1. health and fitness
2. disease
3. heart disease

AMER. ZOOL., 36:537-554 (1996)
Unsteady Mechanisms of Force Generation in
Aquatic and Aerial Locomotion1
MICHAEL H. DICKINSON
Department of Integrative Biology, University of California, Berkeley,
Berkeley, California 94720
adult leeches, all questions in aquatic locomotion
involve, to some degree, the fluid
These are exciting times for the study of
aquatic locomotion. As illustrated by the mechanical mechanisms of force producpapers within this volume, the field is ex- tion. In order to sustain the efforts outlined
panding both in breadth and sophistication. above, researchers will need to tackle ever
There are several reasons for this accelera- more complex views of hydrodynamics.
tion. First, analyses have become much Presently, the analysis of aquatic locomomore comparative, both across and within tion is dominated by "steady-state" models
species. The great diversity of teleost fishes in which the time-variant properties of
provides one of the best backgrounds for force generating mechanisms are ignored.
exploring the evolution of locomotor sys- Although researchers have made enormous
tems at both the physiological and anatom- progress applying steady-state assumptions
ical levels (see papers by Fish [1996], West- to many forms of aquatic locomotion, the
neat [1996], Gillis [1996], and Lauder and current tool kit does not seem sufficient to
Jayne [1996] in this volume). By doing so, keep pace with the diverse new directions
researchers are discovering new and diverse outlined above. In this review, I will atpatterns of locomotory behavior. In addition tempt to summarize some of the latest adto cross species comparisons, researchers vances in our understanding of unsteady
are attempting to focus on the more subtle fluid mechanics as they might apply to the
but equally important changes in swimming study of aquatic locomotion.
Many of the problems encountered in the
behavior within the ontogeny of single species (see paper by Hale [1996] in this vol- study of aquatic locomotion are identical to
ume). The second reason for an alteration those found in the analysis of animal flight.
in perspective is that researchers are not Unfortunately, the separate histories of the
simply satisfied with treating locomotion as two fields serve as obstructions to a useful
a means of getting efficiently from one and powerful synthesis. Non steady-state
point to another, but also as a complex and approaches are currently much more comflexible system of behavior required for mon in analyses of flight, due in large part
feeding, courtship, and escape. In this new to the influence of Torkel Weis-Fogh in the
perspective, maneuverability is often con- 1970s, and more recently by Charlie Ellingsidered a more important design criterion ton and his colleagues. However, many if
than energy efficiency. And finally, the not all of the unsteady mechanisms that
availability of powerful desktop microcom- have been identified in studies on insect
puters makes it possible to develop more flight are immediately applicable to forms
sophisticated mathematical models of of aquatic locomotion. Given the diverse
aquatic locomotion (see papers by Jordan landscape of locomotory behaviors displayed by fishes (Videler, 1993), the oceans
[1996] and Fauci [1996] in this volume).
may provide the most fertile ground for apWhether focused on juvenile salmon or plying the principles of unsteady fluid dynamics to Biology.
INTRODUCTION
1
From the Symposium Aquatic Locomotion: New
Approaches to Invertebrate and Vertebrate Biomechanics presented at the Annual Meeting of the Society
for Integrative and Comparative Biology, 27-30 December 1995, at Washington, D.C.
DEFINITIONS AND OUTLINE
Several excellent overviews of animal
propulsion in fluids have appeared in recent
537
538
MICHAEL DICKINSON
years (Blake, 1983; Daniel et al., 1992; Ellington, 1995; Maxworthy, 1981; Rayner,
1995; Spedding, 1992; Webb, 1975), most
of which discuss unsteady mechanisms to
some degree. In addition, Van Dyke (1982)
provides an excellent collection of flow visualizations that provides graphic exposure
to the structure and beauty of fluid wakes.
This less comprehensive review will attempt to serve as a primer for students and
researchers who (like the author) were not
originally trained in fluid mechanics, yet
find themselves inadvertently sucked into
its messy realm. In order to foster a more
intuitive sense of unsteady fluid mechanics,
I will attempt to form an explicit link between force generation and wake structure—two topics that are often discussed in
isolation. As a fish swims through water, it
generates forces with its body and creates a
distinct wake. These two processes are inextricably linked, since the forces acting on
an animal must be countered by an equal
and opposite change in fluid momentum.
Thus, there are two basic approaches to the
study of aquatic locomotion, either focusing
on the forces produced by the fins, or examining the changes in wake momentum.
Both approaches have advantages and disadvantages, but any complete understanding of fluid mechanics must integrate the
two.
culatory forces, which are sources of both
lift and drag and require the creation of vorticity. Circulatory mechanisms are dependent on the inertial properties of fluids and
are inhibited by viscosity—making them
most relevant to animal propulsion at intermediate and high Reynolds numbers. Consequently, the interesting realm of low
Reynolds numbers will not be discussed.
Using fundamental principles of fluid
mechanics and lessons taken primarily from
the insect and bird literature, I will attempt
to reconstruct the dynamics of wake structure and force generation for swimming animals. I will focus on two simple kinematic
patterns: the undulation of a single rear biofoil, and the beating of paired lateral biofoils. The first mode includes the forward
motion of a whale or killifish, while the later includes the flight of a pigeon and the
pectoral fin swimming of many fish (see
also papers by Lauden and Jayne [1996]
and Westneat [1996] in this volume). Although these are certainly rough distinctions, they will nevertheless serve to illustrate important general principles in the underlying unsteady fluid mechanics. Many
locomotory patterns, such as anguliform
undulation, do not fit easily into this simplified scheme (see papers by Jordan [1996]
and Gillis [1996] in this volume).
The studies of aerial and aquatic locomotion have independently spawned complex nomenclatures that often function to
obscure important similarities in the underlying fluid mechanics. For example, although animals use a variety of specialized
appendages for moving themselves through
a fluid medium including fins, flukes, flippers, and wings, the mechanism underlying
their functions may be identical. For the
sake of generality, I will use the term biofoil when referring to a generic biological
device that creates forces via circulatory
mechanisms. One commonly applied distinction in aquatic locomotion is that between lift- and drag-based mechanisms of
propulsion (Vogel, 1994). As discussed later, even this dichotomy blurs when applied
to unsteady flows, since both can result
from the same underlying phenomenon.
This paper is primarily concerned with cir-
STEADY-STATE MECHANISMS OF
FORCE PRODUCTION
Although the goal of this paper is to offer
a conceptual overview of non-steady state
fluid mechanics in three dimensions, it is
prudent to begin with simple steady-state
circulatory forces in two dimensions. Lift is
defined as the force that acts perpendicular
to the direction of motion. The most common method for calculating the sectional
lift generated by a biofoil is the familiar
'velocity-squared' relationship valid at high
Reynolds numbers:
L' = %CLU2pc,
(1)
where U is the velocity of the biofoil relative to the fluid, p is fluid density, c is the
chord width of the biofoil, and L' is the lift
force per unit length. The dimensionless
force coefficient, CL, is a time invariant dimensionless parameter determined by the
FORCE GENERATION IN LOCOMOTION: UNSTEADY MECHANISMS
biofoil's geometry, the Reynolds number at
which it operates, and the angle of attack
with respect to the oncoming fluid. An
identical equation is given for drag, the
force acting parallel to the direction of motion:
539
blatant problem is that lift and drag coefficients, while extremely useful terms for
comparison, are opaque parameters that offer little insight into the underlying fluid
mechanics. Specifically, they obscure the
connection between force production and
wake structure—an essential interface for
2
D' = %CDU pc,
(2) developing a useful intuition of unsteady
where CD is the dimensionless drag coeffi- fluid mechanics and testing models of locomotion.
cient.
Since dimensionless coefficients are
The alternative and, for present purposes,
time-invariant, the forces generated by bio- more useful formulation of steady-state
foils during locomotion can be derived force production incorporates the important
solely from kinematics and some indepen- concept of circulation. Figure 1 illustrates
dent measure of CL and CD—which is typ- the canonical view of the flow around a bioically performed by studying excised or foil under steady-state conditions at a low
model biofoils in a wind tunnel or flow angle of attack. The oncoming flow sepatank. This approach is called "quasi-steady rates around the biofoil at an anterior stagstate" or "blade element" analysis, and has nation point on the underside, but flows
been used extensively in the study of ani- smoothly off a rear stagnation point that is
mal locomotion (Ellington, 1984c). The located precisely at the trailing edge. The
most straightforward use of this method be- asymmetrical locations of the forward and
gins by determining how lift and drag co- rear stagnation points are only possible if
efficients vary with angle of attack. Next, a fluid velocity is greater above the biofoil
researcher must gather kinematic data and than below. This velocity differential is
reconstruct the velocity and angle of attack most conveniently expressed as a net cirof the biofoil(s) at a series of discrete time cular flow or circulation around the biofoil.
points within each locomotory cycle. The Mathematically, the circulation is quantified
two data sets are then combined using Eqs. by solving a line integral of velocity around
1 and 2 to calculate the forces generated at a closed loop (any closed loop) enclosing
each stroke position. This may be done by the biofoil. Circulation is thus a scalar with
considering average values along the bio- units of velocity times distance or L2T~'.
foil, or by treating the biofoil as a number By Bernoulli's equation of energy conserof thin sections and integrating along the vation, an increase in velocity is concomitotal length. Finally, the net force generated tant with a decrease in pressure, and thus
throughout the stroke is found by averaging the higher the value of circulation, the
the forces determined over the entire stroke greater the velocity differential and the
greater the resultant pressure force sucking
cycle.
the biofoil upward. This relationship beThere are many problems with the quasi- tween the upward lift force and circulation
steady state analysis (Ellington, 1984c). is expressed in the Kutta-Joukowski equaThe simplest and most fatal deficiency is tion (Milne-Thomson, 1966):
that force coefficients are not constant, but
rather display complex time histories. Thus,
(3)
the performance of a biofoil measured un' = P ur.
der steady-state conditions in a wind tunnel where L' is the lift per unit span, p is the
may bear little resemblance to its perfor- density of the fluid, and F is the magnitude
mance while being flapped and rotated by of circulation. We can also derive an exan animal. The main experimental hurdle is plicit relationship between circulation and
to reconstruct the full 3-dimensional kine- the more familiar lift coefficient by commatics of biofoil motion, which is extreme- bining Eqs. 1 and 2, which yields:
ly difficult to accomplish accurately using
the 2-dimensional view of a single imaging
C - ^
(4)
device. (Gibb et al, 1994). Another less
540
MICHAEL DICKINSON
FIG. 1. Steady-state theory of lift. (A) The velocity of fluid flow over the top surface of a biofoil is greater
than that below. By subtracting away the background symmetrical velocity field, this differential can be expressed
as a net circular flow or circulation around the biofoil. (B) The Wagner effect. When a biofoil begins translating
from rest, it takes several chord lengths of travel for the bound circulation to reach steady state levels (indicated
by line thickness). This sluggishness is due to the proximity of the starting vortex. The starting vortex is required
by Kelvin's Law so that the net circulation within the fluid remains zero. (C) At the end of translation the bound
circulation of a biofoil is shed as a stopping vortex, which will be equal and opposite in strength to the original
starting vortex.
Expressed in this way, the lift coefficient
can be viewed as a dimensionless form of
circulation.
The tangential flow at the trailing edge is
termed the Kutta condition, and results
from viscosity which tends to equilibrate
the velocity of the fluid above and below
the biofoil as it approaches the trailing
edge. This phenomenon illustrates the importance of viscous effects within the
boundary layer, the region that surrounds
all fluid-solid interfaces. Even at high
Reynolds numbers where fluids are often
described as 'inviscid,' boundary layers and
viscous properties are critical in determining flows and forces produced by biofoils.
The importance of the Kutta condition also
explains the sharp trailing edges of biofoils,
which provide a long and gradual trajectory
over which the separated fluid may approach a common velocity.
Another important and useful concept for
discussion of both steady and unsteady fluid
mechanics is that of vorticity. The circulation around a biofoil may be represented as
a point vortex, which acts as an infinitely
thin cylinder spinning in the fluid at a constant rate. Because of the fluid's viscosity,
the thin spinning cylinder induces a circular
flow in the surrounding medium. This induced velocity, u,, at a distance, r, from the
center of a point vortex is given by:
Us = KT 1 ,
(5)
where K is the strength of the vorticity. The
direction of the induced velocity is always
tangent to the family of concentric circles
centered on the point vortex. The circulation associated with a point vortex is found
by taking any line integral of the velocity
along a loop that encloses the vortex, which
reduces to:
FORCE GENERATION IN LOCOMOTION: UNSTEADY MECHANISMS
T = 2ITK.
(6)
541
ping vortex is precisely equal and opposite
to that of the original starting vortex.
In accordance with Kelvin's Law, the
motion of a biofoil in two dimensions at a
low angle of attack may be modeled as two
point vortices: one that represents the
bound circulation of the biofoil, and the
starting vortex created at the onset of translation. For an infinite biofoil in three-dimensions, these point vortices become infinite vortex filaments. In order to construct
the wake of a finite biofoil it is necessary
to consult another conservation law, the
Second Theorem of Helmholtz, which determines the behavior of vortex filaments
(Milne-Thomson, 1966). According to
Helmholtz' Theorem, a vortex filament cannot come to an abrupt end within a fluid,
and therefore must either terminate at either
a solid-fluid interface or join with its other
end to form a continuous ring. Therefore,
as a biofoil moves along at a constant velocity, it creates a complete vortex loop
(Fig. 2B). One segment of the vortex loop
consists of the bound vorticity of the biofoil
itself, while the opposite segment consists
of the starting vortex filament. The bound
and starting vortices are linked together by
two tip vortices that extend from the two
ends of the biofoil.
Helmholtz' first theorem states that the
intensity of vorticity is equal at all points
along a vortex filament or loop. Thus, as a
biofoil moves through the fluid at a small
angle of attack, the vortex loop grows in
size but not in strength. The orientation of
the vorticity is such that the velocity of fluid is downward through the center of the
loop. The momentum of a continuous vortex loop, Mv, is given by (Milne-Thomson,
1966):
Because of this simple relationship between
circulation and vortex strength, the terms
are often used interchangeably to describe
fluid flows. Strictly speaking, however, vorticity is a vector quantity, co, that is denned
as the curl of fluid velocity at a particular
point. The vorticity at a point vortex in the
x-y plane is a vector oriented in the z direction and has a magnitude equal to K.
One advantage of expressing lift in terms
of circulation is that it directly relates the
pressure force exerted on the biofoil to the
magnitude and direction of fluid flow. The
extension of this concept to unsteady flows
in three dimensions will help to make a
more explicit connection between force
production and wake structure. Before forging this link, it is first necessary to consider
a series of fundamental conservation laws
concerning the behavior of fluids. The first
is Kelvin's Law, which states that the net
circulation within a fluid system cannot
change (Milne-Thomson, 1966). Before a
biofoil starts to move, it generates no circulation. At the onset of motion, it begins
to produce bound circulation which, by
Kelvin's law, must be accompanied by additional circulation nearby that is of precisely equal and opposite strength. As the
movement begins, this counter circulation
is manifest in a free starting vortex that develops near the original resting position of
the biofoil (Fig. IB). The fact that total circulation is always zero creates an apparent
paradox that might seem incompatible with
the production of circulatory forces by Eq.
3. If the total circulation is zero, why is not
the total force zero as well? The paradox is
resolved because the effect of a vortex on
the pressure distribution around a biofoil
depends critically on its proximity. The vor(7)
Mv =
ticity and resultant circulation that is bound
to the biofoil exerts an influence on velocity where p is the density of the fluid, A is the
and pressure throughout translation. The in- area of the loop, and Y is the circulation
fluences of the starting vortex, on the other around the ring filament (Fig. 2A). The
hand, becomes infinitesimally small as the force associated with the formation of a
biofoil moves farther away. Kelvin's law is vortex ring is found by differentiating Eq.
also manifest when a moving biofoil comes 7 to yield:
to rest (Fig. 1C), at which point all the
bound circulation of the biofoil is shed as
F, = p-(TA).
(8)
a free vortex. The magnitude of this stop-
542
MICHAEL DICKINSON
c
M =pTA
v v
B
F v =pr
dA.
starting
vortex
bound
vortex
tip
vortex
FIG. 2. Forces exerted on biofoils are matched by a corresponding change in wake momentum. (A) The
momentum of a vortex ring is given by the product of ring area, fluid density, and the circulation around the
loop filament. (B) As a biofoil moves, the area within the resultant wake loop increases at a constant rate.
Differentiation of the equation for loop momentum gives the corresponding upward force exerted on the biofoil.
(C) Since the change in loop area with respect to time is equal to velocity, U, times biofoil length, R, the
differentiation of wake momentum returns the Kutta-Joukowski Theorem.
If the strength of circulation is constant,
then the above equation reduces to:
as a concentric series of many vortex
rings—or even more accurately—as a continuous sheet of vorticity. Nevertheless, it is
Fv = pfUR
(9)
important to recognize that the blade elesince the change in loop area with respect ment and wake momentum approaches reto time is equal to the product of biofoil ally do meet in the middle. Force generalength, R, and the velocity, U, at which it tion by fins, flippers, or wings requires a
moves (Fig. 2B). Dividing both sides by R change in fluid momentum that could be
returns the familiar Kutta-Joukowski equa- quantified as a change in wake structure.
tion:
Under steady-state conditions the rate of
momentum transfer is constant, which
(10) means that the force encountered by the
biofoil is constant, but that the size of the
Thus, the forces generated by an biofoil can vortex ring increases at a steady rate.
So far, this description has only briefly
be related directly to the strength and structure of the resultant wake. I have made a mentioned the drag of a biofoil, which has
number of implicit assumptions in the three independent components: skin friccourse of this derivation. For example, the tion, induced drag, and pressure drag. Skin
circulation along the wing was assumed friction results from viscous shear acting on
constant, when in reality circulation is the surface of the biofoil. The drag coeffigreatest around the center of the airfoil and cient due to skin friction may be estimated
lowest at the ends. In order to satisfy Helm- from the following relationship (Schlichtholtz' Theorem, the wake must be modeled ing, 1979):
pru
I- -
FORCE GENERATION IN LOCOMOTION: UNSTEADY MECHANISMS
2.66
VRe'
(11)
Since skin friction is due to viscosity, not
vorticity, its influence attenuates quickly
with increasing Reynolds numbers.
Induced drag results from the altered velocity and pressure distribution around a
hydrofoil caused by the vorticity within the
wake. As seen in Figure 2B, the tip vortices
induce a downward velocity or downwash
that alters the direction of flow around the
biofoil. This alteration of the velocity
changes the pressure distribution so that the
lift vector tilts slightly backward. The rearward component of the resultant force vector is the induced drag. Using the standard
assumptions of thin airfoil theory, the drag
coefficient due to induced drag may be estimated as (Kuethe and Chow, 1986):
r Dl
-£k.
TTM
(12)
where the aspect ratio, M, is the ratio of
wing length to chord length (R/c).
The third component of total drag, pressure drag, is caused by the separation of
flow from the top surface of the biofoil. The
flow separation results in an underpressure
that pulls the biofoil rearward. At angles of
attack less then 15°, where steady-state assumptions are valid, pressure drag is quite
small and is often ignored. Above 15°, however, pressure drag increases rapidly with
increasing angles of attack. Under steadystate conditions, the mechanism giving rise
to lift and pressure drag are usually considered distinct. As described in a following
section, however, this is not generally the
case for unsteady conditions, and we may
consider lift and drag to be two components
of a single circulatory force.
UNSTEADY MECHANISMS OF
FORCE PRODUCTION
The underlying principle of steady-state
mechanics is that the bound circulation of
a biofoil is time-invariant as long as velocity and wing geometry do not change. In
most forms of animal locomotion, however,
biofoils reciprocate back and forth—all the
while changing their velocity and angle of
attack. Under these conditions force pro-
543
duction will not be steady, and the wake
will probably not form a simple single ring
structure. This does not mean that steadystate approaches are necessarily invalid for
biological systems, but any analysis that excludes unsteady mechanisms is turning a
blind eye to a long list of potentially crucial
phenomena.
Two dimensionless parameters provide
some simple indices of whether or not unsteady mechanisms may be important for a
given locomotory behavior. The first is advance ratio, which is the velocity of the
body divided by the velocity of the propulsive biofoil. For an appendage of length R,
flapping with amplitude <J>, at frequency f,
the advance ratio, J, is given by (Ellington,
1984a):
J =
U
(13)
where U is the forward velocity of the
body. This same concept is also commonly
expressed (inversely) as the reduced frequency parameter, <j (Daniel and Webb,
1987):
a =
23>fR
U '
(14)
These two parameters quantify how much
of the fluid flow past a biofoil is due to its
own flapping, and how much comes simply
from the forward motion of the body. At
large J (and low cr) the situation approaches
gliding, and steady-state mechanisms may
be sufficient to account for force generation. At low J (and largeCT),the fluid velocity is due to the continual back and forth
oscillation of appendages that are unlikely
to be under steady-state conditions. Reduced frequency and advance ratio thus offer a quick and dirty assay for the relative
importance of steady-state and unsteady
mechanisms. However, this method is filled
with many implicit assumptions, and must
be viewed with great caution.
Before delving into the complexities of
force generation and wake dynamics associated with unsteady flow, it is necessary to
consider a few key unsteady phenomena
that, again, are best introduced in two dimensions. One important unsteady effect
544
MICHAEL DICKINSON
concerns the growth of circulation when a
biofoil is started from rest. As was discussed above, Kelvin's Law requires that
the bound circulation of a biofoil is balanced by an equal and opposite starting vortex (Fig. 1). At steady-state, this starting
vortex may be safely ignored because it
does not influence the flow over the moving
biofoil. However, the starting vortex does
affect circulatory lift at the onset of translation when it is still close to the biofoil.
Since the starting vortex and the bound vorticity of the biofoil induce fluid velocity in
opposite directions, they reduce each others
strength (Fig. 1A). This reduction in circulation over the first several chord lengths
of travel results in a diminution of lift. This
phenomenon is termed the Wagner effect,
after the German aerodynamicist who first
predicted it in 1931. The influence of the
starting vortex falls off with its separation
from the biofoil. After about 6 chord widths
of travel, the separation is large enough that
the lift approaches its steady-state value.
The Wagner effect has limited importance
for the aeronautics industry, since airplane
wings move hundreds of chord lengths by
the time they reach the end of the runway.
In animal locomotion, however, a biofoil
may only travel a few chord lengths before
stopping and reversing direction. For this
reason, the Wagner effect probably limits
the development of circulatory forces in
many forms of animal locomotion (Ellington, 19846).
FIG. 3. Impulsive start of a biofoil at a high angle of
attack. The rapidly increasing circulation does not remain bound to the biofoil, but rather forms an attached
vortex on the leading edge. As translation proceeds, a
small trailing edge vortex forms as the leading edge
vortex continues to grow. The leading edge vortex is
eventually shed from the biofoil as von Karman shedding commences. Kelvin's Law is maintained throughout. The figure was traced from an actual flow visualization of a model biofoil at a Reynolds number of
192 (Dickinson and Gotz, 1993).
biofoil, it temporarily reattaches to the upper surface in front of the trailing edge to
form a leading edge bubble that encloses a
vortex (Fig. 3). The resultant circulation is
no longer 'bound' as it is under steady-state
conditions, since the rotational center of the
vorticity enclosed within the bubble does
Another important unsteady effect not lie within the center of the biofoil. Nevemerges from considering an impulsive ertheless, the presence of a strong attached
start at a high angle of attack (Fig. 3). Most vortex still results in lift by creating an unman-made wings or hydrofoils are designed derpressure on the top surface of the bioto function at angles of attack less than foil. For animals that oscillate their biofoils
about 15°. Above a critical angle, the flow back and forth, the phenomenon of delayed
separates off the trailing edge increasing stall may have enormous importance. A
pressure drag and attenuating circulatory biofoil that moves only a few chord lengths
lift—a process that is collectively termed in a given stroke could generate lift of
stall. However, the phenomenon termed de- much greater magnitude than could ever oclayed or dynamic stall follows from the fact cur under steady-state conditions (Fig. 4).
that this separation does not occur instan- This mechanism can more than offset the
taneously. Consequently, a biofoil can tran- deleterious effects of the Wagner effect, and
siently produce circulatory forces that are probably explains why animals as different
much greater than those possible at steady as fruit flies and sunfish use extremely high
state. Delayed stall is often associated with angles of attack (Gibb et al, 1994; Zanker,
the formation of an attached vortex. As the 1990).
flow separates from the leading edge of the
Force generation by delayed stall at high
545
FORCE GENERATION IN LOCOMOTION: UNSTEADY MECHANISMS
2.5
2
ooo° o
1.5
V4-4
§ 1
o
•
•
.
0.5
• o
0
0 10 20 30 40 50 60 70 80 90
angle of attack (degs)
FIG. 4. Delayed stall can greatly augment the magnitude of circulatory forces. The instantaneous lift coefficient during an impulsive start of a model biofoil
at a Reynolds number of 192, is plotted against angle
of attack. The open circles show the lift coefficients
measured 2 chord lengths after the start of translation
when a large leading edge bubble was still attached to
the model biofoil. The dark circles indicate the values
after 7 chord lengths of translation, approximating
steady-state conditions. Notice that the curves diverge
at angles of attack over 15°, the point at which the
flows become unsteady. Data are replotted from Dickinson and Gotz (1993).
angles of attack differs in many critical
ways from steady-state mechanisms at low
angles of attack. In conventional steadystate fluid mechanics, lift and pressure drag
are considered to be cleanly separable phenomena. Lift, acting perpendicular to flow,
results from bound circulation, while pressure drag results from separation of flow
from the top surface of the biofoil. As illustrated in Figure 5, this phenomenological
distinction is not valid at high angles of attack. The under pressure created by an attached vortex will produce a force that acts
roughly perpendicular to the plane of the
biofoil, not the direction of motion. Consequently, at a 45° angle of attack, an attached leading edge vortex will contribute
about equally to lift and drag. The two forces are therefore manifestations of exactly
the same fluid mechanics phenomenon. For
this reason, it may be misleading to distinguish between "lift-based" and 'dragbased' modes of propulsion at high angles
of attack. Rather than dealing with separate
lift and drag coefficients, it is simpler to
define a total circulatory force coefficient,
CT, that is related to the conventional force
coefficients by:
(15)
CT = V(C L 2 + CD2).
The orientation of the total circulatory force
with respect to the surface of the biofoil, 6F,
is given by:
9FT = a + a r c t a n f ^ |,
(16)
B
01
3.5
3
2.5
2
1.5
1
0.5
210
'S- 180
3
•'
150
a> 120 . -
oL'
0
•
90
30
60
a (degs)
90
0
30
60
9
a (degs)
FIG. 5. Lift and drag forces are components of a single circulatory force that acts perpendicular to the surface
of a biofoil. (A) At an angle of attack, a, the total circulatory force is oriented at an angle, 8F,, with respect to
the surface of the biofoil. (B) The magnitude of the instantaneous total circulatory force as a function of angle
of attack. The forces were measured after two chord lengths of motion following an impulsive start at a Reynolds
number of 192. Notice that the magnitude increases linearly to angles of attack as high as 60°, where it then
plateaus. (C) At angles of attack above 15°, the circulatory force is almost perfectly normal to the surface of
the model biofoil. Data are replotted from Dickinson and Gotz (1993).
546
MICHAEL DICKINSON
where a is the angle of attack. The behavior
of the total force is rather more intuitive and
straightforward than lift and drag. As indicated in Figure 5, The total force coefficient
rises linearly with angle of attack from 0 to
60°. At angles of attack above 10°, the total
force is oriented almost exactly at 90° with
respect to the surface of the model biofoil.
This is not surprising since pressure forces
act perpendicularly on solid surfaces.
The fact that unsteady lift and drag are
caused by the same mechanisms is also
manifest in their dependence on Reynolds
number. Lift coefficients should rise with
increasing Reynolds numbers since circulation is supported by fluid inertia and attenuated by fluid viscosity. In contrast, one
of the great icons of steady-state fluid mechanics is the monotonic decline in drag coefficients with increasing Reynolds number
(Schlichting, 1979). However, the instantaneous drag coefficients measured during the
process of delayed stall actually grow in
parallel with lift coefficients as Reynolds
number increases (Dickinson and Gotz,
1993). The effect is expected, since under
unsteady conditions lift and drag are simply
arbitrary components of a single circulatory
force caused by an attached vortex. The increasing strength of vorticity at higher
Reynolds number will augment the strength
of attached vortices, resulting in an elevation of both and lift and drag.
Given enough distance, a biofoil translating at a high angle of attack will eventually stall. However, within a biologically
relevant range of Reynolds number (from
about 5 to 200,000), the process of stall itself is quite dynamic. By Kelvin's Law, the
vorticity generated by the moving biofoil
must be matched by an equal and opposite
starting vortex. At a low angles of attack,
the bound circulation grows asymptotically
to a constant value, and all the counter-circulation resides in the starting vortex (Fig.
IB). The essential feature of unsteady flow
at high angles of attack is that the circulation is not stable, but rather continues to
grow throughout translation. At the start of
translation at a high angle of attack, an attached vortex grows on the leading edge
(Fig. 3). As the biofoil pulls away from the
starting vortex, this growing vorticity is bal-
anced by the formation of another attached
vortex that forms on the trailing edge. As
this trailing edge vortex grows, the center
of the original leading edge vortex moves
farther from the biofoil and is eventually
shed into the wake. As the trailing edge
vortex grows, a new leading edge vortex is
formed, in accordance with Kelvin's Law.
The process continues, producing a long
chain of alternating vortices that is called a
von Karman street. Are von Karman streets
important for aquatic locomotion? In most
reciprocating forms of locomotion, biofoils
move over too short a distance to ever generate a full-blown von Karman street. Periodic vortex shedding probably does occur
from the bodies of swimming organisms,
however, and these vortices may influence
force generation on more posterior appendages (Ahlborn et al., 1991). In any event, a
familiarity with von Karman wakes is essential for developing an intuitive sense of
unsteady flows. For example, the 'constant'
steady-state forces measured on a biofoil at
angles of attack greater than about 15° are
actually the time-averages of oscillatory
forces caused by von Karman shedding.
Now we may extend these unsteady phenomenon to three dimensions by considering the force generated and wake formed
by a finite biofoil. As with a low angle of
attack, the translating biofoil will generate
a vortex ring consisting of the starting vortex connected to two tip filaments. At high
angles of attack, however, the vortex along
the biofoil may be attached, rather than
bound. From the work of Charlie Ellington
and his colleagues (Ellington, 1996; Willmott, 1995) it seems that leading edge bubbles grow more slowly on three-dimensional biofoils than one would predict from
two-dimensional models. This delay is
probably caused by extensive length-wise
flow along finite biofoils. Although this
lengthwise flow reduces the magnitude of
circulation, it may also extend the distance
over which force is generated before the
leading edge vortex detaches from the biofoil. In any event, the circulatory forces
generated during a short translation at high
angle of attack can be much greater than
could be stably produced under steady-state
conditions. During a single stroke started
FORCE GENERATION IN LOCOMOTION: UNSTEADY MECHANISMS
from rest, the benefits of delayed stall outweigh the detriments of the Wagner effect.
At the end of the translation, the vorticity
along the biofoil is shed as a stopping vortex filament, and the entire ring is free within the fluid. While translation is occurring,
the relationship between instantaneous
force production and wake structure is still
given by Eq. 8, but modified for time variant circulation as:
(17)
Thus, in order to recreate the instantaneous
forces from the structure of the wake under
unsteady conditions, it is necessary to measure both the change in ring area and change
in ring circulation with respect to time.
Rotational circulation
So far this discussion has only considered
circulation created by linear translation of a
biofoil. However, just like the spinning cylindrical "sails" on Flettner's famous boat,
a biofoil can also generate circulatory forces
through angular rotation (Ellington, 1984fo).
Kelvin's Law is still in play, so any change
in rotational circulation must be accompanied by counter circulation nearby. For example, if a biofoil rotates impulsively from
rest, the bound circulation of a rotating biofoil is accompanied by a free rotational starting vortex, and when rotation ceases, the
biofoil will shed a rotational stopping vortex. Rotational circulation is important in biological fluid mechanics because reciprocating biofoils often undergo a rotation during
each stroke reversal. Depending upon the
precise kinematic conditions, the rotational
circulation may either enhance or hinder the
generation of forces during the subsequent
stroke (Dickinson, 1994).
BASIC KINEMATIC PATTERNS
OF LOCOMOTION
Locomotion with single posterior
appendage
From the palette of simple unsteady
mechanisms described in the last section we
can begin to consider the unsteady force
generation and wake dynamics in real animals. Many fish swim by reciprocal undu-
547
lation of a posterior tail fin (Videler, 1993;
Webb, 1975). In this pattern, the tail moves
a few chord lengths at a high angle of attack, stops, rotates slightly, and repeats the
motion in the opposite direction. During
each stroke, the tail develops a force acting
normal to its surface while generating a
vortex ring (Fig. 6A). At the end of each
stroke, the tail will deposit the old vortex
ring into the wake, reverses direction, and
starts to develop circulation in the other direction. Thus, each complete stroke cycle
will produce a pair of vortex rings moving
in opposite directions. After several undulatory cycles, the wake will consists of a
series of rings that continue to translate
through the fluid under their own induced
velocity, thus spreading the wake laterally
(Fig. 6B). This generalized wake is very
similar to that observed behind slowly
swimming danios (Rosen, 1959; McCutcheon, 1977). Under these conditions, the
highest forces should be produced during
the stroke as the vortex ring grows in size
and strength. As long as the rotation of the
tail during stroke reversal is slow with respect to the lateral translation, the contribution of rotational circulation during
stroke reversal should be small. At faster
speeds, the stopping and starting vortices of
successive strokes may fuse, forming a
linked chain (Rayner, 1995).
Locomotion with paired appendages
Many animals in air and water create locomotory forces using a bilateral pair of appendages. Predicting the forces and wake
created by animals using paired undulating
appendages is extremely difficult, but extensive work on the flight of birds, insects,
and bats can serve as models for aquatic
locomotion using pectoral fins and nippers.
In addition, Freymuth (1990) has experimentally investigated the forces and wakes
generated by a reciprocating biofoil in an
elegant physical modeling study. Figure 7A
illustrates the wake that results when two
paired biofoils beat with a large stroke amplitude separated by only a spindly body.
At the start of each stroke, the two appendages move apart without creating starting
vortices. This peculiarity does not violate
Kelvin's law, because the bound circulations
548
MICHAEL DICKINSON
A
B
oo
oo
oo
oo
°o
FIG. 6. Cartoon illustration of vortex wake expected behind a fish swimming with a reciprocating tail fin. The
sequences starts at the bottom and moves to the right. (A) Each lateral stroke of the tail creates a single vortex
loop, seen in cross-section as a pair of starting and stopping vortices. The direction of the total circulatory force
is indicated by the biofoil outline drawn to the left of each figure. (B) After many stroke cycles, the fish creates
a wake of alternating vortex pairs, each moving laterally under their own induced velocity.
of the two biofoils are equal and opposite. Maxworthy, 1979). During the fling, a tip
This trick of making the starting vortex of vortex forms connecting the bound or atone biofoil be the bound vortex of the other, tached vorticity of the two appendages,
is one advantage of the fling mechanism that thereby creating a horseshoe-shaped vortex
was first described by Weis-Fogh (1973) for loop attached to the body. At the end of the
tiny insects, but may also be important in stroke, the two biofoils come together, shed
many forms of aquatic locomotion. Another their vorticity in a stopping vortex. During
advantage of the fling occurs because the this process, the tip filaments fuse to combiofoils pull apart first along their leading plete a circular vortex loop.
edges. This rotation develops circulation beAfter a sufficient pause and rotation, the
fore the start of translation and thus coun- process would repeat again in the opposite
teracts the Wagner effect to augment the pro- direction. If the pause between strokes is
duction of force during the subsequent sufficient, a complete upstroke and downstroke (Bennett, 1977; Ellington, 1995; stroke cycle would create two vortex loops,
FORCE GENERATION IN LOCOMOTION: UNSTEADY MECHANISMS
549
FIG. 7. Cartoon illustration of the vortex wakes behind an imaginary fish swimming with pectoral fins. (A)
When using a large stroke amplitude, the two fins will make a single large vortex loop during the downstroke.
If the fins clap at the end of the downstroke, another loop may form during the upstroke. (B) If the wings do
not clap at the end of the downstroke, circulation may be inhibited by the cumulative Wagner effect and no
upstroke loop will form. After shedding, the ends of the vortex filaments may slide along the body before fusing
together at the tail. (C) The 'double-loop' wake resulting from the kinematics shown in A. (D) The 'singleloop' wake resulting from the kinematics shown in B. (E) The linked chain wake that results from the fusing
of stopping and starting vortices in a double-loop system (after Brodsky, 1994). (F) The wake resulting from
low stroke amplitude kinematics, in which each wing makes its own vortex system. This mode of kinematics
may be beneficial for maneuverability.
550
MICHAEL DICKINSON
similar to the wake behind a reciprocating
tail fin, except that the successive vortex
rings would be oriented in the same direction (Fig. 7C). Experimentally, detection of
two rings in each full stroke cycle is instructive, since it indicates that the wings
generate circulatory forces during both half
strokes and that the sign of circulation
changes around the biofoil during each
stroke reversal. The wake pattern many
change, however, if the pause between
strokes is short. The stopping vortex of each
stroke has the same rotational orientation as
the starting vortex of the next stroke (Fig.
7E). Consequently, the stopping and starting
filaments may fuse in a combined "stopping-starting" vortex, linking successive
vortex rings together. Brodsky (1991, 1994)
first described this linked vortex chain in the
wakes behind flying insects.
The cumulative Wagner effect
In order to reverse the sign of circulation
and the direction of resultant forces, the
biofoil must undergo an extensive rotation
during each stroke reversal. The bound circulation of this rotation has the same sign
as the previous translational circulation, and
might possibly augment force production
during the last portion of each stroke (Fig.
8A). Once shed, however, the rotational circulation has the same orientation as the
stopping vortex of the previous stroke and
the starting vortex of the next stroke. Consequently, unless the pause between strokes
is long, each stroke reversal may produce a
large combined 'stopping-starting-rotation'
vortex that could have a profound influence
on force generation during the next stroke.
As the biofoil begins to translate in the opposite direction, a cumulative Wagner effect
will operate, since the biofoil now must
fight the large counter vorticity of the combined vortex (Dickinson and Gotz, 1996).
An experimental illustration of the cumulative Wagner effect is shown in Fig. 8B.
The magnitude of this effect will depend
critically on the duration of the pause between half strokes, the speed of rotation,
and the axis of rotation during stroke reversal (Dickinson, 1994).
At the start of each stroke, paired biofoils
may have to start their translation under the
o
B
2.5
2
1.5
1
0.5
0
-0.5
-1
impulsive start
0 1 2 3 4 5 6 7 8
chord lengths
FIG. 8. The cumulative Wagner effect. (A) Before
stroke reversal, a biofoil translates with an attached
leading edge vortex. At the start of the next stroke, the
large stopping vortex combines with the new starting
vortex and hinders the development of translational
circulation. For simplicity, the contributions of rotational circulation have not been included. (B) Experimental verification of the cumulative Wagner effect.
The instantaneous lift coefficient is plotted for 7.5
chords of motion at a 22.5° angle of attack (Reynolds
number of 192). The large peaks at the beginning and
end of each trace are inertial transients. The lift is
greatly attenuated over the first 3 chord lengths of travel if translation is proceeded by a reverse stroke and
rotation (return stroke), compared to that generated following a start from rest (impulsive start). The lift following rotation eventually recovers, and attains an
even higher degree of delayed stall, perhaps because
the circulation develops more slowly. However, it is
the first 3 chord lengths of travel that are most relevant
to biological locomotion. Data are replotted from
(Dickinson, 1994).
influence of a cumulative Wagner effect. If,
however, the wings could be brought close
enough together, their shed stopping vortices might rapidly annihilate one another,
providing a clean slate for the subsequent
stroke. This might be one function behind
the clap behavior that was also described
by Weis-Fogh (1973) for small insects. Together, the clap and fling kinematics provide mechanisms for more efficiently
changing the sign of circulation during
stroke reversal (Ellington, 1995). If the biofoils close first at their leading edges, the
FORCE GENERATION IN LOCOMOTION: UNSTEADY MECHANISMS
clap would also squeeze fluid rearward in a
jet providing an additional source of momentum exchange (Gotz, 1987).
The efficiency of the clap and fling depends critically on the spatial separation between the pair of reciprocating biofoils. In
many animals it is morphologically impossible to clap the biofoils together during one
or both of the two stroke reversals. Since
the circulation generated during one stroke
interferes, via the cumulative Wagner effect, with the generation of circulation in
the next stroke, it may be advantageous under certain conditions to limit the production of circulatory force to one of the two
strokes (Fig. 7D). Indeed, the wakes of
many animals, including many birds and insects, consist of only a single vortex loop
in each stroke cycle, indicating that they
limit the production of circulatory lift to the
downstroke. (Dickinson and Gotz, 1996;
Grodnitsky and Morozov, 1992; Grodnitsky
and Morozov, 1993; Kokshaysky, 1979;
Spedding, 1986; Spedding et al, 1984). For
many of these animals, the wings come
close enough for a clap and fling during the
extreme upstroke position, but are separated
by a large angle during the extreme downstroke position. While the development of
circulation at the start of the downstroke is
aided by the fling, the situation is quite different at the start of the upstroke. Without
the clap to annihilate the vorticity on the
two wings, the development of upstroke circulation is inhibited by the cumulative
Wagner effect. The circulatory forces during the following upstroke are therefore
small and little vorticity is shed into the
wake. On the other hand, the absence of
upstroke circulation is beneficial for force
generation during the downstroke, because
the cumulative Wagner effect will be greatly attenuated. In any locomotory system
with paired biofoils, the inhibitory interactions between the vorticity generated by
successive strokes probably plays a central
role in the evolution of kinematic patterns.
The situation is quite different for a single
reciprocating tail fin where the forces generated by two successive strokes must be
symmetrical. In this case, the animal cannot
alter its kinematics to favor one stroke over
the other.
551
If the stroke amplitude during hovering
is small and there is a large angle between
the paired biofoils at both their ventral and
dorsal extremes, then the vorticity created
by the two biofoils will remain separate.
Under these conditions, Kelvin's law will
be satisfied separately on each side of the
body, and each biofoil will create its own
wake consisting of stacked vortex rings
(Fig. 7F). This double ringed structure has
been identified behind a hovering crane fly
by Brodsky (1991). One would expect the
forces generated by the biofoils during low
stroke amplitude hovering to be comparatively small, because the animal cannot exploit the advantages of the clap and fling
during either stroke reversal. For this reason
there seems little to gain from utilizing low
stroke amplitude kinematics during hovering. However, in many cases maneuverability, and not power efficiency, may exert
the strongest selective pressure in the design of locomotory behavior. By uncoupling the fluid mechanical interactions, the
forces generated by one biofoil become independent of forces generated by the other.
By exploiting this independence, an animal
may produce much more sophisticated motions, enabling it to dodge in and out of
coral heads or around flower stems. In this
regard, it is worth noting that the hoverflies
(which have my vote for the most maneuverable group of organisms on the planet),
are noteworthy in their use of low stroke
amplitude hovering.
Much of the work on hovering has concerned insects and birds that must continuously generate a downward aerodynamic
force to overcome the force of gravity. In
contrast, fish using paired pectoral fins are
mentally buoyant which may free the paired
biofoils for more exclusive use in maneuvering. Another reason that fish may be relatively more maneuverable than flying animals results from their laterally compressed body morphology. While birds and
insects have relatively thin and spindly bodies compared to their wings, the pectoral
fins of many fish are separated by a "wall"
formed by a broad laterally-compressed
body. The presence of a large flat body has
several potentially interesting consequences
for the mechanisms of force production. A
552
MICHAEL DICKINSON
flat body would enhance maneuverability,
since it would physically separate the hydrodynamics of force production by the two
fins. Nevertheless, the animal could still exploit the advantages of the clap and fling to
counteract the Wagner effect at the start of
each stroke. This favorable arrangement is
possible because solid walls within fluids
are analogous to mirrors in Optics. The
large body of the fish would act as an image
plane to mimic the presence of a paired appendage. In fact, researchers have used the
image plane trick in order to construct
physical models of the clap and fling behavior using a single mechanical biofoil
(Bennett, 1977). For these reasons, big flat
bodies may be extremely beneficial for fish
that employ pectoral fin locomotion. It offers them the benefits in maneuverability
that are found in low amplitude hovering
without sacrificing the benefits in force production that arise from the interactions between paired biofoils.
Fast forward locomotion with
paired biofoils
The necessity for birds and insects to
continuously generate lift to counteract
gravity has important consequences for the
transition from hovering to fast forward
propulsion. As discussed above, in some
hovering animals the direction of circulation changes from one stroke to the next in
order to generate an upward force throughout the entire cycle. During forward flight,
when the velocity over the biofoils is determined primarily by the forward velocity
of the body, the sign of circulation must
remain the same during both the upstroke
and downstroke. Thus, the gait transition
from hovering to forward flight will require
a change in the direction of circulation during the upstroke. For hovering animals that
generate circulation only on the downstroke, the transition is less extreme. (Rayner
et ah, 1986). Once the direction of circulation stays constant from one stroke to the
next, the structure of the wake simplifies,
becoming topologically similar to that of an
airplane with two continuous tip vortices
trailing from the bound circulation of the
wings (Spedding, 1987). For fish, penguins,
and other aquatic animals using paired bio-
foils, the functional constraints are quite
different. Because of their buoyancy, aquatic animals need not create lift throughout
the entire cycle. Indeed, in order to maintain a level trajectory they may need to balance the upward force during one stroke
with a downward force during the next. To
accomplish this during forward locomotion,
the circulation must still change sign from
one stroke to the next. The resultant wake
of each paired biofoil would consist of a
series of downward and upward direction
vortex rings, perhaps similar to the pattern
in Figure 7F.
FUTURE DIRECTIONS
The field of Biological Fluid Mechanics
has made great progress in exploring adaptations of locomotion using steady-state
models and other simplifying assumptions.
Why is it all of the sudden necessary to
explore the same problems using a more
difficult and complex analysis? The simplest answer is that many forms of locomotion cannot be explained by steady-state
approximations. This was certainly the case
for insect flight (Ellington, 1984c), and is
probably also true for many forms of aquatic locomotion. In addition, while simpler
steady-state models are satisfying in that
they emphasize unifying general principles,
many important biological questions require that we identify subtle differences in
behavior and physiology. The facility with
which we can resolve these subtleties may
determine how successfully Biological Fluid Mechanics can be integrated with other
disciplines such as Ecology, Evolution, and
Neurobiology. The methods employed must
be precise enough to resolve functionally
significant differences among closely-related species, different points along an ontogenetic series, or animals performing small
but functionally important variations in
steering maneuvers.
The generalized descriptions presented
throughout the paper have relied upon unequal doses of data, intuition, mathematics,
and fantasy. The difficult but essential task
is to measure forces and visualize flows on
real behaving animals. There are three main
avenues currently available for the study of
biological fluid mechanics. The first and
553
FORCE GENERATION IN LOCOMOTION: UNSTEADY MECHANISMS
most important path is the acquisition of
more elaborate flow visualizations and force
measurements from swimming animals. The
ultimate experimental paradigm would allow
direct measurement of instantaneous forces
during locomotion while simultaneously visualizing kinematics and wake structure. At
first glance, such a scheme might seem impossible for most animals. However, the
continuous technological advances in computers, video, radio transmitters, and countless other devices will undoubtedly make
such attempts easier in years to come. Clearly, the measurement of instantaneous forces
is a particularly daunting task. However,
such measurements with swimming animals
are possible. Mark Westneat is currently using a "fish-on-a-stick" transducer to study
the swimming mechanics of sunfish (personal communication).
A second useful approach, is to develop
physical models that more accurately replicate the complex kinematics of real animals. Devices that could re-create all the
degrees of freedom of flapping biofoils
would be extremely useful in testing models of unsteady force production. Charlie
Ellington and his coworkers in Cambridge
have constructed a giant mechanical model
of a hawk moth, which has already proven
of great importance in identifying new unsteady mechanisms of force production in
insect flight (Ellington, 1996; Willmott,
1995). Mimi Keohl and her colleagues have
developed a flapping model of a crustacean
swimming appendage to examine oscillatory flows at low Reynolds numbers (Loudon et ai, 1993). In the future, simple flow
tanks and static models should be replaced
by complex machines flapping through
towing tanks or wind tunnels. The flapping
appendages could be equipped with an array of sensors for measurement of instantaneous aerodynamic forces. The reproducible and programmable motions of mechanical devices would facilitate flow visualizations of wake structure. Unfortunately,
such devices require expense, space, and
development time that may be beyond the
resources of many individual laboratories.
However, a single device, designed with
flexibility in mind, could serve the needs of
a large number of researchers.
The third obvious direction for the analysis of unsteady fluid mechanics is the development and use of sophisticated computer models, as illustrated by the work of
Jordan (1996) and Fauci (1996) within this
volume. Recently, Liu and colleagues
(1996) have used the computational fluid
dynamic method (CFD) to presented an elegant and visually beautiful analysis of the
flows and forces generated by undulating
tadpoles. At the moment, such analyses are
the domain of computational specialists, although the rise in cheap computer power is
already spawning workstations throughout
the labs of many biologists. At the same
time, software companies are offering better
and cheaper environments for simulating
fluid mechanical phenomena. These trends
will make it easier to tackle tough problems
involving time-dependent flows.
Taken together, these three approaches
make up a powerful tool kit for analyzing
the mechanisms of force production in
swimming and flying animals. With these approaches at hand, it should become easier to
incorporate principles of unsteady fluid mechanics in the study of aquatic locomotion.
REFERENCES
Ahlborn, B., D. G. Harper, R. W. Blake, D. Ahlborn,
and M. Cam. 1991. Fish without footprints. J.
Theor. Biol. 148:521-533.
Bennett, L. 1977. Clap and fling aerodynamics—an
experimental evaluation. J. Exp. Biol. 69:261272.
Blake, R. W. 1983. Median and paired fin propulsion.
In P. W. Webb and D. Wells (eds.). Fish Biomechanics, pp. 214-247. Preager Publishers, New
York.
Brodsky, A. K. 1991. Vortex formation in the tethered
flight of the peacock butterfly Inachis io L. (Lepidoptera, Nymphalidae) and some aspects of insect flight evolution. J. Exp. Biol. 161:77-95.
Brodsky, A. K. 1994. The evolution of insect flight,
pp. 229. Oxford University Press, New York.
Daniel, T., C. Jordan, and D. Grunbaum. 1992. Hydrodynamics of swimming. In Advances in comparative and environmental
physiology,
Vol. 11,
pp. 17-49. Springer-Verlag, Berlin.
Daniel, T. and P. Webb. 1987. Physical determinants
of locomotion. In P. Dejours, L. Bolis, C. Taylor
and E. Weibel (eds.), Comparative physiology:
Life in water and on land, pp. 343—369. IX—Liviana Press, Padova.
Dickinson, M. H. 1994. The effects of wing rotation
on unsteady aerodynamic performance at low
Reynolds numbers. J. Exp. Biol. 192:179-206.
Dickinson, M. H. and K. G. Gotz. 1993. Unsteady
554
MICHAEL DICKINSON
aerodynamic performance of model wings at low
Reynolds numbers. J. Exp. Biol. 174:45—64.
Dickinson, M. H. and K. G. Gotz. 1996. The wake
and flight forces of a friut fly, Drosophila melanogasler. J. Exp. Biol. 199:2085-2104.
Ellington, C. P. 1984a. The aerodynamics of insect
flight. III. Kinematics. Phil. Trans. R. Soc. Lond.
B 305:41-78.
Ellington, C. P. 1984fc. The aerodynamics of insect
flight. IV. Aerodynamic mechanisms. Phil. Trans.
R. Soc. Lond. B 305:79-113.
Ellington, C. P. 1984c. The aerodynamics of hovering
insect flight. I. The quasi-staedy analysis. Phil.
Trans. R. Soc. Lond. B 305:1-15.
Ellington, C. P. 1995. Unsteady aerodynamics of insect flight. In C. P. Ellington and T. J. Pedley
(eds.), Biological fluid dynamics. Symp. Soc. Exp.
Biol. 49:109-129. Company of Biologists, Cambridge.
Ellington, C. P., C. van den Berg, A. P. Willmott, and
A. L. R. Thomas. 1996. Leading-edge vortices in
insect flight. Nature (In press).
Fauci, L. J. 1996. A computational model of the fluid
dynamics of undulatory and flagellar swimming.
Amer. Zool. 36:599-607.
Fish, F. E. 1996. Transitions from drag-based to liftbased propulsion in mammalian swimming. Amer.
Zool. 36:628-641.
Freymuth, P. 1990. Thrust generation by an airfoil in
hover modes. Experiments in Fluids 9:17-24.
Gibb, A. C , B. C. Jayne, and G. V. Lauder. 1994.
Kinematics of pectoral fin locomotion in the bluegill sunfish, Lepomis Macrochirus. J. Exp. Biol.
189:133-161.
Gillis, G. B. 1996. Undulatory locomotion in elongate
aquatic vertebrates: Anguilliform swimming since
Sir James Gray. Amer. Zool. 36:656-665.
Gotz, K. G. 1987. Course-control, metabolism and
wing interference during ultralong tethered flight in
Drosophila melanogaster. J. Exp. Biol. 128:35-46.
Grodnitsky, D. L. and P. P. Morozov. 1992. Flow visualization experiments on tethered flying green
lacewings Chrysopa Dasyptera. J. Exp. Biol. 169:
143-163.
Grodnitsky, D. L. and P. P. Morozov. 1993. Vortex
formation during tethered flight of functionally
and morphologically two-winged insects, including evolutionary considerations on insect flight. J.
Exp. Biol. 182:11-40.
Jayne, B. C. and G. V. Lauder. 1996. New data on
axial locomotion in fishes: How speed affects diversity of kinematics and motor patterns. Amer.
Zool. 36:642-655.
Jordan, C. E. 1996. Coupling internal and external
mechanics to predict swimming behavior: A general approach? Amer. Zool. 36:710-722.
Kokshaysky, N. V. 1979. Tracing the wake of a flying
bird. Nature 279:146-148.
Kuethe, A. M. and C.-Y. Chow. 1986. Foundations of
aerodynamics. John Wiley & Sons, New York.
Lauder, G. V. and B. C. Jayne. 1996. Pectoral fin locomotion in fishes. Testing drag-based models using three-dimensional kinematics. Amer. Zool. 36:
567-581.
Liu, H., R. Wassersug, and K. Kawachi. 1996. A computational fluid dynamics study of tadpole swimming. J. Exp. Biol. 199:1245-1260.
Loudon, C, B. A. Best, and M. R. Koehl. 1993. When
does motion relative to neighboring surfaces alter
the flow through arrays of hairs? J. Exp. Biol. 193:
233-254.
Maxworthy, T. 1979. Experiments on the Weis-Fogh
mechanism of lift generation by insects in hovering flight part 1. Dynamics of the 'fling.' J. of
Fluid Mech. 93:47-63.
Maxworthy, T. 1981. The fluid dynamics of insect
flight. J. Fluid Mech. 93:47-63.
McCutchen, C. W. 1977. Froude propulsive efficiency
of a small fish measured by wake visualization. In
T. J. Pedley (ed.), Scale effects in animal locomotion, pp. 339—363. Academic Press, New York.
Milne-Thomson, L. M. 1966. Theoretical aerodynamics, pp. 210-211. Macmillan, New York.
Rayner, J. M. V. 1995. Dynamics of vortex wakes of
swimming and flying vertebrates. In C. P. Ellington and T. J. Pedley (eds.), Biologicalfluiddynamics. Symp. Soc. Exp. Biol. 49:131-155.
Rayner, J. M. V., G. Jones, and A. Thomas. 1986.
Vortex flow visualizations reveal change in upstroke function with flight speed in bats. Nature
321:162-164.
Rosen, M. W. 1959. Water flow about a swimming
fish. University of California at Los Angeles.
Schlichting, H. 1979. Boundary-layer theory, pp. 817.
McGraw-Hill, New York.
Spedding, G. R. 1986. The wake of a jackdaw (Corvus mondela) in slow flight. J. Exp. Biol. 125:
287-307.
Spedding, G. R. 1987. The wake of a kestral (Falco
tinnunculus) in flapping flight. J. Exp. Biol. 127:
59-78.
Spedding, G. R. 1992. The aerodynamics of animal
flight. In R. M. Alexander (ed.), Advances in comparative and environmental physiology, vol. 11,
pp. 51-11. Springer-Verlag, London.
Spedding, G. R., J. M. V. Rayner, and C. J. Pennycuick. 1984. Momentum and energy in the wake
of a pigeon (Columba livia) in slow flight. J. Exp.
Biol. 111:81-102.
VanDyke, M. 1982. An album of fluid motion. Parabolic Press, Stanford.
Videler, J. 1993. Fish swimming. Chapman & Hall,
New York.
Vogel, S. 1994. Life in moving fluids. Princeton University Press, Princeton.
Webb, P. B. 1975. Hydrodynamics and energietics of
fish propulsion. Bull. Fish. Bd. Can. 190:1-158.
Weis-Fogh, T. 1973. Quick estimates of flight fitness
in hovering animals, including novel mechanisms
for lift production. J. Exp. Biol. 59:169-230.
Westneat, M. W. 1996. Functional morphology of
aquatic flight in fishes: Kinematics, electromyography, and mechanical modeling of labriform locomotion. Amer. Zool. 36:582-598.
Willmott, A. P. 1995. The mechanics of hawkmoth
flight. Cambridge.
Zanker, J. M. 1990. The wing beat of Drosophila melanogaster I. Kinematics. Phil. Trans. R. Soc.
Lond. B 327:1-18.