braithwaite 1980). It's likely that plesiosaurs did
something similar in aid of olfaction. According to Cruickshank et al.
(1991), their palates had scoop-shaped internal nares leading through
short ducts to flush-mounted, external nares on the outside of the head.
The internal nares strongly resemble nonprotruding air-intake ducts used
on aircraft in which a groove deepens in the direction of flow until it ends in
a transverse scoop. The most widespread and perhaps dramatic use of such
a device is in what's called "ram ventilation" in fishes that swim with their
mouths open; we'll return to it shortly.
Limitations and Precautions
At the start of the chapter, I made some derogatory remarks about
Bernoulli's principle; by this point the reader may be wondering if these
60
PRESSURE AND MOMENTUM
Figure 4.6. Two natural Pitot tubes: (a) the case and catch-net of the
larval caddisfly, Macronema; (b) the incurrent and excurrent siphons of
the ascidian, Styela montereyensis.
were just gratuitous curmudgeonly mutterings. The principle does
surprisingly good service on many occasions, but it's still dangerous. Don't
forget the assumption of zero viscosity (quite aside from the steady,
incompressible flow assumed in our version of Bernoulli). Ultimately energy
must be conserved, and in fluids viscosity provides a major route for energy
to leave the mechanical domain. Bernoulli's principle gets less and less
reliable as the scale of speed and size go down, quite aside from any issue of
mensurational sensitivity—for slow, small-scale flows viscosity is
particularly significant. It should be applied only along a streamline or within a
streamtube, and even there only when pressure taps are close together and
shear rates are low. The principle is especially unsafe for points along any
traverse normal to the direction of flow, and it's especially perilous in the
velocity gradients near walls. After all, these velocity gradients are caused by
viscosity. Since flow speed drops off near a wall as a consequence of viscosity
and the no-slip condition, static pressure rather than total head remains
constant there. Speed may drop, but energy is converted to heat in the
shear rather than being converted to pressure potential.
I make this point about the nonconstancy of total head with some
passion, since I once narrowly escaped misattributing a set of interfacial
phenomena entirely to Bernoulli's principle. By the way, the heat from shear is
quite real. Some years ago I encountered an annoying problem of drift of a
61
CHAPTER 4
flowmeter being used in a new flow tank. It turned out (after false leads
were chased and the investigator was chastened) that the one horsepower
being transferred to the water by a pump was warming the water by several
degrees per hour. I ended up crudely confirming James Joule's figure for
the mechanical equivalent of heat!
Perhaps the worst abuses of Bernoulli occur when people apply it to
circulatory systems. In these totally bounded systems pressure drops
almost entirely as a result of shear stresses. Flow in capillaries is several
orders of magnitude slower than in the aorta, but pressure falls rather than
rises as blood flows from latter to former. Textbooks of physiology
commonly mention Bernoulli's principle at the start of their sections on
circulation, but most of them (fortunately) never apply it to any specific situation.
It does have some relevance to the operation of heart valves (see Caro et al.
1978), and it's not negligible for certain pathologies such as local aneurysms
(dilations) or coarctations and stenoses (constrictions) of the aorta and
larger arteries (Milnor 1990; Engvall et al. 1991). More on this in Chapter
14.
Pressure Coefficients and Pressure Distributions
Recall that dynamic pressure appeared as a measurable pressure when
fluid was suddenly brought to a halt. That will occur at some point on the
upstream side of any object facing a flow, not just a Pitot tube. This
consistent behavior permits us to define a rather handy dimensionless variable.
Consider a set of pressure measurements taken at a sequence of locations
from upstream to downstream extremities of some solid object in a flow;
each is referred to a small static hole in an adjacent flat plate oriented
parallel to flow (Figure 4.7). The pressure differences will vary widely, not
only with the shape of the object but with such things as the speed of flow
and the properties of the fluid. But somewhere up front the measured
pressure difference will be exactly the dynamic pressure, and the latter can
be calculated from the free stream speed as well as measured. If all the
measured pressure differences are divided by that dynamic pressure, then any graph
of pressure versus location will begin upstream at a value of1.0.
That's very nice—a measure of pressure that corrects for speed and for
at least one of the fluid properties, density. It even puts cases of airflow and
water flow on axes with the same scales. One sees, relatively uncontami-
nated, the effects of shape; and one can always undo the division,
multiplying by dynamic pressure to restore pressure dimensions and the original
measurements. This dimensionless pressure is called the "pressure
coefficient," Cp\
_ 2A/?
C, = 7JF2- <4-10)
62
PRESSURE AND MOMENTUM
Figure 4.7. How to determine the way pressure varies over the surface
of a body exposed to a flow. Pressure differences are measured between
each of a series of tiny holes (only one is shown) and a static hole well
behind the upstream edge of a flat plate oriented parallel to the flow.
Let's use this new variable to look at how pressure varies around some
simple shapes. Consider, first, the pattern of streamlines around a low-
drag ("streamlined") object (Figure 4.8a). Notice how the streamlines
bunch together near the widest part of the body; by the principle of
continuity we know that the speed of flow ought to be greatest there. By
Bernoulli's principle, we suspect that the pressure ought to be lowest, which in
fact is very nearly what occurs (Figure 4.8b). The graph turns out to be
nicely general, with only minor differences between the toy water rocket I
used (to get something of biologically average size) (Vogel 1988b) and a
giant airship (Durand 1936).
The underlying pressures are, of course, forces per unit area; more
particularly, they are forces perpendicular to the surface per unit area.
Calculating drag from such pressure data (adjusted for orientation of each
element of surface) omits what is often a significant component. The kind
of drag that results directly from the effect of viscosity, the kind that we
used in Chapter 2 as a measure of viscosity, has not been taken into account,
and the closer one gets to a flat plate parallel to flow or to perfect
streamlining the more important this component gets.
Using ambient pressure as a baseline, as is usually appropriate, a positive
pressure coefficient represents a net inward pressure, and a negative
pressure coefficient represents a net outward pressure. These inward and out-
63
CHAPTER 4
(a)
-0.25 -| -
upstream downstream
distance along surface
Figure 4.8. The distribution of pressure on the surface of an object
that has a low drag relative to its size as determined by an apparatus such
as that of 4.7. The data were obtained using a toy water rocket with its
fins removed, about 30 mm in diameter, in an airflow of 10 m s_i
ward pressures are perhaps of equal consequence to the drag-related
forward and rearward ones, and I'll take a few pages to talk about some of the
biology that attaches to them.
Pressure Distribution and Lift
Imagine that the streamlined object in Figure 4.8 is sliced longitudinally
in a horizontal plane (a "frontal section" for classically trained zoologists)
and the lower half is discarded. Pressure coefficients on top will remain
negative, that is, pressure will be outward and upward, while pressure
64
PRESSURE AND MOMENTUM
coefficients below will be nearer zero, or ambient. Let's further imagine
that we prevent relief of the resulting pressure difference by flow from
bottom to top by merely placing the bottom on the substratum. Aha—we've
got lift, which is simply to say that pressure above is less than pressure
below. And the lift comes from the operation of Bernoulli's principle. Lift-
producing airfoils will get the full attention of Chapters 11 and 12. For now
let's look just at several situations in which organisms get lift because they
form protrusions from a flat substratum over which water flows.
How the Plaice Stays in Place
Plaice (Pleuronectes platessa) are bottom-living flatfish similar to flounder.
At rest they constitute low, rounded humps on smooth, sandy bottoms. As
convex elevations, they experience lift by Bernoulli's principle; and lift isn't
exactly the blessing the term usually implies. Despite its fine low-drag
shape, a quiescent plaice has some tendency to slip downstream when
exposed to flow. This tendency is offset by its friction with the bottom and
its submerged weight; it's in fact an especially dense fish. But the weight of a
plaice is reduced by its lift, and its lift is ten to twenty times its drag. So
slippage is more a matter of lift than drag, since lift reduces purchase on
the bottom (Figure 4.9a). In practice it remains quiescent in currents up to
a "slip speed" of 0.2 m s~l and beats its posterior median fins in stronger
currents—up to a "lift-off speed" of around 0.5 m s_1. Above that speed its
net weight is zero or less, and it must either dig in or take off. Rays, which
are essentially dorso-ventrally flattened sharks, behave in much the same
way (Arnold and Weihs 1978; Webb 1989).
A Slotted Sand Dollar
A sand dollar resting on a sandy bottom presents a hump similar to that
of plaice or ray. As a result it also experiences lift and must dig in if faced
with currents above a critical value. Several species have slots—"lunules"—
of varying numbers that run radially and connect upper (aboral) and lower
(oral) surfaces. In a current, water is drawn up through the slots by the
reduced pressure on top (Figure 4.9b). Telford (1983) has shown that the
presence of slots in Mellita quinquiesperforata reduces lift sufficiently to raise
the dislodgement speed by about 20%. Again, lift seems to be more of a
problem than drag. Small individuals have lower critical speeds than large
ones, and young sand dollars select especially dense sand grains and use
them as ballast in a kind of weight belt (Telford and Mooi 1987). The slots
may also function in feeding, with the upward flow through them helping
to draw food-laden water up from the substratum (Alexander and Ghiold
1980).
Feeding by drawing water upward with a mound-shaped body has also
65
CHAPTER 4
Figure 4.9. (a) A plaice or flounder, lying on sand and exposed to flow,
develops lift as a result of the operation of Bernoulli's principle,
(b) The same pressure difference draws water up through the holes
in a perforate sand dollar, so it suffers somewhat less lift; the water
movement may be useful for feeding as well.
been suggested for some other kinds of organisms—for sclerosponges, a
relatively poorly known group of Porifera, and for stromatoporoids, a
group of uncertain affinities best known from Devonian deposits (Boyajian
and LaBarbera 1987). On the upper surfaces of the mounds the
sclerosponges have closed canals and the stromatoporoids have grooves
running radially from apices to valleys. In a flow, water is drawn from valleys to
apices by the lower pressure at the apices; stromatoporoids (assuming the
grooves were open in life) could have entrained fluid and directed it
upward from anywhere along their surfaces.
More Hoisting and Heisting
The fish and sand dollarsjust mentioned are motile creatures that live on
shifty bottoms. Even for sessile organisms on hard substrata, lift as a
consequence of protrusion can present serious problems. Mussels often live in
environments subjected to the very high velocities associated with breaking
waves, and they're attached with impressive tenacity by their byssus
threads. But Denny (1987a) measured pressures immediately above and
within a mussel bed in a location whose extremes of wave action and
disturbance history he already knew quite a lot about. He showed that the
predicted lift maxima were adequate to account for initiating the bare patches
that occurred from time to time—by contrast neither drag nor the
unsteady forces associated with the waves were seriously disruptive.
A similar problem of lift faces limpets, and Denny (1989) ran into an
amusing and instructive case when looking at the hydrodynamic behavior
of a bunch of their shells. One shell turned out to have an anomalously low
drag, undoubtedly associated with some aspect of its shape or surface
66
PRESSURE AND MOMENTUM
sculpture. If one shell could do the trick, why hadn't evolution seized upon
the possibility and equipped the other limpets with the facility? Denny
argues, and I'm certainly persuaded, that drag simply doesn't matter, that
no selective advantage attaches to a low-drag shell, that lift is what dislodges
limpets, and since the anomalous shell was in no way unusual in the lift it
suffered, the limpet that made it was neither more nor less fit. I gather that
the particular shell is carefully preserved—though one hesitates to call it a
valuable artifact.
The same problem of lift probably afflicts freshwater organisms as well,
especially those that have flat bottoms attached to hard substrata and form
protruding mounds—water penny beetles (Coleoptera: Psephenidae), for
instance (see Smith and Dartnall 1980). The underlying problem seems to
be a geometric one. A shape that avoids drag by hugging a surface and has a
large area for attachment will almost automatically develop substantial lift.
Pressure Distributions around Flexible Organisms
Most objects with which our technology deals are fairly rigid. Thus shape
is constant no matter what the flow. Organisms are more commonly
flexible; as a general rule nature is stiff only for particular ends. The
consequence is a vast increase in the complexity of interactions between
organisms and flow. Not only do the forces on a living object in a flow depend on
its shape, but its shape in turn depends on the forces it experiences, as first
pointed out (I believe) by Koehl (1977) for sea anemones. On one hand the
added complexity may seem daunting; on the other it gives nature a very
powerful additional variable with which to work her adaptive tricks. The
point will arise again in chapters to come, particularly when we talk about
drag. Here we'll just consider some direct effects of pressure distributions
such as that shown in Figure 4.8b.
The Form of Fishes
Consider a fish swimming rapidly through water. The pressure on the
head, or at least on its forward part, will be above ambient, that is, inward;
the pressure farther back will be subambient, or outward; DuBois et al.
(1974) got about the expected results from 0.6 m-long bluefish moving at a
little under 2 ms-1 (Figure 4.10) through which they'd run catheter tubing
to transmit pressure. Quite a lot of fish morphology and some behaviors
are at least consistent with this pressure distribution. For instance, each of
the fish's eyes is located at a point where the pressure coefficient crosses
ambient pressure. At such a location, uniquely, an eye will be neither drawn
out from the head by reduced pressure or pressed into the head by
increased pressure as the animal swims more rapidly; only there will the
67
CHAPTER 4
upstream downstream
location
Figure 4.10. Pressure distributions (expressed as pressure coefficients)
around a bluefish (DuBois et al. 1974) and a model of a squid (Vogel
1987).
pressure not be a function of swimming speed. Where the fish experiences
inward pressures, it has a compression-resistant skull, analogous perhaps
to the reinforcing battens that are built into the anterior region of every
nonrigid airship. Farther back, where net pressure is outward, a flexible but
minimally extensible skin is adequate to maintain normal shape.
Interestingly, an increase in intracranial pressure in a bluefish, as would happen
in rapid swimming, triggers an increase in heartbeat and blood pressure
(Fox et al. 1990)—a really reasonable reflex.
Most fish ventilate their gills with water that enters through the mouth
up front and leaves through the operculum, located near the point of
maximum thickness. While one certainly cannot prove that natural
selection has been at work, it's certainly a reasonable arrangement to take in
water where pressure is most strongly inward and to eject water where it's
most strongly outward. Ram ventilation—augmentation of ventilation by
the motion of fish—has been known for many years. For some pelagic
fishes such as mackerel it's mandatory—if they stop swimming they
suffocate (Randall and Daxboeck 1984). Other fishes such as trout make a sharp
behavioral switch at some critical speed from active branchial pumping to
ram ventilation, with the shift speed varying inversely with the local oxygen
concentration. The phenomenon of switching permits comparison of the
cost of the two modes; interestingly, the 10% drop in oxygen consumption
68
PRESSURE AND MOMENTUM
attending the switch exceeds the estimated cost of pumping. The
suggested explanation, which seems reasonable, is that a fish achieves a slightly
lower drag when doing ram ventilation—the pressure-induced flow out
the operculum in some way gives hydrodynamically superior flow along
the body (Steffensen 1985).
Refilling Squid
A squid jets rapidly by contracting its circumferential mantle muscles
and forcing water out a nozzle, as we'll discuss in just a few pages. It refills
between jet pulses by drawing in water through a pair of valves on either
side of the head. Part of the inward pumping is provided by a layer of short,
radial muscle fibers that act as mantle thinners; additional pumping is
provided by elastic recoil from the previous contraction. But once under
way it seems to use flow-induced pressures to augment the refilling (Figure
4.10). Most of the mantle is located where, according to measurements on
models, the pressure is most strongly outward, while the intake valves are
located where the the pressure is close to or a little above ambient. So flow
will draw the mantle outward and draw water into the mantle cavity. The
mechanism may provide over half the refilling pressure for a rapidly
moving squid (Vogel 1987). Still, energy saving of the sort we encountered in
ram ventilation isn't really available here—outward pressure simply means
that more squeeze is needed when the mantle is contracting. The benefit is
more likely to be one of quicker refilling and thus shorter pulse cycles and
more rapid bursts of swimming.
Engulfing Whales
The biggest users of flow-induced pressures are certainly fin whales,
investigated by Orton and Brodie (1987). These rorquals feed as they swim
along, first engulfing very large volumes of seawater plus comestibles and
then forcing the water through their baleen plates and out near the hinges
of theirjaws. During engulfing the anterior half of a whale enlarges greatly,
mainly by stretching the longitudinally grooved ventral surface of the
throat. Whales are well streamlined and ought to have pressure coefficients
approximating those in Figure 4.8b. Using those numbers and data from
throat-stretching tests, Orton and Brodie calculated that the combination
of the dynamic pressure at the mouth's opening and the subambient
pressure outside the throat should be just sufficient to expand a whale's buccal
cavity if it swims at a little under 3ms-1, as in Figure 4.1 la. Which, perhaps
not coincidentally, is about what's been estimated for the speed of feeding
fin whales.
69
CHAPTER 4
Figure 4.11. Putting flow-induced pressures to work to (a) draw water
into the mouth of a cruising and feeding fin whale, and (b) maintain an
oxygen-extracting bubble around an attached beetle in a shallow,
torrential stream.
A Beetle's Bubble
The air-breathing adults of an elmid beetle, Potamodytes tuberosus, inhabit
rapidly flowing streams in Ghana. These beetles congregate on submerged
rock surfaces within a few centimeters of the surface of the water. Most of
each beetle is enclosed in a large air bubble supported by the thorax and
forelegs and extending a centimeter or two beyond the end of the
abdomen (Figure 4.1 lb). The bubble appears to act as a physical gill, but it's
permanent only in moving water saturated with air at atmospheric
pressure. In the laboratory, a beetle can maintain a bubble of normal size only
when it's near the surface in rapidly flowing, well-aerated water; otherwise
the bubble shrinks and the beetle dies (Stride 1955). What happens again
reflects the distribution of pressure of Figure 4.8 (with a dash of 5.2).
Pressure in the bubble is an average of the pressures from upstream to
downstream extremities, so the overall pressure inside is subambient. Thus
dissolved gas will diffuse inward, maintaining the bubble—at least if the
pressure is subatmospheric as well as subambient. This latter requirement
limits this curious adaptation to very shallow and very rapid water: the
greater the depth, the greater must be the current needed to
counterbalance hydrostatic pressure. Stride showed that at a depth of only a
centimeter it took a current of over about 0.8 m s~l to get subatmospheric air
pressure inside a bubble. That's pretty shallow and pretty rapid, which
explains why the scheme is so rare.
Forcing Further Flows
When talking about Venturi tubes, I mentioned that organisms used the
same geometric arrangement but commonly used the pressure differences
70
PRESSURE AND MOMENTUM
to pump some of the ambient fluid through an open pipe that took the
place of the manometer. Basically, flow across the opening of a pipe is used
to draw fluid out that opening, with the fluid in the pipe ultimately
replaced from another opening less exposed to flow—taking advantage of
the drop in pressure coefficient when flow is more rapid, as through the
constriction in a Venturi tube (Figure 4.4) or across the wide part of an
object (Figure 4.8). Perhaps the easiest way to envision the system is simply
to shear off the top half of the main pipe in Figure 4.4, treating it now as an
open flow across a substratum that happens to include a bump (Figure
4.12a) as we did with lift-producing protrusions. The direction of flow of
the air or water isn't crucial—a difference in speeds is really what matters.
In effect, an organism living at a solid-fluid interface can use the energy of
the flowing external medium to drive a flow through itself or its domicile
(Vogel and Bretz 1972).
This scheme for flow induction helps prairie dogs ventilate their
burrows (Vogel et al. 1973) or use flow through the burrows for olfactory
assistance; the animals appear deliberately to maintain sufficient
asymmetry between openings to develop adequate pressure differences. Open-
country African termites use it to help ventilate their mounds, with intake
openings around the base and exhaust holes near the apex (Liischer 1961;
Weir 1973). It reduces the cost of filtering water in sponges when, as is
usual, they live in currents; as in the termite mounds, the input openings
(ostia) are peripheral and the output openings (oscula) are commonly
apical or elevated (Vogel 1977). It aids airflow through the thoracic
tracheae of some large insects in flight (Stride 1958; Miller 1966), using not
only the forward speed of flight but the prop-wash of the beating wings. To
make best use of it, suspension-feeding articulate brachiopods orient
themselves in currents (LaBarbera 1977, 1981). And it's essentially what's
happening in the feeding sand dollars, the sclerosponges, and the
stromatoporoids mentioned earlier.
According to Armstrong et al. (1992) the system is used to supply oxygen
to the under-marsh rhizomes of the common reed, Phragmites australis.
Wind across tall, dead, broken-off culms (flower-and-seed stalks) draws gas
out of their passageways, which in turn draws gas from the rhizomes
(interconnecting roots), which in turn draws air into the low leaves. Oxygen
diffusion out of rhizomes is considered desirable as part of good
management of wetlands; evidently Mr. Bernoulli is put to use in getting the
oxygen down there. (At least one other mechanism of bulk flow is at work in
reeds; in general, plants seem far less limited to diffusion for gas transport
than has been traditionally assumed.)
If we're invoking a purely physical pumping system and not even
demanding moving parts, then we're talking about a scheme so simple that it
might crop up even without natural selection. Webb and Theodor (1972)
showed that seawater moving across ripples of sand at a depth of about 3 m
71
CHAPTER 4
(a)
(b)
(0
B
B
f
B
Figure 4.12. (a) If a primary flow speeds up when crossing an
obstruction, reduction in pressure caused by Bernoulli's principle can drive a
secondary flow from A to B. (b) If a small pipe is mounted normal to the
free stream in the velocity gradient near a surface, greater viscous en-
trainment at B will draw fluid from A to B. (c) If the lower opening of the
small pipe is looped under the substratum, the arrangement gets a little
like that of (a), and the two physical agencies get a bit tangled.
induces water movement in and out of the sand—out at the crests of the
ripples and inward in the troughs or the toes of the slopes. This flow
induction has effects on sediment chemistry through its presence in relict
burrows (Ray and Aller 1985). It's involved in a phenomenon called
"breathing" in volcano cones (Woodcock 1987). And it's widely used in the
design of houses and storage structures, ancient and modern, in which
local airflow is used to provide ventilation (Dick 1950; Bahadori 1978).
In most of these cases, though, Bernoulli's principle isn't the complete
extent of the relevant physics—viscosity plays a part as well. As mentioned
earlier, pressure rather than total head remains constant in the velocity
gradients near surfaces. If a tube is placed transversely in a channel with
one end near a wall (Figure 4.12b), then viscous action will move fluid from
wall to free stream. Viscosity, you'll recall, amounts to resistance to rapid
shear rates. At the end of the tube away from the wall, the velocity and thus
the shear rate and thus shear stress will be greater. In response to the shear
stress, fluid will move out of the tube and into the channel. At an equal rate,
other fluid must move into the tube at the wall end. The phenomenon,
rarely of interest to fluid mechanists, is termed "viscous entrainment"
(Prandtl and Tietjens 1934). It's one of the reasons why the static holes in
Pitot tubes must be small (Shaw 1960).
The difficulty of dissecting the physical mechanism of a given induced
flow is evident if the tube in Figure 4.12b is looped around as in 4.12c; the
72
PRESSURE AND MOMENTUM
result looks all too similar to 4.12a! In actuality, some mix of the two
mechanisms probably always operates, and the nature of the mix is not of
overwhelming concern to the biologist (Vogel 1976).
Momentum
Newton quite clearly recognized something basic about a concept of
"quantity of motion," and he saw that it was proportional to neither mass
nor velocity separately but to their product. To this product of mass and
velocity we give the name momentum. Newton's second law, of which we've
already made use, is commonly stated as an equality between force and the
product of mass and acceleration, that is, of mass and the rate of change of
velocity. But for present purposes the law is better regarded as equating
force and the rate of change of momentum.
Every elementary physics book notes that in all collisions between bodies,
momentum is conserved. Energy may be conserved as well, but only in
perfectly elastic collisions can we neglect the conversion of mechanical to
thermal energy. Conservation of momentum is thus the more potent
generalization for purely mechanical problems. To apply energy conservation
and stay within our purely mechanical domain we have to presume the
tacitly self-contradictory notion of an ideal fluid. By contrast, the use of
momentum conservation involves no such risky simplification.
In our macroscopic fluid mechanics, discrete collisions are not usually of
much consequence. No matter—momentum is still conserved, and if the
fluid is isolated, the total momentum will be constant in both magnitude
and direction. But a moving fluid is rarely isolated—it passes walls,
obstacles, pumps, and so forth. One can with great confidence state that any
change in the momentum of such a nonisolated flow must reflect a forceful
relationship with its surroundings. Thus, returning to Newton's second
law,
d(mU) mdU /A . ,.
F = —"—: -=—7— • (4.11)
dt dt ;
The last term is the simpler version; but its presumption of constant mass is
not always appropriate, for instance for jet propulsion.
The main trouble with the equation is that it's stated in terms of mass and
time, neither of them handy quantities when dealing with steady flows of
fluids. We can do better by considering what must go on within a stream
tube. How fast does mass move through the tube? If we consider uniform
flow by taking a small enough tube so transverse velocity differences are
trivial, mass flux (mlt) will equal the product of the density of the fluid, the
cross-sectional area of the tube, and the velocity of flow—which is the same
as the product of density and volume flow rate:
73
CHAPTER 4
-j = PSU = pQ.
Similarly, momentum flux is obtained by multiplying both sides by velocity:
(This combination of variables, pSU2, will occur repeatedly in chapters to
come.)
Now we can state an equation, sometimes called the "momentum
equation," that puts into a practical form the rule relating force and change in
momentum. In terms applicable to a stream tube with entry area dS{ and
exit area dS2, assuming (as usual) steady flow and assuming that everything
of interest goes on in the x-direction,
dF = pdSyU^ - pdS2U22. (4.12)
To get the force F from dF, one just has to add up the contributions of all
the relevant streamtubes that thread their way across the field of flow. The
equation (or slightly more complicated versions) can be applied to a variety
of problems, including such things as the force exerted on a pipe as fluid
passing through it goes around a bend and the force exerted on a surface as
a fluid is squirted at it.
We can use this statement of Newton's second law to define some forces.
Drag becomes the rate of removal of momentum from a flowing fluid.
Thrust is then the rate of addition of momentum to a stream of fluid. And
lift is the rate of creation of a component of momentum normal to the flow
of the undisturbed stream.3 These definitions, of course, say absolutely
nothing about the physical origin of forces such as drag. Drag, that most
awkward subject, will be tackled in the next chapter.
Indirect Force Measurements
But even without really understanding how it happens, we can still talk
about this force we're calling drag. Despite the disparaging remarks in
connection with displacement of plaice and lifting of limpets, drag is a
variable of substantial adaptive significance. Due to the irregular shapes
and flexibility of organisms, it can only rarely be calculated or looked up in
tables and so most often must be measured. Such measurement, though,
may not always be an easy matter under anything approaching normal
conditions. Not all creatures can with impunity be mounted with instru-
1 Note that lift is not defined as upward with respect to the surface of any planetary
body.
74
PRESSURE AND MOMENTUM
ments in flow tank or wind tunnel. But our formal definition of drag, when
combined with the notions of continuity and streamlines, can provide an
alternative. Working out a procedural formula will provide a little
introduction to the use of the momentum equation.
By Newton's third law, if a body exerts a force on another, then the
second body exerts a force on the first of equal magnitude and of opposite
direction. Thus if the moving fluid exerts a drag on a body, then the body must
remove momentum from the fluid at a rate that just balances its drag. If this rate of
removal of momentum is measurable, then it's possible to get the drag of an
organism without detaching or even touching it! The procedure can be a
bit laborious and cannot be reasonably expected to give data of better than
5% accuracy, but sometimes nothing else will work.
Consider an attached object subjected to a flow (Figure 4.13), an object
that (for simplicity) has its bulk concentrated well away from the
substratum, so it's not in a velocity gradient. Consider, as well, two imaginary
parallel planes, one upstream (S{) and one downstream (S2) from the
object, each sufficiently far from the object that flow is normal to its surface.
Consider, finally, a stream tube that leads fluid from part of the upstream
plane (dS{) to a corresponding part of the downstream plane (dS2). Across
the upstream plane the velocity (U{) is uniform, while across the
downstream plane the velocity (U2) varies from a minimal value in the center of
the wake of the object. The momentum flux across dSl will be, from
equation (4.12), pU]2dSx, and that across dS2 will be pU22dS2. The difference
between these fluxes will be the part of the drag acting on that stream tube,
so,
dD = pdSxUx2 - pdS2U22. (as 4.12)
By continuity, though,
dSlUl = dS2U2.
Combining these equations to get rid of dS{ gives
dD = pU{U2dS2 — pU22dS2.
All that remains is integrating over the whole plane, S2. The double
integral just means that we have two dimensions to worry about, that dS2 —
dy2dz2:
D = p J J U2(U{ - U2)dS2. (4.13)
While this last equation may look complex, the complexity is more
apparent than real. Only rarely can we solve it explicitly; instead, it functions
as the procedural formula we've been seeking. It tells us what we have to do
to measure drag indirectly. First, we have to measure the velocity, U2, at a
75
CHAPTER 4
Figure 4.13. An attached object removes momentum from a stream
tube that passes it. Its drag is the rate at which the object removes
momentum from all the nearby stream tubes.
set of points on plane S2. Naturally it's only necessary to do so in the wake of
the object—where U{ and U2 are appreciably different—so the limits of
the plane of integration present no problems. Second, U{ must be
measured somewhere on the upstream plane; it makes no difference where,
since we're assuming that U{ is constant across Sl. Then all that's left is for
each U2(Ul — U2) to be multiplied by its corresponding area, dS2 (or AS2, to
be realistic about it), the products added up, and the result multiplied by
the density of the fluid. Voila, drag!
Several practical matters. First, the location of the downstream plane
poses a slight problem. If you're too close to the object, the assumption that
the flow direction is uniform and normal to the plane may be violated, so
the plane should be distant by at least several times the maximum diameter
of the object. If, on the other hand, you're too far away, viscosity will have
begun to eliminate the wake altogether, to reaccelerate it to free-stream
velocity. A few trial runs are necessary. The requirement for a very large
number of measurements (perhaps as many as a hundred) of velocity in the
wake may seem daunting. But traverses back and forth with the measuring
instrument on a motor-driven manipulator and direct input into a
computer make the procedure more reasonable than it might appear.
Thrust can be measured in much the same manner; only the sign of the
result is changed. Lift determinations require some way of establishing the
deflection of the wake from the free-stream direction. I once measured
the lift of a fixed fly wing by first measuring the airspeed at each of a series
of points behind it; for each point I then centered the wake of a tiny wire on
the airspeed transducer to get the local wind direction. From these data it
76
PRESSURE AND MOMENTUM
was a simple matter to calculate the downward momentum fiux created by
the wing. Further information on indirect force measurements can be
found in Prandtl and Tietjens (1934, pp. 123-130), Goldstein (1938,
pp. 257-263), and Maull and Bearman (1964).
Jet Propulsion
If you hold a hose with a nozzle out of which water is flowing rapidly, you
feel a force in the direction opposite that of flow. A toy water rocket speeds
off impressively when pressurized air forces water out through its nozzle.
And when a squid reduces the circumference of its outer mantle, the water
in the mantle cavity is forced through a nozzle, the siphon, and the squid is
elsewhere with remarkable alacrity. Jet or rocket propulsion is clearly
effective with water as the working fluid.
No small number of animals use jet propulsion (Figure 4.14), and it
seems to have evolved on quite a number of occasions. For that matter, it
may have been the earliest truly macroscopic mode of animal locomotion.
At least one ctenophore, Leucothea, uses it (Matsumoto and Hamner 1988)
to get up to un-ctenophoric speeds of about 50 mm s_1. Medusoid
cnidarians are, I think, entirely ajet set; see, for instance, Daniel (1983) and
DeMont and Gosline (1988). Pelagic tunicates move by jet propulsion;
Madin (1990) provides a general view and references. Scallops, although
they do it only briefly, can achieve speeds of 0.6 m s~l (Dadswell and Weihs
1990). At least one family of fish, the frogfishes (Antennariidae), jet by
ejecting water from a large oral cavity through a pair of small, tubular
opercular openings; their speeds (up to 27 mm s~l), though, aren't
impressive for fish (Fish 1987). And dragonfly nymphs have an anal jetting system
that can get them up to around 0.5 m s~l (Hughes 1958, Mill and Pickard
1975). Of course the best known jetters are the cephalopod mollusks, both
the shelled Nautilus and the shell-less octopuses, cuttlefish, and squids. In
the latter, speeds may reach as much as 8 m s-1, according to what are,
admittedly, somewhat anecdotal accounts (Vogel 1987).
It should be easy to make ajet engine by modifying a ventilatory system,
and muscular systems that squeeze cavities run innumerable hearts and
guts. So perhaps the most interesting question to ask is whyjetting isn't still
more common, why the really big and fast swimmers mostly use quite a
different and less obvious scheme. Considerations of momentum turn out
to provide a fairly good rationalization. The underlying problem, and an
important consideration in other forms of locomotion in fluids, boils down
to a peculiar dissonance between momentum and energy. The force ajet
produces reflects its rate of momentum discharge, loosely the product of
mass per time and velocity. The cost of producing this jet reflects its energy,
proportional to the product of mass per time and the square a/Velocity. Thus
the economical way tojet is to process water at a high rate, maximizing mass
77
CHAPTER 4
Figure 4.14. Jet propelled animals: (a) squid, (b) frogfish, (c) a trio of
salps, (d) dragonfly nymph, (e)jellyfish, and (f) scallop.
per time, and to give the water a minimal incremental velocity. A jet,
though, tends to do the opposite. Since the water has to be contained in
some kind of internal squeeze-bag, its volume is relatively limited.
Increasing the girth of the bag means increasing the volume of the animal,
increasing its drag, and even reducing (by Laplace's law)4 the effectiveness of a
muscular layer in producing the pressure that will generate thrust.
4 Laplace's law is the rule that the pressure inside a closed vessel will be proportional to
the tension in its wall (what muscles do) divided by the radius of curvature. For more
detail, see Vogel (1988a).
78
PRESSURE AND MOMENTUM
Let's discuss the matter with a little more formality. The thrust of a jet is
the product of the mass flux and the difference between the jet velocity
(U2) and the free stream velocity (l/j), so
m
r = 7(t/2-i/,). (4.14)
(We'll stick with the simpler mass per time rather than working through
densities, cross sections, and so forth.) Jet velocity is the maximum speed of
the jet, usually developed some short distance behind the motor. Power
output, then, is the product of the thrust and the speed of travel or free
stream velocity:
/>„„, = ^ (U2 - t/,). (4.15)
Power input is the familiar kinetic energy per unit time:
w
P;n=2t^U^- <4-16)
Dividing output by input gives us, as always, an efficiency:
v^ij-ru-r (4'17)
This particular efficiency is called the "Froude propulsion efficiency" after
William Froude, a nineteenth-century British naval engineer, whose name
will reappear in quite a different context in Chapter 17. As you can see (and
certainly not surprisingly), the jet velocity must be higher than the free
stream speed—but it should be only minimally higher for best efficiency.
(The ideal, giving an efficiency of 100%, would be equal jet and free stream
speeds.) A propeller or a set of external fins or paddles processes a lot of
fluid; it can therefore achieve good thrust (equation 4.14) with only a small
incremental velocity, and it therefore can operate at high Froude efficiency
(equation 4.17). A jet, though, won't do where efficiency is the adaptively
significant criterion.
A few more items connected with jet propulsion. Weihs (1977) suggested
that a pulsed jet, usually what animals use, can do a little better than the
formula for efficiency predicts. He pointed out (and Madin 1990 gives
persuasive pictures) that pulses of rapid water entrain additional water as
they roll up into toroidal vortices, and he calculated a factor of increase of
approximately 30% in efficiency as a result. This entrainment of additional
moving mass is not dissimilar to what modern ducted-fan jet engines do to
reach higher efficiencies than the earliest jets could manage.
The cost of transport for a squid, jetting along, is about three times that
of a trout of similar size or, as a similar comparison, a squid takes twice the
79
CHAPTER 4
power to go only half as fast as the average fish (O'Dor and Webber 1986).
And the culprit is clearly what we've just been talking about—Alexander
(1977) notes that a trout imparts rearward momentum to about ten times
as much water per unit time as does a squid. Still, this odious comparison
may be at least a little bit unfair. Smaller or slower jetting animals have
substantially lower costs of transport than squid. Nautilus gets its mass
around for about a sixth the cost paid by a squid (Wells 1990), and small
salps (tunicates) expend only a little more energy than that (Madin 1990).
O'Dor and Webber argue that the cost of jetting scales differently than the
locomotion of most fish, with jetting looking rather less disadvantageous as
size is reduced.
Solids and Fluids: Quick Comparisons
As large organisms whose density far exceeds that of the surrounding
medium, most of us regard solids as more palpably real than fluids; our
training in physics begins with the solidest of solids, and we forget the
fascination of the two-year-old who pours water repeatedly from one
container to another. Since we're trying to build a similarly intuitive familiarity
with fluids, perhaps it might be useful to list some specific points of
comparison, some analogies between the world of solids and that of fluids.
While the paired items are certainly not synonymous, the items on the right
effectively replace the ones on the left when one deals with fluids instead of
solids. Thus . . .
SOLIDS FLUIDS
mass density
elasticity viscosity
interfaces streamlines
shearing planes velocity gradients
friction drag
conservation of mass continuity
conservation of energy Bernoulli's principle
80
CHAPTER 5
Drag, Scale, and the Reynolds Number
So far fluid mechanics has probably struck the reader as a decent,
law-abiding branch of physics. We've touched on viscosity, continuity,
momentum; we even dragged in drag and how it might be measured—and
all proved to be quite ordinary topics. Only turbulence and streamlining
hinted at deferred peculiarities. I now want to pursue the business of drag
somewhat further, asking in particular about its actual physical basis. With
this most innocent question, Pandora's box springs a leak, and keeping
head above water gets harder. So queer is this aspect of the physical world
that we'll have to defer most of the relevant biology to the next two chapters
in order to spend this one on the physics.
From Whence Drag?
It was easy, in the last chapter, to define drag as the rate of removal of
momentum from a moving fluid by an immersed body. Similarly, it's no
trick at all to get a dimensionally correct formula for drag from that
definition: pSU2. Newton suggested just such a formula from just such
dimensional considerations—but where in it does shape enter? It certainly
matters in practice! Alternatively, one might try to apply Bernoulli's principle,
adjusting equation (4.8) to work for more than just an upstream point by
multiplying both sides by the projected area normal to flow ("face area").
One gets a similar formula: pSU2. In physical terms, though, this second
approach implies a rather curious situation. Fluid particles collide with the
object, in the process losing all their momentum in the x-direction. And
that means they disappear, or they're carried with the object like snow
ahead of a badly designed plow, or they move precisely sideways without
further interactions with the oncoming flow—all distressingly unrealistic
scenarios.
Aristotle, by the way, was even further off the mark. He didn't believe in
drag at all, figuring that an object needed air to give it thrust, that in a
vacuum an arrow would tumble earthward upon leaving the bow. Newton's
first law, a thoroughly counterintuitive idea, lay 2000 years in the future.
We might try Bernoulli (with a little more complete accounting) and the
ideal fluid theorists of the nineteenth century. Theoretical streamlines can
be calculated for steady flow around, say, a circular cylinder with its axis
normal to the flow (Figure 5.1). At upstream and downstream extremities
81
CHAPTER 5
Figure 5.1. Theoretical streamlines for flow normal to the long axis of
a circular cylinder, as shown in perspective in the inset at the right.
are so-called stagnation points where the fluid is locally stationary with
respect to the cylinder. By continuity, the fluid reaches maximal velocity
laterally, where the cylinder blocks the greatest part of the path of flow. As a
result, at the front and back the pressures on the surface will reach H, the
total head. On either side, where streamlines are closest together, the
pressure will be less, H — 2pc/2 to be exact (Massey 1989, pp. 329-331, gives
particulars). However, the whole diagram is symmetrical about a cross-flow
plane along the axis of the cylinder. Not only do the pressures on each side
cancel each other, but those on the front and back do so as well. As a result,
no net pressure at all tends to carry the cylinder with the flow—the cylinder
feels no drag at all! This amazing result can be generalized to cover bodies
of any shape; it's called d'Alembert's paradox, and perhaps is the ultimate
pursuit of a will-o'-the-wisp into a cul de sac.l Lord Rayleigh (1842-1919)
pointed out that according to such theory a ship's propeller wouldn't work
but, on the other hand, wouldn't be needed anyhow. Furthermore,
There is no part of hydrodynamics more perplexing to the student
than that which treats of the resistance of fluids. Acording to one
school of writers, a body exposed to a stream of perfect fluid would
experience no resultant force at all, any augmentation of pressure on
its face due to the stream being compensated by equal and opposite
pressures on its rear. . . . On the other hand it is well known that in
practice an obstacle does experience a force tending to carry it
downstream.
As a first step toward reconciling theory and reality, we can return to the
pressure coefficients introduced in the last chapter and compare our
results for a cylinder with the calculations for an ideal fluid, as in Figure 5.2.
For these particular data, the conditions are equivalent to a sapling in a
1 This phase is borrowed, with affectionate memories, from the late Carroll M.
Williams.
82
DRAG, SCALE, REYNOLDS NUMBER
1
-
-
1
1 1
y \ reality
■1— 1-
• '
/ ideal fluid
1 !
-
-
-
I 1 1 1 1 1 1
0 30 60 90 120 150 180
degrees around from upstream center
Figure 5.2. Two pressure distributions around a long circular
cylinder—that for an ideal fluid following the streamlines of 5.1 and for
a real fluid at a Reynolds number of 10r\ Theory and reality diverge
most at the sides and rear.
light breeze or a human foreleg wading in a very gentle stream. At the
upstream center, the pressure is, as expected, H, so the pressure coefficient
is unity. Theory and data are in fine agreement, which reaffirms the utility
of Pitot tubes, for instance. But proceeding further along the cylinder, the
rest of the decrease in pressure does not entirely materialize; and, at about
70°, the pressure actually begins to rise somewhat. From here rearward,
reality bears almost no resemblance to ideal fluid theory. About the only
point on which theory and data agree is that, for the cylinder as a whole, the
net overall force is negative, so the cylinder will tend to expand rather than
collapse in the current—what we saw earlier in the air bubble maintained
by the beetle Potamodytes.
This difference between the dynamic pressure on the front and the
absence of the predicted counteractive pressure at the rear is what we
subjectively feel as drag. We haven't, by this examination and comparison,
really solved anything, but we have at least located the problem. The flow of
fluid around the cylinder isn't symmetrical, front to back. And it's in the
rear where something strange is happening. Not that this pressure
difference can't be useful—it causes gases to percolate through cylindrical grain
storage buildings (Mulhearn et al. 1976). We'll return to this problem of
pressure distribution after introducing some analytical tools; for now just
bear in mind that for limiting drag in fluids, discriminating design of the
derriere is de rigueur.
Nor is this odd pressure distribution the only peculiarity, even for a case
83
CHAPTER 5
as ostensibly simple as the drag of a cylinder normal to flow. Both the
pressure distribution for such an object and the drag itself vary with speed,
for instance, in queer and irregular ways. At low speeds drag is
proportional to the first power of velocity; at higher speeds it gradually
approaches a second-power dependence; it then drops abruptly; and at still
higher speeds it resumes its second-power behavior but with a lower
constant of proportionality. Clearly our problem isn't simple omission of some
simple component of drag, perhaps some direct function of viscosity.
A New Player: The Reynolds Number
Perhaps at this point we should remind ourselves that one objective of
science is securing the minimum number of rules that account for the
largest variety of phenomena. We don't have and may never have a simple,
universal, and practical rule for predicting the drag even of a shape as
regular as a circular cylinder. But trying to simplify the situation as much as
possible is still worthwhile—anything is better than bludgeoning the
problem with massive empirical tables of how drag varies with shape, speed,
size, and other relevant variables. For each additional independent variable
the number of data will rise by one or two orders of magnitude! And such
tables would be of little use for the shapes a biologist is wont to consider. At
the very least we ought to ask whether some of the variables have the same
effects as others, whether each need be regarded as behaving in a
completely unprecedented fashion.
This latter possibility turns out to be both real and useful. Its pursuit
leads to the peculiarly powerful Reynolds number, the centerpiece of
biological (and even nonbiological) fluid mechanics. The utility of the Reynolds
number extends far beyond mere problems of drag; it's the nearest thing
we have to a completely general guide to what's likely to happen when solid
and fluid move with respect to each other. For a biologist, dealing with
systems that span an enormous size range, the Reynolds number is the
central scaling parameter that makes order of a diverse set of physical
phenomena. It plays a role comparable to that of the surface-to-volume
ratio in physiology.
This almost magical variable can be most easily introduced injust the way
it originated, in the empirical investigation done by Osborne Reynolds
(1883) already mentioned in Chapter 3 in connection with laminar and
turbulent flow. As you may recall, Reynolds introduced a dye stream into a
pipe of flowing liquid. Sometimes the resulting straight streak indicated
laminar flow, and sometimes dispersal of the streak signaled that the flow
was turbulent. The transition was fairly sudden, both in location in the pipe
and as the characteristics of the flow were altered. He found that the flow
could be persuaded to shift from laminar to turbulent in several ways: by
84
DRAG, SCALE, REYNOLDS NUMBER
increasing speed; by increasing the diameter of the pipe; by increasing the density of
the liquid; or by decreasing the liquid's viscosity. Each change was as effective,
quantitatively, as any other; and they worked in combination as well as
individually. The rule that emerged was that when a certain combination of
these variables exceeded 2000, the flow became turbulent. The particular
combination is what we now call the Reynolds number, Re2, with the length
factor, /, taken as the pipe's diameter.
Re^JLL = ^L. (5.l)
|X V
Before going further I simply must point out that, claims in the
biological literature notwithstanding, the specific value of 2000 applies only to
transition in a long, straight, circularly cylindrical tube with smooth walls,
at a decent distance from the tube's entrance. With stringent prevention of
any initiating circumstances it may be raised as much as an order of
magnitude. With roughened tubing, transition can happen at lower values. So
only one figure is at all significant. For external rather than internal flows,
the particular datum of 2000 has no relevance at all.
One of the marvelous gifts of nature is that this index proves to be so
simple—a combination of four variables, each with an exponent of unity. It
has, however, a few features worth some comment. First, the Reynolds
number is dimensionless (as you can verify through reference to Table 1.1),
so its value is independent of the system of units in which the variables are
expressed. Second, in it reappears the kinematic viscosity, v, (after a three-
chapter absence). What matters isn't the dynamic viscosity, |x, and the
density, p, so much as their ratio, v: the higher the kinematic viscosity, the lower
the Reynolds number. Finally, a bit about /, commonly called the
"characteristic length." For a circular pipe, the diameter is used; choosing the
diameter rather than the radius is entirely a matter of convention.3 For a
solid immersed in a fluid, / is typically taken as the greatest length of the
solid in the direction of flow. But it's a very rudimentary measure of size,
which emphasizes the coarse nature of the Reynolds number as a yardstick
when objects of different shapes are being compared. As mentioned in
Chapter 1, the value of the Reynolds number is rarely worth worrying
about to better than one or at most two significant figures. Still, that's not
trivial when biologically interesting flows span at least fourteen orders of
magnitude.
2 Take notice that no apostrophe precedes or succeeds the "s" in "Reynolds," that
"number" isn't capitalized, and that the abbreviation used in text and equations is Re, not
R>-
3 Reynolds used the radius, so he figured the laminar-turbulent transition point as
1000, not 2000. His nicely "round" number gives a better sense of the level of precision
involved.
85
CHAPTER 5
Table 5.1 A spectrum of Reynolds numbers for self-propelled
ORGANISMS.
Reynolds Number
A large whale swimming at 10ms-1 300,000,000
A tuna swimming at the same speed 30,000,000
A duck flying at 20 ms"1 300,000
A large dragonfly going 7 m s~' 30,000
A copepod in a speed burst of 0.2 m s~ ' 300
Flapping wings of the smallest flying insects 30
An invertebrate larva, 0.3 mm long, at 1 mm s_l 0.3
A sea urchin sperm advancing the species at 0.2 mm s~' 0.03
A bacterium, swimming at 0.01 mm s_1 0.00001
Of greatest importance in the Reynolds number is the product of size
and speed, telling us that the two work in concert, not counteractively. For
living systems "small" almost always means slow, and "large" almost always
implies fast. That's why the range of Reynolds numbers so far exceeds the
eight or so orders of magnitude over which the lengths of organisms vary.
At this point looking at some real Reynolds numbers might be useful. Since
we're interested only in orders of magnitude, Table 5.1 is based on rough
approximations for speeds and sizes.
A Physical View of the Reynolds Number
This same dimensionless number can be reached by another path.
Consider what must have gone wrong with the attempt to explain drag with
ideal fluid theory. Ideal fluids have no viscosity, so failure ought to trace to
the neglect of forces associated with viscosity. Certainly viscosity involves
force—we recognized it in the first place as a retarding force on a moving
flat plate that was adjacent to a fixed one. So perhaps two sorts of forces
matter when a moving fluid crosses an immersed body. First are inertial
forces, those derived directly from Newton's second law, which essentially
defines inertia, and expressed by an equation that we obtained easily and
found practically useless:
Fj = pSU2.
What we mean by inertial forces are those attributable to the momentum of
a bit of moving fluid. Thus, at the urging of its inertia, a particle of fluid
keeps on doing its usual thing to the extent that it remains unmolested.
Second are the previously ignored viscous forces. Fluids don't like to be
sheared—if a flow involves shear, then viscous forces will oppose the
persistence of the motion just as friction opposes the movement of one solid
86
DRAG, SCALE, REYNOLDS NUMBER
across another. So viscosity will tend to smooth out any internal
irregularities in the flow. Viscous force was defined when we defined viscosity:
\lSU
/ '
Put somewhat more metaphorically, inertial forces reflect the
individuality of bits of fluid while viscous forces reflect their groupiness. The former
describes the progress of a milling crowd, the latter of a disciplined march.
That, in fact, is exactly the distinction between laminar and turbulent flow.
So let's make what's now only a small intuitive leap and suggest that what
distinguishes different regimes of flow is the relative importance of inertial
and viscous forces. The former keeps things going; the latter makes them
stop. High inertial forces favor turbulence, with the substantial shear rates
inevitably involved. High viscous forces should prevent sustained
turbulence and favor laminar flow by damping incipient eddies and other
irregularities. To take a rough look at the relative magnitude of the two, we need
only divide these equations. What we get is, of all things, the Reynolds
number:
Fv [iSUIl |x • K }
Here we see its lack of dimensions as the natural outcome of dividing one
force by another—the Reynolds number is a ratio of inertial to viscous
forces. I say "a" ratio, not "the" ratio as a reminder that we've done little
more than a loose dimensional accounting. It is, for instance, distinctly
unsafe to assert that a value of 1.0 indicates equality of two entirely specific
forces.
Another point should be made emphatically. If, for example, the
Reynolds number is low, the situation is highly viscous. The flow will be
dominated by viscous forces, vortices will be either nonexistent or nonsustained,
and velocity gradients will be very gentle unless large forces are exerted.
The value of the Reynolds number is what indicates the character of flow,
not the value of dynamic viscosity or even kinematic viscosity per se. A 10-
fold reduction in size (length) will increase relative viscous effects with
precisely the same efficacy as a tenfold increase in viscosity itself. If, in
nature, small means slow and large means fast, then small creatures will live
in a world dominated by viscous phenomena and large ones by inertial
phenomena—this, even though the bacterium swims in the same water as
the whale.
Consider an object of a given shape and orientation immersed
alternatively in two flows. Equality of the Reynolds number for the two situations
guarantees that the physical character of the flows will be the same. A moment's
thought should persuade you that this last statement is powerful but still
quite reasonable, given the origin of the Reynolds number. As we'll see,
87
CHAPTER 5
equality of the Reynolds number doesn't necessarily mean that forces are
unchanged, but it does assure us that the patterns of flow as might be
revealed by locating streamlines will be the same. And that holds even if
one flow is of a gas and the other is of a liquid.
We come, parenthetically, to a mild curiosity and convenience. It was
mentioned back in Chapter 2 that at ordinary temperatures air is about
fifteen times more kinematically viscous than is water. As a result, for an
object of a given size, the same Reynolds number will be achieved at a
velocity fifteen times higher in air than in water. I doubt whether it's more
than coincidental, but in nature flows of air are very roughly fifteen times
as rapid as flows of water—or at least, as a sober assessment, an order of
magnitude more rapid. Thirty meters per second (68 mph) is a hurricane
of air; two meters per second is a torrent of water. A third of a meter per
second in air we find barely detectable; twenty millimeters per second in
water appears to be a roughly similar threshold. In short, organisms of any
given size operate at approximately the same Reynolds numbers whether
they live in air or water, quite a convenience for any biologist who is trying
to understand the relationship between flow patterns and functional
adaptations. Again, though, I emphasize that equality of Reynolds numbers
doesn't imply that the forces of flow are the same—only the patterns of
flow. Forces will be dealt with shortly.
This notion that equal Reynolds numbers imply geometric similarity in
flow patterns is a potent one. A good cross-sectional shape for a bird's wing
ought to be much the same as for the tail of a tuna—the Reynolds numbers
are similar. Eel and spermatozoan may look similar, but one shouldn't
presume that they swim by the same hydrodynamic mechanism—their
Reynolds numbers are too divergent. On the other hand, micturition in
humans (Hinman 1968) and the ejection of ink in some ink-jet printers
(Levanoni 1977) involve approximately the same Reynolds numbers. So
perhaps work on avoiding what are called "satellite drops" by the printer
folk may be helpful in understanding the design of a penile orifice that
squirts with minimal spraying (Chapter 17). Stokes' law (Chapter 15) is a
good rule for sinking rate of pollen and plankton. But if you try to apply it
to sinking of trout eggs, you get an erroneous answer of 11 rather than 88
mm s~l—the predictions of Stokes' law increasingly diverge from reality at
Reynolds numbers over about 1, and a sinking trout egg operates at about
50 (Crisp 1989).
Reynolds Number and the Drag Coefficient
Back to drag. Newton figured that it was proportional to three
quantities: first, to the density of the medium; second, to the projected area of
the body; and third, to the square of the velocity. And drag does indeed
88
DRAG, SCALE, REYNOLDS NUMBER
behave in this manner—but only sometimes, and even then only
approximately. One can do somewhat better, though, with a dimensional analysis,
a mathematical technique mentioned in Chapter 1. (Rouse 1938 has a
particularly illuminating discussion of the approach and its application to
the present problem.) Dimensional analysis leads to an empirical formula
for drag with all the peculiarities heaped upon one composite variable; I'll
not go through the formalities but just sketch the logic.
The analysis begins by identifying the variables upon which drag might
depend—the size of the object, its speed relative to the fluid, and the viscosity
and density of the fluid. By a theorem involving the relative number of
dimensions and variables, it predicts that the formula for drag will be the
product of a term with dimensions of force and a single dimensionless
term. The force term proves to be the dynamic pressure (pc/2/2,
remember) times the area of the object; it essentially accounts for the ordinary
variations that Newton identified. The dimensionless term accounts in toto
for all of the peculiarities in the behavior of drag:
d - (i ,*«,.) (f y. ««,
The dimensional analysis tells us nothing about how this second term
behaves, merely that it is the only term that should matter. The second term
(less its exponent) happens to be the Reynolds number, so the latter has
now appeared in a third context!
Thus the variations in the drag of any object of fixed shape and
orientation is describable as the product of two variables—the dynamic pressure
times the area of the object and some function of the Reynolds number. In
practice, that awkward exponent, a, about which we know nothing a priori,
is replaced by the so-called drag coefficient, Cd.
Cd={Re)«=f(Re). (5.3)
Of course we know as little about Cd—a multiplicative coefficient is just
handier than an exponent. The drag coefficient, then, is a function only of
the Reynolds number and handles all the oddities in the behavior of drag.
This last statement is a powerful one; put another way, it says that equality
of Reynolds number for a given shape and orientation in flow implies
equality of the drag coefficient. That's sometimes called the "law of
dynamic similitude"; as we'll see, it's the basis of almost all testing of model
systems in flow tanks and wind tunnels.
The drag coefficient is most easily envisioned as a dimensionless form of
drag, the drag per unit area divided by the dynamic pressure. Or, as
usually given,
D = I CdPSU*. (5.4)
89
CHAPTER 5
That, of course, makes it closely analogous to the pressure coefficients with
which the last chapter was replete—they were obtained by dividing
measured pressure by dynamic pressure. The only change is the addition of an
explicit area.
Equation (5.4) is most definitely not the formula for drag, no matter what
one sees in the biological literature. It's just a definitional equation that
converts drag to drag coefficient and vice versa. The value of a in (5.2) and
(5.3) wasn't constant, we have no grounds for declaring C(l a constant, and it
doesn't often turn out to be particularly constant. Still, equation (5.4)
represents a huge simplification, the one we've been working toward. Instead
of having to record how drag varies separately with speed, size, viscosity,
and density, we need know only how the coefficient of drag varies with the
Reynolds number. The fact that both drag coefficient and Reynolds
number are dimensionless may take a little getting used to, but after a while one
does acquire a properly contemptuous familiarity with them. As with
pressure coefficients, actual drag can be obtained from a value of the coefficient
simply by recourse to the defining equation (here, equation 5.4).
The inclusion of a factor, S, for area does present a minor difficulty, one
a little trickier than that of picking a length for use in the Reynolds number.
Clearly the drag coefficient one calculates from a datum for drag depends
very much on the choice. The commonest reference area is the "frontal" or
projecting area of an object—its maximum projection onto a plane normal
to the direction of flow. Hereafter, we'll use Sf for that area and Cdf for the
corresponding drag coefficient. Frontal area is particularly useful for non-
streamlined objects of relatively high drag at high and medium Reynolds
numbers, for which drag isn't so very far from dynamic pressure times
frontal area.
Three other areas are used, and each has points in its favor. "Wetted
area" is the total surface exposed to flow; it's commonly used for
streamlined bodies where (as we'll discuss) drag is largely a matter of viscosity and
shear, and what matters most is how much skin is showing; for it, we'll use
Su„ and for the corresponding coefficient, Cdw. For most situations it's more
biologically relevant than frontal area, and it's independent of the specific
orientation of the organism. The main difficulty in using wetted area is the
practical one of measuring it on a real organism. Fish and cetaceans may be
fairly smooth sorts, but consider flying insects, swimming crustaceans, nau-
tiloids, trees—they're what we might call "f ractomorphic," and any figure
for surface area is in part a definitional artifact. One can approximate the
shapes of creatures with cylinders, spheroids, and so forth and from these
calculate areas; but that may obscure features that have mattered to the
selective process.
A third reference, "plan form area" or "profile area" (S^ and Cdf> here) is
commonly used with lift-producing airfoils; it's the maximum projected
area of the airfoil, the area that one would see if the airfoil were laid on a
90
DRAG, SCALE, REYNOLDS NUMBER
table and viewed from above. Like wetted area, it's independent of
orientation with respect to flow. That's important if, say, one draws a graph of drag
coefficient against values of the small angle between wing and wind—it
would be most awkward to have the definition of one variable changing in
response to alterations in the other. One should never change the
reference area used from one datum to the next, even if something is
reorienting in a flow or even if (as with a leaf in a wind) its actual exposed surface
area is changing. Also, anticipating a bit, if one is to compare lift and drag
using lift and drag coefficients, the same reference area (usually plan form)
must be used for both.
Finally, the two-thirds power of volume has been used as reference area,
originally for airships, where volume was proportional to lift. We'll
designate it as Sv and the drag coefficient with it as Cdv. V2/3 is probably the most
appropriate for organisms, with their fitness a matter of guts and gonads
inside; it's also perhaps the easiest to measure, from a simple ratio of mass
to density. But it's also the least common. I'll give a strong pitch for its use,
and not just because it coincides with my initials.
Picking a reference area is more important than it may at first appear. It
can have a major influence on the conclusions one draws, as most cogently
illustrated by D. E. Alexander (1990). He compared the drag of specimens
of two species of marine isopod crustaceans, Idotea wosnesenskn and /. re-
secata. I. wosnesenska is relatively more rotund, 2.9 times as long as wide,
while/, resecata has an aspect ratio of 4.5:1. Using frontal area as reference
he found that the drag coefficient of/, wosnesenskii was 77% of that of/.
resecata. By contrast, using wetted area (and a presumption of oblate
spheroidal shapes), the same comparison came out to 142%. Which has the
higher drag? It depends on how one does the comparison! So why not just
compare drag and ignore the coef fiencients? The problem is that one has
then done no normalization for the considerable effect of size (not to
mention speed); we're usually not so much interested in drag per se as in
comparing the "dragginess" of shapes.
At this point I'll be frankly imperious and declare that no published
figure for drag coefficient is of any value unless the reference area is
indicated. And, I reemphasize, the results of comparisons among different
shapes depend on that choice of area—even for the Cd\ of automobiles.
For instance, if frontal area is used, then the lower car is at a disadvantage—
its lower drag may not result in a lower Cd.
With only two variables for a given shape—coefficient of drag and
Reynolds number—graphs are a great convenience. We have, remember,
lumped all the peculiarities into the relationship between these variables. If
drag behaved as Newton (and much popular literature) believed it should,
then all graphs of C(l as functions oiRe should be horizontal, lines. In its
deviation from such a line, the graph shamelessly dissects out and exposes
the queerness. Figures 5.3 and 5.4 give such plots for cylinders and
91
CHAPTER 5
10000
100000 1000000
Re
Figure 5.3. Drag coefficient (based on frontal area) versus Reynolds
number (based on diameter) for flow normal to the long axis of a long
circular cylinder. The dashed line comes from a fitted formula, equation
(15.7).
spheres, respectively. If your intuition needs a crutch, think of these
graphs as representing, for a given object and fluid, drag divided by the
square of speed (perhaps "speed-specific drag") on the ordinate and speed
on the abscissa.
Biologists may be surprised to find that a simple case such as the drag of a
10000
100000 1000000
Re
Figure 5.4. Drag coefficient (based on frontal area) versus Reynolds
number (based on diameter) for flow across a sphere. The dashed line
comes from a fitted formula, equation (15.2).
92
DRAG, SCALE, REYNOLDS NUMBER
circular cylinder can generate the kind of graph we think of as the special
curse of our more complex systems. If a cylinder is messy, what of a
barnacle? The abruptness of the transitions and the bumpiness of these graphs
of Cd versus Re for simple objects prove to be much more drastic than the
equivalent transitions for complexly shaped "biological" entities. The very
symmetry and regularity of these simple shapes mean that transitions from
one flow regime to another tend to take place synchronously on the whole
object rather than first at one location and then at another.
Flow around a Cylinder Revisited
We began the chapter by asking about the flow around a circular cylinder
and its drag. We saw that the pressure distribution differs markedly from
that expected for an ideal fluid, and we've just seen that the plot of Cd
versus Re is, to say the least, irregular. Let's tie up a loose end by looking at
the actual patterns of flow that occur as the Reynolds number increases,
combining a few paragraphs of verbal description with the illustrations in
Figure 5.5. You ought to refer as well to Figure 5.3 as we proceed.
At Reynolds numbers well below unity, smooth and vortex-free flow
surrounds the cylinder. Flow looks a little like that expected for an ideal
fluid (Figure 5.1), but the resemblance fades on close examination—there's
no fore and aft symmetry here, only the same absence of vortices. Notice
that the presence of the cylinder lowers the velocities of flow for quite a
distance from itself; the cylinder is surrounded by a cloud of retarded fluid
that largely obscures any details of its shape. This wide influence can
corrupt measurements of drag if a solid wall is anywhere around, and a wall's
effect gets worse as Reynolds numbers decrease. Thus for flow across a
circular cylinder, White (1946) found that, at Re = 10~4, the presence of
walls 500 cylinder-diameters away doubles the apparent drag. As the
Reynolds number approaches and then exceeds unity, the volume of the
disturbed fluid gradually diminishes, and we enter a regime in which the drag
curves for different shapes diverge more radically.
At Reynolds numbers between about 10 and 40, the cylinder bears a pair
of attached eddies on its rear. With flow from left to right, the upper eddy
rotates clockwise and the lower one counterclockwise. Above about 40, the
pattern is no longer stable, and the vortices alternately detach, producing a
wake of vortices with each rotating in a direction opposite that of its
predecessor farther downstream. This pattern of alternating vortices is known as
a "Von Karman trail" and will get more attention in Chapter 16. Periodic
shedding of vortices continues up to a Reynolds number of about 100,000,
but with an increasingly turbulent wake behind the cylinder.
Somewhere between 100,000 and 250,000 another transition occurs.
The wide wake of turbulent eddies narrows rather abruptly, and the drag
coefficient concomitantly drops by about two-thirds. The "somewhere" is a
93
CHAPTER 5
Von K£rm£n vortex trail
Figure 5.5. Patterns of flow behind a circular cylinder. Note the
absence of vortices at low Reynolds numbers in (a) and the constriction of
the wake between (c) and (d). This last change is concomitant with the
drop in drag coefficient—the great "drag crisis"—at Reynolds numbers
between about 100,000 and 250,000.
deliberate evasion, for the precise point depends on the circumstances.
The less turbulent the basic flow to which the cylinder is exposed, the
higher is the Reynolds number for the transition. Not only does the drag
coefficient drop, but the drop is usually so sudden that the drag itself
briefly undergoes a paradoxical reduction with further increase in the
Reynolds number. With still further increase the pattern of flow changes
little more—up to numbers so high that compressibility is no longer
negligible and so high as to be biologically irrelevant.
The abrupt drop in drag coefficient and the narrowing of the wake is
associated with a phenomenon of importance at all Reynolds numbers
much above unity—it's called "separation of flow." Recall from Figure 5.2
94
DRAG, SCALE, REYNOLDS NUMBER
that the pressure coefficient increases from the widest part of a cylinder
toward the rear. Just where and how much it increases depends a bit on the
Reynolds number, but there's always some region of increase as one looks
farther rearward around the surface. Thus in moving around the cylinder
from the forward stagnation point, fluid first goes from high to low
pressure, an easy and natural journey. But beyond the point of minimum
pressure, fluid must progress "uphill" to make any more progress toward
the center of the rear. Such motion against a pressure gradient is possible
only at the expense of preexisting momentum, which ordinarily isn't
sufficient for the flow to get to the rear stagnation point. The trouble is that
fluid has been robbed of its momentum by viscous effects—shear at and
near the surface. It's just like a sled that coasts down one hill but, having
experienced friction, can't make it up another hill of equal height. What
happens instead is that at some point the fluid stops following the surface of
the cylinder and instead heads off more or less straight downstream; the
place at which this occurs is called the "separation point." It occurs very
near the point (see Figure 5.2) at which ideal and real pressure curves
diverge most severely.
Downstream from the separation point are eddies, general turbulence,
and, further back, the periodically shed vortices of the Von Karman trail.
The noise you hear when facing directly into a moderate wind is the
turbulence around your ears, which are just downstream from the separation
point for wind blowing around your head (Kristiansen and Petterson
1978). And the coolness felt between the shoulder blades when running
shirtless reflects the general stirring about of the postseparation flow
around your unstreamlined torso. Immediately behind the separation
point, fluid near the surface commonly flows in what's otherwise the
upstream direction (Figure 5.6a). Thus the separation point is (going along
the length of the cylinder) a line of minimal flow near the surface—flows
from front and rear join and move away from the surface. By monitoring
the intensity of flow near the surface with something that erodes more
swiftly in more rapid flow, you can easily detect this line of reduced fluid
motion. Try arranging a piece of ordinary chalk so it protrudes upward
from the bed of a steady stream. After a few hours its cylindrical shape will
have altered enough to indicate the variations in speed of flow near the
surface—in particular, a lengthwise ridge will mark the line of separation.
What happens in the great drag crisis at Reynolds numbers around
100,000 is that the separation point suddenly shifts downstream. At rising
but lower Reynolds numbers it had been slowly moving forward as the fluid
gradually lost its ability to curve around behind the point of maximum
width. Thus at Re = 300 the separation point is 121° around from the
forward stagnation point, while by Re = 3000 it has moved ahead to 96°
(Seeley et al. 1975). But at that transition, turbulence (which has already
95
CHAPTER 5
Figure 5.6. Separation of flow from a surface, (a) A detailed look at the
streamlines near the separation point for flow across a cylinder,
(b) The same for a flat plate broadside to flow. The location of the
cylinder's separation point varies with the Reynolds number while the plate's
doesn't; concomitantly the drag coefficient of the cylinder varies much
more than that of the plate.
been present) invades the region of fluid very near the surface of the
cylinder. The result is an increase in the momentum of the fluid near the
surface, which permits the flow to follow the surface farther around before
separating. And that, in turn, leads to abrupt the abrupt reductions in the
width of the wake and in both drag and drag coefficient.
Spheres do much the same things as cylinders, as is evident from Figures
5.3 and 5.4, except that spheres leave no tidy Von Karman trails in their
wakes. The Reynolds number of a sphere's transition point can, in fact, be
used as a measure of the turbulence of an airstream and thus as the basis of
a figure of merit for a wind tunnel. A lower value for the transition point
indicates greater turbulence and, by the usual criteria, a poorer tunnel
(Pankhurst and Holder 1952).
Shape and Two Kinds of Drag
The peculiarity of drag, especially its relationship to the Reynolds
number, has been blamed on this odd phenomenon of separation. But, as we'll
see, separation isn't the whole story—even without it drag remains real.
The overall drag of an object, what one measures with a force transducer,
can be dissected into two components. Both the "skin friction" and the
"pressure drag" ultimately come from viscous effects, but they do so in
quite different ways.
Skin friction is the direct consequence of the interlamellar stickiness of
fluids, the viscous mechanism by which one plate exerts a force on another
(as shown in Figure 2.2). It always exerts its force parallel to the surface and
in the local direction of flow. As you might expect, it's more significant in
more viscous situations, that is, at low Reynolds numbers. But no Reynolds
96
DRAG, SCALE, REYNOLDS NUMBER
number is so high that skin friction disappears. As long as the no-slip
condition holds, there's a shear region in a fluid near a surface. And the
more surface (skin) a body exposes to the flow, the more skin friction it will
experience.
Pressure drag is also a result of viscosity, but a less direct one. It occurs
mainly because of separation and the consequent fact that the dynamic
pressure on the front is not counterbalanced by an equal and opposite
pressure on the rear. In effect, energy is being invested in accelerating
fluid to get it around the object, but the energy isn't being returned to the
object in decelerating fluid near its rear. Instead, the energy is dissipated
(to heat, eventually) in the wake. While it's commonly regarded as a high
Reynolds number phenomenon, pressure drag is significant at all Reynolds
numbers; only its magnitude relative to skin friction is greater when the
Reynolds number is high. Of the overall drag of a circular cylinder (yet
again), pressure drag constitutes 57% at Re = 10; 71 % at Re — 100; 87% at
Re = 1000; and 97% at Re = 10,000.
Although the occurrence of pressure drag ultimately requires viscosity,
it mainly reflects inertial forces. As such it's roughly proportion to surface
area (or to the square of a linear dimension) and to the square of velocity.
The drag coefficient is inversely proportional to these factors (equation
5.4), so where pressure drag dominates, the drag coefficient will not vary
widely. By contrast, skin friction is a direct matter of viscous forces and
follows the first powers of linear dimensions and velocity as in equation
(2.3). Thus at low Reynolds numbers, where skin friction predominates,
the drag coefficient will be inversely proportional to the Reynolds
number—as you can see in the descending lines at the left in the graphs of
Figures 5.3 and 5.4. In between, at Reynolds numbers of roughly 1 to
1,000,000 are the odd transitions we've just discussed. Airplanes may fly
above all these phenomenological bumps, but the range includes an
awesome diversity of biologically interesting situations.
An example should sharpen the contrast between pressure drag and
skin friction as well as between high and low Reynolds numbers. Consider
two objects, the first with negligible pressure drag at any Reynolds number,
the second with very high pressure drag at all Reynolds numbers. One
object is a long, flat plate of negligible thickness parallel to the flow
(imagine a long, thin wing). The other object is the same flat plate now oriented
perpendicular to flow. Table 5.2 gives data for the drag coefficients of these
objects over a millionfold range of Reynolds numbers, together with the
ratio of the perpendicular to the parallel drag coefficients.
The first thing that emerges from Table 5.2 is the rise in drag coefficient
at low Reynolds number, what we saw for spheres and cylinders. One then
notices the differences from spheres and cylinders—in particular the
absence of the sudden drop in the drag coefficient of the perpendicular plate
97
CHAPTER 5
Table 5.2 Drag coefficients (based on wetted area) of a very
long, flat plate extending across a flow based on wetted area.
C
dw
Reynolds Number
1,000,000
100,000
10,000
1000
100
10*
1*
Parallel to Flow
0.0013
0.0042
0.013
0.042
0.13
1.1
6.2
Perpendicular to Flow Ratio
0.98 740
0.98 230
0.98 74
1.00 24
1.22 9.2
1.9 1.7
9.2 1.5
Sources: Ellington 1991; Thorn and Swart 1940
*Thin airfoils, actually.
at high Re. What causes this omission is the fact that the location of the
separation point is fixed at the outer edge of the plate so it can't move
upstream or downstream. The constancy of the drag coefficient that
results from the fixed separation point can be put to good use. A fiat plate or a
disk perpendicular to flow can be used as an object of known drag so a force
transducer can be used to calibrate the speed of a flow tank or wind tunnel.
One might expect a value of 0.5 rather than about 1.0 from all the talk
earlier about stagnation points; the increase results from a wake that's
actually wider than the plate.4 What happens, as shown in Figure 5.6b, is
that flow moving laterally along the front of the plate is carried by its
momentum a short ways beyond the edge of the plate—it doesn't turn the
corner sharply as it heads downstream again. For that matter, the width of
the wake gives a good first indication of the magnitude of the drag of any
object at moderate and high Reynolds numbers.
More interesting yet are the ratios of the drag coefficients (and thus the
drag data themselves, since the reference area is unchanged). In general, at
low Reynolds numbers, as was noted earlier, skin friction is of great
importance. It depends only a little on orientation. At moderate and high
Reynolds numbers, pressure drag can be overwhelming; but its actual
significance depends very much on the orientation of the plate. Thus orientation
has a far more potent effect on overall drag at high and moderate Re's.
Which brings us to the question of how drag might be reduced, clearly a
vital matter for either nature or the human designer. At moderate and
4 Flat plates that aren't really long have lower drag coefficients—leakage around an
end makes quite a difference. If the plate is only ten times as long as wide, C^,, isjust 0.68
above Re = 103; if less than five times longer than wide, C(ht, is 0.61. A circular disk has a
C^, of 0.56 (Hoerner 1965; Ellington 1991)
98
DRAG, SCALE, REYNOLDS NUMBER
high Reynolds numbers, any object from which flow separates will
experience a relatively high drag; the problem, again, is the energy loss in the
rear. A trick, though, has been recognized at least empirically for a very
long time. If the object is endowed with a long and tapering tail, fluid
gradually decelerates in the rear, little or no separation occurs, and the
object is literally pushed forward by the wedgelike closure of the fluid
behind it. Instead of being lost in the wake, the energy reemerges as a
forward-directed pressure from the rear that nearly counterbalances the
dynamic pressure on the front. The trick is called "streamlining" because,
loosely speaking, an object such as that of Figure 5.7 is shaped to follow (or,
much the same thing, produce) an advantageous set of streamlines.
Especially at high Reynolds numbers streamlining can be immensely effective: a
well-designed airship may have less than 2 percent of the drag of a sphere
of the same frontal area. The sphere, by the way, is an example of a "bluff
body," the term antonymous to "streamlined body." What matters is mainly
the design of the rear. As Hamlet put it, "There's a divinity that shapes our
ends, Rough-hew them how we will."
But as the Reynolds number gets lower, streamlining isn't quite so
overwhelmingly beneficial. The reshaping involved exposes more surface
relative to either projected area or to volume contained. And more surface
inevitably means more skin friction. At the Reynolds numbers
encountered by whales, large fish, and flying birds, skin friction is a minor matter.
At the moderate Re's of the larger flying and swimming insects,
streamlining is still a good thing. At much lower Reynolds numbers the extra surface
exposed may outweigh any reduction in pressure drag, which is
undoubtedly one of the reasons why very small swimming creatures do not look
obviously streamlined.
Perhaps the main disadvantage of streamlining as a scheme for drag
reduction, from a biological point of view, is that it assumes a particular
orientation to the flow. A minor change in wind direction would more than
wipe out the benefits of a streamlined tree trunk. If the direction of flow
can be neither predicted nor controlled, then either a bluff body must be
tolerated or else some other stratagem brought to bear. More about the
latter possibility in the next chapter.
Can anything at all be done about the high drag of bluff bodies such as
spheres and cylinders? Curiously enough, the situation can occasionally be
improved by roughening the surface. Usually, and the point should be
emphasized, roughness is either without effect or it increases drag. At low
Reynolds numbers small bumps will be within the slowly moving fluid near
the surface and will thus be of little consequence. At high Re's, roughness
increases the drag of streamlined objects. But in a certain narrow range
roughening can help: a rough surface promotes turbulence close to a
surface and can thereby postpone separation as the fluid travels around a
99
CHAPTER 5
chord, c '
Figure 5.7. A streamlined shape, with the familiar blunt front and
sharp rear. It may be viewed either as a body of revolution, symmetrical
about its long axis in the plane of the paper, or as a typical cross section
of an elongate body extending far above and below the page. This
particular shape has a "fineness ratio," or chord over maximum thickness, of
about 3.5. Its maximum thickness occurs 30% of the way from front to
rear, that is, at x/c = 0.3.
bluff body, just as does turbulence in the flow itself. The transition to
a lower drag coefficient that we noted for Reynolds numbers between
100,000 and 250,000 can be pushed down to as low as 25,000 for a rough
cylinder or sphere (Figure 5.8). Any resulting increase in drag coefficient
below 25,000 won't matter since down there the actual drag won't be
especially high anyway. If the object is operating with a top Re oi\ say, 100,000,
then any increase in drag coefficient above perhaps 150,000 won't matter
either. Within limits, the rougher the surface, the lower the transition
point, but the higher the drag coefficient both before and after transition.
Some refining is possible by roughening only the portion just upstream of
the maximum diameter, but that requires knowing the direction of flow;
however, if the direction can be counted on, then streamlining is a much
better bet.
The most familiar example of intentional roughness for drag reduction
is a golf ball with its set of dimples. The practice of dimpling originated
with the observation that old, rough balls traveled farther than smooth,
new balls. At the urging of a driver, a golf ball moves at a Reynolds number
of about 50,000 to 150,000, so the scheme is reasonable. One might wonder
whether the rough bark of trees is a similar adaptation: slender branches
are smooth while trunks and large branches are much rougher. I think it is
unlikely—for a trunk or branch 0.2 m in diameter, Re = 100,000 will be
reached at a speed of about 8ms-1, less than 20 mph. Speeds high enough
to be troublesome would automatically exceed the turbulent transition
whatever the degree of roughness, which means that a smooth surface
should be better. In any case, leaves (about which more in the next chapter)
contribute much more to drag under most circumstances than do trunk
10 0
1.25'
1-
Cdp 0.75-
0.25
0.015
0.0125-
'dp
0.01-
0.0075-
0 005
DRAG, SCALE, REYNOLDS NUMBER
0.5- (a) cylinder
—i—
4.5
T
5
Log Re
5.5
Figure 5.8. The effects of surface roughness on the drag coefficient of
streamlined and unstreamlined shapes. At low Reynolds numbers
roughness is inconsequential in either case. For a cylinder (a), but not a
streamlined body (b), surface roughness can lower the drag, but it can do so
only over a limited range of Reynolds numbers. Drag coefficients are
based on plan form area (for a cylinder the same as frontal area).
and branches. Another possible application of purposeful roughening
might be in the design of attached reef or rocky-coastal organisms that are
exposed at times to fairly violent water currents—but Denny's (1989)
caution that lift is often more important than drag, mentioned in the last
chapter, should be kept in mind. At this point I know of no decently
documented case in which a bluff organism in nature achieves drag
reduction through roughening that might be adaptively significant.
Again, streamlining is a much more potent approach to drag reduction.
Quite a number of human cultures have developed streamlined throwing
sticks ("boomerangs"), mostly of the nonreturning genre. These are most
impressive weapons against small prey and might have really caught on if
standardization and mass production had been feasible; not only do they
cut a wide swath with lots of rotational momentum, but they travel far
101
CHAPTER 5
greater distances than any hand-thrown, spherical projectiles. I'm told that
while it's quite difficult to return a ball from the outfield wall, a boomerang
could easily be thrown out of any baseball stadium from home plate. The
difficulty with the streamlined stick, once again, is the necessity for
knowing or controlling the direction of flow—it loses all advantage if it tumbles.
These matters of drag, Reynolds number, separation, and so forth are
particularly well explained by Shapiro (1961), intuitively and anecdotally,
and by Goldstein (1938), more formally and thoroughly.
Modeling and the Reynolds Number
Let's return to two points made earlier. Consider all imaginable
situations involving either rigid objects in flows or flows through the interstices
of such objects. If any pair of objects are geometrically similar, then
equality of the Reynolds numbers of their flows implies (1) geometric similarity
of the pattern of flow and (2) equality of pressure and drag coefficients.
You might just reread that last sentence. Physical science has rarely given
the biologist a better deal or one that's so commonly ignored. What it
means is the following. Say you have a situation involving fluid movement
that you find experimentally awkward by virtue of inconvenient size,
speed, or fluid medium. You are completely free to model that situation
with another one of different speed, size, or medium provided only that
you maintain the original Reynolds number and preserve geometric
similarity. Few restrictions apply to this rule. Mainly, compressibility of the
fluid under the circumstances must be negligible, no fluid-fluid interfaces
may be involved, and the situation must be isothermal. Beyond these, the
rule is both precise and general.
Put in the usual jargon, if the Reynolds numbers for two situations are
the same, the situations are "dynamically similar." Drag may be a complex
phenomenon, inadequate theory may make recourse to measurements
and models mandatory, but dynamic similarity greatly facilitates the
business. If both the original situation and the model system involve the same
fluid medium, then equality of the Reynolds number is ensured just by
maintaining constant the product of length and velocity. If the medium is
changed, the variables are three—length, speed, and kinematic viscosity
(equation 5.1)—the juggling is only a little more complicated and the
options are far wider. Often one can take the same object and test it in a new
medium, adjusting the speed to compensate for the change in kinematic
viscosity. For instance, flow visualization is far easier in water than in air,
especially for small systems, because neutrally buoyant markers are more
readily available and because equivalent flows in water are much slower
(ordinarily fifteenfold, recall). To work on very small organisms, large
models can be made and then tested in a highly viscous medium such as
102
DRAG, SCALE, REYNOLDS NUMBER
corn syrup, various oils, or glycerin—the shift has been found useful in
recent years for several studies on the mechanisms involved in suspension
feeding (see, for instance, Braimah 1987). For inconveniently large
systems, a smaller model may be tested at a higher velocity.
I've used the scheme on quite a number of occasions. One happened
when investigating induced ventilation of the burrows of prairie dogs
(Vogel et al. 1973). Such burrows are unwieldy objects, about 0.1 m in
internal diameter, about 15 m long, and located more than 2000 km from
my laboratory. To look at the effect on internal flow of entrance and exit
geometries, we built a model burrow ten times smaller than a real one, a
model that fit into the local wind tunnel. To achieve the equivalent of an
external wind of 1ms-1 the wind tunnel was simply run at ten times that
speed—about 22 mph. Perhaps a lot of stove pipe could have been buried
in an open field, but no advantage would have justified the heroics.
As part of a more general look at flow induction, with its mix of Bernoulli
effects and viscous entrainment, I wanted to see how the phenomenon
scaled with Reynolds number (Vogel 1976). By combining models of a
range of sizes with a range of speeds in two flow tanks, I could measure the
rate of induced flow over three orders of magnitude. The scheme worked
perfectly—the resulting curve had no discontinuities between segments
obtained on different models.
Much of the work on flow-induced pressures mentioned in the last
chapter was done in wind tunnels despite the fact that all the cases—scallop and
algal thallus refilling as well as the squid—were aquatic (Vogel 1988b). It
just happened that I had equipment for measuring very low pressures only
in air. With scallops I used real shells and so worked at life size, with the
normal fif teenfold speed increase. For the squid I made a model 0.3 rather
than 0.2 m long, which meant that I could reduce speeds a little. Otherwise
I would have exceeded the top speed of the wind tunnel whose use our local
mechanical engineers kindly provided, not to mention tickling the tail of
the dragon of compressibility. 8 m s_1 times 15 is 120 m s-1; reducing that
by 0.2/0.3 brought it down to 80 m s-1, a manageable 180 mph.
One additional topic on modeling needs mention. Recall that equality of
Reynolds number means equality of drag and pressure coefficients. It
doesn't necessarily mean equality of either drag or pressures per se unless
the original object and the model are both exposed to the same fluid
medium. For a shift from air to water, with both at 20°, drag and pressures
will rise, although not enormously—3.5 times if the size of the object
remains unchanged. Water may be 800 times more dense than air, but the
speed decrease in the shift of medium compensates for a lot of that. (800
divided by 15 squared is about 3.5.) But even that 3.5-fold can spell trouble
when dealing with non-rigid objects, not at all an unusual thing for a
biologist concerned with organisms. The behavior in flow of a leaf cannot
103
CHAPTER 5
be presumed the same at 1 m s_1 in water as at 15 m s_1 in air, and
sometimes one prefers not to deal with models.
Sometimes a liquid of appropriate viscosity and density can be picked so
an object that normally experiences airflow can be shifted to a liquid system
without alteration of its actual drag and pressures. Deriving an expression
that permits such a liquid to be chosen isn't at all difficult. With subscript 1
indicating the original air and subscript 2 the unknown liquid, constant
Reynolds number means that
Pl^l _ P'2^2^2
M-l
&2
And constant drag coefficient means that
Dx _ D2
plSlUl
p2S2U22
Using the same object implies that /, = l2 and S{ = S2, so that the S's and /'s
in equations (5.5) and (5.6) cancel. We can then combine the two equations
to eliminate U{ and U2, obtaining a simple and useful result:
D2 _ p^
2
(5.5)
In shifting from one medium to the other, drag will be unchanged ifD2/D{
= 1. Table 5.3 gives D2/D{ values for the transition from air to a variety of
liquids for an object of a given size. Water at 52° would seem a reasonable
choice if the specimen can stand the heat. Alternatively, mixtures of
acetone and water or ethanol or of carbon disulfide and methanol might be
useful; one might have to get out the Ostwald viscometer (Chapter 2) to get
the viscosities of various mixtures.
I found this scheme quite useful when looking at the transmissivity to air
of the antennae of the giant silk moth, Luna, a few years ago (Vogel 1983).
Table 5.3 Increase in force on an object if shifted from air to
the indicated liquid.
Liquid
Acetone
Carbon disulfide
Carbon tetrachloride
Ethanol
Methanol
Octane
Increase Factor
0.421
0.379
2.069
6.331
1.556
1.441
Liquid
O-xylene
Propanol
Toluene
Water
Water, 52° C
Seawater
Increase Factor
3.052
21.957
1.369
3.467
0.994
4.137
Note: Liquids are at 20° C unless otherwise specified.
104
DRAG, SCALE, REYNOLDS NUMBER
While I could measure wind speeds just behind an antenna, I couldn't tell
exactly where to put my tiny anemometer to pick up air that had passed
through rather than gone around the antenna. That called for flow
visualization, but the antennae were too flexible to take the requisite force of
room temperature water at the right Reynolds number. 52° water with dye
markers worked fine as long as I kept immersion times fairly short.
105
CHAPTER 6
The Drag of Simple Shapes and
Sessile Systems
The subject of drag has now been introduced, aspersions cast upon
its origins, and defamatory remarks leveled at its character. In this
chapter and the next we focus attention a bit more specifically on what
organisms do about drag—how much drag they encounter and with what
adaptations they increase or decrease it. After all the complications of the
last chapter, you now appreciate that drag depends on shape in ways so
subtle that no single numerical index can relate the shape of an object to its
drag. Drag coefficients and Reynolds numbers, one must remember, are
mainly devices of practical utility for organizing empirical observations.
A Bit about the Biology of Drag
We talk quite a lot about copying nature, but we've really done it both
deliberately and successfully surprisingly few times (Vogel 1992b). One of
these was the occasion on which Sir George Cayley (1773-1857) devised
what seems to have been the first deliberately drag-minimizing shape.
Cayley recognized that in "spindle shaped bodies," "the shape of the hinder
part of the spindle is of as much importance as that of the front in
diminishing resistance." Since the explanation of the phenomenon seemed so
mysterious, he quite frankly copied nature, picking a biological system—a
trout—in which he guessed that drag was low. He measured the girth of
various cross sections of the trout and then divided by three to get rough
diameters. From the results he constructed the profile shown in Figure 6.1.
Von Karman (1954) has pointed out that Cayley's profile corresponds
almost precisely to that of a modern, low-drag airfoil. The coincidence may
seem somewhat surprising, since the body of a trout is a thrust-producing
form. But fish do glide or coast, and (as Weihs 1974 has shown) they may
improve their energy economy by alternately accelerating and gliding.
The biological relevance of drag needs little belaboring. The relatively
high metabolic cost of locomotion in continuous media—swimming and
flying—are indisputable; and, except for accelerations, the cost is a direct
reflection of the drag that must be overcome. The coincidence in shape
between large and fast swimmers and low-drag bits of human technology is
profound and demonstrable in sink or bathtub (Kaufmann 1974). It
applies to the fuselages of small aircraft, to the bodies of large birds (Pen-
106
DRAG AND SESSILE SYSTEMS
Figure 6.1. The profile of a modern, low-drag airfoil superimposed on
Cayley's data from girth measurements of a trout, the coincidence
pointed out by Von Karman (1954).
nycuick 1972), and to the hull shapes of ducks and boats (Prange and
Schmidt-Nielsen 1970). Only a little less obviously, sessile organisms may
pay a substantial price for incurring drag, a cost less easily denominated in
units of energy, but evident in elaborate and massive supportive systems.
Trees sometimes blow down and corals suffer damage in storms,1 and
trunks and stalks represent as real a metabolic investment as the machinery
and operating expenses of locomotory systems. Indeed, much of the
difficulty of looking at the biology of drag is that we see the solutions rather
than the basic problems, and thus the latter must of necessity be inferred.
Measuring the drag of some natural object yields only a datum. Only
through controlled alterations of the object and comparisons with models
of varying degrees of abstraction can much be said about its design. And
any such statements must be based, as well, on information about the
characteristics of the flow the living system encounters in nature.
The particular relevance of drag depends very much on the
circumstances. For a fish or bird, propelling itself away from a possible
contribution to the next tropic level, drag is clearly a Bad Thing. But for the same
fish, faced with an urgent need to reverse course, or for the bird, about to
land on a fixed branch rather than a runway, drag is undoubtedly most
welcome. Achieving especially low (or, as we'll see, high) drag isn't an end in
itself—drag isn't power, and so it's even further from the real currency of
natural selection, fitness, than is metabolic expenditure. Guts and gonads
have to be fitted into the fuselage of a bird—if drag were all that mattered,
minimization of size would be the main element of successful design.
Important functions may require structures that incur substantial
drag—suspension feeding and photosynthesis are only the most obvious
cases. Thus sea fans on reefs are exposed to strong, bidirectional flow.
They begin attached life oriented quite randomly but gradually grow to be
perpendicular to the currents; apparently this bad orientation is important
in maximizing the feeding efficacy of their tiny polyps (Wainwright and
Dillon 1969). And sometimes survival in the most immediate sense
depends on having sufficient drag, with nothing fixing fitness better than
1 But breakage may be, in part, a reproductive device in corals; so the cost-accounting
isn't as obvious as it seems (Tunnicliffe 1981).
107
CHAPTER 6
parachute or patagium. I once dropped two mice onto concrete from about
15 meters. They certainly seemed no worse from the experience; but as
interesting as the lack of injury was their adoption of a nicely parachutelike
posture during descent, with all appendages spread wide and with no
tumbling at all. Drag maximization clearly looked like a standard feature of
their behavioral repertoire.
Several cautions need mention at about this point. Asking whether drag
per se is what matters is very important, even in a system in which the forces
of flow are obviously substantial. I already mentioned (Chapter 4) a limpet
(Denny 1989) for which under some circumstances lift can be a worse
problem than drag. But that's only the tip of the iceberg. We'll not talk
much about unsteady flows until Chapter 16, but they can't be totally
ignored until then. As a wave passes an attached organism, the organism
experiences not just the drag attributable to the instantaneous velocity of
the water but also a force due to the acceleration of the water crossing it.
Perhaps the simplest way to view this "acceleration reaction" is to realize
that if an object is accelerated in one direction, an equal volume of fluid or
some equivalent of it must be accelerated in the other. So to accelerate an
object, one must pay a force tax proportional to its volume and to the
density of the medium. Wave surge causes a rapid increase in the speed of
flow around an attached organism, so the situation is almost the same as
that when the organism is accelerated—even if the frame of reference
makes the phenomenon oddly nonintuitive. As Denny et al. (1985) point
out, drag will increase with the area of an organism and thus with strength-
dependent variables such as attachment area. But the acceleration reaction
will increase with volume, so it becomes especially significant for large
objects—if the flow has sufficient acceleration it may be the greatest force
that a large organism must withstand. The moral is that it's as unsafe to
ignore unsteady effects as to ignore lift, at least with flows of water. If your
flow fluctuates, look at Chapter 16 or read Denny (1988) and try a few
ballpark calculations.
We'll be concerned here mainly with minimization of drag, and we'll
consider what happens at Reynolds numbers of roughly 100 and higher.
Most of the interesting cases of deliberate maximization of drag occur at
lower Reynolds numbers and are thus in a rather different physical world;
they'll be deferred to Chapter 15. There's human relevance to these
questions about the drag of organisms. Jobin and Ippen (1964) emphasized the
crucial role of drag on snails in irrigation canals. Only with sufficient water
velocity will snails be consistently dislodged and the local inhabitants not be
exposed to schistosomiasis. Considerable effort has gone into determining
optimal planting patterns for plantations of trees in which large-scale
blow-downs ("wind-throw") of evenly aged stands are insufficiently
uncommon. And excessive tendency toward "lodging" (the same thing) must be
108
DRAG AND SESSILE SYSTEMS
avoided when developing high-yield plants; about the matter, see Grace
(1977) and Niklas (1992).
More about Shape and Drag
The distinction between streamlined and bluff bodies has already been
mentioned. For organisms, the functional distinction isn't just a matter of
whether drag is to be minimized, ignored, or maximized. It depends, in
addition, on whether the direction of flow is constant or at least
controllable through passive, behavioral, or morphogenetic reorientation.
Streamlining, again, only works for a rather narrow range of relative orientations
of object and flow.
And streamlining isn't a simple and definitive "cure" for drag, with one
standard shape to replace a cylinder and another to replace a sphere. No—
things are more complicated. First, the optimal shapes depend on the
Reynolds number. At low Re (below 100 or so) minimization of pressure
drag entails a significant rise in skin friction, so one minimizes overall drag
with a somewhat stubbier shape than what would give the lowest pressure
drag. And second, at all Reynolds numbers alternatives of equivalent drag
are available, trading off, as it were, a convexity here for a concavity there;
the further from some ideally absolute minimum the system is (for
whatever perhaps laudatory reason), the more latitude will exist for such
tradeoffs. Finally, what is almost always being optimized will be some complex of
variables. If, for instance, the object in question is a compression-resisting
strut, then it needs a decently high "second moment of area" to resist
buckling without excessive weight (for an introduction to such solid
mechanics, see Gordon 1978). Consequently a wider cross section may permit
a strut that's smaller as a whole, and a higher drag coefficient relative to any
of the areas we defined in Chapter 5 may actually be concomitant with a
lower overall drag.
Still, the tacit assumption just made that the wider strut has higher drag
is reasonably safe at moderate and high Reynolds numbers; it even holds
(except around/?£ = 100,000) for bluff bodies. Putin terms of streamlined
airfoils (for simplicity assume no lift is being generated), the higher the
ratio of thickness to chord (length in the direction of flow), the higher the
drag coefficient based on wetted area, as shown in Figure 6.2. Tricks,
though, are still possible, such as maintaining laminar flow by sucking fluid
in through the porous skin of an object. The latter has been demonstrated
in both streamlined and bluff bodies. Hoerner (1965) cites data for bluff
bodies in which the drag coefficient (on frontal area) was reduced from 1.2
to 0.15 at Re = 100,000, and Riedl (1971 a) suggested that the scheme might
be used by sponges. Most sponges prefer habitats in which the water moves,
and as we saw earlier, they suck in fluid through pores distributed over
109
CHAPTER 6
0.025-
0.02-
0.015-
0.01-
0.005- 1 1 1 1 r
4 8 12 16 20
fineness ratio
Figure 6.2. The variation of drag coefficient, based on plan form area,
with fineness ratio (recall 5.7) for an airfoil section at a Reynolds number
of 600,000. As an exercise, the reader might try to visualize such a graph
with a drag coefficient based on frontal area.
their surfaces. But one attempt to demonstrate drag reduction in a
spongelike model (Susan Conova, pers. comm.) proved unsuccessful. Perhaps
both the volume flow rates into sponges (recall Table 3.1) and the Reynolds
numbers at which they live are too low for much of an effect. If the value of
100,000 is critical, then one is looking for a sponge 10 centimeters in
diameter in a flow of a meter per second, which is pretty big and torren-
tially fast.
Another trick is nearly the opposite of sucking—ejecting high-velocity
fluid in a downstream direction around the normal separation point.
Ejection of fluid near the appropriate location happens when fish discharge
water through their opercula, either actively or by passive ram ventilation
(Chapter 4); and Freadman (1979, 1981) and Steffensen (1985) have
shown in bass and bluefish that ram ventilation, at least, produces flow over
the body with less turbulence, separation, and drag.
Yet another trick is the use of "splitter plates" (Figure 6.3) behind bluff
bodies to reduce the rate at which vortices are shed. Again Hoerner's
(1965) compendium supplies data. Benefits are obtained at Reynolds
numbers between 10,000 and 100,000, certainly a biologically well-inhabited
domain. Chamberlain (1976) suggested that the trailing part of a ceph-
alopod shell acts as a splitter plate, and Dudley et al. (1991) suggested the
same for the tail of a gliding tadpole.
At least one difference between the behavior of streamlined and bluff
bodies is more apparent than real. The transition to turbulent flow appears
1 10
DRAG AND SESSILE SYSTEMS
flow around flat plate
with rear "splitter"
at 10,000 < Re < 100,000
flow
Drag coefficients. Cdf
1.6
1.9
1.4
1.7
0.59
with
"splitter"
1.03
without
"splitter"
Figure 6.3. The influence of "splitter plates" on flow patterns and drag
coefficients of long flat plates and cylinders. Reynolds numbers are based
on maximum dimensions normal to flow and drag coefficients on frontal
areas.
to occur at different Reynolds nu mbers, between 100,000 and 250,000 for
bluff bodies and between 500,000 and 1,000,000 for streamlined ones. But
that's mostly an accident of the convention used to pick the characteristic
length in the Reynolds number. If, as is usual, maximum length in the
direction of flow is used, then the relatively more elongate shape of
streamlined bodies will automatically result in higher Reynolds numbers. The real
difference is what happens at transition—for the bluff body the onset of
turbulence close to the surface leads to a narrower wake, less pressure drag,
and hence less total drag than just prior to transition. For the streamlined
body, separation is absent or minor, pressure drag is quite small, and
turbulence, if it does anything, almost always increases drag.
Streamlined and bluff bodies are best regarded as the extremities of a
continuum of shapes, with degrees of bluffness as well as variations in the
effectiveness of streamlining. With a great enough thickness-to-chord
ratio, the presence of some posterior pointedness will do little to alter the
performance of what is just a minimally disguised cylinder. And the shape
of the anterior of a bluff body makes a considerable difference to its drag,
although in no version would we refer to it as streamlined. Thus rounding
or fairing the upstream edges of a flat-fronted object reduces the width of
the wake and the drag coefficient. Figure 6.4 gives some data for bluff
bodies at moderate and high Reynolds numbers. For instance, a solid
hemisphere with the flat side facing upstream has about the same drag as a
1 1 1
CHAPTER 6
sphere
Cdf
0.47
cylinder
Cdf
1.17
flow
hollow 0.38
hemisphere
hollow 1.42
hemisphere
hollow
half-cylinder
hollow
half-cylinder
1.20
2.30
solid
hemisphere
0.42
half-rectangular 1.55
solid
solid
hemisphere
1.17
long, flat plate 1.98
Figure 6.4. Drag coefficients, based on frontal area, for a variety of
three-dimensional bodies and two-dimensional profiles at Reynolds
numbers between 104 and 10°. All the data may not be precisely comparable
due to variations in the experimental conditions.
circular disk, but it has over twice the drag of a sphere of the same frontal
area or, for that matter, the same hemisphere with the flat side facing
downstream. A flat front is a bad bluff, whatever the implications in poker.
Still bluffer, as you can see from the figure, is a concave front. A hollow
hemisphere facing upstream has about 20% more drag than a solid
hemisphere or circular disk, three times the drag of a sphere, and almost four
times the drag it would have if facing downstream. This last comparison, by
the way, explains why whirling cup anemometers turn. The cup facing the
wind has a higher drag, not due to any air it contains, but because it has a
much wider wake than that of the cup facing downwind.2 A corresponding
difference occurs with long, hollow half-cylinders as well, which have Cdf's
of 1.17 and 0.42 when facing upstream and downstream, respectively. This
last difference is the basis for the operation of a "Savonius rotor," a pair of
half-cylinders on opposite sides of a vertical shaft. It makes an inefficient
2 Notice, the next time you see one, that whirling cup anemometers have three cups,
not two or four. One with two or four cups will work, but it doesn't start as well or run as
smoothly. We ran afoul of the problem when designing a current-driven stirrer for small
chambers to be located on a bay bottom (Cahoon 1988); it was just another thing obvious
only post facto.
1 12
DRAG AND SESSILE SYSTEMS
windmill, drag based instead of lift based; for a time it enjoyed some
countercultural appeal since the open half-cylinders could be nothing
more than the halves of an ordinary oil drum that had been cut lengthwise.
Structures in which a concave side faces upstream occur in organisms, most
commonly in connection with passive suspension feeding. The same
pressure drop responsible for their high drag is what drives fluid through the
filter—suspension feeders can't have their cake and eat it too. Thus the
front of a sea pen (Figure 6.5a) is more or less a hollow half-cylinder (Best
1988), and the cephalic fans of black fly larvae (Figure 6.5b) living atop
rocks in rapid streams are cuplike. These fans contribute about half of the
total drag of the creatures (Eymann 1988).
Very much the same story applies to axisymmetrical bodies oriented with
their axes parallel to flow (Figure 6.4 again). A concave face is worst, a flat
face is somewhat better, a convex hemispherical front is better yet, and a
parabolic nose has about the lowest drag of all. The same ordering holds
for a "rectangular section," a long cross-flow beam with a chord much
larger than the thickness. Rounding the upstream edge, much as the
exposed edge of a stair tread is rounded, can reduce the drag coefficient by
40%. By rounding the front, the drag coefficient of the original
Volkswagen "Microbus" was reduced from 0.73 to 0.44—not the order of mag-
Figure 6.5. Living structures whose concave sides face upstream and
which thus have high drag coefficients: (a) a sea pen, Ptilosarcus;
(b) a larval black fly, Simulium.
1 13
CHAPTER 6
nitude improvement possible with full (and impractical) streamlining, but
certainly no inconsequential difference. Air deflectors atop the cabs of
articulated trucks function in a similar manner. At least the proper ones
do—I've occasionally seen one with a sharp center ridge and the overall
shape of a snowplow that might be worse than none at all.
A streamlined body over which flow goes from trailing to leading edge is
still a kind of streamlined body, but it's not a particularly bad one in terms
of drag. For an ordinary airfoil with a maximum thickness of 12% of the
chord, the drag coefficient (based on frontal area) is 0.06 for normal flow
and 0.15 for reversed flow. While that sounds bad—over twice as much
drag when flow is reversed—it's still only an eighth of that of a cylinder at
the same Reynolds number, which in this case is a million. Decent
streamlined shapes exist with relatively low but equal drags for opposite flow
directions; they have both edges pointed rather than rounded. While
irrelevant for aircraft, which are rarely called upon to fly backwards, they
matter for underwater protrusions on bidirectional boats (some ferries,
for instance) and for the blades of reversing fans. Bidirectional
streamlining may be important in the design of fixed and rigid organisms exposed to
alternating wave or tidal flows.
Most of the data discussed in the past few pages are strictly applicable
only at Reynolds numbers considerably higher than those commonly
encountered by ordinary organisms; they're in the range of about 10,000 to
1,000,000. There seems to be a great dearth of comparable data for the
range of 100 to 10,000 and almost nothing for 1 to 100. And extrapolation
is hazardous—a reasonable way to generate hypotheses but certainly
unsafe as hard evidence. Perhaps we biologists can subvert or convert an
engineer or two, or perhaps we'll just have to settle down to some
exploratory data collection.
Flexibility — Where Drag Depends on Shape
Depends on Drag
The structures thrown up by human technology are mostly stiff ones, at
least by contrast with most of the bits and pieces of organisms—a point that
arose in Chapter 4. While organisms do use stiff materials, there's almost
always some special explanation for their occurrence—bones as levers for
muscles to pull on; especially cheap material such as the calcium salts of a
supersaturated sea used by corals; teeth as cutters, slicers, grinders, and
other stress concentrators; some functional intolerance of sag in a flat
surface; and so forth. The less stiff, more compliant designs of nature are
probably (looked at grandly and globally) more economical of material
when measured against strength achieved, which ought to be argument
enough for their prevalence.
1 14
DRAG AND SESSILE SYSTEMS
Compliance, or flexibility, has major implications for what happens to
objects in flows. It can be either blessing or curse; what's certain is that it
creates complications. And the complications are of at least two sorts. First,
shape is no longer a given, an independent variable upon which
investigator or analyst can rely. Shape becomes an immediate function of flow
speed, and the way drag scales with speed assumes unusual interest.
Second, flows themselves behave in odd ways when moving along surfaces
that don't stay put, and the forces involved may be either increased or
decreased through such interactions.
What must be said with vehemence is that the introduction of flexible
solids into the story of objects in flows destroys the tidy distinction, not
between solids and fluids as such, but between solid mechanics and fluid
dynamics as areas of inquiry. The biologist may be able to ignore
compressible flows and make other simplifying assumptions, but this most
convenient distinction of traditional engineering is very likely to mislead us.
Recognition of the special situation isn't especially recent; Carstens (1968),
for instance, noted at least as a possibility the reduction of drag by
compliance of an organism. But Koehl (1977) really drew our attention to it—so
effectively that at least here at Duke University where she did the work
we've taught solid and fluid biomechanics in a single course ever since.
Even the title of her paper makes the point: "Effects of Sea Anemones on
the Flow Forces They Encounter."
These sea anemones are hydrostatically supported; and, as is commonly
the case for hydroskeletal systems, they're not especially rigid when faced
with bending loads—their flexural stiffnesses are low. Beyond that, they
can deflate and, in extreme cases, retract down against the substratum to
form a hemispherical lump. Deflation is only part of a complex behavioral
response to drag. A tall anemone, Metridium senile, protrudes well into the
mainstream, bends downstream at a constriction just beneath the oral disk,
and filters with its outstretched tentacles (Figure 6.6). Even just
reconfiguration of the tentacles without any alteration in the anemone's trunk
causes major changes in drag. For a rigid model equipped with flexible
tentacles, the drag coefficient is about 0.9. As the oral disk separates into
lobes, the coefficient drops to 0.4; collapse of the oral disk in the manner of
an overstressed umbrella reduces the coefficient to 0.3; and retraction of
the tentacles reduces it to 0.2. All of these coefficients are based on the
original frontal area, so they reflect proportional decreases in drag itself.
We are all too prone to speak of something being deformed or distorted by
the forces of flow, choices of words that carry the bias of a culture with a
penchant for stiff things, where alteration in shape is pathological or
geriatric and most often irreversible. When dealing with biological structures
exposed to flows of varying velocities, I urge the use of the term
"reconfiguration," with a neutral to positive connotation to the simpler but
1 15
CHAPTER 6
v»'V<.
Figure 6.6. Successive changes in the appearance of a large sea
anemone, Metridium, as current increases. Drag coefficients (based on original
frontal area) are 0.9, 0.4, 0.3, and 0.2.
prejudice-laden "deformation." We might prefer it otherwise, but choices
among words affect attitudes and hypotheses, even in science.
A Measure of Reconfiguration
We now have information about how quite a few flexible organisms alter
their shapes and thus their drag coefficients under the influence of the
flows they experience. What may be more useful than a summary is to
suggest how to compare such systems, to give a generally applicable scheme
for putting a number on the degree of reconfiguration. What we need is
some variable that indicates how the change in drag with speed of a flexible
object is different from that of a rigid object, a variable that draws attention
to what's special by correcting for what's ordinary. Drag behaves in such
queer ways that "ordinary" may sound like wishful thinking; still, the
everyday rule that drag is proportional to the square of velocity does work for
some objects on some occasions. Indeed, the rule describes precisely what's
going on whenever the drag coefficient doesn't vary with changes in the
Reynolds number, just by the way the drag coefficient is defined (equation
5.4). For a sphere or cylinder between Re = 1000 and transition and again
above transition, or for a flat plate broadside to flow at Reynolds numbers
above 1000, the rule works—in short, for bluff bodies at moderate to high
Reynolds numbers. For such situations,
Doc U2
(6.1)
and so
C oc u° * Re0.
'd
(6.2)
Plotting drag against speed (6.1) not only doesn't adjust for size, but it
makes any differences inevitably loom larger at higher speeds simply be-
1 16
DRAG AND SESSILE SYSTEMS
cause at low speeds all curves converge. Plotting drag coefficient against
speed or Reynolds number (6.2) avoids this illusion; even better, any
deviation from a horizontal line on such a graph is a telltale sign of something
other than an ordinary bluff body. We have, in such a graph, a rough-and-
ready way of separating the noteworthy from the ordinary.
But the Reynolds number includes a length factor, drag coefficient
incorporates an area factor, and biologically interesting objects come in a
wide variety of shapes. So looking at changes in drag coefficient all too
often gets tangled up in the complications connected with comparisons
among objects that differ in both shape and size. But we can get around
these problems of shape and size, problems afflicting both the rigid and the
flexible, by simply ignoring them. Drag coefficient is by definition
proportional to drag divided by the square of velocity, while Reynolds number is
proportional to velocity. Plotting drag over velocity squared against
velocity gives a graph similar to one of drag coefficient against Reynolds
number. For cases where drag coefficient is independent of Reynolds number,
-^ a u° oc /fc(>. (6.3)
Thus the result will simply be a horizontal line—quite literally, a baseline.
If, on the other hand, the line ascends, that means the object has a
disproportionate drag at high speeds, at least by comparison with our
paradigmatic bluff body. A descending line, by contrast, means that the object has
relatively lower drag at high speeds. Even the shape of the line is of interest.
Thus for small holly and pine trees I found that at low speeds, this D/U2,
what we might call "speed-specific drag," increases, while at higher speeds
it decreases (Figure 6.7). What's happening is that at low speeds the wind
mainly disorganizes the originally fairly parallel or coplanar needles or
leaves, while at higher speeds they reorganize into increasingly tight
clusters (Vogel 1984). Of course at low speeds the real (as opposed to speed-
specific) drag is low enough to matter little—one mustn't lose sight of the
way we've transformed the data.
A further degree of abstraction permits nongraphical comparisons. If
one puts speed-specific drag and speed on a log-log plot, then the slope of a
line (or a piece of a line) is the exponent of the relationship between the
original (nonlogarithmically transformed) variables. One is, in effect,
looking at the value of £ in a general version of proportionality (6.3):
jj$ « UE. (6.4)
The exponent that we're calling E then serves as the measure of
reconfiguration. A value of zero is a baseline, a horizontal line on the graph, a
statement that since drag is proportional to the square of velocity nothing
worthy of comment is happening. A value above zero says that the system is
1 17
CHAPTER 6
00
2
•o
o
<+H
o
21
W5
•o
4>
21
W5
e
S
3
S
<*-
o
c
o
•4—1
o
««
£
1-
0.8-
0.6-
0.4-
0.2 J 1 1 1 1 h
4 8 12 16 20
speed, m/s
Figure 6.7. Speed-specific drag (fraction of maximum) versus speed
for a small branch of a loblolly pine (Pinus taeda). For speeds above 6 m
s~\E = -1.13.
reconfiguring or behaving in a way that makes drag get disproportionately
bad at high speeds; thus a value of + 1.0 indicates that drag is proportional
to the cube rather than to the square of velocity. A value below zero signals a
reconfiguration that produces less than the expected drag at high
speeds—if the value gets down to —2.0, then drag is actually independent
of velocity. Table 6.1 gives a a collection of values of E.
Bear in mind that these values of E are not measures of either drag or
drag coefficient. What they indicate is how drag changes with speed, quite
independent of the specific magnitude of drag or its coefficient—they're a
look at reconfiguration, only half the story, but a particularly interesting
half in the living world of flexible objects.
In addition, a few cautions about using and interpreting them need to be
mentioned. The first is something of a corollary to one made earlier—that
drag isn't an end in itself but merely a factor that may or quite possibly may
not bear on fitness. A low value of E may be the quantitative signature of a
reconfigurational process that lowers drag at high speeds, but that doesn't
automatically mean that drag is what matters. Two separate studies of
suspension-feeding cnidarians provide a useful exposure to cold water.
Harvell and LaBarbera (1985) got nicely negative £-values of around
— 1.28 for a small, branched, colonial hydroid, Abietenana. At the same time
they showed quite convincingly that (1) these creatures would never
encounter flow speeds that posed any mechanical hazard to the colony, and
(2) the way colonies bent in flow prevented the flows to which individual
polyps were exposed from varying nearly as much as the flows facing the
colony as a whole. More recently, Sponaugle and LaBarbera (1991) got
1 18
Table 6.1 Values of E for various systems a
Re or
System Speed Range
Bluff body
Bluff body
Flat plate, parallel to flow
Flat plate, parallel to flow
Streamlined body, laminar flow
Cylinder, axis normal to flow
Hedophyllum sessile (alga)
Nereocystis luetkeana (alga)
Sargassum filipendula (alga)
Laminana (alga) on mussels
Macroalgae, marine
Red algae, freshwater
Pinus sylvestns (pine)
Pinns taeda, 1 m high
Pinus taeda, branch
Quercus alba (white oak), leaf
Quercus alba, clustered leaves
Other broad leaves & clusters
Ptilosarcus gurneyi (sea pen)
Pseudopterogorgia (gorgonian)
Abietenaria (hydroid)
Acropora reticulata (hard coral)
Various limpet shells
Epeorus sylvicole (mayfly larva)
Simulium vittatum (blackfly larva)
Locusta migratona, antenna
<1.0
1000-200,000
10-1000
1000-500,000
1000-500,000
20-120
0.5-2.5 m/s
1.3-2.0 m/s
0.5—1.5 m/s
0.12 to 0.62 m/s
ca. 2.5 m/s
0.2-0.75 m/s
9-38 m/s
8-19 m/s
8-19 m/s
10-20 m/s
10-20 m/s
10-20 m/s
0.11-0.26 m/s
0.13-0.35 m/s
0.025-0.40 m/s
1.5-3.0 m/s
0.15-0.45 m/s
0.4-1.2 m/s
0.1-0.7 m/s
20-120
D SPEEDS, WHERE UE oc D / U2 .
E Source of Data
-1.00
0.00
-0.60
-0.50
-0.50
-0.29
-1.12
-1.07
-1.47
-1.40*
0.28 to -0.76
0.33 to-1.27
-0.72*
-1.13
-1.16
+0.97
-0.44
0.20 to-1.18
-1.14
-1.66
-1.28*
+0.26*
0.0 to +1.2
+0.28*
-0.64*
-0.56
Janour 1951
White 1974
Armstrong 1989
Koehl & Alberte 1988
Pentcheff (pers. comm.)
Witman & Suchanek 1984
Carrington 1990
Sheath & Hambrook 1988
Mayhead 1973
Vogel 1984
n
Vogel 1989
n
n
Best 1985
Sponaugle & LaBarbera 1991
Harvell &: LaBarbera 1985
Vosburgh 1982
Dudley 1985
Weissenberger et al. 1991
Eymann 1988
Gewicke & Heinzel 1980
Note: Asterisks indicate my calculations from published graphs.
CHAPTER 6
very much the same result on a gorgonian (flexible) coral, Pseudopterogor-
gia. The latter, a single unbranched upright cylinder, is quite different in
shape from a hydroid; its £-value, —1.66, is even lower. I don't mean to
sound disparaging; by using their flexibility for reconfiguration, these
cnidarians are doing something distinctly more subtle and neat than mere
drag reduction.
The second caution concerns that baseline of zero—it presumes
exposure to moderate to high Reynolds numbers, free-stream flow, and a fairly
bluff body; to repeat, it presumes situations in which the drag of a rigid
object would be proportional to the square of velocity. At low Reynolds
numbers drag gradually shifts toward a direct proportionality, as we saw in
Figures 5.3 and 5.4. Thus the baseline ought to drop from zero toward
minus one. So the —0.56 of a locust antenna is bestjudged next to the value
for a cylinder at the same Reynolds number, —0.29. Similarly, for a flat
plate parallel to flow, the moderate and high Reynolds number £-value is
— 0.5, not 0.0 (see Table 5.2); the object is essentially a perfectly
streamlined object with skin friction only. So streamlining alone, in the absence of
flexibility, gives negative £-values. Finally, what about organisms that live in
the velocity gradients near surfaces? We'll have a lot more to say about them
in Chapters 8 and 9; for now what matters is how the speed of flow at any
point deep in that gradient varies with free-stream speed. In these velocity
gradients, local speed increases more drastically (even if it's always
absolutely lower) than free-stream speed—when the latter increases, not only
does dU increase, but dz decreases. Thus the gradient region gets thinner
and the gradient steeper, and local speed is proportional to free-stream
speed to the power 1.5. So ordinarily drag ought to be proportional to free-
stream speed cubed rather than squared. Our baseline therefore shifts up
to +1.0. By this criterion, Dudley's (1985) rigid limpet shells don't look
anomalous, and the mayfly larvae (a comment of Weissenberger et al. 1991
notwithstanding) may be doing something special.
A final caution concerns extrapolation. Bluntly, don't. Look at the graph
of speed-specific drag versus speed for a pine branch (Figure 6.7) if you
want to see how shaky are the grounds for extrapolating from high to low
speeds. And just consider the practical limits of nondestructive
reconfiguration if you're tempted to extrapolate to higher speeds. Carrington
(1990) makes the latter point, noting that her£-values, obtained at higher
speeds, are not as negative as those obtained for macroalgae by most other
people.
The Drag of Leaves on Trees
Preeminent among large organisms for whom drag is important are
120
D R A G AND SESSILE SYSTEMS
trees. They may exceed 50 m in height, they have enormous surface area,
they withstand major windstorms, and they constitute a substantial
proportion of the earth's terrestrial biomass. In a sense, trees have been dealt a
curiously awkward hand by the way the evolutionary process works. They
commonly compete among themselves, both intra- and interspecifically,
for their prime resource, sunlight. The tree that gets shaded (at least for
the larger players in the game) is the loser, so taller is better—even though,
with a sun over 100 million kilometers away, no one gets appreciably closer
by all that growth. No anticompetitive treaty seems possible; no way seems
open for an agreement whereby each individual foregoes, say, 10 meters of
trunk, in order that all be less vulnerable to mechanical failure. Thus the
greatest part of their surface area ends up at the greatest distance from the
earth. The result, as shown in Figure 6.8, is a huge lever arm converting
the drag of the leaves to a turning moment about some axis near the base.:*
As Alexander (1971), among others, has pointed out, the local forces
generated in the trunk by winds may far exceed those due to the weight of a
tree—a tree may fall under wind stress but is most unlikely to do so under
its weight alone.
So what is the drag of a tree? The practical problem is formidable. You
can't just measure the drag on a leaf or branch and blithely multiply. If
winds were steady, one might pull on a tree with a force-monitoring cable
and calibrate its own bending; then the observed bending in a wind of
known speed would give the drag, or at least the turning moment. But
winds are rarely steady enough, and (as far as I know), the approach hasn't
been tried. On one occasion a series of pine trees were tested in a large wind
tunnel (Fraser 1962), and drag was measured at speeds between 9 and 26 m
s~l. Sometime later the original data from these experiments were further
analyzed (Mayhead 1973). As speed increased, the drag in all cases
increased, with no discontinuities anywhere in the data—as we'd expect, the
parts of such irregular objects experienced no simultaneous sharp
transitions in flow regime. But as the wind speed increased, the coefficients of
drag (based on original frontal area) decreased since the trees steadily
decreased their exposed area. As a result, the increase in drag was more
nearly proportional to the first than to the second power of velocity. Put in
terms of our measure of reconfiguration, E was —0.72, much closer to — 1.0
than to 0.0.
3 The mechanics of what's called "windthrow" in trees is too complex and
incompletely understood for easy summary here. There seem to be several rather different
schemes for dealing with that evil turning moment. To complicate things further, trees
are much more vulnerable to repeated gusts than to steady winds. Blackburn et al. (1988)
and Niklas (1992) give a lot more information, but I know of no really satisfactory
account of the phenomenon in all its aspects.
12 1
CHAPTER 6
drag
lever
arm
rotation point
Figure 6.8. The principal forces with their lines of action and lever
arms caused by a wind blowing on a stiff tree such as a large oak.
It would be very nice to have a tidy comparison between trees and some
marine equivalent, in view of the higher overall forces to which a body in an
ocean can be subjected (about which more shortly). Most large attached
marine organisms—macroalgae and corals, mainly—aren't very treelike in
shape and loading regime. But at least one alga, the so-called sea palm,
Postelsia palmaeformis (see Figure 6.10), has flat photosynthetic structures
on top of an upright cylindrical stalk over half a meter high and lives in
dense stands on wave-swept shores. Thanks to the work of Paine (1988) and
Holbrook et al. (1991) we know quite a lot about both the ecology and
biomechanics of Postelsia. What we don't know, though, is very much about
its drag when subjected to interestingly high flow speeds—and the relevant
measurements are only moderately heroic.
Quite clearly the major contributor to the drag of most trees is the drag
of the leaves, whether broad or needlelike. A defoliated pine tree barely
moves in a breeze that makes a normal tree sway quite dramatically.
Around here, in the piedmont of North Carolina, broad-leafed trees blow
over much more commonly in summer than in winter—even though in the
summer they enjoy the wind-sheltering effect of the leaves of their
neighbors. Whether a tree avoids uprooting by having a stiff trunk and broad
base, as in the figure, or whether it presses sideways against a deep and
122
DRAG AND SESSILE SYSTEMS
sturdy taproot, the real culprits are the leaves. I was surprised to find, a few
years ago, that no one had looked at the behavior of individual leaves or
clusters of leaves at potentially destructive (to tree or leaves) wind speeds.
So I did a few measurements (Vogel 1984, 1989) ofdrag and areas and took
some pictures (Vogel 1993) of the reconfiguration; it proved to be about
the easiest bit of science I've ever tried.
What might be regarded as the baseline against which broad leaves
should bejudged is the drag of a flag rather than the drag of, say, a rigid flat
plate. And flags turn out to have very high drag. According to Hoerner
(1965), local separation causes flutter, which causes further separation; the
result is a substantial pressure drag that wouldn't occur if the surface were
rigid, as in a weathervane. For fairly ordinary fabric, a flag as long as it is
high, and a Reynolds number of two million, he cites a figure of 10-fold for
the increase in drag attributable to its facility for flapping. I got
comparable, even slightly higher figures, using a 0.1 m square piece of polyethylene
sheet trailing behind a batten—17-fold at 10 m s~l (Re = 65,000) and 12-
fold at 20 ms-1 (Vogel 1989). (However, the£-value for the polyethylene
sheet over this speed range, —0.57, isn't so bad, which reemphasizes the
point about how that variable tells only half the story.)
Put most bluntly, leaves turned out not to be flags. Even in very turbulent
winds most leaves fluttered little, and they reconfigured into cones and
cylinders that became ever more tightly rolled as speed increased up to (in
most cases) 20 ms-1. Leaflike models cut from sheets of plastic or stainless
steel didn't do quite as well but were still much better than flags—the
response clearly involves a lot more than just outline. Shape is obviously a
critical part of the game. Those leaves with long petioles that attached to
their blades distal to the basal extremities of the blades—tuliptree, maple,
sweet gum, sycamore—rolled into cones. It looks as if the free basal lobes
catch the oncoming wind and start the rolling process (Figure 6.9a). Pin-
nately compound leaves—black locust and black walnut—rolled into
cylinders, with the leaflets interdigitating somewhat like the scales on a fish
(Figure 6.9b). Clusters of leaves reconfigured as well, forming bundles that
became tighter as the wind speed increased. Concomitantly, the drag
coefficients of these reconfiguring leaves were very much closer to those of
rigid plates than to those of flags. In general, cylindrically reconfiguring
pinnate leaves had lower drag coefficients than did conically reconfiguring
simple leaves, and clusters had lower drag coefficients than did single
leaves. A complex and versatile hierarchy of reconfiguration must begin
with the individual leaves I looked at and end with the whole trees tested by
Fraser(1962).
Almost all the £-values for leaves, clusters, and trees were negative,
clearly reflecting the reconfigurational process. The aberrant item, single
leaves of white oak, may tell us something as interesting. The latter not only
123
CHAPTER 6
Figure 6.9. The reconfiguration of leaves in high winds: (a) tuliptree or
yellow poplar (Liriodendron tulipifera); (b) black locust (Robinia
pseudoacacia).
had relatively high drag, but the drag increased with approximately the
cube rather than the square of speed—at high speeds it actually exceeded
that of the piece of plastic sheet relative to surface area. Individual oak
leaves, even though they look sturdy, commonly suffered damage at speeds
(15 to 20 m s_1) tolerated by the other leaves. But for one thing, oak
clusters weren't quite so bad; for another, oaks may derive a compensatory
benefit. In modest winds (either outdoors or on a branch in a wind tunnel)
oak leaves maintain their normal, skyward orientations, while others, such
as maples, have begun to turn and flutter.4 In general, some instability at
low speeds seems concomitant with good reconfigurational capability when
winds increase. The extreme example may be the quaking aspen (Populus
tremuloides). I found that the locally available congeneric, white poplar (P.
alba), which with flattened petioles is a proper quaker, does very well at
high speeds, with low drag in clusters and a tolerance even by isolated
leaves of winds above 30 m s-1 Despite a lot of speculation and
measurement on what advantage might derive from quaking, I suspect it's of no
particular functional consequence at the speeds at which it occurs. Rather,
it may be just the extreme case of low-speed instability associated with good
high-speed performance.
Inter tidal Macroalgae Undulating in the Waves
Most of us regard algae as tiny organisms, perhaps pelagic unicells or
small aquatic encrustations. But especially in rich, cold waters, individual
4 So—shift from oak to maple and turn over a new leaf.
124
DRAG AND SESSILE SYSTEMS
plants may grow tens of meters long on rocky coasts in water that's far from
placid. This adds up to lots of drag and to major mechanical problems. Not
surprisingly, they've received a fair bit of attention. In one way at least, the
problem facing most macroalgae is simpler than that of trees—the loads
imposed by drag are almost entirely tensile and avoid all the complications
of more complex bendings and twistings.
For macroalgae, flexibility can be advantageous in a way that escaped
notice until fairly recently (Koehl 1984; Koehletal. 1991); for some
particularly long and flexible algae, the problem of drag may be very much less
acute than one might imagine. If the length of the alga exceeds the product
of average water speed and half the wave period, then the alga may simply
never be fully extended in either direction. Beyond a certain length,
further length contributes nothing additional to the force pulling on the
holdfast—more on this in Chapter 16. Many macroalgae are sufficiently
long for the phenomenon to be significant—one might say that they grow
to great lengths to avoid drag. Leaves and trees can't play the game since
the structures are too small and the speeds and periods far too great.
Can we say anything about macroalgal structure analogous to our
comments on leaf shape? One common feature of large laminate algae is a
ruffled or "undulate" margin (see, for example, Bold and Wynne 1978 and
Figure 6.10). This is the character that results in the folds one sees in the
edges of algal fronds pressed flat in museum collections. Ruffling is minor
in terrestrial leaves, but it occurs in some submerged leaves of stream
vegetation (Sculthorpe 1967). The resulting play on words tempts one to
regard an undulate margin as an adaptation to wave action, but enough
cases have now been investigated so the hypothesis no longer survives close
scrutiny. For instance, Koehl and Alberte (1988) found that the bull kelp,
Nereocystis luetkeana (Figure 6.10), had flat, straplike fronds in areas of
high exposure, with the wider, undulate-margined forms limited to areas
with lower currents. They showed that at a speed of 0.5 m s~l plants from
exposed sites had only a fourth the drag per unit surface area of plants
from more protected sites: the flat, narrow blades stacked tightly in high
flows. On the other hand, they suffered more self-shading, with the
expected effect on photosynthesis. Interestingly, plants from exposed areas
had both lower drag relative to their areas and more negative £-values:
— 1.07 versus —0.80 at speeds up to2ms_1.
Armstrong (1989) got similar results working on a rather cabbagelike
alga, Hedophyllum sessile (Figure 6.10), quite unlike the elongate Nereocystis.
Specimens from sheltered sites had broader, bumpier blades and
experienced higher drag at any specific speed than those from exposed sites;
while both clearly reconfigured, the latter achieved more compact shapes
at high currents speeds. Van Tussenbroek (1989) found essentially the
same thing in Macrocystis pynfera, with thicker and narrower blades from
specimens in exposed sites. Probably the most dramatic intraspecific differ-
125
CHAPTER 6
Figure 6.10. Marine macroalgae that reconfigure in flow. From left to
right: Nereocystis, Postelsia, Laminaria, Hedophyllum.
ences are those measured by de Paula and de Oliveira (1982) in Sargassum
cynosum. In sheltered places plants grew five times as high and had over five
times the biomass, yet had holdfasts with nearly 40% less bottom area;
plants in exposed areas invested 47% of their dry weight in holdfast while
plants from sheltered areas invested about 11%. And a holdfast isjust what
the name implies—a structure whose sole function is to counteract drag.
A reasonable view, one I think was originally suggested by Gerard and
Mann (1979), is that the undulate margin is a device to keep individual
lamellae from getting too tightly appressed to each other, with consequent
deleterious effects on light interception and (perhaps occasionally) uptake
of dissolved substances. But in places when local water movements get
really severe, the arrangement is simply intolerable in that the lamellae
have to clump tightly during periods of high flow for purely mechanical
126
DRAG AND SESSILE SYSTEMS
reasons of drag and strength. What must be emphasized, since algae are
not commonly regarded as especially sophisticated organisms, is the
complexity of the intraspecific tuning of both overall morphology and the
reconfigurational game from habitat to habitat. The magnitudes of the
intraspecific differences seem to be substantially greater even than those
found (as we'll discuss just below) in terrestrial vascular plants.
Growth Alterations in Response to Drag
We live in bodies that adjust their structure in response to patterns of
use, including mechanical loading. We build muscle through exercise, our
bones rearrange their structure after injury or changed loading regime,
and people deprived of gravity for much time suffer all sorts of debilitating
changes in their mechanical equipment. The phenomenon of
readjustment proves to be more widespread and dramatic than one might imagine;
it's as good an illustration as I know that organisms are not just some
mindless (or feedbackless) constructs of genetic blueprints. We've already
seen that macroalgal structure has a wide range of adaptively appropriate
intraspecific variation. In these organisms transplant experiments suggest
that selection, initial development, and regrowth rather than
rearrangement of preexisting structures underlie most of the adjustment.
Terrestrial vascular plants appear to make more use of rearrangement—trunks
and branches are usually long-lived elements even if leaves often are not.
The distinction, though, isn't tidy since much of a large plant is made of
essentially dead material external to the active protoplasts. Thus whether
differential growth through incremental addition to one place rather than
another should be considered rearrangement is in part a definitional
matter. Old observations (Jacobs 1954 is usually cited) show that a bit of
swaying leads to a tree with a thicker trunk, that guyed trees grow more slender,
that exposed, previously sheltered trees very often get blown down after
their neighbors have been cut.
Extreme examples of peculiarities in plant growth in windy habitats
abound, as do nonmechanical explanations involving such factors as
temperature and water availability. Trees at high altitudes where winds are
fairly unidirectional are "flagged"; that is, the branches protrude mainly
downwind. Whatever the chain of causation, these trees do have lower drag
when presented with winds in the direction to which they've been
previously exposed (Telewski and Jaffe 1986). In the tropics, a peculiar kind of
montane rain forest often occurs on exposed ridges, one characterized by a
dense growth of short, gnarled trees with their crowns packed into a low
canopy. Lawton (1982) measured tree heights and winds in and above such
an elfin forest and did the same farther down on the mountain slopes;
either within (despite its density) or above the trees, speeds are much
127
CHAPTER 6
higher in the exposed forest. The suggestion is that energy has to be used
to acquire strength as opposed to height to resist the extra wind. These and
other cases (such as Palumbi's [ 1986] observations on sponges) make what I
think is an important point. To repeat, drag reduction is not an end in
itself. If periods of substantial wind are short and intermittent, then even
very extreme reconfiguration through flexibility may be a fine thing—
cheaper construction justifies giving up photosynthesis or suspension
feeding for short periods, and one should go with the flow. If, on the other
hand, rapid flows are chronic, then the cost of going hypofunctional is
high, and a steadfast stand may be preferable.
The mechanisms by which plants respond to wind have come in for a lot
of attention in recent years; beyond matters of hormones and pathways the
main surprise is just how little perturbation it takes to make a substantial
morphogenetic difference. A little intermittent stroking or bending is all
that's needed. "Thigmomorphogenesis" is the name given to the
phenomenon, with "thigmo-" indicating "touch" (Jaf fe 1980); the primary response
is retardation of lengthwise growth and augmentation of radial growth.
Niklas (1992) gives a good account of the mechanical implications of the
changes in structure. It happens in the great majority of the plants in which
it has been looked for, from herbaceous annual beans to small trees. Thus
Rees and Grace (1980a, b) not only found the expected differences
between pines grown in a low wind growth chamber and ones grown with an
imposed high wind, but they gotjust the same differences when one set of
plants was shaken for 24 minutes per day. And Telewski and Jaf fe (1986)
found the same parallel between intermittent shaking and wind when
looking at Fraser firs in both laboratory and field—increases in Young's
modulus and flexural stiffness of trunks, shorter stems and needles,
reinforcement of branch bases around the stems, and the usual decrease in
height and increase in girth. Six-year-old seedlings treated to a short daily
shaking for a year showed precisely the structural changes of specimens
that had grown at higher altitudes.
Exposure versus Maximum Flow
For both the marine algae and the terrestrial vascular plants, the
prevailing or average flow in their immediate habitat is obviously a matter of
substantial relevance. What's often needed is some really cheap-and-dirty
device that can be used in large numbers to achieve a nice view of spatial
heterogeneity. The high drag of flags was mentioned earlier; that disability
has been put to use in what have been called "tatter flags" to get an index of
exposure to winds following calibration in a wind tunnel (Grace 1977;
Miller et al. 1987).
An equivalent exposure-measuring scheme is available for flows of
128
DRAG AND SESSILE SYSTEMS
water—it looks at the rate at which a standard piece of plaster of Paris
is eroded by water movement (Muus 1968; Doty 1971). Alternatively,
standard hemispheres of carpenter's chalk can be used; I had a flow tank
in which all the joints turned blue from a long series of calibration runs
someone did on such pieces of chalk. Peppermint candy ("Life Savers")
was used to good advantage by Koehl and Alberte (1988). "Exposure"
is a little hard to define precisely—is it looking at an average of speed
or an average of squares of speed, for instance—but for many inter-
habitat comparisons such precise definition is less important than simple
standardization.
What may often be more important for absolute survival (as opposed to
long-term efficient function) are the maximum flows an organism
encounters. In terrestrial situations few macroscopic organisms live where winds
are maximal, for reasons of soil and water that probably transcend
personal mechanics. On rocky coasts, though, even very exposed headlands
support a decent flora and fauna—algae, limpets, and barnacles where
even the snails have fled. Sojust what the maximum flows and flow-induced
forces might be in such places has been a matter of considerable curiosity.
And the techniques by which maxima have been estimated are of
considerable relevance to investigators looking at specific habitats.
The simplest device I know for recording the maximum current is a kind
of drogue or sea anchor (Figure 6.1 la) used by Jones and Demetropoulos
(1968). The device uses a disk normal to the flow trailing behind a spring
scale equipped with an arm that remains in whatever extreme position it
reaches; the whole device is free to swing downstream from its attachment
post. It's inexpensive and can be produced without special equipment from
ordinary components in numbers sufficient to survey a decent number of
sites over periods of months. It does suffer from a relatively long response
time, perhaps a few tenths of a second; it behaves similarly to long, flexible
algae. With a device of their design I strongly recommend determining the
drag coefficient empirically, using a fast flow tank or towing alongside a
boat on a line attached to another spring scale.
The fastest flows reported by Jones and Demetropoulos (1968) were
about 14 m s-1; I think that choice of a more realistic drag coefficient
would raise them to about 16 m s_1. You should be impressed. 16 m s-1
may be only 36 mph, but this is water we're talking about, not air. In water
at this speed a broadside flat plate of about the area of an outstretched
hand with fingers together would experience a drag of 1700 N, nearly the
weight of a 400-pound object. Put another way, if compressibility is
ignored, the same drag would require an airspeed of 450 m s-1 or about
1000 mph.
Quite a different kind of device was contrived by Denny (1983), who has
given this problem of maximum flow-induced forces more attention than
129
CHAPTER 6
(b)
standard shape
rubber
band
smoked plate
Figure 6.11. Meters to record maximum flows, (a) The tethered device
of Jones and Demetropoulos (1968) in which a spring scale is modified by
adding a lever that remains at whatever extreme position it's pushed, (b)
The subsurface device of Denny (1983), consisting of a smoked plate and
a scriber that scratches the plate as flow pushes spring-mounted exposed
object and slider from their resting position.
anyone else. His design (Figure 6.1 lb) uses a smoked plate and a scriber in
a housing beneath an object such as a limpet shell or a standard drag-
suffering shape. It has a much shorter response time, and it records
direction as well as magnitude of forces, but it's a bit more complicated to build
and calibrate. With an admittedly more complex arrangement, he
telemetered continuous records from instruments attached to intertidal rocks
in a very exposed location during serious storms (Denny 1985).
Accelerations of water were high and had to be taken into account—we talked about
them some pages back. On the other hand, the "impact pressures" caused
by large volumes of water suddenly changing speed and direction as a wave
breaks against a rock prove to be no big deal. Impact pressure doesn't
amount to much unless the flow is oriented just normal to the rock and
unless the water entrains none of that compressible stuff, air. Neither of
these conditions is commonly satisfied. All of the work, along with the
relevant background material, is described in his book (Denny 1988); the
ecological relevance of the mechanics of both organisms and water is
summarized in a short article (Denny 1987b).
But direct measurement of maximum flows may now be unnecessary,
which, I suppose, is the ideal outcome of a combination of data, theory, and
analysis. Denny (1985, 1987b, 1988) explains how linear wave theory
permits prediction of velocities from wave heights. And Denny and Gaines
130
DRAG AND SESSILE SYSTEMS
(1990) have developed rules for predicting the probabilities of different
maximal speeds and forces over different time periods—they use a
mathematical approach called "statistics of extremes." From a fairly small
number of measurements one can make surprisingly robust predictions about
maxima.
13 1
CHAPTER 7
Shape and Drag: Motile Animals
For streamlining to be effective, the organism must continuously
face into an oncoming current—which means either weathervaning if
it's attached or locomoting if it's free-living. In addition, the organism must
have a fairly definite, noncompliant shape. As a result, streamlining is
largely the province of animals that propel themselves in continuous fluid
media—the swimmers and fliers of the world. Humans have put a lot of
effort into designing streamlined shapes; if Cayley's trout is no fluke,
nature has done the same. For a sessile organism, resisting drag means
resisting only force; deflections can be restored elastically, and little work need
be done once the system is constructed. For motile creatures, the price of
excessive drag is more immediate and serious; to move, animals must
expend energy at a rate that's the product of drag and velocity. If the drag
coefficient were halved, then surely power output and most likely power
input could be halved. Alternatively, an animal might be able to swim or fly
about 40% faster.
Streamlining and the Comparison of Drag Coefficients
For a start, we must bear in mind that drag coefficients and Reynolds
numbers, while useful, are imperfect bases for comparing different
streamlined shapes. The main difficulty comes from the different
conventions used to define the reference area in the drag coefficient together with
the fact that just what shape gives the lowest coefficient depends on the
choice of reference area. To repeat a bit of Chapter 5, four different areas
are in use: (1) frontal or flowwise projecting area, mainly with bluff bodies;
(2) plan form, profile, or maximum projecting area, for wings; (3) wetted
or total area, for streamlined bodies; and (4) volume to the two-thirds
power, for airships. When considering the quality of a streamlined body
remember that a change in shape can affect the drag coefficient with no
change in drag or the drag with no change in drag coefficient. If drag
coefficient is based on frontal area or V2/'\ then stubbier shapes will look
better; if it is based on plan or wetted area, then more elongate shapes will
give lower drag coefficients.
The diversity of conventions in the literature makes it useful to be able to
convert one sort of drag coefficient into another. Fable 7.1 gives some
conversion factors; in particular, these may help in using Figures 5.3 and
132
SHAPE AND DRAG
Table 7.1 Factors for converting drag coefficients from one
REFERENCE AREA 'I O ANOTHER.
Frontal area
Plan form area
Wetted area
Volume273
Sphere
1
1
0.250
1.208
Cylinder
1
1
0.318
0.932
2:1
1
0.500
0.146
0.762
Prolate Spheroids
3:1
1
0.333
0.102
0.581
4:1
1
0.250
0.078
0.480
Notes: To convert a drag coefficient, multiply its value by the ratio of the factor for the
area in which you want it and the factor for the area in which you already have it.
Flow is perpendicular to the long axis of the infinitely long cylinder; flow is parallel to the
long axes of the prolate spheroids, which are described by the relative lengths of their
axes.
5.4 for comparisons with data for biological objects. For a sphere or
cylinder moving normal to its long axis, frontal and plan form areas are the
same. Streamlined objects are a heterogeneous collection, but the factors
for prolate spheroids ought to prove to be adequate approximations for
rough comparisons. For example, a good low-drag body of revolution (an
elongate, radially symmetrical body) at a Reynolds number in the low
hundred-thousands will have a drag coefficient, based on frontal area, of
about 0.04 (Mises 1945). Using the figures of Mises and of Hoerner (1965),
such a body may be approximated by a spheroid whose length is three
times its thickness (a 3:1 spheroid), so the drag coefficient based on wetted
area will be about 0.004, that based on plan area about 0.013, and on V2/3
about 0.02. From the figures given by Pennycuick (1968) and Tucker and
Parrott (1970) the 4:1 spheroid should provide a reasonable facsimile of a
pigeon and perhaps of other birds. I'm not sure, though, that any of these
figures are trustworthy for bilaterally flattened fish.
Beyond the differences in reference areas and the underlying issue of
choosing biologically relevant references, other matters cloud the skies and
muddy the water. First is the familiar one of lack of physical data. Between
Reynolds numbers of 104 and 107, we have good information on the design
of streamlined shapes. This ought to be a good base for work on birds,
medium and large fish, and so forth. Between 102 and 104 we have almost
no precise information about the shapes of objects designed for minimal
drag. One might guess that just below the transition Reynolds numbers of
around 100,000 for a sphere or cylinder, the fairly forward position of
separation would mandate a very slender shape in order to prevent that
separation. After all, prevention or drastic delay of separation in order to
minimize pressure drag is what streamlining is all about. Additionally, one
133
CHAPTER 7
might guess that as Reynolds numbers decrease further and separation
moves rearward around spheres and cylinders (recall the data of Seeley et
al. 1975 for spheres), the best shapes will be more rotund. In addition, the
increasing role of skin friction at low Reynolds numbers will favor shapes
that expose less skin, again favoring the fatter. But, again, what's best
depends on the objective. Nature is probably more concerned with moving
a volume of organism than with frontal or surface area, and this will also
favor less elongate shapes.
At this point it might be useful to go through a few simple comparisons
of streamlined and nonstreamlined shapes. After all, when considering
claims of drag reduction for one scheme or another, one must bear in mind
just how monumental the effects of shape change can be—in particular of
streamlining. Consider, first, a long cylinder extending across a flow, with
the drag coefficients at different Reynolds numbers from Figure 5.3
converted to wetted area with a factor from Table 7.1. Then squeeze it flat, so
it's now a flat plate with its width parallel to and its length extending across a
flow—the gold standard for streamlining. The drag coefficients at the
same speeds (the conventionally defined Reynolds numbers are larger by
it/2 as a result of the squeeze) can be looked up in Table 5.2. At Re = 100
(for the original cylinder), drag has been reduced by a factor of 8; at Re =
1000, by 19; at Re = 10,000, by 65; at Re = 100,000, by 215; and at Re =
1,000,000 by 180. (The last is a little lower since the cylinder has passed the
turbulent transition.)
Our flattened cylinder, of course, is not practical for many applications:
what of real struts, as might support a wing or some organism elevated
above a substratum? Hoerner (1965) cites data for struts about 6% as thick
as they are wide (width is usually called "chord length") that have drags
within 20% to 30% of the calculated values for flat plates. For struts whose
thickness is 12% of chord length, the drag is about 50% greater than for
ideal flat plates. These are still tiny increases compared to the orders of
magnitude decreases we just saw for streamlining itself. Pushing practical
comparisons farther, though, depends on how the structure we're
considering is used. For an elongate body loaded solely in tension, such as a cable,
all that matters is cross-sectional area, and a fairly low ratio of thickness to
chord will be optimal. For one carrying a lengthwise compressive load,
such as an ordinary strut or perhaps a leg, buckling has to be avoided; and
that pushes the optimum toward a greater thickness-to-chord ratio—a
point made earlier. The specific optima, not surprisingly, depend on
Reynolds numbers.
Analogous comparisons can be made between solid bodies of revolution
—axisymmetrical or radially symmetrical bodies—and spheres. Using the
data in Hoerner (1965), we see that such a body with a length-to-diameter
ratio of 8.0 at a Reynolds number of 100,000 has a drag coefficient (re-
134
SHAPE AND DRAG
ferred to wetted area) of 0.006. For a sphere of the same diameter the
Reynolds number will be 12,500, and the drag coefficient (Figure 5.4) is
about 0.12. So streamlining has reduced the coefficient 20-fold. If we begin
with a circular disk instead of a sphere, then the reduction is fully 90-fold.
As mentioned in the last chapter, this ratio of length to maximum diameter
or thickness is commonly referred to as the "fineness ratio."
These figures, though, are not a sufficiently general guide to the quality
of streamlining. How low is the lowest achievable drag coefficient? Recall
what streamlining accomplishes—it prevents separation aside and behind
an object. Again, the ideal nonseparating flow is that across a flat plate
parallel to the flow, so the drag coefficients for such flat plates provide a
standard of comparison. Fortunately, convenient and fairly trustworthy
formulas are available, and here theory agrees decently with practice.
Where flow across the plate is laminar (the plate being smooth, the flow
itself nonturbulent, and the Reynolds number not too high),
Cdw= 1.33 tor05 (7.1)
using wetted area or both surfaces of the plate for reference. For turbulent
flow across the plate,
Cdw = 0.072 Re-"2. (7.2)
These formulas are given by Goldstein (1938), Hoerner (1965), and many
standard textbooks; Hertel (1966), a widely cited but generally
untrustworthy source, misquotes equation (7.2). The transition to turbulent flow
occurs somewhere between 5 x 105 and 1 x 107, depending on the
circumstances. Equations (7.1) and (7.2) generate Figure 7.1, which illustrates the
advantage of postponing the turbulent transition to the highest possible
Reynolds number. This is very nearly the same thing as postponing
transition to a location as far back as possible from the forward stagnation point
or leading edge of a streamlined object, since the Reynolds number, with its
length, is proportional to that distance.1 The equations and graph also
point out that drag coefficients typically drop with increasing Reynolds
numbers for objects that experience mainly skin friction—something that
came up in the last chapter when we talked about the baseline for
evaluating reconfiguration.
These equations and their graph have been frequently used as a
reference against which the dragginess of organisms is measured, and it's quite a
good reference provided its basis is kept in mind. What's being done is to
replace an animal with a rectangular flat plate of the same surface area (the
Cdu, part) and the same length (the Re part) for purposes of comparison.
1 We're tickling the edge of something called "local Reynolds number," which will be
given proper attention in the next chapter.
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CHAPTER 7
103 i()4 105 106 107
Re
Figure 7.1. Drag coefficient (based on wetted area) versus Reynolds
number for long, flat plates with long axes extending across the flow and
chords oriented parallel to flow. The gentler line presumes turbulent flow
across the plate (equation 7.2) and the steeper one laminar flow (equation
7.1). In practice an object in a flow is on the steeper curve at low Reynolds
numbers but shifts to the gentler one at some transition value—perhaps
500,000.
For instance, the equivalent of a spherical animal would be a rectangle,
parallel to flow, with a width (cross-flow) of ird/2 and a length (streamwise)
of d, where d is the sphere's diameter. (At Reynolds numbers above about
100,000, there's a correction for end effects on the plate; the equations
were derived for drag per unit length of a plate stretching infinitely far
across the flow, essentially a very long wing or strut. But the adjustment is
much less than for a plate broadside to flow and isn't of much consequence
on a log-log plot; for more, see Elder 1960 or Webb 1975.) The other
matter worth emphasizing about a plot using equations (7.1) and (7.2) is
that even the turbulent line for a plate represents a very harsh standard of
comparison against which tojudge any solid body—we're letting a porpoise
compete in a swimming meet or a kangaroo in the long jump.
Drag and Thrust
A motile animal must do more than resist or minimize drag. In
particular, it must produce thrust; and under ordinary circumstances drag and
thrust must be equal. "Ordinary circumstances" means steady speed, hori-
136
SHAPE AND DRAG
zontal motion, an otherwise still medium, and only the horizontal
component of thrust if it's generating lift as well. That equality of thrust and drag
is both opportunity and obstacle for the investigator. On the good side, it
means that you can measure either—if you know one, you know the other,
which is why figures for drag take on some significance in assessing power
output in locomotion. On the other side are all the problems caused by
imperfect separation between thrust-generating and drag-sustaining
structures.
Sometimes the two kinds of structures are decently demarcated, as with
the screw and the hull of a ship. The drag of the hull is nearly (but not
quite!) the same whether the ship is being propelled by its screw or being
towed by some agency entirely out of the water. The situation is worse but
not totally calamitous in such biological systems as flying insects and birds,
and in swimming turtles, swans, penguins, and water beetles. In these, the
non-thrust-producing structures are subjected, not just to a simple and
smooth oncoming flow, but to a complex and temporally varying flow from
wings or legs as well. For fish (except for those that keep fairly rigid bodies),
for invertebrate larvae with ciliated regions, for most microcrustacea—for
all of these the whole body participates in propulsion, and the situation is
catastrophic. For them no experimental separation of thrust and drag
using models, dead specimens, or other devices can be regarded as a useful
dissection of reality.
A number of published studies are irretrievably corrupted by ignoring
the problem in organisms where it just can't be swept under the rug—I
have a little list—but I won't embarrass anyone with details. Perhaps the
way to view a locomoting system is to ask whether the flow pattern around
the body when the animal is self-propelled will be reasonably well
represented by the flow pattern when the fuselage is isolated and its drag
measured. Wu (1977) gives two illustrations of flow around a ciliated
microorganism, Paramecium, one with the cilia propelling the creature through
water and the other of the creature falling under gravity at the same speed.
Beyond the facts that in both cases flow is laminar and goes from anterior to
posterior, I see no similarities! As one might expect, the problem is worst at
low Reynolds numbers; so we'll return to it in Chapter 15.
Put in the usual jargon, what we're talking about is "parasite drag." The
term is used in situations where thrust-producing appendages are
involved, and it normally refers to the combined drag of all non-thrust-
producing structures, in short, to fuselage or body drag. If a lift-based
system is producing the thrust, its airfoils suffer in addition from the
inevitable "profile drag" of any structure in a flow and an "induced drag"
that accompanies production of lift—matters intrinsic to the airfoils.
Parasite drag represents the external drag that the thrust producer has to
offset. Unfortunately, this usage isn't universal, so one has to check the
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CHAPTER 7
definitions used by each author. Thus Pennycuick (1989) uses the present
convention, while Tucker and Parrott (1970) include profile drag as a
component of parasite drag, reserving the term "body drag" for what we're
calling parasite drag.
Which brings up yet one more item in this dragged-out accounting:
"interference drag." Consider measuring the drag of a body in a flow tank
or wind tunnel. The most direct way to do it involves attaching to one side
or the rear of the body a mounting strut that leads to a force transducer of
some kind. Inevitably some portion of the mount, called (without hy-
menopterous allusion) by the engineers a "sting," is exposed to the flow.
Keeping the unshielded portion of the sting as short as practical,
streamlining its profile, and subtracting its drag from that of the body is often
sufficient to discount its influence. But the better streamlined the body, the
worse the problem. For one thing, the drag of the sting gets relatively
higher. For another, the quality of the streamlining is very much a matter
of the details of flow around the body, and the presence of a sting alters the
flow. Thus the presence of a sting can increase the drag of a streamlined
body by more than the drag of the isolated sting. That's the opposite of
what one might expect from their mutual shielding, which suggests an
underestimate rather than an overestimate of the body's drag when the
sting's drag is subtracted. The extra component due to the bad interaction
is what's called "interference drag." The item isn't trivial—Tucker (1990a,
1990b) gave the matter a careful analysis and found that the apparent drag
of the body of a falcon could be overestimated by more than 20% if no
correction for interference drag were applied. While using a wake-traverse
system for measuring drag (Chapter 4), I once minimized the problem by
considering only the half of the wake opposite the mount. In fact, the
phenomenon of interference is a pervasive one that applies even where two
bodies don't actually touch; Hoerner (1965) devotes a whole chapter to it.
On a more positive note, one can sometimes get drag figures for
swimming animals in a very simple way. A fish or squid doesn't swim
continuously; gliding is quite a normal event. If one makes a videotape of an
animal that has just stopped active propulsion, one gets a record of its
deceleration. As always, force is mass times acceleration—here the force,
drag, is rearward, and the acceleration is negative, but Newton's second law
still holds sway. Just one extra factor is relevant, and it's often a minor
correction. The "virtual mass" of the animal via the "acceleration reaction"
now works in the opposite direction from drag, tending to keep the animal
going. To deal with it you need to know the volume of the animal (but you
need the mass anyway) and the added mass factor, which depends on its
shape and can be obtained from Chapter 16. (If you don't correct for
added mass, you'll get slight underestimates for drag.) So drag can be
determined for an unrestrained animal following its own devices rather
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SHAPE AND DRAG
than being towed by yours. While you still don't really know what the drag is
during active swimming, the resulting data should be closer. The scheme
has been used for quite a variety of animals, mostly large ones such as
penguins and marine mammals. For small creatures such as zooplankton,
drag is all too immediate, and deceleration is so dramatic that really
highspeed cinematography is necessary. And the latter involves special cameras
and lots of costly, nonreusable film rather than virtually free videotape. As
Lehman (1977) points out, a 30 millisecond deceleration doesn't partition
well at 24 frames per second. But it does work, and Strickler (1977) gives
practical advice.
However drag is determined, for results to be applicable to more than
just a gliding phase, the drag of the thrust-producing organs must not
interfere with the measurement. A flapping wing or paddling leg, like any
other structure in a flow, certainly experiences drag. But its drag (together
with its lift) appears as thrust—literally antidrag—with respect to the
body; I'll withhold the details of this sleight-of-forelimb until Chapter 12.
For analysis of a glide, the handiest situation occurs when the animal is
accommodating enough to fold or collapse its thrusters. A squid jet is no
problem, a penguin's flipper only a little more so. The problem bedevils
attempts to measure fuselage drag; the simplest solution, tolerable
sometimes, is to remove thrust-producing structures prior to measurement of
drag.
Looking for Streamlined Struts
Earlier we compared cylinders to streamlined struts, as in Figure 7.2a;
the latter in fact prove relatively uncommon among organisms. But they do
occur, and they're a good place to start a survey of streamlining.
The so-called torrential fauna, mostly insects, are inhabitants of exposed
rocks in rapid streams; they appeared briefly as sufferers from
inopportune lift in Chapter 4 and will play a larger role in Chapter 9. While most
are flat and cling closely to rocks, at least one sits a bit off its rock and is well-
rounded and streamlined—the mayfly nymph, Baetis. Long ago (in quite a
fine paper) Dodds and Hisaw (1924) compared a series of species of Baetis
with one another and with relatives from less violent waters. Individuals of
species preferring rapid flows are generally smaller and have larger legs;
they extend downstream from these legs, swinging to and fro if the
direction of flow varies—limited-excursion weathervanes if you wish. From the
present point of view what's interesting is the cross-sectional shape of those
legs. The femurs and tibias are flattened, thicker at the anterior margins,
and thinner posteriorly—in a word, streamlined. While these insects are
only about a centimeter long, they're exposed to rapid flows. For the
habitat of B. bicaudatus, Dodds and Hisaw measured speeds up to 3 m s_1; at
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CHAPTER 7
(a) streamlined strut (b) shark hammerhead
(c) fishing bat toe <d) skimmer bill
Figure 7.2. Low-drag struts: (a) a streamlined section; (b) approximate
cross section of the "hammer" that extends outward from each side of
the head of a hammerhead shark, Sphyrna; (c) the lower bill of a
skimmer, Rynchops nigra; (d) a hind toe of a fishing bat, Noctilio. The last two
extend through the air-water interface and prefer to obey the injunctions
of Chapter 17 instead of those of the present one.
that speed (and the worst case is probably the relevant one) the Reynolds
number for a leg 0.8 mm wide is around 2000, certainly high enough for
streamlining to be effective.
Crab legs may sometimes be arranged as streamlined struts. Quite a few
crabs either walk around or stand erect in water currents, and they must
have only limited purchase on the bottom due to their substantial
buoyancy. And flattened legs are quite common in crabs. The problem is that
many of them swim at least occasionally, with legs serving as paddles and
rudders (see Hartnoll 1971 and Plotnick 1985, for instance). The
distinction between paddles, rudders, and streamlined struts is not one I can
confidently make. A^ good animal to look at might be a shore crab, Pachy-
grapsus crassipes, for which data on underwater walking is given by Hui
(1992). Another case worth a careful look is the hammer of hammerhead
sharks (genus Sphyrna). Figure 7.2b is something of a guess, based on my
recollection of having fondled a preserved specimen and on published top
and front views.
Exposure to spectacularly fast flows of water happens in a pair of
strongly convergent cases involving bats and birds. A few species of each fly
through the air while dipping gaffing hooks into the water. The bats are
fishing bats, genera Noctilio and Pizonyx (of two different families—
another convergence); and their gaffs are hind feet (Figure 7.2c). While
these are high-drag hind feet by bat standards, they're also bigger and
stronger than the common cave-clinging clamp. The drag coefficients of
14 0
SHAPE AND DRAG
the hind legs of both fishing and nonfishing bats have been measured by
Fish et al. (1991). What's initially curious is that even though those of the
fishing bats are lower, about 2-fold at the highest speeds they tried, the
coefficients are still high, around 1.0 referred to frontal area. Equally
curious are the cross sections, which look like backward airfoils with sharp
fronts and blunt backs. We noted already that the drag of reversed airfoils
isn't all that bad, but is such complete reversal an inevitable consequence of
the morphological revolution that redirected the toenails forward? Fish et
al. point out that the hydrodynamic situation is complicated by the fact that
the digits extend through the air-water interface and so suffer from some
of the special problems of surface ships. The sharp leading edge has
positive advantages in reducing drag due to waves and spray, as we'll talk about
more specifically in Chapter 17.
The birds are skimmers (genus Rynchops), whose gaff is the lower and
longer part of the bill (Figure 7.2d). These are major mandibles—a captive
bird grew one 17 cm long. In use, they stick down into the water at an angle
of roughly 60° below horizontal, at speeds of around 6 to 10ms-1. Zusi
(1962) commented on their lateral compression and knifelike streamlined
profiles, clumsy for grasping and holding but great gaffs. Withers and
Timko (1977) measured a fineness ratio of about 6.0 for the lower
mandible—compared with 1.7 for the upper—and give profiles showing
conventional streamlining for the lower part of the bill but sharp leading
edges for the part that encounters the surface. Both sources note that these
bills are ridged in a curious way that might bear investigation. For both
skimmers and fishing bats drag may not represent a big cost in energy—
dipping isn't done continuously and the drag of the gaffs is insignificant
next to body drag—but it generates an awkward turning moment. In
particular, a skimmer with a high drag bill would have to do something special
to avoid upending, which in fact it nearly does whenever it catches a fish.
A Diversity of Streamlined Bodies
Two-dimensional streamlined forms—struts—may be rare in nature,
but three-dimensional streamlined bodies are the hallmark of virtually
every competent swimmer or flyer. The sampling in Figure 7.3 is intended
to emphasize the diversity, with representatives of two classes of each of the
three great phyla of mollusks, arthropods, and chordates. I'll do a flying
survey of drag data, both to illustrate what nature's been up to and to point
a finger at gaps and problems of both measurement and interpretation.
Table 7.2 is a summary of progress, while Figure 7.4 displays data,
converted with any necessary guesswork to wetted area and compared to the
standards we established earlier.
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(a) (b)
(c) (d)
Figure 7.3. Streamlined organisms in a flow that goes from left to
right: (a) a crayfish going rearward in a rapid escape; (b) a large aquatic
beetle; (c) a pelagic fish such as a tuna; and (d) a baleen whale.
Smallish Arthropods
If muscle is muscle is muscle, then animals of all sizes ought to be able to
jump to the same maximum height or range—Galileo first made the
point—but only if air resistance is negligible. Resistance is a nuisance for
baseball or golf ball, reducing the maximum range by 19% and 36%,
respectively (Vogel 1988a). For a jumping locust, the reduction is only about
16%—they may be smaller, but they move more slowly. For a generic flea I
calculated a range reduction of about 83%, so drag is far beyond being
some minor correction and has replaced gravity as the main factor
determining range. Bennet-Clark and Alder (1979) shot fleas, a big one (3.7 x
1.7 mm) and a small one (2.05 x 0.7 mm), from a spring gun, calculating
drag coefficients from the heights achieved. They obtained coefficients,
based on frontal areas, of just about 1.0 at Reynolds numbers between 65
and 205. That's nothing to be proud of, about that of a sphere. Referred to
wetted area and assuming 2.5:1 spheroids, that's a coefficient of 0.12, about
half that of a sphere. Either wetted area or volume273 are probably the
more appropriate reference; the latter also yields a coefficient about half
that of a sphere. So maybe fleas aren't quite as bad as they at first glance
appear. Contriving an elaborately specialized jumping apparatus and then
neglecting aerodynamics altogether certainly would seem odd. If fleas
tumble in local air currents, though, the potential advantages of
streamlining might be hard to realize.
The parasite drag of very small insects is of particular interest. As we'll
see in Chapter 11, airfoil performance deteriorates as Reynolds numbers
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SHAPE AND DRAG
Table 7.2 Drag coefficients for animals that move through fluid media
Fleas
Ctenophthalamus
Hystricopsylla
Fruit fly, Drosophila virilis
Locust, Schistocerca gregaria
Marine isopods
Idotea wosnesenskii
Idotea resecata
Dytiscid beetles
Acilius sulcatus
Dytiscus marginalis
Tadpole, Rana catesbiana
Frogs
Hymenochirus boettgeri
Rana pipiens
Crabs
Callinectes sapidus
Cancer productus
Ducks, various underwater
Cephalopod, Nautilus
Falcon, Falco peregrinus
Fish
Trout, Salmo gairdneri
Mackerel, Scomber
11
Saithe, Pollachius virens
Penguin, Pygoscelis papua
Marine mammals
Sea lion, Zalophus californ.
Seal, Phoca vitulina
Human, swimming
Re
65-205
II
300
8000
2700
5500
8600
15,000
1000-2500
1500-8000
17K-40K
10,000
10,000
420,000
100,000
380,000
50K-200K
100,000
175,000
500,000
1,000,000
2,000,000
1,600,000
1,600,000
cd
0.96
1.02
1.16
1.47
0.08
0.055
0.28
0.33
0.36-0.74
0.11-0.24
0.05-0.06
0.3
0.35
0.028
0.48
0.24
0.015
0.0043
0.0052
0.005
0.0044
0.0041
0.004
0.035
Area
f
f
f
f
w
w
f
f
f
w
w
p
p
w
V
f
w
w
w
w
w
w
w
w
Source
Bennet-Clark 8c Alder 1979
Vogel 1966
Weis-Fogh 1956
Alexander & Chen 1990
»
Nachtigall 1977a
»
Dudley et al. 1991
Gal 8c Blake 1987
II
Blake 1985
»
Lovvorn et al. 1991
Chamberlain 1976
Tucker 1990a
Webb 1975
n
a
Hess & Videler 1984
Nachtigall 8c Bilo 1980
Feldcamp 1987
Williams 8c Kooyman 1985
«
Note: The reference areas vary among the sources: f = frontal; w = wetted; p = plan form; v =
volume273.
drop, and small insects are of the order of magnitude of estimates of the
lower limit for the practical use of lift. And, as with jumping fleas, drag
becomes more important relative to gravity—in part because of the
increasing area-to-volume ratio concomitant with small size. Thus parasite
drag at ordinary flying speed is 18% of weight in a fruit fly, Drosophila virilis,
but only 4% in the desert locust, Schistocerca gregaria. As part of my thesis
project, I measured the parasite drag of this particular fruit fly—the coef-
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CHAPTER 7
I I I L
103 104 105 106 107
Re
Figure 7.4. Drag coefficients, based on wetted area, for motile animals
that might be designed for low drag compared with the long flat plates of
7.1 and equations (7.1) and (7.2). 1, desert locust; 2, beetle, Acyhus; 3,
beetle, Dyttscus; 4, bullfrog tadpole; 5, frog, Hymenochirus; 6, frog, Rana
pipiens; 7, crab, Cancer productus; 8, underwater duck; 9, trout; 10,
mackerel; 11, pigeon; 12, vulture; 13, emperor penguin; 14, gentoo penguin;
15, sea lion; 16, harbor seal.
ficientis 1.16 based on frontal area or (assuming a 2:1 spheroid) about 0.17
based on wetted area at a Reynolds number of 300. For comparison I
measured the drag of several spheres (bearing balls) under the same
conditions. A sphere has a slightly lower drag coefficient based on frontal area,
1.08, but it has a coefficient 60% greater relative to wetted area, 0.27. A flat
plate is much better—the calculated coefficient is 0.077, about half that of
the fly. So the fruit fly is at least so-so as far as its drag, roughly midway
between sphere and plate, and much like the fleas. (Incidentally, reversing
a fruit fly in the airstream increases its drag about 10%—that's a crude test
of design since the reversed fly has the same frontal and wetted areas but
has no reason that I can imagine for low drag.)
The desert locust operates at a Reynolds number of about 8000 and has a
Ctif of 1.47 (Weis-Fogh 1956) and a Cdw of about 0.12 (as a 4:1 spheroid)—
the former is higher than the value for a sphere, 0.47, while the latter is the
same. Locusts are really quite bad; one might wonder about data, but a
photograph by Weis-Fogh of a locust body with smoke markers shows quite
substantial separation with a large and messy wake. Of course the flow with
wings attached and beating might be quite different from that
surrounding an isolated body.
The isopod genus, Idotea, was mentioned in connection with the choice
144
SHAPE AND DRAG
of reference area for drag coefficient. Alexander and Chen (1990) give the
drag coefficients for two species, a slower, smaller, and fatter one and a
faster, larger, and more elongate one. Both have about three times the drag
of the equivalent flat plate; the smaller has two-thirds and the larger about
half the drag of a sphere. For something as poorly streamlined as these
isopods, the sphere might provide the better comparison, so we can at least
restate the obvious, that elongate is better if wetted area is the reference. A
little higher in Reynolds numbers than the low thousands of these isopods
are the dytiscid water beetles investigated by Nachtigall and Bilo (1965,
1975; see also Nachtigall 1977a). Aahus, operating at Re = 8600, has a
typical Cd< of 0.28; Dytiscus, operating at Re = 15,000, has an average Cd< of
0.33. Assuming 2:1 spheroids, these are CdJs of about 0.041 and 0.048;
around 2.6 times lower than that of a sphere; the comparisons of Crfl,'s are
about the same. Relative to flat plates, they're about 3.6 times worse—
again, the animals are about in the middle of the range between spheres
and plates. Note, though, that at these higher Reynolds numbers the gap
between sphere and flat plate is widening, with increasing wiggle room for
nature to play drag reducer. Dytiscid beetles may be as short as 2 mm (by
contrast with the 35 mm length of Dytiscus), with swimming Reynolds
numbers as low as 90. Beetles of smaller species are increasingly round, about
what we'd expect as skin friction becomes more important and pressure
drag due to separation less so.
At this point I ought to repeat my wish for better information on the
optimal design of axisymmetrical bodies—for biologically reasonable
criteria such as volume transported or surface exposed, just how good can one
do? Is drag minimization simply accorded a low priority in the design of all
these arthropods? Or are we looking at their drag in some flawed manner?
Or are they really quite near the practical minima? I just don't know of the
existence of the basic data that might provide some basis for judgment.
Several Undistinguished Swimmers
Again, let's remind ourselves to keep a little distance from drag, to
eschew the prejudice that keeping it down is what life's all about. Perhaps
that's best done by looking at some creatures that aren't really noteworthy
as swimmers, ones that only intermittently bother doing it, ones that carry
other disabling constraints, and ones for whom achieving great speed is
probably of little importance.
Consider frogs, the paradigmatic organisms of generations of biology
students. The larvae come equipped with tails, but otherwise hatch with the
foreshortened form, the low fineness ratio, of adults rather than the
elongate shapes of proper swimming amphibia—their slow swimming
probably isn't too much of a disability since they're not pursuit predators. And
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CHAPTER 7
any fish large enough to eat a tadpole will likely be faster on that account
alone—maximum speed goes up with size in a fairly regular way in any
relatively homogeneous group of swimmers. The adults swim occasionally,
but they also walk and jump; they aren't pursuit predators either. Recently
Dudley et al. (1991) looked at the drag coefficients of bullfrog (Rana cates-
biana) tadpoles; these are relatively large as tadpoles go, around 10 cm in
length, including tails. Cdr values ranged from 0.36 to 0.74 at Reynolds
numbers (based on body length, excluding tail) between 1000 and 2500.
That's equivalent to Cdw (presuming a 1.5:1 spheroid) from roughly 0.1 to
0.2, about four times that of a flat plate and about the same as a sphere—
not very good at all. And things get somewhat worse when forelimbs and
hind limbs erupt.
As part of a study of drag-based locomotion (about which we'll have
more to say) Gal and Blake (1987) measured the drag of the adults of
several species of frogs, removing feet and dropping them, ballasted, in a
column of water. Individuals of the smaller and more aquatic species,
Hymenochirus boettgen (1.5 to 2.5 cm snout-to-vent2), have CdJs of 0.24 to
0.11 at Reynolds numbers between 1500 and 8000. Those of the larger
species, Rana pipiens (7.5 cm), have CdJs of 0.06 to 0.05 at Re's between
17,000 and 40,000. Given the differences in Reynolds numbers, the drag
coefficients aren't notably different. Again the figures are much the same
as the drag coefficients of spheres; they're about seven times higher than
those of flat plates.
Many crabs are swimmers; like frogs many propel themselves with
appendages that do duty as walkers as well, although the particular mix of
mechanisms and modes are quite diverse (for surveys, see Lochhead 1976
and Hessler 1985). As noted earlier, most crabs are negatively buoyant—
sinkers—not inappropriately for animals that do any bottom-walking.
When swimming, lift is needed. Both lift and drag have been measured by
Blake (1985) for several species. At Reynolds numbers above about 10,000,
Callinectes sapidus has a drag coefficient referred to plan form area of 0.3
and a lift coefficient3 of 1.2—a decent lift-to-drag ratio of 4.0. Cancer
productus has a higher Cdp—0.35 and a lower C/—0.65, giving it a lift-to-
drag ratio of only around 2.0. The CdJs are around 0.1; at Re = 10,000
that's about eight times higher than a flat plate but half that of a sphere.
These animals swim sideways; being bilaterally symmetrical they have
equal coefficients for either direction. Going foreward or backward, drag
is approximately doubled.
2 Called the anus by people who lack appreciation of these finer points of amphibian
anatomy.
3 Lift coefficient is defined the same way as drag coefficient Plan form area is almost
always the reference; whenever lift coefficients are used, the drag coefficients should also
use that reference. Details in Chapter 11.
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SHAPE AND DRAG
Ducks fly, swim on the surface, and swim under water; again one suspects
functional compromises. From the regression equations obtained from
underwater towing tests by Lovvorn et al. (1991), I get a Crfl/f of 0.028 at Re =
420,000; that represents a speed of a meter per second. The value is very
close to that of a sphere (after the great drop in drag); it's five times higher
than that of a flat plate over which flow is turbulent and thirteen times
higher than a flat plate in laminar flow. In drag, these ducks are turkeys.
A final group of what must be judged poor swimmers—the shelled
nautiloids and ammonoids—are now represented only by the genus
Nautilus. Chamberlain (1976) towed a Nautilus shell and got a drag coefficient,
based on volume2/:\ of 0.48. At Re = 100,000, that's a bit better than a
sphere before the big drag drop (0.57), but not much. It's thirteen times
worse than a flat plate with turbulent flow and twenty-two times worse than
one with laminar flow. One wonders whether these animals are trapped by
the inflexible geometry of a conical or spiral shell and about what effect the
soft parts, extending downstream, might have on drag. Perhaps moving
fast just wasn't necessary to make the paleo scene.
Fish
Webb (1975) collected a large amount of data from his own and other
measurements along with a useful discussion of the difficulties in getting
reasonable measurements of drag. As even more of a dose of cold water, he
describes the theoretical difficulties of using data for body drag to estimate
the thrust a fish has to produce. In a typical fish, more than perhaps in any
other sort of macroscopic organism, thrust-producing and drag-incurring
structures are inseparable. But fish do glide, so what a nonoscillating body
does isn't entirely artificial. And quite a few fish do maintain fairly rigid
bodies as they swim—some that use pectoral fins for propulsion and a
considerable number of large and fast pelagic fish that move only their
wide tails. For a 30 cm-long trout (Salmo gairdneri), freshly killed and with
paired fins amputated, Webb got a drag coefficient of around 0.015 based
on wetted area at Reynolds numbers between 50,000 and 200,000; that's
about eight times less than that of a sphere before the big drag drop. It's
still, though, around twice the drag of a flat plate even with turbulent flow.
Again one wonders why nature does no better or whether Cayley just got
lucky with his trout.
Webb (1975) quotes a figure obtained by Quentin Bone of 0.0043 for a
small mackerel (Scomber scombrus) at Re = 100,000. That's essentially
perfection, the drag of a flat plate with laminar flow; and mackerel do swim
rapidly with nearly rigid bodies. But whether such relatively low drag
coefficients can be maintained at higher Reynolds numbers isn't clear.
Thus the same mackerel had a Cdw of 0.0052 at Re = 175,000, not the ideal
147
CHAPTER 7
0.0032. Nonetheless they're quite good, and these scombroid fishes
(including the tunas) look more like laminar-flow airfoils (with maximum
thickness fairly far downstream) than almost any others. Hess and Videler
(1984) estimate a Cdu, for a saithe {Pollachius vixens, a gadid fish) of 0.005
from a computation of thrust. Like a mackerel, a saithe is a fast, predatory
fish that swims with a fairly rigid body, and their result is consistent with the
data from towed mackerel. Perhaps it's no coincidence that the lowest
coefficients result from testing the kinds of fishes in which drag data from
towing or gliding is most relevant to performance in nature. Or perhaps
these stiff-bodied fishes are simply better than more flexible, troutlike
forms.
The Strange Case of Bird Bodies
Whether hummingbird or eagle, in flight posture a bird certainly look
slick and smooth. Some years ago, Pennycuick (1968, 1971) measured the
drag on wingless, frozen birds in a wind tunnel; he obtained drag
coefficients, based on frontal area, of 0.43, for a pigeon, Columba hvia, and a
vulture, Gyps ruppelli. These are an order of magnitude apart in frontal
area (36 and 300 cm2); and they were tested over a wide range of speeds,
corresponding to Reynolds numbers of 200,000 to 800,000. Now those are
pretty draggy bodies—CdJs of about 0.06 or roughly twelve times the drag
of flat plates (at Re = 500,000) with turbulent flow—very much in the
sphere's sphere. The very constancy of the drag coefficient over a large
range of Reynolds number suggests the pressure drag of bluff bodies
rather than the skin friction of streamlined shapes.
The question, then, is whether parasite drag, measured in this way,
actually reflects the situation in normal flight. A search for some fly in the
ointment isn'tjust a desperate attempt to keep our adaptationist faith—not
only do birds look streamlined, but a flock of good candidates might
provide experimental culprits. Interference drag has already been
mentioned, and it's good for at least 20% (Tucker 1990a,b). All the caveats about
measuring drag on thrust-producing machines must apply. And
turbulence in the wind tunnels used for testing may also make a difference.
Beyond these is that special avian curse, feathers. Thus Pennycuick et al.
(1988) found that a generous application of hair spray to a bird body in a
wind tunnel reduced the drag by about 15% below Crf/s between 0.26 to
0.38 (at Re's around 300,000) obtained on mallard, snow goose, bald eagle,
and tundra swan. Tucker (1990a) has now obtained a coefficient of 0.24 for
the body of a falcon, Falco peregrinns, paying careful attention to both
interference drag and feather position. But he comments elsewhere that
he was "unable to preen the feathers to lie in the smooth, orderly,
overlapping position they assumed on a living Harris' hawk in flight" (Tucker and
148
SHAPE AND DRAG
Heine 1990). That feathers are major culprits is suggested by his figure for
drag coefficient for a smooth-surfaced model of the falcon, Cd< = 0.14. Not
that the issue is really settled—the flying bird still has feathers, and we
don't know what they might cost in drag. Tucker (pers. comm.) is at this
point pursuing the question with additional models. Real birds in flight
might possibly do even better than featherless models. Dare I say that while
the matter is still up in the air one can't yet see which way the wind is
blowing?
Penguins
The cost of moving a body through a fluid is best expressed as "parasite
power," the product of parasite drag and forward speed. In flying birds,
power output may be divided into parasite power and two other
components. These are "induced power," the cost of producing lift, and "profile
power," best regarded as the remaining aerodynamic cost of moving the
wings; these latter two correspond to induced drag and profile drag,
mentioned earlier. At reasonably economical flying speeds, parasite power is
much lower than the sum of the other two components. So one can argue
that a flying bird might not wish to compromise other functions in order to
drive body drag close to some aerodynamic lower limit. What about a bird
for whom rapid locomotion in a continuous medium is important but
where the medium is 800 times denser? While ducks swim under water,
they're also paddlers, walkers, and excellent fliers. A better kind of bird if
we're interested in how far nature has pushed the matter is a penguin.
Penguins swim with their wings, using an up-and-down motion that
produces thrust by a proper lift-based or propellerlike mechanism at
maximum speeds of about 2.5 m s_1. A similar bird flying six times as fast
(assuming drag is proportional to both density and the square of speed)
would encounter less than a twentieth as much parasite drag.
Penguin hydrodynamics has been the subject of quite a few
investigations. In the first of these, Clark and Bemis (1979) derived drag from
decelerative gliding. For emperor penguins (Aptenodytes forsteri) about a
meter in body length, they got average CdJs of 0.0027 at Reynolds numbers
around a million and a half. That's about two-thirds of the turbulent-flow
drag of a flat plate, although several times higher than the laminar-flow
drag—pretty spectacular for such a high Reynolds number. But they
didn't correct for virtual mass, so the figure ought to be adjusted slightly
upward. And Nachtigall and Bilo (1980), also looking at deceleration, got a
figure of 0.0044 for the Cdu, of a gentoo penguin (Pygoscelis papua) at a
Reynolds number of a million. In towing tests of wingless carcasses, Hui
(1988a) got somewhat higher values, around 0.016 for Humboldt
penguins (Spheniscus humboldti)\ rigid resin models were about 20% lower. My
149
CHAPTER 7
guess at this point is that the gliding data are more realistic, and that these
penguins do about as well as flat plates in turbulent flow, which is to say very
well indeed. Incidentally, Baudinette and Gill (1985) found that the cost of
transport (energy expenditure per unit mass and distance) was lower for
penguins than for any other aquatic endotherms.
Marine Mammals
Like penguins (and ichthyosaurs, plesiosaurs, and others), these are
animals of secondarily aquatic habit, reinvaders of the sea—air breathers that
nevertheless do most of their swimming underwater. They're also
creatures of romance, myth, and (despite the extreme divergence of habitats)
anthropomorphization. So graceful and maneuverable is their swimming
that it seems almost effortless. In fact they're highly muscular and have
high aerobic capacities (= metabolic scope)—the effortless ease is that of a
great ballerina. Over fifty years of looking for "the dolphin's secret" (as one
speculative paper put it) has led, as we'll talk about shortly, to the sober
suspicion that there really isn't any particular secret—just a lot of good
design of a kind that isn't really radical from the point of view of either
engineer or biologist (Fish and Hui 1991; Fish 1992). We'd become captive
of a kind of wishful thinking that was entirely understandable given an
initial and understandable misestimate (Gray 1936), what can only be
described as Pentagon and Kremlin paranoia (both!), the enthusiastic
optimism that characterizes effective program directors and successful grant
applications, and the ever-tempting but rarely realized promise of
generating technology by copying nature (Vogel 1992b).
First, the pinnipeds—seals and sea lions. These look quite similar, with
casual distinction based on the earlessness (of external pinnae, anyway) of
the former. For us the more interesting distinction is that the true seals
propel themselves mainly with their tails, like cetaceans, while the sea lions
produce thrust solely with their pectoral flippers. We now have a little data
on both. Feldcamp (1987) measured the drag of a California sea lion
(Zalophus californianus), obtaining from gliding deceleration Cdw = 0.0041
at Re = 2,000,000—virtually perfection by the standard of turbulent flow
over a flat plate. Williams and Kooyman (1985) measured the drag of
harbor seals (Phoca vitulina) of several sizes using two techniques, towing
and gliding. Interestingly, even though towing involved live, compliant
animals biting a soft mouthpiece that they could freely relinquish, the data
for gliding yielded substantially lower drag coefficients—consistent with
what we saw with penguins. For a gliding adult seal they obtained Cdu, =
0.004 at Re = 1,600,000—just the same as the result with the sea lion,
which, among other things, gives one real confidence in such data. (Wil-
150
SHAPE AND DRAG
Hams and Kooyman also towed a human, who had a drag coefficient of
0.035, about 3.5 times that of a towed seal and almost nine times that of a
gliding seal. We are consummately clumsy and ineffective swimmers.)
And then the cetaceans—dolphins and whales. To start with, while
dolphins (at least) can achieve high speeds, they can do so only for very brief
periods. Lang and Norris (1966) measured a top speed for a bottlenose
porpoise (Tursiops gilh) of 8.3 m s_1 for a duration of 7.5 seconds and a top
speed of 6.1 m s_1 for 50 seconds. For extended periods this strongly
motivated porpoise could manage only 3.1 m s~ *. Using the power output
per unit weight of human athletes, Lang and Norris calculated that the
porpoise had about the drag of turbulent flow over a flat plate.4 From films
of gliding, Lang (1975) measured drag coefficients on two other species of
small cetaceans (Lagenorhyncus obhquidens and Stenella attenuata); again the
coefficients were about those of a flat plate with turbulent flow. Quite a few
other estimates have been made of drag coefficients based on various
models and presumptions. In no case does decent evidence show that a
cetacean achieves a lower drag coefficient than that of a flat plate with
turbulent flow. Au and Weihs (1980) suggested that drag could be partly
evaded and energy saved by repeatedly leaping from the water—
"porpoising." But the suggestion has been criticized, and a respiratory
function has been proposed for such behavior by these obligate air
breathers (Fish and Hui 1991).
Tricks for Reducing Drag—Hope and Reality
Extreme flexibility in flow, streamlining, growth to great length in
oscillating flow, staying close to a solid surface, expulsion of water through
opercula in ram ventilation in fish—these are reasonably well documented
ways in which organisms minimize drag. Beyond them are quite a number
of schemes, popularly believed to be matters of fact or extreme probability,
but for which hard evidence is hardly in evidence. In particular, these
schemes describe ways in which organisms, largely large swimming
animals, can get body drag down below that of a flat plate with turbulent flow.
The potential benefits are large: at a Reynolds number of a million the drag
with turbulent flow is 3.4 times the drag with laminar flow. I'd suggest that
the interested reader keep an eye on extant and forthcoming volumes of
the Annual Review of Fluid Mechanics for developments. Four principal
1 Larger cetaceans can certainly go faster, based on credible reports of 10 m s~' in fin
whales {Balaenoptera physalus) cited by Bose and Lien (1989) But reports of speeds of
swimming organisms include so many egregious overestimates that extreme skepticism
is recommended when dealing with the literature. For that matter, quite unrealistic
figures for the speeds of running and flying are bandied about as if factual.
15 1
CHAPTER 7
arrangements have been espoused; they're discussed rather skeptically by
Fish and Hui (1991), a little less skeptically by Bushnell and Moore (1991),
and with the full faith of the true believer by Aleyev (1977).
Compliant Surfaces
These were, I think, the first proposed solution to what's been known as
"Gray's paradox," alluded to earlier. Specifically, Sir James Gray (1936)
suggested that only if flow were laminar could a dolphin's muscles manage
to move it at its estimated maximal swimming speed. In fact, the paradox
has essentially solved itself. Dolphins don't go quite as fast as Gray thought
(a familiar problem!), and striated muscle turns out to be rather better than
the figures available to him. But paradoxical or not, drag is something to
avoid, so the larger issue is still of interest. The basic idea behind the use of
compliant surfaces is the damping5 of incipient turbulence; it was
proposed and promoted by Kramer (1960, 1965). Cetacean surfaces are rather
different mechanically from what's typical of mammals, but no one has
ever demonstrated that they work in the manner proposed. Moreover,
compliant coatings such as he designed have not proven to yield anything
like the large drag reductions originally claimed (Riley et al. 1988).
Mucus Secretions
Adding long-chained polymers such as mucopolysaccharides or (a little
less certainly, surfactants) to a flow, especially if they're added in the velocity
gradients at surfaces, can reduce drag. And many fish are slimy, and
dolphin skin produces new cells at a high rate. Fish slimes at concentrations of
less than (but approaching) 1% reduce skin friction several-fold, and
slimes from faster fishes seem to be more effective (Hoyt 1975). Other
natural polymers are effective as well (see, for instance, Ramus et al. 1989).
But "effective" most often refers to pipes and rheometers; while not
uninteresting, these are not swimming animals. With real fish it turns out to be a
distinctly iffy business—sometimes yes (Daniel 1981, for instance) and
sometimes no (Parrish and Kroen 1988). It does seem clear that (1) the rate
of secretion of polymers adequate to get the ambient concentrations giving
good effects is higher than fish could reasonably manage, and (2) the cost
of producing such secretions would more than offset any benefit that could
be realized, at least over all but the shortest emergencies. The situation is a
bit like that of snail locomotion, in which the cost is driven upward
enormously by the continuous need to produce nonreusable slime.
3 Not "dampening"; the distinction is lost too often even to let Hy a pun—even if a cold
shower, which this section is intended to provide, both damps and dampens
1 5 2
SHAPE AND DRAG
Surface Heating
Recall that the viscosity of water depends drastically on temperature,
that skin friction depends directly on viscosity, and that viscosity matters
most where shear rates are greatest—immediately adjacent to surfaces. So
a hot surface will have less drag (other things being equal) than a cold
surface. Cetaceans commonly have surface temperatures somewhat above
that of the ambient water, and continuous cutaneous heat loss is necessary
to offset production by very active muscles. But calculations of the effect
on drag suggest insignificance—flows are too fast and temperature
differences too small to transfer enough heat to make a difference (Webb 1975,
Fish and Hui 1991).
Surface Morphology
Drag reduction has been claimed for just about every feature of the
surface of every large and rapidly swimming animal. The present chief
candidate is the ridging characteristic of the dermal scales of sharks. These
are claimed to be lined up with the local flow direction (Reif 1985).
Experiments with analogous physical systems have been successful enough to
result in production of a coating material ("riblets") that has been used on
racing yachts. The ridges have apparently evolved separately in several
lineages of fast-swimming sharks. It should be emphasized that in both
sharks and artificial coating these are tiny ridges, closely spaced—less that
100 |xm apart and still less in height—and that what's involved is a
reduction of skin friction and not postponement of separation (Bushnell and
Moore 1991). Two matters, though, get omitted from popular accounts.
First, no one seems to have any direct evidence that the ridges actually
reduce the drag of sharks or that they work on sharks by the proposed
mechanism. And second, the drag reductions achieved with the artificial
coatings are less than 10%, enough to create excitement in the hypercom-
petitive world of boat racing, enough perhaps to make a difference to
fitness in the competitive world of pelagic predation, but nothing
approaching the difference in skin friction between laminar and turbulent
flows.
Writers of popular material in science are biased toward believing what
scientists claim or even suggest. Perhaps they don't appreciate sufficiently
the difference between the enthusiasm associated with a novel and exciting
hypothesis and the more restrained satisfaction that accompanies decent
confirmation and achievement. But we can't escape by shifting blame; I
think what's needed at this point is a bio-fluid version of Koch's famous
postulates in bacterial epidemiology. A claim of drag reduction should be
viewed with skepticism until it (1) has been tied to a plausible physical
153
CHAPTER 7
mechanism, (2) has been shown to work on physical models under
biologically relevant conditions, and (3) has been shown to work by some direct
test on real organisms under controlled and reproducible conditions. Much
less desirable alternatives to the third are interspecific comparisons of
morphology and correlations of morphological differences with
differences in habit and habitat.
Drag-Based Locomotion
At first suggestion, this sounds like a contradiction in terms. But drag in
one direction is, of course, thrust in the other. Thus all one needs is some
appendage that moves back and forth in the intended direction of
progression whose drag when moved one way is less than its drag when moved the
other. An appropriate difference in drag demands nothing obscure—
recall the data for the drag of flat plates parallel and perpendicular to flow
in Table 5.2 and the high drag of some of the forms shown in Figure 6.4.
Nor is the motion particularly peculiar. Conversion of a terrestrial walk
needs only a recovery stroke close to the body or parallel to flow with a
folded or twisted appendage and perhaps some drag-increasing webbing
exposed during the power stroke. This kind of secondary swimming is very
common in mammals and birds—muskrats, minks, dogs, humans, ducks,
and so forth (Williams 1983; Fish 1992). It's common also among smallish
creatures, especially swimming arthropods, in which setae are arranged to
stick out on one stroke and lie back on the other (Figure 7.5). And it's
commonly used by organisms with a lengthwise series of appendages that
are moved in either a metachronal rhythm or synchronously.6 This meta-
meric arrangement may be a really ancient one among metazoa since it's a
scheme that can make a walker or burrower as well as a swimmer, and since
metamerism is such a widespread morphological arrangement.
Getting high drag for a power stroke is no special trick. Drag coefficients
for paddles (on frontal areas) are typically over 1.0, whether the hind legs
of frogs (Gal and Blake 1988), water boatmen (corixid hemiptera; Blake
1986), or swimming beetles (Nachtigall 1980, 1981). Getting low drag for a
recovery stroke isn't hard either, although information is scanty about just
how low the coefficients might be. For the system to work well, the drag
coefficient for a good recovery stroke ought to be quite a lot lower than that
of the power stroke. Averaged over a cycle, an appendage must move
6 A nonsynchronous, metachronal wave of paddling turns out to be a bit more efficient
than synchronous rowing. The nearly universal choice in metazoans is a forward-
running metachronism (protozoa are far more variable in their metachronal
arrangements), which minimizes both solid and fluid limb interactions (Sleigh and Barlow 1980)
But in racing shells, where several humans have to coordinate their efforts, the
synchronous scheme is still used.
154
SHAPE AND DRAG
Figure 7.5. The rowing stroke of the hind leg of the beetle, Gyrinus,
viewed from behind. The oarlets fold back to make a low-area recovery
stroke in (c) and (d).
forward at the same speed as the body to which it's attached. Thus once the
animal is under way, the speed of the appendage with respect to the fluid
medium must be greater during the recovery stroke than during the power
stroke. Blake (1981) has done a theoretical analysis of drag-based
locomotion, and studies comparing lift-based and drag-based schemes are
available—Davenport et al. (1984) on marine and freshwater turtles and
Baudinette and Gill (1985) on penguins and ducks; in each pair the first
uses a lift-based and the second a drag-based system. But we'll defer this
most interesting comparison to Chapter 12.
155
CHAPTER 8
Velocity Gradients and Boundary Layers
At the interface between a stationary solid and a moving fluid, the
-Z~\. velocity of the fluid is zero. This, of course, defines the no-slip
condition, mentioned back in Chapter 2. The immediate corollary of the no-slip
condition is that near every such surface is a gradient in the speed of flow.
Entirely within the fluid, speed changes from that of the solid to what we
call the "free stream" velocity some distance away. Shearing motion is
inescapably associated with a gradient in speed, so in these gradients near
surfaces, viscosity, fluids' antipathy to shear, works its mischief, giving rise
to skin friction and consequent power consumption. The gradient region
is associated with the term "boundary layer," which I've quite deliberately
postponed mentioning until it could be defined carefully—which is what
this chapter is mainly about. Most biologists who have heard of the
boundary layer have the fuzzy notion that it's a distinct region rather than the
distinct notion that it's a fuzzy region. The essence of the present task is
defining without ambiguity something that's not quite physically distinct.
While these interfacial velocity gradients are inevitable consequences of
the no-slip condition, they were undoubtedly recognized much earlier.
The boundary layer, by contrast, wasn't so much discovered as it was
invented, in the early part of this century, as a great stroke of genius of
Ludwig Prandtl. Recognizing the origin of this notion is crucial. In the
basic differential equations for moving fluids, the Navier-Stokes equations,
some terms result from the inertia of fluids and some from their viscosity.
These equations turn out to be difficult to solve explicitly without
simplifying assumptions, just as we simplified matters by assuming steady flow in
deriving the Bernoulli equation. As we've seen, the Reynolds number gives
an indication of the relative importance of inertia and viscosity; and it can
be used to determine just what simplifying assumptions in the basic
equations might be appropriate to a given situation. At Reynolds numbers
below unity, inertia can be ignored and nicely predictive rules nonetheless
derived—Stokes' law for the drag of a sphere (Chapter 15) is one. At high
Reynolds numbers, one might expect to get away with neglecting viscosity,
in which case the Navier-Stokes equations reduce to the Euler equations,
which are the three-dimensional analogs of Bernoulli's equation. It may
sound neat, but it all too commonly gets us in trouble—results diverge
from physical reality, drag vanishes, and d'Alembert has his paradox.
Prandtl reconciled practical and theoretical fluid mechanics at high
156
VELOCITY GRADIENTS
Reynolds numbers by recognizing that viscosity could never be totally
ignored. What changed with Reynolds number was where it had to be taken
into account; initially it mattered everywhere, but as the Reynolds number
increased well above unity, viscosity made a difference only in the gradient
regions near surfaces. These regions might be small, and they might get
ever smaller (or, more to the point, thinner) as the Reynolds number
increased; but as long as the no-slip condition held, a place had to exist where
shear rates were high and viscosity was significant. Prandtl called the place
in question the "Grenzschicht" or (the English conveys a little less
specificity) the "boundary layer." In general, a higher Reynolds number implies a
thinner boundary layer but a higher shear rate in that boundary layer.
Shear entails dissipation of momentum and energy, so another way of
viewing the boundary layer is as the region near a surface where the action
of viscosity produces an appreciable loss of total pressure head. As
mentioned in Chapter 4, Bernoulli's equation cannot be applied to the
differences in velocity within a boundary layer because the equation assumes
constant total head. Put another way, the velocity variation within a
boundary layer comes about in a way that violates the very starting point from
which Bernoulli's equation comes.
Of course the main question about boundary layers isjust how thick they
are. Here's where the real trouble surfaces. The inner limit is no problem—
the solid-fluid interface. But there's no outer limit in nature to the gradient
region. The speed of flow approaches zero almost linearly at the interface;
by contrast, it asymptotically approaches the free-stream velocity with
increasing distance from that interface, as shown in Figure 8.1a. So the
(a) (b)
8
z
• »
0 U
Figure 8.1. Defining a boundary layer well downstream on a flat plate
parallel to free stream flow, (a) The way the speeds of flow vary with
distance out from the plate, (b) Defining the boundary layer thickness with a
graph of distance from the plate versus local speed—the layer extends
z-ward to where the local speed is 99% of the free stream speed.
157
CHAPTER 8
definition of the outer limit of the boundary layer must be arbitrary since
the layer isn't a naturally discrete region bounded by an identifiable
physical discontinuity. I emphasized (and will repeat) that the choice of an outer
limit depends very much on the circumstances in which one is invoking a
boundary layer—don't lose sight of the fact that it is an invented concept.
The most commonly used outer limit is where the local velocity, Ux, has
risen to 99% of the free-stream velocity, U:
UX = 0.99U. (8.1)
Figure 8.1b puts the matter graphically. Biologists should immediately
recognize that the constant, 0.99, is no more than an artifact of our bipedal
habit and pentadactylic anatomy.
The Boundary Layer on a Flat Surface
Just how thick is this boundary layer? Consider a thin, flat plate oriented
parallel to a flow for which the Reynolds number based on the distance
from upstream to downstream (leading to trailing) edges is under about
half a million. (Yes, we're talking about the situation to which equation 7.1
applies, if some sense of deja vu occurred to you.) We'll call the distance
downstream from the leading edge of the plate x, and the thickness of the
boundary layer as defined by equation (8.1) 8 (delta). An approximate
solution to the Navier-Stokes equations gives a formula for the thickness of
the boundary layer on one side of the flat plate where flow is laminar and
where the boundary layer is defined as in equation (8.1):
8 = 5^- (8.2)
Notice that the boundary layer thickness increases in proportion to the
square root of the distance from the leading edge; its outer limit thus forms
a parabola (Figure 8.2). Again let me emphasize the lack of physical
discontinuity—nothing out there demarcates the parabola, not even a
streamline. Increases in density or free-stream velocity thin the layer, while
increases in viscosity thicken it. Authors differ somewhat on the value of
the constant in the equation—I've seen values ranging from 4.65 to 5.84, so
5.0 should be a reasonable approximation.
And notice the familiar ratio of viscosity to density—kinematic
viscosity—just as in the Reynolds number. One can, in fact, express
equation (8.2) in terms of the Reynolds number, only here one uses a mild
permutation called the "local Reynolds number," Rex. The latter is simply a
version applicable to a specific place on an object, not to the object as a
whole; the only change in figuring it is that the characteristic length is taken
as the distance (x) downstream from the leading edge to the specific place.
158
VELOCITY GRADIENTS
Figure 8.2. The thickness of a laminar boundary layer on a flat plate
and several velocity profiles within it. The z-direction is much
exaggerated; otherwise continuity would require that the flow arrows be tilted
upward. Notice how the velocity gradient near the surface gets gentler
with increasing distance downstream.
The result of this conversion, equation (8.3), shows that the thickness of the
boundary layer relative to the distance downstream (a dimensionless
thickness) depends solely on the Reynolds number at the particular point—low
Reynolds numbers mean (relatively) thick boundary layers:
- = 5Re-l/2. (8.3)
x
Thus at a local Reynolds number of 10,000, the thickness of the boundary
layer is a twentieth of the distance from the leading edge, while at a local
Reynolds number of 100,000, it's about a sixtieth of that distance.
As usual, we have to attach some cautions and conditions to the equations
(8.2 and 8.3). First, the derivation assumes that Uz, the velocity component
normal to the plate and the free stream, is zero. In effect, the equation
presumes that all flow is specified by Ux. Now, if fluid slows down upon
entering the boundary layer, by the principle of continuity streamlines will
diverge from the flat plate, and fluid not entering the layer will swing out
and around it. So the very existence of a boundary layer must create a
nonzero Uz. But if the boundary layer thickness is much less than the
distance downstream from the leading edge, if 8 << x, then Uz will be close
to zero. (Note, though, that while Uz may be zero for all practical purposes,
the rate of change of velocity normal to the plate, dUJdz, will always be
significant within a boundary layer—again, that's what boundary layers are
all about.)
These equations, (8.2) and (8.3), are usable up to Reynolds numbers
around 500,000—to where turbulence invades the boundary layer. To how
low a Reynolds number can they be applied? In my experience, they're safe
down to where the boundary layer thickness is as much as 20% of the
distance from the leading edge, in which case the local Reynolds number is
around 600. (Twenty percent may sound pretty generous, but remember
159
CHAPTER 8
that this isn't a stagnant layer, and the outer half has very nearly free-
stream velocity.) For cruder estimates of boundary layer thickness, they can
be used down to about 100 (Vogel 1962). They work for slightly curved
surfaces almost as well as for flat surfaces, but they break down if the flow
separates at all.
The most common use of these equations is in the calculation of forces;
in fact, equation (7.1) was originally derived from (8.2). The biologist,
though, should have a profound interest in specific conditions within the
boundary layer. Many organisms live partly or entirely within the
boundary layers of either inanimate objects or other organisms. For many
creatures flow matters but free-stream velocity is something they never
encounter; what's critical in determining where and how they live is the
distribution of velocities within a gradient region. As it turns out, the curve
describing the speeds of flow within a boundary layer (Figure 8.1 b) is not a
simply expressed function. Still, our requirements are usually modest; and
if we accept a systematic error of up to 5%, the data cited by Rouse (1938)
permit adequate approximations. Taking (as before) z as the
perpendicular distance from the surface of a flat plate, we'd like to know Ux as a
function of z. If z is half or less than half of the thickness of the boundary
layer as defined in equations (8.1) and (8.2), then the following linear
approximation will serve:
Ux = 0.32 zU V—. (8.4)
If z can be anywhere within the boundary layer, then a slightly more
complex parabolic distribution is necessary, with a little less precise fit
immediately adjacent to the surface:
l~oU z2U2o
Ux = 0.39 zU >/— - 0.038 LJLJL. (8.5)
What if our thin, flat plate sports warts? How rough can a surface be
without materially affecting the flow over it? Goldstein (1938) gives a pair
of formulas that presume the flow is unaffected if the Reynolds number of
a projection is 30 or less for pointed projections or 50 or less for a rounded
one. The permissible heights of roughness (e) relative to distance
downstream from leading edge are then
- < 9.5 Re~3/4 (pointed) (8.6)
- < 12.2 Re-*'4 (rounded). (8.7)
x
(The Reynolds numbers in the equations are based on distances
downstream from leading edges, not on heights of projections.) Organisms are a
160
V E L O CII Y GRADIENTS
bumpy, bristly bunch, and totally smooth surfaces are exceptional in
nature. These formulas should provide handy tests of the applicability of the
equations given earlier.
Clearly, roughness of a given size is more likely to have a disturbing effect
if it's near the leading edge. Comparison of the exponents in these last two
equations with that of equation (8.3) shows us that the higher the Reynolds
number, the deeper within the boundary layer a protuberance must be lest
it disturb the flow. As an example, consider a point a centimeter behind the
leading edge of a flat plate moving through water at 0.1 m s-1. The local
Reynolds number is 1000, and a pointed projection of as much as half a
millimeter is tolerable; this is one-third the thickness of the boundary layer.
By contrast, if the flow is a meter per second or ten times as fast—a
Reynolds number of 100,000—the protrusion can be only about a sixth of a
millimeter or a tenth of the thickness of the boundary layer to be without
effect.
All these formulas and calculations apply only to laminar boundary
layers. Just as we needed a different description of the drag of a fiat plate at
high Reynolds numbers (equation 7.2), we need another pair of equations
for situations above Reynolds numbers of about 500,000, for situations in
which the boundary layer becomes turbulent:
8 = °'376x V^Z7 (88)
- = 0.376 Re~l/*. (8.9)
x
In practice, a boundary layer may be laminar near the leading edge and
then turbulent somewhere downstream, with the particular location of
transition dependent (unsurprisingly) on the local Reynolds number, as in
Figure 8.3. Within the turbulent boundary layer, right near the surface, a
so-called laminar sublayer remains. Incidentally, it's no shear accident that
the exponent in equation (8.3) is the same as that in (7.1) and the one in
(8.9) is the same as the one in (7.2).
A turbulent boundary layer isn't quite the same kind of beast as a laminar
layer, one differing only in the formula for how it happens to thicken in the
downstream direction. Fluid in a laminar boundary layer moves
consistently downstream as individual bits, not just as a statistical milling crowd.
So there's no cross-flow transport of mass or heat as a result of the flow, and
it's thus a semistagnant region in which wastes can accumulate, nutrients
can be depleted, and mixing is restricted. A laminar boundary layer, in
short, is likely to be a substantial barrier to exchange of material or heat. A
turbulent boundary layer might provide a similar refuge against the forces
of the free stream; but the turbulent motion is automatically associated
with cross-flow transport of mass and heat. Thus it provides much less of a
161
CHAPTER 8
\3
turbulent
laminar
laminar
sublayer
Figure 8.3. A boundary layer that's laminar upstream but becomes
turbulent farther along, with a laminar sublayer in the latter region. Again
the z-distances are greatly exaggerated.
barrier to transport. The biological implications of the distinction are great
and will loom large later in this chapter and in the next.
We mustn't forget the arbitrary definition of the thickness of boundary
layers stated as equation (8.1) and implicit in equations (8.2), (8.3), (8.8),
and (8.9). Since he was interested in calculating the forces of skin friction
on a plate, Prandtl took a generous view—he didn't want to exclude any
region where viscosity mattered. For other applications, that limit of 99%
of free stream may be misleading. In particular, for an organism living in a
boundary layer and for a biologist interested in how much of the organism
protrudes from the boundary layer, a somewhat thinner definition may be
more realistic. Consider a point where the speed of flow is 90% that of the
free stream. Something located there is unlikely to be in a region that gets
depleted of whatever is supplied by the oncoming flow. Nor does
something there get much shelter from drag—the latter will typically be over
80% of what would happen out in the free stream. The matter being
ultimately arbitrary, perhaps one should choose 90% of free stream as a
functional outer limit for reduced exposure. The difference of 9% may
not sound like much, but the shape of the velocity distribution in a
boundary layer is a peculiarly asymptotic one (Figure 8.1b again). A boundary
layer bound by Ux = 0.9 U is only 70% as thick as one bound by Ux = 0.99 U.
If this smaller boundary layer is used, then the constant of 5 in equations
(8.2) and (8.3) should be replaced by 3.5.
The trouble with all of these formulas is that they apply, at least strictly, to
situations uncommon in nature. A stone in a stream certainly has a
boundary layer, but the stone is rarely flat and as rarely has a sharp and regular
leading edge. An organism sticks up into a boundary layer; by doing so it
distorts the velocity distribution in that boundary layer; and, of course, it
has a boundary layer of its own. A dense array of organisms raises the
effective level of the surface; somewhere between a single item protruding
z-ward and a solid layer, the location of the surface has to be redefined as we
encounter what's often called "skimming flow" (about which more in a few
pages). The obvious response to such complications is an empirical one—
162
VELOCITY GRADIENTS
to measure rather than to calculate. That's not as difficult as it sounds, since
flowmeters and anemometers can be quite small and since boundary layers
are thick at low Reynolds numbers.
Even where the geometry seems appropriate, caution and skepticism are
needed. Perhaps a few examples—horticultural rather than just hortatory
—will emphasize the point; for others, see Grace (1977). Most leaves are
fairly flat, and (at least at low speeds) the wind might appear to blow
smoothly over them. But ... (1) in a comparison of soybean leaves and
metal models it turned out that the structure of the boundary layer was
comparable only at the lowest wind speeds (Perrier et al. 1973); (2)
turbulence occurred at much lower wind speeds on a poplar leaf than on a flat
plate, and evaporation rates were more than twice those predicted on the
presumption of a laminar boundary layer (Grace and Wilson 1976); (3) the
turbulence of the natural wind led to a lower-than-expected resistance to
heat transfer for a leaf in an apparently smooth and steady airflow (Par-
lange and Waggoner 1972). So don't perpetuate the practice of equation-
grabbing predecessors. Don't blindly adopt the 99% definition; don't use
the formulas unless they demonstrably apply; don't be intimidated by the
prospect of measuring low flows in small places.
A reasonable compromise between full faith in some revealed truth of
physics and the complete agnosticism of empirical measurements is the
generation of semiempirical formulas. These can be derived from
measurements on models and then tested for their applicability to organisms in
nature. The following are some good (and useful) examples, formulas
given by Nobel (1974, 1975) for effective, average boundary layer
thicknesses on various parts of terrestrial plants to use for predicting water loss
and gas and heat exchange.
1. For a flat, leaflike object at Reynolds numbers from 300 to
16,000:
5 = 0.0040 \jj (8.10)
2. For a cylinder at Reynolds numbers from 1300 to 200,000:
5 = 0.0056 y^. (8.11)
3. For a sphere at Reynolds numbers from 400 to 40,000:
8 = 0.0033 V| + ^^ • (8-12)
These formulas, incidentally, are not dimensionally homogeneous; as
given here they presume SI units.
163
CHAPTER 8
Forces at and Near Surfaces
Avoiding the full wind on a beach by lying against the sand is an
experience familiar to most of us. As we've seen, the closer one gets to the surface,
the lower is the local wind, with no wind at all right at the surface. Does this
mean that the force of the wind, the drag, also approaches zero as an object
flattens itself against the substratum? The drag quite clearly diminishes,
but it doesn't actually drop to zero. Bear in mind that at the surface Ux may
be zero, but dU/dz is not. Thus some skin friction must remain.
This minimal drag of a flat object on a surface can be easily calculated
from information already at our disposal. Consider a small spot (not quite a
point!) on a flat plate parallel to flow, a spot well back from the leading edge
and lying beneath a laminar boundary layer. The velocity gradient above it
is just the derivative of equation (8.4) with respect to z:
^ = 0.32 L/^/2p1/2x-1/2fjL-1/2. (8.13)
The drag per unit area is, of course, simply the shear stress:
D = [ dUx '
S L dz -lz=o
(2.4)
We can define a "local drag coefficient," Cdl, for the spot at issue just as we
did the original drag coefficient:
Crf/=2(|)p-'t/-2. (8.14)
We then substitute (8.13) into (2.4) and the combination into (8.14) and get
(throwing in the definition of the Reynolds number as well)
Cdl = 0.64 Re~l/2. (8.15)
This is a formula for the local drag coefficient—the drag of a spot of small
but definable area on a large flat plate. (Hoerner 1965 gives the same
formula but with a coefficient of 0.664.) Incidentally, if one integrates
equation (8.15) from the leading edge of the plate rearward, one gets
equation (7.1). An analogous equation works for turbulent flow (Leyton
1975 and other sources):
Cdl = 0.058 Re~^\ (8.16)
The skin friction of part of a surface, this thing we're calling local drag,
provides a useful baseline with which the measured drag or the force
needed to dislodge an attached organism can be compared. If no explicit
164
VELOCITY GRADIENTS
formula is available for the velocity gradient, it may be measured directly
by determining the speed of flow at a few points near the surface.
Conversely, one might use the shear on a force platform to infer the velocity
gradient of the inner part of the boundary layer or even the boundary layer
thickness.
The local drag on a spot on a surface is not a large force. A quick
comparison with our previous standard for low drag, a flat plate parallel to
free-stream flow, is instructive. Consider a square centimeter of surface,
half a meter downstream from the leading edge of a flat plate, with a free-
stream water flow of a meter per second. The local Reynolds number is
500,000; by equation (8.15) the local drag coefficient is 0.0009. By
equation (8.14), that's a drag of 4.5 x 10~5 newtons. Now take that same square,
1 cm2 on each side, and mount it in the free stream. The Reynolds number
is now 10,000; by equation (7.1) Cdw is 0.0133, which corresponds to a drag
of 0.0013 newtons.1 In this case the square centimeter experiences fully
thirty times less drag when a part of a larger exposed plate than when
exposed alone—the same area, the same free-stream speed, the same
orientation. And that's by comparison with our gold standard for low drag. By
hunkering down against a surface you can't escape drag entirely, but what's
left of it isn't exactly monumental.
Naturally, only a rare organism can get sufficiently intimate with the
substratum to have a drag as low as that given by equations (8.15) and
(8.16). Most creatures protrude to some degree above any surface of
attachment, and most creatures are themselves equipped with protrusions—
eyes, pinnae, nares, and other such excrescences. Aircraft and ship
designers, worried about bolt heads, rivets, and so forth, have given some
attention to the drag contribution of minor surface protuberances. Hoe-
rner (1965) gives values of the drag coefficient between 0.74 and 1.20 for
Reynolds numbers of 20,000 to 50,000 based on the frontal area exposed
to the flow (in C^) and the diameter of the protuberance (in Re); some of
these are given in Figure 8.4. The highest value, not surprisingly, is for a
plate protruding at right angles to both surface and the local flow. In
general, the higher the protuberance, the worse are both the drag and the
drag coefficient. Protuberances with rounded tops are better than flat-
topped ones. If the flow direction is known and the organism arranged to
face it, then improvement of almost an order of magnitude is possible—
streamlining works here also. As we'll see in the next chapter, the trick
seems to be used by some aquatic insects. In fact, a half-streamlined body is
still far from ideal because it suffers quite a lot of interference drag around
1 For that matter, both are pretty low forces. 4.5 x 10-5 newtons is the weight of a 4.5
milligram mass—several fruit flies. 1.3 x 1()_H newtons is a weight of about an eighth of a
gram or a two-hundredth of an ounce.
165
CHAPTER 8
side view front view Cdf
rectangular solid 0.74
erect cylinder 0.76
hemisphere 0.32
streamlined bump 0.07
Figure 8.4. Drag coefficients of protuberances on flat plates, based on
frontal area. The values have only relative significance since, in practice,
they depend on the heights of the protuberances relative to the local
boundary layer thickness, here assumed very thin.
its base, especially in front. It helps to lengthen and flatten the leading edge
of the protuberance; by such a maneuver the drag coefficient can be halved
again, to a minimum of 0.03 at these Reynolds numbers.
These drag coefficients give some idea of the effects of shape on drag for
objects within a boundary layer. But they shouldn't be applied directly to
biological situations without checking the local Reynolds number as well as
the Reynolds number of the protrusion itself. For the coefficients cited, the
local Reynolds number is about two orders of magnitude larger than the
latter, implying a turbulent boundary layer. Still, aquatic insects may
pupate on large rocks, and epizoans and epiphytes afflict large organisms
in rapid flows. In any case, the same general rules most likely apply at more
modest Reynolds numbers: (1) rounded tops are better than flat, sharp-
edged tops; (2) half-streamlined bodies are better yet, and to avoid
separation maximum thickness should be near the upstream rather than the
downstream extremity; (3) smooth fairingof theedgeof the protuberance
into the surface of attachment improves matters; (4) with such fairing, the
distinction in shape between upstream and downstream ends of a
protuberance should be slight—certainly less than that distinction for well-
streamlined bodies in a free stream. Thus good low drag shapes for uni-
fe"
f_\
166
VELOCITY GRADIENTS
directional, bidirectional, and even omnidirectional flows may converge.
Still, altogether too little hard information is available.
Unbounded Boundary Layers
What happens if our flat plate lacks a leading edge? All the best
authorities, antedating even Aristotle, have recognized that the surface of the
earth is quite without such an edge. Some sort of gradient region is
obviously always present—the no-slip condition isn't easily violated; and one
can, as mentioned already, escape most of the wind by lying down even on
an unobstructed beach. In fact, only at altitudes of 500 meters or greater is
the wind unaffected by friction with the surface and the true geostrophic
wind appears (Sutton 1953). Water movement on the bottom of a stream,
river, or ocean presents much the same situation and has to be treated in
very much the same manner. What seems to be quite a diverse lot of cases
reduces to flows across dense arrays of bluff bodies, the latter boiling off
turbulent wakes downstream and upward (Rose 1966). Thus
superimposed on an overall horizontal, turbulent flow, whatever its speed
spectrum, are eddies of various sizes that transport momentum vertically.
These eddies prove to be sensitive to the roughness of the surface even
when well above it. As a result, the way average speed varies with height
depends quite strongly on the character of the surface beneath. All of
which sounds dauntingly complex and resistant to simple summarization
in predictively useful equations.
But the situation isn't quite as turbulent conceptually as it is physically.
Both theory and measurements indicate that the variation of horizontal
flow speed with height above the substratum is a logarithmic function, at
least if buoyant effects from temperature variations are negligible. And
this logarithmic relationship persists up to the winds of destructive gales
(Oliver and Mayhead 1974). The following general formula is usually
cited:
U =UjLln (i^iH (8.17)
k V z() /
It will stand a bit of explanation. We're interested, most often, in how the
horizontal wind, Ux, varies with height, z, above the ground. Those are
variables with which we're already familiar, k (kappa) is the dimensionless
"Von Karman's constant" with an empirically determined value of 0.40. d is
called the "zero-plane displacement"; it accounts for the fact that the
logarithmic profile extrapolates to zero velocity somewhere above the ground,
especially on thickly vegetated surfaces. The so-called roughness
parameter or roughness length, z(), adjusts the steepness of the logarithmic velocity
167
CHAPTER 8
gradient because it's related to the size of the eddies generated at the
surface. Rougher surfaces make bigger eddies, which transfer more
momentum in the z-direction and thus reduce the steepness of the velocity
gradient. U# is called the "shear velocity" or the "friction velocity"; it
indicates the amount of turbulence, and its value is independent of height for a
given surface and free-stream flow.
What's a little odd about equation (8.17) is that the meanings of several
variables diverge from their dimensions and names. Thus U* isn't really a
velocity in any normal sense—it's the square root of the shear stress (recall
equation 2.4) divided by the density of the fluid:
U*= y-. (8.18)
And z() is only a nominal length, just as V2/3 was used as a nominal area. The
equation assumes that the shear stress is constant with height, which turns
out to be reasonable for the first 10 to 20 meters above a surface (in air—
less in water). This constancy is maintained by a constant downward flux of
momentum through turbulent transport of eddies. Figure 8.5 gives a
typical profile, plotted linearly in (a) and logarithmically in (b). It assumes that
the zero plane displacement is negligible (d = 0), which is essentially true
for surfaces without a lot of high protrusions—low grass, rocks with
barnacles, mud flats with occasional worm tubes. If d matters, the most common
procedure (Rosenberg et al. 1983) is to fiddle with its value until the profile
comes out properly logarithmic, which is to say until it plots linearly on a
graph such as that of Figure 8.5b. An alternative is to use the rules of
thumb given by Monteith and Unsworth (1990), applicable to fairly thick
layers of vegetation. They suggest that z() will be between 8% and 12% of
the vegetation height and that d will be 60% to 70% of that same height.
Another alternative is to take values from previous studies; Table 8.1 gives
some representative values.
But we still don't have a convenient way to determine the velocity profile
from the properties of the fluid, some length, and some free-stream speed
as we had for the cases with nice leading edges. (Nor, for that matter, are we
any longer using that limit of 99% of asymptotic free stream—taken
strictly, there's no outer limit with equation 8.17!) U* presents the most
problems; according to Sutton (1953) (concerned with terrestrial systems),
it varies from about 3% to 12% of the mean wind speed a meter or two
above the surface or zero plane, which is pretty unconstant. On the other
hand, the data cited by Ippen (1966) for water flow near the smooth bottom
of a channel 10 meters in depth suggest that 3% is a good figure for a
decently wide range of average velocities. For rougher bottoms about 5% to
15% of ordinary velocity is reasonable, according to Denny and Shibata
(1989). It's just not easy to measure shear stress under appropriate circum-
168
VELOCITY GRADIENTS
50-
40-
op
S
op
S
e
Ux, m/s
Ux, m/s
Figure 8.5. Two views of the distribution of velocities within a
logarithmic boundary layer: (a) a linear plot; (b) a logarithmic plot. Here d,
the zero plane displacement, is assumed zero; zQ, the roughness length is
0.005 m; and U*, the shear velocity, is 0.33 m s~'.
stances. Either one can measure the velocity itself, U, at several heights (z's)
and do a logarithmic regression to determine U# from the slope and z()
from the intercept, fiddling, as mentioned, with d if the thing doesn't plot
out as a straight line (Jumars and Nowell 1984). Or one can use the rules
above for estimating z() and d or use values from previous studies and a
single measurement of velocity at a known height. If one isn't wedded to
Table 8.1 Values of the roughness length, z(), the zero plane
displacement , d. and the shear velocity, u* , for various
substrata.
Surface
Very smooth
Lawn grass, to 1 cm h.
Short grass, 1—3 cm h.
Sparse grass, to 10 cm h.
Thick grass, to 10 cm h.
Thin grass, to 30 cm h.
Thick grass, to 50 cm h.
Long grass, 60—70 cm h.
Tall crop, 1 m height
Forest, deciduous
Forest, coniferous
Over the sea
Desert sand
Z()
0.00001
0.001
0.005
0.007
0.023
0.05
0.09
0.03
0.2
1.0-6.0
1.0-6.0
0.0003
0.0003
d
0.007
0.0
<0.07
<0.66
0.30
0.95
<20.0
<30.0
0.04
0.0
u*@ u
0.16 @ 5
0.26 @ 5
0.33 @ 5
0.36 @ 5
0.45 @ 5
0.55 @ 5
0.63 @ 5
0.50 @ 5
0.46 @ 3
Sources: Sutton 1953; Sellers 1965; Oke 1978
Note: U is taken at 2 meters above the surface or zero plane. In SI units.
169
CHAPTER 8
commercial equipment one can measure U*, at least in unidirectional
aquatic systems or the shear stress, t, which is just as good. Statzner and
Miiller (1989) give the details of a technique for using the threshold of
movement of hemispheres of a graded series of densities across standard
flat plates as a measure of shear stress.
Why bother with all this semiempirical stuff? For one thing, a graph of
how speed, even average speed, varies with height can be a useful thing in
an investigation of the forces on some organism that sticks up from a
substratum (see, for instance, Denny 1988). The present discussion shows
how to get that profile from something less than a full set of measurements
of average speed versus height. And such profiles are relevant to
phenomena that include fluxes of dissolved materials and settling of planktonic
organisms (Nowell and Jumars 1984). For another, U* itself is an
interesting quantity—it is, after all, really a kind of shear stress. Together with the
roughness length, it has immediate relevance to similar erosive and deposi-
tional processes, whether of snow or of terrestrial or submerged sand
(Ippen 1966; Monteith and Unsworth 1990) or of edible particles on a bay
bottom (Jumars and Nowell 1984).
For that matter, one can put this peculiar shear velocity, U*9 to good use
in yet another version of the Reynolds number, this one called the
"boundary roughness Reynolds number," Rer (see, for instance, Jumars and
Nowell 1984). It can give a picture of the character of flow immediately
adjacent to the surface (Figure 8.6). Shear velocity simply substitutes for
free-stream velocity, and / is taken as the height of the roughness elements
—commonly the diameter of particles on the surface. If Rer is less than
about 3.5, then the surface is hydrodynamically smooth with a viscous2
sublayer of the ordinary sort of thickness 11.6 |x/pc/*. So any exchange of
dissolved material has to depend on the phenomenon of molecular
diffusion and the value of the diffusion coefficient of each kind of molecule for
its final stage. (The effective diffusive sublayer is about 20% of the thickness
of the viscous sublayer.) If Rer is greater than about 100, then the surface is
hydrodynamically rough, and no real viscous sublayer exists. For exchange
of dissolved material, what then matters is the eddy diffusion coefficient,
much larger than molecular diffusion coefficients and independent of the
chemical species involved. (For sucrose in water, the molecular diffusion
coefficient is 5 x 10~6 cm2 s_1 compared with the eddy diffusion
coefficient of about 100, a 20-million-fold difference.) If the boundary
roughness Reynolds number is between 3.5 and 100, there's a transitional mess.
The literature on these logarithmic boundary layers is scattered among
subjects such as micrometeorology, coastal engineering, limnology, and
2 The "viscous" sublayer mentioned in connection with logarithmic boundary layers
corresponds to the "laminar" sublayer of the rest of the fluid mechanical literature.
170
VELOCITY GRADIENTS
0.1
0.01-
1
o
1 0.001
W5
00001-
plU*/n = 3.5
transitional
regime
hydraulically rough
plU*/n = 100
hydraulically smooth
0 0001
0.001 0.01
roughness height, m
Figure 8.6. The character of flow right near a solid surface beneath
flowing water, as organized by a plot of shear velocity against the
roughness height of the bed.
oceanography. Good sources of general information, besides those
referred to specifically, are Lowry (1967), McCave (1976), Arya (1988), Gar-
ratt (1992), and Wieringa (1993); a simple and intuitive introduction is
given in a book produced by the Open University (1989) entitled Waves,
Tides, and Shallow-Water Processes.
Flow Right on Bumpy Bottoms
The top of the canopy of a forest, the top leaves of a uniform field of
some crop, the top of a "forest" of tall sea anemones on a subtidal rock
face—each of these forms an effective substratum below the logarithmic
boundary layer about which we've been talking. Within these stands,
matters are complex. Speeds of flow are lower than those above the stand and,
as a rough rule, are inversely related to the local density of solid material
such as leaves, appendages, tentacles, and so forth, although naturally
speeds drop off near the ground (Grace 1977). In open forests with little
understory or in a forest of sea anemones, winds beneath the canopy may
exceed those within (but not above) the canopy.
In considering surfaces that are either uniform, of low roughness with
rare upright elements, or have nearly flow-impervious canopies, we're
talking about extremes. Whether the bumps are living or not, one very often
encounters intermediate situations, cases where protrusions abound but
don't abut. As a rule of thumb, the protrusions have only a minor effect on
the speeds of flow near the surface until they occupy about a twelfth of the
171
CHAPTER 8
(a) Independent flow
y • * * '
>'.*...■.. I ..■•■. '.V.l ,■.'i ..I. .1 .•/. . •.-.■.'
(b) Interactive flow
(c) Skimming flow
Figure 8.7. Flow right near a bumpy bottom for increasingly closely
spaced bumps. At least three separate regimes can be recognized, loosely
(a) independent flow, (b) interactive flow, and (c) skimming flow.
overall (plan or top view) area of the surface. When more than a twelfth of
the surface is covered, then eddies don't penetrate well, and one gets what's
been called "skimming flow" (Jumars and Nowell 1984). In a sense, two
phases of flow of substantially different character then coexist and interact,
much as happens in and above a forest.
As will become evident in the next chapter, what happens right down
amid the bumps is of enormous biological importance. A finer-grained
categorization of phenomena than merely skimming or nonskimming
turns out to be a practical necessity; useful approaches are those of Nowell
and Jumars (1984), Davis and Barmuta (1989), and Carling (1992).
Consider the three rough surfaces of Figure 8.7. If the individual bumps are
well spaced, that is, if their heights are much less than the distances between
them, then each acts essentially alone, with an anterior downflow that
forms a stable vortex and with another vortex behind that also turns (with
flow from left to right) clockwise. If the bumps are closer, with their spacing
only moderately greater than their heights, then interactions between each
rear vortex and the next front vortex create a considerably less stable and
regular pattern of flow. If the bumps are still closer, then skimming occurs,
with stable vortices in the interstices. All of this, of course, presupposes
overall Reynolds numbers high enough so vortices will happen readily and
depths (in water) great enough so the bumps don't act like weirs. And it's a
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VELOCITY GRADIENTS
bit two-dimensional—with individual rocks instead of cross-flow walls or
steep ripples, we encounter horseshoe-shaped vortices wrapped around
obstacles and other such phenomena that will resurface in Chapter 10.
Beyond what happens in and between bumps, considerable flow goes
into and out of porous substrata, driven by differences in hydrostatic and
hydrodynamic pressures on stream beds and terrestrial surfaces; the
matter came up in Chapter 4, and we'll have more to say about it when we
consider interstitial flows in Chapter 13. I again urge the reader to take
some small syringe or baster and a bottle of marker (fluorescein or uranine,
food colorant, or even skim milk) to a stream with some bed irregularity
and to explore the local patterns of flow. Or, if the opportunity presents
itself, toss a bunch of toy parachutes in different directions from the top of
a building while the wind is blowing. An otherwise invisible world of
environmental heterogeneity will reveal itself.
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Life in Velocity Gradients
We're set to consider the physical situations and adaptations of a
vast multitude of organisms—the world of organisms whose heights
above the substratum are less than or of the same order of magnitude as the
thickness of the boundary layer. This world, then, is defined not by some
velocity but by the ubiquity of a spatial gradient of velocity. It's the world of
smallish organisms at the interface between moving air or water and a solid
substratum; it's the world, even, of the occasional creature on a solid
substratum that's moving through a fluid—the mite on the bird, the barnacle
on the whale. If we want to afflict biology with yet another name, we can
easily coin one for this assemblage. Expropriating from the Greeks, we
might call them craspedophihc creatures, from craspedo-, an edge or border,
and phile, fond of—these are the interfacial plants and animals.
In a sense, virtually every organism exposed to a flow lives in a velocity
gradient inasmuch as the logarithmic boundary layers of both terrestrial
and aquatic habitats are universal and are almost always thicker than an
organism is tall. But up to this point we got away with the assumption that
organisms were exposed to free-stream velocity—or at least that the spatial
variations of velocity impinging on them could be adequately represented
by some simple average. Not much heavy weather was made about it, but
the assumption enforced a severe selectivity in the examples I used.
Not only are the inhabitants of velocity gradients diverse in size and style,
but the functional consequences of these gradients are manifold. A
gradient region is both good news and bad news. It's a hiding place from drag,
but it's a barrier to exchange of materials and energy. It may be useful as
insulation—you can get a burn in a sauna if you're exposed to a local wind.
Or it may impede heat loss enough to present serious problems of
overheating—that's a difficulty faced by broad leaves when the breeze fails.
It may afford some mechanical protection from the impact of material
carried by a flow, but on a filter-feeding organism it may cause detrimental
deflection of edible particles around the filter. Flow-dispersed seeds,
spores, and spiderlings are likely to travel less far if released deep in a
gradient region, especially in air, where gravitational settling is a major
factor; the same propagules are likely to remain where they settle if velocity
gradients are gentle.
Let me repeat for the sake of emphasis that the effects of velocity
gradients—and indeed of local currents in general—are diverse. From my
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LIFE IN VELOCITY GRADIENTS
point of view, trying to deal with a vast and disparate literature, it seems
altogether too easy to do comparative studies with flow as the variable
quantity. Comparing different natural habitats, comparing modified and
unmodified habitats, comparing artificial streams indoors and out, we now
have quite a lot of data that correlate flows with who lives where. Sometimes
"who" is intraspecific and sometimes interspecific. Differences in velocity
and velocity gradient are sometimes decently documented but more often
just roughly categorized. My dissatisfaction is that of the functional
biologist looking for mechanisms who is less than satisfied with inference from
correlation. Field studies that merely document the distribution of
organisms cannot explain distributions in any truly causal sense.
Equally, though, a laboratory study is apt to focus too exclusively on a
single facet of multidimensional suites of adaptations. For instance, the
strong preferences of an organism for a specific regime of flow could as
easily reflect factors such as predation and trophic preference as purely
hydrodynamic adaptation. Statzner et al. (1988), for instance, cite
literature that documents apparent effects of the current speed of streams on
(among others) mate choice, posture, case building, compass orientation,
fighting success, net building, phototaxis, rheotaxis, schooling,
territoriality, and respiratory movements.
Drag of the Flat and Not-so-flat
Can you stand a little more about drag, mentioned in each chapter since
number four? In particular, while we talked about the drag of attached
organisms and how many of them contrive to reduce it in Chapter 6, at that
point little was said about drag reduction by hunkering down against a
surface. If you're located well down within a boundary layer, then flow
speed drops steadily as you get closer to the surface; if the layer is a laminar
one on a flat plate (so that equation 8.4 applies), then the drop is very nearly
linear. Thus drag goes down (for^ bluff body, at least) quite dramatically as
you approach the surface, reaching the very low value of the local drag of
your footprint if you have no height. Mention was made (with something of
a promissory note) that in a typical velocity gradient the drag of a bluff
body is expected to vary not with the square of flow speed but with an
exponent a little greater. We can now seejust why. Again consider the inner
part of a laminar boundary layer on a flat plate. By equation (8.4), at a given
point above the plate (a given x and z), the local velocity, Ux, varies more
drastically than the free-stream speed:
Ux*Ui\ (9.1)
And thus for a bluff body at moderate Reynolds numbers (ignoring skin
friction on the surface itself),
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CHAPTER 9
D oc U* °. (9.2)
What's happening, of course, is that higher free-stream speeds mean
thinner boundary layers, so as speed increases, a given location becomes closer
to the outer edge (however defined) of the layer. Since drag is proportional
to the cube rather than the square of velocity, the baseline for speed-
specific drag, D/U2 (equation 6.4), becomes + 1.0 rather than 0 for objects
in the inner portions of laminar boundary layers.
Before passing lightly over local drag, I might note that I've found few
well-documented cases in which the skin friction of something that doesn't
protrude at all takes on biological significance. Conversely, practical
relevance is obvious, since washing without cloth or brush depends on skin
friction to remove any insoluble material. One case that did turn up
involved the shear stress needed to get bacteria off the walls of capillary
tubes: Powell and Slater (1982) found a shear stress of around 50 Pa
necessary to assure removal. Using room-temperature water as the washing
medium, that's a shear rate of 50,000 s~l, the velocity gradient that would
occur between two plates a millimeter apart, moving at the enormous speed
of 50ms_I relative to each other. Other things (speed and scale) being
equal, using hot water would be counterproductive since its lower viscosity
would give a lower shear stress for a given shear rate—recall equation (2.4).
One other matter might be mentioned again: the caution that the lift on
creatures at interfaces may be a worse problem than their drag. Even if lift
isn't (as it sometimes is) greater than drag, it works in a more awkward
direction, outward ("z-ward"), with an awkwardly upstream line of action.
Moreover, any upward motion due to lift near the upstream extremity will
likely increase lift further (for a while) due to the increased angle of
incidence. It will also increase drag and decrease contact area. If that weren't
bad enough, drag loads the contact line in shear, of which adhesive layers
are typically fairly tolerant, while lift raises the possibility of peel failure, a
much trickier business for a glue. (Peeling is the only practical way to get
adhesive tape off a roll.) Substantial lift on creatures on surfaces has been
documented for limpets (Denny 1989), sand dollars (Telford and Mooi
1987), mayfly larvae (Weissenberger et al. 1991), crabs (Blake 1985), flatfish
(Arnold and Weihs 1978), and rays (Webb 1989). And Kirstenjohannesen
(pers. comm.) found that periwinkle snails (Littorina sp.) oriented to flow in
the only position for which they had negative (substrateward) lift—drag
was not the primary consideration.
The " Torrential Fauna"
These animals, inhabitants of streams where the free-stream speeds are
over about 0.5 m s_1, made a brief appearance (in Chapter 7) when we
talked about drag and streamlining of the legs of the mayfly nymph, Baetis.
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LIFE IN VELOCITY GRADIENTS
Even the swiftest streams have vegetation on their stones and carry
suspended comestibles. Despite currents as high as3ms_1 (Nielsen 1950), a
specialized fauna of both grazers and suspension feeders manages to make
a living in such streams without being swept downstream. Most of its
macroscopic members are immature insects: mayfly nymphs (Ephemerop-
tera); caddisfly larvae (Trichoptera); beetle larvae and adults (Coleoptera),
especially of the family Elmidae; black fly larvae (Diptera: Simuliidae); net-
winged midge larvae and pupae (Diptera: Blepharoceridae); larvae and
occasional pupae of some moths (Lepidoptera: Pyralidae); and a few
others. Their diversity is far less than that of the animals that live in the
same streams but avoid the current by hiding beneath rocks, by burrowing,
or by other means. All live in an unusually harsh habitat that demands
substantial adaptations to the rapidity of flow and its capacity for
impressing forces. Merritt and Cummins (1984) provide a very useful
compendium of who lives where, as well as voluminous references; Hynes (1970)
and Ward (1992) put the whole subject in a richly ecological context.
Early in this century, Steinmann (1907) suggested that the insects of
rapid waters are dorso-ventrally flattened in order to reduce drag and to
help them stay attached. The idea sounds reasonable, and it penetrated
generations of textbooks, but it doesn't look quite so tidy on close scrutiny
and has been questioned in almost every serious contemporary paper on
the subject. Part of the problem was that Steinmann viewed the flow as
pushing the flattened insect down against the substratum by striking its
inclined, upstream-facing surface. The notion isn't intuitively
unreasonable, and the nonwhiggish historian bears in mind that modern notions of
lift and the concept of a boundary layer were developed in that decade and
weren't widely appreciated even among physical scientists (Goldstein
1969). In fact, positive rather than negative lift will ordinarily be
developed; to make matters even less obvious, the amount of lift will depend in
part on whether flow separates when passing over the organism. So flatness
is a two-edged sword—on the one hand it affords a location deeper within
the boundary layer of the substratum and thus lower drag plus a greater
surface for attachment, but on the other hand it raises the bothersome
bugbear of lift. Even separation is a mixed bag—less lift but more drag. To
complicate matters further, flatness isn't exactly inappropriate if a creature
is to crawl around a stone and get underneath it.
It ought, then, to be no surprise to find that some of the torrential fauna
are flattened and some are not and that some nontorrential (lenitic)
creatures are nonetheless flattened. The mayfly nymph Baetis holds itself off
the substratum and has a body that approximates a streamlined body of
revolution rather than being appreciably flattened (Figure 9. la). The
species of Baetis that inhabit more rapid flows are smaller, but have larger legs,
smaller gills, and smaller and fewer caudal cerci (long trailing hairs)—all
quite reasonable for organisms found in flows of up to 3 ms-1. And (just so
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CHAPTER 9
Figure 9.1. Creatures of the so-called torrential fauna:
(a) Baetis, a mayfly nymph; (b) Rhithrogena, another mayfly nymph;
(c) Ancylus, a freshwater limpet; (d) Psephenns, a water penny beetle;
(e) Bibiocephala, a pupal blepharocerid fly; (f) Neothremma, a caddisfly
larva.
nothing is too simple) individuals can to some extent flatten themselves
against a substratum (Dodds and Hisaw 1924). Streamlining, as we've seen,
is a potent drag reducer, and Baetis should not be troubled by lift. But it's a
relatively uncommon adaptation among aquatic insects, found otherwise
mainly in beetles (Resh and Solem 1984). Other torrential mayfly genera
such as Iron, Epeorus, Ecdyonurus, and Rhithrogena (Figure 9.1b) are well
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LIFE IN VELOCITY GRADIENTS
flattened dorso-ventrally and live appressed to rocks. Weissenberger et al.
(1991) discovered that Ecdyonurus sometimes had negative lift, apparently
achieved by lowering its head shield and by using its femurs as spoilers.
Mayfly nymphs aren't the only group in moving fresh water to have
evolved flattened forms with strong devices for attachment. One thinks
first, of course, of turbellarian flatworms, but they're really not an
especially good case since flatness is certainly the ancestral condition and since
they tend to avoid places with really high currents (Hansen et al. 1991). Too
bad—for a planarian the skin friction of local drag might be a real
phenomenon. Better examples are several families of freshwater gastropod
mollusks, most notably the Ancylidae, which are low-coned limpets (Figure
9. lc). None of the animals we're talking about is especially large; most are
less than a centimeter long. The same suite of adaptations, though,
happens in even smaller arthropods, the water mites (Hydracarina: Arach-
noidea), among whom a millimeter in length marks a fair giant. Species
living in streams are smaller than others, are relatively more flattened, have
stouter claws, and have fewer protruding hairs (Pennak 1978). Even some
tadpoles play the game—several rheophilic groups have independently
evolved severely depressed bodies and ventral adherent devices (Lamotte
and Lescure 1989).
Perhaps the flattest of the flat are the members of the coleopterous
family Psephenidae, the "riffle" or "water penny" beetles. Late-stage larvae
run around half a centimeter long but are less than a millimeter thick at the
center (Figure 9.1 d). They live both under and above logs and stones; being
negatively phototactic and positively rheotactic they're exposed mainly at
night (and so are sometimes thought to stay burrowed). Smith and Dartnall
(1980) found that blocking normal flow through the marginal lamellae on
each side led to earlier separation of flow on top of the animals; on that
basis they suggested that the animals normally reduce drag by boundary
layer control through suction, as described for conventional airfoils. The
implications for lift are at this point unknown, although the matter has at
least been raised by McShaffrey and McCafferty (1987).
Separation of flow behind the region of maximum thickness seems to be
common among these flattened forms. Some early drawings of flows based
on observations of moving particles, such as those of Ambiihl (1959), don't
show the phenomenon. However, more recent work with much less
intrusive laser-doppler anemometry (Statzner and Holm 1982) shows significant
separation behind the mayfly nymph Ecdyonurus, and really substantial
separation behind the freshwater limpet Ancylus. And, as just noted, Smith
and Dartnall (1980) found separation even in unplugged water penny
beetles. My guess is that separation reflects a price in drag paid to avoid
paying in the harder currency of lift—it ought to reduce lift more
efficiently than any other kind of drag-increasing arrangement.
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CHAPTER 9
Insect pupae occasionally occur where they're exposed to rapid currents.
They don't take on particularly flattened shapes, although some pupae of
torrential Pyralidae (Lepidoptera) live in very flat cocoons (Nielsen 1950);
probably metamorphosis imposes constraints. But pupae are usually
compact and fusiform, and thus, assuming appropriate orientation, they're
reasonably well streamlined. Pennak (1978), for example, provides a
drawing of a Blepharocerid (Diptera) pupa that looks almost identical to one of
Hoerner's (1965) streamlined rivet heads—compare Figures 9.1 e and 8.4).
Most of the torrential caddisfly larvae live in cylindrical or conical cases
that they construct of tiny stones and bits of vegetation (Figure 9.If).
Dodds and Hisaw (1925) suggested that the caddisflies of rapid waters use
stones as ballast; the counterargument is that stones are merely what are
available in faster currents (Resh and Solem 1984). In any case, their
resistance to being detached and carried away by the current is compounded by
active attachment and a passive component, the latter including ballast.
Young larvae otAllogamus auricollis mainly depend on active attachment,
but by the fifth instar active and passive elements are nearly equal and the
overall current that can be resisted has doubled—presumably due in part
to the larger stones incorporated into the case (Waringer 1989). But of
course in the velocity gradients on surfaces the larger organism must suffer
greater ambient current. Some species of caddisflies erect elegant nets for
catching planktonic prey; the larger the caddisfly, the coarser is the mesh of
the net. In addition, nets are coarser in more rapid currents (Wallace et al.
1977). Either larger larvae can eat larger particles and faster currents
suspend larger particles (Cummins 1964), or else larger animals, with
larger nets, must face something closer to free-stream speed, and the
coarser mesh is a way to keep drag from becoming excessive. In short, a
larger net may be in part a scheme to prevent destruction of a net even at
the cost of missing fine particles. But let's defer further discussion until we
get to the subject of filtration, a really low Reynolds number business, in
Chapter 15.
And Some Analogs of the Torrential Fauna
A marine equivalent of this torrential fauna is that of the ectoparasites of
fast swimmers such as whales. Whales commonly provide a substratum for
barnacles of the genus Cryptolepas—on a gray whale, C. rhachianecti occurs
mainly on the head and the anterior part of the back, which I interpret to
mean that it prefers quite a steep velocity gradient. In addition, whales
harbor so-called cyamids or whale lice, amphipods of Cyamus and related
genera. These live mainly around the barnacles or are associated with the
various slits and pleats on a whale's surface—they presumably prefer less
steep gradients (Blokhin 1984). The preference of cyamids for a little
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LIFE IN VELOCI I Y GRADIENTS
shelter may be involved in the genesis of the skin irregularities
("callosities") on the heads of southern right whales (Eubalaena australis), the
patterns of which prove useful (at least to humans) in distinguishing
among individuals. According to Payne et al. (1983), cyamids feed on the
surface layer of skin; not only do they excavate pits and depressions, but
they avoid projections, which would be exposed to higher velocities. So as a
lousy whale makes skin, projections get longer and longer; the degree of
relief of the patches of callosities gives an indication of a whale's age.
Drag on a creature deep within a velocity gradient might matter in air as
well as in water, and at least one case constitutes a fine parallel to the lice of
whales—the feather mites of some large seabirds. Mites are, of course,
arachnids, kin to spiders, rather than crustaceans, as are the whale lice
(proper lice are wingless insects). According to Choe and Kim (1991),
several species otAlloptes (Acari: Analgoidea) live on the flight feathers of
murres and kittiwakes. The mites prefer to be well back from the leading
edge of the wing, they prefer the slightly concave ventral surfaces of
feathers, and they prefer the grooves between the individual barbs that
make up the vanes of feathers. In each choice, then, they select the aerody-
namically more benign location. Nonetheless, they're mighty modified
mites—flattened dorso-ventrally, elongated, with reduced dorsal setae,
with long lateral setae that seem to help keep them attached, with large
suckerlike empodia that also seem important in attachment, and so forth.
Sheets of Water with Really Extreme Gradients
Sometimes flows in nature are essentially all gradient, and not just the
gentle gradients of a breeze across the landscape or the current in a river.
Consider what happens when a thin layer of water, the swash, runs up a
sandy beach and then down again. With the no-slip condition operative,
the layer has to have an extremely high value otdU/dz. The same
circumstance characterizes sheets of water moving across rock faces as parts of
mountain streams. Both situations have called forth rather specific
adaptations, although the two are far from parallel: a rock face affords secure
opportunities for attachment but permits no escape through burrowing,
while the surface of a beach presents exactly the opposite situation.
Ellers (1988) has given the name "swash-riding" to the phenomenon of
tidally cyclic migration by using this water movement up and down the face
of beaches; he's done a thorough study of how a particular 1 to 2 cm-long
clam, the coquina, Donax variabilis, does the trick. Donax are normally inter-
tidal, but on most marine beaches "intertidal" refers to a situation rather
than a fixed location—so the clams must move regularly to maintain that
intertidal position. Which they do. Normally they're within the sand at a
depth just a little beyond easy reach of beach birds (Richard Coles, pers.
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CHAPTER 9
Figure 9.2. The tiny clam, Donax, oriented in the backwash of a wave
on a sandy beach.
comm.). On an incoming tide clams emergejust before the leading edge of
the swash reaches them. On a receding tide they emerge directly into the
backwash. In the swash they're carried upbeach in the extremely chaotic
flow while in the backwash they orient with anterior ends upstream (Figure
9.2) and move down the beach without significant tumbling.
The tiny clams choose exceptionally large waves through acoustic cues,
and they maintain their tidal rhythm of sensitivity to wave sounds and their
emergence and burrowing behavior in the laboratory for a time, but these
neat features of the scheme aren't important in the present context. More
germane is the fact that Donax are structurally rather unusual clams. For
one thing, they're unusually dense, about 13% more so than other bivalves
or 45% above the density of seawater. For another, their anterior ends are
thinner than their posteriors, and maximum thickness is fairly far back.
The combination seems to assure consistent passive orientation and to
prevent tumbling in the backwash with the consequent problem of being
carried too far seaward without getting a foothold. In their normal
orientation in the backwash, separation of flow is fairly severe, although they
experience lower drag than in any other position. And lift is downward,
according to Ellers' measurements on a large model in the gradient region
on the floor of a wind tunnel.
This clam isn't the only swash-rider but merely the one about which we
know the most about the mechanics of the behavior. Besides other species
of Donax and a few other clams, the phenomenon has been described in
mole crabs and amphipods. A whelk (a gastropod mollusk), Bullia digitalis,
uses swash riding to scavenge prey that it has located chemotactically,
according to Odendaal et al. (1992).
The inhabitants of sheets of water flowing across rocks have been called
"hygropetric" organisms; they include some specialized tadpoles, chi-
ronomid (midges) and psychodid fly larvae, caddisfly larvae, and gerrid
bugs (water striders). Not all habitats described as hygropetric involve re-
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LIFE IN VELOCITY GRADIENTS
ally fast flows—continuous exposure to the spray of a waterfall may result
in just a slow seepage—so the designation includes considerable hydro-
dynamic heterogeneity whatever its hygrometric1 consistency. Some of the
chironomid flies seem to have secondarily adopted the thin water films of
the percolating filters of sewage plants as a habitat, where they're called
"sewage filter flies" and can constitute a local nuisance (Cranston 1984;
Houston et al. 1989).
Settling Down in a Velocity Gradient
One shouldn't forget that an organism deep within the velocity gradient
on a surface knows nothing about the speed of the free-stream flow—or
even if a proper free stream exists. What ought to determine whether a
particular site is an appropriate abode is the steepness of the gradient—not
U but dU/dz. Organisms with both sessile and motile stages inevitably face
some kind of choice of velocity gradient, whether behaviorally mediated, as
when a moth picks a leaf for an egg or a spiderling picks a site for a web, or
mechanically determined, as when a spore lands on a slice of bread whose
surface is too rough for it to blow off again. (As incorrigibly motile animals
of the almost invariably motile vertebrate lineage, we may easily forget how
widespread is an alternation of sessile and motile phases, how attractive
organisms have found a distinction between nutritive and dispersal stages.)
Larval Recruitment
In several instances, a proper velocity gradient has been shown to be
something chosen by an actively swimming organism looking for an
attachment site for the next stage of its life history. Barnacles, for example, prefer
rocks in areas of high current; but the specific choice depends on local
velocity gradients. Crisp (1955) passed cyprids of the barnacle, Balanus
balanoides, through glass pipes in which the velocity profiles were known.
(As we'll see in Chapter 13, the velocity gradient at the inner surface of a
circular pipe with laminar flow is an easy thing to predict or manipulate.)
He showed quite clearly that free-stream velocity was largely irrelevant—
instead it was the current 0.5 mm from the surface that determined
whether attachment would occur, a current indicative of the velocity
gradient at the surface. A shear rate of 50 s~l was the minimum that stimulated
attachment, 100 s_1 was optimal, and the cyprids were unable to attach
above 400 s_1.
1 In this business it's worth remembering that the prefix "hydro-" or "hydr-" comes
from the Greek word for water, while the prefix "hygro-" comes from the word for wet or
moist. Thus a hydrometer measures specific gravity (the inverse of buoyancy), while a
hygrometer measures humidity.
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CHAPTER 9
Lacoursiere (1992) found that black fly larvae (Simuhum vittatum)
selected not the greatest speed or even (as with Crisp's barnacles) the steepest
gradient at the surface, but the greatest velocity gradient along the lengths
of their bodies. As he noted, the choice is one that would give the greatest
particle flux through the filters, the cephalic fans, relative to the drag
incurred by the rather wide lower (posterior) part of the larval body
(Figure 9.3). He looked specifically at settlement on cylinders running across a
flow in a laboratory flow tank at Reynolds numbers from 400 to 30,000. (To
get the situation back in mind the reader might refer again to Figures 5.3
and 5.5.) Under most circumstances larvae preferred to settle near the
separation line. If, for example, separation occurred 81° around from the
upstream center, that's where most larvae settled. Fewest settled between
55° to 70°, the region of maximum shear stress and maximum local velocity,
and very few settled behind the separation line. Once settled, larvae would
shift positions if the cylinder was rotated slightly, maintaining their initial
pattern of preference, except that the forward stagnation point was an
acceptable location even if only rarely an initial choice.
Larval settlement or "recruitment"2 has been a subject of considerable
interest in recent years, and quite a few studies have implicated hydro-
dynamic factors—one can get a good impression of both the level of
interest and the size of the literature from Butman (1987). Despite our present
bias, though, hydrodynamics is not all that matters. Some larvae are simply
gregarious; while this sometimes confers a hydrodynamic advantage (as
will be described in the next chapter for black fly larvae), it's probably more
often a device to assure gametic proximity where the forthcoming adults
are sessile. And some larvae prefer surfaces of particular chemical or
physical characteristics—the bases for choice are summarized by Woodin (1986).
So sometimes local flow patterns are decisive in determining settlement
patterns (see, for instance, Havenhand and Svane 1991 on the tunicate
Ciona intestinahs), and sometimes they're not.
Even the hydrodynamic aspect is turning out to be fairly complicated.
For one thing, all the complications peculiar to real surfaces that we noted
in the last chapter are operative on the kinds of surfaces on which larvae
settle in nature. On thin plates set out just above a sea mount, benthic
foraminifera (shelled protozoans) preferred a consistant thickness of
boundary layer (or steepness of gradient), but on thick plates a little
separation eddy near the leading edge complicated matters in a way analogous to
what Lacoursiere found for black fly larvae (Mullineaux and Butman
1990). For another, the velocity gradients on real surfaces have their own
2 "Recruitment," as used by ecologists and population biologists, includes attachment
of pelagic larvae, their postattachment metamorphosis, and survival to some later stage at
which they're subject to notice by the Authorities.
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LIFE IN VELOCITY GRADIENTS
Figure 9.3. Black fly larvae, Simulium, attached to a cylinder at points of
their choosing in a laboratory flow tank.
diversity. Barnacle larvae (in this case Chthalamus fragilis and Semibalanus
balanoides) proved to prefer lower shear when settling on rougher surfaces
(Wethey 1986). He suggested that the larvae might prefer high shear above
the surface, which would assure rapid growth and adult fecundity, but low
shear at the actual microsites of attachment, which would improve their
survival prospects early on.
And this microworld is embedded in a larger one, the world of
logarithmic boundary layers, shear velocities, and the like. Rocks rather than
fine sediment characterize habitats where the ambient velocities are large
and the turbulence is substantial; and rocks in substantial currents provide
auspicious sites for attachment for organisms with sessile adults. As Denny
and Shibata (1989) point out, the speeds of flow at least in surf zones will
vastly exceed the swimming speeds typical of pelagic larvae—0.1 to 10 mm
s~1—and exceed even more drastically the sinking speeds of the spores of
marine algae—0.01 to 0.1 mm s_1. So transport to the surface will happen
as a result of turbulent motion. Any laminar sublayer will be of the same
order of thickness as the larvae, which is to say insignificantly thick. What
matters in surf zones, then, is whether a larva holds on or a spore sticks and
whether a larva does any exploration of the surface prior to attachment.
Not surprisingly, barnacle cyprids of the genus used by Crisp (1955) turn
out to be transported passively up to their initial contact but then explore
the surface a bit before settling for good (Mullineaux and Butman 1991).
Mere ability to hold on isn't necessarily enough. Just as Crisp found a
minimum speed for attachment in his glass tubes, Pawlik et al. (1991) noted
that the larvae of a marine tube worm (Phragmatopoma lapidosa californica)
actively avoided settling when exposed to very low speeds in their flow tank.
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CHAPTER 9
A rough surface adds its own complications, and Eckman (1990) has
taken a brave stab at a theoretical model for passively transported particles
approaching such a surface. In his model, settlement rate should increase
sharply with the areal density of roughness features. Settlement should
also increase (not unexpectly) with increase in the ratio of the settling
velocity of larvae to the shear velocity. Less obviously, settlement should
decrease with increasing height of roughness features relative to boundary
layer thickness; a high value of this last variable implies less effective
mixing immediately adjacent to the surface.
It's just a little frustrating to find that most of the extensive literature on
larval recruitment couches things in terms of flow speeds rather than shear
velocities and velocity gradients. Admittedly the former are easier to
measure, but it really does seem as if the latter are closer to what really matters
to the organisms.
Settling from Air
The most relevant physical difference between air and water is in density.
The main biological difference is the absence in air of very much that's
equivalent to the wide world of pelagic invertebrate larvae. (Denny and
Shibata 1989 remind us that fully 70% of benthic marine invertebrates
have pelagic larvae.) Settling from air might strike one as trivial—what goes
up must come down, as the saying goes. And the terminal velocities of a
large number of pollens and spores are matters of public record (see, for
example, Gregory 1973 but also some cautionary comments on measuring
sinking rates in Chapter 15). Here again, complications rear up before
things settle down.
In still air, spores settle randomly on a horizontal, fiat surface. But with
increasing winds, the efficiency of capture is progressively reduced, with
the least deposition on surfaces parallel to flow. The photographs of Hirst
and Stedman (1971) strongly suggest that deposition is inversely related to
the thickness of the local boundary layer. Several phenomena are probably
involved. First, with a thick boundary layer, fewer particles will enter the
inner reaches from which they may be captured in a given time. And in a
boundary layer of appreciable thickness, as on a leaf at low wind speeds, the
downstream thickening produces a divergence of streamlines from the
surface, which will tend to deflect dropping particles. In addition,
touchdown doesn't guarantee landing, since particles such as pollen and
spores can not only blow off but actually bounce off as well (Paw U 1983;
Aylor and Ferrandino 1985). Not surprisingly, Chamberlain et al. (1984)
found an optimal velocity (and shear velocity) for deposition oi:Lycopodium
spores on a nonsticky, ribbed surface.
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LIFE IN VELOCITY GRADIENTS
Tuning Up Nonmotile Systems
That a barnacle cyprid might poke around a little before attaching for
the rest of its life doesn't strikes one as startling; more noteworthy is
evidence of specific site selection among nonmotile propagules. Still, if both
settler and settlement site have a positive stake in the outcome, a biologist
shouldn't be shocked. Nevertheless, the idea of fluid mechanical tuning to
maximize the chance of contact is novel, even technologically portentous.
But that's just what Niklas (1985) found in pine cones—they're
aerodynamic pollen traps. Some of anemophilous (wind pollinated) plants even
manage some specificity in the trapping mechanism. Niklas and Buch-
mann (1985, 1987) have evidence for several genera suggesting that
individuals of each of two sympatric species preferentially trap conspecific
pollen.
In at least one case, this kind of specificity extends to settlement of an
epiphyte of no demonstrated value to its host. Pearson and Evans (1990)
showed under the controlled conditions of a flow tank that spores of a red
(rhodophyte) alga, Polysiphonia lanosa, settle at higher than expected rates
on the normal host, Ascophyllum nodosum, and at lower than expected rates
on the sympatric Fucus vesiculosus, the latter both large brown (phaeophyte)
algae.
More Direct Effects of Shear
By definition, drag is a force tending to carry an object downstream. In a
velocity gradient, though, rather than simply pushing an object, the flow
will impose a shearing load that will tend to make the object rotate (Figure
9.4; Silvester and Sleigh 1985). Thus an object that doesn't touch the
substrate will rotate as it travels downstream, since what it experiences as a
result of the flow is simply a force in one direction (upstream, although it
can't easily know that) on the wallward side and a force in the other
(downstream) on the streamward side. That's why off-axis blood cells rotate as
they flow along in small vessels (Chapter 14; Vogel 1992a). An object
protruding from or sitting on a surface will be more effectively rolled than
pushed along, quite apart from friction with the substratum. In part, that's
why tumbleweeds tumble (Van der Pijl 1972; LaBarbera 1983), and it's an
important element in the initiation of erosion (Bagnold 1941).
Sometimes the gradient-induced rotation may be quite a serious stress.
Morgan et al. (1976) looked at the effects of severe shear on suspended fish
eggs. For most eggs, shear rates (calculated from their data) around 35,000
s~l are lethal with only a few minutes of exposure. Such shear induces very
rapid rotation, and the resulting centrifugal disruption may be what causes
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CHAPTER 9
velocity gradient ( ) â–º overall
motion
Figure 9.4. Rotation of a solid body induced by the shearing motion
within a velocity gradient.
the high mortality. Such rates are unlikely to occur in natural or even
artificial channels where the turbulence associated with the high shear
rates may be important in keeping eggs suspended. But damaging rates are
approached near the hulls of fast ships and exceeded in their propellers.
They're also of the order of the shear rates occurring in the cooling systems
of nuclear power plants, although in these latter I'd guess that simple
heating is a far more significant hazard.
The shear stress on the surface itself may be far below what might cause
direct mechanical damage, but it proves to be of such enormous
importance so close to home that the system must at least be mentioned here; a
proper discussion will be deferred to Chapter 14. A circulatory system in
which the resistance to flow through the complex manifold of vessels is
minimized was shown by Murray (1926) to be one in which the shear rate
on the walls of every vessel was the same. Fairly recently it's been shown that
the endothelial cells that line blood vessels have a mechanoreceptive
function and that they are specifically responsive to shear stress. So all that's
needed for self-optimization is some specific threshold, above which a
sequence of events leading to vessel enlargement is initiated and below
which a sequence leading to vessel narrowing comes into play.
Suspension Feeding and Local Currents
Suspension feeding most often takes place in the gradient region
adjacent to a surface. (Suspension feeding by some swimming fish, tadpoles,
and baleen whales is an exception; see Sanderson and Wassersug 1990.) So
it's certainly germane to the present discussion—even though discussion of
mechanisms will await Chapter 15, on low Reynolds numbers. Of present
relevance are the ways in which interfacial suspension feeders deal with the
velocity gradients in which they find themselves.
First, however, several terminological notes. "Suspension feeding"
subsumes "filter feeding" as the term was used by older sources. The shift
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LIFE IN VELOCITY GRADIENTS
involves a bit more than fashion. "Filter feeding" implies that food is
obtained by simple filtration or sieving, but emergence of the details of
capture mechanisms has made it clear that the implication is often misleading.
In any case, it refers literally to how food is obtained rather than, as with
"suspension feeding," to the nature and location of the food—particulate
material, typically tiny, typically living, often actively if slowly moving in the
ambient fluid, the latter most often water. As we'll see in the next chapter,
detritus feeding by using a filter is possible if you employ some device to
resuspend particles resting on the bottom; we prefer to exclude these by
using an ecological rather than physiological distinction.
Among suspension feeders a distinction is usually made between
"passive" and "active" ones, with passive suspension feeders dependent on the
motion of the water to move fluid to and through their separation
equipment and active suspension feeders working some energy-consuming
pump. But these turn out to be either idealizations or extremes (depending
on viewpoint or bias), applicable, on the one hand, to a spider or mayfly
larva waiting by its net and, on the other, to a clam in a muddy bottom with
its siphons barely protruding. In between are a wide range of mixed-media
specialists, including whales that swim along with their mouths ajar,
barnacles that sweep their cirri through water that's moving in the opposite
direction, sponges and brachiopods that are quite competent pumpers but
can augment their pumping by using environmental currents (Vogel
1978a). One has to bear in mind just how dilute most natural waters are as
sources of particulate food. A marine sponge, which we generally think of
as an unsophisticated and slow-growing animal, manages to process its own
volume of water every 5 seconds and to extract edible material with almost
no loss down to the size of bacteria (Reiswig 1971, 1975b; Vogel 1977).
Bivalve mollusks do even better, with scallops and mussels pumping
volumes equal to their own volumes every 3 or 4 seconds (Meyhofer 1985).
Incidentally, the cost of processing a lot of water underlies a rather nicely
convergent arrangement occurring in several groups of fishes. The
economic advantage of ram ventilation in fishes and its use by rapid
swimmers were mentioned in Chapters 4 and 7. Some fishes filter food from
water passing through the same mouth-to-operculum stream. Suspension-
feeding fishes commonly prove to be obligate ram ventilators—the
conjunction occurs in anchovies, menhaden, mackerel, and paddlefish.
According to Burggren and Bemis (1992), paddlefish have a minimum
metabolic rate about twice that of other fishes their size because they have
to keep swimming. Still, the respiratory cost of constant swimming is no
additional burden for a filterer; it's really analogous to the cost of
swimming in a pursuit predator.
I think suspension feeding is best viewed as a set of compromises among
conflicting factors. One was mentioned a page or so ago—low flow right
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CHAPTER 9
near the surface to facilitate attachment but high flow slightly farther away
to provide ample supplies of food. That argues for sites with some surface
roughness and sites where boundary layers are thin and gradients steep;
and certainly suspension feeders abound on the blades of marine grasses
and macroalgae, on wharf pilings, on rocky outcrops, and (under the
manipulations of marine ecologists) on small ceramic plates in open racks
exposed to tidal currents. The optimal location certainly depends on how
an animal deals with fluid. Hughes (1975) measured the currents around
erect colonial hydroids (Nemertesia antennina) in about one meter of water
and found, not unexpectedly, that speeds near their tops were generally
above those near the holdfasts. He then made a very thorough catalog of
the numerous species that live on the hydroids, noting where each was
found. The more passive feeders (some smaller hydroids and others) were
more distal or apical; the more active feeders were more proximal or basal.
Really active pumpers—some bryozoans, bivalves, ascidians, and
sponges—were usually found only on the holdfast. Furthermore, the
suspension feeders on the holdfast treated food differently from the higher
and more passive sorts. The former were much less selective in initial
capture but had well-developed mechanisms for subsequent rejection of
unwanted stuff. Clearly there's a secondary disadvantage of being on the
bottom. Suspended plankton is more likely to be nutritive than particles
that have dropped out of the water column and are tumbling along near
the bottom. A large and scattered literature addresses this matter of choice
of habitat by suspension-feeding animals.
Another compromise must lie between maximizing rates of fluid
processing and ensuring that processed, food-depleted fluid is prevented
from reentering the separator. The problem will be most severe where the
suspension feeders live deep within the velocity gradients—where, as at
substantial depths and in sheltered locations, ambient flows are low, and
where flows are both nonturbulent and bidirectional. Both the problem
and one solution seem first to have been recognized by Bidder (1923),
working on sponges. He identified the constrictions on the excurrent
openings (oscula) as nozzles that increase flow speed, and he noted the role
of the apical locations and upward orientations of these openings. The
siphonal openings of many ascidians and bivalve mollusks certainly work as
jets in the same way and for the same purpose, as mentioned in connection
with the principle of continuity back in Chapter 3.
Making a really small nozzle or jet that carries fluid any decent distance is
hard because the area of "contact" between thejet's sides and the main fluid
will be great relative to thejet's volume (and thus its momentum flux). Some
groups of small suspension feeders get around these pernicious problems
of scale and viscosity through cooperative flow management. Some of the
ascidians, the Botryllidae, have a colonial discharge system to get a single,
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LIFE IN VELOCITY GRADIENTS
Figure 9.5. Colonial discharge systems—common jets: (a) an ascidian,
Botryllns; (b) an encrusting bryozoan, Membranipora.
large jet instead of a half-dozen or a dozen individual ones (Figure 9.5a).
More impressive still are the cooperative arrangements of some species of
bryozoans that form thin colonial encrustations on the flat surfaces of
marine macroalgae. Since the individuals are only a fraction of a millimeter
tall and since they form a continuous and fairly smooth layer, these animals
certainly live well within local boundary layers. A neat device has been
described by Cook (1977) and investigated in more detail by Lidgard
(1981). While the individual zooids have no gaps between them, a colony of
Membranipora villosa has regions of nonfeeding, degenerate individuals
every few millimeters (Figure 9.5b). These regions act as excurrent
chimneys for colony wide currents. Water from near the surface of the colony is
pumped downward by the cilia on the tentacles of the individuals. The
water then moves laterally along the substratum between individuals
toward the chimneys. It emerges from the latter as jets of substantial speed
(25 mm s-1) that broaden and themselves coalesce well above the zooids,
above and as well separated from incurrent water as if a collecting manifold
of physical chimneys were present. So, except perhaps in the stillest of still
waters, no zooid reprocesses water previously used by other zooids.
An alternative to a jet is elevation of the filtration apparatus itself.
Wherever external nets or tentacles occur, the device is almost inevitably
concomitant. One can point to the various erect colonial coelenterates and
bryozoa, to sea anemones such as Metridium, whose drag was mentioned in
Chapter 6, to several families of polychaetes living in tubes and extending
feeding appendages upward (see Fauchald and Jumars 1979), to sea lilies
among the echinoderms, to goose barnacles among the Crustacea, to the
stalked solitary or colonial protozoa such as Vorticella and Carchesium, and
to the black fly larvae that keep their cephalic fans uppermost. In some
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CHAPTER 9
members of this diverse assemblage the stalk or other elevation device is
retractile, in others it isn't.
Still another kind of compromise comes to play in an animal's choice of
flow speeds at which to feed. At low (and concomitantly less turbulent)
ambient flows, decreased supply, local depletion, and increased
reprocessing of food should have a detrimental effect on the cost-benefit ratio for
suspension feeding. At high ambient flows, several adverse factors become
more significant. A larger fraction of high density, nonedible material may
be entrained into the medium (a problem for clams at least, according to
Turner and Miller 1991); drag and the consequent difficulty of keeping the
food-trapping device properly exposed will be greater; and the greater
momentum of passing particles may make them harder to capture. And
even where organisms don't seem to take advantage of local currents to
induce internal flow, the rates of suspension feeding often depend quite
strongly on ambient flow. An increase in the rate of active pumping with
increases in local flow was found by Walne (1972) in five species of clams,
mussels, and oysters—perhaps the more rapid the flow, the likelihood of a
richer food resource and the desirability of a greater investment of energy
in pumping. But Walne used flow speeds that seem to have been below 10
mm s~l; at substantially higher speeds, pumping may decrease, as found
for mussels by Wildish and Miyares (1990). A large literature documents
great influence of current speeds on suspension feeders (see, for instance,
Shimetaandjumars 1991), although a search for mechanisms and general
patterns is hampered by inconsistency in choice of variables to measure—
rates of fluid processing, rates of food acquisition, growth rates, and so
forth.
Dispersal and Velocity Gradients as Barriers
Passive dispersal of propagules through the use of winds and water
currents is practiced by at least one member of every major phylum of
animals and every division of plants. Aerosol particles, polluting and
otherwise, likewise enjoy the benefits of the free transit system in the sky. It
looks as if all a propagule has to manage is a sufficiently low sinking rate
through high drag, lift production, upcurrent detection, or density
reduction. In fact such factors address only part of the problem. Somehow a
propagule has to get liberated into the current in the first place, and it
needs to alight on a surface at the end of the ride. In both, velocity
gradients must be crossed, and we turn now to the extraordinary collection of
devices nature has contrived for accomplishing that first step in aerial
transport.
The basic problem may be easiest to see if we consider a situation in
which spores simply sit on a surface in a wind. Grace and Collins (1976) let
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LIFE IN VELOCITY GRADIENTS
Lycopodium (a club moss) spores settle on leaf models of paper, and they
then tried to blow them off again in a wind tunnel. These relatively large
(30-|xm diameter) spores protruded an average of 15 |xm above the surface.
A spore 40 mm from a leading edge needed a free-stream wind of 2.7 m
s_1 to get the necessary 18 mm subthreshold wind speed on its center—
fully 150 times higher. To get 50% of the spores off a piece of paper of
roughness comparable to spore diameter took a wind of 6.8 ms_1, no mere
zephyr. In general, increasing the turbulence level in the wind tunnel
improved matters, even with these tiny particles well down in the boundary
layer. Nonetheless, in nature things might not be quite so bad—Grace
(1978) has shown that boundary layers on leaves will be thinner than
assumed from work in wind tunnels. In a nearly smooth free stream,
turbulent flow across a fluttering white poplar (Populus) leaf will start at 8.6 m s~l,
while in a flow with a turbulent intensity of 20% to 25%, flow over the leaf
will get turbulent at only 3.8 m s~l. The lesson in this is to be cautious in
using the carefully crafted wind tunnels of aerodynamicists, who want
minimum turbulence in order to model systems that ordinarily propel
themselves through otherwise still air. For present purposes their worst
wind tunnel may be our best! When looking at the drag of leaves in high
winds (Vogel 1989) I deliberately worked in the messy flow downstream of
the propeller for just this reason.
To experience flow, getting even a little way up through a boundary layer
helps a lot. Consider a location 40 mm downstream on a flat surface parallel
to flow in a wind of a meter per second. The local speed is 4 mm s~l at 10
|xm above the surface, 40 mm s~l at 100 |xm above the surface, and 400 mm
s~l at 1 mm above the surface (equation 8.4). And elevation is the most
common tactic. Spores, pollen, seeds, and so forth are commonly
presented to the wind on some sort of elevation, whether we consider the
spores of a Penicillwm fungus held less than 100 |xm high or the elaborate
arrangements to expose pollen to wind in many anemophilous higher
plants. The newly hatched first instars of a scale insect or coccid, Pul-
vinariella mesembryanthemi (Homoptera), stand on their hind legs facing
downwind with their bodies well elevated when they decide to enter the air
column for dispersal. Standing erect, a coccid is only 0.34 mm tall and still
in the region of the local boundary layer in which velocity increases linearly.
Washburn and Washburn (1984) tested them in a wind tunnel and found
that free-stream winds of over 2ms-1 were needed to produce enough
drag to tear their hind tarsi from the substratum.
Perhaps these coccids hold on strongly enough to guarantee that they
won't get wafted off by too gentle a breeze to assure effective dispersal—
their sinking rate is fairly high, 0.26 m s-1, even with extended legs and
antennae. The case seems similar to that of a fungus, Helminthosporium
maydis, that afflicts corn. Aylor (1975) found that it took a force of 10~7
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CHAPTER 9
newtons to detach conidia from corn leaves, which may sound trivial but
which requires a free-stream wind of 10 ms-1, both by calculation (using
Stokes' law) and by direct measurement. I think a useful scheme lurks here.
An animal might begin by holding tightly and gradually relax its grip,
thereby picking a high wind if one is available but not being overchoosy if it
can't do as well. Similarly, a wind-dispersed seed might gradually reduce
the strength of its attachment, eventually just coming loose if no decent
wind ever occurred. Alternatively, a propagule of constant tenacity might
be borne on a stalk that gradually grows away from its substratum into
higher flow speeds.
Nor does elevation as a benefaction top out right above the boundary
layer of a leaf or flower—the logarithmic boundary layer is orders of
magnitude thicker. Remember that the tassels are on top of a corn stalk and the
male cones are highest in a pine tree. When I mow my scruffy lawn, the
highest items are the structures bearing the wind-dispersed seeds of grass,
dandelions, and plantains. And both lycosid spider young and gypsy moth
caterpillars are reported to climb as high as they can before spinning their
silk threads and letting themselves go in the breeze (Tolbert 1977).
A logical alternative to getting some height in a velocity gradient is to
pick circumstances in which the gradient is especially severe. According to
Matlack (1989), the seeds of black or sweet birch (Betula lenta) are released
in the winter on days that are below freezing and have substantial wind and
low humidity. In a deciduous forest, winter dispersal gets around much of
the problem of low wind beneath the canopy; and, after falling, these seeds
can slide over the smooth and nonadhesive snow to increase the area of
dispersion more than 3-fold—an especially useful device for a gap-
colonizing species.
Yet another way to defeat the velocity gradient for wind-dispersal is brief
use of an engine for propulsion through it. One-shot engines are fairly
common among plants; most often a gun of some sort (rather than a jet or
rocket) gives an initial push to the seeds or spores. Just among seed-
producing plants twenty-three different ballistic mechanisms have been
described in as many different families (Stamp and Lucas 1983). Among
this array of phyto-artillery we can loosely distinguish two sorts. In some
cases the distance over which the propagules are aerially dispersed
corresponds closely to the range of the gun, and ambient fluid motion is of little
consequence. At the other extreme the only real function of the gun is to
get away from Mama and her semistagnant miasma.
The first arrangement is clearly the most common; but, while quite a lot
is known about the diverse propulsive schemes, little attention has been
given to any aerodynamic specialization of the projectiles. One might
expect that projectiles could deviate from simple sphericity in ways that
enhance stability in flight and, by limiting tumbling, permit streamlining.
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LIFE IN VELOCITY GRADIENTS
Certainly with Reynolds numbers in the thousands, streamlining might
usefully lengthen their range. But whether any such drag-reducing
scheme is used is unclear. Arceuthobium, a dwarf mistletoe, manages a
muzzle (discharge) speed of around 14 m s; it has an ellipsoidal seed, 2.9 x 1.1
mm and thus an initial Reynolds number of around 3000 (Hinds et al.
1963). But the seeds tumble during flight, going at most about 15 meters
rather than the 20 or so that I calculate would be achieved in the absence of
drag.
The other arrangement, using a gun simply to get out of the parental
velocity gradient, is less common for, I think, a very basic fluid mechanical
reason. A good projectile needs relatively low drag and high momentum.
The simplest way to get that combination is to be large, to be much denser
than the medium, to have a high initial velocity, and to operate in air—in
short, to operate at fairly high Reynolds numbers. A particle suitable for
effective dispersal by ambient fluid motion needs almost the opposite
characteristics. It must have high drag so as to settle out very slowly, which
argues for small size, low density, and a preference for water over air. So the
requirements for shooting and settling are substantially antithetical. One
suspected case is that of the ascomycete fungus, Sordaria—at least the
performance of the gun is well established, and one can think of few other
reasons to shoot so hard for such a short range. Sordaria seems to hedge its
bets by shooting between one and eight spores per projectile. An eight-
spore ball is about 40 |xm in diameter and goes about 60 mm (Ingold and
Hadland 1959), from which I calculate a muzzle speed of 30 m s_1 (a
program for doing such computations is given in Vogel 1988a). In street
terms that's a range of two and a half inches consequent to an initial speed
of nearly 70 mph, a catastrophically draggy shot. Concomitantly, the initial
angle for the maximum horizontal range is about 6°, not the 45° one
calculates for projectiles that experience no drag. That, incidentally, is a general
rule—the worse the force of drag relative to the force of gravity, the lower
is the angle that gives the greatest horizontal range. Put in intuitive terms,
where drag causes severe deceleration, a projectile must achieve most of its
horizontal distance while it still has decent speed. Some calculated
trajectories for Sordaria spore clusters are shown in Figure 9.6—parabolic they are
not.
A partial evasion of the problem of reconciling projectile and particle
sizes can be had by shooting a mass that then breaks up. A puffball (Lycoper-
don) works this way. Indenting its thin and flexible wall forces out ajet of air
and spores; in nature raindrops are probably the immediate stimuli
(Gregory 1973) in this bellowslike scheme.
The conflicting requirements, bad enough in air, appear to be truly
disabling in water. Relative particle densities are, of course, much less, and
drag is much greater. Going with the flow is so easy that settling out rather
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CHAPTER 9
0 10 20 30 40
range, mm
Figure 9.6. Trajectories for microprojectiles in air, calculated assuming
the drag of a sphere and the characteristics of an eight-spore cluster of
the fungus, Sordaria. Successive circles on each line mark positions at
approximately equal intervals of time.
than staying aloft is what takes explanation. Conversely, guns work very
badly. The nematocysts of the coelenterates aren't really guns in the usual
sense but rather devices that stab the victim after it touches their triggers.
And even our macroscopic and bellicose species uses self-propelled
torpedoes rather than cannon to convey explosives under water.
Diffusion across the Velocity Gradients at Surfaces
Never is a velocity gradient adjacent to a surface of greater biological
consequence than in the presence of diffusive exchange between the
surface and the flowing medium. And such exchange is enormously common.
Diffusion is the basic transport scheme at the cellular level (cyclosis and a
few other phenomena notwithstanding), but diffusion is pitifully slow over
all but the shortest distances. The almost inevitable concomitant of
elaborate macroscopic and multicellular organization is some scheme to
augment diffusion with convection—in other words with bulk flow, as pointed
out by Krogh (1941). Whether we call the systems circulatory, respiratory,
translocational, or other, they have in common this circumvention of the
speed limitations of diffusion over macroscopic distances (LaBarbera and
Vogel 1982). But such systems only augment diffusion; and each
inescapably retains a final pair of diffusive links. One, of course, is diffusion
through whatever cellular or membranous barrier divides transport me-
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LIFE IN VELOCITY GRADIENTS
dium from utilization site. The other occurs across the innermost portion
of any laminarly flowing transport medium.
Since the steepness of a gradient determines transport rate, whether as a
factor in Fourier's law for heat transfer or Fick's law for diffusion, the
steepness of the velocity gradient takes on special significance. Bird et al.
(1960) do an especially good job on the interrelationships of the transport
of mass across a concentration gradient, of heat across a temperature
gradient, and of momentum across a velocity gradient. Not only are these
transport phenomena interrelated, but often one can be used as a
quantitative model for another—for instance, using heat conduction for oxygen
diffusion in egg masses of snails (Hunter and Vogel 1986).
A simple consequence of diffusion through a velocity gradient, in this
case the boundary layer on a flat plate, can be seen when a photographic
print is developed. As the print is gently agitated in a tray of developing
solution, the image first appears near the edges (Figure 9.7) because the
necessary chemical reactions proceed more rapidly where the steeper
gradient supplies reactants more rapidly. The phenomenon can be made to
serve as a way to measure concentration or velocity gradients at surfaces
(Dasgupta et al. 1993).
Will Diffusion across a Velocity Gradient Limit a Process?
The two diffusive links just mentioned are arranged in series, and more
often than not other diffusive elements extend the chain. The implications
of a sequential array are perhaps easiest to appreciate for transpirational
water loss from leaves. Air within a leaf is water saturated, and water vapor
diffuses through tiny holes of adjustable aperture, the stomata. Thus stom-
atal resistance and so-called boundary layer resistance are additive in the
manner of serial electrical resistors.3 Does wind increase water loss?
Sometimes yes and sometimes no, depending as much on the resistance of the
stomata as on the magnitude of the wind, as Grace (1977) very nicely
explains. Where stomata are closed and their resistance high, wind has
little effect—if resistors of a thousand ohms and often ohms are in series,
decreasing the latter even by an order of magnitude makes little overall
difference. Similarly, opening stomata beyond a certain width usually has
little effect on water loss since boundary layer resistance has become the
dominant element.
I've seen some oddly worded explanations of how the stomata, most
often making up less than 1% of total leaf area (Meidner and Mansfield
3 I'm simplifying by judicious omission. "Cuticular resistance" is an element in parallel
to stomatal resistance that's enough higher than the former unless the stomata are closed
to be of little importance. Then there's "mesophyll resistance" and other series elements.
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CHAPTER 9
Figure 9.7. A photographic image appearing, edges first, in a tray of
gently agitated developing solution.
1968), are equivalent to a surface entirely open to water loss on account of
interactions of the vapor caps outside each stomate—to me what they're
saying is simply that boundary layer resistance is now dominant. In
addition, at high levels of illumination another factor sometimes makes the
situation appear anomalous. Vaporization of water increases with
temperature, and leaf temperatures can get as much as 20° above ambient
when the wind is very low. Thus an increase in wind speed can lead to a
reduction in transpiration by decreasing leaf temperature more than
enough to offset the lower boundary layer resistance. For a leaf, still air and
full sunlight are in any case mutually exclusive conditions because of solar
heating and the convection the consequent temperature difference
induces. Diffusion of water vapor through boundary layers has been of
special concern to plant physiologists, who periodically produce reviews of the
state of the art. For further information, I suggest consulting Meidner and
Mansfield 1968, Leyton 1975, Grace 1977, Rand 1983, or Monteith and
Unsworth 1990.
Sometimes the resistance imposed by velocity gradients may be
unimportant as rate-limiting elements. For instance, Tracy and Sotherland
(1979) showed that the boundary layer of bird eggs is of little consequence
in water loss; under no circumstances is its resistance more than 10% of the
total.
The Minute Currents That Matter in Water
The diffusive resistance associated with interfacial velocity gradients
may be important in air, but it's overwhelming in water. The diffusion
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LIFE IN VELOCITY GRADIENTS
coefficients of small molecules through water are about 10,000 times lower
than through air. An alveolus in a lung is of the order of a hundred times
the diameter of a pulmonary capillary (the square root of 10,000, in
accordance with Fick's law); that's the difference it takes to ensure that transport
in air and in blood are reasonably balanced. Concomitantly, flow makes
more difference in water, and much lower currents have noticeable effects
on exchange between an organism and its environment.4
Very low flow rates may make such a difference that still water is no
simple thing to achieve as a control. Schumacher and Whitford (1965)
measured both respiration rates (in the dark) and photosynthetic rates
(with light) for four classes of attached algae; in all cases slight currents
greatly increased these rates. In Spirogyra, a filamentous green alga, a
current of 10 mm s~l increased the photosynthetic rate by 18% compared to
the rate in still water. Even lower speeds were found effective by Westlake
(1967), who could detect an increase in photosynthetic rate in Ranunculus
pseudofluilans (an aquatic dicot) when the current was raised from 0.2 to 0.3
mm s_1. The latter is about a meter per hour, well below what we'd
normally notice as moving. Perhaps we should regard truly still water as an
idealization comparable to a completely rigid body.
Conversely, if high flow rates increase productivity over rates found with
more moderate flows (as has sometimes been found for large marine
algae), the explanation cannot be assumed to be augmentation of diffusive
transport without some more specific demonstration or systematic
rejection of alternatives. Changes in exposure to light, reduction in fouling with
both living and nonliving material and other less tidy explanations must
first be given decent burial.
We saw that morphological adaptations were clearly central in avoiding
excessive drag; only a little less certain is that another suite of structural
features may alleviate the problems of diffusion through velocity
gradients. Small leaves, which would on the average have thinner boundary
layers, are variously reported to have higher photosynthetic rates than
large leaves. Aquatic plants often have smaller, thinner, or more highly
dissected leaves than their terrestrial relatives (see Sculthorpe 1967 or
Haslam 1978). The possibility of reduced damage from flow forces comes
immediately to mind, but many of these plants never experience rapid
currents. So their shapes and sizes may well be primarily adaptations to the
1 Nonetheless, diffusion is slow even in air—the classroom demonstration using the
spread of perfume is a fraud, with convective transport in the ubiquitous air currents
conveying the aroma over all except the last small fraction of a millimeter at the nasal
epithelium. Indeed, a decent demonstration of diffusion takes great effort to achieve
even in a liquid system, much less in air—convection can't be turned off without
stringent control of temperature and without a long wait to let any history of motion become
adequately ancient. I've usually resorted to a little agar or gelatin in the water to keep the
medium from stirring itself up.
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CHAPTER 9
much greater difficulty of diffusive exchange in an aqueous medium. But
by contrast with drag, here current helps matters. As mentioned earlier, at
least some of the insects of torrential currents have gills of reduced area
(Dodds and Hisaw 1924); when rapid flow is normal, then gills of ordinary
size would just be a drag.
Closer to home phylogenetically than photosynthesis in aquatic plants is
cutaneous respiration in amphibians. Quite a few of these animals stay
submerged for long periods (overwintering submerged in some cases), and
most deliver blood to the skin from a branch of the aorta that supplies the
lungs as well—skin is a major element in oxygenating blood. For bullfrogs
(Rana catesbeiana) the external velocity gradient constitutes a highly
significant 35% of the total resistance to oxygen uptake in an ambient flow of 50
mm s~l. In even slower water the situation is yet worse—at 1 mm s_1 it
makes up fully 90% of the resistance. Pinder and Feder (1990), whose
figures I quote, point out that putting an immobilized frog into still water is
equivalent to putting it in fully anoxic water. In still water bullfrogs do a
certain amount of spontaneous and seemingly undirected movement,
which turns out to make quite a lot of difference to oxygen availability.
Another frog, Xenopus laevis, not only moves about more in still water but
makes voluntary dives of less than half as long as in moving water. Lung-
less, gill-less salamanders (Desmognathus quadramaculatus in particular) have
an especially drastic way of coping with the problem. When they get into
stream microhabitats with low flow rates they cope with the chronic
hypoxia by substantially reducing their metabolic rates (Booth and Feder 1991).
Diffusion through velocity gradients is important in the world of cell
biologists as well. I can't resist citing one iconoclastic experiment (Stoker
1973). On a culture of fibroblasts a strip was removed, leaving a denuded
patch. Cells at the border of this denuded zone increased their growth and
invaded it. Conventional wisdom would assume the growth to be caused by
reduction of contact or diffusional inhibition between cells. But Stoker
showed that one needn't invoke so biological an explanation—nothing
more is responsible than higher concentration gradients in the stirred
medium at the edges of the patch due to the steeper velocity gradients.
The " Unstirred Layer"
Dropping down in organizational level from our usual concern with
whole organisms we encounter another important aspect of velocity
gradients. Textbooks of physiology often discuss something called the
"unstirred layer," something that (in some cases at least) sounds a little like the
distinct region beneath a surface of discontinuity that I fussed about at the
start of the last chapter. But the unstirred layer turns out to be yet another
useful fiction ("still water," again). What it really amounts to is the region
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LIFE IN VELOCITY GRADIENTS
adjacent to a surface through which transport is diffusive rather than
convective, the region where there is a significant concentration gradient
of whatever material is under consideration. Which is to say that it's what
happens in a laminar velocity gradient near a surface.
As explained by Barry and Diamond (1984), a gradient in the relative
significance of bulk flow or stirring can be modeled for analytic purposes as
a completely flowless region outwardly bounded by an abrupt transition
plane. In the present context especially, it's important to appreciate that
this unstirred region is defined in terms of concentrations and
concentration gradients rather than velocities and velocity gradients. Obviously the
two are related, as we'll get to just ahead, but definitionally at least the
relationship is incidental. The specificity of the notion is perhaps best put
with a graph (Figure 9.8) and a formula. Using 8W for the thickness of this
idealized unstirred layer adjacent to a surface, ch and cm for the
concentrations of the solute in the bulk solution and at the interface (the
"membrane"), respectively, and (dcldz)m for the concentration gradient at the
interface, the thickness of the unstirred layer is defined as
u (dc/dz)m' { }
As the graph illustrates, what's being done is to extend the surface
concentration gradient linearly until the concentration reaches that of the bulk
solution. That coincidence point is then taken as the location of the
transition plane between unstirred layer and bulk solution.
One could, of course, define a velocity boundary layer in an analogous
way, extending the velocity gradient at the surface out to where it reaches
the speed of the free stream. The relative thickness of such a laminar
boundary layer would still vary inversely with the square root of the
Reynolds number as in equation (8.3). But it would be considerably thinner
than one whose outer limit is reached at a speed of 99% of free-stream
speed. From equation (8.4) it's a simple matter to calculate the thickness of
this velocity boundary layer—it comes out to about 60% of the thickness of
the conventionally defined layer. Coincidentally, this is about the thickness
of a layer whose outer limit is set at 90% of free-stream speed and whose
use was recommended in the last chapter. Thus the "diffusive boundary
layer," as the term has been defined in the studies of cutaneous respiration
in amphibians (see also Feder and Pinder 1988), corresponds to the
"unstirred layer" as defined by Barry and Diamond (1984) and to the 90%
velocity boundary layer as defined here.
Besides referring to a model with a discontinuity rather than a reality
without one, the term "unstirred layer" can be a little misleading in another
way. One is likely to presume that substantial mixing takes place outside the
layer, that all else is "stirred" with some form of cross-flow bulk transport.
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CHAPTER 9
N
O
W5
•3
[Cm!
concentration, [c]
[cbl
Figure 9.8. Defining the thickess of the "unstirred layer" for diffusion
adjacent to a membrane across which fluid is moving—the concentration
gradient at the membrane is extrapolated to the concentration of the
unaffected bathing solution.
If the unstirred layer corresponded to the laminar sublayer in turbulent
flow, the image might be fairly realistic, but the concept is ordinarily used
for physiological situations in which Reynolds numbers are sufficiently low
that flows are assuredly laminar. The region of "mixing"—one really
should put quotes around it—doesn't necessarily have cross-flow transport
of mass and momentum by turbulent eddies. It's just a region with a free-
stream concentration of solute and no gradient. With exchange across a
membrane happening in a particular place rather than along an infinitely
long pipe or surface, free-stream concentration can be maintained
wherever flow from offstage is sufficient to offset diffusive depletion. Beware of
explanations that presume or specify turbulent mixing outside the
unstirred layer—smile and calculate the Reynolds number for yourself.
A useful index for deciding whether exchange in a given situation is
dominated by diffusive or by convective transport, the Peclet number
(sometimes known as the Sherwood number), will be introduced in
Chapter 14.
Interactions among Organisms and the Gradients
near Surfaces
By now the notion that the presence of an organism almost inevitably
alters the local flow field needs no belaboring—the idea is implicit in all
that's been said about topics as diverse as interference drag, colonywide
currents, the location of ectoparasites, and variations in shear velocity and
zero-plane displacement. From the collection of references from which
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LIFE IN VELOCI TY GRADIENTS
I'm working, I find I can assemble a very large pile devoted to flow in and
among . . . well, everything from forests to eel-grass communities in
marshes. Each of these points out that the alteration of flow resulting from
the presence of some kind of large organism has a major influence on
community structure. I have another group of references that document
flow-mediated interactions among organisms—cases in which competition
for food, light, attachment space, and so forth takes the form of
competition (both intra- and interspecific) for adequate flow; schemes to use
someone else's flow (with or without detriment to the host); arrangements in
which the forces on some organism depend very much (positively or
negatively) on epiphytes or neighbors; cost-benefit analyses of relative location
with respect to flow; and systems in which various forms of chemical (and
even physical) communication are flow mediated.
Perhaps the appropriate point to emphasize at the end of what has
turned out to be quite a lengthy chapter is one that emerges from the wild
diversity of these ecological and behavioral phenomena. We've come a long
way from some idealized diagram of the distribution of flow speeds on a flat
plate as far as physical complications, consequences, and implications go.
What ought to be transparently clear without further examples or
belaboring is the still greater complexity of the biology that's built on these
complications, consequences, and implications. The central message I'd push is
that without some appreciation of the world of flow one is unlikely to
recognize what organisms are up to in this world of velocity gradients.
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CHAPTER 10
Making and Using Vortices
At one level we all know about vortices. Water doesn't just drain
IT\. downward from the bathtub, but instead goes around and around
before disappearing, like a dog circling a time or two before lying down.1
Much larger in scale are various violent storms—hurricanes, typhoons,
tornadoes—called cyclonic on account of their vortical motions. Clearly
organisms are exposed to vortices—quite beyond Edgar Allen Poe's horror
story about a boat sucked into a maelstrom. And some creatures even
contain vortices, for instance in the larger and more rapid parts of internal
fluid transport systems of large animals. In addition, they deliberately
either make vortices or use naturally occurring ones for quite a number of
useful purposes. In particular, flying and the more macroscopically
popular forms of swimming are inseparably linked to the existence and behavior
of vortices.
Like just about every other aspect of fluid mechanics, this business of
vortices is a distinctly peculiar one, although I suspect that the oddness
comes mainly from our usual lack of familiarity with the details of the
motion of transparent media. A vortex is quite unlike a rotating solid.
Neither fluid nor solid minds going in circles; indeed it can be argued that
truly noncircular motion must be a special case in any bounded system. But
the solid stolidly rotates as a solid, while the rotion of a vortex gets more
vortiginous as one approaches the axis. Let's take a few pages to build a
picture of the events and concepts surrounding vortices. Incidentally, an
entertaining and eminently penetrable book (Lugt 1983a) describes how
vortices lace together just about everything in the larger subject of moving
fluids.
Two Ways to Go in Circles
If a solid body rotates about an axis that passes through it, all bits of the
body have the same angular velocity. Consequently the tangential velocity
of any bit is proportional to its distance from the axis of rotation. You can
1 The bathtub vortex, commonly ascribed to the Coriolis pseudoforce that results, off-
equator, from the rotation of a spherical earth, is supposed to rotate counterclockwise in
the northern hemisphere and clockwise in the southern. In fact, the expectation is
realized only with the kind of painstaking circumvention of confounding factors that
precludes making relevant observations in an ordinary (Archimedean) bathtub. With no
204
M A K I N G AND USING VORTICES
hit a golf ball or a baseball hardest with the longest club or bat, assuming the
same angular velocity of swing. That's because angular momentum (the
product of mass, angular velocity, and radius squared) varies in proportion
to the square of the distance from the axis of rotation. Since angular
momentum is ordinarily conserved, a figure skater rotates more rapidly
when hands and feet are brought closer to the axis of rotation; with mass
remaining constant, any decrease in radius must be accompanied by a great
increase in angular velocity.
A body of fluid may certainly rotate about an axis in just this manner.
Spin a bowl of water and, after a little time for viscosity to get the water up
to the angular velocity of the bowl, the system rotates without any
particular pattern of internal motion. A tiny boat on the surface of the water will
go around with it, facing first east, then south, then west, then north (or, of
course, vice versa). The surface, as in Figure 10.1, is higher around the
periphery, with a shape that looks spherical but is really a paraboloid. Its
parabolic section is a result of the pressure differences caused by that
square of the radius in the formula for angular momentum. We say that the
rotating water constitutes a vortex.
But a body of fluid may move about an axis in another way, one rather
hard to persuade a solid to do. Fix the bowl of water, insert into it a cylinder
with a vertical axis, and make the cylinder rotate at a steady rate, as in
Figure 10.2. After a time, viscosity and the rotation of the axial cylinder
induce motion throughout the bowl (assumed, for simplicity, to be very
large or without outer walls). A very different situation has been set up. As
would be expected from all that was said in the last two chapters, a velocity
gradient extends outward from the inner cylinder's surface. So the angular
velocity of a bit of fluid isn't constant but now decreases with distance from
the inner cylinder. Likewise, tangential velocity now drops with distance
from the center. It does so asymptotically, specifically in a hyperbolic
fashion. As with fluid rotating without internal shear, the height of the water,
indicating relative pressure, is lowest nearest the middle and highest
peripherally. Here, though, the shape of the surface is hyperbolic, not
parabolic; and this kind of vortex looks a lot more like a whirlpool or a bathtub
drain.
What's perhaps most startling about this latter vortex is the motion it
gives a tiny boat on its surface. No longer does the boat face each of the
compass points in turn. Instead it keeps facing in the same direction, as it
moves around a short distance outward from the inner cylinder. What
small effort, Shapiro (1962) in Boston and Trefethen et al (1965) in Sydney obtained the
proper opposite rotations, but both used shallow tubs almost 2 meters across and had to
wait about 24 hours before residual motion subsided sufficiently for a consistent
direction to be obtained.
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CHAPTER 10
Figure 10.1. Making water go around in a cylindrical bowl by rotating
the bowl. At right is the water's surface profile across the bowl.
we've made is something called an "irrotational vortex." To realize that
motion around and around might be appropriately called "irrotational" is
a little jarring, but the designation is no more than proper recognition of
the difference between translational and rotational motion. In translation,
an object changes its location—it moves from place to place. Nothing in that
definition restricts the object from periodically revisiting an earlier
position. In rotation, by contrast, an object changes orientation, whether or not
it ever comes back to a place it previously occupied. That the objects here
are just arbitrarily small elements of a fluid makes no difference to this
definitional distinction.
Translation in a circle without rotation isn't something that only a fluid
can do. While the wheels and cranks of a bicycle rotate, the pedals just
translate. The flagella of bacteria truly rotate; those of protozoa merely
translate in circles. For that matter, translation around some repetitious
orbit is an almost inevitable feature of appendage motion in legged
locomotion.
That hyperbolic variation of tangential velocity is the critical item in
rendering a vortex irrotational. Consider an object some distance from the
axis of such a vortex (such as our boat). For it to maintain its orientation, the
water nearer the axis must be moving faster than the water farther from
the axis. More specifically, it can be shown that for the irrotational condition to
occur, the product of tangential velocity and radius must be constant—hence the
hyperbola.
But a constant product of velocity and radius implies that velocity
approaches infinity as the radius approaches zero—not an agreeable
prospect, although one that victims of hurricanes and tornadoes might find
credible. In practice, however, any real vortex in which the fluid extends
inward to the axis of rotation has a rotational core. Viscosity sees to that,
206
MAKING AND USING VORTICES
Figure 10.2. Making water go around in a cylindrical bowl by rotating a
smaller cylinder sticking coaxially down into it. At right is the water's
surface profile across the bowl.
since the shear rate gets greater and greater as the axis is approached. In a
sense, our original driving cylinder in the bowl has some physical reality;
one can usefully imagine the cylinder as hollow, with a rotational vortex
inside and the irrotational vortex surrounding it—essentially the
distribution of speeds shown in Figure 10.3. The hurricane does, after all, have an
eye, whose low winds give a respite before it moves farther and hits you with
a great gust from the opposite direction.
o
JO
13
>
c
60
distance from center of vortex
Figure 10.3. The variation of tangential velocity with distance from the
center of an ordinary vortex—irrotational but with a rotational core.
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CHAPTER 10
More about Irrotational Vortices
In moving fluids these irrotational vortices are the really interesting
ones. Not that rotational ones or intermediates don't occur, but these
irrotational vortices crop up under an amazingly wide range of circumstances
and are rather curious creatures. For one thing, they can be made in several
ways. A rotating cylinder will naturally surround itself with an irrotational
vortex in any fluid with finite viscosity; in fact, any other rotating object
such as a flat plate will do likewise, the only difference being the greater
complexity of the vortex it makes. Velocity gradients can make them. And,
as we'll see in the next chapter, even some nonrotating objects such as
inclined flat plates and airfoils can induce irrotational vortices.
From the skater's scheme to increase the rate of spin comes yet another
way to make such a vortex. Angular momentum, remember, is the product
of mass, angular velocity, and the square of the radius. If a bit of fluid with
some small angular momentum, that is, a little spin on it, moves inward in a
rotational vortex, it will increase in angular velocity—it will spin faster. If
lots of bits of fluid move inward, a gentle rotational vortex can be converted
into a substantial irrotational one. Remember that at any point in the
motion of a bit of fluid, its tangential velocity will be the product of its
angular velocity and the radius. Thus constancy of angular momentum
generates the requisite condition for the irrotational vortex, a constant
product of radius and tangential velocity.2
Since fluids, in our domain, are incompressible, simultaneous inward
motion of bits of fluid from all directions sounds pretty unlikely. But that's
only if one lives in Flatland (Abbott 1885)—in a properly three-
dimensional world, fluid can move axially through the middle of the
vortex. It may then either leave the system entirely, as through the drain
beneath the bathtub vortex. Or it may move peripherally again somewhere
else, perhaps to be recycled in toward the axis again, as when cold milk is
gently added to the middle of a stirred cup of hot coffee. The axial and
upward component of flow is perhaps the worst feature of the most violent
of all meteorological phenomena, tornadoes.
What happens to such a vortex if it's left to its own devices? In a friction-
less, inviscid world any vortex, irrotational or rotational, ought to persist
indefinitely. Galaxies, though perhaps irrelevant here, are not short-lived
phenomena. But friction is the great enemy of vortices, and it's at its worst
for irrotational ones. A patch of fluid spinning as if solid need only suffer
2 For instance, if inward movement halves the radius, constancy of angular
momentum means that the angular velocity will increase 4-fold With radius down 2-fold and
angular velocity up 4-fold, tangential velocity will by increased 2-fold The latter will just
offset the 2-fold decrease in radius to maintain the constant product. We are, of course,
making our usual presumption of constant density.
208
M A K I N G AND USING VORTICES
interaction with its surroundings; by contrast, in an irrotational vortex
shear is ubiquitous since angular velocity varies with radius. So it takes
energy to keep an irrotational vortex going—to offset the pernicious
effects of viscosity quite beyond the cost of any interaction with its
surroundings. Deprived of an adequate energy source, an irrotational vortex
deteriorates toward a rotational one as the original radial variation of angular
velocity decreases. Thus the distinction between rotational core and
irrotational periphery is gradually lost—in effect, the core grows at the expense
of the periphery as overall angular momentum drops.
This requirement for energy to counteract the shear stresses of viscosity
has another consequence. Where viscosity is greatest, relative to
momentum, sustaining a vortex is hardest. Therefore vortices at low Reynolds
numbers have a proportionally larger rotational core and a greater hunger
for energy—cut off their supply and they quickly grind to a halt. And at
really low Reynolds numbers proper vortices simply don't happen. I think
the minimum Reynolds number at which vortices have been produced is
about 10~2. Organisms manage to crowd the lower limit. Vorticella, a
stalked protozoan (Figure 10.4) beats its cilia in a way that puts it in the
center of a large, food trapping vortex (Sleigh and Barlow 1976); the
Reynolds number is of the order of \0~l.
One can view fluid motion as having three domains. In one, the world of
turbulent flow, vortices form so readily that there's little else around. In a
second, vortices occur but need substantial provocation and are relatively
orderly and free of smaller vortices within themselves. In the third, the
phenomenon of a vortex is simply out of the range of practicality except
inasmuch as something might be immersed in a larger one. Note that the
latter two are both characterized by laminar flow—under conditions of
laminar flow vortices are often but not always possible. In a sense they're an
example of the kind of quasi-periodic state that characterizes the transition
region between order and chaos in many physical domains (Van Atta and
Gharib 1987).
So, high Reynolds numbers are propitious for vortices. What would
happen at a Reynolds number of infinity, in that impossible but
occasionally revealing idealization called an inviscid or ideal fluid? In a world of
unbounded inviscid fluids vortices would have a curious sort of quasi-
reality. We used viscosity to get ours started, and it turns out that nothing
else will do it. On the other hand, once somehow started, without viscosity a
vortex would last forever since there's also no way to stop it. Since no
inviscid fluids occur anyway, the fact that such a fluid couldn't generate a
vortex might be said to make the situation no less real; and quite a bit of
analytical work has been done on such vortices in inviscid fluids. In fact,
many of the conclusions prove quite useful in understanding what goes on
in the world of real, viscous fluids, so we're not just snickering at games
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CHAPTER 10
Figure 10.4. The unusually small toroidal vortex with which the stalked
protozoan, Vorticella, draws edible particles within reach.
played by people whose fascination with mathematics distracts them from
science.
In an unbounded, inviscid fluid a vortex not only has no temporal end
but it has no spatial end either. The only way to stop the thing this side of
infinity is to loop it back on itself in the form of a closed loop—whether a
tidy torus, some pair of elongate whirls, or a complex snarl is immaterial
(Figure 10.5a). That's very much what happens in the real world, where
vortex rings (smoke rings and so forth) are much more the rule than the
exception. A droplet of colored water released just above the surface of a
larger body of water will usually form a vortex ring as it falls. A pulse of
fluid forced out of a sharp-edged orifice into a larger body of fluid will
almost always make a vortex ring. If you imagine that some system contains
a vortex, you ought immediately to ask about its ends. If you observe a whirl
in some two-dimensional view, then you ought to be alert to the location of a
twin that makes ends meet. And, as reference to the figure or a moment's
consideration should reveal, the members of any such pair of vortices
should have opposite directions of circulation (whoops, I almost said
rotation!)—a pair is really just a slice of a vortex ring.
Since vortices (at least when viewed in section) commonly occur in pairs,
one is naturally curious about interactions between the members of such a
pair. As it happens, a pair with opposite circulations will repel each other.
On the sides facing each other, the flows will be in the same direction, so the
relative speed of flow will be minimal; by Bernoulli's principle, pressure
will therefore be high. The closer they are, the greater the repulsion, so a
vortex ring will tend to form a proper circular torus.
A toroidal vortex, besides rounding up nicely, likes to move through the
210
MAKING AND USING VORTICES
<
<
Figure 10.5. Vortex rings: (a) a single torus; (b) a travelling torus;
(c) a pair of vortices, one of which has just passed through the other.
fluid in which it lives. What drives the motion is shear in the extensive
region of contact between the periphery of the vortex and the rest of the
fluid—shear is diminished if the vortex moves axially in a direction such
that the fluid on its outside is going at a speed closer to that of the rest of the
fluid, as in Figure 10.5b. Thus fluid on the inside of the torus moves
forward relative to the movement of the torus as a whole, while fluid on the
outside moves backward relative to the overall movement. So this inner fluid
moves in the same direction as the torus to which it belongs, but it moves at
a higher speed. To put the matter another way (and one that will take on
relevance when we talk about soaring), in the inner part one could fly or
swim against the current indefinitely without getting out of the toroidal
vortex of which the current is part. This kind of vortex ring does slow down
over time if not maintained by some force such as might be provided by
buoyancy—it gets bigger by entrainment of surrounding fluid and slower
by loss of its original impulse, as described by Shariff and Leonard (1992).
Despite their predilection for rings and loops and hence, in effect, pairs
of opposite circulation, a vortex can sometimes encounter another that has
the same direction of circulation. Here, instead of being repulsive and
essentially stabilizing, the interactions are attractive, even salacious. Flows
are opposite on the sides that adjoin, and the biologist can easily recognize a
basis for attraction. But the biological analogy doesn't completely apply—
two small vortices enwrap each other but the normal result is merger into a
single larger one. A particularly odd interaction is what's been called
"leapfrogging" (Shariff and Leonard 1992). If two vortex rings have the same
rotational sense they will travel in the same direction. If one follows
another of the same size, the rearward one will catch up and attempt to pass
21 1
CHAPTER 10
through the forward one. If it succeeds (without merging), then the new
follower will in turn catch up and try to squeeze through the middle of the
new leader. (See Figure 10.5c.)
Making and Using Vortices near Interfaces
Biologists needn't worry much about whether or how vortices might
form in an unbounded fluid—for most of us what matters is the immediate
vicinity of some solid substratum or air-water interface. In such places,
provided Reynolds numbers are decently above unity, vortices are
extraordinarily easy to make. Indeed, they're the rule rather than the exception,
and what takes careful design is avoidance rather than instigation.
Streamlining, for instance, is in one sense a matter of preventing the formation of
vortices associated with and just downstream from a separation point—
bluff bodies form and shed more and bigger vortices than streamlined
ones. As a general rule, anywhere there's a z-ward gradient of velocity one
ought to be alert to vortex formation. The matter came up in the last
chapter for spores rolling across leaves, where the side of a particle farther
from the surface met a higher flow speed, and the whole particle was
therefore set into rotational motion. The situation applies to bits of fluid in
quite the same way, and it applies whether or not either chunk or bit
actually touches the surface. Very simply, when the lines of action of
resistance to motion (resulting from the no-slip condition) and the main
motion-inducing push (from the free stream) don't coincide, a turning
couple is produced, as we saw in Figure 9.4. Let's look at some of the
possible vortex-inducing circumstances at interfaces and the forms of the
vortices to which they give rise—all have potential biological relevance
even where documented cases are at this point lacking.
Elongate Grooves
Consider flow across a long furrow or trench in the substrate (Figure
10.6a). The resistance of the fluid to shear between what's inside (and
initially stationary) and what's outside (and moving) will cause the fluid
inside to rotate in a vortex that runs the length of the furrow. The situation
came up briefly in Chapter 4 (specifically, in Figure 4.12) in connection
with viscous entrainment and the movement of fluid out of a hole in the
substratum, and again in Chapter 8 in connection with the issue of
skimming flows. Here attention is directed to the existence of persistent
vortices, with all the attendant possibilities of particle resuspension, enhanced
material exchange, induced flow in and out of porous substrata, and so
forth. Such vortices should be expected whenever air or water flows across
corrugated surfaces, whether plowed fields or rippled sand on dunes or the
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MAKING AND USING VORTICES
Figure 10.6 Places where vortices happen: (a) flow across a furrow
(b) flow making a turn upstream from a sharp corner; (c) flow just
behind a sharp cross-flow edge; (d) flow across and within a pit or cup;
(e) internal circulation in a fluid drop falling through another fluid;
(f) secondary flow just downstream from a bend in a cylindrical pipe.
bottoms of streams or oceans If the furrow or trench is narrow but deep,
the vortex can even generate another beneath it, one circulating in the
opposite direction. The phenomenon isn't limited to long trenches in
otherwise smooth surfaces across which fluid flows; the same vortex induction
can take place in a sharp corner, where a vortex (or occasionally a set of
vortices of diminishing diameters) is generated by fluid making a gentle
turn (Figure 10.6b). And it can take place behind a sharp corner as a form
of separation of flow (Figure 10.6c); this last case is relevant to flow around
the rear edges of airfoils tipped substantially with respect to the free
stream.
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Cups and Funnels
A trench makes a nicely two-dimensional case; with the inescapable
three-dimensionality of a hole in the substratum (as in Figure 10.6d) things
get a little more complicated. Blow gently across a full cup of coffee with a
little unstirred milk in it to mark the flow. Coffee will move downwind
across the part of the surface that's widest in the direction you're blowing.
But it may move upwind on each side of that widest portion, indicating that
something other than a vortex with a horizontal axis has been set up. What
happens is that the area of contact between fluids is most extensive in the
middle, so there intei fluid shear is especially effective in making the coffee
move. Conversely, the portions lateral to it are influenced by shear with the
walls of the cup as well as by the motion of breath and midstream coffee. As
a consequence the circulatory axis bends upward on either side, forming a
torus that's sliced off at the coffee-exhalation interface.
This pattern in which flow is downstream across the (free-streamwise)
widest portion of a cup and upstream to either side is particularly easy to
form in a conical cup. According to Brodie and Gregory (1953), it's put to
use by a lichen, Cladonia podetia, which bears its propagative elements,
soredia, on the insides of upright cups. The cups don't just keep soredia
from falling to the ground—the wind-induced vortices in the cups are
important in getting the soredia out. They found that a free-stream wind
of 2 m s~1 removed soredia from cups but was inadequate to blow them off
a flat surface. Using model cups and lycopodium powder (especially
appropriate for an investigation of lower plant dispersal) they found that the
system cared little about whether the wind was smooth or turbulent and
that a cone with walls diverging about 60° was most effective. The same
system seems to work in at least one slime mold, where wind at no more
than 0.5 m s~l proved adequate to blow spores out of cupulate sporangia.
These shear-driven vortices in cups and funnels are sites of enhanced
deposition of low-density suspended material—they're self-full-filling
feeding troughs; the process has been described by Yager et al. (1993) and
will reappear in a few pages in connection with a mayfly larva.
The Insides of Droplets
A complete and regular torus occurs in a fluid droplet if another,
immiscible fluid blows across it. That happens, among other instances, when a
raindrop falls through air at Reynolds numbers above about 10 (Figure
10.6e). The passage of air upward around the periphery of the drop
induces upward flow of water at the sides of the drop; that in turn forces
downward flow in the center. The overall effect, aside from the mixing, is a
little like a violation of the no-slip condition at the droplet's surface—the
surface itself goes with the flow. And that very slightly decreases the drag
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MAKING AND USING VORTICES
relative to what a solid body would experience. The effect is more
substantial for a droplet of gas ascending in a liquid; as we'll see in Chapter 15, the
internal toroidal circulation can reduce the drag by up to a third from what
Stokes' law predicts. In a large bubble of gas this internal circulation
induced by passage through the water may be sufficient to deform it into a
proper torus.
The Wakes of Jets
As mentioned earlier, another way to form a toroidal vortex is by
expelling a pulse of fluid in a jet into an otherwise stationary body of the same
fluid—one can make something that initially looks quite a lot like ajellyfish
with an eyedropper of milk and a glass of water. The fact that the structure
appears a little ways from the dropper is exploited, it appears, by cuttlefish
and other ink-squirting cephalopods. A persuasively writhing blob is a
particularly nice target to offer a predator if you're decently distant from it
and if that distance is progressively increasing. And increase it will, as a
result both of the opposite momenta of blob and creature and of the
normal progression of a vortex ring.
Bent Pipes
Quite a different situation in which interfacial velocity gradients
generate vortices involves flow through pipes that go around curves. In a pipe,
flow is (as will be elaborated in Chapter 13) fastest along the axis and
nonexistent at the walls. That means that momentum is greatest on the
axis. So in a bend the axial fluid turns less readily than the peripheral fluid,
and it moves toward the wall on the outside of the curve. That forces
peripheral fluid centripetally, and thus fluid leaves a bend with a pair of
vortices distorting the normally parabolic distribution of speeds across a
pipe's radius (Figure 10.6f). Pipes with curves are certainly not uncommon
items in organisms—you need look no further for an example than your
own aorta. A good introduction to vortices in and downstream from bends
is given by Berger et al. (1983).
The Rears of Cylinders
Back in Chapter 5 we considered flow around circular cylinders in
connection with the cylinders' oddly bumpy curves of drag coefficient versus
Reynolds number. Vortices are generated on either side of a cylinder
whose axis extends across a flow as part of the separation process (Figure
5.5)—at least above a Reynolds number of about 10. The pair of vortices
are of equal strength (we'll get to the matter of quantifying the intensity of
a vortex shortly) and of opposite signs. But under ordinary conditions this
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CHAPTER 10
nicely symmetrical system is unstable above Reynolds numbers of around
40, and the vortices are alternately shed into the wake to form the Von
Karman trail mentioned earlier. (Goldstein 1938 has a good discussion of
how the instability comes about.) So this is yet another way of generating
vortices, one that might be important on account of the prevalence of
cylindrical elements in organisms, but one that, at least for a stable pair,
seems limited to a narrow domain.
In fact, stable, paired vortices behind cylinders may occur at higher
Reynolds numbers than general books on fluid mechanics state. Goldstein
(1938) cites experiments where testing in narrower channels postponed
the start of shedding—for instance, where the channel width was only ten
times the diameter of the cylinder, shedding didn't occur below Re = 62.
More or less equivalent to a narrower channel would be other cylinders
with parallel axes across the flow. In addition, shedding is discouraged
when a cylinder extends upward from a solid surface through a velocity
gradient (Lugt 1983a). Either way (or both) the proximity of surfaces has a
stabilizing effect. In casual observations in a small flow tank, I've found that
tipping a protruding cylinder back with respect to the flow has a further
stabilizing effect. So one catches the odor of biological relevance—a wider
range near interfaces, Reynolds numbers appropriate to the situations of
quite a few organisms, an erect cylindrical structure that we commonly
find, a posture consistent with what unstiff structures would normally do in
a flow, potential for communal advantage, and so forth.
A cylinder sticking up from a substratum but still largely located in the
boundary layer of that substratum has another feature of interest both
fluid mechanically and biologically. Flow in its vortices has a strong upward
component in addition to its circulating motion—fluid enters near the
substratum and leaves near the top. Vortex shedding may be limited to the
top, where it can be anything from a conventional Von Karman trail to a
pair of vortices bending rearward and detaching in tandem, as in Figure
10.7a.
And creatures demonstrably use these ascending, paired vortices. We
talked about suspension feeding quite a lot in the last chapter, but we've not
yet put in a word for another way in which an animal can acquire minimally
motile, particulate food. That's detritus feeding, taking advantage of the
fact that many things worth eating are a little denser than water and tend to
settle out on any available substratum. The main problem facing the
detritus feeder, the rate of deposition of its resource, is at least partly
circumvented in a flow by transport across the substratum. Such transport in and
by the slow flow just above the surface may have the additional benefit of
sorting lower-density edibles, higher up in the flow and thus faster, from
higher-density inorganic particles, lower and slower. Several cases are now
known in which erect, tubular organisms resuspend low-density particles
2 16
MAKING AND USING VORTICES
Figure 10.7. (a) Ascending vortices behind a cylinder that protrudes
from a substratum, (b) How a larval black fly uses an ascending vortex to
feed on detritus with one cephalic fan while suspension feeding with the
other.
in the vortices, which bring the food up to structures that one would
ordinarily regard as suspension-feeding devices. One (worked out by
Carey 1983) involves a marine terrebellid worm, Lanice conchilega, that
makes a tube a few centimeters high and about half a centimeter wide
emerging from a sandy substratum. The head of the worm, with an array
of tentacles of which Medusa would have been proud, sticks out the upper
end of the tube; and the worm feeds on suspended material delivered by
the flow. At the same time, it takes food from the ascending vortices—
detritus feeding and suspension feeding with the same equipment. A pho-
ronid worm (not an annelid), Phoronopsis viridis, operates similarly
(Johnson 1988), again using suspension-feeding equipment to feed on detritus.
These creatures live in dense populations, just short of densities that would
produce skimming flows; and, according to Johnson (1990), the communal
arrangement enhances turbulence and upward water flow and thus the
feeding of all the individuals.
An analogous but fancier case (also including communal augmentation)
is that of the black fly larvae (Simulium vittatum) of rapid streams that have
appeared earlier in several related contexts. These attach to rocks at their
posteriors and stick up in the form of tapering cylinders that terminate
with two cephalic fans. Chance and Craig (1986) showed that they twist
lengthwise and bend so that the anatomically paired cephalic fans end up
not on either side in the flow but instead above and below each other
(Figure 10.7b). The top fan is a normal suspension feeder, but the bottom
one (still near the top of the erect larva) feeds from a vortex that has
ascended from the substratum and will subsequently detach into the wake.
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CHAPTER 10
Matters are complicated a little by the fact that larvae are bent a bit across
the flow (as well as downstream) so the paired vortices are above and below
as well as side by side—only the upper one passes through the lower fan.
Besides their role in larval feeding, the vortices are put to respiratory use
by the pupae, which stick their gill filaments into them (Eymann 1991).
The Vorticity of Velocity Gradients
At the start of this chapter we distinguished between rotation and
translation, noting that any element of a vortex not including the axis (outside
the core) usually translated in circles without rotation even though the
vortex as a whole was obviously rotating. That was possible because fluids
permit internal shearing motion, in this case slip between concentric tori.
Under another circumstance as well the occurrence of internal shear
implies that vorticity is present in a fluid. Well over a century ago, Sir George
Stokes explained the matter as follows. One suddenly solidifies a tiny
sphere of fluid and looks to see whether it's rotating; if so, the fluid is said to
have vorticity. Consider an ordinary velocity gradient of the kind that
occurs adjacent to planar solid surfaces. As mentioned a few pages back,
because of the shearing motion in the gradient, such an element in such a
gradient will be subjected to a turning couple (Figure 9.4). Thus vorticity is
normally associated with a velocity gradient.
The presence of a vortex implies that a fluid has vorticity (even if the
vorticity is concentrated as a vortex line running along the axial core of the
vortex); on the other hand, the presence of vorticity isn't a perfect
indicator of vortices. While vortices don't happen without vorticity, with vorticity
they are likely but not inevitable. Do boundary layers generate vortices?
Quite often they do, with all sorts of effects on bed erosion, cross-flow
mixing, particle transport, and so forth. Much of the puffiness of gentle
breezes at the surface of the earth represents the passage of what might be
called "rollers"—vortices where the boundary layer has, as it were, rolled
up. The resulting periodicity is certainly conspicuous in the motion of a
field of tall grain or grass. One might naively expect that in an eastbound
breeze (Zephyrus by name) the flow near the actual surface might head
westward, but that's just a misapprehension caused by a stuck-in-the-mud
frame of reference. Remember that the rolling vortex was formed as a
result of variation in the intensity of the eastbound flow, so even the lower
part will normally head that way, albeit more slowly. Light-wind lake sailors
know that as a rolling vortex approaches, an especially calm moment
typically precedes the puff.
We noted a bit earlier that vortices didn't easily form ends within a fluid.
Together with the present business about vorticity near a surface, this
creates some very odd phenomena (or sorts out some very ordinary
phenomena that are otherwise hard to rationalize, depending on your point of
2 18
MAKING AND USING VORTICES
view). Imagine vortices rolling (or vorticity flowing) along a surface from
which something protrudes. To get around the protrusion requires an
interruption of the vortex. But vortices don't like to make even temporary
ends any more than a skier (as in a famous Charles Addams cartoon) can
pass a tree with one track on either side. So they get tangled around
protrusions, with the part just upstream of the protrusion staying in place. As a
result snow counterintuitively fails to accumulate on the upwind side of a
tree—the trapped vortex has a locally downward direction at the tree's
surface and scoops out a hollow. Similarly, if you stand facing the ocean in
the swash on a beach, after passage of a few episodes of backwash you
notice your heels sinking in. The receding water, with a severe velocity
gradient, has scooped out sand from the upstream side of your personal
protrusions. Somewhat less obviously under most circumstances, the
vortex that's doing the digging bends around tree or leg to take on the shape of
an elongated horseshoe, as in Figure 10.8, with regions of scour (upstream
and laterally rearward) and regions of deposition (just downstream and
farther laterally). Such horseshoe vortices, with erosion and deposition,
happen around scallops in scallop beds, with implications for passive
transport and other processes (Grant et al. 1993).
One consequence of such vortices is that an erodible bed beneath moving
air or water may not necessarily be stabilized by worm tubes or emergent
vegetation. Only if the protrusions are sufficiently dense will skimming
flow, mentioned earlier, have a stabilizing effect. In practice, "sufficiently
dense" means a cross-sectional area parallel to the substratum more than
about a tenth or twelfth of that of the substratum (Eckman et al. 1981;
Nowell and Jumars 1984). At lower densities, protrusions usually have a
Figure 10.8. The action on an erodable bed of the "horseshoe vortex"
that's associated with a protruding object in a shear flow, here taken from
data of Grant et al. (1993) showing what happens beneath a sedentary
scallop.
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CHAPTER 10
destabilizing effect. And the destabilization generates exceedingly
heterogeneous conditions on the surface, with—as Eckman (1983) has shown—
major effects on both nonbiological processes such as sedimentation and
on biological ones such as larval recruitment.
Nor are such general considerations the extent of the biology associated
with these horseshoe vortices. Soluk and Craig (1988) have described the
behavior of a mayfly larva (Ametropus neavei) that lives in the surface layer of
unstable, sandy sediment on river bottoms. The insect seems to require a
flow in order to feed; given a flow, it faces upstream and excavates a shallow
pit in front of its head. A vortex forms in the pit, its formation aided by the
postures of forelegs, head, and antennae; and the larva suspension feeds
from the vortex (Figure 10.9a). The vortex seems to assist feeding in
several ways—it deflects fluid downward, enabling the animal to remain
largely within the substratum and sheltered from hydrodynamic forces
that might otherwise preclude life on a soft bottom. It also increases the
residence time of suspended particles within reach of the forelegs and
their filtering setae. Also under some circumstances the combination of pit
and vortex may act as a depositional trap and resuspension device so
Figure 10.9. Two mayfly larvae that take advantage of the upstream
(center) part of a horseshoe vortex. Ametropus (a) makes a pit for its
vortex, which then traps edible suspended particles. Pseudiron (b) arranges
itself so the vortex digs the pit while it eats any excavated prey. It moves
slowly backward (downstream) so excavation proceeds continuously.
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MAKING AND USING VORTICES
suspension-feeding equipment can effectively detritus-feed as well, as we
saw earlier with terrebellid and phoronid worms and the black fly larva.
Another mayfly larva (Pseudiron centralis) of a similar habitat plays a
different variation on the horseshoe vortex theme. It feeds on larval chi-
ronomid flies that live in the sands, but it hasn't any decent digging
appendages. According to Soluk and Craig (1990), the insects are dependent
on erosion to expose prey, but they make sure that erosion happens when
and where they want it. With adequate current, larvae arch their bodies
and lower their heads and thus create the conditions for vortices to dig out
upstream pits. From these they seize any prey that get exposed (Figure
10.9b). Excavation is a continuous process, so as a Pseudiron larva moves
slowly backward it leaves a shallow groove as a trace. By contrast with
Ametropus, Pseudiron lets the current do all the work.
Thermal Vortices
We talked a few pages ago about toroidal vortex rings, but mainly about
small ones. They exist on much larger scales as well; indeed, the mushroom
cloud of a nuclear explosion is a partially formed vortex resulting from the
interaction of rapidly rising hot gas and the surrounding atmosphere.
More benignly, ascending vortex rings commonly form when winds are
light and the ground heats the lowest part of the atmosphere. Such hot air
beneath cooler air is unstable. Here and there a bubble of hot air will detach
itself and rise, forming, in a way now familiar, a vortex ring that ascends
because of its buoyancy and the normal axial motion of such a structure.
"Here and there" may not be entirely random. A tree in otherwise low
vegetation can generate a convective updraft as a result of the heating of its
leaves (Gates 1962 describes the process); that updraft can initiate these
thermal vortices (usually simply called "thermals"). A plowed field, more
absorptive of solar radiation than surrounding vegetation, provides a
common site for initiation of thermals. Highways through vegetated areas are
similar initiation sites. The events involved in starting a thermal vortex are
described in most books on microclimatology (see the references on
planetary boundary layers in Chapter 8) or where these structures take on
biological relevance, as in Pedgley's (1982) Windborne Pests and Diseases.
The most conspicuous users of thermals are large terrestrial birds that
soar. The process (Figure 10.10) is nicely described by Cone (1962) and
Pennycuick (1989). Since air in the inner part of the circulating and
ascending torus is moving upward faster than the overall system, a bird can be
descending with respect to the local air while still ascending with respect to
the ground. What it must do is to keep turning with a sufficiently narrow
radius to stay in that inner, locally ascending, part of the torus; large
raptorial birds such as hawks, vultures, and eagles are quite good at such
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CHAPTER 10
Figure 10.10. Thermal soaring. A bird (here a bit larger than life) can,
by circling within a rising vortex ring, descend with respect to the local
air but remain within the ring and ascend with it.
maneuvers. Human-carrying gliders may have better overall lift-to-drag
ratios (more about these in the next chapter), but they travel faster and
can't turn as sharply, so they either require larger thermals or suffer a
shorter residence time in each. Where I live, thermal soaring is common,
especially among vultures; and the sight of one or more often marks the
location of a freshly cleared or plowed field beneath. Major highways,
cutting swaths through our local forests, may provide a bonanza for such
birds—a bird flying along such a road on an appropriately sunny day of
low wind will sooner or later encounter a thermal. It could then take the
elevator up and glide slowly downward until running into the next
thermal—all the while watching for the road-kills thoughtfully provided
by the same highway.
These thermal vortices also seem to be used for transport by quite
different kinds of creatures. Some spiders and moths (most notoriously the
gypsy moth, Lymantria dispar) exude long silk threads out into a breeze
when in their first instars (the immediate posthatching stage). Under
particular meteorological conditions they then let go of the substratum and
drift off—the phenomenon is called, a bit inappropriately, "ballooning."
They can go remarkable distances—Darwin noted a landing of spiders on
the Beagle 60 miles from the coast of South America. Now no amount of
thread can produce true buoyancy—what these tiny spiders and
caterpillars are gaining is drag. Which they get in abundance: adding a thread of
less than 0.1% of body weight quadruples the drag of the system
(Humphrey 1987; Suter 1991). Release into rising air might just be a chance
occurrence, but quite clearly it's not. After climbing as high as possible
(Tolbert 1977) they wait for the right sort of atmospheric instability. One
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M A K I N G AND U S I N C, VORTICES
preferred set of conditions is low wind and a rapid rise in temperature on a
sunny day after a cold night. These are conditions, as Vugts and Wing-
erden (1976) pointed out, that favor the production of thermals. Neither
spiders nor caterpillars select especially high winds, as they would if simple
wind-drift were the objective: ballooning ceases when winds exceed about
3 meters per second.
To bring the story around to some topics that came up earlier, ballooning
spiders find still another condition auspicious for flight—a steep vertical
wind gradient, that is, a high value ofdU/dz (Greenstone 1990). That implies
turbulence near the ground and, more particularly, rolling vortices with
horizontal axes. Greenstone suggests that the loose silk strand ought to
permit a spider contemplating takeoff to determine the local wind
direction and thus pick the side of the vortex (the rear) in which air is ascending.
But a somewhat more startling suggestion was made to me by Lloyd Tref-
ethen. He figures that a silk strand won't follow streamlines in a vortex since
it resists extension. He also points out that thermal vortices have some
horizontal motion around their axes as well as the toroidal circulation. In
combination, spiderlings ought to be drawn in toward the center, where air
is ascending faster than the overall vortex, and they ought to be able to do
the same trick as the soaring vulture—to descend locally while ascending
globally. The stunt should work, I think, in Greenstone's rolling vortices as
well, with the spider acting as the gravitational analog of a sea anchor
beneath a thread that gets wound into and pulled along with the vortex,
whose axis is always just ahead.
Something like ballooning in aerial arthropods may happen in marine
mollusks. A young postlarval mussel (Mytilus eduhs) produces a byssus
thread of the kind that will later be used for anchorage (the "beards" one
cleans off mussels before cooking), but a single long one instead of a whole
pelage of tie-downs. A mussel half a millimeter long may produce 70
millimeters of thread with a diameter of about a micrometer. These
threads are clearly used to increase drag during passive dispersal
(Sigurdsson et al. 1976), and mussels may be capable of rapid deployment
of thread and quick detachment into upward currents (Lane et al. 1985). A
gastropod (Lacuna spp.) may manage something similar using a stretchy
mucous thread and active foot-raising to release itself into oscillating water
currents (Martel and Chia 1991).
On a very much larger scale, locust swarms usually move downwind.
Given the normal vortical motions of the atmosphere, that will move them
toward low-pressure areas, whose updrafts generate a rain and new
vegetation (Rainey 1963; Pedgley 1982). Such movement, though, seems to
require ground cues; and just how locusts (either the desert locust, Schis-
tocerca gregaria, or the migratory locust, Locusta migratoria) accomplish the
requisite navigation (Baker et al. 1984) is still unclear.
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Figure 10.11. Benard cells, a packed array of density-gradient vortices.
You can make them yourself by mixing a little pearlescent liquid soap
with water in a dark frying pan and, after allowing flow to stop, heating
the pan very gently and uniformly from beneath.
On a very much smaller scale, density-gradient vortices occur in small,
shallow vessels of liquid. If a pan of water is very gently and uniformly
heated from below, the same kind of instability occurs that made thermals
in the atmosphere. A pattern of point upwellings and peripheral sinkings
such as that in Figure 10.11 often develops—they're called "Benard cells"
(or "Rayleigh-Benard cells"). In addition to thermal gradients, cultures of
motile microorganisms can generate these cells. What's required is for the
organisms to be negatively buoyant so they passively sink, but at the same
time to be either negatively geotactic or positively phototactic so they
actively swim upward (Pedley and Kessler 1992). Laboratory cultures thus
develop local areas of concentration, rather like what happens in the Lang-
muir circulations near the surfaces of lakes and oceans (Chapter 17).
Sometimes these "bioconvective" structures can be seen in dense algal blooms in
small natural puddles.
Circulation and Vorticity
If something is constant, it gets named. That's no more than a reflection
of the way we look for elements of order and rationality in a superficially
messy and irrational universe. The constant product of speed and distance
in an irrotational vortex is given the name "circulation," and its value
provides a measure of the intensity of such a vortex. Put properly,
circulation (capital gamma, T) is the product of circumference and tangential
velocity; thus for any streamline surrounding the core of an irrotational
vortex,
T=2irrUr (10.1)
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MAKING AND USING VOR'I ICES
Circulation has dimensions of distance squared per unit time.3 It turns out
that the value of the circulation doesn't depend on going around the core
on a single streamline. So one can give a more formal and more general
definition of circulation as the line integral of the component of velocity on
and tangential to a closed curve lying entirely within the fluid:
r = J Utdl. (10.2)
The value of the circulation defined in this way comes out the same for
any closed loop that encloses the core of an irrotational vortex. And it
comes out to zero for any closed loop that doesn't encircle the core. Very
tidy, except that Ut isn't the most convenient thing to measure inasmuch as
it requires determination of both speed and direction without a handy
shortcut to either. I can't recall any instance in which a biological
investigation used it as a primary datum. Still, the notion of circulation as a proper
quantitative variable is important, and the concept plays on a major role in
explaining the origin and magnitude of lift.
Another odd notion, vorticity, came up a little earlier. It, too, is defined
as a properly quantitative variable, one even more abstract and general
than circulation. In explaining how a boundary layer, with shear but with
all flow going in the same direction, could have vorticity, we invoked Stokes'
device of "freezing" a tiny bit of fluid and looking at whether it rotated.
That's the basis of one definition of vorticity—the angular velocity of
matter at a point in space. In slightly different terms, vorticity is the circulation
around an infinitesimal circuit divided by the area of that circuit4 (and
hence it has dimensions of time-!). Vorticity is an enormously important
concept in theoretical fluid mechanics, but it's inseparable from a level of
mathematical formality quite beyond both the level of this book and the
sophistication of its author; in short, we won't make specific use of it, and
the interested reader is referred to the standard textbooks of fluid
mechanics.
The Origin of Lift
Let's return to our rotating cylinder sticking down into a body of fluid
(Figure 10.2). In an otherwise stationary fluid it will be surrounded by an
irrotational vortex whose speed distribution was given in Figure 10.3 and
the strength of whose circulation was defined by equation (10.2). Now
3 As do both kinematic viscosity and the diffusion coefficient. No particular
relationship between the three is implied, and the coincidence is best regarded as coincidental.
4 This latter, the formal definition, makes vorticity equal to twice the mean angular
velocity of particles at a point.
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CHAPTER 10
imagine superimposed upon this circulation an additional motion, a
translation in a line, either by moving the rotating cylinder sideways through the
fluid or by moving the fluid past the cylinder. No surprise—the cylinder
now incurs a certain amount of drag. But the force on it has an important
difference from most of the cases of drag that we've previously considered.
Apparently as a consequence of the rotation—for what else could be
responsible?—the direction of the force is no longer exactly in a direction
opposite that of the motion. For convenience (that's all, really) we can
resolve the force into two components. One is the familiar drag, the force
component opposite the motion that pushes the cylinder downstream; the
other is a component normal to the flow, what we define as lift, pushing it
across the flow. (It's important to keep in mind this definition of lift as a
force normal to flow. In the vernacular, "lift" implies an upward, contra-
gravitational direction that is often at variance with the present, technical
use. We really need two terms, and we'll use "upward force" for the
everyday version of lift, reserving "lift" for use as just defined.)
A look at the resulting streamlines (Figure 10.12) clarifies what's
happening in this superposition of rotation and translation of a cylinder. On
one side of the cylinder the two motions in the fluid oppose one another, so
the velocities are lower and the streamlines are farther apart. On the other
side, the motions are additive, velocities are increased, and the streamlines
are closer together. By Bernoulli's principle pressure will be elevated on the
side where flow speeds are lower and will be reduced on the side where the
speeds are higher. Thus a net pressure or force will act in a direction
normal to the free-speed flow—in short, lift. Note, though, that while
circulation is fundamental in generating lift, no actual fluid particle need
travel all the way around the cylinder. A limited amount very close to the
surface may do so as a result of viscosity, but not enough to merit more than
mention.
Upon what does the magnitude of this lift depend? Within certain
practical limits, it's directly proportional to the rate of rotation of the cylinder—
to the circulation, which is really why the concept was introduced. It's also
directly proportional, again within limits, to the speed of the cylinder's
translational motion. A formal statement, the Kutta-Joukowski theorem,
puts it in tidier form: "If an irrotational air stream surrounds a closed
curve with circulation, a force is set up perpendicular to that air stream,
and the force (per unit length) is the product of the fluid's density, the free-
stream velocity, and that circulation":
- = pc/r. (10.3)
This phenomenon, the lift of a rotating cylinder moving through a fluid, is
called the "Magnus effect," after H. G. Magnus (1802-1870).
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MAKING AND USING VORTICES
translation
circulation
+
lift
t
Figure 10.12. If a solid body such as a cylinder rotates as it translates
(here right to left) through a fluid, the resulting asymmetry of flow
generates a force normal to the free-stream flow. We call the force lift.
The Magnus effect (at a little lower intensity) works for spheres as well as
for cylinders. It's a really big deal in sports in which spheres are thrown, hit,
or otherwise put into motion since (except for a golf slice) a confusingly
nonstraight course is distinctly meritorious. Two pleasant books on such
contemporary compulsions are Sport Science, by P. J. Brancazio (1984), with
good references, and The Physics of Baseball, by R. K. Adair (1990), with
more on the Magnus effect specifically. Incidentally, according to the latter,
outfielders find balls with some spin more predictable than those
without—baseballs have stitched seams that protrude a little, and so the
paths of nonspinning ones can be even worse.
Flettner Rotors
The Magnus effect can be put to more serious use. Aircraft have been
designed with rotating cylinders sticking out of their fuselages in place of
conventional wings. Rotation of the cylinders so that the upper side moves
rearward will generate—engines willing and plane moving forward—an
upward force to sustain the craft in the air—if conventional airfoils didn't
yield so much lift with so little drag we might use such aircraft. Actual
application of the Magnus effect was made in several ships that had one or
more large, rotating, vertical cylinders in place of masts, built in the early
1920s by Anton Flettner. The last of these, the Buckau, displaced 550
metric tons and had fore and aft rotors 3 meters in diameter and 16 meters
high that rotated 100 times per minute (specifications and pictures are
given by Herzog 1925 and in Flettner's fascinating autobiography, 1926). A
sailboat that required an engine to sail sounds like the worst of both worlds,
but the idea would have been quite viable twenty or thirty years before.
Hybrid sail and screw freighters were at that earlier time quite common,
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CHAPTER 10
since coal was bulky and not cheap at out-of-the-way ports. A Flettner ship
used a lot less coal than a propeller-driven steamer; at the same time the
sailing apparatus required far fewer crew (with attendant space and
facilities) to operate. Flettner claimed that in a storm the rotors caused a ship to
heel less than did bare, sailless rigging. What made Flettner's design
unattractive were the easy availability of liquid fuel and a shipping glut in the
'20s; it has since been consigned to textbooks to provide a little levity in the
long series of abstractions leading to lift and airfoils. There is, by the way,
nothing special about the use of a cylinder; Hoerner (1965) says that a rod
of X-shaped cross section works as well.
And when, as McCutchen (1977a) has pointed out, Nature anticipated
Flettner, she hasn't used cylinders. Flat plates, sometimes with lengthwise
twist, take their place; and the necessity of a rotating joint is avoided by
having the entire craft rotate about its long axis. These rotorcraft are
certain of winged seeds (or samaras), gliding down and away from the
parent tree by autorotating. In the tree-of-heaven (Ailanthus altissima) the
samaras are blades about 3 to 4 cm long, twisted lengthwise, with the seeds
in thickenings in the center (Figure 10.13a). When released they rotate
about their long axes and either travel away from the parent or descend in a
wide helix slowly enough to achieve some dispersal by wind. The trick
requires little in the way of specialized structure—if you drop a small index
card it will almost always begin rotating about its longer center line and will
thereafter move sideways as well as downward. Cutting the card lengthwise
to increase the length-to-width ratio gives earlier rotating and a bit better
glide angle; cutting the card a bit diagonally makes it descend along a
helical path. According to Lugt (1983b), who gives a good review of
autorotating flat plates, an aspect (length-to-width) ratio greater than 5.0 is
best, and a plate needs certain minimum moment of inertia to autorotate.
Two other autorotating samaras common where I live are those of ash
(Fraxinus) and tuliptree or yellow poplar (Linodendrori). Each has its seed
not in the middle but at one end, and descent is inevitably in a tight helix
that looks very much like what's done by a maple (Acer) samara (Figure
10.13b). The maples, though, get their lift from conventional airfoils with
identifiable front and rear (leading and trailing) edges, while these
autorotating samaras are symmetrical front to rear. Both sorts "autogyrate"
downward, essentially gliding along a helical path as if on a twisted sliding
board. In addition, though, the symmetrical ones autorotate, getting lift as
Flettner rotors by Magnus effect. In comparisons of performance in either
nonhelical gliding (McCutchen 1977a) or helical descents (Green 1980),
the symmetrical autorotators seem a little inferior to the asymmetrical
nonautorotators. It's been both argued (Green 1980) and disputed
(Greene and Johnson 1990b) that the symmetrical ones are, on the other
228
MAKING AND USING VORTICES
Figure 10.13. Two samaras that work as Flettner rotors: (a) Ailanthus
rotates about its long axis and thus moves with some horizontal
component instead of descending directly, (b) Fraxinus autorotates similarly but,
with the seed at one end, its centers oflift, drag, and weight no longer
coincide; so it descends in a tight helix, gaining longer exposure to
ambient winds rather than direct dispersal distance.
hand, more stable and do better when turbulence or obstacles complicate
descent.
Many long, thin leaves act as Flettner rotors when they're shed. Around
my place the willow oaks (Quercusphellos) are particularly conspicuous auto-
rotators, and their leaves are carried well beyond the tree even with no
noticeable breeze. It's not clear whether the behavior has any adaptive
significance—perhaps excessive deposition of leaves above the roots is in
some way detrimental—or is merely an incidental consequence of a leaf
shaped by selection for other functional characteristics.
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CHAPTER 11
Lift, Airfoils, Gliding, and Soaring
In our discussion of vortices and circulation, we explained lift in a
way far removed from wing of bird, bat, insect, or even pterosaur. To
explain how any of these (and airplanes, too) produce circulation-based
lift, we still have several sticky problems left to resolve:
1. Things other than rotating devices can be observed producing lift. I
refer, of course, to airfoils1 or wings that stay properly fixed—
wings that may translate and thus transport but that quite obviously
don't rotate while doing so. Do these also work by the superposition
of circulation and translation, or do they depend on a different
physical mechanism?
2. For a flat, inclined surface to deflect an airstream and produce lift
seems intuitively reasonable. But a curved surface with its convex
side upward commonly produces an upward force even if its
leading (upwind) and trailing (downwind) edges are at the same
horizontal level, with no inclination at all.
3. For rotating cylinders lift is proportional to the first power of trans-
lational velocity (equation 10.3), while for airfoils lift is more nearly
proportional to the second power of the translational velocity. Is the
Kutta-Joukowski theorem underlying the equation applicable to
nonrotating, lift-producing devices?
Circulation and Airfoils
The route around these awkward points requires that we explore the
behavior of this thing we're calling an airfoil, the relevant terminology for
which is given in Figure 11.1. According to F. W. Lanchester (1868-1946),
an airfoil is a device that can produce circulation in its vicinity without itself
actually rotating. Just how can this happen? The fact that the lift for an
airfoil is proportional to the square rather than the first power of the free-
stream speed suggests that the circulation itself must be proportional to the
speed of the oncoming airstream. If so, then somehow the shape and
orientation of the airfoil must interact with the oncoming air to produce
1 All of the material to follow applies to lift production in water as well as air. Using
"airfoil," "airstream," "wind," and so forth should be regarded as just a linguistic
convenience or ancient prejudice of the author
230
LIFT, AIRFOILS, GLIDING, SOARING
leading
ecige
oncoming
wind
cross section or profile
chord, c
trailing a
e thickness
lift resultant force
aerodynamic forces
oncoming wind
span, b
Figure 11.1. The terminology for lift-producing airfoils.
the circulation—the faster the air, the more intense the circulation. A
rotating cylinder, of course, generates its circulation quite independently
of any translating airstream.
What determines the value of the circulation and hence the lift was first
realized by Joukowski (or Zhukovskii) early in the present century. He
pointed out that for an airfoil with a sharp trailing edge, only one pattern
of flow permitted the air to slip off the rear of the airfoil without any
discontinuity or without having to turn the sharp corner at the trailing
edge. This pattern had to determine the amount of circulation and lift.
Pressure might be lower on top than bottom because of the sharp trailing
edge and the rapid rearward flow on top—but fluid, with momentum, just
doesn't find it feasible to sneak around and relieve the difference. So
instead everything else shifts, creating a net circulation, as shown in Figure
11.2. It's only a net circulation, of course—fluid doesn't actually travel
around the airfoil—but it's still a flow pattern equivalent to that which
would be generated by the superposition of translation and circulation. In
short, a wing can be viewed as surrounded by a fictitious vortex (fictitious
because it's only a net vortex), and the strength of that vortex (its
circulation) is proportional to the airfoil's translational velocity. Hence the
dependence of lift on U2—one U is that of translation itself, the U in equation
(10.3); the other U comes from the circulation, since r <* U.
In the last chapter, vortices were made with the aid of viscosity, whether
by rotating a cylinder in a liquid and capitalizing on the no-slip condition at
231
CHAPTER 11
translational component 4- circulatory flow = overall lifting flow
Figure 11.2. The overall flow over an airfoil that's producing lift
consists of a translational component across it and a circulation around it.
its surface, or by using the vorticity of the velocity gradientsjust above the
ground, themselves a result of viscosity. In an ideal fluid, circulation can be
neither created nor destroyed. While real fluids are more accommodating,
the apparent paradox does have some reality and consequences. Prandtl
and Lanchester first recognized the existence of a "starting vortex," equal
in strength and opposite in direction to the "bound vortex" of a wing
(Figure 11.3). Thus the system as a whole need have no net circulation, and
the phenomenon of lift (unlike drag) needn't depend on a distinction
between real and ideal fluids. In the 1920s Prandtl even made movies
showing starting vortices; the movies also showed that when an airfoil was
stopped, the bound vortex slipped out from around the airfoil and became
a free and obvious "stopping vortex." The reality of these starting and
stopping vortices gives added credibility to the notion of a bound vortex, a
reality sometimes uncomfortably evident: they have the unfortunate habit
of hanging around airports on still days until eaten up by viscosity,
buffeting small craft that are landing or taking off.
For reasons that will become clear shortly, a wing of finite length sheds
"tip vortices" at its ends that amount to continuations of the bound vortex
or circulation about the wing. These tip vortices extend back to the starting
vortex, completing a vortex ring like those already considered. Naturally,
the vortex ring is almost always much longer than wide, about 100,000
times so for a large plane completing a transcontinental flight. (Of course,
in such a situation, as a result of viscosity the starting vortex gets dissipated
long before the stopping vortex is shed.) But we needn't postulate any
discontinuities in the flow, and ideal fluid theory (of which Bernoulli's
principle as the basis for lift is a part) isn't seriously abused.
No great mystery then attaches to the business of getting lift from curved
plates whose leading and trailing edges are at the same level with respect to
the free-stream motion. If a plane is convex on the side facing upward, and
if the flow goes smoothly from leading to trailing edges both above and
below it, then the streamlines will be squeezed together above and drawn
232
LIFT, AIRFOILS, GLIDING, SOARING
Figure 11.3. The bound vortex around the wings, the tip vortices, and
the starting vortex for a gliding aircraft. In all, the vortices form a
complete vortex ring.
apart below. That means, by Bernoulli's principle, lower pressure above
and higher pressure below and therefore an upwardly directed net force
acting normal to the free stream—in short, lift. Higher velocity above and
lower below—that's what we recognized as equivalent to a superposition of
translational and circulatory components of flow. Thus we've used the
notion of circulation to get a consistent view of lift, one in which airfoils and
Flettner rotors in free-stream flow develop their forces by the same
mechanism, the mechanism that we noted back in Chapter 4 for flow across a local
elevation of a substratum. Autorotating Ailanthus seed, gliding bird wing,
and bottom-resting flounder all develop their normal forces by the same
process.
Incidentally, this use of an upwardly convex surface to develop "upward
lift" (not, remember, a tautologous term) explains why the contour of the
upper surface of a lift-producing airfoil is generally of more consequence
than that of the lower surface. Flat lower surfaces work aerodynamically
about as well as concave ones; being flat below mainly gives an enclosed
volume useful for fuel tanks, bracing, muscles, control equipment, and so
forth. A sharp trailing edge, as already noted, is crucial for the functioning
of an airfoil as a lift generator as well as for drag minimization. And a
rounded leading edge discourages separation, quite as important for
producing lift as for limiting drag. The asymmetrical streamlined cross section
of airfoils useful at moderate and high Reynolds numbers and subsonic
flows is thus rationalized.
The distribution of lift on the surface of a wing is of considerable
interest, whether viewed as part of the circulation argument or as a practical
233
CHAPTER 11
matter of the location of the overall force vector that passes through the
wing. The detailed distribution of lift can be determined in the same way as
can the pressure distribution around a cylinder (Chapter 4)—a series of
holes is drilled in the surface and led by internal tubes to a manometer. The
only difference is that one must now look specifically at the component of
force normal to the free stream. One finds (1) that more of the lift comes
from the reduced pressure on the top than from the excess pressure on the
bottom, something quite consistent with our notion of the role of the
convex upper surface; (2) the center of lift is relatively near the leading
edge of the airfoil, usually at or in front of the point of maximum thickness;
and (3) as the angle at which air meets wing (the angle of attack) increases,
the center of lift shifts forward, with consequent alteration of pitching
moments and effects on stability.
Lift Coefficients and Polar Diagrams
The lift of a wing is approximately proportional to the square of its
velocity through the air. It's also nearly proportional to the area of the wing
and to the density of the air. All of this sounds very much like the way drag
behaves at moderate and high Reynolds numbers. So we can conveniently
de-dimensionalize lift in just the same way we did drag (Chapter 5),
dividing lift per unit area by dynamic pressure to obtain a dimensionless
coefficient, this one called the "lift coefficient" and analogous to the drag
coefficient of equation (5.4). Similarly, one shouldn't be misled by regarding this
as a "formula for lift" since it does nothing more than standardize the
interconversion of lift and lift coefficient. Thus
L = |c/PSf/2. (H.l)
The convention for designating an area, 5, for the equation is fairly
specific—plan form or profile area, the area one would see if viewing from
above an airfoil lying on a horizontal surface. Like the drag coefficient, the
lift coefficient is a function only of shape, orientation, and Reynolds
number. The two coefficients, though, depend on these three factors in rather
different ways, and the consequent interplay of lift and drag underlies
much of the subtlety of airfoil design.
For a given airfoil at a given Reynolds number, orientation determines
both lift and drag coefficients. Orientation, in practice, hinges mainly on
the angle between a line from leading to trailing edges (the "chord") and
the direction of the oncoming wind—what was defined as the "angle of
attack" in Figure 11.1. Lift (and its coefficient) increases from zero at an
angle of attack near 0° (how far from zero depends on the top-to-bottom
asymmetry of the airfoil) to some maximum at an angle of attack of around
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LIFT, AIRFOILS, GLIDING, SOARING
c
&
8
1.25-
1-
0.75-
0.5-
0.25-
0
-0.25-
slope =
max L/D
0
0.05 0.1
drag coefficient
0.15
0.2
Figure 11.4. A polar diagram for an airplane wing. The lift coefficient
is plotted against the drag coefficient, both referred to plan form area
and usually with different scales, and the angles of attack are noted on
the curve. Any line that passes through the origin is a line of constant
lift-to-drag ratio. The one tangent to the curve gives the maximum
obtainable ratio, and it touches the curve at the angle of attack that gives
that maximum ratio.
20°. It then drops off again, either sharply or gradually, depending on
airfoil and circumstances. Drag is of course never zero, even at 0°, and it
increases continuously up to angles of attack so high that lift has long since
become (relative to drag) negligible.
One might look at these variables by drawing graphs of lift and drag
coefficients against angle of attack. But such a pair involves an unhelpful
and unnecessary redundancy since the choice of angle of attack determines
both lift and drag. Gustav Eiffel (the famous tower builder) had a better
suggestion. Lift coefficient (as ordinate) can be plotted against drag
coefficient (as abscissa), with the angle of attack treated parametrically and
merely noted on the curve. Figure 11.4 gives an example of such a "polar
diagram"2—a fast, easy view of several of the characteristics of an airfoil.
Besides the obvious horizontal and vertical tangent lines that define
maximum lift and minimum drag, two most interesting data practically jump
out. First, a line through the origin that's tangent to the curve is a line whose
2 Polar diagrams, a little confusingly, do not use polar coordinates.
235
CHAPTER 11
slope gives the maximum ratio of lift to drag possible with that airfoil. As
we'll see, maximizing that ratio is often a primary objective in airfoil design
and operation. And second, the point at which that tangent line touches
the curve gives the angle of attack at which the maximum LID occurs.
To achieve its promised directness and utility, a polar diagram must be
constructed from lift and drag coefficients based on the same reference
area. So the convention is consistent (and thus rarely mentioned)—plan
form or profile area is used for both, with the same maximum projected
area however the airfoil might be tilted.
What Determines Airfoil Performance?
An intimidating number and diversity of variables influence the
performance of a lift-producing airfoil, determining (among other things) the
form of its polar plot. But none really flies in the face of an intuitively
reasonable explanation, so we'll work our way through them on a verbal
and semiquantitative level with an intuitive "feel" rather than real analytic
rigor as objective. At a next-higher level, the reader might look at Mises
(1945) for a purely physical and technological view and at McMasters
(1986), Pennycuick (1989), and Norberg (1990) for integration with the
relevant biology.
Aspect Ratio
Figure 11.1 includes a definition of the "aspect ratio" of a wing—in
simplest form it's the ratio of tip-to-tip length ("span") to average width
("chord"). Thus long, skinny wings have high aspect ratios. Since wings,
especially those of organisms, vary widely in shape, it's usually handiest to
multiply span over chord by span over span and thus slightly redefine
aspect ratio as span squared over area. Even so, the definition has a certain
looseness, since for aerodynamic as well as mechanical reasons wings taper
in various ways from center to tip; but it proves adequate for practical
purposes. Aspect ratio is far more important than one might guess, and its
consequences need a little explanation.
Measuring the lift and drag of something approximating an infinitely
long wing can be done: one just puts big end plates on a model or extends
the model from one side wall to the other in a wind tunnel. If real wings,
with aspect ratios much lower than infinity, are compared to an infinite (or
"two-dimensional") airfoil, the real ones are inevitably the poorer
aerodynamically—they produce less lift and suffer more drag. What a
finite wing has that an infinite wing lacks are tip vortices. And, indeed, they
turn out to be serious culprits—intuitively quite reasonable since the
stubbier the wing, the more it ought to be influenced by what happens at its tips.
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LIFT, AIRFOILS, GLIDING, SOARING
The Cost of Lift
But the more basic issue underlying the inferiority of finite airfoils is one
that came up in Chapter 4 in connection with jet propulsion. Recall that
power output was the product of thrust and jet velocity, while power input
was kinetic energy per time as delivered by an engine. The ratio of output
to input is the Froude propulsion efficiency,
Ti'WT^y (4-")
where Ul is the free stream velocity and U2 is the jet velocity.
Now consider what the fixed wing of an airplane does. It produces an
upward force, lift, by creating downward momentum at an adequate
rate—by making a downward jet. But since no downward component is a
preexisting part of the free stream (U{ = 0), the Froude propulsion
efficiency for staying aloft must be zero. That may strike you as odd, but after
all no power is absolutely required to produce a force. A properly designed
machine shouldn't use any energy to stay aloft; certainly the chain holding
up the chandelier consumes no power and requires no fuel. On the other
hand, helicopters and hovering hummingbirds are profligate consumers
of energy.
The sad circumstance that determines why power is required is that the
spans of the bird's wings and of the helicopter's blades are not infinite. The
reasoning is essentially the same as that used to derive the propulsion
efficiency. To produce downward momentum at a given rate one might use
either a large mass of air per unit time and give it a small downward velocity
or one might use a small mass per time and give it a large downward
velocity—all that matters is the final mUlt. But the power input to make that
momentum flux is energy per unit time, proportional to mU2lt. So keeping
the downward velocity as small as possible is best, which can be done by
dealing with a large amount of air. Ideally, with mlt infinite, no power at all
would be needed to stay aloft. Thus, since the long, narrow wing intercepts
more air per time than does the short broad wing, the former consumes
less power. The limit, again, is the infinite wing, which takes an infinite
mass per time and imparts to it a negligible downward velocity.
Induced Drag
How does a finite wing signal its requirement for power? The only way it
can do so is by having more drag than an infinite wing and thus demanding
that the engine (assuming for ease of explanation a separate propeller)
impart more momentum per time to the passing (horizontal) airstream.
The extra drag is termed the "induced drag"; it is the drag incurred as a
237
CHAPTER 11
]jft effective lift
(a) infinite span (b) finite span
Figure 11.5. The origin of induced drag in a wing of finite span. Since
the wing gives the air a significant downward push in (b), the net local
wind direction crossing the wing is no longer the same as that in (a) or of
the free stream. Thus the lift vector is tipped back and resolves into a
slightly lower vertical force ("effective lift") and the induced drag.
consequence of producing lift with a less-than-infinite wing. A useful way of
looking at the induced drag is to recognize that in giving air a significant
downward velocity, the airstream crossing a wing is deflected from the
horizontal, as in Figure 11.5. The lift vector, normal to the local airstream,
is no longer normal to the overall free stream or (in level flight) to the
horizon. It can be resolved into two components—an upward (literal lift)
vector normal to the free stream and the induced drag, a horizontal drag
vector parallel to the free stream. Referring to a polar diagram, reduction
of the aspect ratio pushes a wing's curve toward the right and toward the
abscissa, with less lift, more drag, and a lower maximum lift-to-drag ratio.
Put another way, induced drag times free stream velocity is the price in
power paid for staying aloft with a wing of less than infinite span or aspect
ratio—the measure of the energetic inferiority of short, broad wings.
The induced drag and the resulting induced power vary in an odd but
not unreasonable way with the speed of the airfoil through the fluid. As
speed increases, the airfoil comes in contact with more air per unit time.
Since its requirement for lift is nearly constant (equal to the weight of the
craft), the airfoil needs to give the air passing it less of a downward
deflection at higher speeds. If lift is kept constant as speed increases, the local
airstream gets more nearly horizontal, the lift vector gets more nearly
vertical, and the induced drag is less. The induced drag, then, is greatest
when speed is least; it's almost inversely proportional to the square of speed
of the airfoil relative to the undisturbed fluid. This dramatic increase in
induced drag at low speeds is one of the main reasons why flying slowly is so
expensive.
For a wing of a given aspect ratio protruding directly outward from a
fuselage, the induced drag is minimized if the wing has a particular ellipti-
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LIFT, AIRFOILS, GLIDING, SOARING
cal plan form. It ought to have its maximum chord at the base and taper
outward toward the tip, with most of the taper very near the tip. And the
wings of many low-speed aircraft have shapes that approximate this ideal
ellipse. Natural airfoils are more complex structures, especially those
called on to produce thrust as well as lift (Chapter 12), and no elliptical plan
form is ordinarily obvious. Recently, though, Burkett (1989) has shown
that a tapered wing with its tips swept backward can experience an induced
drag under reasonable operating conditions about 4% less than that of an
elliptical wing. By contrast with elliptical wings, tapered, aft-swept wings
are conspicuous in nature; and they occur in creatures for which high
speed seems a primary consideration—in the lunate tails of the fastest
marine fish, in cetaceans, in swifts, and in many seabirds (Figure 11.6).
Body Lift
How low can the aspect ratio be and still develop some useful lift? The
question may seem impractical if the design of wings is at issue, but it's of
relevance in a slightly different context. Rather than being symmetrically
streamlined, the bodies of many flying animals are somewhat more convex
on their upper surfaces. Hocking (1953) seems to have been the first to
suggest that an appreciable fraction of the lift of a fly could come from
airfoil action of the body; more recent workers have viewed with some
skepticism his claim that up to a third of total lift could be fuselage
generated. After all, the aspect ratios for most flying animal bodies are well less
than unity, which is pretty horrid. And Jensen (1956) showed that a desert
locust, an elongate creature with a great wing area, gets no more than 5% of
its lift from the body and has a lift-to-drag ratio for the body of 0.8.
Similarly, Dudley and Ellington (1990b) found that at its top speed of 4.5 m
s_1, a bumblebee worker's body has a lift of just 4.7% of its weight (see
Table 11.1). Nachtigall and Hanauer-Thieser (1992) found a small but
significant body lift in honeybees; they give a useful summary of other such
data.
But these low figures shouldn't be regarded as definitive since they were
obtained on isolated fuselages, and thus they ignore any effect on body lift
of the complex airflow generated by attached wings. An extreme case of
body lift occurs in swimming rays where, without interaction between
beating wings and a pitching and rebounding body, insufficient lift would be
generated to produce enough thrust to account for the speed of swimming
(Heine 1992).
Body lift may play a substantial role in the "swooping" flight of small
birds, in which the wings are periodically folded and held close to the body.
In zebra finches, according to Csicaky (1977), as much lift as drag is
produced when the body is pitched head up at about 20°. Body lift should be
239
CHAPTER 11
Figure 11.6. Natural airfoils whose tips are swept back: (a) the flukes of
a whale (from the top); (b) the tail of a shark (from the side); and
(c) the wings of a swift.
mainly significant in descending body-gliding. If a bird keeps the initial
heading that gave it a negligible body angle at the start of a body glide, its
increasingly downward trajectory would increase the body angle and lift as
the glide or swoop progresses. Rayner (1977, 1985) has analyzed the
phenomenon from a theoretical viewpoint. Another case in which body lift
proves useful is in extending the flight of ski jumpers. Leaning forward
over the skis and inclining them at an angle of attack of about 25°, ajumper
can achieve a lift-to-drag ratio of about 0.3 (Ward-Smith and Clements
1982).
Profile Drag
Besides induced drag, a queer sort of drag that's produced only when a
finite airfoil is developing lift, there remain the two familiar sorts of drag—
pressure drag and skin friction—that were introduced in Chapter 5.
Together they're usually called "profile drag," as mentioned in Chapter 7.
Like induced drag, profile drag increases sharply with increases in angle of
attack. Like parasite drag (but unlike induced drag) it increases nearly in
proportion to the square of speed. As a result, if lift is kept constant by
appropriate adjustment of the angle of attack, the overall drag of an airfoil
(profile plus induced) passes through a minimum value as speed increases.
The power required to offset that drag passes through a minimum value at
a speed just a little lower (due to the extra velocity factor in power) than that
giving minimum drag, as in Figure 11.7. The energy expended per unit
distance traveled (the cost of transport) will be minimized at a slightly
higher speed than that giving the minimum power or energy expended per
240
LIFT, AIRFOILS, GLIDING, SOARING
0.16H ' ' ' ' h
OH 1 1 1 1 h
0 2 4 6 8 10
Flight speed, m/s
Figure 11.7. Profile, induced, and total power as functions of speed for
the wing of a bat that's producing a constant amount of lift, from the
data of Norberg (1990). (Power is the product of speed and the thrust
needed to overcome drag.)
unit time. In short, while total drag goes through a specific minimum, just
what constitutes the optimal speed of flight depends a bit on the criterion
used (Pennycuick 1989).
One other factor complicates consideration of the relative importance of
profile drag and induced drag, and that's (yet again) the Reynolds number.
As a consequence of increasing skin friction, drag coefficients begin to rise
markedly when the Reynolds number drops below about 1000. Since skin
friction is part of profile drag, the latter increases at low Reynolds numbers
and may become substantially larger than induced drag. As Ennos (1989a)
perceptively pointed out, the long, skinny wings that by minimizing
induced drag give best performance in conventional (large scale) fliers will
not be optimal for smaller craft such as gliding insects and seeds,
statements in Vogel (1981) to the contrary notwithstanding. The problem is that
long, skinny wings develop more skin friction since more of their area is
near the leading edge where velocity gradients are steepest. So wings of
somewhat lower aspect ratio in small natural gliders are entirely
reasonable.
Perhaps at this point we ought to categorize all the kinds of drag that
have now been considered for lift-producing systems:
1. Parasite (body) drag, made up of skin friction and pressure drag
241
CHAPTER II
2. Airfoil drag (not a common designation)
a. Profile drag, made up of skin friction and pressure drag
b. Induced drag, the consequence of producing lift
3. Interference drag, the consequence of hooking airfoils on bodies
Stall
Assume you want your aircraft to take off at a fairly low speed. Lift at
least a little greater than weight is required, and the obvious way to
compensate for the low lift associated with low speed (as in equation 11.1) is
with an increase in angle of attack. This might be accomplished by rotating
the wings lengthwise with respect to fuselage, by changing the effective
pitch of wings with trailing ailerons, or by raising the nose and thus
increasing the pitch of the craft as a whole. Lift is nearly proportional to angle of
attack, and a poor lift-to-drag ratio matters little in the short run if
sufficient power is available for a short takeoff run. So operation at an angle of
attack well above that giving the greatest LID is a fine idea for takeoffs. But
lift increases with angle of attack only up to a point. Figure 11.4 shows that
at angles above some critical value, lift drops; it may crash abruptly. In
physical terms, the flow has begun to separate on the upper surface of the
airfoil, the circulation is much reduced, and the airstream is not so
effectively deflected downward. So the stall point, the angle of attack giving
maximum lift, has considerable practical relevance, and aircraft are
commonly equipped with stall-warning consciousness-raisers.
The stall point, though, is not inviolate, and a wide variety of devices for
postponing stall to higher angles of attack have been proposed, tested, and
even used; Mises (1945) has a particularly useful discussion of stall-
deterring and other high-lift devices on low-speed aircraft. Not
surprisingly, candidates for antistall devices have been proposed for living
fliers. The most frequent of these is the "alula" of bird wings, a group of
feathers attached to the anatomical thumb on the leading edge of the wing.
Nachtigall and Kempf (1971) found that when the angle of attack is high
(30° to 50°), as just prior to landing, the alulae are commonly erected and
give lift increases of up to 25%. Their photographs of flow show the
expected mechanism—air is persuaded not to separate so readily as it flows
across the top of a wing at high angles of attack. The alulae seem to have
little effect on lift or drag at lower angles of attack. Blick et al. (1975)
reported that leading-edge barbs copied from those of great horned owl
wings (and previously presumed associated with the especially quiet flight
of owls) could eliminate the sharp drop in lift at an angle of attack of 12°,
replacing it with a gentle leveling of lift (at a somewhat submaximal value)
out to at least 25°.
242
LIFT, AIRFOILS, GLIDING, SOARING
Wing Loading
How might lift be varied either in addition to changing angle of attack or
without leading to submaximal lift-to-drag ratios? Adjustment of wing
area is obviously attractive, so much so that even our stiff technology uses
the device. Modern jet transports use some remarkable area-increasing
excrescences during landings and takeoffs. But even these are minor
compared to the dramatic alterations of wing area in birds as their speeds
change—watch one try to keep lift equal to body weight as it lands on a
zero-length runway. Pennycuick (1968) has carefully documented the
phenomenon in gliding pigeons. For that matter, wing area is altered
continuously during the flapping cycle of virtually all birds and bats.
Quite beyond any use as a controlling variable, wing area is an important
factor in determining the performance of an aircraft. Slow craft have large
wings and fast ones have small wings in order to produce the requisite lift
without operation at uneconomical angles of attack. A common measure of
the relative area of a wing is the so-called wing loading—the weight of the
craft divided by the area of the wings. Successful people-pedaled planes
operating at 5 or 6 m s-1 have the lowest values among human-built
aircraft, about 20 N m~2. A 747 jet transport operating at about 240 ms_1 has
a wing loading of about 6000 N m~2. Wing loading, though, is a peculiar
kind of variable. It does give a direct look at how much lift a given area of
wing must produce since lift has to equal weight in steady-state flight, and it
bears strongly on the mechanical stresses and structural requirements of a
wing. But it can't be considered a direct indication of wing quality—only at
constant speed might a higher wing loading be taken to reflect a better lift-
producing design.
Also, wing loadings for conventional aircraft aren't directly comparable
to those of flying animals since the wings of flapping fliers produce thrust
as well as lift and since small flappers beat their wings much more
frequently than do large ones. But, for the record, an Andean condor has a
wing loading of about 100 N m~2 (McGahan 1973), a wren of 25 N m~2, a
bumblebee of 50 N m~2 (Greenewalt 1962), and a fruit fly of 3.5 N m~2.
McMasters (1986) gives lots of other examples.
Consideration of wing loading does, though, draw attention to a basic
problem of scaling. The lift needed by a flying animal ought to be
proportional to its body weight and thus to the cube of a typical linear dimension.
But an airfoil ought to produce lift in proportion to its area (equation 11.1,
again) and thus to the square of a linear dimension. Constant wing loading
therefore requires that shape change with size and implies that the larger
flier will need disproportionately large wings. Among animals, larger ones
do in fact have relatively larger wings than do smaller ones, but the
differences aren't sufficiently great to keep wing loading constant, as indicated
243
CHAPTER 11
by the figures just cited (Alexander 1971). The main compensation is in
flying speed—larger creatures typically fly a bit faster, and since lift varies
with the square of speed even a small speed increase goes a long way. An
incidental consequence is the greater difficulty of takeoff in larger fliers.
This scaling problem implies that angels must be either of unusually low
mass, supersonic, or buoyed up by contemplation of their divine mission.
The Effects of Reynolds Number
Conventional aircraft airfoils are designed or have been selected (by a
process resembling natural selection) for operation at Reynolds numbers,
based on chord length, above about a million. Birds, bats, and insects
operate at values below about half a million—about 500,000 for large
birds, 50,000 for smaller birds, 5000 for large insects, 500 for medium-
sized insects, and 50 for tiny insects. We've already noted that because of
increasing skin friction, profile drag gets worse at lower Reynolds
numbers. In fact, just about everything about a lift-producing airfoil gets worse
as the Reynolds number goes down through this biological range. The
ultimate culprit is viscosity through its effects on velocity gradients. With
gentler gradients, the carefully contrived shapes of airfoils are increasingly
obscured by a cloud of low-speed fluid. With increasing shear stress for any
given gradient, the rotational cores of vortices are larger and circulation is
more difficult to maintain. So the maximum obtainable lift coefficient
decreases. With greater skin friction, profile drag and thus total drag gets
greater. Consequently the maximum lift-to-drag ratios achievable get
lower. For an airfoil whose best LID (at infinite aspect ratio) was 80 at Re =
6,500,000, the best LID dropped to 47 at Re = 310,000 (Goldstein 1938).
Even the stall angle gets lower. An airfoil for which the stall angle was 18° at
Re = 3,300,000 had a stall angle of 12° at 330,000 and 9° at 43,000.
To a large extent this deterioration is real and probably unavoidable, but
some of the data of the kind just cited must simply be the result of using
specific airfoils under conditions for which they're inappropriate. Some
effort has gone into the design of airfoils for model airplanes (Schmitz
1960), which might serve better as comparisons for the wings of birds. At
Re = 42,000, a moderately cambered flat plate whose thickness was 5% of
its chord proves far better than a conventional airfoil—the cambered plate
has a best lift-to-drag ratio of about 11.0 while the conventional airfoil
reaches only 4.5. Only in minimum drag coefficient are they similar.
Conversely, at Re = 168,000, the conventional airfoil is better in all respects.
Further comparisons and discussion of the matter are provided by Lissa-
man (1983)—but the reader should bear in mind that when it comes to
Reynolds numbers, "low" may mean quite different things to engineers
and biologists.
244
LIFT, AIRFOILS, GLIDING, SOARING
At high Reynolds numbers, even minor surface irregularities may have a
great effect on the performance of an airfoil, and competitive glider pilots
take fastidious care of the tops of the wings of their craft. But airfoils such
as insect wings, selected for operation at low Reynolds numbers, are
nothing if not irregular. While exceedingly thin (a fruit fly wing has a mass of
about a gram per square meter), insect wings are usually somewhat
corrugated. Not only do veins protrude, but the veins are not coplanar, giving
greater stiffness to these light structures that move at several meters per
second and change directions several hundred times each second. In large
insects eddies may form in the pleats, as suggested by Newman et al. (1977)
for dragonfly wings. Still, considering the magnitude of the irregularities,
their effects seem usually to be fairly subtle. Nachtigall (1965) measured
only small differences in the polar plots of butterfly wings with and without
scales. And Rees (1975) compared the performance of a model of cross
sections of a hover fly (syrphid) wing to a smooth model representing an
envelope around the corrugations. Again little difference appeared, and
his polar plots are nicely consistent with my data (Vogel 1967b) on fruit fly
wings and the data of Dudley and Ellington (1990b) on bumblebees.
Apparently at his Reynolds numbers of 450 and 900, the flow treats the folds
as if filled.
The Limits of Circulation
At some point the deterioration of airfoil performance as Reynolds
numbers drop will preclude flight using this circulation-based scheme for
producing lift. As we'll see shortly (and is obvious by a glance forward to
Table 11.1), the maximum achievable lift-to-drag ratios are substantially
lower in smaller creatures. To how low a Reynolds number can flight by the
present mechanism be pushed? The only data I know of for airfoils at
really low Reynolds numbers are those of Thorn and Swart (1940) on
cambered airfoils at Re = 10and/?£= \.AtRe= 10, their best LID was only
0.43—the drag was over twice the lift—and it occurred at an angle of attack
of 45°. At Re = 1, the best LID was 0.18, again at about 45°. Despite the very
high angles of attack, the deflection of the wake of the airfoil was never
more than about 11°.3 In this world, fluid is highly resistant to being set into
circulation by any fixed (non-Flettner) airfoil. And, as far as I know, the use
of the Flettner type of rotating airfoil has never been investigated at
Reynolds numbers this low. Nor can I think of a biological case. Flettner
devices seem to exist only among plants, and really small wind-dispersed
3 Still, one ought to bear in mind that the airfoils used in this investigation were chosen
fairly arbitrarily, certainly with none of the fastidious testing of natural selection working
on a major functional feature.
245
CHAPTER 11
seeds apparently prefer drag maximization to lift production as a means of
slowing their descents.
Other mechanisms for generating lift are available at Reynolds numbers
around unity, but most of the possibilities are unpromising. We'll return to
the issue in Chapter 15.
More on Biological Aireoils
The discussion so far has looked at lift-producing airfoils in general; let's
now turn more specifically at the performance of those of organisms.
They're quite a diverse group. Their Reynolds numbers extend over four
or five orders of magnitude; the equivalent for all human-carrying
subsonic aircraft is less than three orders. And they serve a wide range of
functions, with the production of lift through circulation about the only
common feature—in at least one case we'll see that even the generation of
force is irrelevant. Finally, they've evolved from a wide variety of
precursors in a large number of animal and plant groups. I mean to emphasize
these latter points to suggest that many biological lift-producing airfoils
have escaped identification as such because of insufficient appreciation of
how they might look and what they might do.
A Lift-producing Sand Dollar
First, a brief look at what must be the least obvious use of an airfoil to
have emerged so far. A certain sand dollar, Dendraster excentricus, living
subtidally on the Pacific coast of North America, routinely adopts a most
un-sand-dollar-like posture when exposed to currents. It buries its anterior
edge in the sand and erects its posterior in the flowing water above—
instead of lying, half-buried, like a bump on the bottom. And it likes
company: these sand dollars cluster at densities of 200 to 1000 per square
meter (Figure 11.8b).
While this peculiar habit was well known, its function remained
enigmatic until O'Neill (1978) showed that each animal formed a lifting body. A
sand dollar, flat or slightly concave on its lower (oral) surface and convex on
its upper (aboral) surface, develops a substantial circulation (Figure 11.8a).
Since one end is in the sand, it has only one tip vortex and an effective
aspect ratio about double that of a dollar in a free stream. The lift, of
course, is directed horizontally, and as a force proves to be of little interest.
Nor is drag of direct consequence. What mainly matters is that the
circulation developed by the cambered profile brings food particles closer to the
tiny feeding appendages on its oral surface. Aggregation of individuals
turns out to be valuable through mutual improvement of feeding currents;
usually the oral surface of one individual faces the aboral surface of its
246
LIFT, AIRFOILS, GLIDING, SOARING
Figure 11.8. Flight isn't the only use for lift, (a) At least one sand dollar
stands on its anterior edge, (b) Dense groups of these dollars orient with
respect to the flow and each other to operate as a colonial, multiwinged
craft and bring feeding currents closer to their oral surfaces.
nearest neighbor, much like the wings of a multiwinged aircraft. The sand
dollars seem to regulate their density to maintain optimal gaps between
individuals, adjusting their spacing from 10 to 100 mm in proportion to
the square of the average surge velocity. As hydrofoils with identical
leading and trailing edges that get good circulation at near-zero angles of
attack, they're well suited to take advantage of bidirectional wave surge
through their mutually formed channels. Their tuning to flow is
impressive—populations from sheltered locations are slightly more
cambered and have slightly higher lift coefficients than those from more
exposed localities.
Wings
Figure 11.9 gives the polar plots and Table 11.1 the salient aerodynamic
characteristics of a variety of animal airfoils, mainly obtained from
measurements on isolated but real specimens in wind tunnels. Several of the
points made earlier can be seen in these data. For the wings,4 the greater
minimum (profile) drag at lower Reynolds numbers is clearly evident, as is
the increase in the angle of attack that gives the best ratio of lift to drag. But
1 Insects have two pairs of wings. In locusts, most of the lift and thrust are produced by
the hind wings; in bees, each functional wing consists of fore- and hind wings hooked
together; in beetles, the elytra, or forewings, commonly function as fixed wings ahead of
the propellerlike hind wings; in flies, the hind wings have been modified into tiny clublike
organs that have no direct aerodynamic function or consequences.
247
CHAPTER 11
c
.a
o
o
1.25-
1-
0.75-
0.5-
0.25-
U"
0.25-
~ 30°
20°^"—o
/ locust
f hindwing 2Q0
10°/ /*
£ / 20° / 10°
o°<
/io° /
> / /
<
>0° \
6 0°
,-10°
30°
40°
bumblebee
wing
i
40°
50°
i
â–
-
50°
fruit fly
wing
-
-
i
0
0.25
0.5
drag coefficient
0 75
Figure 11.9. Polar plots for several insect wings: a locust (Schistocerca
gregaria) hind wing (Jensen 1956), the linked fore- and hind wings of a
bumblebee (Bombus terrestris) (Dudley and Ellington 1990b), and a fruit fly
(Drosophila virihs) wing (Vogel 1967b).
the data have to be viewed with a certain cautious skepticism. Bird wings in
wind tunnels usually do rather badly by any reasonable aerodynamic
standards—the maximum lift-to-drag ratios of 3.8 for hawk and duck and
those of 2.7 obtained on another duck and a sparrow by Nachtigall and
Kempf (1971) try the adaptationist faith of this biologist. The problem
almost certainly comes from the subtlety of feather configuration and its
better feather management by bird than biologist—something that came
up earlier (Chapter 7) in connection with the parasite drag of bird bodies.
As we'll see in a few pages, such data can be estimated from tracks of live,
gliding birds in the field or wind tunnel; and these versions look quite a lot
better. Maximum steady-state lift coefficients around 1.5 to 1.6 have been
measured on gliding birds and bats (Pennycuick 1989; Norberg 1990).
Maximum lift-to-drag ratios of 8.5, 10.4, 13.7, 15.5, and 18 have been
reported for petrels (Pennycuick 1960), falcons (Tucker and Parrott 1970),
condors (McGahan 1973), vultures (Pennycuick 1971), and albatrosses
(Pennycuick 1982), respectively. Moreover, these latter figures are for
entire animals and so are deflated by inclusion of parasite drag in their
denominators.
By contrast, the data for insect wings are as tidy a set of numbers as one
might wish, especially since they represent the results of as many investiga-
248
LIFT, AIRFOILS, GLIDING, SOARING
Table 1 1.1 Characteristics of biological airfoils.
Buteo hneatus
(hawk) wing
Aix sponsa
(wood duck) wing
Chaetura pelagica
(swift) wing
Schistocerca gregana
(locust) hindwing
Tipula oleracea
(crane fly) wing
Bombus terrestris
(bumblebee) wing
Melalontha vulgaris
(beetle) elytron
Drosophila vinhs
(fruit fly) wing
Petarus breviceps
(flying "squirrel")
A musium japonicum
(scallop) shell
Bombus terrestris
(bumblebee) body
Velella velella (sailor-by-
the-wind) sail
Alsomitra macrocarpa
seed-leaf
Acer diabohcum
(maple) samara
Re
10,000 to
50,000
a
a
4000
1500
1240
1100
200
100,000
79,000
6200
1000
4000
1400
AR
6.0
6.2
7.8
5.6
6.9
6.7
4.2
5.5
1.0
1.3
0.3
1.4
3.5
4.3
Cd min
0.074
0.096
0.030
0.06
0.10
0.13
0.17
0.33
0.10
0.036
0.34
0.2
0.04
0.1
Ctmax-
1.0
0.90
0.80
1.13
0.82
0.78
0.83
0.87
0.6
0.85
0.22
0.95
0.87
1.6
-@ot
25°
20°
8°
25°
35°
30°
25°
30°
34°
25°
40°
35°
14°
19°
LIDmax-
3.8
3.8
17.0
8.2
3.7
2.48
1.9
1.8
2.0
5.47
0.33
0.23
4.6
3.3
-@a
6°
8°
5°
7°
13°
15°
15°
15°
34°
12°
32°
10°
—
12°
Source
1
1
1
2
3
4
5
6
7
8
4
9
10
11
Sources: (1) Withers 1981; (2) Jensen 1956; (3) Nachtigall 1977b; (4) Dudley and Ellington 1990a,b;
(5) Nachtigall 1964; (6) Vogel 1967b; Zanker and Gotz 1990 on D melanogaster is similar; (7)
Nachtigall et al. 1974, Nachtigall 1979b; (8) Hayami 1991; Millward and Whyte 1992 on A
pleuronectes is similar; (9) Francis 1991; (10) Azumaand Okuno 1987; (11) Azumaand Yasuda 1989.
tors as species and they span some thirty-five years. Still, even these data are
likely to represent underestimates of actual flight performance. For wings
that normally flap, steady-state measurements lack several crucial elements
of reality. Even beyond nonsteady effects (to get attention in the next
chapter), flapping causes the wind direction to vary spanwise, something
quite hard to simulate in static measurements. I'd make a strong pitch for a
presumption of asymmetrical bias in such experimental results. For a ma-
249
CHAPTER 11
chine designed by the evolutionary process, the only appropriate operating
conditions are its natural conditions, and data obtained under other
circumstances are more likely to show substandard than superior performances.
In comparing the five insect wings, we see the pernicious effect of low
Reynolds number on performance discussed earlier. Drag gets much
worse relative to lift as the Reynolds number decreases and the drag
becomes almost all profile drag (as indicated by C(/mm) rather than being
about half induced drag. Thus making longer and more slender wings is of
little help, although it might still give benefit by reducing the interaction of
the wind of the beating wings ("prop-wash") and the fuselage.
Concomitantly, the best lift-to-drag ratio drops. More interestingly, this shift of
polar curves to the right pushes the best operating point up to higher
angles of attack, essentially to the stall point for an insect the size of a fruit
fly. The benefits of stall-resisting arrangements ought to be greater; and,
indeed, some evidence points to such devices in small insects. A fruit fly
wing doesn't stall in the usual sense; its lift just levels off above an angle of
attack of about 20°. What seems to be happening is that with so much drag,
very little momentum flux is left behind the wing to be deflected
downward. But a flat or cambered plate at the same Reynolds number does stall
(although with less dramatic effect than for wings of larger craft), and the
lift coefficient drops at angles above about 25°. Just what's different about a
real wing isn't clear, although some indirect evidence implicates the tiny
hairs (microtrichia) on the surface of the wings (Vogel 1967b). One way or
another, such a wing clearly doesn't quite work like an amorphous paddle
in a cloud of attached air.
Lifting Bodies
Table 11.1 also gives the characteristics of a gliding phalanger, Petaurus
breviceps, an engaging marsupial of extraordinarily close resemblance to
the North American flying squirrel, Glaucomys volans; its polar plot is quite
similar to that for the fruit fly wing in Figure 11.9. These data were
obtained on whole, performing animals and are probably quite reliable. The
interesting point about this airfoil is its very low aspect ratio (Figure
11.10a), a rather bad state of affairs if these animals are viewed as purely
aerodynamic devices. The low ratio limits maximum lift and increases
induced drag—what pushes its polar curve to the right is increased
induced drag rather than profile drag as in small insects. The result, though,
is the same—the best operating point is essentially the stall point. And
anything that either postpones stall to higher angles of attack or reduces
the suddenness of separation and stall should be of substantial benefit.
Nachtigall (1979a,b) has some evidence that the phalanger has special fur
with such a function.
250
LIFT, AIRFOILS, GLIDING, SOARING
Figure 11.10. Lifting bodies in nature: (a) a gliding phalanger, Petaurus
breviceps; (b) a scallop, Amusium japonicum; (c) the by-the-wind-sailor, Ve-
lella velella; (d) the seed-leaf of the Javanese cucumber, Alsomitra macro-
carpa; (e) the fruit of an elm, Ulmus.
The table gives, as well, the performance of a scallop (Figure 11.10b).
Scallops can swim briefly but surprisingly rapidly (well above half a meter
per second for some) and are denser than seawater. Swimming is by jet
propulsion through apertures near the hinge; they make some use of
vectored thrust at the gape to get the upstream end initially elevated. Some
scallops have markedly asymmetrical shells, with the functional upper one
rounded and the functional lower one flat (anatomically these are left and
251
CHAPTER 11
right valves), and these asymmetrical scallops are in general the better
swimmers (Millward and Whyte 1992). The best seem to be several species
of Amusium, although not by any overwhelming margin. Amusium both
looks and behaves like quite a competent airfoil; the main limitation on its
performance is the low aspect ratio, a variable that none of the scallops
seem to go to any great adaptive lengths to improve relative either to each
other or to other bivalves. According to Hayami's (1991) data and
calculations, Amusiumjaponicum could manage level swimming without any
downward jet thrust at a speed of only 0.45 m s~l and an angle of attack of just
5.3°.
And then there's the bumblebee fuselage as a lifting body. We talked
about body lift a few pages back; it seems to be significant without being a
really big deal. What ought to be mentioned is that a bumblebee can't just
pick a body angle (pitch) that gives the best performance as a lifting body.
In many insects, including bumblebees, the pitch of the body is adjusted to
set the pitch of the plane of beating of the wings and thus the ratio of lift to
thrust. At high speeds, where body lift might be greatest, the body is
pitched down to a nearly horizontal posture (Dudley and Ellington 1990b).
There is also Velella, perhaps the most off-beat of animal airfoils. Velella
(Figure 11.10c) is a pelagic, colonial coelenterate, a bit like the better-
known Portuguese man-of-war, Physalia. But instead of an emergent
balloon, it has a thin, leaflike sail obliquely mounted above a less conspicuous
float and skirt, beneath which dangle its tentacles. Velella reportedly can sail
as much as 63° off the direction of the wind as a result of the lift of the sail
and its asymmetry above the float—it's about the closest thing to a sailboat
that nature has contrived. But, as Francis (1991) has shown, it's not a really
great sailboat; on the other hand, what tack can one take to reach for
adaptive advantage for good cross-wind sailing?
Two botanical airfoils are included in Table 11.1, one the seed-leaf of the
Javanese cucumber, the other a samara (the fruit) of a maple. The former is
a simple glider and will be discussed shortly; the latter is an autogyrator
and will reappear in the next chapter.
Gliding and Soaring
An unpowered airfoil can't travel horizontally through still air at a steady
speed—that's why engines and muscles are so useful. Before putting
power sources to work, though, let's consider the possibilities of getting a
free ride, first from gravity and then from atmospheric motion. We'll
follow the common practice of calling the first scheme "gliding" and the
second "soaring." Gliding, to be explicit, is defined as a situation in which
an airfoil moves through still air, while losing altitude just rapidly enough
to maintain a steady course by producing a vertical force the same as its
252
LIFT, AIRFOILS, GLIDING, SOARING
weight. What's important is that no forces are unbalanced, so the
aerodynamic resultant force, the combination we ordinarily separate into lift and
drag, precisely balances the weight of the craft.
How to Glide
Knowing that weight and aerodynamic resultant are equal and opposite,
the angle at which the craft descends can be easily derived, as shown in
Figure 11.11. Both resultant force (R) and weight (W) are obviously
vertical. The oncoming wind (U), equal and opposite the path of the craft, and
the drag (D) are both obviously perpendicular to the lift (L). By elementary
geometry the angle between resultant and lift vectors must therefore be the
same as the angle between horizon and wind. The former is the angle
whose cotangent (the reciprocal of the tangent) is the ratio of lift to drag,
while the latter is the angle of descent, called the "glide angle." In short, the
glide angle, 6, is set by the lift-to-drag ratio:
cot6 = ^ = ^. (11.2)
Isn't it nice that something as simple and familiar as the lift-to-drag ratio
determines something as critical as the glide angle—to minimize the glide
angle, just maximize the lift-to-drag ratio. Incidentally, we now know how
to determine lift and drag coefficients on freely gliding birds. One needs
only glide angle, flying speed, wing area, and weight; and by equations
(11.1) and (11.2) the coefficients fall out.
If the air is otherwise still, then minimizing the glide angle maximizes the
distance a simple glider will go before reaching the ground. An albatross,
with a lift-to-drag ratio of 18, released 1 km above ground, could glide a
horizontal distance of 18 km. A phalanger (Petaurus) with a best lift-to-drag
ratio of 2.0 would go at most 2 km. Long, skinny wings and the consequent
low-induced drag are clearly advantageous (at least where Reynolds
numbers are high enough so profile drag isn't a big concern). Whether bird or
person-carrying craft, good gliders typically have wings of high aspect
ratio. One high-performance sailplane with an aspect ratio of 20 has a lift-
to-drag ratio of 39, and so can go 39 km horizontally for each kilometer of
descent (Tucker and Parrott 1970).
This business of distance traveled versus descent is counterintuitive in at
least one important respect. For moderate and high Reynolds numbers,
speed affects both lift and drag in much the same manner, so their ratio
isn't especially speed dependent. As a result, the glide angle is also largely
independent of speed, and a very heavy glider descends along nearly the
same path as a light one! Weight, though, must be balanced by lift, and the
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CHAPTER 11
R
L A
A
descent path
f
Figure 11.11. The relationship among the glide angle (6), the lift, the
drag, and the weight supported by an airfoil in a steady glide.
latter is roughly proportional to the square (or a little less) of speed. While
the heavier glider may fly the same path, it must of necessity travel faster, so
it covers its horizontal distance in less time. A rapid descent is usually not
desirable, since gliders are typically looking for opportunities to soar.
Therefore most gliders, living or not, avoid excessive weight. Sometimes,
though, staying aloft poses hazards. Military gliders were usually heavily
loaded affairs, balancing the pitfalls of high-speed impact against the
inefficiency of carrying only light loads and the insecurity of a long, slow flight
over hostile territory.
The relationship between glide angle and lift-to-drag ratio explains, in
part, why gliding animals are fairly large. Small size means relatively more
profile drag and, as we've seen, a lower maximum LID. So small size almost
invariably implies a higher (worse) glide angle. An albatross may descend
at about 3°; a falcon can do no better than 5.5°, a pigeon about 9.5°. Small
insects are far worse—even if the fuselage contributed no drag at all, a
bumblebee would descend at 22° and a fruit fly at nearly 30°. Simple
gliding, based on circulation and lift, just isn't very good at low Reynolds
numbers. In general, the larger members of most animal groups are the
more recently evolved—large size constitutes a specialization, and likely
ancestors tend to be unheroic in stature. Exceptional are those groups that
engage in active flight—birds, bats, and insects—for a reason that the
present discussion makes persuasive. If in each case active, powered fliers
evolved through ancestral forms that were exclusively or mainly gliders,
then those ancestors ought to have been fairly large.
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LIFT, AIRFOILS, GLIDING, SOARING
The Glide Polar
A convenient way of viewing the performance of a glider is with a so-
called glide polar, a curve on a graph that has flying speed (airspeed, not
necessarily ground speed) on the abscissa and sinking speed (increasing
downward) on the ordinate, as in Figure 11.12. On such a graph, any
straight line extending from the origin is a line of constant lift-to-drag
ratio. The one tangent to a craft's curve (the line of least slope) gives, at the
tangent point, the flying speed and sinking speed at the minimum glide angle.
Craft are characterized by curved rather than straight lines mainly as a
result of the way drag varies with speed—again at low speeds induced drag
dominates, while at high speeds profile drag dominates, with an overall
minimum (and hence maximum Ct/Cd) in between. Increasing the weight
of the craft would shift its curve rightward and downward without much
change in the slope of the tangent line.
The glide polar permits easy comparison of several other factors as well.
Time aloft may be more important than maximum still-air glide distance.
A horizontal line tangent to the curve gives, at its tangent point, the flight
speed at which sinking speed is minimized and thus time aloft maximized; at its
intersection with the abscissa it gives the specific sinking speed. Notice that
the greatest time aloft is achieved at a lower speed than the least glide angle,
so a slow glider can do as well as a faster one that has a better lift-to-drag
ratio. If time aloft is the important factor (as we'll see that it is in soaring),
then birds don't look quite so bad when compared with sailplanes. By this
criterion even large insects aren't disastrous gliders. A desert locust,
despite a lift-to-drag ratio of only 8.2 for a hindwing (and less for the entire
insect) descends at 0.6 m s~l—in the same range as a vulture or sailplane
(Jensen 1956). A monarch butterfly has an overall lift-to-drag ratio of 3.6
and therefore a steep glide angle of 15.5°, but it travels at only 2.6 m s~l
and consequently descends at only 0.68 m s~ l—again in the same range
(Gibo and Pallett 1979).
Another variable that emerges from the glide polar is the minimum glide
speed. It's the left end of the curve, the point at which combination of the
maximum wing area (if variable) and the maximum lift coefficient provides
just enough lift to balance the weight. One can encapsulate the conditions
for minimum glide speed in a formula recognizing that lift must equal
weight and using the definitions of wing loading (WIS) and lift coefficient:
/ 2W/S \1/2
V P^lmax '
Just where an animal might choose to operate on its glide polar depends
in part on what the wind is doing. If the animal wishes to maximize distance
traveled and is assisted by a tail wind, a lower flight speed than that which
255
CHAPTER 11
73
K
60
.s
0
0
0.5
1 -
1.5 -
2 -
2.5
air speed, m/s
10 15
20
25
10
30
* % % * - m
* * *% *-*.
* * *
» » *»
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- (e) \ \
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•\ (b)\
(c) \
\
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\
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(d)
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falcon \
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40
U
20
Figure 11.12. Glide polars for several craft—SHK sailplane, vulture,
and falcon (Tucker and Parrott 1970). Also shown are single data points
for some other gliders: (a) Andean condor; (b) desert locust; (c) monarch
butterfly; (d) flying phalanger; and (e) Javanese cucumber seed-leaf.
would maximize LID is the best tactic. If it faces a head wind less than its
flying speed, it's best off picking a higher flight speed than that maximizing
LID. If it faces a head wind greater than its flying speed, the best course is to
land as quickly as possible (Tucker and Parrott 1970).
The same considerations, of course, hold for plants. The most famous
simple glider is that of the seed-leaf of the Javanese cucumber, Alsomitra (=
Zanonia) macrocarpa (Figure 11.1 Od), for which data are given in Table 11.1.
The gliding seed formed the model for a successful series of gliders around
the turn of the century (Bishop 1961)—the death of Otto Lilienthal in a
birdlike hang-glider focused attention on the need for stability and the
clear instability of birds.5 The Alsomitra seed-leaf, with a span of about 140
mm and a weight of 210 mg , glides at 1.5 m s~1 as it descends at 0.41 m
s_1—at a lift-to-drag ratio of 3.7 and an angle of 15°, according to Azuma
and Okuno (1987). They found that a seed-leaf could reach LID = 4.6 if
forced to elevate its nose a bit, but it then suffered substantially reduced
5 The instability of a bird is almost certainly concomitant with its maneuverability and
must have evolved in parallel with highly sophisticated controls, according to the
aerodynamicist-turned-evolutionary biologist, John Maynard Smith (1952). A seed leaf,
by contrast, must be intrinsically stable. But the model didn't lend itself to the addition of
an engine, and the design was returned to historians and botanists.
256
LIFT, AIRFOILS, GLIDING, SOARING
stability. Of particular interest in the present context is that normal
operation, at Cf = 0.34, is between the points on polar plot and glide polar that
give minimum glide angle (C, = 0.27) and minimum rate of descent (C, =
0.53).
But simple gliding is a lot less common among wind-dispersed, lift-
producing seeds than are autogyrating, autorotating, or their
combination. McCutchen (1977a) argues (and I see no reason to disagree) that
simple gliders are inferior in the presence of any turbulence. So they
should be useful mainly in the still air of the interior of a rain forest. In fact,
smaller and less elegant plain, winged seeds with some central
concentration of mass are not all that uncommon (see, for instance, Augspurger
1986, 1988); examples are the fruits of the hoptree, Ptelea trifoliata, and of
the elms (Ulmus) (Figure ll.lOe). These seem to be simple gliders, if
smaller and less stable than Alsomitra.
Reduction in sinking speed by lift production is, in any case, a great deal
less common among plants than is reduction by drag production. The two
schemes seem to represent a major fluid-mechanical dichotomy since lift is
most useful when drag is least; that is, a high LID makes the present scheme
work best. Matlack (1987), looking at the situation from a comparative
viewpoint, pointed out that plumed drag maximizers are more common at
weights under 45 mg, while samaras and other lift maximizers more
commonly weigh more than 45 mg. In addition, drag maximizers are more
common among herbs and lift maximizers among trees, perhaps reflecting
differences in optimum investment of mass per dispersal unit. That larger
size makes gliding work better is, of course, the same argument made
earlier to rationalize the evolution of small active fliers from large ancestral
gliders.
Gliding and Parachuting
Gliding of some sort is practiced by many animals other than birds, bats,
and insects—a few are shown in Figure 11.13. Flying squirrels and pha-
langers have already been mentioned; detailed information about the
flight-related morphometries of the former has been collected by Tho-
rington and Heaney (1981). In addition there are flying lemurs (Cy-
nocephalus), several separate lineages of flying frogs (Rhacophorus, Phry-
nohyas, etc.), a colubrid snake (Chrysopelea), and several geckoes and lizards,
among which the best is certainly the aptly named Draco volans (Norberg
1990). Gliding ability is quite variable, and Oliver (1951) suggested that a
distinction be made between gliding and parachuting, based on whether
the still-air descent angle is greater or less than 45°. The point of division
isn't entirely arbitrary—where data exist, it appears that animals with
descent angles over 45°, such as the arboreal lizard, Anohs carohnensis (68°),
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CHAPTER II
Figure 11.13. Gliding animals from groups that don't go in for active,
flapping-wing flight. A frog, Rhacophorus; a lizard, Draco; and a fish, one
of the Exocoetidae.
and the tree frog, Hyla venulosa (57°), don't have particularly obvious
aerodynamic structural adaptations (Oliver 1951).
Draco, the flying lizard of the Philippines, has a broad patagium
supported by rib extensions. Hairston (1957) estimated its glide angle at 11°,
certainly well within the range of flying squirrels. A Malayan flying frog,
Rhacophorus nigropalmatus, with huge, webbed hands and feet, glides at 25°
to 37°, moving about 5ms-1 (Emerson and Koehl 1990). Curiously, gliding
vertebrates are especially diverse and common in Indo-Malaysia; Dudley
and DeVries (1990) argue that gliding is especially useful in the unusually
high dipterocarp forests found there.
Probably the best gliders among animals that don't also actively fly are
flying fish. Some (the Exocoetidae) use only their pectoral fins as wings.
Larger ones (such as Cypselurus) use pelvic fins as well to form a staggered-
wing biplane; according to Davenport (1992), the main contribution of the
pelvic fins is to assure pitching (fore and aft) stability since the pectoral fins
are in front of a fish's center of mass. Cypselurus, among others, initially
leaves the hypocaudal lobe of the tail in the water, beating fifty to seventy
times per second. During this "taxiing" phase, it goes from emersion speed
of about 10 ms-1 (in a large specimen) to takeoff speed of 15 to 20. Single
flights may be up to 50 meters long, and repeated flights in a sequence can
258
LIFT, AIRFOILS, GLIDING, SOARING
extend up to 400 meters. The aspect ratios of flying fish are usually high (3
to 20); estimates of lift-to-drag ratios also run high but are almost certainly
unreliable. One can't do a quick calculation from glide angle since these
creatures don't do anything close to simple gliding. They lose little altitude
during a glide, since they don't have much to start with, and probably lose
speed instead. They glide close to the ocean's surface, so ground effect
probably augments performance considerably. In addition, they may
practice slope soaring, taking advantage of local updrafts upwind of waves.
Much of this information comes from Fish (1990), the other modern
source on the subject.
Soaring
Atmospheric motion, of course, includes much more than just
horizontal wind. And for organisms, the speeds of air movement are of the same
order as or higher than their own flight speeds. The combination must, on
one hand, enforce respect for local air movement but, on the other, permit
all kinds of schemes for soaring in the moving medium. Their general
utility is obvious—air in motion with respect to the ground provides a
source of energy; and flight, per unit time, is the most energy-intensive
form of animal locomotion. In simple gliding, an organism heads
earthward, the only real question being its time, speed, and place of arrival. In
soaring, as commonly defined, altitude is maintained or even gained by
using the energy of the wind. The problems involved in soaring can
perhaps best be appreciated by reading an instruction manual for sailplane
pilots such as Conway (1969).
Two general forms of soaring can be usefully distinguished. In the
simpler, "static soaring," a craft or creature takes advantage of a region in
which air is moving upward. Somewhat trickier is "dynamic soaring,"
which involves no upward air movement but only a temporal or spatial
gradient in wind velocity from which an animal extracts the power
necessary to stay aloft. Good accounts of soaring are given by Cone (1962),
Pennycuick (1972, 1975), and Norberg (1990); I'll just briefly describe the
possibilities in order to emphasize their diversity.
The simplest sort of static soaring is "slope soaring," or flying in air that's
moving upward along the side of a hill. The updraft may extend above the
crest of the hill and may have a fairly complex structure, but the successful
craft finds a place where it can sink continuously with respect to the local air
without sinking with respect to the earth—that is, where the magnitude of
the vertical component of the wind is equal to or greater than the sinking
speed of the craft (Figure 11.14). Slope soaring is the usual kind practiced
by hang-gliders. Petrels and albatrosses do it along ocean waves, as do
(probably) the flying fish mentioned earlier (Pennycuick 1982). Sometimes
259
CHAPTER 11
Figure 11.14. Slope soaring—if air moves up a hill, then a bird can
descend with respect to the local air without descending with respect to the
earth. Thus it can maintain position without active propulsion.
a version of slope soaring is possible in waves and vortices in the lee of a
range of mountains, and it may be possible in the lee of ships and
elsewhere. Birds have been reported to move long distances along ridge lines,
almost certainly by slope soaring. The most careful analysis of free-flying
birds is that of Videler and Groenewold (1991), who measured wind pro-
Hies as well as bird performance for kestrels hanging in fixed positions over
a Dutch sea dike; they calculated that by hanging rather than actively
flying, a kestrel can reduce its energy expenditure by fully two-thirds.
A second kind of static soaring is "thermal soaring"; it was described in
the last chapter when we considered vortices. Yet another is what might be
called "sea anchor soaring," suggested for petrels by Withers (1979).
Petrels get their name (from "Peter") from their apparent ability to walk on
water. What they're more likely doing is getting enough drag from their
feet in the water to be pushed only slowly downwind as they face into the
wind with outspread wings. A bird, then, can gain enough lift (with a little
ground effect helping) to get the body (except the feet) entirely out of the
water—it operates much like a kite (the toy, not the bird), which stays aloft
only as long as a tether is provided to allow it to maintain airspeed. The
trick may also be used by another kind of seabird, the prions (Klages and
Cooper 1992), which suspension feed on copepods—they get their bodies
nearly out of the water with the lift of spread wings while keeping heads
submerged for feeding.
Of the possible types of dynamic soaring, the best understood is that
done in the altitudinal wind gradient near the surface of the ocean. Again,
note that a vertical gradient in wind speed is used, not a vertical component
260
LIFT, AIRFOILS, GLIDING, SOARING
A
Figure 11.15. Dynamic soaring in a wind whose horizontal speed
increases with altitude. The bird alternately ascends and descends,
extracting energy from the gradient. The bird reverses its heading at the
marked points.
of velocity. This "gradient soaring," as done by an albatross, has two
alternating phases—a downwind, downward glide and an upwind, upward
glide (Figure 11.15). A bird well above the ground or ocean glides
downwind, gaining airspeed as it loses altitude and encounters slower, lower air.
Nearly at the lowest point it turns into the wind, and dives a bit further. It
then reinvests its high airspeed in the upwind ascent, which brings it into
air that's moving increasingly rapidly with height. The bird then turns
downwind again, and repeats the process. The overall motion is a series of
vertical loops that progress downwind.
Other kinds of dynamic soaring are also possible—a bird ought to be
able to use a temporally unsteady wind to soar by flying through gusts.
Similarly, it ought to be able to fly in and out of breeze fronts and other such
irregularities. A large amount of energy is dissipated in the lowest part of
the atmosphere, and extracting some of it is possible with appropriate
machinery and behavior.
261
CHAPTER 12
The Thrust of Flying and Swimming
The origin of drag came in for attention quite a few chapters back;
the origin of lift occupied much of the last two chapters. Our tour de
force has one other major variable yet to explain. Real airfoils can make lift
only by paying some price in drag. So what we now have to do is to make
some antidrag to counteract its nefarious (except for parachutes)
influence. The polite name for antidrag is thrust.
Thrust from Flapping
The notion of an oncoming wind that wasn't horizontal proved central in
explaining how simple gliding works. In gliding, though, that oncoming
flow was never far from either the horizontal or the line of progression of
the craft. Generating thrust requires that an airfoil encounter a flow
decently distant from the line of progression of the craft. As will become clear
shortly, a craft can't produce thrust without some cross-course wind. Such a
situation can be made to happen if the airfoil moves with with respect to the
craft—it must rotate, flap, wiggle, undulate, or otherwise move about. But
if it does move in an appropriate manner, then, mirabile dictu, thrust is
exerted on the craft by a structure that, in its own frame of reference,
always suffers drag, usually produces lift, and never feels thrust.
The Origin of Thrust
Before turning to reciprocating airfoils such as flapping wings and
beating tails, let's first look at a simpler case. Consider an element of a rotating
propeller, the latter turning about an axis that happens to coincide with the
overall movement of the craft, as in Figure 12.1. What we mean by an
element is a cross section normal to the long axis of the propeller, usually
drawn as if the craft were behind the page, the propeller sticking through
it, and the craft progressing from right to left—so the free-stream flow is,
as usual, left to right. The oncoming wind seen by this blade element, Uw, is
the resultant of two component winds, U^ the wind due to the craft's
forward motion (the free-stream flow) and U,, the wind due to the
rotational (here downward) motion of the propeller blade.
If the airfoil, this propeller blade element, is set at an appropriate angle
to the line of progression of the craft (usually called either the "pitch" or
262
THRUST OF FLYING AND S W I M M I N G
Uw Ur
Figure 12.1. The origin of thrust. If the overall oncoming wind strikes
a blade element from a sufficiently nonhorizontal angle, then the
aerodynamic resultant may be tipped forward (upstream) of the vertical. If so, it
has a component opposite the free stream flow—negative drag, or thrust.
The thrust component appears explicitly when the resultant is
reanalyzed, as at the right.
the "angle of incidence"), it can have an angle of attack with respect to the
resultant oncoming wind that gives it good lift and reasonably low drag.
But, it must be emphasized, such lift and drag are defined relative to the
oncoming wind—drag parallel and lift normal—and the oncoming wind is
anything but horizontal. So we immediately combine the lift and drag,
concentrating on what's real: the resultant force on the airfoil as a result of
its shape, pitch, and motion.
The resultant force may (as in the illustration) be tilted forward, which
means that it has a component opposite the progression of the craft. By any
reasonable definition such a component is thrust. Thus, even though the
airfoil knows only lift and drag, the craft as a whole gets pushed forward by
thrust. Again, only when the oncoming wind relative to the blade element
has a component coming from above or below the direction in which the
craft is moving can thrust be produced—just as a conventional sailboat
can't make headway directly into the wind.
What we've done is to re-resolve the resultant of lift and drag on the
airfoil section into a pair of forces on the craft—a thrust and a residual
component normal to the thrust. This latter component can be viewed in
several ways. If the blade element is moving downward, then it's a kind of
lift. But in a rotating propeller, downward movement of one blade element
is always balanced by upward movement of some other blade element (or
elements). So in an aerodynamic sense, this other force isn't any useful kind
of lift. What it does is to determine the cost of spinning the propeller, since
it directly opposes that spin. Thus the component (L' in Figure 12.1)
multiplied by its distance from the axis of rotation of the propeller (behind the
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CHAPTER 12
page) gives the torque that the engine must supply to keep that element of
the blade moving. If the analysis is extended to all elements of all blades,
the sums give (1) the overall thrust that (at steady speed) must just balance
the drag of wings, fuselage, and all else except the propeller, and (2) the
torque required of the engine.
A practical propeller is twisted along its length from hub to tip. The
reason is simple enough: nearer the hub, the spinning propeller
encounters a wind mainly due to the craft's progression, while nearer the tip it
meets a wind whose direction reflects more of its own rotation. So the
farther from the hub, the more the wind a propeller encounters will
deviate from the direction of motion of the craft as a whole. Thus, to maintain
an optimal angle of attack along its length, the propeller ought to be twisted
with its flattened faces more nearly aligned with the craft's motion near the
base and with the plane of the propeller's motion near the tips. Assuming a
constant rate of rotation, little twist will be needed at low flying speed but
more will be necessary as speed increases. Aircraft of the nonliving sort do
a crude approximation of such variable twist by lengthwise rotation of their
rigid propellers—they vary the pitch. Flying organisms with nonrigid
beating wings—insects and hummingbirds most clearly—manage to change
the degree of lengthwise twist, even (as necessary of course) while reversing
the direction of twist between downstrokes and upstrokes (see, for
instance, Ennos 1989b). Ennos (1988) showed very neatly that one needn't
invoke special machinery in the insect thorax—the wings are designed to
bejust sufficiently flexible in torsion so the inertial and aerodynamic forces
of the wingbeat are sufficient to cause the twisting. But not all fliers do the
twist. It seems to be nearly or entirely absent in some wings that operate at
low Reynolds numbers—for example, fruit fly wings (Vogel 1967a) and
maple samaras (Norberg 1973). At low Reynolds numbers, the lift-to-drag
ratio just isn't so sensitive to small changes in angle of attack, so twist
matters less, as one can see from Figure 11.9.
The Plane of Flapping
In birds, bats, and insects, flapping wings combine the functions that
airplanes divide between fixed wings and propellers—in a sense, they're
closer to helicopters than to airplanes, and it's all too easy to be misled by
our habit of calling the propulsive appendages "wings" rather than
"propeller blades." But they aren't quite like ordinary propellers either, since
flapping wings produce both thrust and lift directly, rather than producing
thrust directly and getting lift by diverting some of the thrust to pay for the
drag of fixed, lift-producing wings. That composite function, as well as
their reciprocating rather than rotational motion, means that the motion
of flapping wings is inevitably complex. In general, the stroke takes the
264
THRUST OF FLYING AND SWIMMING
form of an inclined ellipse or figure eight if projected onto the surface of a
sphere, or a saw-tooth pattern if projected (to account for forward motion)
onto the surface of a cylinder, as in Figure 12.2. The downstroke moves a
wing forward as well as downward and produces mainly upward force but
usually some rearward force as well. The upstroke goes backward as well as
upward, producing mainly rearward force but often some upward force.
For a flying animal we need several variables to deal with items that are
invariant in an airplane. For one thing, the airplane's propeller turns on a
shaft that closely parallels the direction of flight, so the plane in which the
propeller turns is always one that's normal to the craft's progression. In a
flying animal, what's called the "stroke plane" is almost inevitably tilted—as
just described, its top is farther aft than its bottom. That tilt, though, is
quite variable, ranging from almost horizontal to nearly vertical. In fact,
the tilt of the stroke plane is often a function of flying speed, with a near-
horizontal plane (small angle) characterizing hovering and with larger angles
typical at faster speeds of flight.1 Early work on tethered insects suggested
that the stroke plane angle was the principal variable used to set the ratio of
lift to thrust in hummingbirds and in insects such as fruit flies but not in
others such as hover flies and locusts. Now that more data are available
from freely flying animals, things look less simple—stroke plane angle is
only one of the variables that can adjust that crucial ratio of lift to thrust
(Ellington 1984a; Ennos 1989b); the main alternative is really subtle
alteration of the phasing of lengthwise wing rotation during the stroke. Ennos,
in fact, found that a hover fly, Eristahs tenax, can hover with either a
horizontal or a near-vertical stroke plane. Horizontal-stroke hovering is
energetically cheaper, but he figured that vertical-stroke hovering permits
quicker transition to fast forward flight—like a sprinter on starting blocks
instead of a distance runner standing upright. Of course in horizontal-
stroke hovering wings don't really go up or down on upstroke or down-
stroke; a classical anatomist would call the half-strokes dorsad and ventrad
and never get confused.
Hovering is in almost every instance the most expensive kind of flight an
animal can do, and work on hovering (especially the monumental treatise
of Ellington 1984a) has been of the greatest importance in understanding
the special features of animal flight. The high cost is concomitant with that
high induced drag at low speeds discussed in the last chapter. With no
component due to forward speed, wind over the wings is slower, and lift
coefficients have to be higher to get enough lift to offset body weight. Also
(as you can see in Figure 12.3), in hovering a wing has to rotate lengthwise
1 Confusion can arise all too easily between this stroke plane angle, the angle between the
plane in which the wings beat and the horizontal, and the stroke angle, the amplitude of
wingbeat, about which more shortly.
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CHAPTER 12
Figure 12.2. The wing motion of a insect flying forward, as the path of
the wingtip might trace a path on the surface of a cylinder. (We're
neglecting movement off that surface resulting from the forward and
backward motion of a wing of fixed length.) The dashed curve uses the
insect's body as a frame of reference, while the solid line refers to the
earth as the insect flies along. Notice that the curve gets nearer the top
than the bottom of the cylinder—opposite wings nearly meet at the top
of the stroke.
more at the top and bottom of each stroke to achieve decent angles of attack
in the two half-strokes—Uw is almost entirely a matter of the Ur
component, and the latter reverses between half-strokes. So a larger fraction of
the stroke must be spent in such shifts. With a limit of about 250 W kg"1
(Weis-Fogh and Alexander 1977) for the power output of muscle, hovering
for more than very brief periods isn't possible for creatures much larger
than hummingbirds (Weis-Fogh 1977)—wing loading is higher, so a larger
downward velocity has to be imparted to the air, so the Froude propulsion
efficiency is lower, so the cost of sustaining body weight is higher. Still, quite
a few larger birds and bats hover at least briefly (incurring oxygen debts) in
connection with feeding, landing, and other intermittent activities (Nor-
berg 1990).
Advance Ratio
Obviously no thrust can be generated if the relative wind striking a
rotating propeller is too close to the axis of progression—the aerodynamic
resultant (Figure 12.1, again) will tip backward, not forward. That's almost
certainly why many insects (mosquitoes and crane flies, for instance) have
wings that are narrow near their bases and broad farther out, and why
maple samaras put most of their area well away from their axes of rotation.
More importantly, the necessity of having a resultant wind well off the
direction of progression sets an upper limit on the speed of a craft. What's
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THRUST OF FLYING AND SWIMMING
. , _ _ "downstroke"
"upstroke'
â–
Figure 12.3. The wing motion of an insect hovering with a horizontal
stroke, a little idealized to fit on a flat page as in a Mercator projection of
a global map.
critical is the ratio of the forward speed of the craft to the speed of the
blades of its propellers. The usual index of this quantity is the "advance
ratio,"/:
•j-3- (12J)
Uf is the craft's forward speed, d is the diameter of disk swept by the
propeller blades, and n is the rate of revolution of the propeller. For
ordinary airplanes, advance ratios are generally under 4.0—if you want to go
faster, you should increase either the length or the rotation rate of the
propeller. For a propeller without provision for pitch adjustment, the
angles of attack (and thus the L/Ds) of its blade elements will vary with the
advance ratio, and so its operating efficiency will depend very strongly on
flying speed—it's common to plot propeller efficiency as a function of
advance ratio. Mises (1945) gives a very good introduction to the blade
element view of propellers.
Flying animals use reciprocating rather than fully rotating wings, so
equation (12.1) proves inadequate. Every rotating propeller blade makes
360° per revolution, but a reciprocating blade or wing can be operated over
a range of stroke amplitudes. So both amplitude and frequency must be
specified. (Typically an insect wing, for instance, goes up and down
through an arc of around 100°, or 200° per full stroke cycle.) Using/? for
the length of a wing, § for the amplitude (usually referred to as "stroke
angle"), and n for wingbeat frequency, Ellington (1984a) gives the
following formula for the advance ratio:2
2 This formula, now in general use, differs by a factor of tt from that given in the
earlier edition. Thus the figures I gave earlier should be divided by tt. I suggest
reconverting the present figures (multiplying by tt) if one wants to compare them to data for
airplanes.
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CHAPTER 12
/ = f (12 2)
Using this formula, a bumblebee (Bombus terrestris) reaches an advance ratio
in free flight of 0.66 (Dudley and Ellington 1990a), a black fly (Simuhum sp.)
0.50, and a fruit fly (Drosophila melanogaster) 0.33 (Ennos 1989b). All are
figures for insects in free flight; tethered ones typically give somewhat
lower values. Below about bumblebee size, the smaller the insect, the lower
is its best advance ratio—with higher profile drag, it takes more wing
beating to put a decent forward component on the aerodynamic resultant
force.
The wingbeat frequencies of small insects are, in fact, very high. Fruit
flies beat their wings around 200 or so times per second, mosquitoes
around 300 to 600 (Greenewalt 1962), and midges up to about 1000. The
all-time record for rate of reciprocation of any animal appendage is held
by a midge that had its wings almost completely amputated to reduce their
moments of inertia—it achieved 2218 wingbeats per second (Sotavalta
1953). On the other hand, the higher frequencies are almost entirely offset
by shorter wings—tip velocity, the denominator in (12.2), doesn't change
all that much with size.
The advance ratio provides a handy way to make crude estimates of the
top speeds of insects. Onejust assumes a value of/ appropriate for the size
of the insect together with a moderate amplitude (say 2 radians). Wing
length is easily measured, and data on frequency have been collected by
Greenewalt (1962), so Uf can be calculated from (12.2). The results may be
crude, but they're a lot better than some of the grotesque guesses buzzing
about in the entomological literature!
Four Kinds of Moving Airfoils
The present explanation of how beating wings work began with airplane
propellers, noted the addition of variable amplitude and variable stroke
plane, and then drew on helicopters to deal with hovering. For a better
perspective, let's look at the entire realm of moving airfoils that deal with
thrust and power. Besides propeller and helicopter blades, there are
windmill and autogyro blades—four types in all, of which three have clear
biological examples. Figure 12.4 considers a blade element of each of these
devices, first generating an aerodynamic resultant as the consequence of
the relative wind striking it, and then re-resolving that resultant into
components parallel and normal to the axis of rotation. We have, then,
1. A propeller with (ideally) a horizontal shaft, which inserts power
into a horizontal airstream;
2. A windmill, also with a horizontal shaft, but which extracts power
from the horizontal airstream;
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T H R U S1 OF FLYING AND SWIMMING
(a) propeller blade (b) windmill blade
(c) helicopter blade (d) autogyro blade
Figure 12.4. Four kinds of moving airfoils, analyzed as in 12.1. Ui is the
wind component induced by the action of a helicopter blade as it pushes
air downward; Ud is the wind component due to the descent of the
autogyro. Other symbols are the same as in 12.1. Variables in quotes refer to
the earth rather than the airfoil, and the desirable components of the
resultants are boxed.
3. A helicopter rotor (here hovering) with a vertical shaft, which
inserts power into the airstream, taking air from above and thrusting
it out below the plane of the rotor; and
4. An autogyro rotor (here descending) with a vertical shaft, which
extracts power from the airstream, taking air from below and
retarding its passage upward through the plane of the rotor.
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CHAPTER 12
We've already mentioned the propeller (Figure 12.4a); its operation is
essentially the same whether it rotates as an airplane propeller, oscillates as
a beating wing, or reciprocates as the dorso-ventrally elongate (lunate) tail
of a mackerel or tuna. The flow comes from ahead, and the aerodynamic
resultant generates a force that engine or muscles must counteract. Notice
that to keep a positive angle of attack, an airfoil that's not symmetrical top
to bottom must have its convex side facing more upstream and its concave
or flat side facing more downstream. Only when one considers the
direction of the oncoming wind does the arrangement appear logical. The rule,
by the way, is general and quite useful for all axial fans and blowers in home
or lab: blades should be concave downstream. You can't reverse the direction in
which a fan blows by reversing the fan hub on its shaft; all the operation
does is reduce the fan's effectiveness because an inverted airfoil has a lower
L/Z)max!3 To reverse a fan either the motor must be reversed as well as the
hub or the lengthwise twist and chordwise curvature (camber) of each
blade must be reversed. Reversing fans usually blow these latter problems
under the rug by using airfoils that are symmetrical top to bottom and have
no lengthwise twist.
A windmill (Figure 12.4b), on the other hand, must have the "lower" flat
or concave side of its airfoils facing the wind (which is why it's lucky that the
sails of a sailboat are naturally concave upwind). Thus a propeller makes a
poor windmill unless the hub is reversed on the shaft or the flow direction
is reversed. If the hub is reversed but the flow direction kept the same, the
propeller-now-turned-windmill will, of course, revolve in the opposite
direction. In windmills, the wind produces a resultant force that can be
separated into a torque (per unit radius), now the useful component that
drives whatever the windmill is connected to, and a drag parallel to the
overall wind. The latter is irrelevant to power generation and matters only
in judging the necessary strength of blades, bearings, and tower. A
windmill, incidentally, works as much because its feet are on the ground as
because its rotor is in the air—a velocity difference is what it's taking
advantage of.
I know of no close biological analog of a windmill. Wheel-and-axle
arrangements are unknown in organisms except among bacteria, so we
ought to cast about for reciprocating analogs of beating wings. Leaves such
as those of aspens and poplars rotate a little as well as translate in circles in a
wind, restrained by petioles that can't rotate freely, but nothing suggests
that useful power is being extracted in the process. Aquatic organisms are
probably better candidates than terrestrial ones, and probably the power
3 I once took an ordinary ventilating fan and reversed the hub on the shaft,
whereupon the speed dropped from 0.8 to 0.2 m s-1. The direction of flow, naturally, was
unchanged.
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THRUST OF FLYING AND SWIMMING
extracted would have to be invested in something such as induction of fluid
movement that requires only a minimum of transducing machinery.
Biological windmills certainly do exist in the general sense of devices that
extract energy from velocity gradients in fluid media, as mentioned in
Chapter 4. But they don't seem to include any quixotic devices that do so by
direct application of lift-producing airfoils. Birds performing dynamic
soaring extract energy from velocity gradients, but again the mechanism is
a long way from a windmill.
Helicopter rotors (Figure 12.4c) are the closest technological analogs of
hovering fliers. By contrast with propellers and windmills, the wind comes
from above, so these airfoils must be concave downward and downwind,
with their leading edge elevated. During pure hovering, rotation produces
an upwardly tilted resultant; the latter resolves, in turn, into a vertical force
(lift with respect to the earth) and a torque (per unit radius) that, as with a
propeller, the engine must counteract. Since the only downward
component of the wind is that produced by the rotor, helicopter rotors should be
as long as mechanically practical in order to intercept the maximum
volume of air per unit time and thus minimize their induced-power appetite.
This preference for long blades is the main reason for the impracticality of
aircraft in which propellers used for fast forward flight are made to face
upward for hovering or takeoff. That's of course the same argument
(ultimately based on Froude propulsion efficiency) we used to explain why a
flying animal capable of reasonably fast forward flight finds it so expensive
to hover.
Autogyros (Figure 12.4d) may be less familiar to the reader than the
previous three devices, but they're completely competent flying machines.
Superficially they look like helicopters, but the overhead rotor isn't
powered, another propeller is necessary for level flight, and they can't hover or
ascend vertically. An autogyro, moving forward, has the plane of its passive
rotor tilted back a little so that the approaching air encounters the
underside of that plane of rotation (a helicopter uses the opposite tilt). Air passes
upward through the rotor and, in being retarded, it generates lift and drag
with respect to the craft. The (separate) propeller counteracts that drag
and so keeps the craft moving. Unpowered, an autogyro descends slowly
with the rotor passively turning in a horizontal plane; in effect it's a set of
airfoils (the blades) gliding earthward in a tightly helical path. As in any of
these devices, air must strike the concave side of the blades, so the blades
have to be concave downward (as in helicopters). But in contrast with
helicopter blades, the leading edges of the blades of the autogyro must be
depressed rather than elevated.
If a helicopter loses power, it can revert to autogyration by rapidly
shifting the pitch of the blades to get their leading edges downward. Otherwise
the rotor will turn backwards, and its profile will be wrong for action as a
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CHAPTER 12
good airfoil. If pitch is adjusted, though, both camber and lengthwise twist
are proper, unlike the shift from propeller to windmill.
The Flight of Samaras
An unpowered autogyro, again, is a kind of glider that happens to
descend along a helical path. Like any glider, it can have no unbalanced forces
under steady-state conditions. Initially, the aerodynamic resultant is
tipped forward, pointing in the direction of rotation, and produces a
torque that accelerates the blade or blades, as shown in Figure 12.4d. As the
blades speed up, the oncoming wind becomes more nearly horizontal, and
the resultant tips back until it's vertical and just equal to the weight of the
craft.
Autogyros of this kind are quite common in nature as seed dispersal
devices ("samaras") in plants; as mentioned in Chapter 10, some are
Flettner-rotating autogyros, but more are simple lift-producing airfoils.
These latter range in blade length from about 10 to 180 mm, have from
one to three blades, and span several orders of magnitude in mass. The
best-known samaras are those of the maples (Aceraceae) (Figure 12.5), but
they also occur in some tropical Juglandaceae and Leguminoseae as well as,
among conifers, in many Pinaceae and Cupressaceae. The definitive study
of the flight of samaras is that of Norberg (1973); for more on the
aerodynamic characteristics of their wings, see Azuma and Okuno (1987) and
Azuma and Yasuda (1989), and on their flight see Seter and Rosen (1992).
Quite a few people have recently looked at performance variation and its
ecological implications—for instance Green (1980), Guries and Nordheim
(1984), Augspurger (1986, 1988), and Greene and Johnson (1989, 1990b),
most of whom have been mentioned earlier.
For an autogyro, a low sinking speed is obviously desirable, which implies
that a low rather than a high advance ratio indicates good performance. A
fairly large, one-bladed samara of the Norway maple, Acerplatanoides, with
a functional blade length of 37 mm, falls at 0.9 m s_1,and revolves thirteen
times per second; its advance ratio (equation 12.1) is 0.9. The smaller
samara of the Norway spruce, Picea abies, has a blade length of 11 mm, falls
at 0.64 m s, and revolves twenty times per second; thus it has an advance
ratio of 1.5 (data from Norberg 1973). I measured an advance ratio of 1.3
for red maple (Acer rubrum) samaras, whose functional blade length is
about 17 mm. That's as it ought to be—the advance ratio should increase as
the Reynolds number decreases because the poorer lift-to-drag ratios at
low Reynolds numbers mean relatively less force is available to spin a
samara and more drag opposes the spin. These advance ratios don't vary
greatly, so just as one can estimate insect flight speeds from wingbeat
272
THRUST OF FLYING AND SWIMMING
Figure 12.5. (a) The autogyrating descent of a maple samara,
(b) The low cone it describes as it rotates about the seed as an apex.
frequencies, one can make good guesses of spinning rates (and Reynolds
numbers and so forth) from measurements of falling speeds.
If the advance ratio stays fairly constant, then sinking speed and rotation
rate will remain in the same proportion. That's really a version of the
statement made in the last chapter about the insensitivity of the glide angle
in simple gliding to the weight of the craft. For a constant velocity descent,
the upward force will equal weight. Lift, of course, is nearly proportional to
the square of airfoil speed—to both sinking speed and rotational speed—
and overall upward force will nearly follow airfoil lift. Thus sinking speed
should be proportional to the square root of weight. And that's about what
happens. For a bunch of red maple samaras of different degrees of
dryness, with masses from 11.2 to 57.2 mg (5-fold), I measured sinking rates
ranging only from 0.62 to 1.15ms-1 (less than 2-fold). In fact (as you can
tell from these numbers), the sinking-rate variation is even a little less than
just predicted since faster sinking means higher Reynolds number and an
improved (lower) advance ratio. This relative insensitivity of sinking rate to
weight is a rather nice feature of the autogyrating scheme!
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CHAPTER 12
Momentum Flux and Actuator Disks
In talking earlier about drag, we took several viewpoints on the
interaction of fluid and solid, ultimately equivalent but each with different
perspectives, insights, and applications. We looked at the particulars of skin
friction and pressure drag after looking at the more global business of
momentum flux. One can do the same for a thrust-producing or power-
extracting rotor through which fluid passes. Up to this point we've taken a
close-up view, a largely qualitative glimpse of what's properly called "blade
element analysis." For some problems that's unnecessarily particular, and
returning to a consideration of momentum gives more insight—specifically,
looking at the rate at which the momentum of the fluid changes as it passes
through the thrust-producing device.
To look at momentum flux, it's customary to set up an ideal "actuator
disk," a plane extending across the airstream with an area equal to that
swept by blades or beating wings, as in Figure 12.6. Matters can be
simplified further by assuming that the disk accelerates fluid only axially, that
is, normal to its plane; that fluid velocity is uniform across the disk; and that
a complete, frictionless discontinuity exists between fluid that passes
through the disk and the rest of the fluid. In passing through the disk, a
stream of fluid is accelerated; and therefore, by the principle of continuity,
the stream contracts. According to the momentum theory of propeller
action, the velocity of the fluid passing through the actuator disk is just
halfway between that well upstream and well downstream. Consequently
half the cross-sectional contraction occurs upstream and half downstream
from the disk.
The main utility of analysis using momentum flux through actuator
disks is in estimating such important items as induced velocity and power.
To hover, for instance, an animal must balance its weight by making
downward thrust, and the latter is a matter of the rate at which it imparts
downward momentum to the airstream. Thus if you know wing length, you
know disk area; from the latter you can figure the velocity increment (the
induced velocity) that the wings must impart to the air—at least given the
simplifications implicit in the approach. A useful variable in this regard is
what's called "disk loading," weight over disk area, the equivalent for a
thrust-producing propeller or beating wing of wing loading (Chapter 11)
for a lift-producing airfoil. Induced velocity, U-, turns out to be
proportional to the square root of the disk loading,4 W/S(l; more specifically,
4 One can combine this statement with the earlier one about the sinking speed of a
samara being proportional to the square root of its weight, declaring (as did Norberg
1973) that the sinking speed of a samara is proportional to the square root of its disk
loading.
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THRUST OF FLYING AND SWIMMING
Figure 12.6. Replacing the beating wings of a hovering animal with an
"actuator disk" that inserts momentum at a certain rate into a downward
airstream.
Ui
/ J4^ \ 1/2
\29Sd
(12.3)
Since induced power is equal to weight times induced velocity, then
W?> \ 1/2
P.- =
2pS,
(12.4)
And this can be read as yet another way to say that low weight and long
blades are best for hovering, whether in hummingbird or helicopter. But
I'll declare further details beyond our present scope, suggesting reference
to Mises (1945), Ellington (1984a), and Norberg (1990).
Ellington, incidentally, recommends against using a fully circular disk
for animal flight, replacing it with two pie-shaped sectors to account for the
less-than-full-circle stroke angle and the tilted stroke plane.
A Wake of Vortices
Another view of flapping flight focuses on the vortex structures left
behind in the wake. We've already talked (Chapter 11) about bound
vortices, tip vortices, and starting vortices, and the way in a fixed-wing aircraft
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CHAPTER 12
the three are linked in an elongate vortex ring. We've noted (Chapter 10)
that a vortex ring will progress through the fluid on account of its own
circulating motion—a ring with its outside going downward and its inside
upward, such as a thermal vortex, will move upward as a whole, even if it's
not buoyant. Similarly, a ring with its inside moving downward will move
downward as a whole—and that's the kind made by an airfoil producing
lift. And lift, asjust noted, must be associated with downward movement of
fluid. So there's a nice congruence among these matters.
Where the vortices get really interesting is in flapping flight. Consider,
again, a hovering flier with a near-horizontal stroke plane. Reciprocating
wings have a problem quite unknown among rotating propellers.
Reversing the direction of wing motion, as happens twice during each stroke,
without at the same time reversing the direction of thrust requires that the
circulation about the wing must also reverse. What was on the downstroke
the airfoil's upper surface, with net circulatory flow from leading to trailing
edge, becomes on the upstroke its lower surface, with circulation from
trailing to leading edge. Reversing the bound vortex requires shedding the
old vortex as a stopping vortex after each half stroke; since successive half-
strokes have opposite circulations, successive vortices shed into the wake
will turn oppositely. The wake of a hovering flapper, as pointed out by
Ellington (1978) and Rayner (1979), must contain a downward-moving
stack of vortex rings, each one generated by a half wingbeat, as in Figure
12.7a. Looking at hovering in terms of actuator disks and momentum flux
ignores all these periodic effects and thus gives what amounts to a
minimum estimate of induced velocity, thrust, and power.
A bird flying rapidly beats its wings mainly up and down, getting lift on
both strokes but more on the downstroke than the upstroke. It leaves a
wake that's closer to that of a fixed-wing craft, a single vortex ring with tip,
starting, and bound vortices (Figure 12.7c). The variation in lift leads to an
in-and-out wobble of the tip vortices, closer together on the upstroke and
farther apart on the downstroke. A bird flying more slowly leaves
something in between in its wake—a zigzag ladder of vortices, with horizontal
cross-pieces between tip vortices corresponding to each reversal of wing
direction and the concomitant reversal of circulation (Figure 12.7b). In
addition, the direction of rotation of the tip vortices reverses with each
half-stroke.
The reality of these vortices is well established. Kokshaysky (1979) was
the first to persuade birds to fly through clouds of particles and then make
photographs of the vortices. Spedding (1986, 1987) then did a very much
more elaborate analysis of such moving particles for a variety of birds
under a variety of conditions. Ellington (1980b) showed that cabbage white
butterflies (Pieris brassicae) flung vortex rings earthward as they beat their
broad wings up and down in slow flight. More recently, Brodsky (1991)
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THRUST OF FLYING AND SWIMMING
(a) (b) Cx (c)
'oft fc^< CU5 V W^fc**^
Figure 12.7. The vortices behind flying birds, (a) A hovering bird
leaves a vertical stack of separate vortex rings beneath itself, (b) A bird
flying slowly forward leaves a zigzag ladder of vortices, with tip vortices
reversing direction between half-strokes and with adjacent rungs of the
ladder having opposite directions of circulation. The stronger vortex
rings correspond to the lift-producing downstroke. (c) A bird flying
rapidly forward leaves a pair of tip vortices that wobble inward when shed
during upstroke and outward on downstroke—since downstroke
produces more lift.
found quite a complicated set of coupled and uncoupled rings, depending
on flying speed, in another butterfly, Inachis io. These vortices constitute a
record of what airfoils have just done, and we're learning to make them
visible and to read their messages. This wake-vortex view of flapping flight
has proven most fruitful in the dissection of the roles of various nonsteady
phenomena involved in such flight—to which we'll now turn. It seems to
hold considerable promise, as well, for analyses of swimming in fishes.
Nonsteady Effects in Flapping Flight
The difference between a propeller and a pair of beating wings, again, is
that the wings reciprocate, changing direction and pitch twice in each
stroke. And therein lies a set of problems for analyses of animal flight even
more vexatious than anything so far considered.
On one hand, if an airfoil starts suddenly from rest, full circulation about
it doesn't develop for the first two or three chord lengths of travel. In this
"Wagner effect," the starting and bound vortices initially interact
destructively, so the lift doesn't develop right away (Weis-Fogh 1975). Since wings
may start and stop a hundred times each second and since they travel no
more than a few chord lengths before reversing, the lift actually available in
flapping flight may be considerably overestimated by steady-state measure-
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CHAPTER 12
ments on isolated wings in wind tunnels. On the other hand, if the angle of
attack of a wing is suddenly increased above an angle that would produce
stall under steady-state conditions, the wing can travel several chord
lengths before separation and stall occur; and lift can reach values around
50% greater than the steady-state maximum ("delayed stall"; Ellington
1984b).
Of what significance are such nonsteady effects? They have long been
invoked as a kind of deus ex machina to cover oversimplified views,
inadequate data, and ex cathedra statements about bumblebee lawlessness. The
first really relevant measurements were made by Jensen (1956), working
with Weis-Fogh. Jensen painstakingly showed that forward flight in desert
locusts could be fully accounted for by appropriate summing of the force
coefficients derived from steady-state measurements on isolated wings—
what's called a "quasi-steady analysis." But at least two troublesome matters
kept this from settling the issue, both emphasized by (among others)
Ellington (1984a, b). First, Jensen's work failed to find only that nonsteady
phenomena needed to be invoked and didn't disprove their involvement;
and second, his work didn't even do that for the more demanding case of
hovering flight.
Estimating the average lift coefficients needed to support a hovering
creature's weight isn't especially difficult, although the exact figures
depend a bit on the particular simplifications used—one needs to know only a
few things such as body weight; wing length and area; and wingbeat
frequency, stroke angle, and stroke plane angle. The results in all too many
cases are steady-state lift coefficients well in excess of those such as we saw
in Table 11.1 —maximally only 1.0 or a little more for insects and perhaps
up to 2.0 in birds. And bear in mind that an animal need hover only very
briefly to make trouble, since we're talking about short-term lift, not
aerobic metabolism. Ellington (1984a) and Ennos (1989b) got values ranging
between 1.2 and 4.4 for quite a wide range of hovering insects. Norberg
(1975, 1976) estimated a coefficient of about 5.0 for the pied flycatcher,
Ficedula hypoleuca, and over 3.0 for a bat, Plecotus auritus. The most extreme
cases are animals hovering with inclined rather than horizontal stroke
planes since they produce most or all of their lift during downstroke.
Delayed stall, the most conventional tactic, apparently doesn't play a
major role in achieving high lift coefficients. Of more significance is a
scheme that not just avoids the Wagner effect of delayed onset of
circulation but that gets especially strong circulation going right at the start of the
downstroke, where high lift coefficients matter most. Weis-Fogh (1973)
first proposed this "clap-and-fling" mechanism for very small insects such
as the tiny wasp, Encarsia, and fruit flies. (The original evidence included
some pictures I had published without having noticed anything of
significance!) The scheme is really quite simple, once one is accustomed to think-
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THRUST OF FLYING AND SWIMMING
Figure 12.8. Weis-Fogh's "clap and fling" mechanism for the initiation
of circulation through interaction of a pair of wings. The wings are
shown cut chordwise about halfway along their spans. At the start of
downstroke the two wings part first in the front, and air passes around
the leading edge to fill the gap. That air goes from what will be the
undersides to what will be the tops of the wings, so the direction is
appropriate for lift-producing circulation.
ing in terms of circulation and vortices. Recall that as it first develops lift, a
wing becomes surrounded by a bound vortex and leaves behind a starting
vortex of the same magnitude but opposite spin—vorticity (at least in the
short run) is conserved. What a beating wing can do is to use its opposite
number on the other side to account for the opposite vortex, rather than
having to leave one at the starting gate—each wing aids development of the
bound vortex of the other. What happens is that at the end of the upstroke
the wings "clap" together above the thorax, either touching or coming very
close. They then rotate ("fling"), peeling apart from their leading edges,
pivoting about the trailing edges, as in Figure 12.8. The space between
them must be filled with air, and that air can most easily come around the
leading edges—so it constitutes a circulation about each wing, well begun
at the very start of the stroke. This circulation is created quite
independently of the classical scheme described in the last chapter. What
determines its magnitude is the angular velocity of the lengthwise rotation of the
wings, not their chordwise, translational speed—the Flettner scheme
momentarily reenters the picture. So here at least the low wind speeds of
hovering flight impose no handicap.
The clap-and-fling mechanism has been found in all small insects thus
far examined, as well as some (but not all) larger creatures, including some
moths and butterflies and the hind wings of locusts in climbing flight
(Ellington 1984b). Birds that can hover only briefly, such as the pied
flycatcher (see above) and doves and pigeons, commonly bring their wings
together with an audible clap—Weis-Fogh (1975) noted Virgil's allusion to
the phenomenon in the Aeneid—and are almost certainly using the
mechanism.
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CHAPTER 12
A second rotational lift mechanism occurs farther into the stroke. The
leading edge moves downward faster than the trailing edge—as the wing
accelerates downward, the relative wind gets more nearly vertical, so
keeping an appropriate angle of attack requires lengthwise rotation as well as
the basic downward motion of the beat. This wing rotation causes
additional circulation of air around the wing, and thus generates additional lift
(Ellington 1984b). At this point there's no reason to believe that all
unconventional mechanisms of lift production have been uncovered; indeed,
people working in the area are upbeat about the chance of recognizing
others.
Swimming
Swimming, the other mode of locomotion in a continuous medium, is far
more common among animals than is flying. It's certainly easier, whether
in terms of power expenditure relative to body mass or in terms of
minimum necessary modification of, say, a walking creature. A swimmer can at
any point stop swimming (with at most a minor problem of buoyancy or
respiration), while in flight the consequences of sudden termination of
powered motion are more immediate and dramatic. Conveniently, for
most of the ways that animals swim, the principles developed for flying
ought to be applicable.
Swimming may be easier to do, but it has not proven easier for analysis.
Part of the difficulty is the greater morphological diversity in equipment
used for swimming, but the greater obstacle is an all-too-common
characteristic of the way macroscopic swimmers operate. The problem was
alluded to back when we talked about the drag of motile animals—thrust-
producing and drag-inducing structures cannot easily be separated.
Perhaps these swimming systems come especially close to an ideal, what's been
called a "zero momentum wake" (Elliott 1984), in which drag and thrust
cancel locally as well as globally, the Froude efficiency is unity, and nothing
is left behind to indicate that an animal has swum by. A fish such as a trout
bends its body in producing thrust to overcome the drag of its body. As
pointed out in Chapter 7, one can tow a dead fish and measure the drag,
but the figures come out unreasonably high, and the fish in any case
wobbles unrealistically. Stiffening the fish leads to lower figures for drag (Webb
1975), but what assurance have we that these latter are relevant to active
swimming and not just to decelerative gliding? Considerations of
momentum transfer suggest that they underestimate the force that thrust must
offset, and this argument is consistent with the common alternation of
active swimming and stiff-body gliding (Webb 1988).
This "burst-and-coast" behavior, in fact, can be used to get a crude
estimate of swimming drag. Videler and Weihs (1982) showed that saithe
280
THRUST OF F L Y I N G AND SWIMMING
(Pollachius vixens) and cod (Gadus callanas) choose initial and final velocities
for bursts of swimming close to the values expected if swimming drag is
three times greater than coasting drag. For a saithe about a third of a meter
long swimming at five body-lengths per second, they found that burst-and-
coast swimming is about 2.5 times cheaper than continuous swimming. A
really small fish, though, should gain no advantage—stiff-body drag is too
high relative to swimming drag. As he predicted, Weihs (1980) found a
gradual transition from continuous to burst-and-coast swimming as larval
anchovies (Engrauhs mordax) grew larger.
For fish alone—much less other aquatic creatures—swimming modes
are diverse. Various classifications have been used, mostly derivative of the
scheme of Breder (1926); summaries are given (with less than total
consistency) in quite a number of places, including Lighthill (1969) and Webb
(1975). A mode is most often named after an exemplar of that mode, which
implies a certain amount of pure memorization for the nonichthyologist—
presumably at the start no one else was expected to have any interest in fish
swimming! Thus passing bending waves down a body one or more
wavelengths long is referred to as "anguilliform," after the eel, Anguilla (Figure
12.9a). Waves move backward both with respect to the fish's body and with
respect to the free stream. The mode is also used by some largely or
entirely limbless amphibians, by aquatic snakes, and by the smooth-bodied
polychaete worms that lack conspicuous parapodia (Clark and Hermans
1976). The cost of transport (power per unit distance covered) appears to
be higher for this mode than for the ones that follow (Webb 1988). Eel
elvers may make long migrations; but, according to McCleave (1980), they
have to take advantage of water currents.
What we usually regard as the ordinary mode offish swimming is termed
"carangiform," after Caranx, the jack (Figure 12.9b)—and is used by the
perches, trout, cichlids, mullets, and so forth. It's the mode regarded as the
best generalist arrangement, combining versatility in possible movements
and control, good starting acceleration (using the "C-start" and other
devices about which more shortly), good maximum speed, and good Froude
propulsive efficiencies of between 50% and 80%. A wave of bending is
passed backwards, but these fishes are shorter relative to the wavelength
than the eel-like ones. (Sometimes "subcarangiform" is used for cases
where the body is more than half a wavelength long.) The wave increases in
amplitude as it passes, from negligible at the head to maximal at the tail.
Both necking just ahead of the tail and bilateral flattening of the body
minimize recoil and thus reduce body oscillation relative to tail oscillation.
In both anguilliform and carangiform locomotion, vortices are passed
down the body, one on each side per full wave, and shed with their
associated momentum into the wake—thus propelling the fish forward.
A useful distinction separates the ordinary carangiform mode and
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CHAPTER 12
(a) (b) (c)
Figure 12.9. Modes of swimming in fishes: (a) the "anguilliform"
motion of an eel; (b) the "carangiform" motion of a trout; (c) the "thun-
niform" motion of a tuna.
what's most often called "carangiform with lunate tail" (Figure 12.9c) or
"thunniform" after the tuna (Thunnus). Practitioners of the latter are
distinctive in shape, with a tail that extends across the flow a considerable
distance beyond the body. In addition, they have somewhat stiffer and
rounder bodies and a more substantial narrowing (the caudal peduncle)
between body and tail. Trunk muscle seems arranged primarily to power
the tail beat rather than to bend the body. Thus for all practical purposes
the tail functions like an oscillating propeller or a pair of beating wings; the
main difference is the fluid-mechanically minor one of a pair of hydrofoils
moving laterally and together rather than swinging above and below their
separate articulations. Aspect ratio is once again a relevant parameter, and
these fishes have fairly high values of 4.0 and above (Chopra 1975). This
latter group includes many large pelagic fish that go especially fast both
with respect to top (anaerobic burst) speed or sustainable (aerobic) speed—
the mode appears to be the most force- and power-efficient system for
high-speed swimming. Besides these tunas, mackerels, marlin, and the
like, the arrangement is used by some sharks and by whales and dolphins—
these latter with dorso-ventral rather than lateral tail oscillation; and it's
clearly indicated by the shapes of fossil ichthyosaurs. A good analysis of
how the tail of a fin whale (Balaenoptera physalus) works as a propeller was
done by Bose and Lien (1989). The flukes of their 14.5-meter whale had an
aspect ratio of 6.1; it got its best propeller efficiency at an impressively high
advance ratio of 5.0 to 6.0.
These three modes are far from exhaustive of the arrangements that
have been described and named. What they (and most of the others)
certainly have in common is the business of generating and shedding vortices.
Almost certainly they produce the zigzag vortices observed for birds flying
282
THRUST OF FLYING AND SWIMMING
forward (Figure 12.7). Since they're making almost pure thrust and doing
it with (except cetaceans) side-to-side oscillations, any ladders ought to have
vertical rather than horizontal cross-flow rungs. Visualizing these vortices
has proven even more difficult than for birds; the pictures published by
McCutchen (1977b) are crude but very nearly all we have. Recently Charles
Pell (pers. demonstration) has begun producing models out of flexible
plastic that, when forced to oscillate, produce lots of both thrust and
vortices and swim in a persuasively fishy fashion; if these prove good models of
what fishes do they could really teach us a lot by permitting far easier wake
visualization.
We'll return to fish swimming in Chapter 16, which will deal with
unsteady flows and the phenomenon of "added mass" or "acceleration
reaction," alluded to earlier in Chapter 7. For now, just bear in mind that when
locomoting in a medium of density comparable to that of the craft, the
resistance of the medium to acceleration can be of the same order of
importance as its momentum once it is in motion.
Drag-based versus Li ft-based Thrust
Consider two ways of using paired, flattened, elongate appendages to
produce thrust. They might be moved back and forth along the axis of
progression, oriented broadside to flow on a backward power stroke and
parallel to flow during a forward recovery stroke. The backward stroke
develops much more drag, and rearward drag is equivalent to forward
thrust—a proper analysis of the scheme is provided by Blake (1981). You
can easily row a boat this way, feathering the oars on the recovery stroke
without taking them from the water. It's also very close to what a paddling
duck does with its webbed feet. Alternatively the appendages might be
moved up and down, that is, in a plane normal to the axis of progression. As
we've seen in this chapter, by rotating the appendages periodically along
their length an appropriate angle of attack can be maintained to generate
enough lift via circulation so there's a net forward thrust. That's what a
swimming penguin does with its short wings.
Clearly nature makes no absolute choice between these mechanisms, so
superiority is likely to be circumstance-dependent. Thus a comparison of
their respective performances might be enlightening—it might even have
evolutionary implications. One can, in the manner of classical comparative
biologists, make guesses about function based on the different life-styles of
what we'll call paddlers and flappers; but models and mechanical analyses
seem a surer basis for understanding.
Asa start, let's therefore consider a very crude pair of models. Imagine a
pair of rectangular plates, each 10 mm by 100 mm, arranged to extend
outward from a fuselage in water. They thus have an effective aspect ratio
283
CHAPTER 12
power stroke
^
l
!\
downstroke
\
(a) drag-based thrust
(b) lift-based thrust
recovery stroke
7
t
7
T upstroke
=7
Figure 12.10. The simplified motions of paddles and hydrofoils
assumed in our comparison of (a) drag-based (top view) and (b) lift-based
(front view) thrust making.
of 20 and each has a plan form area of 0.001 m2. As in Figure 12.10, they
might move back and forth or up and down. (We'll ignore any swinging
about a point of articulation.)
Case (a): Drag-based Thrust
The plates go back and forth relative to the free stream, oriented
perpendicular to flow going backward and then parallel to flow going fore-
ward. We assume a speed of movement, relative to the craft, of 2 m s~l and
a duty cycle of 50%—that is, the recovery stroke occupies half the time.
We'll also assume that the drag coefficient, Cd,, during the power stroke is
2.0 (Ellington 1991) and that drag during the recovery stroke (negative
thrust) is negligible. At Reynolds numbers in the present range of 10,000 to
20,000, that's not unreasonable: the drag of an elongate plate parallel to
flow is only a few percent of that of the plate when perpendicular to flow
(Table 5.2). With the craft stationary in the water, the paddles have a water-
speed of 2 m s-1 on both strokes; if the craft moves forward at 1 ms-1, then
the power stroke has a water speed of 1ms-1 and the recovery stroke a
speed of 3 m s~l; if the craft moves at 2 m s~l, no thrust can be produced
since the power stroke has no motion relative to the water. So the system
has an absolute upper speed limit, even if no parasite drag is imposed. How
284
THRUST OF FLYING AND SWIMMING
speed, m/s
Figure 12.11. The results of our simple models of lift- and drag-based
swimming, looking at the time-averaged thrust produced as a function of
swimming speed.
does its thrust vary with craft speed? Figure 12.11 shows that it's terrific
when the craft is stationary but drops off rapidly toward its limit. In fact, it
might do even better at the start if we didn't do such an unrelievedly steady-
state analysis. For the first power stroke perhaps we ought to drop the duty-
cycle discount and thus double our calculation of thrust, and the
acceleration reaction (Chapter 16) will give a substantial bonus.
Case (b): Lift-based Thrust
The plates go up and down, keeping the angle of attack adjusted to
maximize the lift-to-drag ratio. We assume a speed of movement, relative
to the craft, of 1 m s~ l\ the duty cycle is 100% (no glide phase or reversal
delay); the lift-to-drag ratio is assumed 10:1 with a lift coefficient of 0.8 at
that LID. With the craft stationary in the water, the resultant force tilts
maximally forward, which is good; but the relative speed of water
encountering paddles is only 1 m s~I, which is not so good. If the craft moves, say, 1
m s~ *, then the resultant has a relatively smaller forward component, not
so good; but the relative water speed has increased to 1.4ms-1, and the
magnitude of that resultant has thus doubled, certainly good. So one thing
offsets another, and thrust continues to quite a high speed (or advance
ratio, if you prefer). This gives a more complex relationship between the
craft speed and the thrust available, as you can see on the graph. Thrust is
distinctly submaximal at zero speed (the problem of hovering, again), as-
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CHAPTER 12
cends to a maximum, and then drops (even as parasite drag must be
increasing rapidly) at still higher craft speeds.
Interesting comparison! Even without considering power requirements,
the drag-based system is very much better when the craft is stationary but
the lift-based system is clearly superior at any decent forward speed. Of
course we might have picked different conditions, but a very high flapping
rate would be needed to make the lift-based system do as well at zero speed
or a very high paddling rate to make the drag-based system better at high
speeds. I think the comparison tells us a lot about who ought to use one or
the other. If initial acceleration is what matters, then a drag-based system is
clearly advantageous. A lot of animals are wait-and-lunge predators who
might well prefer high acceleration to high steady speed. If speed matters,
then flapping is much better than paddling. But two further
considerations ought to be mentioned. The first is that no rule disallows combining
both systems, even using the same appendages. An animal just has to use a
little more complicated stroke, and to vary the stroke parameters with
speed in a fancier manner. (And such mixed strokes appear to be used by
many noncetacean swimming mammals, according to Fish 1992.) The
second goes back to the material on streamlining and the shape of airfoils.
The lift-based system is considerably less forgiving with respect to the
cross-sectional shape of the plates if a decent LID is to be achieved. Getting
a high ratio of normal drag to parallel drag can be done with cruder
equipment. Thus multifunctional appendages ought to be more likely to
use the drag-based system.
So what happens in the real world? Penguins clearly flap their wings up
and down, propelling themselves by developing lift (Hui 1988a). Sea lions
(Otariidae) get thrust solely with pectoral flippers (by contrast with true
seals, the Phocidae, which use their tails) with a lift-based system (Feldcamp
1987). Sea turtles, as well, use lift for swimming (Davenport et al. 1984).
Most fish propel themselves with their tails, as described earlier, but a few
use paired pectorals as lift-based principal propulsors. For instance, the
shiner seaperch, Cymatogaster aggregata, normally swims with pectoral fins
only, using a lift-based system (Webb 1973). The ratfish (so called because
of its rat-tail-like tail) Hydrolagus colhei, and most rays (Myliobatoidae), both
cartilagenous fishes but otherwise distant, do likewise. Far distant, phy-
logenetically, from all of these is a pteropod mollusk, the sea butterfly,
Clione limacina, which beats its "wings" in an up-and-down, lift based system
(Satterlie et al. 1985). And it appears that both portunid crabs (extant) and
euripterids (extinct) use lift as well (Plotnick 1985).
Lift-based systems may be widespread, but they're not universal. Ducks
paddle with their legs, using a drag-based arrangement (Prange and
Schmidt-Nielsen 1970; Baudinette and Gill 1985), as do muskrats (Fish
1984), freshwater turtles (Davenport et al. 1984), nereidiform polychaetes
286
THRUST OF F L Y I N G AND SWIMMING
(Clark and Tritton 1970), and large water beetles (Nachtigall 1980);
further examples can be found in Blake (1981), Braun and Reif (1985), and
Fish (1992). At least one fish uses a drag-based system: Blake (1979a) found
an angelfish (Pterophyllum eimekei) doing it with its pectoral fins when
swimming slowly. Overall propulsive efficiency is low—around 0.18—but, as
Blake comments (and is consistent with our present argument), the
efficiency of lift-based propulsion ought to be even lower at such low speeds.
As expected, many of the drag-based swimmers occur where there's a
less than single-minded focus on swimming as a way to get from place to
place. Polychaete worms burrow; and ducks, muskrats, and freshwater
turtles all walk on the same legs as are used for swimming. By contrast, sea
turtles walk only minimally and crudely on their flippers.
Acceleration as a consideration is a little more complicated since lift-
based swimmers often use some non-lift-based assist for starts. But it must
be important. Webb (1979b) noted that a lift-based system in a fish needs
quite a lot more muscle to get comparable accelerations even though the
fish can achieve much higher final speeds. Similarly, Davenport et al. (1984)
found that, while sea turtles could swim six times as fast as freshwater
turtles, sea turtles produced only twice as much static thrust (the stuff from
which acceleration is made) when tethered. Ordinary fish do accelerate
very well, but they usually use special, high-amplitude (drag-based)
bends—the "C-start," as shown by Webb (1976, 1978), Harper and Blake
(1989), and Frith and Blake (1991). A diversity of fish can achieve
accelerations of about 40 or 50 m s~2, according to Webb (1975). Some do
much better: Harper and Blake (1989) measured a peak acceleration in a
pike, Esox lucius, using a C-start of 245 m s~2, about twenty-five times
gravity!
Perhaps the most extreme development of this initial drag-based body bend
is in crayfish and similar crustaceans, which begin their impulsive rearward
motion with a powerful flexing of the abdomen. Webb (1979a) measured an
acceleration of 51 ms~2 in a crayfish (Orconectes virihs); Daniel and Meyhofer
(1989) got 100 m s~2—about ten times gravity—in a shrimp (Pandalus
danae); and Spanier et al. (1991) got up to 5 m s-2 in a lobster (Scyllandes
latus). Since maximum accleration decreases with size for basic reasons of
scaling (Vogel 1988a), these are probably comparable performances.
At least one partly drag-based system is used in flight: broad-winged
butterflies throw periodic vortices earthward with such a scheme (Ellington
1980b).
Fillips and Flourishes in Flying and Swimming
A few incidentals, to tie up loose ends of lift-based locomotion as
practiced in both air and water . . .
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CHAPTER 12
Formation Flight in Birds
A bird flying rapidly sheds a continuous tip vortex from each wing, with
air on the sides of the vortices toward the bird going downward and air
outboard going upward (Figure 12.7c). So a bird flyingjust behind another
should be at a disadvantage. Conversely, a bird flying behind and to one
side of another ought to be able to take advantage of the upwardly mobile
air to reduce its cost of flight. A good aerodynamic argument thus favors
flying in a diagonal line or a vee-formation in a horizontal plane. (While the
leader clearly has to work harder than the followers, the leader need work
no harder when followers are present than when they're absent.) Lissaman
and Shollenberger (1970) calculated that by flying in formation twenty-five
birds could get an increase of 70% in distance traveled for a given
expenditure of energy. Badgerow and Hainsworth (1981) figured an energy
savings of up to 23% for Canada geese (Branta canadensis) flying in formation.
But achieving best economy demands precise coordination of position and
the frequency and phase of wing beating, and the variation that
Hainsworth (1989) found implies that the full, calculated savings in induced
power are unlikely to be realized in practice.
Schooling in Fish
If birds can do it, why not fish? Weihs (1973) has described a diagonal
diamond pattern with which swimming fish can save energy. The vertical,
diagonal diamond amounts to a filled-in vee; fish, of course, need little net
lift, so one can find a good site directly behind another as long as it's
sufficiently far back. Also, lateral neighbors give additional benefit, as long
as they're sufficiently far aside. While fish clearly school for nonhydrody-
namic reasons, the appropriate spacings and synchronizations for
energetic benefit do seem to occur.
Ground Effect
If an airfoil moves just above a solid surface, its performance is improved
by a phenomenon termed "ground effect," mainly as a result of a decrease
in induced drag. Withers and Timko (1977) have pointed out that
skimmers (Rhynchops nigra), by flying close to the surface of calm water, can
achieve about a 20% reduction in their requisite power (assuming an
average wing-to-water distance of 70 mm). A skimmer, as pointed out back in
Chapter 7, flies with its lower mandible partly submerged, but the
mandible is well streamlined, and its hydrodynamic drag is negligible compared
to the aerodynamic forces on the body. Fishing bats (Fish 1990) probably
derive similar assistance from ground effect. And the phenomenon came
288
T HRUST OF F L Y I N G AND SWIMMING
up in the last chapter in connection with the sea anchor soaring of petrels
and prions and the gliding of flying fishes. The scheme sounds like a
wonderful way to check ground or water for prey, but I suspect that flying
speeds are ordinarily too high to allow enough time for the predator to take
action before moving well beyond the prey. Ground effect does somewhat
reduce the minimum speed of flight, and that ought to be important for a
bird (especially a large one) landing on ground or water without an
awkward run or roll.
Ground effect seems to be used by fishes that swim just above the bottom,
as many bottom-feeders do. Blake (1979b) calculated that it enables a
mandarin fish (Synchropus picturatus), a negatively buoyant fish that hovers near
the bottom, to reduce its power output 30% to 60%.
Combining Clap-and-Fling and Ground Effect
At the end, a speculative note. Flatfish, whether skates and rays or
flounder and plaice, begin swimming upward from the bottom by elevating their
leading edges. This might be especially effective in initiating the vortices
associated with their production of thrust—using the hydrodyamic
advantage of the fling or peel mentioned in connection with flapping wings.
Together with the ground effect just described and the mechanical
advantage of having a solid substratum to push against (used according to Webb
1981 in the fast start of a flatfish, Citharchthys stigmaeus), it could provide
considerable assistance in starting. The matter was mentioned as a
possibility for rays by Heine (1992) but work on the relevant hydrodyamics
hasn't been reported. Another place where something of this sort might
work is in penguins swimmingjust beneath the water's surface.
Underwater swimming is better than surface swimming, as we'll see in Chapter 17,
and being deeper is better than being barely submerged. But moving
flippers downward from near the air-water interface might help start
circulation and improve their efficiency. And the same possibility might be of
consequence in dolphins and other air-breathers that often swim near the
surface.
289
CHAPTER 13
Flows within Pipes and Other Structures
Quite a few chapters have come and gone since we last encountered
situations in which the moving fluid was on the inside and the solid
side of the interface was on the outside. Pipes, in particular, were put aside
after the discussion of continuity and Bernoulli's principle—with the aid of
streamlines, imaginary pipes could fill any field of flow. But organisms are
filled with pipes and channels through which fluids flow, so it's worth
plumbing the intricacies of how fluids go through such conduits. We'll
mainly concern ourselves with fully laminar flow, where at every point both
instantaneous and time-averaged velocities are the same. What we'll find is
a domain far tidier than those to which we applied crude devices such as
boundary layers and force coefficients. The tidiness, though, isn't
principally a result of dealing with internal rather than external flows—although
that does simplify things a bit. Mainly, it reflects crossing a line defined by
the Reynolds number. The external flows of interest so far have been either
transitional or turbulent; that is, we had to contend with vortices,
separation, and true turbulence. Here we're dropping down into fully laminar
flows for the first time, simply because internal flows of biological interest
are mostly in that domain.
Basic Rules for Laminar Flow
Consider for a start a portion of a long, straight, unbranched pipe, one
with rigid walls and a cross section of circular shape and constant size. Fluid
has entered the pipe a long way upstream and flows steadily through it.
Continuity works with a vengeance—the amount flowing through any
section is the same as that passing through any other section. What can we
say, in quantitative terms, about flow through it?
Velocities across a Pipe
One nice thing to know would be the way in which velocity varies across
the pipe—the equivalent of the velocity variation with depth in a boundary
layer. An engineer usually uses a velocity distribution as a step toward some
other calculation; for a biologist the distribution itself is of no slight
interest. Obviously the velocity at the walls of the pipe will be zero; concomi-
290
FLOWS WITHIN PIPES
tantly the velocity along the axis must be maximal. We're a long way down
the pipe, far enough from any entrance or other complication so that the
no-slip condition will have exerted its influence throughout the cross
section. In effect, the pipe will contain nothing but boundary layer, so the
notion will be of no utility and can be discarded. At the same time the
principle of continuity vetoes any chance that the skin friction of the pipe's
walls might continually slow the flow—friction has to reduce the pressure,
but it can't slow the flow. Thus we might expect that some velocity
distribution across the pipe will, once established, persist farther downstream—
like the logarithmic boundary layer on the earth's surface rather than the
ever-thickening layer behind the leading edge of a flat plate.
Assume a piece of pipe of length / and radius a, as shown in Figure 13.1.
What keeps fluid moving through it? It must be a drop in pressure, A/?,
between the ends of the piece, exerted as a uniform cross-sectional push.
We needn't worry about variations in pressure across the pipe associated
with differences in velocity—Bernoulli, remember, is strictly applicable
only along streamlines and assumes no frictional losses, which is exactly
what is causing the present velocity variation. Thus pressure, not total
head, will be constant across the pipe, just as across a boundary layer. What
will resist the motion? The only relevant agency is the skin friction of the
walls, dependent on the shear stress at the walls, again as in a boundary
layer (and as in the definition of viscosity in Chapter 2). In a steady flow, the
push and the resistance must just balance.
Since flow will be axisymmetric for a circular pipe, we can consider the
piece of pipe as filled with concentric cylinders, each with a particular
radius, r, measured from the center. These cylinders will slide past one
another, with the center one going fastest and the outer one not moving at
all. The force pushing a cylinder and each cylinder within it will thus be the
pressure drop over that length of pipe times the cross-sectional area:
The force resisting the push will be the shear stress (t) times the surface
area of the side walls of the cylinder, and shear stress is the product of
viscosity and velocity gradient:
F, = T(2irr7) = |ul ^ (2-rrr/).
ar
These two forces will be equal and opposite, so
A/nrr* = "^^ (2irr7).
Canceling and rearranging, we get
29 1
CHAPTER 13
(Ap = pi - p2)
P2
t
Figure 13.1. Conventions and symbols for describing flow through a
length of cylindrical pipe.
dU = -
A/?r dr
And integrating with U and r as the variables,
U =
frdr= — ——T- + C.
2[il
4 yd
The no-slip condition requires that U = 0 where r = a, so
_ a2A/>
And the whole expression becomes
U, = $ («» - r»).
(13.1)
How nice! Equation (13.1) is exactly what we sought—a formula for how
the velocity varies across the pipe. The distribution, as shown in Figure
13.2, is a particular parabola. At the center (r = 0) the velocity is maximal,
while at the walls (r = a) it's zero. Note that the velocity approaches its zero
value at the walls more nearly linearly than asymptotically. The equation
says other agreeable things. It asserts that if the pressure drop per unit
length (A/?//) doesn't change, then the velocity distribution won't either
and, of course, vice versa.
Total Flow
Consider the same piece of pipe filled with concentric cylinders of
flowing fluid, but now with a face area, dS, on each cylinder in the cross-
sectional plane of the pipe. Each of these face areas is
dS = 2irr dr.
292
FLOWS WITH IN PIPES
\
I
9
/
*
m<m^~
^ *
*
*
Figure 13.2. Velocities at a series of points across the diameter of a
long, straight, circular pipe for laminar flow a long distance from the
pipe's entrance. Lengths of the arrows are proportional to flow speeds at
their bases; the dashed line marks the parabola of both calculation and
measurement.
The volume flow (Q, volume per unit time, as opposed to U, distance per
unit time or speed of flow) through each face, dQ, will be
dQ = UrdS = U, (2irr) dr.
The overall volume flow will then be
Q= I c/, (2-rrr)dr.
â– M)
The integral might look awkward, but we can now have equation (13.1) for
U, and so can plug 'n chug to get
irA/?a4 _ irApd4
0.
8jjl/ 128 yd
(13.2)
(The d in the final term is pipe diameter.) This expression is known as
"Poiseuille's equation" or the "Hagen-Poiseuille equation" after its
independent discoverers. As Prandtl and Tietjens (1934) pointed out, Hagen
has precedence by a year (1839 vs. 1840) but he expressed his results in
obscure units and for many years missed recognition. (Message from a
reviewer of all too many papers: stick with SI.) As for Poiseuille, opinion in
the English-speaking world varies on an appropriate mispronunciation.
Equation (13.2) makes the famous statement that volume flow is
proportional to the fourth power of radius or diameter. A larger pipe of the same
length carries much more fluid for a given pressure drop than does a
smaller one. Thus halving the diameter of a pipe without change in volume
flow rate entails a 16-fold greater pressure loss. Remember, though, that
two factors co-conspire to give the drastic exponent—the larger pipe has
both greater cross-sectional area and less wall area relative to its cross
293
CHAPTER 13
section. So dividing a big pipe into a pair of smaller ones without changing
total cross section will, if the pressure drop is unchanged, only halve the
overall volume flow rate. Also bear in mind (as some popular literature on
coronary arteries does not) that pipes are commonly used in serial arrays,
and alteration of the size of one element of an array will almost never
change either flow or pressure drop in the way that casual application of
equation (13.2) suggests.
Notice, mirabile dictu, the lack of nonsense about arbitrary constants,
empirical coefficients, Reynolds numbers, or other bits of deus ex machina.
Indeed, the main reason for presenting a derivation of (13.1) and (13.2)
was to expose this uncharacteristic absence of chicanery. We did assume a
few things—a long, straight pipe, laminar flow, uniform pressure, and
viscosity—but we got unique and simple solutions. And these theoretical
results correspond very well to reality—so well that (as Massey 1989 points
out) the coincidence constitutes one of the better arguments justifying the
presumption of a no-slip condition at walls.
Resistance
In laminar flow through a pipe, we've just shown that volume flow is
proportional to pressure drop per unit length. The constant of
proportionality can be called the resistance (R), as in its analog, Ohm's law, for
electrical conduction. Thus
A£ = _M
Q ira4 v
This quantity, resistance, can be used to characterize some pipe or system
of pipes just as the electrical resistance characterizes an ordinary ("ohmic")
conductor. It will usually be independent of the particular pressure drop,
total flow, or velocity at which it was determined. For laminar flow in a
circular pipe beyond the "entrance region" (to be explained shortly), the
final term in equation (13.3) can be used to calculate resistance; for resistive
elements in general, the middle term applies. For arrays of pipes, the usual
(electrical) rules for serial and parallel hookups apply: resistances in series
add directly, while for resistances in parallel the sum of the reciprocals of
the resistances gives the reciprocal of system resistance. Notice the fourth
power of radius in the denominator: resistance, like volume flow, is
exceedingly sensitive to small changes in pipe bore. Note also that resistance has
peculiar dimensions: force times time divided by length to the fifth power.
Power
Because of the inevitable shear stresses, pushing a fluid through a pipe
that has material walls takes work. It's a simple matter to calculate the
294
FLOWS WITHIN PIPES
power required either from the dimensions of the system along with
velocity, volume flow, or pressure drop—or power can be obtained from some
prior resistance measurement and a datum for volume flow or pressure
drop. As with resistance, the case is formally the same as that of electrical
conduction, with volume flow and pressure drop taking the place of
current and voltage, respectively:
P = QAp = QW = <^. (13.4)
Power expenditure, as we'll see in the next chapter, is quite a serious
consideration for such important arrays of pipes as those of circulatory and
filtration systems.
Average and Maximum Velocities
Still other useful relationships fall into our laps from equations (13.1)
and (13.2). The average velocity, U, of flow in a pipe is the total flow, Q,
divided by the cross-sectional area, S. From (13.2), then,
The maximum velocity, occurring along the axis of the pipe, can be
obtained from (13.1) by setting r = 0:
t/„,ax = ^J (13.6)
These are truly splendid results! They tell us that the maximum velocity
is precisely twice the average velocity, something exceedingly useful. If you
mount a calibrated flow probe on the axis of a pipe you can obtain average
velocity, total flow, resistance, and power—all from a single datum without
the bother of a lateral traverse, integration, or iffy approximation.
Conversely, if you can measure total flow, perhaps by catching the output of the
pipe over a measured time, you can calibrate a flow probe located along its
axis, or you can tell the maximum rate at which some item, in solution or
suspended, can possibly be transported.
On several occasions, I've used equation (13.6) to ease awkward
problems of measurement. In a transparent tube of liquid, one can readily
follow the progress of an advancing front of dye and thus obtain a datum
for axial velocity. Combined with the dimensions of the tube and the
viscosity of the liquid, it gives the pressure drop in the tube. The setup, then, is a
kind of pseudomanometer, useful for measuring very low pressure
differences in liquid systems. One need only connect two points in a flow by an
appropriate tube and inject a bit of dye. As an example, consider a water-
filled tube of 2 mm internal diameter and 20 cm length with a measured
295
CHAPTER 13
axial flow of 2 mm s_1. The corresponding pressure drop is 1.6 Pa or 16
microatmospheres, a really low pressure. But it's the difference between
static and dynamic pressures in a water current of 57 mm s-1, a speed of
flow that's really in the biological mainstream. As a general rule when doing
such pseudomanometry, the Reynolds number, based on the diameter of
the tube, should be kept under about 50 in order to avoid bias from
entrance phenomena. Also, the flow in the tube should be less than about a
tenth of the external current being measured, or else flow through the tube
may appreciably relieve the pressure difference being measured. The
latter requirement makes little trouble since one can always slow the flow by
using a longer tube.
Roughness
The relative irregularity of the inner wall of a pipe has only a small effect
on the volume flow or pressure drop—at least where flow is laminar. The
same kind of criteria for tolerable roughness apply that were cited for flow
over a flat plate (Chapter 8)—the critical thing is the Reynolds number of
the heights of projections. Above 30, for pointed projections, and above
50, for rounded projections, roughness appreciably affects the shearing
stress and character of the flow. The permissible heights (e) of roughness
elements, according to Goldstein (1938), are then
- < 4 Re-"* (pointed) (13.7)
LA/
- < 5 Re~1/2 (rounded). (13.8)
LA/
(The Reynolds numbers in these formulas are based on the diameters of
the pipes and not on the heights of the projections.) We see that the
permissible roughness heights relative to pipe radius decrease as the Reynolds
number increases—the flow becomes more sensitive to disturbances such
as might be caused by bumps and bristles. The limits are very generous,
though; even at a Reynolds number as high as 1000 (as with a water flow of
0.1 m s_1 through a 10 mm pipe) a pointed protrusion can be up to an
eighth of the pipe's radius without making much difference. And for lower
Reynolds numbers considerably larger protrusions are tolerable. The
main effects, then, of protrusions are reduction in the overall cross-
sectional area of a pipe and alteration of the circular cross-sectional shape.
The "Entrance Region"
All the previous formulas assume that a steady-state parabolic velocity
distribution has been established somewhere upstream. We turn now to
296
FLOWS WITHIN PIPES
V
—^1
H
—z?
Figure 13.3. Gradual development of the parabolic profile of velocity
in the entrance region of a cylindrical pipe.
what happens at that "somewhere upstream." As fluid enters a pipe from,
say, a reservoir, its speed will be nearly uniform across the pipe; the velocity
profile is called "plug flow" or "slug flow." As shown in Figure 13.3, a
gradient region gradually develops, beginning at the walls of the pipe and
thickening downstream as does a boundary layer until the final parabola is
achieved; the flow is then said to be "fully developed." In this entrance
region, mass and momentum are transferred from the periphery toward
the axis of the pipe—flow in the middle has to speed up as flow near the
walls diminishes. The kinetic energy of the fluid increases with distance
from the entrance, and the pressure drop per unit length is therefore
greater than that predicted by the Hagen-Poiseuille equation (13.2).
The immediate question about the entrance region concerns its length.
Here we have to resort to an approximation—seeking the end of an
asymptotic curve is a little like looking for the end of a rainbow. The usual
approximation, a relatively stringent criterion, calls the flow fully
developed when it achieves 99% of its final axial velocity. Just as the thickness
relative to downstream distance of a boundary layer depended only on the
local Reynolds number, so does the "entrance length" (L') expressed in
units of pipe diameters (d)\
^r = 0.058 Re = 0.058 ^. (13.9)
a |x
(Some sources give slightly different numerical constants.) For many
purposes it's unnecessary to get so close to full development, and a constant
about half that of equation (13.9) may be ample. In fact, the Hagen-
Poiseuille equation gives a decent estimate of the pressure drop almost to
the very entrance. Conversely, whatever the outcome of applying (13.9), it's
best to allow a length of at least one pipe diameter downstream from an
^
-*
-H
s
?
/
297
CHAPTER 13
entrance for the parabolic profile to develop, no matter how low the
Reynolds number.
Another way of looking at development in the entrance region is in
terms of the time needed for the process. Since length is average velocity
times time,
t = 0.058^-. (13.10)
The time needed for a flow to become fully developed is independent of its
velocity; it varies inversely with either dynamic or kinematic viscosity of the
fluid and varies directly with the cross-sectional area of the pipe.
The Limits of Laminarity
The classic experiments of Osborne Reynolds (1883), described in
Chapter 5, are again of direct relevance. Reynolds established a value of around
Re — 2000 for the transition from laminar to turbulent flow in a circular
pipe (using as characteristic length the pipe's diameter). Below 2000, a
disturbance will not persist but will damp out through viscous action.
Above 2000, a disturbance, once started, propagates throughout the fluid
as it travels down the pipe. While the transition is commonly sudden—that
is, the transitional Reynolds number is well defined—transition doesn't
necessarily occur at that exact value. Much depends on the smoothness of
entry upstream, the distance from entry or upstream branch point, the
straightness of the pipe and smoothness of its walls, and so forth. With
meticulous care, transition can be postponed to Reynolds numbers as
much as 10-fold higher.
If you want a quick indication of the limit of laminar flow in a pipe, using
Re — 2000 as the criterion, just remember that for a 10 mm pipe it's about
0.2 rns-1 in water and about 3 m s~1 in air. Only rarely within organisms
are these limits exceeded—the flow through most of our internal pipes is
decently laminar, even if many organisms find themselves in situations
where turbulent pipe flow is relevant.
Laminar Flow between Parallel Plates
The general tidiness of the equations for laminar flow through circular
pipes reflects the laminarity of flow rather than the specific and simple
geometry of the pipes. One can do equivalent derivations to obtain
equations that apply to other geometries—perhaps the most useful of such
equations is a set that applies to a channel between a pair of flat plates
(Figure 13.4). By assuming that the plates are very much closer to each
other than the channel is wide, we grant ourselves license to ignore the
298
FLOWS WITHIN PIPES
f
PI
Figure 13.4. Flow between a pair of wide but closely spaced parallel
plates—conventions and another parabolic profile.
edges of the plates, and we can again look at flow far enough downstream
to presume full development. To keep matters as similar as possible to the
case developed for pipes, let a now represent half the distance between the
plates and h define locations from the center (h = 0) to either surface (h =
a). The analog of equation (13.1) is the following:
U
h
Ap(a2 - h*)
2 m-/
(13.11)
Things look very much the same! But not exactly—the change from four
to two in the denominator means that, while the profile remains parabolic,
the parabola is a slightly different one.
A similarly obtained equation for volume flow has the same near but not
complete identity with the one for circular pipes. Wejust treat the flow as a
stack of parallel fluid plates bounded by the solid ones. Using w for the
width of the channel and d for the distance between plates (= 2a), we get an
analog of the Hagen-Poiseuille equation (13.2):
o.
2Apwa* _ kpwd*
3|x/
12 [il
(13.12)
Note that the fourth power of the radius of the pipe is replaced by the third
power of the shorter cross-sectional dimension of the channel times the
first power of the longer cross-sectional dimension. The former, of course,
provides all (in our derivation) the shear stress that resists the flow.
While at it, we can extend the parallel between circular pipes and parallel
plates by giving equations for average and maximum velocities:
U
Apa2
3|x/
(13.13)
299
CHAPTER 13
Figure 13.5. The sieve units of the gill of a tuna, usefully idealized as a
set of closely spaced parallel plates. The view is a frontal section, in the
plane indicated by the dotted line in the insert at left.
U
max
"2JU
(13.14)
The maximum velocity, instead of being twice the average, is now one and a
half times the average. Again, though, the ratio comes out to an engagingly
round number.
Quite a few biological situations involve closely spaced and parallel flat
plates—the gills offish and many invertebrates, the book lungs of spiders,
various nasal passages, and so forth. It should make little difference if, as in
nasal passages, the channels and its walls are rolled as long as channel
depth is roughly constant. Equation (13.11), for instance, was used by
Stevens and Lightfoot (1986) and (13.12) by Hughes (1966) in work on flow
through the gills of fish. A gill of an ordinary bony fish has four slots
between its arches (Figure 13.5); in tuna (where they're relatively narrow) a
slot (called a "sieve unit) is 127 |xm by 20 |xm in cross section and about 1.5
mm in the direction of flow. The Reynolds number is under 100 (Stevens
and Lightfoot 1986).
Basic Rules for Turbulent Flow
Turbulent flow is less straightforward than laminar flow, both physically
and mathematically. In turbulent flow, momentum is continually being
transported across a pipe, so the notion of a set of concentric sliding
cylinders applies only to a time average of the instantaneously jumbled motion.
The precise parabola disappears; and the velocity distribution, even as an
300
FLOWS WITHIN PIPES
average, resists simple description. The speed of flow is still zero at the
walls, but flow along the axis is less than twice the average speed. And the
roughness of the inside wall of a pipe, of little effect in laminar flow,
becomes a major determinant of the pipe's resistance.
Friction Factor
To deal with turbulent pipe flow it's the practice to heap all eccentricities
onto a dimensionless variable analogous to the drag coefficient. C(h you'll
recall, could be defined as the ratio of drag per unit surface to the dynamic
pressure. Similarly, a "friction factor,"/, can be defined as the ratio of the
shear stress to the dynamic pressure. The former is what impedes flow in
pipes; the latter is what promotes it. Alternatively, the friction factor can be
viewed as the ratio of the pressure drop per unit length times pipe
diameter to the dynamic pressure:1
f=2Apa /pU* = 4Apa
1 1/2 p/{?2- U >
Just as the drag coefficient is a function of the Reynolds number and the
shape of an object, the fraction factor is a function of the Reynolds number
and the roughness of the lining of the pipe. And in just the same way, we
can plot the friction factor as a function of the Reynolds number, as in
Figure 13.6.
This graph of friction factor against Reynolds number has several
noteworthy features. The left-hand line refers, of course, to laminar flow,
where the friction factor is simply 64/Re, a result following directly from
equations (13.2) and (13.15). (In practice one ignores both friction factor
and Reynolds number for laminar flow and just uses equation 13.2.) The
right-hand lines are a little more complicated. They begin well above
nearest values for laminar flow, indicating that the transition to turbulence
involves an abrupt rise in the resistance of a pipe to flow. For smooth pipes
the friction factor decreases slowly and steadily with increases in the
Reynolds number over quite a range. For rougher pipes that decrease ceases
sooner; in all cases the friction factor doesn't change much at yet higher
Reynolds numbers.
1 Taken literally, defining friction as shear stress over dynamic pressure generates a
coefficient with a numerical constant of 1 rather than 4 Some sources use the constant of
4, as here, but others don't. If it isn't obvious which is intended, find the coefficient that
applies to laminar flow. If the latter is 16/Re, then the constant is 1; if it's 64/Re, the
constant is 4. The names given the dimensionless index vary sufficiently to lack reliability
in the matter—pressure drop coefficient, pipe resistance coefficient, Fanning friction
factor, and so on. Most recent American sources use the 4 and call the result the friction
factor.
3 01
CHAPTER 13
0.12-
0.08
c
o
;c 0.04
0
1
A Re < 2000
- f = 64/Re\
Re > 2000
i. .....
roughest
y ***-«■rougher
f= 0.316 Re'1/4
-
smooth
T â–
1000
10000
Reynolds number, Re
100000
Figure 13.6. A plot of friction factor,/, versus Reynolds number for
flow through cylindrical pipes of varying degrees of wall roughness. Note
the transition at Re = 2000 and the fact that roughness matters only
above the transition.
Velocity Distribution
As mentioned, the onset of turbulence in a pipe is accompanied by a
drastic alteration of the distribution of velocities across the pipe. Not only
does the regular and consistent parabolic distribution characteristic of
laminar flow disappear, but, even worse, the actual distribution for
turbulent flow depends on both wall roughness and Reynolds number. For
smooth pipes, the ratio of maximum to average velocity diminishes slowly
with increases in the Reynolds number, eventually reaching an asymptote
of about 1.25 (Prandtl and Tietjens 1934). For rough pipes, this ratio is
nearly constant above the transition range, with the value depending on
the roughness. The subject of velocity distributions and friction factors is
somewhat more complex than this and is of great technological (if not
biological) consequence; it's considered in detail in most textbooks of fluid
mechanics.
Entrance Length
The entrance length for turbulent flow is less than that for laminar flow.
This results from the lateral transport of momentum in eddies and from a
steady-state velocity profile closer to the slug flow of the entrance. Caro et
al. (1978) give the following equation for estimating entrance length:
^r = 0.693 Re
a
1/4
(13.16)
302
FLOWS WITHIN PIPES
As with the formula (13.9) for laminar flow, the criterion is a fairly stringent
one and might be relaxed substantially for many purposes.
Flow through Circular Apertures
Two geometrically similar but physically disparate situations share this
designation—first, flow of a liquid out a bunghole into a gas and, second,
flow of a fluid through what is essentially a pipe of negligible length from
one reservoir into another. We'll postpone any consideration of the former
to Chapter 17 since it involves fluid-fluid interfaces, focusing here on the
latter, sometimes called "flow through a submerged aperture." Flow
through such orifices isn't uncommon in biological systems—consider pits
and stomata in plants, perforate vessel walls in various places such as the
glomeruli of kidneys, and various other filtration devices such as the
dermis of sponges. Still, in all such cases one must carefully distinguish
between situations in which net transport is osmotically or diffusively
driven and those involving true bulk flow forced by hydrostatic pressure
differences.2
For a single aperture, reasonably far from other apertures, a formula
analogous to the Hagen-Poiseuille equation (13.2) is quoted by Happel and
Brenner (1965):
The practical difficulty with equation (13.17) is that it's dependable only up
to a Reynolds number of about 3. Above that, edge effects complicate
matters, the shape and sharpness of the lip become important, and vortices
are generated as fluid leaves the aperture.
For higher Reynolds numbers, another and more commonly cited
formula has to be used:
Q= C„ira2 y]^-. (13.18)
In (13.18) a dimensionless "orifice coefficient," C0, is needed. The value of
the coefficient varies with the Reynolds number but may be taken as 0.61
above about Re = 30,000 as a reasonable approximation. Do not, as I once
inadvertently did in a lecture, refer to the orifice coefficient as "just
another bugger factor." This orifice coefficient has to be distinguished from
the so-called coefficient of discharge, which more commonly (usage is
inconsistent) applies to nonsubmerged apertures—but sometimes the same
2 Even if the hydrostatic pressure difference has ultimately been generated by some
osmotic potential.
303
CHAPTER 13
formula applies. Also, whatever the Reynolds number, bear in mind that
these formulas presume flows that are hydrostatically driven without any
push from preexisting momentum. In practice that means that flow within
the orifice must be at least several times faster than the flow upstream of it.
Formulas for Flows at Low versus High
Reynolds Numbers
Both for fully developed flow in circular pipes and for flow through
sharp-edged circular apertures, we've needed two kinds of formulas. One
applies when flow is not only laminar but vortex-free as well; the other is
used for other situations—laminar flow with discrete vortices and fully
turbulent flow. The contrast between the two kinds of formula is a central
and important one in fluid mechanics. It's at least as important to biological
applications of fluid mechanics as anywhere in technology because living
systems produce and encounter flows on both sides of the transitions. The
contrasting character of equations applicable to the two sides of the
transition—beyond the geometric details—is worth some comment.
Table 13.1 draws attention to the features that mark this great divide by
expressing the relevant equations in equivalent form. Four of the
equations come from this chapter—(13.2), (13.15), (13.17), and (13.18). A fifth
is equation (5.4), rearranged so the drag on a body (we'll presume a sphere
and use frontal area) is expressed as a rearward pressure. The sixth is the
analog of the fifth for low Reynolds numbers, Stokes' law, which will appear
later as equation (15.1). Looking at the differences between the two
columns . . .
1. Pressure drop depends on viscosity rather than on density at low
Reynolds numbers; it depends mainly (one can't make a stronger
statement with one of those coefficients present) on density at high
Reynolds numbers.
2. While pressure drop varies directly with velocity at low Reynolds
numbers, it varies essentially (again hedging a little because of the
coefficients) on the second power of velocity at high Reynolds
numbers.
3. If all terms are multiplied by area, one sees that force varies directly
with some linear dimension at low Reynolds numbers but with the
square of a linear dimension at high Reynolds numbers.
4. All of what we're calling "high Reynolds number" versions have the
form of some dimensionless coefficient times dynamic pressure.
That kind of formula, of course, is fully general because of the
inclusion of a universal variable constant—a dimensionless
coefficient providing a scapegoat for all irregularities. But it's something
304
FLOWS WITHIN PIPES
Table 13.1 Equations for pressure drop, Ap, (a) through a
SECTION OF CIRCULAR PIPE WITH FULLY DEVELOPED FLOW; (b) THROUGH
A SHARP-EDGED CIRCULAR SUBMERGED APERTURE; AND (c) AVERAGED
OVER THE FACE AREA FOR UPSTREAM VERSUS DOWNSTREAM SIDES OF A
SPHERE. IN (c) THE AVERAGE VELOCITY REFERS TO FREE STREAM FLOW.
Low Re High Re Transition Re
8|x/tf fl_ pU*_ 2000
~a~5~ 2a ~2~
Stt^O 1_ p£/2 3
T~ Q* ~2~
6\xU pf/2 0.5
~a~ C(i ~2~
of a last resort, as emphasized back in Chapter 5. By contrast, the
low Reynolds number versions are blessedly unambiguous.
5. The transition points are the maximal Reynolds numbers to which
the "low" versions can be safely applied. By engineering standards,
even what we're calling "high" are really very low Reynolds
numbers. But biology lives with a curious asymmetry with respect to this
general divide. For apertures and spheres, the divide corresponds
fairly neatly to what we'd ordinarily regard as microscopic and
macroscopic worlds. For circular pipes, both macroscopic and
microscopic systems are mostly on the low Reynolds number side, and we
encounter the other domain mainly in either field work or in the
design of apparatus such as flow tanks. The message, to repeat, isn't
that pipe flows aren't intrinsically tidy and external flows naturally
messy (by analogy perhaps with lab and field science) but rather that
we're ordinarily on the side of the angels in one case and too often
bedeviled in the other.
Flow through Porous Media
Whether flow goes through a pipe or between parallel plates, if it's
laminar the average velocity varies directly with the pressure drop per unit
length. That's what Gotthilf Hagen and Jean Louis Poiseuille determined
in 1839 and 1840, respectively. In 1856, the same basic relationship was
shown for what might appear quite a different situation—flow through
porous media—by Henri Darcy, as what's now called "Darcy's law" (Rouse
and Ince 1957). Its applications are wide, including seepage underneath
dams, flows beneath beaches and streams, and filtration by flow through
packed columns. On one hand, flows are inevitably slow and passages nar-
(a) Circular pipe
(b) Circular aperture
(c) Sphere
305
CHAPTER 13
row, so laminarity is certain. On the other hand, the passageways are
complicated and irregular, so one can talk only in global or statistical terms
about flow and pressure—velocity profiles of the kind we derived earlier
can't be easily specified.
Flow though a porous medium made up of packed particles is equivalent
to flow through a statistical spaghetti of pipes with ramifying and
anastomosing passages. Clearly the constant of proportionality between
pressure drop and flow must depend on the average size, the distribution of
sizes, and the arrangement of the solid particles. Despite the complications
the so-called Kozeny-Carman equation, embodying Darcy's law, turns out
to work quite well in practice. Its problem is inclusion of several variables of
variable degrees of awkwardness. (Massey 1989 gives a straightforward
derivation of the equation and its constituent variables, and Leyton 1975
discusses its biological relevance.) Assume a channel of constant overall
cross section filled with particles through which fluid flows under the
influence of a specific pressure drop per unit length. One needs to know (1) the
volume of the solid particles, Vs, and (2) the voidage or porosity, e, the
fraction of the total volume of the pathway made up of the fluid phase. One
also needs (3) the total surface area, S, of the particles. All can be
determined or approximated from fairly simple measurements. All other
relevant variables are lumped into the Kozeny function, k—the tortuosity of
the passages, the effective surface area of the particles since some solid
surfaces contact others, and so forth. Specifically, then,
A/? e3 1
U= — . (13.19)
/ (1 - e)2 yM(S/Vs)2 )
In practice, according to Massey, packed granular materials of uniform
particle size have values of e between 0.3 and 0.5. As particle size becomes
more heterogeneous, porosity drops sharply—voids between large
particles are increasingly filled with small ones. Immediately adjacent to walls,
porosity is greater—a wall prevents contacting particles from interdigitat-
ing as effectively. Thus in a narrow packed column or one with a lot of wall
relative to cross section, flow may be somewhat greater than calculated.
That's rather the opposite of what happens in open columns, but in packed
columns the particles rather than the walls constitute the dominant
resistance.
Kozeny's function, k, normally ranges between 4.0 and 6.0; and a value
of 5.0 is commonly assumed for calculations with equation (13.19).
Considering the abundant and diverse fauna living in porous sediments
(as well as the use of separation columns in the laboratory), the Kozeny-
Carman equation ought to see more biological application than I've been
able to uncover. For macrofauna, penetration of dissolved oxygen will
depend strongly on even slight local flows; and for local flow around inter-
306
FLOWS WITHIN PIPES
stitial microfauna, overall bed resistance is clearly crucial. On a larger scale,
Riedl (1971b) and Riedl and Machan (1972) have found that the net flow of
water into beaches driven by the gravitational pressure head of the swash
has a typical speed of 0.5 mm s~l, and that the resulting flow into the ocean
is greater than that of all the world's rivers combined.
Just as the forces of flow affected the shapes of flexible objects, which in
turn affected those forces, flow through a porous medium may affect the
porosity of the medium, which can then alter the flow. In particular, if
water flows upward through a bed of particles at a sufficient rate (which
may still be very slow), the particles may "unpack" to some extent. In effect
they're supported to some extent by fluid motion as well as by each other; as
a result of this "fluidization" the properties of the bed can change
dramatically. The most common phenomenon that can be blamed on such upward
flow and slight unpacking is so-called quicksand, a sandy bed that looks
solid but behaves more like a fairly viscous liquid. I wouldn't be at all
surprised to learn that some infaunal organism makes quicksand either
incidental to respiratory pumping, as part of a burrowing scheme, or as a
feeding device. But at this point I can't cite specific investigations.
307
CHAPTER 14
Internal Flows in Organisms
The past chapter considered little more than the physical
characteristics of internal flows and the various equations that have proven
useful in dealing with them. While choices of scale and regime were made
with living systems in mind, the specifics of these systems were deferred.
These biological systems are both diverse and complex; a whole book
rather than just a chapter could profitably be devoted to the subject. In any
case, I'll use the title of the book tojustify an episodic rather than a
comprehensive look. And these systems are pretty fancy. Most circulatory systems
of animals involve pulsatile flow of non-Newtonian fluids in pipes of time-
varying cross-sectional areas and shapes; here we have space only for a stab
at the high points. The vessels through which sap ascends in vascular plants
are tiny, interrupted by barriers, and filled with water at wildly subzero (not
just subatmospheric) pressures—their fluid statics have proven sufficiently
enigmatic to loom larger in the literature than their fluid dynamics.
Circumventing the Parabolic Profile
The velocity distribution across a circular pipe in fully developed
laminar flow is parabolic, with an axial velocity precisely twice the average
velocity. In many biological situations either material or heat is exchanged
across the walls of pipes; a moment's consideration should persuade you
that a parabolic profile isn't exactly ideal for exchange. Too much of the
material or heat passes down the middle of the pipe and too little moves
along the edges, near the site of exchange. This places a heavy burden on
molecular diffusion or conduction, weak reeds in a macroscopic, non-
metallic world. And nature has repeatedly found it prudent to arrange
something better than a parabolic profile.
How Far Is Flow from a Wall?
Before revelling in adaptive tricks, it's handy to have an expression for
the distance of the average particle of flowing fluid from the walls of a pipe
or channel. A convenient device is a dimensionless, size-independent
index that compares the average distance of the flow from a wall with the
radius of a pipe or the depth of a channel—we might call it the "distance
index," Di.
308
INTERNAL FLOWS IN ORGANISMS
Consider flow through an axisymmetrical circular pipe. To get the mean
distance, the pipe again needs to be viewed as a series of concentric
cylinders, each with an annular face. The area of each annulus (2irr dr) must be
multiplied by the local velocity (Ur) and the distance from the wall (a — r)
and divided by the total flow (ira2U). The result is then integrated across the
radius to get the mean distance, which is divided by the radius of the pipe.
Written out as an integral, the procedure can be summarized as
2 f° 2ir fa
Di = . UAar - r2)dr = — UAar - r2)dr. (14.1)
a3U •>() Qa Jq
Either form of this expression can be solved by either of two approaches. If
an explicit expression is available for the velocity distribution, that is, for Ur
as a function of r, the expression can be substituted for Ur and the integral
evaluated by the usual mathematics. Alternatively, an experimentally
determined set of measurements of velocity at different radii can be used.
One by one, each is multiplied by its appropriate dr (now a Ar) and by (ar —
r2); the sum of the results is then multiplied by the factors outside the
integral sign.1
For a uniform velocity (plug or slug flow) across the pipe (Figure 14. la),
U = Ur. The remainder of the integral is easily evaluated, and everything
cancels out after integration except a numerical coefficient. The latter is
the desired distance index, 0.333. Flow is, on the average, a third of the way
from wall to axis.
For the parabolic profile of fully developed laminar flow (Figure 14. lb),
Ur is given by equation (13.1), so one just has to insert that along with
equation (13.2) for Q into (14.1). After we do a little more bookkeeping, a
numerical coefficient again emerges; this time the distance index is 7/15,
or 0.467. Flow, on the average, is 40% closer to the axis than in the
preceding case; for exchange of material across walls, parabolic flow is
substantially worse than slug flow.
What if the parabolic profile were reversed? That's about what should
happen if what is forcing the flow rather than what is resisting the flow is
located along the walls. The distance index one calculates depends on the
conditions assumed; consider for an example a case in which axial flow is
exactly zero, wall flow is maximal (assuming that the gradient region
enforced by the no-slip condition is thin enough to ignore), and the parabola
is the inverse of the previous one (Figure 14. lc). The distance index comes
out to 0.20—flow is 40% closer to the wall than even for slug flow.
To get some examples of the use of equation (14.1) with empirical data, I
1 Note that this procedure isn't universally applicable. The Romberg method,
conceptually a little more complex, would require fewer measurements for a given level of
precision; for details, see Pennington (1965).
309
CHAPTER 14
(a)
(b)
(c)
(d)
Figure 14.1. Velocity profiles across a cylindrical pipe: (a) plug or slug
flow, with uniform speed across the pipe; (b) our familiar parabolic
profile; (c) a reversed parabolic profile, with zero speed on the axis and
highest speeds almost at the walls; (d) an experimentally determined profile
for a pipe containing an array of diametric wires.
determined the velocities across a real pipe (about 75 mm in diameter),
using it in several arrangements. For an unobstructed pipe, I got a distance
index of 0.447—the flow was not quite fully developed (Di = 0.467). When
I filled the same pipe with wool fibers, the distance index dropped to
0.387—much closer to slug flow figure of 0.333. The effect of the filling is
to distribute the resistance to flow across the entire volume of the pipe and
thus to reduce the steepness of the parabola. But doing still better was easy.
I threaded a set of wires across the pipe, with each wire passing
diametrically and thus crossing the axis normally (Figure 14.Id). In effect, the
scheme put an axial plug in the pipe. As a result, the distance index
dropped to 0.311 —better than slug flow. And the pressure drop caused by
the wires was substantially less than that of the wool stuffing. It's uncertain
whether any organism grows hairs that, by extending to or beyond the axis
of a pipe, push the average flow toward the walls and increase exchange
between fluid and walls. It's also uncertain, to me at least, whether anyone
has looked for such a phenomenon. Don't sniff at the notion—who knows
what nasal hairs might do!
The general ways to circumvent parabolic flow, to reduce the distance
index, and to improve intimacy of flow with walls are worth a little
exploration.
Using Noncircular Cross Sections
From the point of view of exchange, a circular section is the worst
geometry, whether we consider slug flow or parabolic flow, whatever its
compensating advantages in cheapness of construction, mechanical robustness in
the face of pressure differences across the walls, and minimum pressure
drop per unit of total flow. Flow between parallel flat plates is better. An
310
INTERNAL FLOWS IN ORGANISMS
analog of equation (14.1) for this latter geometry (using the half channel
from center to one wall) is easy to contrive, and the correct parabolic flow is
described by equation (13.11). The result is a distance index of 0.625 for
parabolic flow and 0.50 for slug flow—slug flow is again better. While both
figures sound high, that just reflects the different geometry and a slightly
inappropriate comparison. One does a bit better by taking the squares of
these figures to shift from a one-dimensional to a two-dimensional
channel, getting 0.391 and 0.25 (to compare to the 0.467 and 0.333 earlier).
Large pipes through which exchange takes place are, in fact, often closer
in cross section to parallel plates than to circles. A few examples (some
illustrated in Figure 14.2) are the internal gills offish (mentioned in the last
chapter) and many other animals; nasal passages, especially those working
as countercurrent heat exchange systems (Schmidt-Nielsen 1972); the
intestine of an earthworm, with its large dorsal invagination, the typhlosole;
the "spiral valve," a spirally-coiled fold that fills most of the small intestine
of a shark; and the passages of the open parts of the circulatory systems of
the bivalve and gastropod mollusks and of the arthropods. But while most
pipes are nicely circular for reasons well rooted in a firm combination of
solid mechanics and geometry, parallel plates are only approximately
parallel, constrained merely by the present considerations of fluid mechanics
and exchange processes. So specific analyses can rest less easily on the
equations in the last chapter. Nasal passages, for instance, may be closer to
parallel plates than to circular pipes, but a look at Morgan et al. (1991)
should persuade anyone that the flows within them are bogglingly complex.
Making the Pipes Very Small
Strictly speaking, this doesn't reduce the distance index since the latter is
size independent, but it certainly will improve exchange. Indeed, the use of
very small pipes must be the most common way to improve exchange,
evident in capillary beds, the parabronchi of bird lungs, renal and Mal-
pighian tubules, and numerous other cases in which exchange occurs as
fluid moves through internal conduits. Inevitably (as mentioned in
Chapter 3) the total cross-sectional area of arrays of small pipes is huge and the
velocities in them are therefore low, providing time for exchange as well as
area across which exchange can take place. These low flow speeds also keep
the cost of pumping fluid through small pipes from becoming inordinately
high, something to which we'll return shortly.
Eddies and Turbulence
Turbulent flow has a flatter velocity profile than does laminar flow and
thus is associated with lower distance indices. But in the presence of lateral
bulk transport of fluid, the distance index either becomes irrelevant or
31 1
CHAPTER 14
Figure 14.2. Examples of pipes with noncircular cross sections:
(a) the intestine of an earthworm, with the large dorsal typhlosole
protruding downward; (b) the "spiral valve," in cutaway view, which
runs the length of a shark's intestine; (c) a cross section of the nasal
passages of a kangaroo rat.
takes on quite a different meaning—turbulence will greatly augment
exchange. A minor concomitant is the greater cost of producing turbulent
flow associated with the greater pressure drop per unit length of pipe. A
more important limitation is the impracticality, and often impossibility, of
getting turbulent flow at the low Reynolds numbers set by the sizes of most
biological pipes. Eddies can be generated by protrusions in a pipe, and
these will improve exchange, but the Reynolds number still has to be
reasonably high—at least above about 30. The nasal passages of mammals
look like reasonable candidates for such deliberate generation of
turbulence; the latter would then improve olfaction and heat exchange.
Periodic Boluses
If a solid plug is passed through a pipe, then flow immediately fore and
aft of the plug should be more nearly linear than parabolic in profile. We
send red blood cells through capillaries at a great rate, and the cells have to
deform somewhat to squeeze through. Thus exchange between plasma
and capillary walls might well be improved as a purely physical
consequence of the presence of red blood cells. In fact, putting boluses through a
pipe as close together as are the red cells at a concentration ("hematocrit")
of over 40% ought to generate an interesting pattern of flow between them.
As Caro et al. (1978) and most other standard sources point out, flow
should be toroidal in the spaces between the cells; what resistance is present
is still along the walls, so the flow of plasma (relative to the average flow, not
the walls) should be forward along the axis and rearward along the walls
3 12
INTERNAL FLOWS IN ORGANISMS
lUil/l L
T7////7
blood flow
wall's relative motion
TTTTTTTTTTTTrmTTTrr
Figure 14.3. Red blood cells moving through a capillary. To view the
toroidal flow shown between cells one needs an unusual frame of
reference—imagine that the red cells are stationary and the capillary
is moving past them from right to left.
(Figure 14.3). Either slug flow or such toroidal flow is obviously better for
exchange than is parabolic flow, as suggested by Prothero and Burton
(1961).
But there's a problem—one of significance well beyond the particulars of
red blood corpuscles in capillaries. When looking at augmentation of
diffusion, what's important is the magnitude of the proposed mechanism of
augmentation relative to that of diffusion—just as with many problems
involving several physical agencies. For this one, where the agencies are
convective bulk flow and diffusion, the relevant ratio is the so-called Peclet
number, Pe,
Pe
IU_
D
(14.2)
where D is the diffusion coefficient of the dissolved substance of interest. A
low Peclet number means less importance for convection; for oxygen in
blood in a capillary it should be about unity, which is pretty low. In effect,
the time for circulatory motion between red cells is long (low U) relative to
the time involved in purely diffusive exchange (high D and low /), so the
model underlying the textbook bolus effect isn't really appropriate (Duda
and Vrentas 1971; Middleman 1972). On the general issue of convection
versus diffusion, I suggest consulting Lightfoot (1977).
You may recall that the issue of diffusion versus convection arose earlier
(Chapter 8) in connection with diffusive boundary layers. It's always a
matter of concern for systems of cellular dimensions. At least in animals,2
2 Plant cells are typically larger than animal cells and commonly have considerable
internal bulk flow ("cyclosis"), so the coincidence of physical and biological shifts isn't so
tidy.
313
CHAPTER 14
the boundary between cellular and supercellular domains constitutes an
approximate boundary between two physical worlds. In the small world of
cells, diffusion can provide the main agency of molecular transport; in the
larger supercellular world, augmentation by bulk flow is the nearly
universal way to get around the slowness of diffusion over longer distances. I've
dwelled on the point elsewhere (Vogel 1988a, 1992a) so I won't make more
noise about it here.
Pumping at the Wall
The derivation of the Hagen-Poiseuille equation assumed that the walls
provide the resistance to flow, and this assumption generated a parabolic
profile. If, by contrast, ciliated walls provide the pumping and the
resistance comes from elsewhere in the system and is expressed across the pipe,
then the profile will be quite different. A distance index of 0.2 was
calculated earlier for the case of an exactly reversed parabolic profile. Of course
the no-slip condition must still apply, but here the gradient from zero
speed at the wall to the maximum speed of flow will be very steep, occurring
over a tiny distance adjacent to the basal part of the the ciliary layer. In fact,
a bit of backflow occurs just above the surface, presumably as a result of the
return strokes of the cilia (see, for a profile of flow, Figure 15.2a). A
reversed parabolic profile is a real phenomenon in at least one biological
situation—Liron and Meyer (1980) calculated that it should occur in a
layer of fluid above a ciliated surface, and they verified its presence adjacent
to the mucociliary membrane of the upper palate of a frog.
Cilia or flagella form propulsive coatings in the flagellated chambers of
sponges, the gastroderm of many coelenterates, the gills of many mollusks,
the excretory tubules of some flatworms, the oviducts of various animals,
and other places. (Curiously, motile cilia are apparently absent in
nematodes and arthropods.) In general, the arrangement seems to be more
common where exchange processes are taking place and less common
where simple propulsion of fluid is the primary mission. While the very
steep velocity gradients inevitably associated with ciliary propulsion in
tubes certainly improve the efficacy of exchange, those same steep
gradients must mean that the cost of operating ciliated pipes as pumps is
relatively high—a point made earlier in Chapter 3, and to which we'll return
shortly.
With ciliated walls, fluid can be made to flow in a single-ended circular
pipe; pushing it one way along the walls guarantees flow in the other along
the axis. So the notion of a situation in which the flow profile is even more
deviant from the basic parabola than a full inversion isn't, one might say,
entirely off the wall.
314
INTERNAL FLOWS IN ORGANISMS
Bernoulli vs. Hagen-Poiseuille
Textbooks of physiology commonly and commendably begin their
treatments of circulatory systems by giving a short introduction to fluid
mechanics. However, all too often the first thing presented is Bernoulli's
equation, which is then never invoked for any specific application to circulation.
The problem, alluded to in Chapter 4, is the implied assumption of fric-
tionless flow. Still, Bernoulli is sometimes useful for internal flows—
Venturi meters, for instance, are thoroughly trustworthy devices. In a
sense, the problem is that taken straight, Bernoulli's principle and the
Hagen-Poiseuille equation make diametrically opposite predictions.3 If
faster flow is associated with a decrease in pressure (Bernoulli), then a
flexible pipe will tend to collapse as speed increases. If faster flow requires a
greater pressure difference (Hagen-Poiseuille), then the pipe ought to
expand. Blow through a cylindrical balloon with its far end cut off and it
collapses, with noisy instability. Let a heart push blood faster through an
artery, and the artery swells—in taking your pulse you're counting systolic
and not diastolic phases.
Comparison between Bernoulli's equation in a simple form (Equation
4.7) and a version of the Hagen-Poiseuille equation (13.5) is useful in
deciding how far to trust the former—at least for laminar flows. Giving
Bernoulli the benefit of the doubt we set Ut = 0 in (4.7) and make an index
(called Pih for "tendency to pinch, laminar flow") with the pressure drop
due to Bernoulli as the numerator and that due to Hagen-Poiseuille as the
denominator (assuming U = for U simplicity):
Ptl " 8 yduia? " TelT/' (14-3)
But for the numerical coefficient and the extra length factor, we've
managed to reinvent the Reynolds number! What we see is that an increase in
speed is associated with a drop in pressure (Pit > 1) at highish Reynolds
numbers (short of the turbulent transition, of course), and for pipes that
are wide (large d) and short (small /). For a piece of vein, excised from a
circulatory system, with a diameter of 2 mm, a length of 100 mm, and a flow
speed of blood of 0.1 m s~ ^P^ is only about 0.1—so the vein will bulge, not
pinch, on account of flow. In fact, all normal vessels of the circulatory
system will bulge rather than pinch—even a large artery faces the down-
* It's possible to combine Bernoulli and Hagen-Poiseuille equations, using the latter as
a term for loss of momentum or energy as a result of pipe resistance. Synolakis and
Badeer (1989) urge this for physicists; a specific application of the combination is proved
by Kingsolver and Daniel (1979).
3 15
CHAPTER 14
stream resistance of smaller vessels. About the only time pinching might
happen would be in a very local constriction of a large vessel, where d is
high and / is low—a stenosis or coarctation—and there it's probably not
very much. Hence Engvall et al. (1991) quite reasonably found a pressure
drop in a narrowing of the aorta and just as reasonably only slight recovery
of the pressure just downstream from the constricted region.
An analogous situation ought to occur in turbulent flow. Using equation
(13.15) instead of (13.5) and assuming a value of 0.03 for the friction factor
gives a "tendency to pinch for turbulent flow," Pit:
Pl' = j = ~r- (14-4)
Again, the short fat pipe will tend to collapse while the long, thin pipe will
expand—automobile carburetors and laboratory faucet aspirators are
wide and not especially long. Once flow becomes turbulent, alteration of
the Reynolds number should have little effect. Probably the instabilities
and odd effects of downstream conditions found with rubber tubes of
flowing water by Ohba et al. (1984; as reported by Matsuzaki 1986) derive
from this transition—I calculate an Pit value of about 3.0 for a pipe where
they encountered instability, compared with 0.5 and 0.4 for the two that
gave none.
Of course, a pipe leading to an orifice must at some point get close
enough to the orifice so the remaining length is low enough for Pit or Pit to
rise above unity. Hugh Crenshaw has suggested to me that a urethra ought
to fulfill this condition. But the pulsating instability of the balloon with its
end cut off seems in practice not to afflict urethras—perhaps the most
distal region is sufficiently braced and damped against collapse and flutter.
On the other hand, flow limitations appear in forced micturition as well as
exhalation (Matsuzaki 1986)—perhaps we're arranged to stay normally
just on the safe side of Bernoulli-induced mischief. I do wonder if
organisms ever make use of a high Pi for deliberate pulsation, spraying, sound
production or something. Burton (1972) suggested that the instability is
what's involved in snoring; I wonder about purring in cats.
At least one case of vessel shrinkage associated with faster flow clearly
can't be blamed on the value of our pinchiness index. Faster ascent of sap in
a tree is associated with slight shrinkage of the trunk (Macdougal 1925)—
that's due to greater suction at the top and the consequent reduction of
static pressure in the vessels.
Efficient Branching Arrays of Pipes
Amid the obvious complexity and diversity of living systems, the
biologist ought to treasure any common feature. And the wider the taxonomic
range of organisms that show the feature, the more powerful a generaliza-
3 16
INTERNAL FLOWS IN ORGANISMS
tion its recognition represents. The present book is intended to be read, in
part, as an argument that the unavoidable imperatives of the physical
world underlie much of biological design. I write these grand words at this
point prefatory to introducing one of the nicest generalizations about
living systems ever derived from considerations of fluid flow.
Murray's Law
Construction, maintenance, and operation of a circulatory system are
part of the price of being a large and active animal. The price isn't trivial—
simply keeping your blood moving around accounts for about a sixth of
your resting metabolic rate. How might a circulatory system be arranged to
minimize the cost? The question is an old one; the classic analysis (but not
the first) is that of Murray (1926). He considered two factors—keeping the
blood going against the pressure losses consequent to the Hagen-Poiseuille
equation, and some additional construction and maintenance cost
proportional to the volume of the system. The first is certainly reasonable
according to the material in the last chapter. The second is sensible because blood
has to be made and replenished and because the walls of larger vessels are
proportionately thicker (a consequence of Laplace's law), so wall volume is
proportional to contained volume rather than surface area.
Murray derived an expression, now called "Murray's law," for the
optimal design of a circulatory system based on minimization of his two cost
factors; a clearer and more modern derivation and discussion is given by
Sherman (1981). In the simplest form it can be stated as
Q=ka*, (14.5)
or, volume flow through a vessel should be proportional to the cube of the
radius of the vessel. While this may look like a contradiction of the Hagen-
Poiseuille equation, it isn't anything of the sort. The latter declares volume
flow proportional to the fourth power of the radius, other things—viscosity
and pressure drop per unit length—being equal. Murray's law says that for
minimum cost, other things shouldn't be equal. In particular, since
viscosity is assumed constant, A/?// should vary inversely with radius. Another
way to put the rule is to invoke the principle of continuity and express it in
terms of the relative radii of vessels before and after a bifurcation—the
cube of the radius of the parental vessel should equal the sum of the cubes
of the radii of the daughter vessels. Or in terms of a branching manifold in
which we can look at any two generations or ranks in the array,
a()3 = a{* + a23 + . . . aw3. (14.6)
Herea() is the radius of the parental vessel, and a{, a2, etc., are the radii of
the daughters, or great-granddaughters, or whatever, as illustrated di-
agrammatically in Figure 14.4.
3 17
CHAPTER 14
]
Figure 14.4. A pipe with two daughters and four granddaughters
drawn so their diameters have the relative sizes specified by Murray's law.
The branching angles have no particular significance here. Note that di-
chotomous branching is incidental—the four granddaughter pipes would
have the same size if the original pipe had a 4-fold branch without
intermediate daughters.
Murray's law says that when a vessel bifurcates symmetrically into two
others, each of the others should have a radius or diameter 79.4% of that of
the first. It thereby says that (1) the cross-sectional area of each of the two
daughters should be 63.0% of that of the parent—the sums of both radii
and areas will be greater for daughters than parents. It also says, by the
principle of continuity, that (2) flow will be slower in the daughters by 26%.
Good—smaller pipes ought to have slower flow in order not to have
inordinate resistance. More specifically it says that (3) the average and axial
velocities in any vessel should be directly proportional to the vessel's radius.
That follows from the assertion that total flow is proportional to radius
cubed and cross section is proportional to radius squared. Furthermore, if
velocity is proportional to radius, then (4) a kind of isometry prevails—the
same specific parabolic profile characterizes every vessel. And that
(recalling the derivation of equation 13.1) means that (5) the velocity gradient at
the wall (dU/dr at r = a) will be the same throughout the system. Since (for
constant viscosity) shear stress is proportional to shear strain or velocity
gradient, then (6) all the walls in the system ought to be subjected to the
same shear stress.
3 18
INTERNAL FLOWS IN ORGANISMS
Do Real Systems Really Follow Murray's Law?
The predictions above are audacious in both specificity and generality. If
the size, density, and blood velocity of capillaries are set by the exigencies of
diffusion, oxygen transport, and metabolic rates, then the geometry of
virtually the entire remainder of a circulatory system should follow in
consequence. In fact, the predictions are only a little less powerful than this
ideal. Table 14.1 gives the relevant data for the human circulatory system.
The largest deviations from Murray's law occur in arterioles and
capillaries, the main places where the flow profile isn't really parabolic. But all
values are within an order of magnitude, which isn't bad for the cube of a
very variable variable's value. For comparison, using the squares of radii
leads to values that vary by three orders of magnitude, and using fourth
powers of radii gives a variation of four orders. LaBarbera (1990) points
out that his analysis is a bit flawed by using the reported averages of radii—
the cube of averages isn't quite the same as the average of cubes. He cites
less inclusive studies on cats, rats, and hamsters in which nonaveraged
measurements of the diameters of specific vessels give almost exactly the
predicted exponent of 3.0.
What about systems quite remote from any mammalian circulation?
LaBarbera (1990) has made a thorough search for appropriate data
against which to test Murray's law. On one hand, very little currently exists;
on the other hand, a good set is available for the water passages in sponges,
the macroscopic metazoa most distant from mammals. These are
consummate water pumpers, processing volumes equal to their own volumes about
every five seconds, so cost ought to matter a lot. And the law works about as
well for them as for mammalian circulation, with less than an order of
Table 14.1 The average radii and total numbers of the
conventional categories of vessels of the human circulatory
system along wit h the sums of the cubes of their radii.
Vessel
aorta
arteries
arterioles
capillaries
venules
veins
vena cava
Average Radius
(mm)
12.5
2.0
0.03
0.006
0.02
2.5
15.0
Number
1
159
1.4 x 107
3.9 x 109
3.2 x 10»
200
1
Zr?
(mm x 10~3)
1.95
1.27
0.382
0.860
2.55
3.18
3.38
Source: LaBarbera 1990.
319
CHAPTER 14
magnitude variation in the sums of the cubes of the radii for a range of
vessel radii of over three orders of magnitude. The exceptions are again
the elements in which flow is not parabolic—for sponges the choanocytic
chambers, where flagellar pumping takes place, and the short apopyles
that lead fluid out from them.
Murray's law fails in several instructives cases that argue that it's not
just some geometrical constraint unrelated to flow. Sherman (1981)
reminds us of Krogh's (1920) discovery that the branching tracheal system of
a larval moth preserved its overall cross-sectional area (or radius squared
rather than cubed)—just what's expected in a system relying on diffusion
rather than bulk flow. And LaBarbera (1990) mentions systems in which
pumping is done by ciliated walls, in particular the gastrovascular
transport system in corals (Coelenterata) and the coelomic circulation of crinoids
(Echinodermata).
What about all those conduits through which sap ascends in vascular
plants? My initial guess was that Murray's law shouldn't work. First, the
cells lining the vessels are dead by the time they conduct fluid (a factor
whose relevance will be rationalized shortly). Second, because of the very
high fluid tensions, large vessels seriously risk cavitation. Finally, little
metabolic power is spent in moving water upward: the main pump, an
evaporative lifter, uses solar power directly. In fact, the large vessels of tree trunks
are smaller than expected from Murray's law. But Canny (1993) has now
shown that the law works very well for small vessels in a case that suggests
generality. These are vessels with diameters from 4.0 to 25 |xm in the veins
of sunflower leaves. Here the radii vary by 6-fold, the cubes of the radii vary
almost 250-fold, but the sums of the cubes vary by the remarkably low
factor of 1.3. To guess again, I'd say that construction rather than
operating cost is the key here, made all the more important by the short lives of
such leaves.
Self-optimizing Systems
What to me is the most exciting aspect of the renewed appreciation of
Murray's law is the way it links functional performance, initial
development, and subsequent adjustment. Recall that when the law is obeyed, all
the walls in the system will experience the same shear stress. So in order to
make a branching array achieve the most efficient arrangement, only a
single local command need be given. Endothelial cells lining vessels need
only proliferate (along with the underlying tissues) until the shear stress on
each in the widening vessel drops to a preestablished set point. A variety of
studies (reviewed by Davies 1989, LaBarbera 1990, and Hudlicka 1991)
have shown the necessary cellular behavior in systems ranging from entire
animals surgically altered to cultured endothelial cells subjected to changes
320
INTERNAL FLOWS IN ORGANISMS
in the flow over them. One needn't invoke some fantastic genetic
preprogramming of the details of a circulation; the minimal needs are just
some general instructions for building vessels, some chemical released by
mildly anoxic tissues to determine where the vessels grow, and the set point
of a Murray's law system to determine their relative sizes. And the system
can be continuously retuned as the demands (such as the level of aerobic
activity) upon it change.
The Ascent of Sap in Trees
The various vessels used to move sap in vascular plants must add up to
the greatest total length of biological conduit through which fluid moves.
Almost certainly sap is moved in larger volume—despite the slow speeds—
than all animal fluids combined. The pressures involved are more extreme
by several orders of magnitude than those in any other multicellular
systems. The components of these systems may be unimposing, but these are
by any measure nature's prime fluid movers. As an earnest hemlock once
noted, the sap also rises. By "sap," incidentally, I mean the liquid that rises
in the xylem from roots to leaves, a liquid insignificantly different
(physically) from pure water; how fluid is transported in the phloem is yet
another story.
What's attracted the most attention and controversy is the origin of these
monumental pressures. The tallest trees approach 100 meters; since the
hydrostatic pressure against which any sap-lifter works is pgh, that's about a
million pascals— 10 atmospheres. To that must be added the pressure drop
from flow through vessels whose diameters are inevitably less than half a
millimeter. And that sum must be increased by the pressure needed to
draw water from the interstices of soil—in nearly dry soils that's another
enormous factor. The combined pressures, then, can approach 80
atmospheres. But nowhere in a tree can such high pressures be detected. Roots
can generate considerable pressures through an osmotic mechanism, but
really high root pressures don't generally occur in tall trees. Nor is staged
pumping a viable possibility—the vessels through which sap passes, as
mentioned a page or so back, are dead and devoid of moving parts and are
therefore completely passive pipes.
What lifts sap is overwhelmingly a pull from above rather than a push
from below. Evaporation of water in the interstices of leaves draws water up
the pipes—the system is open at the top, but the openings (in the cellulose
feltwork of cell walls) are small enough so surface tension proves sufficient
to prevent entry of air. Physical scientists react quite negatively upon being
told that the vessels contain fluid under millions of pascals of negative
pressure. But the matter is as near to settled as anything ever is in biology.
Evaporative pull was suggested a hundred years ago; circumferential
32 1
CHAPTER 14
shrinkage of tree trunks when sap speed increased was shown in the 1920s;
the magnitude of the negative pressures were determined by putting cut
twigs in pressure bombs and subjecting them to sufficient pressure to move
sap back to the cut end in the 1950s; and by a variety of methods, water has
been shown to have much more than adequate tensile strength to permit
such pressures. The story is a fine one; it's told by Zimmermann (1983)
about as well as I've heard it, but Niklas (1992) or any textbook of plant
physiology will give the essentials. I'll skip further detail since it's a tale of
fluid statics and this is a book about fluid dynamics. But the basic situation
matters here—a gravitational pressure gradient of minus 10 kPa ( — 0.1
atmosphere) per meter of height, starting at plus 100 kPa (+1.0
atmosphere) at the ground. Thus pressure is negative (water is in tension) above
a height of, at most, 10 meters.
The xylem elements that carry water upward in trees are small pipes
even in tall trees—a 0.25 mm (diameter) vessel in an oak is reckoned large.
Individual vessels may be as long as 0.6 m (grapevine, maple) or 3 m (ash).
Neither total cross-sectional areas nor vessel diameters change much with
height above ground or location in trunk, branches, or twigs. Water moves
from vessel to vessel through various pores, pits, and perforated plates,
and these may provide substantial resistance to flow. The relative resistance
of the pores can be judged by measuring vessel diameter and overall
resistance and then subtracting vessel resistance as determined from the
Hagen-Poiseuille equation (using equation 13.3). For vines, pore resistance
is negligible, and conductivity is fully 100% of that predicted by the Hagen-
Poiseuille equation. For a wide variety of trees, including both conifers
(with smallish tracheids rather than largish vessels) and hardwoods,
conductivity is only around half the calculated value. The figures cited here
and elsewhere are a bit rough, though—any calculation based on the
Hagen-Poiseuille equation (with its a4) magnifies uncertainty in
measurement of diameter. Thus reduction of vessel diameter by only 16% will by
itself reduce calculated conductivity by 50%. And vessels are not at all
internally smooth pipes of constant diameter, nor are all vessels in a tree of
quite the same length and diameter (Zimmermann 1983). Nor are they all
operative all the time.
So, once water is sucked out of soil, two agencies resist its upward flow—
the static gravitational gradient of 10 kPa m~l, and the resistance to flow
per se. Flow isn't trivial, with speeds of over 0.1 m s~l common in
hardwoods such as oaks, which is why trees get detectably skinnier when they're
rapidly evaporating water ("transpiring") through their leaves. 0.1 m s-1,
70% of ideal (Hagen-Poiseuille) conductivity, and a diameter of 0.25 mm
give a pressure drop of 7.3 kPa m~l, only a little less than the gravitational
gradient. Interestingly, just as the gravitational pressure drop is
independent of species and both vessel and pore size, so, roughly, is the maximum
pressure drop due to flow—but the first is unavoidable physics, while the
322
INTERNAL FLOWS IN ORGANISMS
latter depends on a certain congruence between vessel size and flow speed.
Trees that raise water more slowly do so with narrower xylem elements. Of
special interest is the similarity of the two figures. Xylem elements seem to
be made about as small as nature can get away with without being stuck with
a pressure drop due to flow in excess of the drop due to gravity. Or, to put
the matter the other way around, little additional flow will be realized by
using still larger vessels and reducing resistance further inasmuch as the
gravitational pressure drop cannot be altered—a variable resistance is in
series with a fixed resistance. The advantages of using many small pipes are
probably the safety of numbers and redundancy, greater resistance to im-
plosive collapse (by Laplace's law), and greater resistance to cavitation. A lot
of water is being stretched, and embolisms appear to be a significant
hazard—according to Tyree and Sperry (1988) vascular plants generally
operate quite near the point of catastrophic failure. Good reviews of the
present state of the evaporative sap-lifter of plants (with lots of references)
are those of Tyree and Sperry (1989) and Tyree and Ewers (1991). Quite
another view of how the sap rises will be found in Masters and Johnson
(1966).
Low, herbaceous plants never encounter negative pressures, so their
higher rates of flow and higher pressure gradients might be taken as an
indication of a price paid for extreme height. Their pressure gradients due
to flow and gravity, instead of being about 20 kPa m~l, run up to nearly 100
kPa m_1 (Beggand Turner 1970), and flow speeds reach not 0.1 m s_1 but
at least 0.25 m s~ l (Passioura 1972).
A Pump Primer
Up to this point, the pressure needed to push fluid was provided by a
kind of deus ex machina or, rather more literally than is usual, a power
from above. We considered pressure drop, volume flow, and resistance,
noting the analogous variables in simple, ohmic, electrical circuits. When
talking about thrust production, we were in fact talking about pumps—
they may have worked on external rather than internal flows; but
momentum flux, actuator disks, and propulsion efficiency were all aspects of the
process of expending power to move fluid in ways that the fluid would (on
account of inertia or viscosity or both) prefer to avoid. Only putting the
thrust producer in a duct is needed to make a recognizable pump, in which
power is expended to produce a pressure drop and a volume flow.
Pressure Drop versus Volume Flow
Indeed, that's the basic game for any pump—power output, P, is the
product of pressure drop and volume flow, as explained in the last chapter
(R is resistance):
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CHAPTER 14
P = QA/> = Q2R = "—^. (13.4)
What's important here is the implicit trade-off. One pump might produce a
high pressure difference from input to output but deal with a relatively low
through-put of fluid; another pump might exert the same power but invest
it in a high volume flow with only a small increment in pressure. Note
carefully that efficiency doesn't enter. We're just looking at how output
might be apportioned.
This apportionment of output is rather severely tied to the design or
choice of pump. If you use a pump that is a good pressure producer for an
application that needs mainly a high volume flow, you won't get the desired
result. I say this with the passion peculiar to one who learned the hard way.
Quite a few years ago I built a flow tank using a two-horsepower centrifugal
pump, the largest such pump that could be operated on the available 115-
volt, 15-amp circuit. The pump moved a miserable 2 (U.S.) gallons per
second—a maximum flow of a third of a meter per second through a
section only 15 cm square. A few years later I wised up and used a marine
propeller to push the water around (Vogel and LaBarbera 1978; Vogel
1981). A half-horsepower motor moved 33 gallons per second to give a full
meter per second through a section 35 cm square. That's almost two orders
of magnitude better with respect to the variable—volume flow—that
matters in a flow tank. A centrifugal pump is swell for high-head use such as
lifting water a substantial height or making it squirt a long distance
through a nozzle. On the other hand, while it imparts only a little pressure,
a propeller handles a far higher volume flow.
Nature's pumps are a varied lot—valved hearts, peristaltic tubes, ciliary
layers, paddles, evaporative lifters, and others. What isn't sufficiently
appreciated is that they operate at widely varying points in this trade-off of
pressure drop versus volume flow. For pressure drop, the undisputed
champion must be the evaporative sap lifter of tall trees or desert shrubs—
Ajfr's of up to 8 MPa (80 atmospheres) or so. For volume flow, suspension
feeders with internal filters, such as sponges, clams, brachiopods, and so
forth, must be the best things going. We're in between. A human
circulatory system moves about 5 liters per minute at rest, which is about 0.15% of
body volume per second, with a pressure drop (left ventricle to right
atrium) of around 13 kPa (100 mm Hg).4 As noted earlier, the cost of our
4 The systemic heart of a resting squid (Lohgo pealei) does about the same—about
0.18% of body volume per second (Bourne 1987) at about half our blood pressure. While
squid have the typically low resting metabolic rate of cold-blooded animals, the oxygen-
carrying capacity of their blood is also low, so they have to make a relatively large
investment in moving blood around. A further consequence is a poor capacity for increasing
aerobic metabolism during activity.
324
INTERNAL FLOWS IN ORGANISMS
circulation is not insignificant, yet a marine sponge (Foster-Smith 1976;
Reiswig 1974) or a mussel (Meyhofer 1985) may have a volume-specific
pumping rate several hundred times ours.5 Where Q is only a little less than
one's body volume per second, one can't afford a high-resistance system
that needs a lot of A/?!
Impedance Matching
A productive way to look at pumps is in terms of their "impedance," the
general term for the resistance just invoked. A pump that gets a high
volume flow from a given investment in power is spoken of as a "low-
impedance" pump. The equality of power with (X2R in equation (13.4) is
the key—high Q demands low R. Conversely, one that produces a high
pressure from a given power is a "high-impedance" pump; from the
equality of power with (Ap2)/R in (13.4) high A/? demands high R. The
evaporative sap lifter is obviously an extremely high impedance pump while the
mussel's filtration system operates at very low impedance.
The usual technological practice is to divide pumps (only a few don't fit)
into positive displacement pumps and dynamic (or fluid dynamic) pumps
(see, for instance, French 1988 or Massey 1989). The former enclose fluid
in a chamber, and by reducing the volume of the chamber cause the fluid to
leave through some prearranged opening. Most often volume is reduced
with a piston; the game is really one of fluid statics, not dynamics, which is
why no more will be said about their operation here. They're very high
impedance devices, with pulsating outputs and an intolerance for
suspended particles as their main handicaps. The latter—dynamic pumps—
ordinarily operate at lower impedance; they're in turn divided into
centrifugal and axial types depending on whether they fling fluid outward or
force fluid forward. In general, pumps with centrifugal impellers (such as
"squirrel-cage" fans and centrifugal pumps) operate at higher impedance
than those with axial impellers (most ventilating fans and propeller
pumps). That's the difference in which my nose was rubbed with my first
flow tank. (Jet engines get high impedance out of axial fans, but they do so
by using a bunch of them in a series, alternating rotor blades on a shaft with
stator blades sticking in from the walls.)
Organisms make use of both kinds of pumps as well; for them these
matters of types of pumps, operating conditions, and examples are
summarized in Table 14.2. The best known positive displacement pumps are
valve-and-chamber hearts. These often get especially high overall pressure
' Volume-specific pumping rates for freshwater sponges (Frost 1978) and bivalve
mollusks (Kryger and Riisgard 1988) are lower than those of marine species, although
within the same order of magnitude.
325
CHAPTER 14
Table 14.2. Various kinds of biological pumps.
Type
Evaporative
Osmotic
Valve/chamber
Peristaltic
Piston
Valveless/chamber
Drag-based paddles
Lift-based propellers
Ciliary layer
Helical/sine wave
Category
Positive displ.
Positive displ.
Positive displ.
Positive displ.
Positive displ.
Positive displ.
Fluid dynamic
Fluid dynamic
Fluid dynamic
Fluid dynamic
Impedance
Highest
Very high
High
High
Medium
Medium
Medium
Low
Low
Very low
Examples
Leaf sapsucker
Root sap pusher
Heart, bird lungs, squid
jet
Intestine, some hearts
Some tubicolous worms
Jellyfish jet, mammalian
lung
Crustaceans in burrows
Hive ventilating
honeybees
Bivalve gills
Sponge choanocytes
drops by using serial chambers—contractile vena cavae, atria, ventricles,
and various aortic enlargements, for instance. Fish gills (except for the
obligate ram ventilators described in Chapter 4) and amphibian
buccopharyngeal lung inflators also use valve-and-chamber pumps. Another
kind of positive displacement pump is close to the piston pumps of our
technology. It's used by a variety of polychaete worms to irrigate their
burrows—the paradigmatic polychaete Nereis, and the really fancy-looking
worm Chaetopterus (Figure 14.5), both of which have open-ended U-shaped
burrows of low resistance. Peristaltic pumps (see Jaffrin and Shapiro 1971;
Liron 1976) are also positive displacement devices, and they're quite
common in digestive systems and as hearts—peristaltic hearts are quite the
usual thing among annelid worms, occasionally supplemented by valves
and contractile chambers. The especially ugly lugworm, Aremcola, moves
water through its burrow—in either direction—by passing peristaltic
waves down its body (Wells and Dales 1951). One shaft is filled with sand,
and thus the burrow has a substantial resistance; Aremcola is a detritus
feeder whereas the previously mentioned polychaetes are suspension
feeders and require a higher volume flow. Another such pump is the
evaporative sap lifter—water is removed by evaporation and can be replaced
only by upward flow. As far as I know, all the jet engines of animals are
positive displacement devices. Some have valves (squid, for instance);
others (such as jellyfish) don't. In general, for best operation the input
orifice ought to be bigger than the output since the devices ordinarily make
fluid speed up. That can be accomplished with a single, bidirectional orifice
by changing its diameter—medusae and jellyfish clearly do so.
The usual kind of fluid dynamic pumps are hard to recognize in organ-
326
INTERNAL FLOWS IN ORGANISMS
(a)
Figure 14.5. Polychaete annelids in burrows through which they pump
water: (a) the lugworm, Arenicola; (b) the parchment worm, Chaetopterus.
isms inasmuch as both the axial and centrifugal impellers of our
technology inevitably use an "unnatural" wheel-and-axle. That doesn't reflect any
scarcity, though. All the lift-based thrusters of Chapter 12 as well as the
drag-based paddles of Chapters 7 and 12 are really fluid dynamic pumps.
The latter, at least, do service within pipes, for instance in tube-occupying
crustaceans such as the amphipod Corophium, which pumps water by
beating its pleopods (Foster-Smith 1978). Other examples of fluid dynamic
pumps are bodies that pass helical or sinusoidal waves backward, and yet
others are all the ciliary devices whose operation will be explored in the
next chapter.
The impedance range over which either positive displacement or fluid
dynamic pumps operate is wide, but the same general distinction made in
our technology is apparent—the former are higher impedance devices
than the latter. Thus evaporative sap lifters, as already mentioned, achieve
pressures of a spectacular 8 MPa. The valve-and-chamber heart of a giraffe
can do about 40 kPa (Warren 1974), as can the valve-and-chamber jet of a
squid (Trueman 1980), both above the 25 kPa or so that the heart of an
exercising human achieves.6 The peristaltic pump of Arenicola achieves
around 1200 Pa (Foster-Smith 1978), while the piston pumps of Nereis and
Chaetopterus manage only 80 and 90 Pa, respectively (Riisgard 1989, 1991).
Fluid dynamic pumps encompass a generally lower range of values,
although at least one ciliary device gets into a realm more typical of positive
displacement pumps—a burrowing echinoderm, Echinocardium, achieves
about 90 Pa. But it puts a lot of cilia in series rather than in the more
common parallel array, limiting (comparatively) the volume flow it can
manage (Foster-Smith 1978). Paddles and propellers in ducts are rare in
6 But jets aren't automatically high-pressure generators—a hydromedusan jellyfish,
Polyorchis, achieves only 40 Pa (DeMont and Gosline 1988), three orders of magnitude less
than a squid. The anal jet of a dragonfly nymph is in between, at 3 kPa (Hughes 1958).
327
CHAPTER 14
nature, but one can cite the beating pleopods of the amphipod crustacean,
Corophium, which produce about 40 Pa (Foster-Smith 1978). Such paddles
may seem an odd choice; perhaps the best way to view Corophium is as a
creature constrained by accident of birth into a phylum that doesn't know
that cilia can be motile. Nest ventilation by wing beating in stationary
honeybees clearly uses a propeller pump, but I don't know of data on
pressures achieved. The ciliary and flagellar pumps hold down the low end
of the impedance scale. We have quite a lot of data on them, since bivalve
mollusks are of both ecological and culinary importance. For such animals
as soft-shell clams (Mya arenaria) and mussels (Mytilus edulis) maximum
pressures are about 50 Pa (Foster-Smith 1978; J0rgensen and Riisgard
1988); for marine sponges (Foster-Smith 1976) they run up to about half
this value.
A useful way to view the diversity of devices and their operating
conditions is as a graph of pressure drop versus volume flow, as in Figure 14.6. In
an ideal world, one with no mechanical limitations and perfect (or
constant) efficiency, the product of the two variables, power, would be
constant; and the graph would come out hyperbolic. In the practical world, the
maximum performance of a pump is instead described by quite the
opposite kind of curve, with a specific maximum for pressure head even with no
flow, and a maximum volume flow even with no pressure head to work
against. An organism can use a pump anywhere within the envelope
described by that performance line and the axes. The organism, though, can
be characterized by a resistance—more flow takes more pressure or vice
versa. That resistance generates a line sloping up from the origin—either
straight, if the resistance is linear or "ohmic," or slightly curved otherwise.
The intersection of the maximum pump performance line and the
resistance line marks the best the system can ordinarily do.
Foster-Smith (1978) pointed out a curious feature of this best operating
point for at least the feeding pumps of marine invertebrates. They can
typically generate pressures quite a bit higher than those they actually use.
Thus the 90 Pa of Chaetopterus contrasts with the 15 Pa it uses (Riisgard
1989), and the 50 Pa of Mya and Mytilus with the 1 Pa of normal operation
(j0rgensen and Riisgard 1988). That needn't imply serious energetic
inefficiency, but one wonders about accidents of ancestry, secondary functions,
and so forth. At least wasting potential pressure is more reasonable than
wasting potential volume flow, given the function of these pumps. Systems
that waste potential volume flow are a lot rarer, enough so that I haven't a
handy example.
Measuring either pressure drop or volume flow in systems of very low
internal resistance can be quite a tricky business. Not only do the animals
not always perform (sponges in captivity are notoriously recalcitrant) but
transduction must not impose any significant resistance of its own. For
328
INTERNAL FLOWS IN ORGANISMS
pressure head
against which
pump works
volume flow pumped
Figure 14.6. Pressure drop versus volume flow for a pump. The pump
can operate anywhere within the circular curve. The line from the
origin—what it can do with a load of fixed resistance—intersects the
curve at the maximum output of the pump for that load. The line's slope
reflects the impedance of the pump, here fairly low. Notice that in low
impedance systems maximal volume flow isn't especially sensitive to the
particular pressure drop with which the pump must contend.
pressure drop, old data are scarce; for volume flow, only modern data
obtained with tiny flow probes or through calculations from clearance rates
should be trusted. To make matters worse, one can't assume a parabolic
profile emerging from some excurrent opening and calculate from a
single, axial speed, as shown by Charriaud (1982). Further difficulties arise
when one tries to determine the relationship between maximum pressure
drop and maximum volume flow, the outer envelope in Figure 14.6. Some
biological pumps can be impelled to operate under abnormal
combinations of the two variables while some cannot. The most direct approach is to
adjust the hydrostatic head against which the pump works—for instance,
exposing the output side of a U-shaped burrow to either a higher or lower
water level than the input side or, as I once did (Vogel 1978b), cannulating
an output orifice and connecting it to a device to adjust pressure and
measure flow.
Transformers
The ostensible mismatch makes one wonder, as well, about whether
nature has come up with anything equivalent to the transformers we use in
electrical systems. In a fluid transport system, such a device would allow
efficient interconversion of pressure drop and volume flow. Our
technology uses quite a range of such devices. Ducted fan-jet engines use their
329
CHAPTER 14
basic jet turbines to move additional air around the engines; they thus get
greater flow and less overall pressure drop. As a result of this impedance
reduction their Froude propulsion efficiencies are better than early jet
engines that lacked a fan and peripheral ducting. A so-called hydraulic
ram is a device that raises the impedance of an internal flow and makes
water flow uphill—a sudden surge of flow downhill drives a smaller surge
to a higher level than that of the original reservoir. For that matter, the
combination of the fixed wing and propeller of an ordinary airplane
amounts to an impedance reducer—the propeller gives a small stream a
high speed and thus a high pressure increase to make its thrust. The wing
converts some of that horizontal thrust to an upward force by making a lot
of air flow just a little bit downward. In the process it shifts from a high
pressure drop, low volume flow to a low pressure drop, high volume flow.
After all, the ambient vertical wind is negligible, so a high speed (or
pressure) and low volume flow downward would incur an unnecessary cost—
something to which a lot of talk was devoted a few chapters back. And these
aren't the only examples of impedance-adjusting arrangements.
What seems to me quite curious is the scarcity of such transformers in
nature. We do encounter lots of devices that trade off the components of
volume flow—cross-sectional area and velocity—using the principle of
continuity. But that's not a transformer in the present sense. It'sjust
equivalent to altering the gauge of a wire, trading current density against cross
section. Odd omission—unless I'm missing something or simply looking at
these systems in some inappropriate way.
But enough about pipes and pumps, lest their efflux develop an effluvial
odor; I offer mild apologies for the absence of such items as nice nectar
nippers (for instance Kingsolver and Daniel 1979) and bad bloodsuckers
(as an example, see Daniel and Kingsolver 1983).
330
CHAPTER 15
Flow at Very Low Reynolds Numbers
At the end of Chapter 13 I made a distinction between flows at
x\^ moderate or high Reynolds numbers and those at low Reynolds
numbers. In the former, drag typically varies with the square of speed, the
value of density is of more consequence than that of viscosity, and formulas
for drag or pressure drop need a dimensionless coefficient. At low
Reynolds numbers, by contrast, drag varies directly with speed, viscosity is the
prime variable, and ugly coefficients are unnecessary. We entered this
more orderly world in the past two chapters in the context of internal flows;
now we turn to their counterparts in which the solid phase is on the inside
and the fluid on the outside.
This is the world, as Howard Berg put it, of a person swimming in asphalt
on a summer afternoon—a world ruled by viscosity. It's the world of a
glacier of particles, the world of flowing glass, of laboriously mixing cold
molasses (treacle) and corn (maise) syrup. Of more immediate relevance,
it's the everyday world of every microscopic organism that lives in a fluid
medium, of fog droplets, of the particulate matter called "marine snow."
After all, the Reynolds number can be as well reduced by decreasing size
as by decreasing speed or increasing viscosity. "Creeping flow" is the
common term in the physical literature; for living systems small size rather
than (or as well as) low speed is the more common entry ticket. And it's a
counterintuitive—which is to say unfamiliar—world.
At very low Reynolds numbers, flows are typically reversible: a curious
temporal symmetry sets in, and flow may move matter around but in doing
so doesn't leave much disorder in its wake. Concomitantly, mixing is
exceedingly difficult—spreading two miscible fluids amid each other need
not actually mix them. Stirring three times clockwise can undo the results
of stirring three times counterclockwise—try it by injecting a bit of colored
glycerin beneath the surface of a beaker of clear glycerin and stirring with a
rod, first in one direction and then in the reverse.
Inertia is negligible compared to drag: when propulsion ceases, motion
ceases. Berg (1983) has calculated that if a swimming bacterium (Re =
10~5) suddenly stopped rotating its flagellum, it would coast to a stop in a
distance much less than the diameter of a hydrogen atom. Propulsion in
fluid media is possible, but not by imparting local rearward momentum to
the fluid any more than we impart rearward momentum to part of a floor
when we walk on it.
33 1
CHAPTER 15
Separation behind bluff bodies is unknown—take a look again at the
streamlines around a cylinder in Figure 5.5. Separation results from
inertia, the tendency of a fluid to continue to move downstream rather than
curve around the rear of an obstacle; and where inertia is negligible, fluid
oozes around curves and corners with magnificent indifference. Velocity
gradients are what the fluid abhors. Streamlining is a fine way to increase
drag—the extra surface exposed in the process incurs extra skin friction
and, without any separation to be prevented, pressure drag drops little.
Shape matters, but to a lesser extent and in a different way than what we've
become used to.
Boundary layers are thick because velocity gradients are gentle, and the
formal definition of a boundary layer has little or no utility; I urge that the
term be avoided for very low Reynolds numbers. Moving with respect to a
fluid alters the motion of the fluid a long distance away, and the drag of a
body moving through a fluid may be substantially increased by walls
around the fluid a hundred or more body-diameters away.
Nor can one create appreciable circulation about an airfoil. Vortices exist
only in the upper end of this realm; they're very regular, mostly core, and
they dissipate rapidly if their voracious appetite for energy isn't constantly
appeased. Turbulence, of course, is unimaginable.
And Galileo must recant even more emphatically than at high Reynolds
numbers. An object of density greater than that of the medium still falls,
but a large object falls faster than a small one virtually from the time of
release—terminal velocity is reached almost immediately. After that, the
balance of weight and drag is all that sets speed.
While this queer and counterintuitive range is of some technological
interest, its biological importance is enormous. Most often being small
means being slow—if for no other reason than that drag is a function of
surface and thrust is a function of engine size, or volume. Smallness and
slowness lower the Reynolds number in concert; since the vast majority of
organisms are tiny, they live in this world of low Reynolds numbers. Flow at
very low Reynolds numbers may seem bizarre to us, but the range of flow
phenomena with which we commonly contend would undoubtedly seem
even stranger to someone whose whole experience was at Reynolds
numbers well below unity.
How the tactics of living depend on size and Reynolds number is perhaps
best illustrated by some calculations given in a charming essay by Purcell
(1977). Consider a bacterium about one micrometer long, swimming
through water at about 30 |xm s~l and thus at a Reynolds number of about
3 x 10~5. The bacterium can swim, but should it? Diffusion brings its food
to it; to increase its food supply by 10% Purcell calculated that it would have
to move at 700 |xm s~1(1). The only reason to swim is to seek a more
concentrated patch of food—greener pastures. We've encountered the
33 2
FLOW AT LOW REYNOLDS NUMBERS
equivalent of a casual cow who, after eating, just waits for the local grass to
regrow.
But while these slow, small-scale flows may seem peculiar, they're orderly
(Purcell calls them "majestic") and far more amenable to theoretical
treatment than the flows we've previously considered. The most sacred Navier-
Stokes equations, obeisance to which was pronounced earlier, contain inertial
and viscous terms. A very low Reynolds number implies a preponderance
of viscous forces, giving license to ignore the inertial terms. With this
simplification, explicit solutions for many cases are possible, and the
advent of large computers has permitted approximate solutions to others. In
practice, then, we're able to rely more on equations and less on semiempiri-
cal graphs and coefficients. The relevant fluid mechanics is well treated in
detail by Happel and Brenner (1965), with clarity and brevity by White
(1974), and with a more biological perspective by Hutchinson (1967).
Drag
At high and moderate Reynolds numbers inertial effects were all-
important, and we found that drag depended mainly on the product of
projecting area and dynamic pressure. A dash of Reynolds number
dependence accounted for the vagaries of flow and reflected the residual action
of viscosity close to the surface. At Reynolds numbers below 1.0, drag
depends mainly on viscous forces and ought to be largely independent of
fluid density—since inertial effects are very much less than the effects of
shear. The customary drag coefficient, Cdj, as defined by equation (5.4), is
often still used, mainly to maintain consistency with practice at higher
Reynolds numbers. Using Cd, incurs no sin, since it's just the result of a
definition and carries no automatic phenomenological implications. But it
may be a bit misleading. Thus the drag coefficient of a sphere can be
figured as 24/Re, combining equations (5.4) and (15.1), up to Re = 1; as
White (1974) points out, this implies a Reynolds number effect where none
really exists. Dragjust varies with the first rather than the second power of
speed. Reynolds number is an unnecessary complication when simple
formulas can describe the drag of ordinary objects.
Spheres
Certainly the most useful formula for drag at low Reynolds number is
Stokes' law for the drag of a sphere of radius a2:
1 Consider feeding on a sugar such as sucrose, whose diffusion coefficient (D) is about
5 x 10-1() m2 s_1. The Peclet number (UUD), used last when we talked about flow and
exchange in capillaries, is thus about 0.06; so diffusion is a much more potent agent of
exchange than bulk flow.
2 But for a droplet of gas in a liquid medium, consider using equation (15.9) instead.
333
CHAPTER 15
D = fayuiU. (15.1)
Stokes' law is trustworthy up to Reynolds numbers of about 1.0—with
errors of only a few percent as that upper limit is approached. Above this,
various theoretical treatments are available, but the most useful formula
I've seen is a curve-fit equation given by White (1974):
The drag coefficient is the conventional one (5.4) that I just maligned; the
characteristic length in Re is the diameter of the sphere. The equation is a
useful approximation of the line in Figure 5.4 up to a Reynolds number of
about 2 x 105, that is, up to the great drag crisis where the boundary layer
becomes turbulent.
Of the drag predicted by Stokes' law, two-thirds comes from skin friction
and one-third from a fore-and-aft pressure drop. The negligibility of
inertia doesn't imply that flows aren't associated with pressure differences.
Persuading a fluid to move from one place to another still takes a difference
in pressure. In fact, relatively more pressure is needed at low Reynolds
numbers because of the greater retarding effects of viscosity, as one can see
in the Hagen-Poiseuille equation (13.2).
Circular Disks
Exact solutions to the basic equations of flow are also available for the
drag of circular disks. For a disk of radius a that faces the oncoming flow,
one for which flow is parallel to the axis of rotation,
D = \6iiaU. (15.3)
For a disk with its faces parallel to the flow and the axis of rotation normal
to the flow,
D = 10.67 y,aU. (15.4)
These last formulas contrast interestingly. They apply to orientations with
the most and the least drag, respectively, and they apply to an object that
appears about as anisotropic with respect to drag as anything we're likely to
devise. First, the difference, though, is only 1.5-fold, the same as we noted
for long, nearly flat plates at a Reynolds number of 1.0 (Table 5.2). Second,
the values of the drag of a flat plate for the two orientations do not
converge further with further decrease in the Reynolds number. Third, while
shape (here in the guise of orientation or by comparison with equation
15.1) certainly affects drag, the effect is scarcely overwhelming; indeed, it's
counterintuitively modest. Fourth, referred to projected area, a disk
334
FLOW AT LOW REYNOLDS NUMBERS
broadside to flow has less drag (16 vs. 6tt) than a sphere, quite the opposite
of what we saw at higher Reynolds numbers; only when referred to wetted
area does the broadside disk have more drag. The latter, though, is still a
relevant comparison—streamlining may not have its usual meaning, but
evidently elongating a body fore and aft can still lower the drag per unit of
surface.
Cylinders
As we'll see shortly, the drag of cylinders takes on great importance in
explaining propulsion by flagella and cilia. As it turns out, calculating the
drag for cylinders is slightly messier than for spheres and disks—cylinders
are treated as very long prolate spheroids with axes either parallel or
normal to flow. According to Cox (1970), the drag of a cylinder of length /
and radius a extending parallel to the flow is
D = 2^Ul (15 5)
ln(lla) - 0.807' K }
Long cylinders parallel to flow—that sounds like silk strands above
ballooning spiders. Suter (1991) compared the predictions of (15.5) with
careful direct measurements; he found that real strands had about 2.5
times the expected drag. The likely explanations are innocent enough. For
one thing, flexible objects may have more drag than rigid objects (recall the
leaves in Chapter 6), especially when one is assuming the minimum-drag
orientation. For another, these draglines are really a fused pair of
cylindrical strands, so exposed surface is much greater than the minimum figured
from mass per unit length of a simple cylinder.
The drag of a cylinder with its axis of rotation normal to flow is
D = 4^Ul (15 6)
ln(lla) + 0.193' { }
One occasionally runs across analogous equations in which the numerical
constants in the denominators are —0.5 and +0.5, respectively; but, as
Brennen and Winet (1977) point out, these are solutions derived for zero
Reynolds number, and inertia always makes a small but quite real
contribution. And the assertion is occasionally made that a cylinder normal to flow
has exactly twice the drag of one parallel to flow. That's nearly right but
would really be reached only with a cylinder that's infinitely long. For a
length-to-diameter ratio of half a million, the ratio of the drags is still only
1.86. Still, that's significantly above the ratio of 1.5 that we saw for disks in
their orientations of greatest and least drag. Oddly enough, at low
Reynolds numbers cylinders are more anisotropic than are disks. Or maybe
it's not so odd: the steepness of velocity gradients is what matters, and the
335
CHAPTER 15
further downstream, the gentler; a cylinder parallel to flow has about the
maximum downstream surface relative to what's upstream.
For a circular cylinder normal to flow, White (1974) offers the following
empirical formula for Reynolds numbers from unity to 1 x 105:
Cdf= 1 + 10.0/te"2'3. (15.7)
It gives about the same results as the line derived from direct
measurements in Figure 5.3, and it was found useful by Greene and Johnson
(1990a) who looked at the sinking rates of plumed seeds.
Spheroids
Many biological objects may be reasonably approximated by either
oblate (pancake or discus) or prolate (cigar or football) spheroids, as
mentioned in Chapter 6 in connection with drag at higher Reynolds numbers.
The extreme cases are movement with axes of rotation normal or parallel
to the free stream; two classes of spheroids and two orientations give four
situations of interest. These fill in the gaps between spheres and disks and
cylinders. Loosely speaking, a disk is the ultimate result of squeezing a
sphere into an oblate spheroid, while a cylinder is the final result of
stretching a sphere into a prolate spheroid. Happel and Brenner (1965) give
appropriate, if mildly messy, equations for the drag of all these cases;
surface areas of spheroids can be calculated from formulas for areas and
eccentricity in Beyer (1978).
Visualizing what's happening to drag in these cases is easiest if we look at
two kinds of coefficients for drag: drag over the product of speed and
viscosity relative to (1) the square root of surface area, and (2) the cube root
of volume—with these last replacing the radius as used in equations (15.1),
(15.3), and (15.4). Table 15.1 gives these coefficients for a variety of
spheroids as well as for the sphere, disk, and cylinder already considered. A
number of items are noteworthy. First, the coefficients are not particularly
shape-dependent. Again, shape matters, but far less than at high Reynolds
numbers. Second, orientation matters, but again not very much—no
change in orientation can make even a 2-fold difference in drag. More
specifically, relative to surface area, least drag is incurred by a rather long
prolate spheroid traveling lengthwise or a flat disk traveling edgewise; a
circular cylinder moving crosswise incurs the most drag. Relative to
volume, least drag is suffered by a prolate spheroid twice as long as wide and
moving lengthwise. Both it and a l:2-oblate spheroid moving edgewise
have less drag relative to their volumes than does a sphere. The difference
isn't great, but it's enough to embarrass glib claims that spheres have least
drag at low Re (Zaret and Kerfoot 1980).
336
FLOW AT LOW REYNOLDS NUMBERS
Table 15.1 Calculated drag factors for simple bodies at very
low Reynolds numbers.
Circular disk
Oblate spheroid
Oblate spheroid
Oblate spheroid
Sphere
Prolate spheroid
Prolate spheroid
Prolate spheroid
Circular cylinder
lid
1
1
1
1
1
2
3
4
50
50
4
3
2
1
1
1
1
1
Flow Parallel of Axis
$112
6.383
6.174
6.012
5.794
5.317
4.896
4.762
4.734
6.598
of Rotation
yi/3
36.60
16.10
14.82
13.34
11.69
11.17
11.39
11.77
24.33
UIUS
0.319
0.726
0.789
0.877
1.000
1.047
1.027
0.993
0.480
Flow Normal to Axis
t
$112
4.255
4.857
4.950
5.074
5.317
5.608
5.858
6.093
10.448
if Rotation
yi/3
24.40
12.67
12.14
11.68
11.69
12.80
14.01
15.15
38.53
UIUS
0.479
0.923
0.963
1.001
1.000
0.914
0.835
0.772
0.304
Notes: The axis of rotation is what a biologist would call the axis of radial symmetry, lid:
length along axis of revolution over maximum width; 51/2: drag divided by speed,
viscosity, and the square root of surface area; V1/3: drag divided by speed, viscosity, and
the cube root of volume; UIUS: terminal velocity relative to that of a sphere of equal
volume and weight.
Fluid Spheres
If a sphere of fluid rises or falls through a fluid medium, the passage of
the medium will induce a toroidal motion within the sphere, as was shown
in Figure 10.6. The result will be lower drag on the sphere, as if the no-slip
condition were partially relaxed. The phenomenon depends on the
relative viscosities of sphere and medium: for a droplet of water in air the
effect is negligible, whereas for a droplet of air in water there is, in effect,
perfect slip. Happel and Brenner (1965) give the following formula:
1 + (2/3)(fA,ext/fA,im)
D = 67T|xextac/
1 + (^ext^int)
(15.8)
where |xext is the viscosity of the medium and |xint is that of the sphere. For
air in water or any gas in a liquid the ratio of the viscosities is essentially
infinite, and the equation simplifies to
D = 47T|xcxta{7,
(15.9)
which is to say that drag is just two-thirds of that given by equation (15.1)
for a rigid sphere. It's not at all uncommon to find this distinction missed.
On the other hand, Batchelor (1967) comments that if the interface
between a sphere of gas and the surrounding liquid is dirty, surface motion
337
CHAPTER 15
will be much reduced, and equation (15.1) is likely to be closer to reality
than is (15.9).
The phenomenon has quite another implication. A reasonable way to
develop thrust at low Reynolds numbers is to move a membrane around a
sphere or spheroid, extruding it at the front and absorbing it behind. That
will happen either if toroidal motion can be created inside, through cyclosis
or something similar, or if the membrane can be pulled upon. It's a direct
application of the no-slip condition and should work in any decently defor-
mable medium. Some such scheme seems to be used by, among others,
sporozoan trophozoites (Jahn and Bovee 1969).
Orientation
At intermediate and high Reynolds numbers, a moving body that's
symmetrical about each of three mutually perpendicular axes (cylinders,
spheroids, rectangular solids, etc.) will either tumble or take up an
orientation with the maximum cross-sectional area normal to the direction of
motion. Typically this will maximize its drag, for better or worse. A proper
arrow requires feathers to maintain its desired orientation. Shot off at a
Reynolds number of about 100, a Pilobolus sporangium, according to the
late Robert Paige, tumbles in flight. By contrast, for symmetrical objects
moving through fluids at low Reynolds numbers, any orientation is stable.
Released into an unbounded medium, a particle retains its original
orientation if it has uniform or at least symmetrically varying density. But as at
higher Reynolds numbers, providing protrusions to ensure that an object
seeks a preferred orientation following any rotational displacement isn't
difficult (McNown and Malaika 1950; Hutchinson 1967).
Wall Effects
Organisms frequently move near solid surfaces or air-water interfaces,
and every measurement in a wind tunnel or flow tank involves the
imposition of a wall somewhere near the test object. So a lot depends on just how
close a wall can be without significantly affecting the flow pattern and the
forces on an object. The problem is most acute at low Reynolds numbers,
where a body influences the flow a great distance lateral to its outer surface.
This lateral influence is, of course, just what makes flow through a pipe
develop its parabolic profile nearer an entrance at lower Reynolds numbers
(equation 13.9). The lower the Reynolds number, the larger must be the
ratio between the distance to a wall and the diameter of an object if
interference is to be kept acceptably low. An earlier "gee whiz" example bears
repeating. At a Reynolds number of 10~4, the presence of a wall 500
338
FLOW AT LOW REYNOLDS NUMBERS
diameters away from a cylinder doubles the effective drag; at Re = 10~3, a
wall 50 diameters away dominates the determination of drag (White 1946).
Happel and Brenner (1965) give a general treatment of wall effects, but
for most experimental purposes a quick index is adequate. Robert Zaret
suggests using the following rough-and-ready guide, derived from White
(1946), to be reasonably sure that wall effects can be ignored:
y->^ (15 10)
/ Re9 l j
wherey is the distance to the nearest wall, and / is a characteristic length of
the object. The formula is usable only at Reynolds numbers below unity
and is a fairly stringent criterion that could be shaved a bit for nondemand-
ing applications.
The same wall effects will influence the response of hot wire and hot
probe flowmeters when these are used near walls, so appropriate caution
should be exercised (White 1946). And walls add significantly to the drag
experienced by swimming microorganisms (Winet 1973).
Unless an object moves axially in a cylinder or midway between plates, a
wall introduces some asymmetry in the drag of even a symmetrical object.
A sphere moving parallel to a wall will rotate in the same direction as if it
were rolling along the wall. The rotation may, in turn, generate other
forces. For instance, constriction of streamlines between moving object
and wall will produce an attractive force, a lift as we've defined it earlier,
drawing the body toward the wall. These wall effects are pernicious; their
presence should be suspected (or even presumed) in any study that doesn't
specifically deal with them. As I'll argue shortly, they corrupt quite a lot of
data in the biological literature.
Terminal Velocity
If an object whose density differs from that of the medium is released, it
accelerates either upward or downward. As noted, at low Reynolds
numbers the period of acceleration is brief and the distance traveled is short; the
object very soon reaches (asymptotically, of course) a constant, "terminal"
velocity. Not that terminal velocities don't exist at high Reynolds numbers;
they're just less commonly of biological relevance and they take more time
and distance to achieve. As in gliding, the absence of acceleration implies a
balance of forces—weight minus buoyancy (net body force) against drag
(resistive force).
Consider a solid sphere of density r falling in a medium of density p0.
Gravitational force is gravitational acceleration times the difference
between the masses of sphere and displaced fluid, (p — p())(4/3)(7ra3g). Drag is
given by Stokes' law (equation 15.1)—6ir^aU. If we equate the two and
339
CHAPTER 15
solve for velocity, we get the very well known equation for terminal velocity,
U = 2fl2g(P " Po). (15.11)
This work for about the same conditions as does Stokes' law—up to a
Reynolds number of about 0.5. Notice that terminal velocity is
proportional to the square of the radius—a bigger object falls faster, not because it
has less drag (it has more), but because for a given density it has more
volume and hence a larger net weight. On the other hand, if the investment
in mass rather than density is fixed, then a larger sphere will fall more
slowly than a smaller one, at least in a medium such as air whose density is
much lower than that of any solid sphere. For a given balloon, the greater
its inflation, the slower it will fall—although here the Reynolds number is a
little high for equation (15.11) to apply strictly.
(Above Re = 0.5 a somewhat more complex formula can be derived from
equation (15.2) and the definition of the drag coefficient, and at still higher
Re's from Cd alone. In effect, the dependence of terminal velocity on radius
decreases as the Reynolds number rises, from variation with the square of
radius where Stokes' law applies through a direct proportionality just
above that, to the square root of radius where the drag coefficient has
leveled off (as in Figure 5.4). Bigger things may in general fall faster, but
the dependence of the rate of fall on size is less. Thus Brooks and
Hutchinson (1950) found for largish narcotized microcrustacean water fleas
(Daphnia), where drag coefficients ought still to be falling with Reynolds
numbers, that sinking rate varied directly with length. A little surprising is
what Dodson and Ramacharan (1991) found for Daphnia—the sinking rate
varied with length to the power 0.58, so something more than passive
sinking must have been involved. Incidentally, at constant densities (of both
object and medium) proportionality with the square root of length or
radius implies that terminal velocity will vary with the sixth root of mass—a
much less drastic dependence than one might imagine. "The bigger they
are, the harder they fall" is still credible, as long as one recognizes that
"harder" alludes more to momentum or kinetic energy than to velocity.)
For objects of other shapes, one can calculate terminal velocities from
equations (15.3) through (15.6) or from the data in Table 15.1. For higher
Reynolds numbers, some approximation of Cd is needed (such as equations
15.2 or 15.7); since Cd depends on velocity, the nuisance of an iterative
solution is hard to avoid.3 The underlying rule remains that weight minus
buoyancy equals drag. For objects of negligible displacement such as disks,
or for objects moving in a medium whose density is much less than their
own, buoyancy can, of course, be neglected.
3 Don't, repeat don't, even think of using Stokes' law and equation (15.11) above Re —
1.0. KvRe- 1.6 the latter overestimates speed by a factor of about 1.2;at/te= 10 by 1.8;
at 100 by 5-fold; at 1000 over 25-fold.
340
FLOW AT LOW REYNOLDS NUMBERS
Stokes' Radius
Free descent or ascent at terminal velocity is a most ordinary
phenomenon among tiny biological objects—consider dispersing seeds, spores,
baby spiders, gypsy moth caterpillars, and many nonmotile planktonic
creatures. The shapes of such small objects are extremely diverse; to
separate the effects of shape from those of size and density, a measure of
effective size may prove useful. Since most of these objects are either quite
small or have elaborate filamentous outgrowths of very small diameter,
Reynolds numbers are fairly low. A reasonable measure is the radius of the
sphere of the same mass that would ascend or descend at the same rate; let's
call it as, the "Stokes' radius."4 To obtain this radius, one just needs to
measure the mass and terminal velocity of an object and solve the following
equation for as:
4
mg ~ 3 Pogiraf = 67T|xa,/7. (15.12)
For solid objects falling through air, the middle term can be safely
dropped. The density of the object has been eliminated, so one needn't
prejudge the object's effective size. The resulting Stokes' radius can be
viewed as a measure of the effectiveness of any protrusions in increasing
effective size.
(Several alternatives to the Stokes radius are in use. The most common,
generally called the "coefficient of form resistance," is the ratio of the
terminal velocity of a sphere of the same density and volume to that of the
objector organism at issue [Hutchinson, 1967]. In effect, it's an extra factor
hooked on to the right side of equation 15.1.)
Interactions among Falling Objects
What happens if two objects, each going up or down at its terminal
velocity, get close to one another? As one might guess from the earlier
discussion of wall effects, the behavior of a particle may be substantially
altered by another anywhere nearby. A moving particle moves a lot of fluid
along with it, and any other particle will be carried along. If two particles
are sinking, the one behind will thus tend to fall faster and catch up to the
one ahead—even if in isolation the latter might have a slightly lower
terminal velocity. Two particles falling in tandem will go faster than either
alone, whether they fall one behind the other or side by side. The closer
4 It's been called the "nominal radius" by Kunkel (1948), but I think the name "Stokes'
radius" is less ambiguous—Stokes' law and terminal velocities are so commonly
considered concurrently.
341
CHAPTER 15
they travel, the greater the interaction and the faster (by up to 30% or so)
they go.
Once one gets used to thinking about these viscosity-dominated systems,
intuition becomes a decent guide to quite a few phenomena. For instance,
if two spheres sink side by side, each will induce some rotation in the other;
and the sides of the particles facing each other will move in the direction of
the overall sinking motion. That's reasonable since fluid between the two
spheres will to some extent be carried with them, so the retarding effects of
passage though fluid will be felt most strongly on the sides farthest from
each other. The direction of rotation in this case is opposite that noted for a
sphere falling parallel to a wall—here the two solids are moving in the same
direction, while in the case of the sphere and wall, the directions are
opposite. Again, Happel and Brenner (1965) provide a wealth of additional
information in a properly quantitative treatment.
Clearly a group of particles will fall faster than would the individual
particles, and such a group will tend to maintain coherence since ones
behind will catch up and ones in front will be more strongly retarded. That
may explain some puzzling data of Chase (1979), in which aggregated
particulates in both fresh and sea water fell at up to ten times the rates
predicted by Stokes' law—"a drop of water containing high concentrations
of aggregates [was] released into the charging port." At the same time, a
cloud of particles that fills a container may fall more slowly than a single
particle or a group near the center. If the particles carry fluid downward by
viscous interaction, then somewhere an upward flow must compensate to
satisfy the principle of continuity. This means trouble for the all-too-
common practice of timing descent rates of clouds of organisms to gain
some statistical view of sinking and to permit the use of photometric
detectors. In short, even the sign of the error introduced by using groups of
organisms isn't certain from an ex post facto look at the literature.
Furthermore, a descending cloud may tend to concentrate or spread out over time
and distance, complicating any correction.
Measuring Terminal Velocity
By this point I hope I've engendered a conviction that existing data must
be assumed to be seriously in error unless sufficient procedural detail is
given to demonstrate otherwise. What looks at a facile glance to be the
simplest of procedures isn't anything of the sort. If walls are far away,
convection in the test chamber is quite likely to confound matters, and
observations are difficult; if walls are nearby, then they slow the falls. So
what is the honest investigator to do? And old but still useful correction is
cited by various sources (for instance, Sprackling 1985) for a solid sphere
rising or falling in the center of a cylindrical column of fluid:
342
FLOW AT LOW REYNOLDS NUMBERS
n = Apparent ,j5 j gx
^truc (1 +2.4 a/R)' { }
where a again is the radius of the sphere and R is the radius of the
cylindrical column. The equation can be applied to equation (15.11) either for
viscosity determinations or for correcting terminal velocities obtained in
columns for unbounded systems (R = °°). A lot of questionable data on
sinking rates in the biological literature can be restored to health by
therapeutic use of the equation, at least where the author specifies the size of the
column. One merely assumes that the viscosity in the denominator of
equation (15.11) is the apparent viscosity (as defined by 15.13) and substitutes
[|xtrue(l + 2Aa/R)] for it.
But that'sjust a start. The correction is inadequate if an organism moves
substantially off-axis in the chamber. Perhaps worst is convection in the
chamber—again, the problem gets increasingly severe as chamber size is
increased in order to avoid wall effects. Very slight motion is all it takes to
contaminate data when, for instance, the sinking rate of a diatom is about 5
|xm s-1. That's over three seconds to go a mere millimeter. Even slight
irregularity in illumination can make real trouble, as can the residual
effects of whatever scheme was used to put the players on stage. Arranging a
very gentle thermal gradient in any liquid system to help offset convection
is probably a good idea. Booker and Walsby (1979) describe a technique
and the appropriate correction formulas for using a sucrose density
gradient to control convection.
I'd like to suggest a return to looking at the descent of individual cells or
organisms—decontaminating data for unnaturally concentrated
assemblages is just too problematic. One might permit an acceptably low density
of particles to descend in a thermally jacketed column of generous size,
viewing the center of the column with a microscope equipped with optics
permitting long working distances. A video camera and recorder can pick
up even an occasional item that descends in the right place, illumination
can be intermittent or brief, checks for convection can be made under
equivalent circumstances with dye or particles of known behavior, and so
forth.
When Terminal Velocity Matters
For applications at low Reynolds numbers one thinks immediately of
nonflying airborne particles—pollen grains, fungal spores, and the like—
for which dispersal distance has a very direct bearing on fitness. By the
principle of continuity, as much air must come down as goes up; by the
principle of gravity, any passive object denser than air must always be
sinking with respect to the air around it. Other things being even vaguely
343
CHAPTER 15
comparable, the smaller the propagule the more effective will be wind
dispersal. Ingold (1953) cited, as a record of sorts, 0.5 mm s_1 for the rate
of fall of the 4.2-|xm diameter spore of Lycoperdon pyriforme, but he
considered 10 mm s~l as a more typical rate for fungal spores. Gregory (1973)
provided a wealth of data on size, density, and terminal velocities of spores.
Good discussions of the flight of pollen can be found in Burrows (1987)
and Niklas (1992); the latter, for instance, calculated a sinking rate of 123
mm s~l for a 64 |xm-diameter, 0.137 |xg pollen grain of the hemlock, Tsuga
canadensis. For neither airborne spores nor pollen is density easy to
measure (see Figure 15.1a), so it's simpler (neglecting air density, of course) to
use a version of equation (15.11) based on mass:
o7T|xa5
The interaction of these sinking particles and atmospheric motions (on a
variety of temporal and spatial scales) gets beyond the scope of this book;
quite a lot is known and can be found under topics such as micrometeorol-
ogy, biometeorology, and plant (or seed) dispersal.
Certainly the biologists with the greatest interest in terminal velocity are
the oceanographers and limnologists. Whether in marine or freshwater
systems and whether living, organic but nonliving, or inorganic, most small
items are "negatively buoyant"—sinkers. That's quite a spectrum of
possibilities, and a large literature concerns itself with sinking speeds—of fecal
pellets of microcrustacea (Komar et al. 1981, for instance), of the
macroscopic aggregates a few millimeters long of a mixture of living and dead
material that's come to be called "marine snow"(Riebesell 1992), of marine
bacteria (Pedros-Alio et al. 1989) and fungal spores (Rees 1980), of plank-
tonic ciliates (Jonsson 1989), of various larvae (Bhaud and Cazeux 1990)
and embryos (Quetin and Ross 1984), and, in greatest profusion, of phy-
toplankton such as diatoms.
Living planktonic organisms are almost always denser than their
ambient media—even with the use of devices such as gas vacuoles (particularly
in freshwater forms), fat droplets, or vacuoles in which ammonia is
accumulated and potassium and divalent anions excluded. But they have
various ways of reducing sinking rates. Some, such as water fleas (Dodson and
Ramacharan 1991) and ciliates (Jonsson 1989), swim upward ("negative
geotaxis"). A worm larva, Lanice, makes a mucus thread about 15 mm long
(Bhaud and Cazeux 1990). Diatoms adjust their composition in response to
such factors as light and nutrient concentration (see, for instance, Davey
1988). Some desmids are enveloped in mucilage (Lund 1959). As
Hutchinson (1967) points out, even a negatively buoyant coating can reduce
sinking rate if the density difference between organism and coating is more
than twice the density difference between coating and medium. And the
344
FLOW AT LOW REYNOLDS NUMBERS
Figure 15.1. Small sinkers: (a) a grain of the anemophilous pollen of a
pine tree; (b) the diatom, Thallassiosira.
irregular shapes of many planktonic organisms are commonly interpreted
(at least in part) as evidence that they go to great lengths to sink more
slowly.
Unfortunately, the measurements (as obtained) of sinking rates in this
literature—especially those on slow sinkers such as diatoms and bacteria—
are of very uncertain reliability and applicability to natural circumstances.
Indeed, the comments on measurement of sinking rates a page or so back
were prompted by my sinking spirits when working through a large
number of relevant papers. Some attention has been given to convection in test
chambers (Bienfang and Laws 1977) and to wall effects (Quetin and Ross
1984), but little if any to interactions among sinking particles, despite the
almost universal practice of measuring terminal velocities of fairly dense
suspensions.
Quite a few papers assert an inverse relationship between irregularity
(e.g., nonsphericity) in shape and sinking rate; that an increase in relative
surface area slows sinking would certainly seem reasonable. But here
again, reliable evidence is scanty. The most direct test of the notion was
done by Walsby and Xypolyta (1977); they found that removing the long
chitan fibers protruding from a diatom, Thalassiosira fluviatilis (Figure
15. lb), increased sinking rate from 3.8 to 6.6 |xm s~1 even though the fibers
were much more dense (1495 vs. 1112 kg m~3) than the rest of the cells.
Their measurements of sinking rates, though, were made on a turbid layer
in which the concentration of cells approached 5%. Some sources give plots
showing an inverse relationship between sinking rate and surface-to-
volume ratio. The problem with these is that the ratio, S/V, has a residual
dimension of inverse length; and thus an axis along which S/V increases
345
CHAPTER 15
may represent decreasing size instead of increasing irregularity—or, for
that matter, some combination of the two. I reanalyzed one set of such data
for diatoms, using a dimensionless (and hence size independent) surface-
to-volume ratio, Sl/2/Virs, and found that the ostensible effect of
irregularity disappeared entirely. Thus the variation in sinking rates in this
particular set of data reflects variation in size rather than shape; correcting for
size eliminated the effect.
Since, other things being equal, increased surface area must slow
sinking, other things must not be equal. Probably (at least in this one case) the
density of the diatoms, the main unmeasured variable, varied
systematically with size. Perhaps irregularity is at least in part a device that permits
less fastidious control of density in the process of regulating sinking rate.
That's at least consistent with the observation of Sournia (1982) that small
cells are less irregular than are large cells.
Yet another problem. Why should sinking per se be so ubiquitous even
among phytoplankton? Achieving neutral or positive buoyancy should
present no special difficulty, even in fresh water—fat droplets and gas
vacuoles will always be less dense than water. And sinking takes a photo-
synthetic organism away from its critical resource, light. The most
common argument is that sinking is important for obtaining nutrients, that a
small organism that didn't sink would soon deplete the local larder. I'm
unpersuaded on account of several counterarguments. At least for actively
swimming predators such as ciliates and water fleas, augmenting diffusion
with convection in this way is unlikely to matter much. We need to ask how
fast sinking would have to be in order to make a difference to diffusional
exchange. The diatoms that are the main concern in the literature are very
slow sinkers—4 |xm s~l is about a foot a day. The cells are around 10 |xm in
diameter. The Peclet number, used in the last chapter in connection with
exchange adjacent to capillary walls and in this one to get a quick view of
Berg's (1983) argument against bacterial wandering, again gives an
appropriate index of what motion accomplishes. Assuming that carbon dioxide,
with a diffusion coefficient in water of 1.5 x 10~9 m2s_1, is what matters
for a diatom, the Peclet number (UlID) comes out to less than 0.03. That's
clearly well below the value of unity (see, for instance, Sherwood etal. 1975)
around which the convective augmentation of diffusion might become
significant. Put another way, a diatom would have to sink about 30 times
faster for sinking to have a significant effect. The argument isn't a new one;
in looking for a reason for sinking, Munk and Riley (1952) did some
calculations that for such small cells ruled out obtaining nutrients.5
5 But for a ciliate protozoan, Tetrahymenapynformis, with / = 40 |xm and U - 450 |xm
s_1 (data from Plessetetal. 1975), trying to get oxygen, Pe - 10. Its movement (whatever
its other uses) makes convection distinctly nonnegligible. The argument against moving
doesn't extend to the whole microscopic world.
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FLOW AT LOW REYNOLDS NUMBERS
Why, then, sink? I'd like very tentatively to suggest that sinking might be
a device to keep away from the air-water interface. We're painfully aware
that large organisms can't ordinarily support themselves on an air-water
interface (Vogel 1988a)—the Bond number, the ratio of the force of
gravity to that of surface tension is too high (or its inverse, the Jesus number, is
too low). We're less concerned with the boundary condition on the other
side, that affecting small organisms and loosely defined by the Weber
number. The latter, about which more in Chapter 17 (and defined by
equation 17.6), is the ratio of inertial force to surface tension force; in its
numerator is length and the square of velocity. Small, slow things can't get
enough IU2 to get loose, and the interface is a very special habitat for which
ordinary plankton must be poorly adapted. So, as eloquently pointed out
by D'Arcy Thompson (1942), it can snare as well as sustain: "A water-beetle
finds the surface of a pool a matter of life and death, a perilous
entanglement or an indispensable support." Positive buoyancy may be perilous—
better to sink slowly or to counteract sinking by swimming upward to just
short of the interface. And (as is commonly noted), natural bodies of water
aren't still, so even continuous sinking needn't mean steady and
irreversible movement away from the surface.
Propulsion at Low Reynolds Numbers
Consider, again, the central message of the chapter—in no biologically
relevant area is the counterintuitive character of fluid mechanics more
evident than at low Reynolds numbers. The swimming of spermatozoan
and eel may have a certain superficial similarity, but the underlying
physical mechanisms turn out not to be at all the same. Put a bit crudely, the eel is
imparting local rearward momentum to the fluid, making use of the fluid's
inertia. By contrast, the spermatozoan (assume a free-swimming aquatic
one, not the sort produced by male readers) is pushing against the fluid
more in the manner of a snake slithering on a solid but slightly slippery
substratum, using the fluid's viscosity to permit purchase. It's a radically
different world, and nowhere is its special character more evident than in
the business of propulsion.
The Peculiar Problems and Possibilities
Back in Chapter 11 and early in this one I noted that circulation-based
lift isn't worth much at low Reynolds numbers. Thorn and Swart (1940), the
only explorers of this unpromising territory of whom I know, got a best lift-
to-drag ratio of 0.43 at Re = 10 and of 0.18 at Re = 1.0. Thus tiny organisms
insistent on swimming must resort to some drag-based scheme. As it turns
347
CHAPTER 15
out, drag-based propulsive arrangements encounter special problems of
their own at low Reynolds numbers.
To begin with, the reversibility of these flows presents a subtle but very
real impediment. Consider a simple propulsive system in which
appendages are repeatedly moved fore and aft. They move steadily forward in a
slow recovery stroke and steadily rearward in a rapid power stroke. The
appendages (perhaps elongated cylinders) present the same shape and
area to the fluid in either stroke, and the body from which they protrude
doesn't move through the fluid. Does the reciprocation generate a net
thrust? Clearly the faster power stroke generates more force than the
slower recovery. But what matters in the overall balance sheet is the product
of the force and the time over which it is exerted (strictly, force integrated
over time)—what's called the impulse. The change in momentum of a body
will equal the impulse of the force on it.
At high Reynolds numbers drag will be roughly proportional to the
square of velocity; since the velocity of the stroke is inversely proportional
to its duration, the impulse of a stroke will be proportional to the speed of
movement of the appendages. Thus the thrust of the fast power stroke will
more than offset the drag of the slow recovery, and a net thrust will be
produced, even if inefficiently.
By contrast, at low Reynolds numbers drag is directly proportional to
speed. With F <* U and / <* 1/U, Ft will be constant whatever the speed of the
appendage during the stroke. In short, no juggling of speeds of power
against recovery strokes will enable them to do anything other than cancel
each other's efforts. Not only does low Re imply high drag, but the reduced
dependence of drag on speed is a special drawback.
Still, at least two approaches to drag-based locomotion remain possible.
Formulas for drag at low Reynolds numbers such as Stokes' law (equation
15.1) have three usable variables, speed (U), shape (for instance, 6tt), and
area (in the guise of a); and we've ruled out only interstroke manipulation
of the first. Organisms, in fact, use both changes in the orientation
(effective shape) and changes in the effective area of moving appendages
between power and recovery strokes.
Changing the orientation of a reciprocating appendage looks almost as
inefficient as using circulation-based lift. The change in drag is minimal for
even a most extreme case, a long flat plate shifted from perpendicular to
parallel to flow; at Re = 1.0 the decrease is only about a third (Table 5.2).
That doesn't bode well for a healthy recovery stroke! In fact, as we'll see
shortly, propulsion based on the difference in drag between two
orientations is widespread; but it's based on a scheme other than the kind of
rowing and feathering implied here. For that matter, I can think of no
system in which low Reynolds number propulsion is based purely on
orientation changes of a long, reciprocating paddle.
348
FLOW AT LOW REYNOLDS NUMBERS
200 ' j ' ' '
i 150 \
ai" \
<-> \
t: \
s ioo \
* \
0) \
I 50 \.
0 i i — 1 1
-10 12 3
flow speed, mm/s
Figure 15.2. (a) The velocity gradients near a ciliated surface. 15 (Jim-
long gill cilia of a mussel, Mytilus eduhs, are propelling water through a
200 |xm-wide channel (from Nielsen et al. 1993).
Yet another problem. At low Reynolds numbers velocity gradients are
wide and gentle, so a body towed through or falling through a fluid carries
a lot of fluid with it, and free-stream velocity is approached only quite far
away from the body. But to propel oneself forward one must push against
the free stream. Thus an effectively propulsive appendage must protrude
through and beyond the gradient region. If such an appendage is short (as,
for instance, a cilium), then the gradient must be short and steep, as in
Figure 15.2. That means a lot of work must be done against the viscous
bugbear of skin friction, and some large portion of an organism's surface
ought to be equipped with such appendages—any surface not so equipped
will severely offset their action.
The common use of extensive bands of cilia as well as fully ciliated
surfaces is thus entirely reasonable. At the same time it raises some
awkward problems for the investigator. Say one is interested in the pattern of
flow around a ciliated protozoan such as Paramecium. You can easily ballast a
Paramecium-shaped object so it falls at the right Reynolds number in a very
viscous liquid. But you most certainly won't get a pattern equivalent to that
of a swimming animal—the velocity gradients will be far too gentle for real
cilia (allowing for scale) to protrude through them. And you'll vastly
overestimate the lateral extent of the disturbance caused by passage of a
swimming animal. Queer as it sounds, a swimming Paramecium will disturb the
surrounding fluid less than one passively sinking at the same speed. The
difference is strikingly evident in Wu's (1977) photographs.
Nor can you conclude much about propulsion from looking at the flow
Held around a tethered creature in still water. Emlet (1990) found, not
349
CHAPTER 15
unexpectedly, that attached molluscan larvae moved less water but
disturbed the flow field farther away than ones around which water moved at
speeds typical of swimming. Furthermore, the problem we met earlier of
using drag data to predict thrust in fish is even more pernicious. One can
say almost nothing about the force or power needed for propulsion from
measurements of the drag of a model of, say, a ciliated invertebrate larva.
Being assured that thrust balances drag gives no easy clue as to how to
measure either.
Changing the Area of Appendages: Paddles and Bristles
The real specialists here are small arthropods, mainly aquatic ones but
some of the tiniest fliers as well. The general scheme involves bristles (or in
larger creatures, oarlets) that extend during a power stroke and fold back
during recovery. Nachtigall (1980) has described very clearly how such a
combined system works in the swimming appendages of aquatic beetles.
The outer segments of the propulsive legs are little more than tubes to
which bristles attach. These are spread by the thrusting stroke of the legs
until they lie in a plane; if the leg is a flattened one, they lie in the plane of
flattening. Two-thirds to three quarters of the thrust is attributable to the
bristles; even at these not-so-very-low Reynolds numbers, they produce up
to 54% as much thrust as would an equally broad solid surface. During the
recovery stroke, the hairs rotate back against each other and against the leg
segments. In addition, the legjoints are flexed during the recovery stroke.
At least one beetle, Gyrinns, has flattened blades in place of cylindrical
bristles; spread during the thrust stroke, these overlap like drawn Venetian
blinds, giving about 90% of the thrust of a fully solid surface (Figure 7.6).
Thus the propulsive legs can be drawn forward with minimal exposure to
the free stream both by minimizing effective surface and by keeping them
close to the body.
In at least three orders of insects, small ones have converged toward
wings similar to these beetle legs (Figure 15.3), as noted by D'Arcy
Thompson (1942), who felt that this structure allowed the insects to "row"
through the highly viscous air. Ellington (1980a) has shown that the wing
fringes of the most common of these bristle-winged groups, the thrips
(Thysanoptera), can be actively locked in either an "open" or a "closed"
position by forces greater than the aerodynamic and inertial forces on the
beating wings.
The scheme will work, of course, at all Reynolds numbers. Its rarity in
the really macroscopic world must reflect mainly the existence of better or
easier alternatives such as circulation-based lift and recovery strokes in
which orientation alone is altered. Indeed, if we don't restrict ourselves to
unwebbed bristles, the general scheme is better at higher Reynolds num-
350
FLOW AT LOW REYNOLDS NUMBERS
Figure 15.3. Animals with propulsive appendages that lack a
continuous membrane: (a) a cladoceran Crustacea; (b) a diving beetle;
(c) a thrips; (d) a mymarid wasp; (e) a moth. All but the beetle
are very small. The first two swim; the latter three fly.
bers, emphasizing the special difficulties of being slow and small. Williams
(1991) has looked at changes in swimming pattern over the course of
development of brine shrimp (Artemia) larvae (those crustaceans whose eggs are
sold in pet stores as fish food). The earliest stage (Re = 2) moves with a
ratcheting motion using a single pair of appendages—forward with a little
backslip during recovery. By Re = 5 a bit of a glide follows the power stroke;
35 1
CHAPTER 15
by Re = 13 the glide is more pronounced and the backslip is about gone.
(Still larger larvae, with Re's between 20 and 40, shift to sequential or
"metachronal" strokes of a series of appendages, complicating the
comparison.)
What does change with Reynolds number is the nature of the propulsive
"surface." In swimming beetles we noted that bristles form dense sheets
and may be flattened and overlapped to make a plane surface. By contrast,
in the smaller and slower world of a cladoceran crustacean, hairs are spread
farther apart; velocity gradients are gentler, and fewer bristles suffice. In a
model devised by Zaret and Kerfoot (1980), the propulsive action of the
second antenna of the cladoceran, Bosmina, turns out to be rather
insensitive to the precise position and gap between the bristles. Further
information on how small, swimming arthropods use such bristle-paddles can be
found in Nachtigall (1974), Hessler (1985), Morris et al. (1985), and Blake
(1986).
Changing the Orientation of Appendages: Cilia and Flagella
As mentioned already, the scheme is widely used despite the fact that the
drag of a flat plate parallel to flow isn't much less than the drag of one
normal to flow. The devices used aren't flat plates but, oddly enough,
cylinders; the appendages most often involved are cilia and flagella.
Nothing fundamental distinguishes cilia from flagella in eukaryotic cells,
although cilia are usually shorter (5 to 10 |xm in length) and more
numerous on each cell. A cell rarely has more than a few flagella, but these
may be up to 150 |xm long (Jahn and Votta 1972). Often "flagellar action"
refers to helical or undulatory wave propagation while "ciliary action"
refers to a reciprocating beat, with no implication of any other difference.
Both cilia and flagella are about 200 nm in diameter and are musclelike
devices with motile proteins inside. Bacterial flagella, however, are
something quite different; they're composed of a single, nonmotile filament,
and they're only 20 nm in diameter (Berg et al. 1982).
The arrangements and actions of these organelles are diverse. An
organism (often a single cell) may be covered with cilia or have distinct bands
of them (the latter are especially common among marine invertebrate
larvae; see Chia et al. 1984). It may have one or more flagella on the
anterior that pull, one or more on the posterior that push, or even a
flagellum extending laterally that moves the creature in a helical path.6
The motion of a flagellum may take the form of planar, undulating waves
either from base to tip or tip to base; a flagellum may propagate helical
6 Swimming along a helical path is, in fact, very common among microorganisms of
just about every group that goes in for active locomotion. For consideration of both its
mechanism and functional significance, see Crenshaw (1993).
352
FLOW AT LOW REYNOLDS NUMBERS
| power stroke | recovery stroke |
Figure 15.4. The motion of a cilium, shown as a left-to-right sequence,
pushing water from right to left. It's extended during the power stroke
(left) and flexed down near the surface for recovery (right). The cilium
thus moves more distant from the substratum and normal to its long axis
during the power stroke and closer to the substratum and parallel to its
long axis during recovery. Both differences contribute to net thrust
generation.
waves from base to tip; or it may produce spiral waves. By contrast, ciliary
coats or bands beat more like our description of the swimming appendages
of beetles or microcrustaceans—an extended thrusting stroke is followed
by a flexed recovery stroke in which the cylinder moves lengthwise close to
the surface of the organism, as in Figure 15.4. Groups of cilia are usually
coordinated, and a host of patterns of coordinated beating have been
described (and named). Bacterial (prokaryotic) flagella, rigidly helical, are
driven in a circular path around the long axis of the helix by proper rotary
motors to which they attach; they're the only truly rotational (wheel-and-
axle) machines known in organisms (Berg and Anderson 1973). Eukaryotic
flagella with helical motion translate in circles.
A common physical mechanism for propulsion really does underlie all
this diversity. The basic setup is best introduced by considering what
happens if a long circular cylinder is pulled through a viscous medium (Figure
15.5a). If it's pulled lengthwise, equation (15.5) applies; if it's pulled
broadside, equation (15.6) applies. For a cylinder 50 times as long as it is wide (IIa
— 100) the ratio of the two drags is 1.75; for one 150 times as long as wide
(Ha = 300) the ratio is 1.80—either figure is better than the 1.5 ratio for a
long flat plate parallel versus normal to flow. Cilia, beating rather than
undulating or rotating, take direct and obvious use of the difference in
drags, moving almost broadside on a power stroke and moving almost
lengthwise for recovery. And once the ciliated organism is in motion, drag
during the recovery stroke is kept down by the additional device of moving
the cilia close to the surface of the creature. Even so, such recovery strokes
are far more costly than their high-Reynolds number equivalents. The
motile algal cell, Chlamydomonas, swims in a kind of breast stroke with two
similar flagella; during a recovery stroke it loses about a third of the
distance gained in a power stroke (Ruffer and Nultsch 1985).
What undulating and rotating systems do is considerably more subtle.
The fact that the drags are substantially different means that a rod that's
353
CHAPTER 15
(a)
drag
(b)
resultant
drag normal
D
motion
drag parallel
weight or pull
resulting motion f
Figure 15.5. (a) Pulling a cylinder through a viscous medium; drag is
almost twice as high when it's pulled crosswise, (b) An obliquely towed
cylinder wants to slew off to one side, much like a boat towed by a rope
fixed to one side a little back from the bow.
pulled not quite broadside (obliquely) will slew a bit sideways—it moves
lengthwise relatively more easily than it moves broadside. In effect, the
difference in drag seen in the two extreme orientations generates a
resultant force that's not quite opposite the imposed force trying to move it
obliquely (Figure 15.5b). On a long tether, the cylinder will slew off much
like a water skier with skis askew or a wedge that persists in going
lengthwise even if hit by a glancing blow. You can see this poor sort of gliding in
action by dropping an obliquely oriented fine wire in a tank of water.
According to Happel and Brenner (1965), the best orientation for slewing
is with the long axis of the cylinder about 55° to the direction of pull; it will
then slew about 20° off the direction of the pulling force. That glide angle
of 70° isn't impressive, and the angle of attack of 35° may seem extreme, but
this is something quite different from circulation-based lift.
Sir G. I. Taylor was, I believe, the first to point out (1951) that the
phenomenon can be used for propulsion. It works injust the same way that
thrust is generated from the lift of an airfoil moving with respect to a craft
(Chapter 12). What matters isn't the origin of the lift but simply that some
component of a force is developed in a direction opposite the motion of the
craft. Move an appendage back and forth, changing its orientation
appropriately, and a pulsing component of force will push an organism unidirec-
tionally; the other components will cancel, causing at worst some side-to-
side wobble, as shown in Figure 15.6a. If we expand our view from such a
paddlelike short segment of a cylinder to a long, axially rotating helix, we
can see how rotation of the latter will give steady forward propulsion
354
FLOW AT LOW REYNOLDS NUMBERS
resultant
slewing force
flow due tT>>>>
to slewing /^
net oncoming
flow j
Figure 15.6. (a) Moving a oblique portion of an appendage sideways
generates a force normal to the motion, which may be used for
propulsion ("slewing," locally). If so, the oncoming flow is deflected a bit, as with
a poor glider (recall 11.11). (b) A turning helix will generate just such a
normal force over its entire length.
(Figure 15.6b). Each element of the cylinder is moving obliquely sideways
and thus contributes thrust. That the organism will counterrotate as a
result of the remaining force component is either tolerated or offset by
another such rotating helix. The game is much like screwing into a cork
except that some slip occurs; elements of the helix do not quite advance
lengthwise but (in the case described earlier) move 55° — 20° = 35° to the
long axis of each. It makes little or no difference whether the helix truly
rotates (bacterial fiagella) or just translates around its long axis (eukaryotic
flagella).
Nor does it make much difference if the undulation is in a plane. Pieces
of cylinder are still oriented obliquely and move laterally. In planar
undulation, though, not all of a flagellum can contribute all at once because some
of its length is "wasted" at the upper and lower ends of each wave. On the
other hand, a propagated planar undulation doesn't throw the rest of the
organism into rotation in the opposite direction.
Neither ciliary nor flagellar propulsion can really be described as
efficient. A ciliary tip velocity of 4 mm s~l produces an average flow past the
cilium of only about 1 mm s~l (Sleigh 1978). Still, propulsion seems to take
only a small part of the metabolism of these animals, and any competitor
must pay the same price. Ciliary propulsion, in particular, may have fringe
benefits. As noted earlier, a swimming Paramecium is surrounded by a
steeper velocity gradient than even an equivalent passively sinking body. So
it's minimally "noisy" in a sense that we'll consider shortly. Furthermore,
swimming (by protozoa but not bacteria) can augment diffusion by raising
355
(b)
* motion with respect
to organism
flow due
to pull
CHAPTER 15
the Peclet number decently above unity; swimming with cilia ought to be
especially effective in improving exchange on account of the especially
steep velocity gradients.
Usually animals using ciliated surfaces for locomotion are both larger
and faster than those using a few flagella. Where sizes overlap, Sleigh and
Blake (1977) point out that the ciliated forms go about ten times faster than
the flagellated ones. But, as they also note, propulsive efficiency drops with
increasing size for either, and no large swimmer in a hurry uses either
scheme—muscles have the competitive advantage of another order-of-
magnitude increase in speed. The largest creatures using cilia are
ctenophores ("comb-jellies"), typically around a centimeter or so long; but
they use them in platelike bundles of a kind that occur nowhere else (Tamm
1983). The largest animals swimming with simple cilia appear to be
rotifers, an order of magnitude smaller, and they seem notably inefficient
(Epp and Lewis 1984).
The diversity of ways in which cilia and flagella are used is only hinted at
here. They're commonly used to run pumps involving both water and
mucus, as mentioned in Chapter 14. Some flagella have pinnately
sequential appendages on them (mastigonemes) and propagate waves in the
direction in which the organism is going rather than in the normal, retrograde
direction (Brennen 1976)—just as do swimming polychaetes with para-
podia sticking out laterally (Chapter 7). The density of cilia implies
substantial interaction and some degree of artificiality to any analysis of single
organelles. Useful sources range from purely descriptive ones
emphasizing biological diversity such as Jahn and Votta (1972), to ones of real
mathematical splendor such as Wu (1977) and include Brennen and Winet
(1977), Roberts (1981), and the various papers in Wu et al. (1975).
Filtration
Feeding by using a filter of some sort ("suspension feeding" usually)
almost always involves flow at low Reynolds numbers; edible particles
suitable for capture are rarely over a millimeter in diameter, and the capture
elements likewise are small. Flow speeds at the filter are most often slow,
partly, one may guess, because fine filters may not be mechanically robust
and partly because a large filtration surface will, by the principle of
continuity, be associated with low speeds. The only high Reynolds number fil-
terers may be the baleen whales, and little seems to be known about their
hydrodynamics (Sanderson and Wassersug 1990).
A Few Problems
The preeminent fluid mechanical issues in filtering are those of ensuring
contact between edible particles and filtration surfaces and of keeping the
356
FLOW AT LOW REYNOLDS NUMBERS
resistance of the filter low enough so fluid goes through it. These
requirements are substantially antithetical—the finer the mesh of the filter, the
greater the resistance to flow. When building a rake, it's all too easy to end
up with a paddle, as Cheer and Koehl (1988) put the difficulty. The viscous
character of low Reynolds number flow makes matters worse—if you can
bail with a sieve, you can't easily filter with it. And it's remarkable just how
low the Reynolds numbers of filter-feeding devices can get. Both the tiny
microvilli that make up the collars of sponge choanocytes and their intervil-
lar gaps are 0.1 to 0.2 |xm across, and water passes through them at about 3
|xm s_1 (Reiswig 1975b); this gives a Reynolds number of 4 x 10~7!
The basic problem can be illustrated in what may seem quite a different
system—flow through the large pinnate antennae that male saturniid
moths use to detect the odorant ("pheromone") molecules of females who
may be kilometers away, truly an olfactory sensation. In this context the
problem is that exposing more receptors to the flow increases the fraction
of the flow that will evade the antenna entirely. I found (Vogel 1983) that
only 8% to 18% of the air directly upstream went through; the rest went
around. No one should be surprised (by this point in the chapter) that the
transmitted fraction increased steadily with free-stream speed. Nor that
viscosity proves such a serious impediment to transmission—by contrast
with air, 43% of a light beam penetrated an antenna. Calculations of the
efficiency of antennae as pheromone detectors should incorporate a
correction for viscosity (for a general discussion, see Futrelle 1984).
The problem of flow evading a filter entirely is one with little parallel in
conventional technology in which pumps and ducts ensure 100% trans-
missivity and what matters is the power required to do thejob. But evasion
is common among suspension feeders. It's been documented in such
diverse cases as barnacles (Trager et al. 1990), zooplankton (Hansen and
Tiselius 1992), caddisfly nets (Loudon 1990), and black fly larvae (Lacour-
siere and Craig 1993). In each case the fraction passing through increased
dramatically with Reynolds number. To make matters a little more
complicated, the filtering structures usually adjusted their configurations as flow
speed increased, spreading wider apart at higher flows. The effect is quite
the opposite of what would compensate for viscous effects. Forces and
optimization of capture in a world of size-varying particles seem more
relevant than keeping a constant transmitted fraction. This general issue
of transmissivity has been dealt with by Cheer and Koehl (1988).
Mechanisms
The simplest and most familiar kind of filter for suspended particles is a
sieve, and the filters of suspension feeders have classically been viewed as
sieving devices—particles of diameter greater than mesh size or interfiber
distance got caught. As it happens, sieves are relatively unusual biological
357
CHAPTER 15
filters; and most suspension feeders mainly catch particles smaller than
could be sieved out by their filters. The turning point in how we viewed the
process came with a paper by Rubenstein and Koehl (1977), who adopted
and adapted analytical models that had proven useful to engineers dealing
with aerosols. Besides simple sieving, they identified five ways in which
particles might be captured. Each of these latter particles can be smaller
than mesh or interfiber distance, and they approached the problem by
looking at the interaction of a particle and a fiber. One of the five,
electrostatic attraction, appears unlikely to work in seawater and hasn't yet been
shown in any living system, so perhaps it may be put aside. And then there
were four, shown diagrammatically in Figure 15.7:
1. In "direct interception" a particle passes within a particle radius of
the collecting surface; only in this mode need a particle not deviate
from a streamline.
2. In "inertial impaction" flow curves off away from the collector, but a
particle keeps going because of its excess density.
3. In "gravitational deposition" a particle similarly deviates from
streamlines on account of excess density, but the direction of
deviation is always vertical.7
4. In "diffusional deposition" or "motile particle deposition" a
particle encounters a collector as a result of its Brownian motion or its
(assumed random) swimming activity.
Koehl and Rubenstein provided indices for the efficiencies of each of
these mechanisms as functions of such things as mesh or fiber size, particle
size, flow speed, and viscosity. Recently, their approach has been extended
in an impressively clear, penetrating, and iconoclastic analysis by Shimeta
and Jumars (1991). They point out that encounter rate is as important as
encounter efficiency and that the two don't change with particle size or
ambient velocity in the same manner. Taking a few liberties with their
quantification, I've summarized that part of the Shimeta-Jumars analysis
in Table 15.2.
Shimeta and Jumars also emphasize the distinction between encounter
—touching—and retention. Gut content analyses of suspension feeders,
commonly carried out and certainly of ecological interest, give the results
of both processes, which can't easily be teased apart from the data.
Retention depends on such matters as the excess density, size, and drag of
particles, which determine whether they'll bounce off or be swept away, and thus
on some of the same factors that determined encounter. But it depends on
them in different ways and depends as well on the adhesion-determining
chemical and physical characteristics of the surface.
7 The distinction between inertial mass and gravitational mass, a fundamental and
historic problem in basic physics (see Einstein and Infeld 1966), has thus emerged in this
rather odd context!
358
FLOW AT LOW REYNOLDS NUMBERS
(a) sieving (b) direct interception (c) inertial impaction
(d) gravitational deposition (e) diffusional deposition
Figure 15.7. Five mechanisms for suspension feeding shown
diagrammatically.
Other Matters Left Suspended
Anyone thinking about saying anything about the mechanisms of
suspension feeding ought to read the last three papers cited here and perhaps
also Silvester (1983) and LaBarbera (1984). And don't lose sight of the way
the enormous diversity in collector geometry has been swept under the rug
in this broad-brush view. Despite a huge literature, we're still a long way
from a full appreciation of how animals tune their filters to particular ends
such as specialization on a range of prey size and even farther from
understanding the schemes by which potential prey can minimize their chance of
capture. Some animals suspension-feed from upstream or downstream
vortices, as described in Chapter 10. And flows over suspension feeders
commonly oscillate back and forth, with substantial consequences as yet
Table 15.2 Four mechanisms for particle capture by suspension
feeders and how their effectiveness (both rate and efficiency)
varies with particle size and speed.
Mechanism
Direct interception
Inertial impaction
Gravitational deposition
Diffusional deposition
Particle Size
versus Encounter
Rate
+
+
+
Efficiency
+
+
+
Ambient Velocity
versus Encounter
Rate
+
+
o
(+)
Efficiency
1 1 + o
Note: "+" indicates a direct relationship, "-" an inverse relationship, and "o" no
relationship.
359
CHAPTER 15
barely touched upon (see, for instance, Hunter 1989 and Trager et al.
1990).
Finally, an important portion of what have commonly been considered
suspension feeders are somewhere between that and active predators.
Where prey are capable of active evasion and predators grapple and then
either accept or reject prey, one encounters a world of encounters for
which the preceding discussion approaches irrelevance. This world has
been most clearly revealed by the work on copepods of Rudi Strickler and
his collaborators; a good bridge to it from ordinary suspension feeding is
provided by Koehl and Strickler (1981) and Strickler (1985). But it's really
worth special consideration on its own terms.
Information Transfer by Disturbing the Environment
A recurring theme in this chapter has been the gentleness of velocity
gradients in a viscosity-dominated domain and the consequent distance
over which motion of a solid within a fluid disturbed the fluid. At the very
least, any observations on confined animals (as under cover-slips) should
obviously be viewed with skepticism. So far, though, attention has been
limited to single organisms or, at most, assemblages of like-minded ones.
What about interspecific interactions? What's quite clear is that disturbing
the flow by moving through it constitutes a signal that can be detected by
another organism. What's almost as sure is that a moving organism can
detect a nearby nonmoving one by the resulting asymmetry of forces and
flows as it swims. Or, in less fancy terms, swimming (or even sinking) both
announces one's own presence and provides part of the machinery to
detect someone else's presence (Zaret 1980).
What can we say about this peculiar world? First, swimming with a
coating of cilia ought to be "quieter" (in a nonacoustic sense) even than simple
sinking inasmuch as it involves (as noted earlier) steeper and thus more
spatially restricted velocity gradients. Second, swimming steadily ought to
be "quieter" than the jerky swimming associated with swinging a single pair
of appendages in the manner of cladocerans such as Bosmina and Daphnia
(Figure 2.5). Thus the predatory dipteran larva, Chaoborus, about 15 mm
long, attacks at a certain threshold of mechanical disturbance. For Daphnia,
that occurs at a distance of about 3.1 mm, but Diaptomus, a steadily
swimming copepod (similar to the one in Figure 3.8), can get a lot closer
undetected (Kirk 1985). And the steadily swimming copepod, Epischura, swims
up the wake of a prey animal and then attacks from a short distance above
it; by contrast a pulsing swimmer, Cyclops, makes a longer-distance strike
with a high acceleration (Strickler and Twombly 1975).
In that high acceleration, estimated by Strickler (1975) at 12 ms-2, lurks
a partial evasion of this peculiar world. A predator slowly and steadily
360
FLOW AT LOW REYNOLDS NUMBERS
pursuing its prey may not only alert the prey but will to some extent push
the prey forward before actual contact; a burst of speed can briefly drive
the Reynolds number up and reduce this awkward effect of viscosity. One
copepod briefly manages to get up to Re = 500, reaching several hundred
body lengths per second in what may be a record for any swimming
creature (Strickler 1977). Similarly, moving away from a predator will tend to
draw the predator along. A sessile protozoan, Zoothamnium (a colonial
version otVorticella, shown in Figure 10.4), seems to minimize the problem by
contracting its stalk with extreme rapidity and thus achieving a respectably
high Reynolds number. Contraction of the "spasmoneme" in the stalk
turns out to be the most rapid shortening of any contractile element in any
animal (Weis-Fogh and Amos 1972). Whether for predator or prey, these
low-Reynolds-number environments aren't exactly the easiest of all
possible worlds. While "creeping" may be inappropriate for life within them,
"creepy" has some appeal.
361
CHAPTER 16
Unsteady Flows
Back in the second chapter we deliberately limited our purview to
' "steady flows"—ones where speed didn't change over time at any
location fixed with respect to an unmoving solid. (Speed was permitted, of
course, to change from place to place, so fluid could speed up or slow down
as it progressed.) That limit was only occasionally breached, as when
talking about the "clap-and-fling" of beating wings in flight, when considering
the use of drag-based propulsors for rapid acceleration, or when
mentioning the problem of suspension-feeding in bidirectional flows. Other places
where the limitation was constraining were either quietly passed over or
explicitly deferred. In reality, unsteady flows prove to be of major
biological consequence in quite a few situations—appreciation of their relevance
has simply been limited and recent.
Added Mass and the Acceleration Reaction
Drag, by definition, is a force that resists motion. Strictly speaking (as we
ought), drag resists acceleration only to the extent that acceleration
involves motion. The drag of a body accelerating from rest (as in the initial
stage of free fall) begins at zero and increases continuously. Drag as a result
of motion, though, isn't the only kind of resistance that a fluid imposes on
an accelerating body—another is a consequence of acceleration per se. The
phenomenon may be introduced with an extreme example, adapted from
Birkhoff(1960).
Consider a spherical bubble of air, 1 mm3 in volume, in water at
atmospheric pressure, much like a bubble of C02 at the bottom of a glass of beer.
It is suddenly released and accelerates upward because of its buoyancy—
gravitational acceleration times the difference between the masses of
displaced water, 1 x 10~6 kg, and bubble, 1.2 x 10~9 kg. Since the bubble's
weight is negligible, the buoyant force is 9.8 x 10~6 N. The initial
acceleration of the bubble can be calculated as that buoyant force divided by the
mass of the bubble—8200 m s~2 or all of 830 times that of gravity. That's
disturbingly high.
But for the bubble to advance upward, some water must move
downward. Paths of water and gas are different, and the water movement
involves velocity gradients and no distinct volume, so the analytical problem
362
UNSTEADY FLOWS
is far from trivial. But at least for some regular shapes such as the present
sphere and assuming an ideal fluid, the details have been worked out (see,
for instance, Batchelor 1967). To the mass of the bubble must be added an
"added mass" (sometimes called "apparent additional mass") that, for
linear acceleration of a sphere, equals half the mass of the displaced fluid. We
can now recompute the acceleration of the bubble, taking its effective mass
as 0.5 x 10~6 instead of 1.2 x 10~9 kg. The result is a value only twice that
of gravity, something a lot more credible.
(In what follows, I'm mainly following the explanations given by Daniel
1984 and Denny 1988.) We can now recognize that accelerating a body in a
fluid involves a force with three additive components. The first is our old
friend drag. The second is the force needed to accelerate the mass of the
body forward. And the third is the force needed to accelerate the added
mass of fluid backward. How much mass must be added depends on the
volume and shape of the object; the shape-dependent part is expressed as
the "added mass coefficient," Ca:
F = | C(/pSU* + ma + CapVa. (16.1)
(V is, of course, the volume of the body; density in the third term refers to
the density of the surrounding fluid.) The force expressed by the last term
is commonly called the "acceleration reaction"; the mass of the body plus
the added mass (m + CapV) is often called the "virtual mass."
Sometimes virtual mass is used synonymously with added mass, but the
meaning is usually clear from the context.
While the added mass coefficient looks rather like the drag coefficient, it
proves to be considerably less troublesome—mainly because volume, with
which it's multiplied, isn't subject to as many interpretations as is area.
Lamb (1932) gives the numbers that everyone else uses. They're calculated
for an ideal fluid, but the assumption of an ideal fluid usually isn't too bad.
One typically works with the initial stage of movement from rest (where a is
high and U is low), and many of the queer correlates of viscous flow such as
separation and circulation just haven't had time to get going. And when
drag and velocity are high, a is low, and the correction for added mass is
small.
For a sphere (as just mentioned), the coefficient is 0.5. For a circular
cylinder moving normal to flow it's 1.0. In general, bodies more slender in
the streamwise direction have lower added mass coefficients than fatter
bodies; Figure 16.1 (from Daniel 1984) summarizes the situation. Thus
shapes that minimize drag are also reasonable choices if the acceleration
reaction is a substantial concern. In one way, though, the graph is a little
misleading—it suggests that a circular flat plate broadside to flow has an
363
CHAPTER 16
c
'o
8
o
£
4>
73
73
a
fat
fineness ratio
thin
Figure 16.1. Added mass coefficient versus fineness ratio (flow-wise
over cross-flow lengths, as in 5.7) for spheroids and elliptical cylinders.
Spheroids with fineness ratios less than 1.0 are oblate, while ones with
ratios above 1.0 are prolate. Spheres and circular cylinders have ratios of
just 1.0.
infinite added mass coefficient. That's mainly an accident of definition—
the flat plate has zero volume, and the effective product of volume and
added mass coefficient (CaV) is s/s r3 (Batchelor 1967). For an elongate
rectangular flat plate, accelerated normal to its surface, the added mass is
equal to the fluid mass of a circular cylinder whose diameter is the width of
the plate and whose length is that of the plate (Ellington 1984a): CaV is tt/4
lw2. That's the same as the value for a circular cylinder of the same length
and width, as in Figure 16.1.
Bear in mind that both velocity and acceleration are vectors; while they
act in the same direction when velocity is increasing, they act in opposite
directions when velocity is decreasing. Drag always slows a body down; the
acceleration reaction (like the body's mass) always opposes any change in
speed, up or down. So during (positive) acceleration, the three terms of
(16.1) are arithmetically additive, while during deceleration the two
acceleration terms oppose the drag term.
The Beginning and End of a Swim
The acceleration reaction demands that a swimming animal work
harder to get going, exerting an additional force that's proportional to its
volume in order to achieve a given acceleration. For many escape responses
and lunging predatory strikes, the matter is not negligible. Perhaps the
most extreme case so far uncovered occurs in the escape response of a
364
UNSTEADY FLOWS
crayfish, Orconectes (Webb 1979a). It flexes tail and abdomen and goes
rearward with a maximum acceleration of 51 m s~2. Drag turns out to be
only around 10% of the resistance, with 90% caused by the masses of
crayfish and water—as is reasonable for a high acceleration to a fairly low
final speed. The relative importance of the acceleration reaction isn't
strongly correlated with body size for movements. It's been shown
significant in a small copepod, Acanthocyclops (Morris et al. 1990); in shell closing
of hinged brachiopods, small creatures looking superficially like bivalve
mollusks (Ackerly 1991); in a water boatman bug, Cenocorixa (Blake 1986);
in an angelfish, Pteraphyllum (Blake 1979a); in a frog, Hymenochirus (Gal and
Blake 1988); and in pike, Esox (Frith and Blake 1991). Drag may be only
indirectly associated with body volume per se, but the acceleration reaction
is very closely tied to it—Webb (1979b) argues that reduction in mass is an
important phenomenon in acceleration specialists.
In stopping, the acceleration reaction keeps an animal moving longer
than it otherwise would. I haven't heard of a case in which this has
biological relevance; where it does matter, though, is in determination of drag by
observation of decelerative gliding. Except where deceleration is very
gentle, you'll get a significant underestimate of drag if you neglect the
correction, as mentioned in Chapter 7. To correct, one needs some figure for
added mass coefficient. Such figures aren't as widely available as would be
desirable, and estimation from the data of Figure 16.1 is a reasonable
approach.
For systems in which impulsive swimming is the normal mode, the
acceleration reaction is of special importance. For jet-propelled creatures
accelerating from rest, it represents over half the total resistive force—Daniel
(1984) got the result for a salp, a dragonfly larva, several coelenterate
medusae, and a squid; the accelerations ranged from about 0.2 to 20 m s~2.
Again, we're looking at something that may seem odd but certainly isn't
trivial.
Making Paddles and Tails Work
I once had a small wind-up toy fish whose rigid tail swung back and forth
relative to its body. If one does a crude analysis of the forces acting on the
tail fin, assuming symmetrical motion and so forth, one comes to the
conclusion (and not just at very low Reynolds numbers) that all forces should
balance and the fish should get nowhere. As Figure 16.2 shows, forces are
forward and lateral on half-strokes moving toward the midline, with the
lateral ones canceling between such half-strokes. Forces are rearward and
lateral on the half-strokes moving away from the midline, with the lateral
ones again canceling. Unfortunately, the forward and rearward ones
cancel as well. That's the result of a "quasi-steady" analysis, ignoring the accel-
365
CHAPTER 16
/ ^/
force
motion
(a) quasi-steady view
\ ^\
11 in iv
4 force
(b) unsteady view \s/\ \^\
\ \
Figure 16.2. Forces generated as a rigid tail swings back and forth:
(a) the forces anticipated in a quasi-steady analysis; note that all
ultimately cancel so no net thrust is generated; (b) the forces predicted by
considering the acceleration reaction in an unsteady analysis, as
explained by Daniel (1984). Here each quarter stroke produces a forward
force, while lateral forces again cancel.
eration reaction. As Daniel (1984) points out, a proper "unsteady" analysis
comes to a different conclusion. To the forces on the tail previously
considered, it appends a forward force from the accelerative movement toward
the midline—forward because the acceleration is rearward. This latter isn't
offset; instead, decelerative movement away from the midline also gives a
forward force—forward because while movement is forward, it's now
decelerative. Since the toy fish does swim, legitimization is comforting. That's
not entirely how real fish make caudal fins push, but it shows how the
acceleration reaction can be put to use to generate thrust.
A similar unsteady analysis shows how a dytiscid beetle can get thrust by
swinging its paired hindlimbs rearward and forward without any
asymmetry in speed or configuration (Figure 16.3). What's crucial here is that
each hindlimb swing back and forth, not about an axis perpendicular to the
long axis of the body, but about one tilted backward—in short, that the legs
come closer together in the rear than in the front. Both accelerative
rearward and decelerative forward leg movement will produce thrust near the
forward extreme of the stroke cycle. The equivalent forces near the
rearward extreme of the cycle will be mainly lateral—inward, in fact—and thus
won't greatly detract from the thrust. Daniel (1984) calculated that the
optimal angle about which the legs should swing is 120° back from the body
axis, which corresponds very nicely to what Nachtigall (1960) found. Again
it isn't the whole story of how thrust is made, but the animals are certainly
doing what we think they ought to if acceleration reactions matter.
366
UNSTEADY FLOWS
/ . \
legs accelerate II I legs decelerate
rearward stroke forward stroke
Figure 16.3. Forces generated as paddles are swung backward and
forward of vertical planes 120° back from the free stream. Both rearward
acceleration and forward deceleration near the extreme forward position
produce forward force—net thrust. The analogous speed changes near
the rearmost position produce mainly lateral forces that cancel.
We should remind ourselves that a drag-based, paddling accelerator
encounters the acceleration reaction in two essentially opposing ways. The
reaction will mean that for a given stroke the paddles will exert more force
than would be the case if they only generated drag. In addition it takes
more force to get the body moving than if drag alone counted.
Accelerative Forces on Stationary Objects
Consider what happens when an attached organism is suddenly
subjected to a surge of water—perhaps as a result of a nearly breaking wave.
Again the acceleration reaction comes into play, but its operation is slightly
different from what we saw for impulsive locomotion. As Denny (1988)
commented, simply dropping the middle term of equation (16.1) seems
reasonable since the object itself no longer is accelerated. As it happens, an
equivalent term takes its place. As he explains it, if the object weren't there,
a body of water of its volume would be accelerated along with the
remaining water; thus the presence of the body is equivalent to accelerating its own
volume in the opposite direction. The mass of the body, in that middle
term, must consequently be replaced by the mass of water it displaces:
F = 1 CdpSU2 + pVa + CapVa
or
F = ^ CdpSU2 + (1 + Ca)pVa. (16.2)
A really fine argument about the relative relevance of the two terms of
(16.2) has been made by Denny et al. (1985). They pointed out that the drag
367
CHAPTER 16
of an attached organism, proportional to projected area, scales with the
square of a linear dimension; by contrast, the acceleration reaction,
proportional to volume, scales with the cube of a linear dimension. Attachment
tenacity ought to scale with attachment area, so, assuming geometric
similarity, the large organism is no worse off than the smaller with respect to
drag (assuming thin boundary layers and so forth). On the other hand, the
large organism is distinctly worse off with respect to the acceleration
reaction—with increasing size the latter increases relative to the capacity to
hold on. They applied the argument to such creatures as limpets, snails,
barnacles, and sea urchins, calculating dislodgement probabilities for
various sizes, fluid accelerations, and time durations. In general, the results
agree rather nicely with observations on maximum size versus relative
exposure in the habitats of the organisms. At least in these extreme
environments, nature seems to care very much about the forces specifically
caused by the unsteadiness of flow.
Added Mass in Air
Air is eight hundred times less dense than water. Does that mean that
added mass is insignificant for creatures that live in air rather than water?
After all, density of the medium appears in both terms of equation (16.2),
so changes in it shouldn't affect the ratio of either term to their sum. In
fact, three considerations minimize the impact of the acceleration reaction
in air. First, the speeds of flow are much higher in air, whether we consider
average ambient winds, extreme winds (Vogel 1984), or locomotory
speeds; and speed squared appears in the first term of (16.1) and (16.2).
Second, in all but a few rare instances accelerations aren't especially
impressive,1 whether of organisms taking off or of wind gusts suddenly
striking. Finally, the mass of organisms living in air is far higher than the mass of
the air they displace, so virtual mass is little different from actual mass. To
put this last point another way, mass certainly matters when starting up, but
it matters directly and without any aerodynamic chicanery. In short, the
last term of equation (16.1) will rarely be of consequence.
But once in a while the acceleration reaction does make some difference.
My first publication (Vogel 1962) concerned some variation in the wing-
beat frequency of fruit flies. Frequency in many insects is set by the
stiffness of the thorax and the moment of inertia of the wings. Changing air
density might affect the latter if it alters the effective volume of wings plus
boundary layer or the virtual mass (really virtual moment of inertia) of this
rapidly oscillating system. In fact, calculations bore out the predictions:
1 Such as the sporangium of the fungus Pilobolus when shot from its hypha, flea
jumping, and a few other cases; see Vogel (1988a) for data.
368
UNSTEADY FLOWS
added mass is appreciable relative to wing mass (terms 3 and 2 respectively
in equation 16.1). Still, added mass is significant only in small flapping fliers
with their light wings and high wingbeat frequencies. While it seems always
to be less than wing mass, in many insects it's well within the same order of
magnitude (Ellington 1984a; Ennos 1989b). To emphasize the lightness of
these structures, I note that a kilogram of the wings of Drosophila virilis (a
biggish fruit fly), laid end to end, would extend from Boston to Washington
or London to Glasgow. More relevant is the fact that these insects devote
only half of one percent of body mass to wings.
Consequences in the Wake of Vortices
No rule prohibits steady flows, even through and across rigid objects,
from generating unsteady wakes. And a messy wake may come home to
haunt you—your troubles have not necessarily been left behind. In
particular, vortices shed regularly impose periodically varying forces on the
objects that shed them. Such forces may break an object, especially a fairly
rigid one; worse, perhaps, such forces may induce motion in the object that
increases the shedding and the forces it generates—so-called flutter and
galloping instability.
Vortex Shedding: The Von Kdrmdn Trail
In Chapter 5 we saw that rows of vortices were shed behind bluff bodies
such as cylinders at Reynolds numbers above about 40. These vortices were
left behind in the wake in alternating positions in parallel, streamwise
rows; each vortex rotated in the opposite direction of the preceding and
succeeding ones (Figure 5.5). Any object in (or crossing) such a wake will be
buffeted by these vortices. In an array of closely spaced cylindrical objects
exposed to a flow, the foremost ones may be less subject to damage than
those farther back.
But even the body that sheds the vortices is affected. Recall the
discussion of circulation in Chapter 10. Any circulation created in one place has
to be balanced by an equal and opposite circulation somewhere else.
Consider any bound vortex around body and the vortices the body has already
shed. That bound vortex must be equal in net strength and opposite in net
direction to the combined circulation of those shed vortices. The
implication is curious: every time a vortex is shed, the circulation around the body
must reverse direction! Circulation, of course, produces the transverse
force that we've called lift. So the body is shaken normal to flow as a direct
result of the shedding of vortices, and the frequency of shaking must be
precisely one-half the frequency with which vortices are shed.
Figure 16.4 shows the normal spacing of these vortices that, taken to-
369
CHAPTER 16
flow
Figure 16.4. The spacing of vortices in the Von Karman trail behind an
elongate bluff body such as a cylinder. The ratio of 3.56:1 is stable.
gether, constitute what's called a "Von Karman trail" or "Von Karman
vortex street." The trails are stable if the distance between successive
vortices on each side of the street is 3.56 times the distance between the center
lines of the two rows of vortices. This ratio doesn't change with speed or
with the way the body produces the wake, although the vortices do diverge
a bit well downstream of the body. If the body is free to move transversely to
the free stream, the width of the trail will increase in proportion to this
increased effective width of the body—but the streamwise spacing between
vortices will increase as well, maintaining the 3.56 ratio. Lateral motion of
the body will, as well, increase drag, circulation, and lift; and it will reduce
somewhat the frequency of vortex shedding (Steinman 1955). Incidentally,
the vortices aren't stationary but move in the same (relative) direction as
the body at a slower rate. Their speed is proportional to their circulation, so
one can calculate the lift and drag of a body from measurements of the
dimensions and speed of its trail of vortices, as described by Prandtl and
Tietjens(1934).
What determines the frequency with which vortices are shed? As with so
many situations in fluid mechanics, general guidance takes the place of a
properly precise and universal formula. As mentioned earlier, both the
Reynolds number and the drag coefficient can be obtained by dimensional
analysis, assuming that four variables (length, velocity, density, and
viscosity) are germane. If the same sort of analysis is done with the addition of a
fifth variable, frequency (dimensions of T~l), another dimensionless index
emerges. In the present application this variable is called the "Strouhal
number,"
S*=jj> (16.3)
where n is the frequency of a periodically varying flow; / is a characteristic
length of a solid object, usually the diameter or the distance perpendicular
to both flow and the long axis of the object; and U is the free-stream
velocity. The Strouhal number serves as a dimensionless frequency, just as
370
UNSTEADY FLOWS
the drag coefficient works as a dimensionless drag. And, like the drag
coefficient, it's a function of shape and Reynolds number and can be
conveniently plotted as an ordinate against the Reynolds number; a specific
shape gives a characteristic line. The frequency of vortex shedding can
easily be obtained from the Strouhal number, the size of an object, and the
free-stream speed.
Among biologically interesting shapes, data are available for cylinders
and for fiat plates normal to flow (Figure 16.5). Fortunately, the Strouhal
number varies little with Reynolds number in the range that's most likely to
concern us, even though the physical character of the flow changes.
Consider, for example, a cylinder. At Reynolds numbers below 40, no vortices
are shed, so the Strouhal number is effectively zero. From 40 to 150 the
train of vortices is laminar, and the Strouhal number rises smoothly from
0.1 to 0.18. From 150 to 300, the vortices gradually become internally
turbulent, and turbulence persists up to about 3 x 105 with a nearly
constant Strouhal number of 0.20. Above this drag crisis the wake becomes
narrower, the boundary layer is turbulent, and the vortices are
disorganized. Above about 3 x 10(\ a discrete vortex street is reestablished; and
it persists up to at least Re = 101(). This last figure, by the way, is no wind
tunnel determination but comes from a satellite photo of a cloud-marked
vortex trail behind the mountain peak of Guadalupe Island (Simiu and
Scanlan 1978). Other useful references on Strouhal number are Bishop
and Hassan (1964), Goldburg and Florsheim (1966), Blevins (1977), and
Van Atta and Gharib (1987).
Von Karman trails are everyday occurrences. For example, the way
suspended electrical wires sung aeolean tones in the wind was what drew
Strouhal's attention to vortex shedding in 1878 (Massey 1989). But little in
the way of biological mischief has been laid at their door. Not that vortex
shedding doesn't happen: in a larch plantation in which trunk diameters
were about 0.4 m and wind about 0.6 m s~l the predominant frequency of
turbulent eddies was, as expected, about 0.3 Hz (Grace 1977).
Perhaps it's of some importance that organisms such as trees don't shed
vortices at the wrong frequencies. The shaking of an object induced by
vortex shedding might be of considerable consequence when the object
is either flexible or flexibly mounted and the rate of shedding is close
to coincidence with some natural oscillatory frequency of the object. In a
20-m s_1 wind, a 0.2-m tree trunk will shed vortices at 20 Hz, fortunately
too high a frequency to have any bearing on sway or wind throw. Holbo et
al. (1980) found a sharp compliance peak in 0.35 m Douglas firs at 0.3 Hz,
well below such high-wind vortex shedding frequencies but well above
typical gust frequencies of about 0.05 Hz.2 For a given Strouhal number,
2 And that's about what happens with the loblolly pines (Finns taeda) in my front yard
during storms.
37 1
CHAPTER 16
a
73
C
73
O
i_
00
Reynolds number
Figure 16.5. The relationship between Strouhal number and Reynolds
number for flow normal to a cylinder and a long flat plate.
frequency will be proportional to flow speed divided by body diameter. So
the tapering of a tree trunk together with any skyward increase in wind
speed ought to further reduce the chance of getting shaken up by
shedding vortices. Similarly, a cat's whisker of 0.3 mm diameter in a 1-m s_1
wind would shed vortices at 1000 Hz. But the whisker is neatly tapered
from thick base to fine point, so it isn't likely to hum or purr.
Self-excited Oscillators and Aeroelasticity
Vortex shedding at a frequency determined by the Strouhal number
generates a purely aerodynamic or hydrodynamic forced vibration; that is,
the periodic force driving the vibration of a structure exists whether or not
the structure actually moves. To this mechanism must be added another of
somewhat different origin, but which may act in concert with it. This
second mechanism also releases a trail of alternating vortices, but the
frequency at which they're shed isn't such a simple function of the free-stream
velocity.
Steinman (1955) describes the mechanism of these self-excited
oscillators in the following way. Consider a half-cylinder with the flat face
upstream, as in Figure 16.6a. If the half-cylinder is moving laterally as a wind
strikes it, the net or relative wind will approach obliquely. The forward
stagnation point will be offset from the center of the face, and more of the
flow will move around the leading edge of the cylinder. This difference in
flow will produce a circulation around the object, and that in turn will
372
UNSTEADY FLOWS
(a) (b) (c)
Figure 16.6. Self-excited oscillators: (a) circulation and lift on a half-
cylinder moving across a flow; the lift will augment the preexisting
movement; (b) a pendulum will swing back and forth in a flow as a result of
such circulation; (c) a spring mount can substitute for gravity in
providing the restoring force necessary for such an oscillator.
generate a lifting (cross-stream) force that will tend to keep the half-
cylinder moving in the same direction. In other words, the flow will amplify
any initial perturbation from zero cross-stream speed.
All that's needed now to get a proper oscillation is some restoring force
that increases with distance traveled laterally from the original position.
Steinman describes a version he calls a "Steinman pendulum" (Figure
16.6b) in which the aerodynamic force is eventually offset by gravity, only
to reappear in the next half-cycle and keep the pendulum swinging. I use
the device for a demonstration of both the fluid-mechanically driven
pendulum per se and of circulation and vortex shedding—every time the
direction reverses a vortex is shed into the wake and a new bound vortex of
the opposite spin begins, just as in flapping flight. A rigid flat plate works as
well as a half-cylinder broadside to flow. Mine is about 3 by 20 cm,
suspended beneath a bearing (it must be prevented from downstream
deflection), and gets half-immersed in a flow tank. It does nothing until tweaked
but then oscillates quite persistently, shedding vortices that are easy to
mark with dye in the water or sawdust on the surface. Steinman points out
that a spring might alternatively provide the restoring force (Figure 16.6c);
the scheme then takes on a more distinctly biological odor. I've watched
white poplar (congeneric with quaking aspen) leaves flutter both on the
tree and in a wind tunnel (Vogel 1989, 1992c). They move side to side on a
flattened petiole (Figure 16.6c), and it certainly appears as if they're
playing some version of the present game.
The most famous case of self-excited oscillations must be the failure of
the Tacoma Narrows suspension bridge in Washington State in 1940 after
less than half a year of service. It was designed for static wind loads of 2400
Pa, but in a mild gale that produced a static load of only around 240 Pa it
373
CHAPTER 16
developed spectacular torsional oscillations and, while spectators watched
and movies were made, the roadway broke from its cables and dropped
into Tacoma Bay. Such oscillation (excluding the rigid Steinman
pendulum) depends as much on the character of the solid body as on that of the
moving fluid. In particular, it depends on the lack of stiffhess of the solid—
how it behaves as the spring to provide the restoring force. The subject is
usually called "aeroelasticity," and it's of great importance to people who
design airplanes and large buildings as well as bridges. (The original
explanation of the collapse involved only Von Karman vortex shedding; it now
appears that aeroelastic coupling was the principal culprit—Petroski 1991
gives a nice account of the controversy. On aeroelasticity in general, one
might look at Dowell et al. 1989.)
So two things matter—stiffness and shape. The former is a matter both
of stiffness in the strict sense (especially the torsional modulus of elasticity)
and of damping and resilience. As for shape, an entire cylinder (broadside
to flow) is a relatively stable profile, whereas a half-cylinder is stable when
the flat side faces downstream, not upstream. As mentioned, a flat plate
normal to flow is unstable. A beam shaped like a "T" in cross section is
stable if the lower arm faces upstream but unstable if the lower arm faces
downstream. Thus a bridge with a cantilevered external sidewalk (the
lower arm of the T) is more stable than one with a solid upright truss
outboard of the sidewalk.
The cylindrical profiles of trees, large sea anemones, and large algal
stipes (as in the sea palm, Postelsia) probably limit accidental application of
the mechanism, but I'd expect nature to have put it to use here and there,
as perhaps she does in arranging for the leaves of aspen to quake. A likely
place for this kind of flow-induced, self-excited oscillation is in a scheme by
which seeds or spores are preferentially released when a wind is blowing.
But at this point I know of no specific case that's been the subject of
investigation.
Self-excited oscillation associated with aeroelasticity isn't limited to
external flows. Cut the end off a cylindrical balloon and blow through it. The
balloon will fill and empty irregularly while making socially unacceptable
noises. What seems to be happening here is that increasing flow decreases
the pressure, as described by Bernoulli's equation (4.1); low pressure then
permits the balloon to collapse inward, increasing its resistance and
decreasing flow, which in turn permits expansion (the basis for the
phenomenon was described in Chapter 14). It happens in nature, indeed inside some
of us. Jones and Fronek (1988) found that a constriction in a flexible pipe
generated vibrations in the flow, and these in turn lowered the Reynolds
number at which turbulence set in. They used conditions approximating
pathological stenoses (narrowings) in the human circulation. And flutter in
374
UNSTEADY FLOWS
collapsible tubes provides a persuasive model for the generation of
respiratory wheezes (Gavriely et al. 1989).
Still, flexibility isn't always a hazard, imposing risks of aeroelastic flutter
and such. Koehl et al. (1991) describe how flexibility can provide a way to
escape drag under quite reasonable circumstances. Imagine a
symmetrically bidirectional, oscillating flow. A really flexible sessile organism will
follow the flow and experience mainly tensile forces. If the organism is
longer than the distance the fluid travels in a cycle, some distal portion of
the organism will never be pulled upon! The criterion, then, for the point
at which drag evasion begins is the ratio of the length of the organism to
half the wave period times the average current. For long, marine macro-
algae the point is commonly reached—they can get up to around 10 meters
long. In fact, long algal blades can actually experience less force than
shorter ones The samejust won't happen on land: with wind gusts 10 or 20
seconds apart and speeds of 10 or 20 m s-1, a leaf would have to be a
hundred or so meters long to benefit.
More Ways to Get Unsteady Flow
We've seen several ways in which an initially steady flow could interact
with some solid structure to produce oscillation—vortex shedding behind
a rigid object and self-excited (aeroelastic) oscillation of a flexible object.
These certainly don't exhaust the possibilities, and some others should be
kept in mind:
1. Flow through a smooth pipe faces a sudden increase in resistance
when turbulence sets in, as one can see from Figure 13.6. Assume
flow from a constant-head source at a Reynolds number of around
2000. Faster flow will generate turbulence, increasing resistance
and slowing itself; flow will then become laminar, decreasing
resistance and speeding up. What results is the production of a series of
turbulent "plugs" that appear as spurtings from the end of the pipe.
Massey (1989) gives a little more detail if you want to contrive a
demonstration.
2. The great drag crisis at Reynolds numbers around 100,000 can be
made to produce an analogous oscillation. Consider a sphere
suspended as a pendulum in a wind so it can swing back and forth
(streamwise) at about that value. It can be pushed backward by the
higher drag of laminar flow at the lower relative wind concomitant
with its backward motion. As it slows down the relative wind will
increase, and it can move forward with the lower drag of turbulent
flow; as it starts to swing down and back, its drag will again increase.
375
CHAPTER 16
3. Aircraft can follow an oscillating path. Speeding up due to descent
or a local wind gust generates more lift, which stops the descent and
slows the craft, which reduces the lift, and so forth. The wavelike or
looping path is called a "phugoid" oscillation; it's all too easy to
obtain with model airplanes and gliders (Von Karman 1954; Sutton
1955). This phenomenon and the one below depend on the
pitching moment and thus the location of the lift vector as well as on the
magnitude of lift—I'm ignoring some complications.
4. Another lift-based oscillation, delayed stall, was dismissed in
Chapter 12 as of little consequence for animal flight. If the angle of attack
of an airfoil is raised near the stall point, the increase in lift doesn't
follow immediately. But when it does appear, it elevates the craft
and increases the angle of attack beyond stall. That in turn
decreases lift and causes the craft to descend, decreasing the angle of
attack back beneath the stall point. In effect, the separation point is
moving fore and aft on the top of the airfoil. This (or something
closely analogous to it) seems to be what happens when leaves flutter
up and down in a mild wind (Perrier et al. 1973).
When Is Flow Unsteady Enough to Matter?
Obviously no flow is ever perfectly steadyjust as no solid is ever perfectly
rigid. Unsteadiness is clearly important for animal flight, especially for
small fliers and during hovering (as we talked about in Chapter 12). Other
areas have received much less attention. Some rules of thumb can help
decide when one might safely ignore unsteady effects—when one might
get away with averages over time or with a quasi-steady analysis based on a
sequence of steady-flow situations. Lighthill (1975) gives a dimensionless
criterion, the "aerodynamic frequency parameter" (sometimes the
"reduced frequency") for oscillating systems such as beating wings:
/„ = ^ (16.4)
(n is wingbeat frequency and c is wing chord.) It amounts to a a ratio of
chordwise flow speed to free stream speed. (Some sources omit the factor
of 2ir.) Other things being equal, it will be highest during hovering; it will
be higher for short, broad wings than for long, thin wings and when wing-
beat frequencies are high. If the parameter exceeds 0.5, unsteady effects
are likely to be significant. In full forward flight values usually come out to
about that number—a locust forewing operates at about 0.25 and a hind-
wing at about 0.5; a fruit fly wing works at about 0.5 also. So they're
awkwardly close to that critical value, and the value is very much higher in
slow flight and hovering.
376
UNSTEADY FLOWS
A slightly different test for steadiness is used for pulsating flow in
circulatory systems (Womersley 1955). The dimensionless test parameter for
pipes is called the "Womersley number," Wo:3
VV0 = aJI™£. (16.5)
Here n is the frequency of a sinusoidally applied pressure gradient. If Wo is
less than unity, viscosity predominates and the profile of flow across a pipe
is essentially parabolic; flow is then said to be quasi steady. If Wo is above
unity (favored by large pipes and frequent pressure cycles), then inertial
forces distort the profile and the Hagen-Poiseuille equation cannot be
accorded full faith. It's another test, along with Reynolds number and
entrance length, that a system must pass for the assumption of proper
parabolic flow to bejustified. In our aorta, Wo is 10 or more, but that's about
as high as we ever get, so the Hagen-Poiseuille equation isn't in very serious
trouble in most of our arterial system on that account. For comparison, Wo
is around 0.001 in the capillaries (Caro et al. 1978). The Womersley
criterion also finds use in the analysis of flow in respiratory airways (see, for
instance, Moslehi et al. 1989).
Finally, I should reemphasize the importance of unsteady flows. Crops
wave in the wind, with effects on water loss and gas exchange (Finnigan and
Mulhearn 1978). In storms trees sway before they fall, and falling results
more from the dynamic forces associated with gusts and swaying than from
the static loading of steady winds on rigid objects (Grace 1977). Waves
striking shores impose unsteady flows on every attached organism there.
And, of course, sound production, whether by humans, other animals, or
musical instruments, always involves unsteady flow; but that's something
about which this book will remain silent. One simply can't do everything.
3 It might be of some interest to note that the square of the Womersley number equals
tt/2 times the product of Reynolds and Strouhal numbers.
377
CHAPTER 17
Flow at Fluid-Fluid Interfaces
In this chapter we will relax another of our initial assumptions.
Except for the briefest of allusions, a fluid has been permitted to make
interfaces only with a solid—even if the solid hasn't always been
incorrigibly stiff and unyielding. Let's talk now about situations in which two fluids
make contact without mixing—an interfluid interface—and where one
fluid moves across the other.
What's required to permit such contact without mixing is that the inter-
molecular cohesion of at least one of the fluids be greater than the adhesion of
the molecules of one fluid with those of the other. Since intermolecular
cohesion in gases is negligible, we don't find proper interfaces between
gases. What we do get are gas-liquid and liquid-liquid interfaces, where the
condition that cohesion exceeds adhesion can be met. If it's met, work is
needed to create surface area—a substance that's free to flow will
spontaneously minimize the area it presents to the other substance. If we have two
substances, a and b, we can represent the energy per unit area that we'd
have to supply to create surface as ya and 7^—these are the two "works of
cohesion." Similarly, we can represent the intermolecular adhesion as y(lh.
liyaf, exceeds (7,, + yh), then the fluids mix and no interface occurs. If yah is
less than (ya + 7^,), then a fluid-fluid interface will form. If so, and if 7,, > yh,
then substance b will preferentially surround a.
Surface Tension
For our purposes, one fluid, say a, will almost always be water and the
other, say b, will be air; that means that yh = 0 and yah — 0. We then only
have to worry about the value of ya. ya generates an additional material
property, one that's relevant to a fluid-fluid interface. It is, of course, surface
tension. How do we get the latter from work (or energy) per unit area? In
fact, we have it already—work per area is both dimensionally and
practically the same as force per distance, and ya, henceforth just called 7, is the
surface tension. It's a property of a liquid, more particularly of a liquid
making an interface with another liquid or with a gas; its dimensions are
force per distance (or MT~2), and its units are newtons per meter.
A note about the peculiar dimensions. We're more accustomed to
"tensile force," with dimensions of force, and to "tensile stress," as force per
area of cross section than to "tension," as force per distance. If you pull on,
378
FLOW AT FLUID-FLUID INTERFACES
say, a sheet of rubber, tensile stress is a good measure of what you're
doing—it's the force you apply divided by the product of the width and
thickness of the sheet. But when you pull on an air-water interface, making
more surface, you're pulling on something that really has no thickness in
the normal sense. Or, to put it another way, as you extend the interface,
more water molecules move out to the interface, creating more of it—the
interface isn't a material, and it doesn't stretch out and get thinner as a
result of the pull. So force per distance rather than per distance squared is
entirely reasonable for surface tension.
Again, the interface we care about is that between water and air. As might
be expected from its basis in intermolecular attraction, the value of surface
tension decreases as temperature increases. AtO°C, it's 0.756 N m-1; at 10°
C, it's 0.742; at 20° C, it's 0.728; at 30° C it's 0.712. Adding inorganic salts
increases the surface tension a little; thus for seawater (at a salinity of 35
°/oo), the value is about 0.78 at 20° C.
In addition to density and viscosity, we now have a third property of
fluids relevant to flows of biological importance, this one applicable to
liquids only and not to gases. I don't mean to dwell on all the biological
phenomena to which surface tension is relevant—plastron respiration in
aquatic insects, the prevention of air embolisms in the xylem of leaves,
surfactants in lungs, and so forth; I touched on many of these in an earlier
book (Vogel 1988a). Here we're concerned with fluids in motion and are
ignoring problems of fluid statics—waves and locomotion are what matter.
Waves
A truly smooth air-water interface is fairly unusual in anything other
than small bodies of liquids (hence, I suppose, the expression "tempest in a
teacup"); more common are disturbances, and the most common of
disturbances are periodic waves. These surface waves are such ordinary things it's
easy to forget just what a queer business they are. For one thing, all three
fluid properties that we've talked about conspire to prevent them. The
density of water under the urging of gravity opposes any nonhorizontal
surface, surface tension opposes any surface of nonminimal area, and
viscosity opposes the kind of shearing internal motion that waves inevitably
involve. For another thing, while waves inevitably move, what travels across
the interface is no net volume of liquid but rather a kind of ghost—the
form of a disturbance. Disturb the interface in one place by compression
(pushing down on it) or by shear (blowing across it), and the disturbance
moves laterally for quite a remarkable distance in the form of a traveling
elevation difference, a surface wave.
What's going on in a wave is an orbital oscillation of water beneath the
surface (Figure 17.1). Water travels in the direction of propagation of the
379
CHAPTER 17
wave propagation
t - i - t
CJ t - i - t
_ t - t - t
.•■â–
w t - ♦ - t
(a) (b)
Figure 17.1. Two views of what happens as a wave moves across a deep
body of water: (a) the orbits of several particles of fluid as a wave passes;
note that these are not vortices shearing where adjacent—bits of water
are just translating in circles in synchrony with ones above and below;
(b) an instantaneous view of the water velocities beneath a passing
wave to emphasize the point made above.
wave when it's near the crest and in the opposite direction when it's near
the trough. Thus as a wave passes, a particle of water travels in a circle. The
radius of the circle reflects the amplitude of the wave, and the period
needed to make the circle reflects (inversely) the speed of movement of the
wave across the surface. With increasing depth beneath the surface, the
radii of these orbits decrease. It's easy to imagine from a diagram showing a
few such orbits that a series of vortices exits beneath each wave, but that's
not really the case. Instead a whole vertical sheet of water is moving
upward, forward, downward, and backward. Adjacent sheets, in the direction
of wave propagation, do likewise but at phases that increasingly lag in time.
Thus adjacent sheets shear across each other during their upward and
downward motion. In shallow water (defined arbitrarily as depths of less
than half the wavelength) the circles become increasingly elliptical with
depth—right near the bottom water goes only back and forth. Bascom
(1980) gives a good general introduction to these phenomena.
We need to define a variable for this motion of a train of disturbances.
Velocity as we've used it is a little misleading since almost no net fluid
movement is involved. Instead what's used is a variant of velocity called the
"celerity," defined as distance per time, but where the distance is that
380
FLOW AT FLUID-FLUID INTERFACES
between adjacent wave peaks and the time is that needed for one peak to
replace the previous one at a given point—thus celerity is wavelength (X)
over period:
X
7
(17.1)
As it happens, the celerity depends only negligibly on wave height but
quite substantially on wavelength. What sets the relationship between
celerity and wavelength are the agencies tending to restore the flat and
horizontal equilibrium interface—gravity and surface tension. That an
increase in either will make a wave move faster isn't too hard on the intuition
(think of a stiffer spring), but somewhat queerer is the fact that the two
interact with wavelength in opposite ways. Without troubling with a proper
derivation (see, for instance, Denny 1988 if you wish), the relationship
among gravity, wavelength, and celerity is
v2ir-
And that among surface tension, wavelength, and celerity is
(17.2)
'2jry
Xp
(17.3)
S/UI '
elerity, c
<-»
0.4-
0.3-
0.2-
0.1-
o-
°-23Vs-»*^.
m/s "V
i
real waves^.
17 mm
i
pure capillary
^-«^_^waves
T—
•
•^pure gravity
waves
-
-
....
0
0.02 0.04
Wavelength, A,, meters
0.06
0.08
Figure 17.2. The relationship between celerity and wavelength for real
waves, as well as for gravity waves (were surface tension zero) and
capillary waves (were gravity zero). Notice that for pure water on earth real
waves have a particular minimum celerity and that the minimum celerity
corresponds to a specific wavelength.
381
CHAPTER 17
Since gravity and surface tension work concomitantly, the celerity is really
set by their combination:
c = Jg± + *Z1 (17 4)
This combination generates the peculiar relationship between
wavelength and celerity shown in Figure 17.2. It's no surprise that equation
(17.3) gives a good approximation of reality for very short wavelengths
("capillary waves") and (17.2) does as well for very long ones (" gravity
waves")—which is why (17.3) is relegated to footnote status in books on
ocean waves. What is a little startling are the implications that (1) no real
train of waves of water on this earth can have a celerity of less than 0.23 m
s_1, and (2) the minimum celerity corresponds to a wave of the decently
finite and bioportentous wavelength of 17 mm.
Surface Ships
I think the main thing our analysis of locomotion at the air-water
interface has to explain is why it's so uncommon in nature. We swim, albeit
poorly, at the air-water interface; and we've been building superbly
successful surface-swimming ships for a few thousand years. But nature prefers
submarines—despite the facts that the densities of organisms are well
matched to operation at the surface and that many of the submarine
swimmers are air breathers. Nonetheless a few animals, ranging from whirligig
beetles to ducks, do get around at the surface, so no absolute prohibition is
enforced.
One might expect that a ship would gain efficiency by traveling at a
surface simply because a large part of both surface area and frontal area
faces air, with a resistance almost three orders of magnitude lower than that
of water. But this gain is offset by an additional component of resistance. A
ship moving on the surface generates waves, which means increasing the
surface area of the interface and lifting water above its equilibrium height.
So generation of surface waves means that work has been done. Thus to
skin friction and pressure drag must be added wave-making resistance,
and the latter may be no small matter. A moving ship usually produces two
waves, one at the bow and one at the stern. Thus the waterline length of the
ship ("hull length") almost fully determines the wavelength of the system of
spreading waves.
Wavelength, though, determines celerity. For a large ship, where gravity
waves are the main thing, celerity is proportional to the square root of
wavelength (equation 7.2). And wave celerity proves to have a fairly direct
connection to the practical speed of a ship, as illustrated in Figure 17.3. For
a ship moving more slowly than these waves, the bow wave moves out of the
382
FLOW AT FLUID-FLUID INTERFACES
Figure 17.3. The surface waves associated with the passage of an
ordinary ship with a displacement hull. This particular ship is a "rubber
ducky" ("Ernie's Genuine Playschool") towed in a flow tank, just slower
than hull speed.
way in front of it, at least to some extent, and the stern wave may even
elevate the rear. Consequently the ship faces a level or perhaps a slightly
downhill course. For a ship moving faster than these waves, things are not
so nice. The ship faces an increasingly steep bow wave, while it outruns its
stern wave. As a result, forward progression is an uphill battle in the most
literal sense—the ship is ever trying to climb its own bow wave. If, instead, it
plows through, then it's caught by an alternative problem. With a high bow
wave, more water must be shouldered aside in front. Pushing water normal
to the hull means that some component is being counterproductively
pushed against the ship's motion, even with a fairly sharp prow and narrow
beam. At the same time, less water pushes back in on the converging stern
to counteract the work done by the ship at the bow.
Among other matters, this problem of bow waves leads to a substantial
divergence in form between surface and underfluid craft—compare the
bulbous shape of modern submarines, of blimps, and of whales with the
sharp-prowed narrow profiles of fast ships. Even a strut that penetrates
the air-water interface does best when it has a sharp upstream edge. Which
is what Fish etal. (1991) found for the gaffing feet of fish-catching bats and
what Withers and Timko (1977) found for the part of the lower bill of a
skimmer that's exposed to the interface. In both cases the shape seems to be
determined more by interfacial exigencies than by those of the air above or
water beneath—the interface is the serious problem. (The phenomenon
was noted in connection with streamlined struts in Chapter 7.)
For a ship, then, the cost of propulsion increases severely because of
wave-making resistance when it exceeds a critical speed that corresponds to
the celerity of a wave of its hull length. Herein lies the problem. This speed
will be proportional to the square root of hull length, making it difficult for
383
CHAPTER 17
a short ship to travel rapidly. How short and how fast? We can make a
rough-and-ready estimate from equation (17.2) by substituting hull length
for wavelength and speed for celerity. We get a prediction that the ship
should hit the wall, so to speak, when speed squared equals 1.56 times hull
length—on earth and with SI units. As we'll see, that turns out to be about
right.
The Froude Number
In Chapter 5 the most useful of all the dimensionless indices, the
Reynolds number, was introduced as the ratio between inertial forces and
viscous forces. Since then, we've touched on other indices that represented
other ratios. We now encounter what is probably the second most useful of
these indices, the Froude number, named after William Froude (1810—
1878), a British naval engineer. Froude was interested in hull design for
surface ships and the possibility of using models for testing. If gravity waves
are what matter most, then a reasonable ratio is that between inertial force
(ma) and gravitational force (mg). The ratio, designated Fr, is thus
U2
Fr = ^-r. (17.5)
gl
(Sometimes the square root of the expression on the right is used as the
Froude number.) In fact, the scaling scheme, while useful, is imperfect.
The difficulty isn't the neglect of surface tension—practical technology is
well above its scale. Rather it's the neglect of viscosity, and the continued
relevance of the Reynolds number. As a little manipulation will convince
you, making a scale model that maintains both correct Reynolds and
Froude numbers just doesn't work. So while the Froude number is used,
models are as large and used as close to normal speed as is practical.
The speed limit alluded to just above can be expressed as a value of the
Froude number by inserting our U2 = 1.56/ into equation (17.5)—the
result is Fr = 0.16. And that turns out to be a reasonable value for real
waves and real hulls. A ship that's ten meters long ought to be able to go
about 4 m s~l, or 7.7 knots. A hundred-meter ship should go 12.5 ms_1,or
24 knots, as in Figure 17.4. What's biologically interesting is what happens
down at the lower left of Figure 17.4b. A 0.3 m hull ought to be able to do
only around 0.7 m s~l before the cost begins to rise rapidly. That's a pretty
poor speed by the standards of vertebrate swimmers. (A 0.1 m hull should
go about 0.4 m s-1, which I once verified by towing a rubber duck in a
trough.)
Some data are now available on surface swimmers. Prange and Schmidt-
Nielsen (1970) found that a duck with a hull length of 0.33 m will go no
faster than 0.7 m s-1, as expected. Its metabolic rate is at that point no-
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FLOW AT FLUID-FLUID INTERFACES
0.0125-
0.01-
g 0.0075 H
o-
0
T
skin friction
1 r
0.04 0.08 0.12
Froude number
(a)
0.16
hull length, m
(b)
Figure 17.4. (a) The resistance (expressed as drag coefficient based on
wetted area beneath the water line) versus Froude number for a
displacement hull of fairly ordinary shape, (b) "Hull speed," or Fr = 0.16, as a
function of hull length—the practical speed limit for ordinary surface
ships.
where near what it can reach in flight; apparently the investment in leg
muscle needed to go much faster hasn't proven evolutionarily cost
effective. Fish (1982, 1984) looked at muskrats on a pond and found much the
same thing. Few of them ever exceeded Fr = 0.16, which for them
corresponds to 0.63 m s_1.
While data for the drag of swimming animals gained from towing tests
have their problems (Chapter 7), comparison of data from surface and
submerged tows is still of some interest. Williams (1983) towed minks and
got figures for drag se/en to ten times greater at the surface than
submerged. With Humboldt penguins, Hui (1988a) found that parasite drag
increased much more rapidly with speed (velocity exponent of 2.55) for
carcasses towed on the surface than for ones towed submerged (exponent
of 2.0), although no sharp break at a particular Froude number was
evident. Stephenson et al. (1989) calculated aerobic efficiencies three times
greater during diving than while surface swimming for ducks (lesser
scaup). From these data, together with a lot more that are summarized by
Videler and Nolet (1990), clearly (1) surface swimming is more costly than
submerged swimming, and (2) the cost of surface swimming increases with
speed more drastically than does the cost of submerged swimming. Put
another way, speed-specific drag, D/U2 (Chapter 6), increases with speed
for surface swimming while it decreases with speed for both aerial and
aquatic, noninterfacial systems. Or yet another way—Hui (1988b) found
that the cost of transport (mass moved per unit distance) was about the
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CHAPTER 17
same for penguins at the surface and submerged, but the submerged ones
went a lot faster.
Admittedly, these comparisons may overstate the case for underwater
swimming at least a little. The best shapes for keeping drag down when
submerged aren't the same as those for keeping it down when swimming at
the surface. If an animal does both kinds, any natural advantage to
underwater swimming ought to prompt specialization of body shape for that
mode at (inevitably) the expense of surface swimming efficacy.
In at least one instance, near-surface swimming may have an energetic
advantage; not surprisingly it involves very large creatures and works best
(by calculation) for the largest of these. In a force-5 sea (one beneath a 10 m
s~l wind) a fin whale (Balaenoptera physalus) ought to be able to get as much
as 25% of its power for propulsion from waves when facing them and up to
33% when going with them, according to Bose and Lien (1990).
And our technology has inadvertently provided an analogous opportunity
—whether it ever occurs without human contrivances is unknown. That's
the bow-wave riding practiced by dolphins. Without any obvious locomo-
tory movements, they "ride" for long distances in a positionjust in front of
ships and just beneath the surface. While they're certainly taking
advantage of some inhomogeneity in the local flow, the exact mechanism has
been a subject of considerable contention; see, for instance, Fejer and
Backus (1960).
More Surface Transportation
Most ships and everything thus far considered use what are called
"displacement hulls"—they're Archimedean floaters. These don't exhaust the
ways of getting around while keeping one's head above water. I know of
several more of some biological relevance; still others may exist. But just
what's going on is pretty speculative at this point—which mean we're
talking about things ripe for investigation. Thus . . .
Planing
With an appropriate hull shape—a bow that's a bit flattened below and
slopes gently downward in the aft direction—and sufficient power, a boat
can quite literally climb out of the water and skim along on the surface. The
weight of the boat is offset in large part by lift, a downward momentum flux
of water. Several agencies contribute to the lift, including the acceleration
reaction that we encountered in the last chapter. For a given speed, planing
incurs less drag and thus requires less energy expenditure than does
propulsion of a hull displacing its own weight of water. But it doesn't work at
386
FLOW AT FLUID-FLUID INTERFACES
low speeds, and a boat must ordinarily achieve planing speed as a
displacement device.
Planing is put to only limited use by organisms, and it doesn't appear
associated with very obvious hydrodynamic specializations. Aquatic birds
quite clearly plane as they land. Since flight speeds are almost inevitably
higher than swimming speeds and since hovering is costly, one would be
surprised if they didn't plane. I've heard that by planing for short distances
as they become airborne some swimming birds can manage to evade the
hull-speed limit. That certainly seems to be what's happening in movies
that I've watched—takeoff is no easy matter for large waterfowl since the
water surface severely limits initial wingbeat amplitude. At least one sea-
bird is reported (Klages and Cooper 1992) to plane steadily—a broad-
billed prion, Pachyptila vittata, extends its wings to get lift and paddles into
the wind with its body only barely touching the water. Meanwhile it sticks its
head under water and either seizes or filters food, as mentioned earlier.
Slapping
Recall from the last chapter that a flat disk accelerating broadside to flow
has a substantial added mass-volume product, 8/3 r3. As Batchelor (1967)
pointed out, that's what's at work when you hit the surface of a body of
water with hand or hammer. Water is massy stuff, and fights back when
asked to accelerate at a high rate. Thus upward force is exerted. In at least
one instance the phenomenon is routinely used for locomotion. A genus of
iguanid lizards of the new world tropics, Basiliscus, popularly known as
"Jesus Christ lizards," have large, webbed hind legs and commonly run
across streams and other small bodies of water with entirely airborne
torsos. The legs enter the water rapidly, moving downward and then
backward; they almost certainly make substantial use of the acceleration
reaction and little if any of surface tension. An Asian species of agamid lizard,
Hydrosaurus pustulatus, does very much the same thing. At this writing, the
scheme is under active investigation by James Glasheen (Glasheen and
McMahon 1992); I'm reporting mainly what I've heard from him.
Using Surface Tension to Move
Neither planing nor slapping can provide static support in the way
floating does. But floating is only one of two systems that can provide static
support. Just as waves can be dependent on either weight (gravity waves) or
surface tension (capillary waves), static support can use either system. And
some creatures (Figure 17.5) press surface tension into service for
propulsion.
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CHAPTER 17
(a) (b)
Figure 17.5. Animals that take advantage of surface tension for
support and (sometimes) propulsion: (a) a water strider, Gerris; (b) a
whirligig beetle, Dineutus; (c) a springtail, Podura; (d) a fishing spider,
Dolomedes.
Consider, referring to Figure 17.6a, how the foot of an animal such as a
water strider (a gerrid bug) presses against the surface of a pond or stream.
Forces must balance, which means that the upward force due to surface
tension must equal the downward force of weight. The force exerted by the
interface will be equal to the surface tension of water times the wetted
perimeter of the foot. That force will have a line of action tangent to the
water's surface, so its upward component will be its magnitude times the
cosine of the angle between its line of action and the vertical.
How can the system be used for locomotion? What a leg has to do is push
backward against the water (Figure 17.6b). A leg presses downward and
backward, making a dimple in the water's surface. The dimple moves
backward, resisted by forces from the water's surface tension, inertia, and
viscosity, while the animal accelerates and moves forward. Thus the dimple
becomes asymmetrical. And how can we tell what's going on? That
asymmetry records the forward force on the leg. Since the surface tension of
water is known, the animal is in effect walking on a continuous force-
monitoring platform. Not that we're talking about something easy—
neither the analyses nor the technical problems of continuous three-
dimensional recording of the surface contour are trivial matters.
Moving around on the surface of water has other interesting aspects.
Nothing rules out using a combination of displacement and surface tension
for support—that seems to be what adult whirligig beetles (Gyrinidae) do.
Dineutus, about which we have some information, is supported about half
by buoyancy and half by surface tension (Tucker 1969). Nor are
underwater oars and paddles ruled out as propulsion devices, since beneath the
388
FLOW AT FLUID-FLUID INTERFACES
Figure 17.6. Left: Forces involved when a leg stands on the water's
surface. A heavier load pushes the leg in farther, which makes the line of
action (tangent to the surface) more nearly vertical, which compensates for
the heavier load. Right: Pushing rearward as well as downward against
the water's surface. The surface dimple is asymmetrical, with a net
forward component that offsets (and thus permits) the rearward push. Tilt
of the leg at right is incidental.
contact line of the air-water interface the animal is submerged in every
functional sense. A nudibranch mollusk (a shell-less snail) can move
around while hanging from the surface at least in a laboratory sea table; the
contact line extends around a considerable perimeter. They seem to be in
some way using mucus threads for locomotion, but I don't know of any
specific investigation of what, mechanically, is going on.
The size range of animals that can use surface tension for support and
locomotion is severely limited—the water striders, whirligigs, and spring-
tails whose habits are nicely described by Milne and Milne (1978) range
from only about a millimeter to a few centimeters long. I think the problem
is that two different bits of mechanics set top and bottom of what proves to
be a small window of opportunity. If the downward force of weight scales
with length cubed while the upward force of surface tension scales directly
with length, then an upper size limit will certainly be imposed. The central
issue if you're large is static support.
A lower limit alluded to (in a different context) in Chapter 15 may be
imposed by the functional stickiness provided by surface tension. Surface
tension is a little like viscosity—it opposes motion. A body on a surface is
pulled in all horizontal directions equally, so whichever way it moves it will
be opposed by some tensile force. Consider the ratio of inertial force to the
force of surface tension, called the "Weber number":
We =
plU
(17.6)
A small, slow body will find it difficult to generate enough force to do much
moving around—to offset surface tension—as we can see from the length
389
CHAPTER 17
and velocity factors in the numerator. Staying up may be easy, but getting
around is awkward.
Swimming on the surface is quite an odd thing if you're small. The
whirligig beetle mentioned just above makes no bow wave when swimming
slowly—real waves must go at least 0.23 ms-1 (Figure 17.2). The hull speed
barrier for a centimeter-long beetle comes out to 0.16 m s~l, using
equation (17.5) and Fr = 0.16; that's not very different from the corresponding
point for gravity waves in Figure 17.2. But the limitation doesn't apply, and
such beetles have been observed to hit 0.4 m s~l. The summed line in the
figure is what's relevant, and a beetle is swimming in the range where
wavelength is mainly determined by surface tension. At its size, surface
swimming isn't quite such a bad thing! Nevertheless, the range is narrow;
and adult North American whirligigs range from about 3 to 15 mm in
length (Merritt and Cummins 1984)—not exceeding the wavelength
corresponding to minimum celerity. What may be the oddest feature is that,
since wavelength sets hull speed, the smaller beetle should have the higher
speed limit.1
If you have a hill, you might arrange to slide down it. Which is what the
sport of surfing is all about. Using such a potential difference has a curious
analog in the world of surface tension. Small staphylinid beetles of the
genera Dianous and Stenus normally walk slowly on the interface. But they
can move rapidly shoreward under duress by ejecting a secretion from the
tips of their abdomens that locally reduces the surface tension. Higher
surface tension in front then pulls them forward, and they skim across the
surface, at 0.6 to 0.7 m s_1 (Jenkins 1960). They're using a version of a
common demonstration in which a tiny paper boat is propelled by a speck
of soap on its stern. Veliid bugs (such as Velia) can do the trick as well,
moving rapidly when alarmed by discharging surfactant saliva from their
beaks (Linsenmair and Jander 1963). Hynes (1970) has a nice summary of
all these adaptations.
Sailing
Organisms that support themselves by either buoyancy or surface
tension ought to be in a position to do a bit of sailing—using air movements to
move about. And sailing can be either lift based, drag based or some
combination. With this choice of mechanisms, what's a little surprising is how
little use is made of it. Perhaps it's another of those things that, like going
about in displacement hulls, work best for a size range larger than that of all
but a few organisms. Here one can point to the association of wind and
1 It's too easy to forget that these are adult insects with a complete metamorphosis in
their life cycle. Thus they've completed their growth, and one needn't worry about any
size-based geriatric slowdown
390
FLOW AT FLUID-FLUID IN FERFACES
waves and suggest that with a really useful wind waves would be awkwardly
large.
Still, sailing creatures exist. The most notorious is the Portuguese-man-
of-war, Physaha, a colonial coelenterate with a large and beautiful float
above ferociously armed tentacles. With an bulgy float about 20 or 30 cm
long, it's probably a purely drag-based sailor. Another coelenterate, the by-
the-wind-sailor, Velella, was mentioned in Chapter 11; Francis (1991)
found that its sail got a reasonable lift and noted reports that it could sail as
much as 63° off the direction of the wind. So it's at least to some extent a lift-
based sailor. With a wide skirt on the surface of the water beneath the sail, it
may make very slight use of surface tension. But the skirt is more likely to
oppose tipping over as a result of its static displacement and virtual mass—
a little like the slapping of basilisk lizards except that it's never free of the
surface. At least one surface-tension supported animal is a sailor. On windy
days, a fishing spider, Dolomedes, has been reported (Deshafy 1981) to lift its
second pair of legs from the surface and be carried across the water even
against currents beneath.
Another form of sailing seems not to have been recognized in any formal
sense, yet it's clearly a case of wind-forced movement at the air-water
interface. Wind certainly blows the surface of water around—just drop sawdust
on the water in any basin and blow across it to see how responsive the
surface is. Any organism hanging beneath the surface by local surface
tension will certainly be carried along. Such organisms include more than
just spores and pollen. Hydra, the common freshwater coelenterate, often
hangs down from the surface. If it's carried along with a wind-blown
surface it will experience a velocity gradient along its length; the latter may
expose Hydra's tentacles, extending downward, to prey, just as if it actively
pumped water or swam around. Mosquito larvae hang down from the
surface as well, so they might be blown about the same way.
Communication at the Surface with Waves
Communication has been given little attention in this book, appearing
mainly in connection with the wide disturbances to flow at low Reynolds
numbers (Chapter 15). Accustomed as we are to communicating with both
acoustic waves and electromagnetic waves, the use of surface waves for
communication ought to come as no surprise. To some extent it must be
just a matter of capitalizing on the unavoidable. If you make waves when
you swim, you've left a trace that no predator should ignore.
For a detector at a fixed point, frequency is probably a more directly
relevant variable than wavelength, and people working on communication
usually use the latter. The frequency and wavelength of surface waves
interconvert in just the same way as acoustic and electromagnetic waves—
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CHAPTER 17
frequency, f, s"1
Figure 17.7. How the celerity of spreading surface waves varies with
the frequency with which a specific place is forced up and down. The
graph is basically that of 17.2 with a recalculated abscissa.
frequency is propagation speed (here celerity) divided by wavelength. So
the graph in Figure 17.2 can be redrawn with frequency rather than
wavelength on the abscissa, as in Figure 17.7.
How can a predator tell what's prey? Falling debris produces trains of
concentric waves of short duration, while wind makes nonconcentric trains
mainly at frequencies below 10 Hz and at high amplitudes. So high-
frequency (10 to 50 Hz), low-amplitude, moderate-duration, concentric
wave trains indicate edibles such as struggling insects that have been
trapped on the surface. High frequencies attenuate with distance more
severely than lower frequencies; at least one fish (Aplocheilus) uses relative
attenuation to judge distance. Both a back swimmer, Notonecta, and the
sailing fishing spider, Dolomedes, as well as surface-feeding fish are capable
of frequency discrimination—the general picture is of highly tuned
predators who can tell a lot both about the direction and distance to a signaling
food source and about its character. Surface-foraging bats also use surface
waves as signals; they detect them through echolocation.
Water striders make by far the fanciest use of surface waves so far
uncovered. These waves provide them with the critical cues involved in courtship,
copulation, and even postcopulatory behavior. They produce and detect
waves of specific frequencies in trains of specific lengths in highly species-
specific patterns.
All of this information on communication comes from reviews by
Bleckmann (1988) and Wilcox (1988).
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FLOW AT FLUID-FLUID INTERFACES
Fuzzy Interfaces
At least one other fluid-fluid interface is of considerable consequence to
organisms, but it's one that's not anywhere near as familiar. Not that it's at
all hard to make in a model system—the phenomenon isjust subtle because
the same fluid is on both sides of the interface and the boundary is
indistinct. A situation of surprisingly great temporal stability may occur when
less dense fluid lies above more dense fluid. Thus an "atmospheric
inversion" with hot air above cold traps the effluvia of contemporary life in the
unpleasant form of haze or smog. And a lake may be severely stratified
during the summer, with the surface region depleted in nutrients as
material sinks through the so-called thermocline into the cold water beneath.
Winds in the atmosphere or blowing across a lake's surface ought to stir
things up and put such matters right. Still, inversions often persist for days,
and many lakes aren't mixed until they "turn over" as the surface cools in
the fall.
The density difference may trace to variation in composition—most
commonly salinity—as well as in temperature; that's common in estuaries,
where fresh water comes in above salt water.
But flow can disestablish such interfaces. A useful test is available to
predict whether these density gradients will remain stable or whether flow-
induced instability will produce a layer of mixing vortices and eventual
dissipation, as shown in Figure 17.8. The test involves another dimension-
less index, the "Richardson number," Ri:
gjdpldz)
p(dujdzY' K }
The expression is less complicated than it may appear. The ratio (dp/dz) is
the vertical density gradient, which keeps the layering stable; and the ratio
(dUJdz) is the vertical velocity gradient, which promotes mixing through
shear. The two can be obtained experimentally from sets of measurements
of temperature or salinity and velocity versus depth. Stability is obviously
favored by higher Richardson numbers, and in practice the Richardson
number must be below 0.25 for instability to set in. Negative values occur,
by contrast with all the other dimensionless numbers we've used so far.
Further information on the meaning, use, and (to a limited extent)
biological relevance of Richardson numbers may be obtained from Hutchinson
(1957) or Mortimer (1974) for lakes, from Fischer (1972) or Dronkers and
van Leussen (1988) for estuaries, and from Scorer (1978) or other
meteorological texts for the atmosphere.
Wind over water can do other things besides prevent layering—in
particular it can give rise to peculiar "windrows," vorticesjust beneath the surface
393
CHAPTER 17
â–º flow
flow^
Figure 17.8 The relative stability of density gradients subjected to
shear. Vortices will form in such stratified systems if the density gradient
is sufficiently gentle relative to the velocity gradient. The result will be
mixing and a decrease in the steepness of both gradients.
whose long axes run in the direction of the wind. They're often marked by
windward streaks of accumulated debris. These vortices are called "Lang-
muir circulations," and each vortex in the (cross-wind) sequence
circulates in the opposite direction from its neighbors (Leibovich 1983 gives a
good account of the phenomenon). The distribution patterns of a lot of
planktonic organisms are sensitive to such vortices; see, for instance, Ham-
ner and Schneider (1986). At least one, the scyphomedusan coelenterate,
Linuche unguiculata, seems deliberately to use the patterns to aggregate in
the windrows (Larson 1992).
Squirting and Spraying
Textbooks of fluid mechanics usually spend a page or so on the way fluid
(usually a liquid) flows through a sharp-edged orifice into another fluid
(usually a gas) as a free jet, partly because the case provides a classic
application of Bernoulli's equation. The velocity at the place of minimum cross
section of the jet (the vena contracta) turns out to be the square root of the
product of twice gravitational acceleration and the height of the liquid in
the reservoir. The latter, of course, can be expressed as pressure (recall
394
FLOW A 1 FLUID-FLUID IN IERFACLS
from Chapter 4 that Ap = pgz), which puts the formula in more general
terms:
u = y/2gz = y—• (17-8)
For real fluids, with viscosity and surface tension, the velocities are a few
percent lower than predicted by the equation. And the vena contracta has a
lower cross section than that of the orifice itself—61% to 66% is the
ordinary range, although surface tension can raise the value to as much as 72%,
according to Massey (1989). Discharge from a reservoir through a sharp-
edged orifice isn't a common occurrence in organisms, but it does have
relevance to laboratory setups.
More common, if still not exactly ubiquitous, is discharge of liquid into
air from a nozzle, a considerably more complicated business in which
viscosity and surface tension play major roles. When coming from a small,
circular nozzle, a cylinder of liquid soon breaks up into droplets. The
primary agency promoting breakup is surface tension—any place where
the cylinder is even a little reduced in diameter feels a greater strangling
effect of surface tension. What's responsible is Laplace's law, the rule that
pressure is proportional to tension divided by radius—tension is most
effective in generating pressure (here inward) where the radius is smallest.
So the smooth cylinder is unstable.2
Breakup of the cylindrical jet happens at an increasing distance from the
nozzle as its velocity increases—its persistence time doesn't change much—
up to a Reynolds number that's usually between 500 and 3000. Then
there's a transition from laminar to turbulent flow, whereupon the breakup
length decreases, and ascends again thereafter. This region of turbulent
breakup has a different character, though—the stream begins to wander
about ("sinuous" breakup) rather than strangling into fairly regular
droplets ("varicose" breakup). The ratio of length before breakup to jet
diameter is described by several mildly different formulas (cited by Blevins 1984)
involving the two dimensionless numbers that we'd expect to matter here,
the Weber number and the Reynolds number. A combination, the square
root of the Weber number divided by the Reynolds number, anticipates the
2 A very pretty demonstration of this effect of surface tension is quite easy to arrange.
Push a pipette through the middle of a 4 to 8 cm sphere of styrofoam or similar light
material, and mount it so the sphere rests loosely on the surface of a small, upturned
loudspeaker Connect the pipette to a hose coming from an elevated reservoir of water
and the loudspeaker to an audio-frequency oscillator set to a few hundred hertz. An
almost inaudible sound is sufficient to shake the pipette enough to regularize droplet
formation; the latter can then be viewed as if in slow motion with a repetitive
stroboscope Lord Rayleigh did this (with considerably less convenience) over a hundred years
ago.
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CHAPTER 17
Figure 17.9. (a) Longitudinal (sagittal) section of a human penis,
showing the anteterminal enlargement called the "navicular fossa." (b) Water
coming out of two glass tubes, one with and one without an anteterminal
enlargement.
character of the broken jet. A low value of this so-called Ohnesorge
number indicates that breakup will be varicose (McCarthy and Molloy 1974).
When a circular liquid jet breaks up, it often forms droplets of two widely
different size ranges, the main ones and what are called "satellite droplets";
the phenomenon was first noted by Lord Rayleigh in 1896 (Bogy 1979
gives a nice review). Interest in satellite droplets has increased in recent
years in connection with the development of ink jet printers, for which
they're not at all a good thing. Half of us may be arranged to minimize the
production of satellite droplets. The human penis (and many other
mammalian penes) have an enlargement of the urethra just upstream of the
final orifice, the "navicular fossa" (Figure 17.9a). Its function is unclear,
although enough congenital abnormalities are known for its association
with a smooth stream of urine to be fairly certain (Jordan 1987). I
compared the stream of water coming from a tube with such a preterminal
enlargement with one that lacked it (Figure 17.9b); the stream from the
latter broke up earlier and produced more satellite droplets.
But any functional significance is at this point pure guesswork. Perhaps
quadrupedal animals that urinate while standing shouldn't spray and thus
mark themselves with odorant, either as predators or as prey. One wonders
about other squirters—such as archer fish (Toxotes), which come to the
surface and squirt a jet of water up to a meter long that can knock an insect
into the water. Existing descriptions (such as Waxman and McCleave 1978)
are silent on the geometry of the narrow groove along the roof of the
mouth through which water is forced. And still further, one wonders about
396
FLOW AT FLUID-FLUID INTERFACES
Figure 17.10. A jet of fluid is inconveniently drawn along a surface as
the so-called Coanda effect—despite gravity and surface tension. A
hydrophobic surface recruits the latter and helps a little. Providing a sharp
edge that the fluid would have to turn around brings the fluid's inertia
into corrective action.
animals such as cats that can either squirt or spray urine—what
adjustments are involved?
While micturition might be the most memorable ending for a chapter on
interfaces and a book on flow, one more phenomenon, something called
the "Coanda effect," is aptly named to serve as coda. If a jet of fluid passes
close to a solid object, it tends to adhere and, if the curvature of the solid
isn't too severe, to follow the surface. What's happening is that the stream
of fluid draws along ("entrains") fluid from around itself, but the supply of
such ambient fluid is limited on the side of the jet near the surface. The
resulting drop in pressure draws the jet inward, as in Figure 17.10. The
phenomenon is put to use in various fluid logic ("fluidic") devices. An
analogous attraction occurs with a jet of liquid in a gas; here viscosity and
surface tension must also play a role. The effect is a nuisance for teapot
spouts, but it can be alleviated to some extent by making the surface
hydrophobic—buttering it. The Coanda effect may have relevance in
some cardiac pathologies. Of more interest to the biologist is its use, shown
by Eisner and Aneshansley (1982), by bombardier beetles. A beetle sprays
(usually at ants) a hot jet that it aims by pointing a hind leg, along which the
jet of spray is guided.
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Do It Yourself
Whew. I'm winded by all these words and worry that their thrust has
left the reader, with head barely above water, more draggy than
uplifted. One more topic, though—something of a peroration or
admonition, a return to a subject briefly mentioned in the first chapter. With this
book, I mean not only to provide an introduction to what has been done in
biological fluid mechanics but to what can be done as well. That's an odd
intention, given the almost automatically retrospective nature of a fact-
laden book in any area of science. After all, while the student goes to the
library in hope of finding material on a topic of interest, the intending
investigator really prefers to find nothing at all. Most of us wear both hats.
We begin with the former and, once launched, discover that the latter fits
better. Eventually some of us later revert to the student's hat, finding that,
as with so many things, vicarious pleasures are better than none.
At the least, I have tried to promote the ideas that adaptation to fluid flow
underlies much biological design and that such adaptation is of relevance
to limnologists, marine biologists, natural historians, paleontologists, ecol-
ogists of diverse persuasions, comparative and environmental
physiologists, and, indeed, all biologists who study organisms that either contain or
are exposed to fluid flow. My intention hasn't been to convert unsuspecting
biologists into biofluidmechanicians. Rather, it has been provision of an
easy introduction to the subject for those who suspect that it might bear
relevance to their projects but whose time and vigor is unequal to
immersion in a lot of engineering and applied mathematics simply to test that
suspicion. Put another way, it's intended as a bit of consciousness-raising
for people in other fields who will remain wedded to those fields, but who
might perhaps come to regard flow as fascinating and relevant instead of
fearsome and peripheral.
But my larger intention has been instigational. I'm trying to lure people
into actually doing a little biological fluid mechanics for themselves, to
persuade people that one can usefully get one's feet wet without getting in
over one's head, without taking a full-time, full-immersion bath. Showing
relevance is only one part, the easier or at least the more traditional task.
The other is showing tractability, showing that, once one is familiar with
fluid behavior and can envision matters from the viewpoint of the fluid,
one can without difficulty generate testable hypotheses. What, then, ought
398
DO IT YOURSELF
to be said about doing biological fluid mechanics rather than just reading
about it?
First, much of the subject is substantially counterintuitive. That
counterintuitive character means that only by knowing a little about fluid
mechanics can you realize what your organisms must contend with and
recognize the opportunities for them to be the cleverly adapted rascals in whom
we take delight. Equipped with notions such as Reynolds number,
continuity, separation, velocity gradient, circulation, advance ratio, propulsion
efficiency, and acceleration reaction, you've got the activation energy to
make it over that hurdle. But bear in mind that quite a lot of the biological
literature has been produced by people still mired in misconception and
downright heresy: with a properly jaundiced eye you'll at least be amused;
with luck, you'll gain some of that self-confident audacity that permits your
creativity to flourish. I keep a small file of conceptually flawed papers for
use in teaching—a wonderful sight is an early-stage student newly
empowered by discovering the ability to deconstruct the published word.1
Second, the idea that fluid mechanics and complex mathematics must
walk hand in hand is, I believe, a great misapprehension. Admittedly, a lot
of fancy mathematics underlies the simple equations you've seen. And, so
I'm informed, the more mathematics one can manage, the more problems
one can solve. But I do believe that up to this point our subject has made
more progress through experimental work than through purely
mathematical analyses and that experimental work needn't require great
mathematical sophistication. The real sine qua non for the biologist in this game is
insight into the operative physical processes. While this is a complex area in
which we're second-guessing nature, complex systems are a biologist's
stock in trade—the simplest of our systems makes the most byzantine type
of flow look like child's play. As a rule of thumb, the more complex the
system, the greater the likelihood that the assumptions needed to make
equations manageable will also render them inapplicably unrealistic.
Third, no magic formula points to success in unscrambling adaptations
to flow. Ideally one lets the question determine the approach, although in
practice one's own experience and investigative prejudices play major
parts. The investigator is almost inevitably faced with two gradients of
opposite sign. Both extend from work on real organisms in their normal
habitats, through work on real organisms under controlled circumstances,
and through work on physical models in flow tanks or wind tunnels (or,
rarely, in the field), to numerical simulation or equations. But along one
gradient, running from reality to abstraction, relevance gradually
decreases, while along the other, running from abstraction to reality, such
1 But I'd really prefer to keep the list private. Embarrassing people is certainly not the
intention and quite as certainly would be the result of dissemination.
399
CHAPTER 18
complication gradually accumulates that tractability is lost. Where some
crucial insight will emerge is predictable, if at all, only on a case-by-case
basis. The ideal is a multifaceted study, hitting the problem from enough
directions so a solution just can't stay hidden.
Finally, don't be intimidated by an unfamiliar literature, whether your
background is in biology, engineering, or perhaps geology, physics, or even
medicine. If you insist on limiting your questions to areas in which you're
fully equipped and ready to go, you'll not go far. And speaking of
literature, the biological one is especially scattered, making trouble for anyone
with the temerity to try to keep widely current or even to detect a
mainstream. Going through the present bibliography, I find that no journal
represents even 10 percent of the citations, and only one has over 3
percent. To give specific numbers,
Journal of Experimental Biology 67
Limnology and Oceanography 21
Canadian Journal of Zoology 18
Journal of Experimental Marine Biology and Ecology 14
Biological Bulletin 11
Hydrobiologia 9
Journal of Theoretical Biology 9
Again, let me emphasize that this book should be considered merely as a
starting point by anyone who intends to pursue a flow-related
investigation. In illustrating the diversity of points at which fluids and organisms
come into proximity, it has of necessity done full justice to neither. I've
suggested sources of additional information in the text, to the extent that
I'm familiar with them; but most investigations will probably need to move
beyond these at a fairly early stage.
For biologists such as me, the engineers themselves are enormously
useful. The fact that fluid flow is more in the province of engineers than
physicists has put the subject into a tradition with a strongly practical bent,
one with a pragmatic willingness to deal with complexity one way or
another. That's a great boon to the baffled biologist—even if our problems
are often viewed as some kind of comic relief. The biologist, though,
shouldn't assume that someone doing physical fluid dynamics has any idea
that a lot of really good work has been done on biological problems since
the two literatures tend to be quite distinct. Furthermore, the bemused
engineer needs a little introduction to our approach to design. Scion of a
practical tradition, the engineer designs things or devises rules by which
efficient devices may be designed. By contrast we start with the assumption
that our organism is well designed (bearing in mind, of course, the
constraints intrinsic to natural selection), and we try to figure out just why its
design is a good one.
400
DO IT YOURSELF
This book began with an exhortation, and I mean for it to end with the
same. Biologists seem afflicted with a great faddishness in the choice of
items for investigation. One might argue the merits of the situation—that
breakthroughs are made by concentrating one's troops. But I'd assert that
the history of science indicates the opposite—that the major conceptual
advances usually came before their areas became particularly well
populated. We're told that the rate of scientific progress has never been so great;
we're also told that never have so many been actively doing science. Is itjust
possible that the rate of progress per investigator might be at a low point,
with faddishness a contributing factor? Devil's advocate and curmudgeon
that I am, I frequently mention the matter to undergraduates awed by the
attention given to the most highly funded areas of biology. What matters is
one's chance of doing something that really makes a difference to how
people view something, and a lot of people chasing a few problems ought to
be taken as a counterindication.
Here we have an area in which people needn't trip over each other, where
problems are more abundant than investigators. To appropriate the ecolo-
gist's jargon, this is an r-selected rather than a K-selected field. Certainly
much work has been done in biological fluid mechanics; at the moment I'm
in danger of drowning in references. But I remain impressed with the
diversity of interesting and accessible questions yet unplumbed, with great
gaping lacunae where the hand of an investigator has not yet set foot. Don't
wait to see which way the wind is blowing—get in the swim.
401
List of Symbols
a acceleration
a radius, half of width (elongate
slot)
<2V Stokes' radius
AR aspect ratio
b span (of wing)
C arbitrary constant
Ca added mass coefficient
Cd drag coefficient
Cdj drag coefficient referred to
frontal area
Cdl local drag coefficient (for
point on surface)
Cdp drag coefficient referred to
plan form area
Cdv drag coefficient referred to
volume2'3
Cdw drag coefficient referred to
wetted area
Cf lift coefficient
Ctt orifice coefficient
Cp pressure coefficient
c chord (of wing)
c celerity (of surface wave)
ch concentration in bulk solution
cm concentration at membrane
D drag
D diffusion coefficient
d diameter, of pipe or disk swept
by propeller
d zero-plane displacement (of
logarithmic boundary layer)
Di distance index (internal flows)
E exponent for variation of
D/U2withU.
F force
/ friction factor, Fanning
friction factor
/„ aerodynamic (or reduced)
frequency parameter
Fr Froude number
G shear modulus
g gravitational acceleration
H total head of pressure
h location from center of slot
toward either plate
J advance ratio
k constant of unspecified value
403
LIST OF SYMBOLS
k Kozeny function (porous
media)
L the dimension length
L lift
L' entrance length (for pipe)
/ length, characteristic length
M the dimension mass
m mass
n oscillation frequency,
revolution rate of propeller
P power
Pi induced power (propeller)
Pit tendency to pinch, laminar
flow
Pit tendency to pinch for
turbulent flow
p pressure
Pe Peclet number
Q volume flow rate
R wing length
R resultant force
R resistance to flow (in pipe)
r radial location outward from
center of pipe
Re Reynolds number
Rer boundary roughness Reynolds
number
404
Rex Local Reynolds number (at
place on surface)
Ri Richardson number
5 surface area
S^ area of disk swept by beating
wings
Sj frontal area
S^ plan form area
Sv two-thirds power of volume
Sw wetted area
5/ Strouhal number
T the dimension time
T thrust
/ time, period
U speed, velocity, free-stream
velocity
U+ shear velocity or friction
velocity
U average (mean) velocity
Ud wind component due to
autogyro descent
Uj- wind on blade element due to
free stream flow
Uh velocity at particular place
between plates
Ui wind induced by propeller
action
[/max peak velocity (in pipe or
channel)
Ur wind on blade element due to
its rotation
Ur velocity at particular radial
position in pipe
Ut tangential velocity (in vortex)
Uw net wind impinging on blade
element
Ux local velocity (component in
x-direction)
Uz local velocity (component in
z-direction)
V volume
Vs volume of particles (porous
media)
W weight
w width of channel or slot
We Weber number
Wo Womersley number
x distance or direction along
surface and with flow
y shortest distance to nearest
wall
z distance or direction normal
to surface and flow
z0 roughness parameter or
roughness length
LIST OF SYMBOLS
a angle of attack
T circulation
7 tension, surface tension,
surface energy per area
A prefix for difference between
two values of variable
8 boundary layer thickness
8U thickness of unstirred layer
e height above surface of
protrusion or roughness
element
e voidage or porosity (porous
media)
% Froude propulsion efficiency
6 angle of deformation
6 glide angle
k Von Karman's constant
\ wavelength
(x viscosity, dynamic viscosity
v kinematic viscosity
p density
t stress, shear.stress
<|> amplitude of wingbeat (stroke
angle)
405
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Index
(More complete entries are given under common names than under scientific names.)
Abietenaria (hydroid), 120
Acanthocyclops (crayfish), acceleration
reaction, 365
acceleration: air bubble, 362; crayfish
escape response, 365; crustaceans, 287;
to evade viscous effects, 359; fish
swimming, 287; freshwater turtle, 287;
jetting animals, 365; of gravity, 34;
scaling, 287; sea turtle, 287
acceleration reaction, 138, 362—69; in
air, 368; and body volume, 365;
defined, 108; direction, contra drag, 364;
and dislodgement probabilities, 368;
fruit fly wings, 368; in jet propelled
animals, 365; scaling, 108; scaling
contra drag, 368; stopping, 365;
swimming with tail or paddles, 366; wave
surge on fixed object, 367. See also
added mass
accuracy, 11-12
Acer (maple), 228, 272
acetone, properties, 23 (table)
Acilius (water beetle), 145
actuator disk, 275 (fig.); disk loading,
274; in hovering, 276; induced
velocity, 274
added mass, 362—69. See also acceleration
reaction; swimming
added mass coefficient: and decelerative
gliding, 365; defined, 363; vs. fineness,
data, 364; use, 363; values, 363
advance ratio: vs. angle of attack, 267;
animal flight, 266, 267; defined, 267;
propeller, 267; samaras, 272; vs. size,
best, 268; vs. top speed, 268; whale
tail, 282
aeolean singing of wires, 371
aerodynamic frequency parameter,
defined, 376
aeroelasticity, 372, 374-5
aerosol particles, 192
Ailanthus altissima (tree-of-heaven),
228
air: density vs. temperature, 22; incom-
pressibility justified, 54; kinematic
viscosity, 25, 88
air-water interface. See interface, air-
water
airfoil: angle of attack, 234; area
adjustment, 243; aspect ratio, 236-39;
autogyro blades, 268; biological, 246-52,
249; blade element as, 262; camber,
270; Cayley's profile, 106; chord, 234,
239; components of drag, 242; cross
section, 233; drag, 109; drag vs. Re,
241; elliptical plan form, 239; vs. Flett-
ner rotor, 233; flow pattern for lift,
231; helicopter blades, 268; infinite
wing, 236; leading edge, 233; lift
distribution, 233; lift production, 230-
34; lift vs. drag, 234; operation of
various (figs.), 232, 269; polar diagram,
235 (fig.), 248 (insects); power vs.
aspect ratio, 238; propeller, 268; and Re,
244-46; vs. Re, 142; reference area,
90; reversed, 141; separation, 233;
separation and vortices, 213;
separation point oscillation, 376; shape, 107
(fig.); stall, 242; surface irregularities,
245; swept back tips, 240 (fig.);
terminology, 231 (fig.); thickness-to-chord
ratio, 109, 114; tip sweepback, 239;
trailing edge, 231, 233; very low Re,
245; vortices, 233 (fig.); windmill,
268; wing loading, 243-44. See also
wings
airship, 63, 68; streamlining, 99
albatross: glide angle, 254; lift-to-drag
ratio, 248; slope soaring, 259
algae: Benard cells, 224; current and
photosynthesis, 199; sinking speeds,
185; spore settling, 187
algal cells, as flow markers, 44
algal thallus, modeling, 103
Allogamus auricollis (caddisfly larva), 180
Alloptes (feather mite), 181
441
INDEX
Alsomitra macrocarpa (Javanese
cucumber), 256
alula, as anti-stall device, 242
Ametropia neavei (mayfly larva), 200
amphibia: cutaneous respiration and
flow, 200; limbless, anguilliform
swimming, 281
amphipod: fluid dynamic burrow pump,
327; pleopods as propeller, 328
Amusium (scallop), 252
anchovy, swimming behavior, 281
Ancylus (snail or freshwater limpet), 40,
179
anemometer, 7, 43, 112
anemophilous plants: exposing pollen to
wind, 193; pollen trapping, 187
aneroid barometer, 56
aneurysm, 62
angel, wing loading, 244
angelfish: acceleration reaction, 365;
swimming with drag, 287
angle of attack, 231 (fig.)', vs. advance
ratio, 267; blade element, 263; defined,
234; and delayed stall, 376; and lift,
242; vs. lift and drag, 235; and
pitching moment, 234; propeller, 264; and
stall, 242
angle of incidence, 263
Anguilla (eel), 281
Anolis carolinensis (lizard), parachuting,
257
antarctic animals, blood and circulation,
28
antenna: E-value, locust, 120; modeling,
105; transmissivity, 356
aorta, 8, 35, 55, 62, 316
Aplocheilus (fish), prey detection with
waves, 392
apparent additional mass. See added
mass
Aptendytes forsteri (emperor penguin), 149
aquatic plants, 199
Arceuthobium (dwarf mistletoe), 195
archer fish, squirting, 396
area of reference. See reference area
Arenicola (lugworm), 326
Aristotle, 81, 167
Artemia (brine shrimp): flow markers, 44;
swimming, 350
arterioles, 8, 319
arteries, 315
ascidian: Botryllus, 191 (fig.); jet
propulsion, 77, 80; as Pitot tube, 60;
preventing recirculation, 190; Styela, 61 (fig.)
ash: autorotating samara, 228; vessel
length, 322
aspect ratio, 236-39; 231 (fig.); of
autorotating plate, 228; and cost of lift,
237; defined, 236; flying fish, 259; and
induced drag, 238; lunate fish tail,
282; phalanger, 250; sand dollar
airfoil, 246; scallop, 252; and tip vortices,
236; whale tail, 282
aspen, quaking and reconfiguration, 124
aspirator, 58, 316
atmospheric instability, and thermal
soaring, 222
autogyrating: vs. autorotating, 228; vs.
gliding, 257
autogyro, 269, 272
autogyro rotor: as airfoil, 269; blade
contour, 271; origin of upward force, 271;
plane of rotation, 271
autorotating: of Flettner rotors, 228; vs.
gliding, 257
bacteria: coasting, 331; flagellar rotation,
206; local drag on walls, 176; Re, 87;
sinking speeds, marine, 344; special
flagella, 351
Baetis (mayfly nymph), 139, 177
Balaenoptera physalus (fin whale), 282,
386
Balanus balanoides (barnacle), 183
ballasted case, caddisfly larva, 180
ballistics of seeds, 194
balloon, 12
ballooning by spiderlings and
caterpillars, 12,222-23
barnacle: filtration, 356; settling, 183,
185
Basiliscus (lizard), slapping, 387
bass, ram ventilation, 110
bat, fishing: echolocation of surface
waves, 392; leg cross sections, 141,
383; legs as struts, 140
bat, flying: lift coefficient, 278; power,
241
bathtub vortex, 204, 208
beach: interstitial flow, 307; swash, 307;
swash-riding, 182
bee: advance ratio, 268; alleged flight
442
INDEX
lawlessness, 278; body lift, 239, 252;
flight speed, 13; glide angle, 254; wing
loading, 243; wing polar diagram, 245
beetle: air bubble, 70; elmid, 70 (fig.);
surfactant propulsion, 390
beetle, bombardier, Coanda effect, 397
beetle, diving, 350 (fig.)
beetle, water, 142 (fig); drag coefficients,
145; rowing stroke, 155 (fig);
swimming, 137, 155,287,350,366
beetle, water penny, 67, 178, 179
beetle, whirligig, 387 (fig.); hull support,
388; size range, 390; wave making, 390
Bernoulli, Daniel, 52
Bernoulli's equation 5, 156; in boundary
layer, 5, 157; and orifice, 394; for Pitot
tube, 58
Bernoulli's principle, 21, 52-62, 80, 53
(fig.); and circulatory systems, 62;
derivation, 52; and drag, maledictions,
81-82; and internal flows, 314; and
lift, 65, 232; limitations and
precautions, 60; and pressure-induced flow,
72; and rotating cylinder, 226; use in
flow measurements, 57-60; use in
pressure measurements, 54; vortex
interactions, 210
Betula lenta (birch), 194
Benard cells, 224 (fig.)
bill, skimmer's as strut, 140-41
bioconvection, 224
biometeorology, 5, 10, 221, 344
bird: blood viscosity in antarctic, 28;
body drag, 106, 148-49; body lift,
239; downwind flight, 13; egg
boundary layers, 198; flight, 137; formation
flight, 288; gliding, 21; interference
drag, 148; parasite drag, 248; planing
in landing, aquatic, 387; skimmer bill
as strut, 140, 141; soaring, 221, 260;
stroke plane, 276; swooping flight,
239; tip vortices, 276; vortex wakes,
43, 276; wing aerodynamics, 248; wing
area adjustment, 243; wing feathers,
248
bivalve mollusk, preventing recirculation,
190
black fly, advance ratio, 268
black fly larva, 113 (fig.); cephalic fans,
184; drag, 113, 184; feeding from
ascending vortex, 217; filter elevation,
191; filtration, 356; settling, 184-85;
shape of fans, 113
black fly pupa, gills in vortices, 218
blade element: making thrust, 262;
torque component, 264
Blepharoceridae (fly), 180
blood: as non-Newtonian fluid, 20;
viscosity, 28-29
blood cells, rotation in shear flow, 187
blood flow, cost, 317
blood pressure, 55; dinosaurs, 56; fish,
68; giraffes, 55; horses, 55; mammals,
55; reptilian, 56; systolic, 55
blood vessels, no-slip condition, 19. See
also arteries, etc.
bluefish: intracranial pressure, 68;
pressure distribution, 68; ram ventilation,
110
bluff body, 99, 167; drag, 111; drag in
boundary layer, 175; drag vs. speed,
117; splitter plates, 110; vortex
shedding, 212
body drag, 138. See also parasite drag
body lift: birds, 239; bumblebees, 252;
insects, 239; ski jumpers, 240;
swimming rays, 239; swooping flight, 239
body of revolution, 133; drag coefficient,
135; thickness-to-chord ratio, 134
Bombus terrestris (bumblebee), 268
Bond number, 347
boomerangs, 101
Bosmina (cladoceran crustacean), 351, 359
Botryllidae (ascidians), colonial jets,
190
bottom. See substratum
bound vortex, 232
boundary layer, 5, 19, 156-71 (figs., 157,
159); bird eggs, 198; drag in, 164-67,
175-81; empirical formulas, 163; and
evaporation, rate, 163; vs. free stream,
162; and heat transfer, 163; laminar
flow, 158-61; laminar sublayer, 161;
leaves, 163; logarithmic, 167—71; in
nature, 162; origin, 157; and principle
of continuity, 159; rotation in, 188
(fig.); and settling, 186; and shear rate,
157; speeds within, 160, 193; and
surface roughness, 160; thickness, 157—
60, 162; thickness vs. leaf size/shape,
199; turbulent flow, 159, 161-62;
unbounded, 167-71; vs. unstirred layer,
443
INDEX
boundary layer (cont.)
201; as vortex generator, 218; at very
low Re, 332. See also velocity gradient
boundary layer resistance, 197, 198
bow-wave riding, dolphins, 386
brachiopod: filtration flow patterns, 44;
shell closing acceleration, 365;
suspension feeding, 189
branching of pipes, 216-21
Branta canadensis (Canada geese),
formation flight, 288
brine shrimp: as flow markers, 44;
swimming vs. Re, 350
bristles: spacing vs. Re, 351; for
swimming, 350; use in flight, 350
Brownian motion, in filtration, 357
bryozoa: colonywide currents, 44, 191;
Membranipora, 191 (fig.)
bubble, 8; air, of beetle*, 70; air, of
beetle**, 70
bug, surfactant propulsion, 390
bumblebee, polar diagram, 248
buoyancy, 14; air bubble, 362; ascending
vortices, 221
burrows, induced flow through relict, 72
butterfly: lift-to-drag ratio, monarch,
255; monarch, 13; vortices in flight,
276,287; wing scales, 245
byssus thread, 66, 223
C-start, fish swimming, 287
caddisfly larvae (figs., 61, 178): ballasted
case, 180; catch nets, 180, 356; hy-
gropetric, 182; its Pitot tube, 60
caddisfly pupae, on river bottoms, 40
Callinectes sapidus (crab), 146
Cancer productus (crab), 146
capillarity, 22
capillary, 55, 62; aggregate cross section,
37; blood shear rate, 28; bolus flow,
312; diameter, 28; diffusion in, 313;
flow in, 313 (fig.); lungs, 37; Murray's
law, 319; Peclet number, 313; size and
flow rate, 35; toroidal flow, 312; total
number, 37; transmural exchange,
312; Womersley number, 377
capillary waves, 382
Caranx (jack), 281
carburetor, 58, 316
Carchesium (protozoan), 191
cat, micturition, 397
catch net, caddisfly larvae, 180
caterpillar, thermal soaring, 222
caudal peduncle, 282
Cayley, Sir George, 106
celerity: defined, 380; equation for, 283;
vs. gravity, 381; minimum possible,
382; vs. surface tension, 381; vs.
wavelength, 381, 382
cells, viscosity, 18
Cenocorixa (water boatman bug),
acceleration reaction, 365
cephalopods; ink vortex, 215; jet
propulsion, 77; shell as splitter plate, 110
cetaceans: compliant surfaces, 152; drag,
151; drag coefficients, 151; drag
reduction by heat release, 29;
porpoising, 151; surface heating, 153; tail
sweepback, 239; wetted surface, 90
Chaetopterus (polychaete worm), 326, 327
Chaoborus (dipteran larva), signal for
attack, 359
characteristic length, 111
chemical communication, 6
Chironomidae (midges), 182
Chlamydomonas (algal cell), flagellar
swimming, 352
choanocytes, 356
chord: airfoil, 234; strut, 134
Chrysopelea (colubrid snale), gliding, 257
Chthalamus fragilis (barnacle), 185
cilia: arrangements, 351; associated
velocity gradient, 348 (fig.); bands and
covered surfaces, 349; of bivalves, 39;
in burrow pump, 327; efficiency, 354;
motion, 352 (fig.); operation, 351-55;
of Paramecium, 137; as wall pumps,
314
ciliates: flow, swimming vs. sinking, 349;
sinking speeds, 344; upward
swimming, 344
Ciona intestinalis (ascidian), 184
circular aperture, 303-4; biological
examples, 303; laminar flow, 303; orifice
coefficient, 303; transition Re, 305;
turbulent flow, 303
circular cylinder: across velocity gradient,
216; added mass coefficient, 363;
boundary layer, 163; drag at very low
Re, 335-36, 352, fig., 353; drag
coefficient vs. Re, 93; drag coefficient, data,
92, 112; drag coefficient, equation,
444
INDEX
336; drag vs. orientation, very low Re,
335; drag vs. roughness and Re, 101;
E-value, 120; Flettner rotor, 227; flow
around, 83, 93-96, 94 (fig.); lift when
rotating, 226; Magnus effect, 226;
pressure distribution, 81, 83; pressure
drag, 97; protruding from substratum,
216-18; rotating, 230; slewing
obliquely, 353; vs. streamlined form,
134; Strouhal numbers, 371;
theoretical streamlines, 82; turbulent
transition, 94; very low Re streamlines, 332;
vortices behind, 315; wall effect on
drag, low Re, 339
circular disk, drag, 112, 334
circulation: dimensions, 225; and lift,
226, 230-31; nonsteady, 277; physical
variable, 7; plus translation, 226;
reversal in vortex shedding, 369; of self-
excited oscillator, 373; starting with
clap-and-fling, 279; and translation,
230, 231; and vortex speed, 370; and
vortices, 210, 224; and vorticity, 224-
25; at very low Re, 347; wind driven, 6.
See also circulatory system
circulatory system, 5, 8, 36 (fig.);
antarctic animals, 28; Bernoulli vs. Hagen-
Poiseuille, 314; and Bernoulli's
principle, 62; in blubber, 29; data, 36, 319;
features, 308; iguana, 28; instabilities,
374; Murray's law, 319; output,
human, 324; power, 28; pulsating flow,
377; shear flow, 188; temperature
effects, 28; vessel expansion, 315;
Womersley number, 377
Citharchthys stigmaeus (flatfish), ground
effect, 289
cladoceran, 350 (fig.); jerky swimming,
359
Cladonia podetia (lichen), 214
clam: and continuity, 38 (fig.); pump
pressure, 328; swash-riding, 181, 182
(fig-)
clap-and-fling mechanism, 278-79, 289
Clione limacina (sea butterfly), swimming
with lift, 286
Coanda effect, 397 (fig.); bombardier
beetles, 397; mechanism, 397
coarctation, 62, 316
coccid (scale insect), 193
cod, burst-and-coast swimming, 281
coefficient of discharge, 303
coefficient of drag. See drag coefficient
coefficient of lift. See lift coefficient
coefficient of pressure. See pressure
coefficient
coelenterate gastroderm, cilia on, 314
coelenterate medus, acceleration
reaction, 365
colonywide surface currents, bryozoa, 44
Columba livia (pigeon), 148
communication, chemical, 6
communication with waves, 291—92;
frequency vs. wavelength, 391; signal
diversity, 392
compliant surfaces, drag reduction, 152
compressibility, 102; vs. speed of flow, 21
compressible flow, 5
condor, lift-to-drag ratio, 248
condor, Andean, wing loading, 243
conifers: autogyrating samaras, 272;
cones, airflow patterns, 43
conservation of mass. See principle of
continuity
continuity, principle of. See principle of
continuity
continuum assumption, 20
convection, 6; Benard cells, 224; vs.
diffusion, 196, 313; imprecision, 11; and
sinking speeds, 343, 345; free tree, 221
copepod: acceleration, 360; acceleration
reaction, 365; filtration vs. active pre-
dation, 359; jerky vs. smooth
swimming, 359
copying nature, 106, 150
coral: drag, 107; E-value, 120; gastro-
vascular system, 320
coral, black, on seamounts, 40
Coriolis force, 204
corners, vortices in, 213
Corophium (amphipod), 327
corselet, 29
countercurrent exchange systems, 311
crab: drag coefficient, 146; legs as struts,
140; lift, 146, 176; swimming, 146, 286
craspedophilic organisms, 174
crayfish, 142 (fig.); acceleration, 287,
365; escape response, 365
creeping flow, defined, 331
crinoid, coelomic circulation, 320
crustacean, wetted surface, 90. See also
specific creatures
445
INDEX
Cryptolepas (ectoparasitic barnacle), 180
ctenophores (comb-jellies): ciliary plates,
355; jet propulsion, 77
cup: of lichen, 214; vortices in, 214
Cupressaceae, autogyrating samaras, 272
current speed: effects of, 175; and pho-
tosynthetic rate, 199; and suspension
feeding, 192
cutaneous respiration, amphibians, 200
cuttlefish. See cephalopods
Cyamus (whale lice), 180
cyanobacteria, density vs. light intensity,
30
cyclomorphosis, Daphnia, 30
Cyclops (copepod), jerky swimming, 359
cyclosis, 5, 196
cylinder. See circular cyliner
Cymatogaster aggregata (seaperch),
swimming mode, 286
Cynocephalus (flying lemur), gliding, 257
Cypselurus (flying fish), gliding, 258
d'Alembert's paradox, 82, 156
damping, with compliant surfaces, 152
dandelion, 13 (fig.)
Daphnia (water flea), 31 (fig.), 340;
cyclomorphosis, 30; sinking rates, 30, 340;
swimming, 30, 344, 359
Darcy's law, 305
Deborah number, 18
deer fly, reported flying speed, 21
deformation, shear. See shear
deformation
Dendraster excentricus (sand dollar), 246
density 7, 10, 11, 22-23, 80; constancy
with speed, 21; dimensions and units,
10, 22; in kinematic viscosity, 25;
measurement in plankton, 30; regulation
by diatoms, 346; sorting in vortices,
216; and wave formation, 379
desmids, mucilage coat, 344
Desmognathus quadramaculatus
(salamander), 200
detritus feeding, 189; from ascending
vortices, 216-18
Dianous (beetle), surfactant propulsion,
390
Diaptomus (copepod), steady swimming,
359
diatoms: fig., 345; composition
adjustment, 344; density regulation, 346;
fibers and sinking rate, 345; sinking
speeds, 344
diffusion, 47; air vs. water, 198; alveoli
vs. capillaries, 199; in capillaries, 313;
in cellular domain, 313; coefficient,
170, 313; vs. convection, 196, 313; vs.
distance, 35; eddy, 170; exchange
while swimming, 355; Fick's law, 47,
197; intracellular transport, 196;
molecular, 170; Peclet number, 313;
speed of, 196; vs. swimming, very low
Re, 332; and transmural exchange, 35;
in transpiration, 197; and "unstirred
layer," 201; and velocity gradients,
196-202; in viscous sublayer, 170;
water vapor, 197
diffusional deposition, filtration by, 357
diffusive boundary layer, and unstirred
layer, 201
diffusive vs. convective transport, 202
dimensional analysis, 89
dimensional homogeneity, 8, 11
dimensionless coefficients, role at high
Re, 304
dimensionless numbers, 9. See also specific
numbers such as Bond, Froude, Peclet,
Reynolds, Weber, Womersley; also drag
coefficient, friction factor, etc.
dimensions, 7—9; in dimensional analysis,
89; fundamental, 8; table, 10
Dineutus (whirligig beetle), hull support,
388
dinoflagellates, seasonal polymorphism,
30
dinosaurs, blood pressure, 56
dipteran larva, signal for attack, 359
direct interception, filtration by, 357
discharge coefficient, 303
discharge, rivers, 39
disk, as speed calibrator, 98
disk loading: defined, 274; and induced
velocity, 274
dispersal: ballooning, 222; byssus thread
drag, 223; mucous thread, 223; propa-
gules, 344; seed, 4, 6, 12, 245, 257,
272, 344
displacement ships. See surface ships
distance index, 308. See also pipes
disturbance as signal, very low Re, 359
Dolomedes (fishing spider): prey detection
with waves, 392; sailing, 391
446
INDEX
dolphin: bow-wave riding, 386;
"dolphin's secret," 150; drag, 151; Gray's
paradox, 152; thunniform swimming,
282
Donax (coquina clam), 182 (fig.); swash-
riding, 181
Draco volans (lizard), gliding, 257
drag, 7, 74; in accelerating motion, 363;
airfoil, 109; axisymmetrical bodies,
113; on beds of mussels, 66; in
bidirectional flow, 114; biological relevance,
106-9; black fly larva, 113, 184; bluff
body, 111; in boundary layer, 164-67,
175-81; categories summarized, 241;
cetaceans, 151; change by flattening,
134; circular cylinder, 97, 335, 352;
circular disk, 112, 334; coccids on
leaves, 193; coefficients for low Re,
336; coral, 107; crabs, 146; dead or
towed fish, 280; dependence on speed,
84; dimensional formula, 81; and drag
coefficient, 89; energy cost, 132;
equality with thrust, 136; and fairing, 111;
falling mice, 108; fish, 147-48; and
fitness, 107; flag, 123; flat plate, vs.
orientation, 351; and flexibility, 114-27,
335; fluid sphere, very low Re, 337;
flying deer fly, 21; as force, 14; as
friction, 80; frogs, 146; fruit fly, 10; glide
length as measure, 138-39; golf ball,
142; and growth, 127-28; hollow half-
cylinder, 112; hollow hemisphere, 112;
imprecision, 11; indirect
measurement, 75; insect, 15; jumping flea,
142; leaves, 14, 120-24; with lift, 226;
vs. lift, sessile organisms, 101; vs. lift at
interfaces, 176; limpet shell, 66; low Re
factors, 337 (table); macroalgae, 125;
marine mammals, 150-51;
measurement, pine tree, 121; motile animals,
132-54; mucous thread, 223;
oscillation at transition, 375; as part of
thrust, 139; penguins, 149-50; plaice,
65; from pressure data, 63; during
ram ventilation, 69; reduction by local
heating, 29; reduction by roughness,
99; reduction by streamlining, 98-99;
sand dollar, 65; scaling contra
acceleration reaction, 368; sea anemone, 115;
sea fans, 107; and separation, 96, 99;
sessile systems, 106-28; and shape,
96-98, 106-27, 132-54; silk strand of
spiderling, 12; silk, thread, 222; skin
friction vs. pressure drag, 96-98;
snails in irrigation canals, 108; solid
hemisphere, 111; sources, 6, 7, 102; vs.
speed, very low Re, 333, 348; speed-
specific drag, 117; sphere, 112, 333;
spheroid, 336; spider silk strands, 335;
spot on surface, 164, 165; of sting,
138; from streamlines, 43; strut, 109,
134; surface swimming, 385; surface
swimming, speed-specific, 385; and
suspension feeding, 192; in swimming,
280; tadpoles, 145; for thrust, 283-87;
torrential fauna, 176-80; tree, 14,
107; turbulent flow vs. laminar flow,
151; at turbulent transition, 94;
unsteady flow, 375; and viscosity, 63;
from wake width, 98; wall effect, very
low Re, 339; from waves, 141. See also
drag coefficient; E-value; induced
drag; parasite drag; pressure drag;
skin friction
drag coefficient, 89; vs. angle of attack,
235; area conversion table, 133; body
of revolution, 135; cetaceans, 151;
change by flattening, 134; circular
cylinder, 92 (data), 336 (equation);
conversion factors, 132; crabs, 146; desert
locust, 144; drogue, 129; emperor
penguin, 149; vs. fineness ratio, graph,
110; fishing bat legs, 140; flat plate,
907, 98 (table); flat plate vs. motile
animals, graph, 144; frogs and tadpoles,
146; and front fairing, 111; gentoo
penguin, 149; from glide angle, 253;
harbor seals, 150; Humboldt penguin,
149; isopod, 145; jumping fleas, 142;
leaves, 123; mackerel, 147; "Microbus,"
113; motile animals, table, 143;
Nautilus shell, 147; paddles, 154; pigeon,
148; pine tree, 121; power strokes,
154; protrusions, 165, 166; and Re,
102, 117; reference areas, 90-91;
saithe, 148; sea anemone, 115; sea
lion, 150; and separation, 96; and
shape, 132; sphere, 92 (data), 334
(equation); streamlined animals, 132;
streamlined body, 133; strut, 109;
swimming human, 151; and terminal
velocity, 340; and thickness-to-chord
447
INDEX
drag coefficient (cont.)
ratio, 109, 114; trout, 147; underwater
ducks, 147; variation with velocity,
116; various bodies and profiles, 112
(fig.); very low Re, 333; water beetles,
145
drag coefficient, local, 164-67
drag maximization, seeds, 246, 257
drag reduction: air deflectors, 114; in
boundary layer, 175; boundary layer
suction, 179; by compliance, 115;
compliant surfaces, 152; ejecting high-
velocity fluid, 110; flattened torrential
fauna, 177; mucus secretions, 152;
overall role, 127; porpoising, 151;
reality checking, 153; shark scales, 153;
splitter plates, 110, 111; by
streamlining, 134; suction, 109; surface heating,
153; surface morphology, 153;
swimming, 151-54; vs. thrust production,
136
drag, induced. See induced drag
drag, interference. See interference drag
drag, profile. See profile drag
drag-based locomotion, 146, 154-55
dragonfly: eddies in wing pleats, 245;
flying, 86; nymph as jet, 78 (fig.)
dragonfly larva: acceleration reaction,
365; jet propulsion, 77
droplet, vortices inside, 214
Drosophila (fruit fly), 143, 268, 369
duck: flying, 86; hull length, 384; hull
shape, 107; lift-to-drag ratio, 248;
maximum swimming speed, 384;
rubber, 383; surface swimming, 283, 385;
swimming with drag, 286; underwater
drag coefficient, 147
dye marker, 173
dynamic pressure, 53, 58, 62, 69, 97, 99;
cylinder, 83; in drag coefficient, 89; in
friction factor, 301
dynamic similarity, 88, 102
dynamic similitude, 89
dynamic viscosity, 18, 23-25; cells, 18;
and character of flow, 85; vs. density,
24; dimensions and units, 10, 23, 24;
glaciers, 18; glass, 18; vs. kinematic
viscosity, 24, 25; symbols used, 24; table,
23; tar, 18; vs. temperature, 24, 27-31,
153; in viscometry, 26. See also viscosity
Dytiscus (water beetle), 145
E-value: circular cylinder, 120; extrapla-
tion, 120; flat plate, 120; flexible coral,
120; hydroid, 120; kelp, 125; leaves,
123; locust antenna, 120; low, 120;
macroalgae, 120; pine tree, 121
polyethylene sheet, 123; in velocity
gradients, 120. See also speed-specific drug
earthworm, intestine, as pipe, 311-12
Ecdyonurus (mayfly nymph), 178, 179
Echinocardium (echinoderm), burrow
pump pressure, 327
ectoparasites, 180
eddies: attached, 93; and separation
point, 95; in turbulence, 87; vertical
momentum transport, 167
eddy viscosity, 24, 47
eel, 88; anguilliform swimming, 281
efficiency: cilia and flagella, 354; phe-
romone detection, antenna, 356
Eiffel, Gustav, 235
elevation, to increase wind exposure, 193
elm, winged seed, 251, 257
elphin forest, 127
embolism, in xylem, 38, 323
Encarsia (wasp), 278
endothelial cells: and Murray's law, 320;
and shear stress, 188
energy: conservation, 809; cost of drag,
132; dissipation in boundary layer,
157; dissipation in vortex, 209;
dissipation in wake, 97; in jet propulsion, 77;
vs. momentum, 73, 77; and surface
tension, 378
Engraulis mordax (anchovy), 281
entrance region, 297 (fig.); laminar, 296;
turbulent, 302
Epeorus (mayfly nymph), 178
Epischura (copepod), steady swimming,
359
Eristalis tenax (hover fly), 265
erosion, 19; shear flow, 187; shear stress,
188; by vortices, 219
Esox (pike): acceleration reaction, 365;
C-start acceleration, 287
Eubalaena australis (whale), 181
Euphausia embryos, sinking rates, 30
euripterid, swimming with lift, 286
evaporation rate, and boundary layer,
163
exhalation, flow limitation, 316
exposure, 128-29
448
INDEX
fairing: in boundary layer, 166; and
drag, 111
Falco peregrinus (falcon), 1458
falcon: glide angle, 254; interference
drag, 138; lift-to-drag ratio, 148;
parasite drag, 148
Fanning friction factor. See friction
factor
feathers, and parasite drag, 148
fecal pellets, sinking speeds, 344
fibroblasts, velocity gradient in medium,
200
Ficedula hypoleuca (flycatcher), 278
Ficik's law, 47, 199
filter feeding: and continuity, 39; pumps,
324; vs. suspension feeding, 188; and
temperature, 29; and viscosity, 29. See
also suspension feeding
filtration: vs. advice predation, 359;
collector geometry, 358; direct
interception, 357; electrostatic attraction, 357;
encounter vs. retention, 357, 358;
filter resistance, 356; high Re, 355; iner-
tial impaction, 357; mechanisms, 356—
58; motile particle deposition, 357;
rake vs. paddle, 356; range, 356;
sieving, 356; transmissivity, 356; very low
Re, 355-59
fineness ratio, 135; vs. drag coefficient,
graph, 110
fir, thigmomorphogenesis, 128
fish: antarctic "ice fish," 29; aspect ratios
of flying fish, 259; drag, 106, 147-48;
drag and opercular ejection, 110;
flying, 258 (fig.)', form vs. pressure, 67;
gill ventilation, 68; gills, 29, 68, 300
(fig.), 311; high body temperatures,
28; jet propulsion, 77; lift-to-drag
ratios of flying fish, 259; lunate tail
sweepback, 239; mucus secretion, 152;
ram ventilation,60, 68, 110; schooling,
288; shear flow and eggs, 187; slope
soaring of flying fish, 259;
streamlining, 106; suspension feeding, 189;
swimming, 67, 137, 137-48, 286; swet-
ted surface, 90
flag: drag, 123; tatter, 128
flagella: arrangements, 351; bacterial,
206, 331, 351, 352; of Chlamydomonas,
352; efficiency, 354; making thrust,
352—55; mastigonemes, 355; motion,
351, 352; operation, 351-55; as
pumps, 314; rotation of bacterial, 206;
of sponges, 39
flagging, and tree drag, 127
flapping, 262—64. See also flight
flatfish: ground effect, 289; lift, 176
flat plate: added mass coefficient, 364;
as airfoil, 244; autorotating, 228;
boundary layer, 157—63; drag
coefficient, 97-98, 136 (graph), 98 (table);
drag for laminar flow, 135; drag for
turbulent flow, 135; drag vs.
orientation, 351; E-value, 120; flow around,
98; low drag paradigm, 135;
separation point, 98; as speed calibrator, 98;
splitter plate and drag, 111; Strouhal
numbers, 371; turbulent boundary
layer, 162 (fig.)i turbulent transition,
135
flat plates, parallel 310
flattening: feather mites, 181; lift vs.
drag, 177; mayfly nymphs, 178;
torrential fauna, 178; turbellarian flat-
worms, 179; water mites, 179; water
penny beetles, 179
flea, drag in jump, 142
Flettner, Anton, 227
Flettner rotor, 227-29, 245; vs. airfoil,
233; tree seeds, 228
Flettner ship, 227
flexibility: deformation vs.
reconfiguration, 115; drag and, 114-27, 335, 375;
macroalgae, 124-27; moth antennae,
105
flexible organisms, pressure
distributions, 67-70
flight, 13; advance ratio, 266; birds, 137;
with bristle wings, 350; clap-and-fling
mechanism, 278, 279; estimating top
speed, 268; evolution, 254; formation,
birds, 288; ground effect, 288;
hovering, 265, 278; insects, 137; nonsteady
effects, 277-80; origin of thrust, 262-
64; quasi-steady analysis, 278; samaras,
272-73; swooping, 239; vortex wakes
in flapping flight, 276; wing motion,
266 (fig.)
flounder, 66
flow, fully developed, 297
flow markers, various, 44
flow meter, 7
449
INDEX
flow regimes. See laminar flow; turbulent
flow
flow tanks, 7, 13, 14, 62, 98, 103
flow through porous media, 6
flow visualization, 41, 44; liquids vs.
gases, 44; water vs. air, 102
flow, compressible. See compressible flow
flow-induced forces, maximum, 129
flow-induced pressure, 69, 103. See also
pressure distribution
flowmeter, 62; and wall effects, 339
flows, maximum. See maximum flows
fluid: defined, 16, 17, 80; particle, 21,
47; statics, 5
fluidization, porous media, 307
flutter and separation, leaves, 376
flycatcher, lift coefficient, 278
fly pupa, 178, 180
foraminifera, settling, 184
force: dimensions, 9; indirect
measurement, 74-77; inertial, 86; viscous, 86.
See also drag; lift; thrust
forest. See tree
forest canopy, wind within, 171
form resistance coefficient, 341
frame of reference, 12-14, 21;
acceleration, 108; rolling vortices, 218; thrust
and drag, 262
Fraxinus (ash), 228
free-stream velocity, 20, 76, 79, 156, 157,
158
frequency, vortex shedding, 370
fresh water: hydrostatic pressure, 51;
properties, 23 (table)
freshwater limpet, 178 (fig.)
friction factor, 7, 301-2
friction velocity. See shear velocity
frog: acceleration reaction, 365; drag,
146; gliding, 257-58; leapfrogging,
211; velocity gradient and oxygen
uptake, 200. See also tadpole
frogfish, jet propulsion, 77-78
frontal area, defined, 90
Froude, William, 79, 384
Froude number, 284-85; vs. drag, 385
(graph); and maximum speed, 284;
use in modeling, 384
Froude propulsion efficiency, 237; bird
wings, 237; carangiform swimming,
281; drag-based swimming, 287;
ducted fan engines, 330; fixed wing,
237; force vs. power, 237; helicopter
blades, 271; and hovering, 266; jet
propulsion, 79; with zero momentum
wake, 280
fruit fly: acceleration reaction, wings,
368; advance ratio, 268; aerodynamic
frequency parameter, 376; clap-and-
fling mechanism, 278; drag, 10; glide
angle, 254; parasite drag, 143; polar
diagram, 245, 248; untwisted wings,
264; wingbeat frequency, 268, 368;
wing loading, 243; wing mass, 245,
369; wing stall, 250
fundamental dimensions, 10
fungal spore, sinking speed, 344
fungus, detaching conidia, 194
furrow, flow across, 212
Gadus callarias (cod), 281
Galileo, 142
gas, 16; vs. liquid, 22; rarefied, viscosity,
19
geese, formation flight, 288
Gerridae (water striders), 182
gills: cilia on, 314; fish, 29, 300, 311;
positive displacement pump, 326; reduced
area in rheophiles, 200; ventilation,
fish, 68; in vortices, 218
giraffes, blood pressure, 55
glaciers, viscosity, 18
glass, viscosity, 18
Glaucomys volans (flying squirrel), 250
glide angle, 253-55; and lift-to-drag
ratio, 253; vs. size of animal, 254; and
speed, 253
glide ploar, 255-56
gliding, 252-62; animals, 257-59; and
aspect ratio, 253; vs. autogyrating and
autorotating, 257; biological gliders,
258 (figs.); bird, 21; decelerative, 139;
flying fish, 258; flying vs. sinking
speed, 255; Javanese cucumber seed-
leaf, 256; to measure drag, 138;
minimum speed, 255; vs. parachuting, 257;
phalanger, 250; size of animals, 254;
speed vs. weight, 254; time aloft, 255;
turbulence and seeds, 257
glycerin, properties, 23
golf ball: drag, 142; role of dimples,
100
goose barnacles, filter elevation, 191
450
INDEX
gorgonian coral: E-value, 120; on sea-
mounts, 40
grapevine, vessel length, 322
gravitational deposition, filtration by, 357
gravity: in Bernoulli's equation, 53; vs.
drag in jump, 142; estimating
acceleration, 34; and hydrostatic pressure, 51;
sinking Paramecium, 137; and wave
celerity, 381; and wave formation, 379
gravity waves, 382
Gray's paradox, dolphin swimming,
152
ground effect, 260; with clap-and-fling,
289; flight, 288; swimming, 289
growth and drag, 127
gun: to get through velocity gradient,
195; for propagule propulsion, 194; in
water, 196
Gyps ruppelli (vulture), 148
gypsy moth, thermal soaring, 222
GHyrinus (water beetle): oarlets, 350;
rowing stroke, 155 (fig.)
Hagen-Poiseuille equation, 5, 293, 297,
314, 334; applicability to unsteady flow,
377; and cost of circulation, 317; and
sap ascent, 322
half-cylinder, as self-excited oscillator,
372
hawk, lift-to-drag ratio, 248
hawk, Harris', feathers in flight, 148
hearts, 55, 56; maximum pressure, 327;
output of human, 35; positive
displacement pumps, 325; valve-and-chamber
pumps, 327; valves, 62
heartbeat, fish, 68
heat exchanger, 28
heat transfer: and boundary layer, 163;
convective, 6
Hedophyllum (macroalga), 126 (fig.); drag
and structure, 125
helicopter rotor, 269-71; as airfoil, 269;
and hovering flight, 271; origin of
vertical force, 271
hematocrit, 312
hemlock, sinking speed, pollen, 344
holly, speed-specific drag, 117
honeybee. See bee
Hookean material, 17, 18
hoptree, winged seed, 257
horse, blood pressure, 55
horseshoe vortex, 219; digging, 221;
mayfly larva, 220
hover fly: stroke plane, 265; wing
corrugations, 245
hovering: animal flight, 265; Froude
propulsion efficiency, 266; induced drag,
265; kinds of analyses, 276; lift
coefficient, 278; vs. size, 266; stroke, plane,
278; vortex wake, 276; and wing
loading, 266; wing motion, 267 (fig.)
hull length: vs. hull speed, 285 (graph);
and maximum speed, 384; and
wavelength, 382
hull shape, ducks, 107
human: circulation and continuity, 35-
37; heart output, 35; instabilities in
circulation, 374; micturition, 88; penis as
nozzle, 396; pressure drop, circulation,
324; respiratory wheezes, 375;
swimming drag coefficient, 151; volume
flow, circulation, 324
hummingbird: energy consumption,
237; hovering, 266
Hydra, sailing, 391
Hydracarina (water mites), 179
hydraulic ram, 330
hydraulics, 5
hydroid, colonial: commensal suspension
feeders on, 190; E-value, 120; flow
around, 190
Hydrolagus colliei (ratfish), swimming
mode, 286
Hydrosaurus pustulatus (lizard), slapping,
387
hydrostatic paradox, 51
hydrostatic pressure, 51, 70
hydrostatics, 5
hygropetric organisms, examples, 182
Hyla venulosa (tree frog), parachuting,
258
Hymenochirus (frog), 146; acceleration
reaction, 365
ice cream freezer, 17
ice fish, blood, 29
ichthyosaurs, 150; thunniform
swimming, 282
ideal fluid, 82, 86; deprecated, 52; lift,
232; vortices, 209
Idotea (isopod), 91
iguana, circulation, 28
451
INDEX
impact pressure, 130
impedence: of pumps, 325-29; shifting
with transformer, 329
imprecision, 11
impulse, 348
Inachis io (butterfly), 277
incompressibility, 21, 208; and principle
of continuity, 33
indirect force measurements, 74—77
induced drag, 137, 237-38, 242; and
aspect ratio, 238; in hovering, 265;
insect wings, 250; phalanger, 250; on
polar diagram, 238; vs. speed, 238
induced power, 149; vs. speed, 238
induced velocity, and disk loading,
274
inertial force, and pressure drag, 97
inertial impaction, filtration by, 357
information transfer, at very low Re,
359
insects: aquatic, as protrusions, 165;
body lift, 239; drag, 15; flight, 86, 137;
flight speed, 15; parasite drag of small,
142; polar diagrams, 248; wetted
surface, 90; wingbeat, 15; wing
performance, 250
interfaces, 21; lift vs. drag, 176; solid-
fluid, 18,71, 156, 157
interface, air-water, 19, 141, 378-92,
394-97; Bond number, 347; clap-and-
fling mechanism, 289; Coanda effect,
397; as hazard, 347; Jesus number,
347; Weber number, 347
interfaces, fuzzy, 393; atmospheric
inversions, 393; Richardson number, 393;
salinity gradient, 393; temperature
gradient, 393
interfaces interfluid, 378-79
interference drag, 138, 242; bird bodies,
148; and drag measurement, 138;
falcon, 138; half-streamlined body, 165
internal flows. See pipes; pumps
internal fluid transport systems, pipe
sizes, 35
interstitial flow, beaches, 307. See also
porous media
Iron (mayfly nymph), 178
irrotational vortex, 206, 207 (fig.), 225
isopod: drag, 91; drag coefficient, 145;
reference areas, 91
isotachs, 44-46
Javanese cucumber seed-leaf, 251 (fig.);
airfoil, 252; as glider, 256
jellyfish, as jet, 78 (fig.)
Jesus number, 347
jet: as positive displacement pump, 326;
to prevent recirculation, 190; of puff-
ball, 195; of sponges, 39; vortex wake,
215
jet propulsion, 73, 77-80; acceleration
reaction, 365; accelerations, 365; ceph-
alopods, 77; ctenophore, 77p
dragonfly nymphs, 77; efficiency, 79, 237;
energy, power, thrust, 77; frogfish, 77;
medusae, 77; momentum in, 77;
pulsed jet, 79; scaling, 80; scallops, 77,
251; squid, 69; tunicates, 77, 80
jets, colonial, in suspension feeding, 190
Joukowski, N. E.,231
Juglandaceae, autogyrating samaras, 272
kelp, drag, 125
kestrel, slope soaring, 260
killer bee, 13
kinematic viscosity, 23-25; air, 25, 88;
and boundary layer, 158; dimensions
and units, 10, 25; vs. dynamic viscosity,
24; local lowering by heat, 29; in
modeling, 102; vs. temperature, air, 25; vs.
temperature, water, 25; in viscometry,
26; water, 25, 88
Kozeny-Carman equation, 306
Kozeny function, 306
Kutta-Joukowski theorem, 226, 230
Lacuna (gastropod snail), 223
Lagenorhyncus obliquidens (cetacean), 151
lakes, 6
laminar flow, 5, 46-49, 84, 87, 290;
circular aperture, 303; drag of flat plate,
135; maintaining, 109; parallel plates,
298-300; pipes, 290-98; transition,
11; vortices in, 209; wall pinching
index, 315
laminar sublayer, 161, 185
Laminaria, 126 (fig.)
Lanchester, F. W., 230, 232
Langmuir circulations, 394
Lanice (polychaete worm), 217; mucus
thread, 344
Laplace's law: for jet propulsion, 78;
surface tension and nozzle flow, 395
452
INDEX
larvae: antarctic echinoderm, 29;
swimming speeds, 185
larval recruitment. See recruitment,
larval, 40
leaves: aquatic plants, 199; area, 37;
boundary layer resistance, 197;
boundary layers, 163; drag, 14, 120-24, 100;
E-values, 123; as Flettner rotor, 229;
flutter as separation shifting, 376;
reconfiguration in wind, 91, 123-24; as
self-excited oscillators, 373; size/shape
and boundary layer thickness, 199;
transpiration, 197
leg of fishing bat as strut, 141
Leguminoseae, autogyrating samaras,
272
lemur, gliding, 257
length: characteristic, 85; dimension, 8
Leucothea (ctenophore), 77
lichen, vortices in cup, 214
lift, 7, 74, 225-27 (figs. 227, 232); and
airfoils, 230; and angle of attack, 242;
on beds of mussels, 66; and
Bernoulli's principle, 65, 233; and circulation,
226, 230-31; cost vs. aspect ratio,
237; crabs, 146, 176; from curved
surface, 230, 232; distribution on airfoil,
233; vs. drag at interfaces, 176; vs.
drag, sessile organisms, 101; finite
wing, 237; flatfish, 176; Flettner rotor,
227; flow pattern over airfoil, 231;
ground effect, 260; in ideal fluids,
232; indirect measurement, 76; and
induced drag, 137, 238; Kutta-
Joukowski theorem, 226, 230; vs. lift
coefficient, 234, 244; limpets on
surface, 66, 176; making thrust, 137, 263,
283-87; mayfly larvae, 176, 179;
negative, 176, 179; nonsteady effects, 277;
and orientation, snails, 176; oscillation
near stall, 376; and peel failure, 176;
of plaice, 65; and pressure
distribution, 64-67; from protusions, 65; of
ray, 65; rays, 176; reduction by
separation, 179; reduction by spoilers, 179;
from rotating cylinders, 230; of sand
dollar, 65, 176; scaling, 243; in seed
dispersal, 246; of self-excited oscillator,
373; vs. separation, 177; streamlined
object, 64; unconventional
mechanisms, 280; at very low Re, 347; as wall
effect, very low Re, 339; on water
penny beetles, 67
lift coefficient, 234-36; vs. angle of
attack, 234; bat, 278; flycatcher, 278;
formula, 234; fruit fly wing, 250; from
glide angle, 253; for hovering, 278;
from lift, 234; reference area, 91, 234;
steady vs. non-steady, 278
lift, body. See body lift
lift-based locomotion, 155
lift-to-drag ratio, 238, 242, 243, 244,
245, 248, 253; and angle of attack,
264; flying fish, 259; gliders, 255;
insect wings, 250; and lengthwise wing
twist, 264; ski jumper, 240
limpet: E-values, shell, 120; lift, 176; lift
and drag of, 66; local drag, freshwater,
179; separation, freshwater, 179
Linuche unguiculata (coelenterate),
aggregation in windrows, 394
liquid, 16,22
liquid-gas interface. See interface, air-
water
Liriodendron (tuliptree), 228
Littorina (periwinkle snail), 176
lizard: glide angle of Draco, 258; gliding,
257, 258 (fig.); slapping locomotion, 387
local drag: bacteria on walls, 176;
freshwater limpets, 179; planarian, 179
local drag coefficient, 158, 164-67; and
boundary layer thickness, 159; and
transition, 161
locomotion, drag-based, 146, 154—55
locust: aerodynamic frequency
parameters, 376; body lift, 239; drag
coefficient, 144; E-value, antenna, 120;
hindwing polar diagram, 248; lift-to-
drag ratio, 255; quasi-steady flight,
278; swarm movements, 223
Locusta migratoria (locust), 223
lodging, 108
logarithmic boundary layer. See
boundary layer, logarithmic
lugworm, 327 (fig.); burrow pump, 326-
27
Luna (moth), 104
lunate tail, 282
lung, amphibian, positive displacement
pump, 326
lung, mammalian: capillaries, 37; data,
36; volume flow rate, 37
453
INDEX
lunules, of sand dollar, 65
Lycoperdon (puffball fungus), 195, 344
Lycopodium (club moss) spores, 193;
powder as marker, 214; settling, 186
Lymantria dispar (gypsy moth), 222
mackerel: drag coefficient, 147; ram
ventilation, 68; thunniform swimming,
282
macroalgae, 126 (figs.); bryozoans on,
191; drag and flexibility, 124-27; drag
and wave period, 125; drag in
unsteady flow, 375; E-values, 120;
undulate margins, 125
Macrocystis pyrifera (macroalga), 125
Macronema (caddisfly), 60, 61 (fig.)
Magnus, H. G., 226
Magnus effect, 226, 228
mammals, blood viscosity in antarctic, 28
mandarin fish, ground effect, 289
manometer: Chattock gauge, 56; devices,
55 (fig.); inclined-tube, 56; inexpensive
multiplier, 55, 56, 60; for Pitot tube,
58, 60; for Venturi meter, 58
manometric height, 54, 55
manometry, 54-57
maple: samara, autogyrating, 228, 252,
272; vessel length, 322
marine mammals, drag, 150—51
marine snow, 331; sinking speeds, 344
marlin, thunniform swimming, 282
mass: vs. density, 22; dimension and
units, 8
mastigonemes, on flagella, 355
maximum flows, 128-30; measurement,
129-30, 130 (fig.); prediction, 130
mayfly larva, 178 and 220 (figs.);
E-values, 120; flattening, 178;
horseshoe vortex, 221; leg struts, 139; lift,
176; separation, 179; streamlining,
177
medusae, jet propulsion, 77
Mellita quinquiesperforata (sand dollar),
65
Membranipora villosa (bryozoa), 191
mercury, properties, 23 (table)
metachronal rhythm, 154
Metridium, 115, 116 (fig.); filter elevation,
191
microcirculation, 37. See also capillary
micrometeorology. See biometeorology
micturition, 88; flow limitation, 316;
squirting vs. spraying, 396, 397
midges: hygropetric, 182; sewage filter
flies, 183; wingbeat frequency, 268
migration, monarch butterfly, 13
mink, drag at surface, 385
mistletoe, dwarf, seed as projectile, 195
mites, feather, specializations, 181
mites, water, flattening, 179
mitosis, 5
mixing, and unstirred layer, 202
modeling: constant force, 103; flow-
induced pressure, 103; media shifts,
data, 104; moth antenna, 105; prairie
dog burrows, 103; and Re, 102-5; sea
anemone, 115; suspension feeding,
103; using highly viscous media, 102
molecular viscosity, 24
molecules, 19, 20; intermolecular
cohesion, 378
mollusks, bivalve: cilia, 39; continuity, 39
momentum, 33, 73—77; angular, 205,
208; conservation, 73; dissipation in
boundary layer, 157; vs. energy, 77;
flux, 75; flux in hovering, 276; flux of
beating wing, 273; flux through
actuator disk, 273; and force, 73; and
impulse, 348; in jet propulsion, 77;
momentum equation, 74; near flat
plate, 98; and pressure gradient, 95;
and propulsion efficiency, 237;
transfer in turbulent flow, 47; at very low
to, 331
mosquitoes, wingbeat frequency, 268
moths, 350 (fig.); modeling flow through
antenna, 104; pheromone detection,
356; tracheal system, 320
motile particle deposition, filtration by,
357
motion, relative. See relative motion
mucous threads: drag and dispersal, 223;
use by nudibranch mollusks, 389
mucus, as non-Newtonian fluid, 20
mucus secretion for drag reduction, 152
Murray's law, 188, 317-21; for
bifurcation, 318; circulatory systems, 319;
coral gastrovascular system, 320;
crinoid coelomic circulation, 320; and
endothelial cells, 320; leaf vessels, 320;
moth tracheal system, 320; sap
conduits, 320; and self-optimizing system,
454
INDEX
321; and shear stress on walls, 318;
speed vs. vessel radius, 318; sponges,
319; velocity gradients, 318
muskrat: Froude number, 385;
maximum swimming speed, 385; swimming
with drag, 286
mussel: dispersal via byssus thread drag,
223; lift on beds, 66; pumping rate,
189; pump pressure, 328
muzzle speed: mistletoe seed, 195; Sor-
daria projectile, 195
Mya arenaria (clam), 328
Mytilus edulis (mussel), 223, 328
nasal hairs, 310
nasal passages, 312 (fig.); as parallel
plates, 311
nautilus (cephalopod), 77, 80; shell drag
coefficient, 147
navicular fossa, 396
Navier-Stokes equations, 156, 158; very
low Re, 333
negative pressure, 38. See also sap
conduit
Nemertesia antennina (colonial hydroid),
190
Nereis (polychaete worm), 326
Nereocystis (kelp), 125, 126 (fig.)
Newton's first law, 81
Newton's second law, 52, 73, 74, 86, 138
Newton's third law, 75
Newtonian fluid, 18, 20
no-slip condition, 18-20, 156, 167, 314;
and gas droplet, 214; and pipe flow,
292; thrust from shear, 338
Noctilio (fishing bat), 140
non-Newtonian fluids, 5, 20; viscometry,
26
nonsteady effects, 277-80; clap-and-
fling, 278; insect flight, 249; stall delay,
278; Wagner effect, 277; wing
rotation, 280
nonsteady flows, and suspension feeding,
358. See also unsteady flows
Notonecta (back swimmer), prey detection
with waves, 392
nozzle, 77, 395-97 (figs., 396); breakup
of stream, 395; elevation as, 40; flow
and Ohnesorge number, 396; flow and
Re, 395; flow and Weber number, 395;
and Laplace's law, 395; mammalian
penes, 396; minimal size, 190; power
loss, 33; and principle of continuity,
33; satellite droplets, 396; of sponges,
39; squid, 79; and surface tension 395
nudibranch mollusk, use of mucus
threads, 389
oak: E-values, leaves, 124; sap ascent, 37;
vessel size, 322
oarlets, for swimming, 350
ocean currents, 5
octopus. See cephalopods
Ohnesorge number, 396
olfaction, 356; flow through burrows, 71;
plesiosaurs, 60
Orconectes (crayfish): acceleration, 287;
escape response, 365
orifice: circular aperture, 303; and
continuity, 33; flow through, 394; and
surface tension, 395; vena contracta, 395.
See also nozzle
orifice coefficient, 303
oscillators, self-excited, 372-75
Ostwald viscometer, 104
owl: quiet flight, 242; wing barbs for
anti-stall, 242
Pachyptila vittata (prion), 387
paddles: acceleration reaction, 366; am-
phipod pleopods, 328; drag
coefficients, 154; as fluid dynamic pump,
327
Pandalus danae (shrimp), accelerations,
287
parabolic flow. See pipes
parachute, 12
parachuting, 257. See also gliding
Paramecium (ciliate protozoan), 349; flow
around, 137; sinking, 137; velocity
gradients around, 354
parapodia, on polychaetes, 355
parasite drag, 137, 240, 241; birds, 148-
49, 248; falcon, 148; and feathers,
148; measurement, 139; small insects,
142-43
particles, aerosol, 192
pascal, 23
pascal second, viscosity unit, 24
pathlines, 41-45
penguins: drag, 139, 149-50; drag
coefficients, 149; drag at surface, 385;
455
INDEX
penguins (cont.)
maximum speeds, 149; swimming,
137, 283, 286; transport cost, surface
swimming, 386
Penicillium (fungus), spore elevation, 193
penis, 396 (fig.); as nozzle, 396; orifice,
88
peristaltic pumps, 326
Petaurus breviceps (phalanger), 250
petiole, 123
petrel: sea anchor soaring, 260; slope
soaring, 259
Peclet number, 202, 313; capillaries, 313;
phytoplankton, 346; swimming
protozoa, 355
phalanger, 251 (fig.); gliding, 250, 257;
lift-to-drag ratio, 253; polar diagram,
250
pheromone, detection by moth, 356
Phoca vitulina (harbor seals), 150
Phoronopsis viridis (phoronid worm), 217
photographic image and velocity
gradient, 198 (fig.)
photosynthesis: cost in drag, 107; vs.
current, 199; and drag, 127
Phragmatopoma lapidosa (tube worm),
attachment, 185
Phragmites australis (reed), 71
Phrynohyas (flying frog), gliding, 257
phugoid oscillation of aircraft, 376
Physalia (Portuguese-man-of-war), sailing,
391
phytoplankton: Peclet number, 346;
sinking speeds, 344; why sink?, 346
Picea abies (spruce), 272
Pieris brassicae (butterfly), 276
pigeon: drag coefficient, 148; drag as
spheroid, 133; glide angle, 254; wing
area adjustment, 243
pike: acceleration reaction, 365; C-start
acceleration, 287
Pilobolus (fungus), tumbling sporangium,
338
Pinaceae, autogyrating samaras, 272
pine: cones as pollen traps, 187; drag
coefficient, 121; E-value, tree, 121;
pollen, 345 (fig.); speed-specific drag, 117,
118 (fig.); thigmomorphogenesis, 128
pinnipeds, drag and swimming, 150
pipes: avoiding parabolic flow, 308—14;
Bernoulli vs. Hagen-Poiseuille, 314-
16; biological applications, 308-23;
branching arrays, 33, 316, 321, 318
(fig.); capillaries, 311, 312; with ciliated
walls, 314; continuity, 290; distance
indices, 308—11; earthworm intestine,
311; entrance region, laminar, 296,
297 (fig.); entrance region, turbulent,
302; flow measurement, 295; flow
profiles, 310 (fig.); friction factor 301, 302
(graph); Hagen-Poiseuille equation,
293, 297; laminar flow, 290-98;
Laplace's law, 317; Malpighian tubules,
311; mean vs. max flow, turbulent,
301; mean vs. max speeds, laminar,
295-96; nasal hair role, 310; noncircu-
lar cross sections, 310, 312 (fig); non-
material, 41; no-slip condition, 19;
parabronchi of bird lungs, 311;
periodic boluses, 312; plug or slug flow,
297; power, laminar, 294; principle of
continuity, 32-33; proximity of flow to
wall, 308-11; pseudomanometry, 295;
pumping at walls, 314; pumps, 323-
29; renal tubules, 311; resistance,
laminar, 294; resistance, at transition, 301;
shear stress on walls, 291; size vs.
pressure drop, laminar, 294; speed vs.
x-section, turbulent, 301; of sponges,
38; total flow, 292-94, 295; transition,
290, 298, 305; transmural exchange,
308, 311, 312, 314; turbulent
spouting, 375; using very small, 311; velocity
vs. x-section, laminar, 290-92, 293
(fig.); velocity vs. x-section, turbulent,
300; vortices in bends, 215; wall
roughness, laminar, 296; wall
roughness, turbulent, 301; Womersley
number, 377
pitch: aircraft, 242; blade element, 262;
wings, 242
pitching moment, and angle of attack,
234
Pitot tube, 58-60, 59 (fig.), 72, 83; living,
61 (fig.)
Pizonyx (fishing bat), 140
plaice, 66 (fig.); lift and drag of, 65
planarian, local drag, 179
plan form area, 90
planing, 386-87
plankton: irregular shape, 345;
measuring density, 30; shape vs. temperature,
456
INDEX
30; sinking rates, 29, 88; size vs.
temperature, 30. See also phytoplankton,
zooplankton
plant: cells, 8; growth and wind, 127;
lodging, 109; pressure gradients, 323;
sap speeds, 323; thigmomorpho-
genesis, 128
plates, parallel: biological examples, 300;
laminar flow, 298-300, 299 (fig.);
mean vs. max speed, 300; nasal
passages, 311; speed vs. x-section, 299;
total flow, 299
Plecotus auritus (bat), 278
plesiosaurs, 150; olfaction, 60
Pleuronectes platessa (plaice, flatfish), 65
plug flow, 297, 309
poise, 24
Poiseuille's equation. See Hagen-
Poiseuille equation
polar diagram, 234-36, 235 (fig.); for
airfoil, 235; bumblebee wing, 245;
fruit fly wing, 245; insect wings, 250;
lift-to-drag ratio, 238; phalanger,
250
Pollachius vixens (saith), 148, 281
pollen, 345 (fig.); exposure to wind, 193;
sinking speeds 88, 344; trapping 187
pollination, 43
polychaetes: anguilliform swimming,
281; parapodia on, 355; swimming
with drag, 286
polychaetes, tubicolous, filter elevation,
191
polymers, drag reduction with, 152
poplar, white, leaf reconfiguration, 124
Populus alba (white poplar), 124, 193
Populus tremuloides (aspen), 124
Porifera. See sponges
porous media, 305—307; fluidization,
307; Kozeny-Carman equation, 306;
Kozeny function, 306; quicksand, 307;
voidage or porosity, 306
porous substratum, flow in and out of,
173
porpoise, top speed, 151
porpoising, 151
Portuguese-man-of-war, sailing, 391
Postelsia (sea palm), 122, 374, 126 (fig.)
Potamodytes (beetle), 70, 83, 70 (fig.)
power: finite wing, 237; vs. flying speed,
149, 237, 240, 241 (fig.); loss in nozzle,
33; pipe flow, 294; in pumping, 323; of
pumps, 328; to stay aloft, 238
prairie dog: burrow ventilation, 71;
modeling burrow flow, 103
Prandtl, Ludwig, 156,232
precision, 12
pressure, 21, 50, 51 (fig.); across airfoil,
233; atmospheric, 54; in beetle's
bubble, 70; in Bernoulli's equation, 53;
blood, see blood pressure; dimensions
and units, 8-10; vs. flow speed, 56;
and hydrostatic paradox, 51; impact,
130; manometric height, 55;
manometry, 54—57; measurement of low
pressure, 56; negative, 38; Pitot tube, 58-
60; root, 321; units, 55; Venturi meter,
57—58. See also static pressure;
dynamic pressure
pressure coefficient, 8, 62-64, 67, 90;
near protrusion, 71; and Re, 102;
streamlined object, 63; whale, 69
pressure distribution, 62—70 (figs., 63,
64, 68); beetle's bubble, 70; circular
cylinder, 82, 83; fish, 67-69; grain
storage buildings, 83; squid, 79;
whales, 69
pressure drag, 97, 109, 111, 133,242;
airfoil, 240; inertial force in, 97; and
separation, 97; vs. skin friction, 97,
109; in Stokes'law, 334
pressure-induced flow, 71-73; diagram,
72; prairie dog burrows, 71; reed, 71;
relict burrows, 72; ripples of sand, 71;
sand dollars, 71; sclerosponges, 71;
storage structures, 72;
stromatoporoids, 71; suspension-
feeding brachiopods, 71; termite
mounds, 71; tracheae of flying insects,
71; volcano cones, 72
principle of continuity, 5, 32-40 (figs.,
33, 34), 53, 63, 80, 82; and actuator
disk, 273; and beating wing, 273; and
boundary layer, 159; contracting
column of liquid, 34; and filter feeding,
39; human circulation, 35—37; and
internal fluid transport systems, 35-39;
pipes, 290; rivers rising, 39; and sap
movement, 38; and sponges, 38-39;
streamlines, 40-41
prion: planing, 387; sea anchor soaring,
260; suspension feeding, 260
457
INDEX
profile area, 90
profile drag, 137, 240-241, 242, 244;
gliding, 255; insect wings, 250; vs. size,
268
profile power, 149
projectiles, form, 194
propagule: elevation, 193—94; as
projectile, 194
propeller, 269 (fig.); advance ratio, 267;
as airfoil, 268; airstream contraction,
273; angle of attack, 264; blade
contour, 270; blade element analysis, 267;
and continuity, 34; disk swept by
blades, 267; in ducts as pump, 328;
efficiency, 79, 282; in ideal fluid, 82;
induced velocity, 274; lengthwise twist,
264; like beating wing, 264; local shear
flow, 188; lunate tail as, 282; making
thrust, 262; torque, 263; as volume
pump, 324; whale tail, 282; as
windmill, 270
propulsion: efficiency of cilia and fla-
gella, 354; via effective area change,
348, 350—51; irrelevance of drag data,
350; via orientation change, 348, 351,
355; surface slapping, 387; vs. tethered
flow, 349; using cilia and flagella, 351-
55; using surface tension, 387-90; at
very low Re, 347—55; from waves,
whale, 386. See also flight, swimming
propulsion efficiency. See Froude
propulsion efficiency
propulsion, jet. See jet propulsion
protozoa: Peclet number, swimming,
355; rapid stalk contraction, 360; stalk
for filter elevation, 191
protrusions and protuberances: density
and flow, 171; drag coefficients, 165,
166; to get eddies, 312; organisms as,
165; in pipes, 296; pressure
coefficients, 71; producing lift, 65; Res of,
165; skimming flow, 220; and Stokes'
radius, 341; streamlining, 165; and
vortices, 172, 219; zero plane
displacement, 168. See also roughness
Psephenidae (water penny beetles), 179
Pseudiron centralis (mayfly larva), 221
pseudomanometry, 295
Pseudopterogorgia (gorgonian coral),
120
psychodid fly larva, hygropetric, 182
Ptelea trifoliata (hoptree), winged seed,
257
Pterphyllum (angelfish): acceleration
reaction, 365; swimming with drag, 287
pteropod mollusk, swimming with lift,
286
Ptilosarcus (sea pen), 113 (fig.)
puffball'sjet, 195
pulmonary artery, 35
Pulvinariella mesembryanthemi (scale
insect), 193
pumping rate: bivalve mollusks, 189;
sponge, 189
pumps, 323-29; biological examples,
324, 326 (table); cilia on walls, 314;
ciliary and flagellar, 39; evaporative sap
lifter, 324; filter-feeders, 324;
flagellated chambers, 314; impedance
matching*, 325; impedance
matching**, 329; vs. local flow, 192;
maximum pressures, 328; measuring
output, 328; peristaltic, 326; of poly-
chaete worms, 326; power output, 323;
pressure drop vs. flow, 329 (graph);
resistance, 323; in suspension feeding,
189; transformers, 329-30; valve-and-
chamber, 325
pumps, fluid dynamic and positive
displacement, 325; biological examples,
326-28; impedance, 325; pressure
range, 327
Pygoscelis papua (gentoo penguin), 149
Pyralidae (Lepidoptera), 180
quaking, role in aspen, 124
quasi-steady analyses: of flight, 278;
swimming, 365; when adequate, 376
Quercus. See oak
Quercus phellos (willow oak), 229
quicksand, 307
ram ventilation, 60, 68, 110; and
suspension feeding, 189
Rana (frog), 146,200
range, Sordaria projectile, 195
Ranunculus pseudofluitans (aquatic plant),
199
rate of shear. See shear rate
ray: body lift, 239; ground effect, 289;
lift, 65, 176; pitching body, 239;
swimming, 239, 286
458
INDEX
Rayleigh, Lord: in drag, 82; jet breakup,
396
recirculation, preventing, 190-92 (fig.,
191)
reconfiguration: E-values, 118; flexible
bodies in flows, 116—27; Hedophyllum,
125; leaves, 123-24 (fig., 124); and
speed-specific drag, 117
recruitment, larval, 40. See also settling
red blood cells, 28; as boluses, 312
reduced frequency, 376
reed, pressure-induced flow, 71
reference area: in drag coefficient, 90—
91; frontal, 132; in lift coefficient, 234;
plan form, 132, 236; in polar diagram,
236; profile, 132, 236; volume to the
two-thirds power, 132; wetted, 132
reference frame. See frame of reference
relative motion, 12-14 (figs., 13, 14)
reptile, blood pressures, 56
resistance: of pipes, 294; of pumps, 323,
328
respiratory system, Womersley number,
377
Reynolds, Osborne, 47, 84, 298
Reynolds, number, 5, 84-86, 87, 89;
acceleration to raise, 360; vs. airfoil drag,
241; and airfoils, 142, 244-46; and
airfoil stall, 244; Baetis legs, 140; and
Bernoulli vs. Hagen-Poiseuille, 315; vs.
best aspect ratio, 241; and best
thickness-to-chord ratio, 134;
biological range, 86; birds, 244; vs. boundary
layer thickness, 157; and character of
flow, 87; and circulation, 245; desert
locust, 144; and drag coefficient, 88-
93, 117; dynamic similarity, 88, 102;
and E-value, 120; examples, table, 86;
and flow pattern, 102; and friction
factor, 301; and gliding, 254; high vs. low,
244, 304, 305, 331; imprecision, 11;
inertia vs. viscosity, 156; insects, 244; vs.
lift coefficient, 234, 244; and
modelling, 102-5; and nozzle flow, 395; and
optimal shape, 109; and organism's
medium, 88; and organism's size, 87;
physical meaning, 86—88; pipe
roughness elements, 296; and pressure drag,
97; and profile drag, 244; and
propulsive bristle spacing, 351; of
protrusions, 165; and skin friction, 96; and
streamlining, 99; vs. Strouhal
numbers, 371; tadpoles, 146; and terminal
velocity, 340; trout, 88, 147; and
turbulent transition, 85, 95, 111, 135,
290; useful precision, 85; very low Re,
see Reynolds number, very low; and
vortex shedding, 369; and vortices,
209, 216; water beetles, 145
Reynolds number, boundary roughness,
170
Reynolds number, local, 166
Reynolds number, very low, 331—60;
absence of separation, 332; biological
importance, 332; boundary layers, 332;
characteristics, 331; circulation and
lift, 347; diffusion vs. swimming speed,
332; drag, 338-38; drag coefficients,
333; drag vs. orientation, 334; drag
and shape, 332, 334; drag-based
propulsion, 347-48; flow reversibility,
348; inertia vs. drag, 331; information
transfer, 359—60; negligible
circulation, 332; propulsion, 331, 347-55;
stirring and mixing, 331; streamflines,
332; suspension feeding, 355-59;
terminal velocity, 332, 339-47; velocity
gradients, 332, 348; vortices, 332; wall
effects, 332, 338-39
Rhacophorus (flying frog), gliding, 257
rheophilic tadpoles, 179
rheotaxis, water penny beetles, 179
Rhithrogena (mayfly nymph), 178
Rhynchops nigra (skimmer), 141, 288
riblets, drag reduction, 153
Richardson, L. F., 47
Richardson number, 395
ripples, vortices in, 212
river, 5; and continuity, 39
root pressure, 321
rotation: bacterial flagellum, 206, 352;
vs. circular translation, 206; falling
particle pairs, 342; solid vs. fluid, 204;
as wall effect, 339. See also vortices
rotifers: seasonal polymorphism, 30;
swimming with cilia, 355
roughening, and transition Re, 100
roughness: for drag reduction, 99; and
friction factor, 301; length, 167, 169,
parameter, 167; protrusion heights,
160; Re of elements, 160; vs. Re and
drag coefficient, 101; rocky-coastal
459
INDEX
roughness (cont.)
organisms, 101; and settling, 186; and
spore liberation, 193; surface, and
eddies, 167; tree bark, 100
sailing, 390-91; biological examples,
391; mechanisms, 390; with surface
currents, 391; Velella, 252
saithe: burst-and-coast swimming, 281;
drag coefficient, 148
Salmo gairdneri (trout), 147
salps: acceleration reaction, 365; jet
propulsion, 78, 80
samara: advance ratio, 272; as autogyro,
272; autorotating 228, 229 (figs.); as
Flettner rotor, 228; flight, 272-73 (fig.,
273); lengthwise twist, 264; sinking
speeds, 273
sand, pressure-induced flow through, 71
sand dollar: airfoil, 246, 247 (fig.);
circulation and feeding, 246; effective
aspect ratio, 246; lift and drag of 65, 66
(fig.), 176; pressure-induced flow, 71
sand dollar larvae, swimming speeds, 31
sap ascent, 37-38, 321-23; and
continuity, 38; flow speeds, 38; measuring, 37;
pumping, 308
sap conduits, 321-23; embolisms, 323;
flow speeds, 322; gravitational
gradient, 322; and Hagen-Poiseuille
equation, 322; Laplace's law, 323; Murray's
law, 320; negative pressures, 308, 321;
pore resistance, 322; size vs. flow
speed, 322, 323; suction and trunk
shrinkage, 316; vessel lengths, 322
sap lifting pump: positive displacement,
326; pressures, 324
Sargassum (seaweed), form vs. site, 126
satellite drops, 88
Savonius rotor, 112
scale insect: sinking speed, 193; standing
on hind legs, 193
scales, on butterfly wings, 245
scaling: acceleration reaction, 108, 368;
drag, 108; drag of flexible forms, 115;
lift and weight, 243
scallop (figs., 78, 251): as airfoil, 251,
aspect ratio, 252; asymmetrical shells,
252; jet propulsion, 77, 251; modeling,
103; pumping rate, 189; swimming,
251; vortices around, 219
460
Schistocerca gregaria (desert locust), 143,
223
schistosomiasis, 108
schooling, fish, 288
sclerosponge, 71
Scomber scombrus (mackerel), 147
Scyllarides latus (lobster), acceleration, 287
sea anemone, 116 (fig.); drag, 115
seabirds, wing sweepback, 239
sea butterfly, swimming with lift, 286
sea fan, drag and orientation, 107
sea lily, filter elevation, 191
sea lions: drag coefficient, 150; drag and
swimming, 150; swimming with lift,
286
seals: drag and swimming, 150; drag
coefficients, 150; swimming with lift, 286
seamounts, as nozzles, 40
sea palm, biomechanics, 122
sea pen, 113 (fig.)
sea turtles. See turtles
seasonal polymorphism, 30
seawater: hydrostatic pressure, 51;
properties, 23 (table), 25
seed dispersal. See dispersal, seed
seeds: autorotating samaras, 228—29;
ballistic mechanisms, 194; drag
maximization, 257; sinking rates, 14;
sliding on snow, 194. See also samara
self-excited oscillator, 372-75;
mechanisms, 372, 373 (fig.); roles of stiffness
and shape, 374; seed or spore release,
374; Steinman pendulum, 373; Tac-
oma Narrows suspension bridge, 373
Semibalanus balanoides (barnacle), 185
separation, 94, 102, 111, 133, 143;
airfoil, 233; Donax, 182; and drag, 99;
flattened stream fauna, 179; and lift
vs. drag, 177; nonsteady effects, 278;
and pressure drag, 97; and stall, 242;
and vortices, 213
separation point, 95, 96 (fig.); black fly
larval preference, 184; detecting, 95;
flat plate, 98; head and torso, 95
oscillation on airfoil, 376; below
transition, 95; at turbulent transition, 95
sessile organisms, settling, 183
setae, thrust from drag, 154
settling: from air, 186; algal spores, 187;
ascidian, 184; barnacles, 185; for-
aminifera, 184; hydrodynamic factors,
INDEX
184-85; pollen and spores, 186; rough
surfaces, 186; velocity gradient, 183—
86
sewage filter fly, 183
shape: and drag, 106-27; and drag
coefficient, 132; low Re, 134; of
streamlined bodies, 133; vs. skin friction,
134; variation with force, 67
sharks: dermal scales and ridges, 153;
spiral valve of intestine, 311; thun-
niform swimming, 282
shear: deformation, 16, 20; spinning a
vortex, 211
shear flow: blood cells rotation, 187;
endothelial cells, 188; erosion, 187; fish
eggs, 187; rotation, 188 (fig.)"> tum-
bleweed rotation, 187
shear modulus, 17
shear rate, 18, 25, 87, 153; for barnacle
settling, 183; in boundary layer, 157;
in vortex, 207
shear strain, 17
shear stress, 17, 20, 24, 50, 62, 72, 168;
direct effects, 187-88; in friction
factor, 301; and Murray's law, 318, 320;
on pipe wall, 291; removing bacteria,
176; and shear velocity, 169
shear velocity, 168-69
Sherwood number. See Peclet number
ships. See surface ships
shock absorbers, 18
shrimp, acceleration, 287
SI, 9, 10,22,23,25, 163
sieving, filtration by, 356
silk thread: ballooning in vortices, 223;
drag, 222; telling wind direction, 223
Simulium vittatum (black fly larva), 184,
217
sinking rate. See terminal velocity
ski jumper: body lift, 240; lift-to-drag
ratio, 240
skimmer: bill as strut, 141; ground
effect, 288; lower bill shape, 383
skimming flow, 162, 172, 219
skin friction, 96, 99, 153, 156, 162, 164,
242; airfoils, 240, 241; and drag
coefficient, 135; vs. optimal shape, 134; in
pipes, 291; near propulsive
appendages, 349; and Re, 96; in Stokes' law,
334; and viscous force, 97
slapping locomotion, 387
sliding on snow by seeds, 194
slime mold, vortices in cup, 214
slimes, 5
slug flow, 297, 302, 309
slurries, 5
snails: lift and orientation, 176; mucous
thread drag, 223; on river bottoms, 40
snake: anguilliform swimming, 281;
blood pressures, arboreal, 56; gliding,
257; heart position, arboreal, 56
soaring, 252, 259—61; dynamic, 259,
260; gradient, 261; sea anchor, 260;
slope, 259, 260 (fig.); static, 259;
thermal, 221,260
solid, 16, 17,80
Sordaria (fungus), 195; muzzle speed,
195; range of projectile, 195;
trajectories, 196 (fig.)
soredia, of lichens, 214
sound, speed of, 21
sound production, unsteady flow, 377
spasmoneme, of protozoan, 360
specific volume, 22
speed: estimating flight maxima, 268;
flying deer fly, 21; and glide angle,
253; porpoise, 151; sinking, of
samaras, 273; wing loading vs. flight,
243. See also terminal velocity
speed of flow, measurement, 54
speed of sound, 21
speed-specific drag, 117-19; pine, 118
(fig.); and reconfiguration, 117; vs.
speed, 117; trees, 117
spermatozoan: Re, 86, 88; swimming,
347
Spheniscus humboldti (Humboldt penguin),
149
sphere: added mass coefficient, 363;
boundary layer, 163; drag, 112, 333;
drag coefficient, 92, 112, 334; drag
oscillation at transition, 375; flow
patterns, 96; interactions in sinking, 342;
Magnus effect, 227; Stokes' law, 333;
terminal velocity, 339; transition point,
96; transition Re, 305. See also Stokes'
law
sphere, fluid: drag at very low Re, 337;
induced toroidal motion within, 337
spheroid, drag vs. shape and orientation,
low Re, 336
sphygmomanometer, 56
461
INDEX
spider: ballooning, 12; drag of silk
strand, 335; thermal soaring, 222
spider, fishing, 387 (fig.); prey detection
with waves, 392; sailing, 391
spider silk, strength, 10
spiral valve, of shark intestine, 311, 312
(fig-)
Spirogyra (alga), 199
splitter plates and drag, 110-11
spoilers, mayfly nymph, 179
sponges, 38—39; and continuity, 38;
filtering choanocytes, 356; flagellated
chambers, 314, 320; maintaining
laminar flow, 109; Murray's law, 319; pipe
sizes and flow speeds, 36, 39; pressure-
induced flow, 71; preventing
recirculation, 190; pumping rate, 38, 189, 319;
pump pressure, 328; sclerosponges,
66; suspension feeding, 189
sporangium, of slime mold, 214
spore, liberating and settling, 186, 192
sporozoan trophozoites, thrust from
shear, 338
spot on surface, drag, 165
spraying vs. squirting, 88
springs, 18
springtail, 387 (fig.); use of surface
tension, 389
squid: acceleration reaction, 365; cost of
transport, 79; as jet, 69, 77, 78 (fig.);
modeling, 103; pressure distribution,
68,69
squirrel, flying, 257
squirting and spraying, 88, 394—97
stagnation point, 82, 95
stall, 242; angle of, 244; anti-stall
devices, 242; fruit fly wing, 250; non-
steady effects, 278; vs. Re, 244;
repeatedly delayed, 276; and
separation, 242
starting vortex, 232
static hole, 62
static pressure, 51, 53, 58, 61
steady flow, 21; and streamlines, 42
Steinman pendulum, 373
Stenella attenuata (cetacean), 151
stenoses 62, 316; and turbulent
oscillation, 374
Stenus (beetle) surfactant propulsion, 390
Stichopathes (coral), 40
stiff materials, 114
sting, and interference drag, 138
stirring, at very low Re, 331
Stokes, Sir George, 25, 218
stokes (unit), 25
Stokes' law, 5, 27, 29, 88, 156, 194, 304;
equation, 333; gas droplet, 215; origin
of drag, 334; and propulsion, 348; Re
limit, 334; and terminal velocity, 340
Stokes' radius, 341
stomata, 197
stopping vortex, 232
strain, 9; shear, 17
streaklines, 41-44 (fig., 44)
streamlined bodies, 64 and 100 (figs.);
bidirectional, 114; biological, 141-51;
drag coefficients, 133; drag vs.
roughness and Re, 101; lift, 64; pressure
coefficients, 63; reversed flow, 114;
shape, 133
streamlines, 40-41, 42 (fig.), 61, 63, 80,
81, 99; and airfoil lift, 232; and
continuity, 41; and lift origin, 226; locating,
41; pathlines vs. streaklines, 42; source
of quantitative information, 42-44
streamlining, 98-99, 101, 109, 114, 120;
directional sensitivity, 99; fly pupa,
180; half-streamlined bodies, 165, 166;
ideal, 135; mayfly nymph, 177; motile
animals, 132-36; organisms, 142
(figs.); pressure coefficients, 63;
pressure drag, 133; protrusions, 165; vs.
Re, 99; struts, 139-41 (fig., 140); as
vortex prevention, 212
streams, 5; effects of current, 175;
torrential fauna, 139
streamtubes, 41, 61, 74, 75
stress, 9, 50; shear, 17
stroke angle, beating wing, 267
stroke plane: beating wings, 265; hover
fly, 265; hovering, 278
stromatoporoid, 66, 71
Strouhal number, 370-71, 372 (graph);
circular cylinders, 371; as dimension-
less frequency, 370; flat plates, 371; vs.
Re,'ill
struts: across air-water interface, 383;
biological, 139-41 (fig., 140); crab legs,
140; drag, 134; fishing bat legs, 140;
mayfly nymph legs, 139; skimmer bills,
140; streamlining, 139-41, 140 (fig.);
thickness-to-chord ratio, 134
462
INDEX
Styela montereyerisis (ascidian), 60, 61 (fig.)
submarine, 382, 383
substratum: flow next to, 171—73; flow
patterns near, 172 (fig); shear velocity
vs. roughness, 171 (graph)
surface currents, 391
surface heating, drag reduction with,
153
surface ships, 141, 382-387;
displacement vs. planing hulls, 386; Flettner,
227; hull drag, 137; hull length, 382;
maximum speed, 383, 384; planing,
386
surface spot: drag, 164; local drag
coefficient, 164; shear on, 165
surface swimming, 384—86; cost, 385;
drag, 385; maximum speed, 390, 390;
planing, 387
surface tension, 8, 19, 378—79; biological
relevance, 379; dimensions and units,
10, 378; and nozzle, 395; and
propulsion, 387-90; size vs. speed limit, 390;
and surface currents, 391; useful size
range for support, 389, 390; and wave
celerity, 381; and wave formation, 379;
Weber number, 389
surface waves. See waves
surface-to-volume ratio, 9; dimensionless
version, 346; and sinking speeds, 345
surfactant propulsion, 390
suspension feeding, 4, 216, 355-59;
active vs. passive, 40, 189, 190; colonial
jets, 190; and drag, 107, 127; fish, 189;
high Re, 355; and local flow, 192;
modeling, 103; preventing recirculation,
190-92; prion (bird), 260; shape of
structures, 113; in turbulent flow, 192;
in velocity gradients, 188-90; whales,
189
swash: beaches, 307; velocity gradient in,
181,219
swash-riding, 181-82; clam, 181; whelk,
182
swift, wing sweepback, 239
swimming, 13, 280—83; acceleration,
287; acceleration reaction with tail or
paddles, 366; added mass, 364-67; an-
guilliform, 281; bacterium coasting,
331; brine shrimp, vs. Re, 350; bristles,
very low Re, 350; burst-and-coast, 280;
C-start, 287; carangiform, 281; car-
angiform with lunate tail, 282; cilia vs.
flagella vs. muscle, 355; crabs, 146;
ctenophores, 355; Daphnia, 30; vs.
diffusion, very low Re, 332; drag-based,
154, via drag production, 286; ducks,
147, 283; eel vs. spermatozoan, 347;
fish, 137, 147-48; fish schooling, 288;
fish, pressure on, 67; flow disturbance,
very low Re, 349; frogs, 146; glide to
measure drag, 138-39; ground effect,
289; heat production, 29; irregularity
as signal source, 359; lift vs. drag
based, 283-86; modes, 281, 282 (fig.);
paddles, very low Re, 350; paddling vs.
flapping, 283-86; penguins, 137, 283;
quasi-steady analysis, 365; rays, 239;
sand dollar larvae, 31; scallops, 251;
sea lions, 150; seals, 150; vs. sinking as
signal source, 359; speeds, fin whales,
69; speeds, pelagic larvae, 185; squid,
69; subcarangiform, 281; surface,
384-86; surface vs. submerged, 382;
tadpoles, 145; thunniform, 282;
turtles, 137; unsteady analysis, 366; at
very low Re, 347-55; vortices in, 281,
282; wall effects, microorganisms, 339;
water beetles, 137; zero momentum
wake, 280. See also surface swimming
swooping flight, birds, 239
Synchropus picturatus (mandarin fish),
ground effect, 289
synovial fluid, as non-Newtonian, 20
syphon, 55
systematic error, 11
Systeme Internationale. See SI
tadpole: drag, 145; flattened rheophilic,
179; hygropetric, 182; Re, 146; tail as
splitter plate, 110
tar, viscosity, 18
tatter flags, 128
Taylor, Sir G. I., 353
technology, 7
temperature, 9; animal bodies, 27; vs.
blood viscosity, 28; and convective flow,
6; vs. density, 22; and dynamic
viscosity, 24, 27-31, 153; vs. kinematic
viscosity, 25
tendency to pinch: laminar flow, 315;
turbulent flow, 316
tensile strength, water, 322
463
INDEX
terminal velocity, 339; biological
significance, 343-47; ciliates, 344; diatoms,
344; and drag coefficient, 340; Eu-
phausia embryos, 30; falling object
interactions, 341-42; fecal pellets, 344;
fungal spore, 344; hemlock pollen,
344; marine algae, 185; marine
bacteria, 344; marine snow, 344;
measurement pitfalls, 342-43, 345; particle
groups, 342; phytoplankton, 344;
plankton, 88, 29; pollen, spores, 88,
186; reducing sinking rates, 344; and
seasonal polymorphism, 30; vs. shape,
340; and shape irregularity, 345;
sphere, 339; Stokes' law, 340; Stokes'
radius, 341; and surface-to-volume
ratio, 345; trout eggs, 88; very low Re,
332, 339—47; wall correction formula,
342; vs. weight, samaras, 273; why
sink?, 346
termite mound, ventilation, 71
Thalassiosira fluviatilis (diatom), 345
thermal soaring, 221-23 (fig., 222); and
atmospheric instability, 222; spiders
and moths, 222
thickness-to-chord ratio, 109, 111; body
of revolution, 134; and buckling, 134;
strut, 134
thigmomorphogenesis, 128
thrips (Thysanoptera), 350 (fig.); flight,
350
thrust, 14, 74, 262, 263 (fig.); blade
element analysis, 262; from drag, 154—
55; drag vs. lift based, 283-87;
equality with drag, 136; from flapping, 262-
64; indirect measurement, 76; jet
propulsion, 79; from lift, 137, 149, 263; at
low Re, 354 (diagram); via no-slip, very
low Re, 338; production as pumping,
323; production vs. drag reduction,
136; rotating propeller, 262
Thunnus (tuna), 282
time, 8, 9
timelines, 44
tip vortices, 232; and aspect ratio, 236;
flying bird, 276
torrential fauna, 139, 178 (figs.); dorso-
ventral flattening in order, 177; drag
in boundary layer, 176, 180
total head, 59, 61, 72, 82; in boundary
layer, 157
Toxotes (archer fish), squirting, 396
trajectory, Sordaria spore clusters, 195,
196 (fig.)
transformers, biological, 329-30
transition point: as flow quality indicator,
96; laminar to turbulent, 11
transpiration, 38; oak, 37; role of
diffusion, 197; vs. wind speed, 197
transport: diffusive vs. convective, 202;
phenomena, 197
transport cost: vs. flying speed, 240;
squid vs. trout, 79
trapping pollen, 187
tree: autorotating leaves, 229; autorotat-
ing seeds, 228; bark roughness, 100;
and continuity, 38 (fig.); drag, 14, 107,
117; drag of leaves, 100, 120-24;
elphin forest, 127; "flagged," 127;
high-altitude forms, 127; leaf area, 37;
making free convection, 221; negative
pressures, 321; pipe data, 36; sap
ascent, 37-38, 321-23;
thigmomorphogenesis, 128; trunk shrinkage, 322;
turning moments, 1231—22; vortex
shedding by trunk, 371; wind-throw,
108
tree-of-heaven, autorotating seed, 228
trench, flow across, 212
trout: Cayley's profile, 106, 107 (fig.);
cost of transport, 79; drag coefficient,
147; eggs, sinking, 88; ram ventilation,
68
Tsuga canadensis (hemlock), 344
tuliptree, autorotating samara, 228
tumbleweed, rotation in shear flow, 187
tuna, 142 (fig.); corselet, 29; Re, 86;
swimming, 282
tunicate. See ascidian
turbulent flow, 11,21, 46-49 (fig., 48),
84, 87, 290; biological relevance, 48; in
boundary layer, 159; circular aperture,
303; cross-flow transport, 161;
diffusion analogy, 47; distance index, 311;
and drag, 111; drag of flat plate, 135;
intermittent, in pipe, 375; and local
momentum, 96; and momentum
transfer, 47; noise, 95; on leaves, 163;
oscillation at transition, 375; pipes, 300-
303, 375; shear velocity, 168; speed vs.
x-sec, 301, 311; and spore liberation,
193; near surface, 95; suspension feed-
464
INDEX
ing in, 192; transition, 11, 110; vortex
shedding, 371; vortices in, 209; vs.
unsteady flow, 47; wakes, 167; wall
pinching, 316; in wind tunnels, 193
turbulent intensity, 24
turbulent transition: cylinder, 94; drag
change, 134; drag coefficient, 94; drag
and shape, 133; flat plate, 135; and Re,
85; sphere, 96
Tursiops gilli (porpoise), 151
turtle, swimming, 137, 286
typhlosole, of earthworm intestine, 311
Ulmus (elm), winged seed, 257, 251 (fig.)
undulate margin, algae, 126
units, 9-11
Universal Variable Constant. See dimen-
sionless coefficient
unsteady analysis, swimming, 366
unsteady flows, 21, 372-77; acceleration
reaction, 362-69; added mass, 362-
69; aerodynamic frequency parameter,
376; aeroelasticity, 374—75; drag and
flexibility, 375; flutter, 369;
periodically delayed stall, 376; reduced
frequency, 376; respiratory wheezes,
375; rubber tube, 374; self-excited
oscillators, 372—75; sound production,
377; sphere at turbulent transition,
375; Strouhal number, 370; turbulent
spouting in pipe, 375; virtual mass,
363; Von Karman trails, 369, 370-72;
vortex shedding, 369-74; waves, 108;
Womersley number, 377. See also non-
steady flows
unstirred layer, 200-202 (fig. 202); and
diffusive boundary layer, 201; and
mixing, 202
urethra: navicular fossa, 396; pinching
vs. expansion, 316
urination. See micturition
vein, flow through, 315
Velella (by-the-wind-sailor), 251 (fig.);
airfoil, 252; sailing, 391
Velia (bug), surfactant propulsion, 390
velocity, 8; free stream, 20; tangential,
204, 208, 224
velocity gradient, 5, 12, 19, 20, 25, 61,
72, 80, 156-71, 174; on airfoils, 244;
in ciliary pumps, 314; ciliated surface,
348 (fig.); cylinder across, 216;
diffusion across, 196-202; as dispersal
barrier, 192-96; and E-values, 120; and
ectoparasites, 180; fibroblast growth,
200; life in, 174-200; and oxygen
uptake, frogs, 200; around Paramecium,
354; between plates, 176; propulsion
through, 194; and propulsive
appendage length, 349; rotation, 188 (fig.);
settling, 183-86; from shear
measurement, 165; and soaring in vortices,
223; and suspension feeding, 188-90;
swash, 181, 219; vs. temperature, 29;
the "unstirred layer," 200-202; very
low Re, 332, 348; vortex induction,
212; and vortices, 208; vorticity, 218-
21
vena contracta, 394
vent vs. anus, 146
Venturi meter, 57-58 (fig., 58)
vessel. See pipes; sap conduits, capillary;
etc.
virtual mass, 138, 149; vs. actual mass, in
air, 368
viscoelastic solids, 20
viscometer, 17, 26-27 (fig., 27), 104
viscosity, 5, 7, 18, 21, 46, 50, 80, 156; and
airfoils, 244; and antennal trans-
missivity, 356; and Bernoulli's
principle, 61; bloods, 28; and boundary
layer thickness, 158; and circulatory
systems, 28; and drag, 63; eddy, 24;
and falling particle interactions, 342;
and filter feeding, 29; gases,
measurement, 27; glycerin, 26; hot water, 176;
ideal fluid, 52; increase with polymers,
29; measurement, 26; molecular, 24;
and Newtonian fluids, 20; and no-slip
condition, 20; and pressure-induced
flows, 72; rarefied gases, 19; sugar
syrups, 26; at very low Re, 331; and
waves, 379. See also dynamic viscosity;
kinematic viscosity
viscous entrainment, 72, 103
viscous force, and skin friction, 97
viscous sublayer, 170. See also laminar
sublayer
visualization of flow, 41, 44
volcano cone, 72
volume, specific, 22
volume flow rate, 33, 293
465
INDEX
Von Karman, T., 216
Von Karman's constant, 167
Von Karman trail, 93, 95, 96, 216, 369-
72 (fig., 370), 374; shedding frequency,
370; vortex spacing, 370
vortex ring. See vortices: rings
vortex shedding: cat's whisker, 372; and
circulation reversal, 369; frequency,
370; and Re, 369; shaking body, 369;
tree trunks, 371
Vorticella (protozoan), 191; vortex, 209,
210 (fig.)
vortices, 11,12, 204-27; of aircraft, 233
(fig.); at apertures, 303; ascending
pairs, 216; and bed erosion, 219; Be-
nard cells, 224; bird wing-tips, 276;
behind black fly larva, 217 (fig.); bound,
232; behind butterflies, 277; butterfly
flight, 287; cephalopod ink, 215;
circulation, 210, 224; clap-and-fling
mechanism, 279; in corners, 213; in cups,
214; behind cylinder, 215, 217 (fig.);
density-gradient, 224; inside droplets,
214; ends, 210, 219; examples, 213
(fig.); feeding in, 216-18, 220-21;
fictitious, 231; in fish swimming, 281;
behind flying bird, 276; and formation
flight, 288; horseshoe, 173, 219; in
ideal fluid, 209; induction in velocity
gradient, 212; interactions, 211; irrota-
tional, 206, 208-12; in jet propulsion,
79; in laminar flow, 209; Langmuir
circulations, 394; leapfrogging, 211; non-
steady, 277; pairs, 210; periodic
shedding, 93; in pipe bends, 215;
around protrusion, 219; and Re, 87,
93, 209, 216, 332; respiratory role,
218; rings, 210, 211 (fig.), 232, 276;
rolling, 223; around rotating cylinder,
208; rotational, 204, 206, 207; in
scallop beds, 219; behind self-excited
oscillator, 373; shear driven, 214;
shedding, 216, 369-74 (see also Von
Karman trail); and splitter plantes,
110; starting, 232; stopping, 232; and
streamlining, 212; near surface, 172;
in swimming, 282; thermal, 221-24;
tip, 232; toroidal, 210; in trenches,
212; in turbulent flow, 209; use in
suspension feeding, 358; from velocity
gradient, 208; viscosity and energy,
209; vortex rings, 232, 276; and vor-
ticity, 218; in wake, 93; wake of jet,
215; wakes of, 369-74; in wakes of
fliers, 275-77 (fig., 277); windrows,
393; in wing pleats, 245
vorticity, 218, 225; and circulation, 224-
25; velocity gradients, 218-21; and
vortices, 218
vulture: lift-to-drag ratio, 248, 255;
thermal soaring, 222
Wagner effect, 277
wake: energy dissipation, 97; turbulent,
167; vortices in, 93, 369-74; width and
transition, 111; width as drag
indicator, 98; zero momentum, 280
walking underwater, 140
wall effects: attractive force, 339;
correction formula, 342; drag asymmetry,
339; falling circular cylinder, 93;
induced rotation, 339; quick index to
significance, 339; swimming
microorganisms, 339; very low Re, 332, 338-39
wasp, clap-and-fling mechanism, 278
wasp, mymarid, 350 (fig.)
water: density vs. temperature, 22;
kinematic viscosity, 25, 88; properties,
table, 23; surface tension, 8; tensile
strength, 322
water boatman bug: acceleration
reaction, 365; swimming, 154
water flea. See Daphnia
water strider, 387 (fig.); communication
with waves, 392; hygropetric, 182;
propulsion, 387; use of surface tension,
389
wave, waves, 5, 22, 379-86, 390-92;
acceleration reaction for surge, 367;
amplitude, 380; bow, 382-83; capillary,
382; celerity, 381; celerity vs.
frequency, 392 (graph); at density
gradient, 394 (fig.); drag, 141; frequency vs.
wavelength, 392; gravity, 382; and hull,
383 (fig.); metachronal, 154; as moving
disturbance, 379; and mussel beds, 66;
as orbital oscillations, 379; particle
orbits, 380 (fig.); period, 380; as
resistance source, 382; stern, 382; and
unsteady flow, 108; use in
communication, 391-92; use for propulsion, 386;
vs. water depth, 380; wavelength, 381
466
INDEX
wave period, and drag of macroalgae,
125
Weber number, 347; defined, 389;
nozzles, 395
wetted area, 90
whale (figs., 70, 142); drag, 151;
ectoparasites, 180; pressure distribution,
69; propulsion from waves, 386; Re,
86, 87; shape, 383; suspension
feeding, 189, 355; swimming speed, 69; tail
as propeller, 282; thunniform
swimming, 282. See also cetaceans
whale lice, 180
wheat, flow in root vessels, 38
wheezes, respiratory, 375
whelk, swash riding, 182
white poplar, leaf fluttering, 193
willow oak, leaf as Flettner rotor, 229
wind, 6, 12, 13; and leaf temperature,
198; and plant growth, 127;
pollination, 43, 187; propagule dispersal,
344; and transpiration rate, 197; and
tree falling, 121
wind-driven circulation, 6
windmill blade; as airfoil, 268; contour,
270; operation, 269 (fig.); origin of
torque, 270; quixotic, 271
windrows, 393
wind throw, 108
wind tunnel, 7, 11, 13, 14, 15, 98,
103
wing, 21; amplitude of beating, 267;
beating, as actuator disk, 273-75;
beating, as propeller blade, 264; birds,
performance, 248; blade element
analysis, 273; bristles and fringes, 350;
finite and propulsion efficiency, 237; as
fluid dynamic pump, 327; form of
stroke, 264; infinitely long, 236;
insects, 15, 250; lengthwise twist of
beating, 264; lift production, 230-34;
mass, fruit fly, 369; momentum flux of
beating, 273-75; pitch, 242; plane of
beating, 264; power need of finite,
238; scaling, 243; stroke angle of
beating, 265, 267; surface irregularities,
245; thrust, 14, 15; twist vs. speed,
264; unsteadiness in beating, 376;
wingbeat frequencies, 268. See also
airfails
wingbeat frequency, 268
wing loading, 243-44; Andean condor,
243; bee, 243; vs. flight speed, 243;
fruit fly, 243; and hovering, 266;
pedalled planes, 243; pedalled planes,
243; as scaling problem, 243; wren,
243
Womersley number, defined, 377
worm, phoronid and terrebellid, tube
and feeding, 217
Wormaldia (snail), 40
worms, polychaete: burrow pumps, 326;
pressures of burrow pumps, 327; tube
and feeding, 217
wren, wing loading, 243
Xenopus laevis (frog), 200
xylem. See sap conduits
Young's modulus, 17
Zalophus californianus (sea lion), 150
Zanonia macrocarpa (Javanese cucumber),
256
zebra finch, body lift, 239
zero momentum wake, 280
zero plane displacement, 167-69
zooplankton, filtration, 356
Zoothamnium (sessile protozoan), rapid
stalk contraction, 360
467