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First published online December 28, 2007
Journal of Experimental Biology 211, 187-195 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.008128
Research Article, Biomechanics of Swimming |
Escaping Flatland: three-dimensional kinematics and hydrodynamics of median fins in fishes
1 Department of Biology, University of Maryland, College Park, MD 20742,
USA
2 Department of Organismic and Evolutionary Biology, Harvard University,
Cambridge, MA 02138, USA
* Author for correspondence (e-mail: tytell{at}umd.edu)
Accepted 24 May 2007
Summary
Fish swimming has often been simplified into the motions of a two-dimensional slice through the horizontal midline, as though fishes live in a flat world devoid of a third dimension. While fish bodies do undulate primarily horizontally, this motion has important three-dimensional components, and fish fins can move in a complex three-dimensional manner. Recent results suggest that an understanding of the three-dimensional body shape and fin motions is vital for explaining the mechanics of swimming, and that two-dimensional representations of fish locomotion are misleading. In this study, we first examine axial swimming from the two-dimensional viewpoint, detailing the limitations of this view. Then we present data on the kinematics and hydrodynamics of the dorsal fin, the anal fin and the caudal fin during steady swimming and maneuvering in brook trout, Salvelinus fontinalis, bluegill sunfish, Lepomis macrochirus, and yellow perch, Perca flavescens. These fishes actively move the dorsal and anal fins during swimming, resulting in curvature along both anterio-posterior and dorso-ventral axes. The momentum imparted to the fluid by these fins comprises a substantial portion of total swimming force, adding to thrust and contributing to roll stability. While swimming, the caudal fin also actively curves dorso-ventrally, producing vortices separately from both its upper and lower lobes. This functional separation of the lobes may allow additional control of three-dimensional orientation, but probably reduces swimming efficiency. In contrast, fish may boost the caudal fin's efficiency by taking advantage of the flow from the dorsal and anal fins as it interacts with the flow around the caudal fin itself. During maneuvering, fish readily use their fins outside of the normal planes of motion. For example, the dorsal fin can flick laterally, orienting its surface perpendicular to the body, to help in turning and braking. These data demonstrate that, while fish do move primarily in the horizontal plane, neither their bodies nor their motions can accurately be simplified in a two-dimensional representation. To begin to appreciate the functional consequences of the diversity of fish body shapes and locomotor strategies, one must escape Flatland to examine all three dimensions.
Key words: fish, locomotion, hydrodynamics, flow, kinematics, dorsal fin, anal fin, caudal fin
Introduction
`I exhorted all my hearers to divest themselves of prejudice and to become believers in the Third Dimension.'E. A. Abbott, Flatland: A Romance of Many Dimensions, 1899
In the classic book Flatland, E. A. Abbott describes a world of
two dimensions, a world in which the inhabitants live in ignorance of the
third dimension, a world in which it is impossible to distinguish circles from
squares, and squares from pentagons – everything looks like a line in
the two-dimensional world of Flatland
(Abbott, 1899
). Then, a
disruptive Sphere enters to make inhabitants aware of a third dimension,
causing consternation and ultimately sending the narrator to jail. The
Flatland of Abbott's book is an appropriate metaphor for much of the current
literature on undulatory locomotion in fishes, which is largely devoted to
analyses of fish swimming in the horizontal (two-dimensional, 2D) plane. This
2D world has, so far, been an appropriate place to conduct our studies of fish
locomotion: fewer cameras are needed to acquire data, analyses are simpler,
and theoretical models can be generated more easily (e.g.
Schultz and Webb, 2002).
But fish are three-dimensional (3D), with prominent median and paired fins that project into the water, and with clearly defined edges that demark the dorsal, ventral and caudal body surface. Although thinking in three dimensions may complicate our lives, it is nonetheless a critical next step that experimental and computational analyses of fish undulatory locomotion are only now starting to grapple with. Like the residents of Flatland, we must learn to deal with the 3D world.
The 3D nature of fish functional design is clearly seen in the enormous
diversity of body shapes and swimming modes in fishes, but the precise
advantage of one shape or mode over another is less clear. Generalizations
from hydrodynamic theories combined with observations of typical swimming
behavior have led to much speculation on the adaptations of certain
morphologies to different situations
(Lighthill, 1975;
Marshall, 1971). Tunas, for
example, are highly specialized in many ways, many of which are probably
adaptations to their active, pelagic lifestyle
(Block and Stevens, 2001).
However, eels, which have a different body shape and swimming mode from tunas,
also migrate thousands of kilometers without feeding
(van Ginneken et al., 2005).
Presumably, the different body shapes and swimming modes of these fishes
confer different advantages, but the exact functional benefits of their
morphologies remains unclear.
In Flatland, most of these morphological differences are not evident. In
horizontal cross-section, even eels and tunas are not unlike each other: they
both have streamlined bodies with high-aspect ratios, and they propel
themselves with an undulatory wave that increases in amplitude near the tail.
This similarity among fishes was a boon for classic theories of fish swimming
(e.g. Gray, 1933;
Lighthill, 1960;
Taylor, 1952;
Wu, 1971), because a 2D
undulating plate or cylinder serves as a good initial approximation of most
fishes. The use of this simplification allowed much progress in understanding
the basic mechanisms of fish swimming.
In recent years, the empirical understanding of fish swimming has advanced
beyond the classic theories to produce a general description of flow patterns
during fish swimming. These advances have been achieved mostly through the
technique of particle image velocimetry (PIV)
(Willert and Gharib, 1991).
PIV has established the fluid flow patterns near the body
(Anderson et al., 2001) and in
the wake of many fishes (Drucker and
Lauder, 1999; Müller et
al., 2001; Müller et al.,
1997; Nauen and Lauder,
2002a; Nauen and Lauder,
2002b; Tytell,
2004; Tytell and Lauder,
2004). The basic propulsive flow structure is a set of alternating
jets, oriented laterally and posteriorly. The wake structure behind eels seems
to be somewhat different from this basic pattern
(Müller et al., 2001;
Tytell and Lauder, 2004),
which we will argue below is a sign of the importance of the 3D eel body
shape. Even the wake of eels, though, is not so different as to indicate
dramatic differences in propulsive mechanisms from those of other fishes. In
general, the overall picture that has emerged is one of consistency, from eels
to mackerel.
The consistency may be, in part, a result of the 2D approximation. PIV is a
fundamentally 2D technique: it only permits measurements of flow velocities in
a plane. The underlying assumption of most PIV studies of fish swimming has
been that the flow in a plane along the horizontal midline accurately captures
the 3D flow around the fish. Some studies have used multiple orthogonal planes
to establish a better approximation of the 3D flow structure
(Drucker and Lauder, 1999;
Nauen and Lauder, 2002a), but
even these make the assumption that flow in a midline plane (whether
horizontal or vertical) captures the structure of flow in parallel planes
offset from the midline. However, Tytell's results in bluegill sunfish suggest
that flow in a horizontal plane at the level of the dorsal or ventral tips of
the caudal fin may be quite different from flow in the horizontal midline
(Tytell, 2006).
Our view is that the current understanding of undulatory propulsion, from empirical, theoretical and computational standpoints, has progressed to the point that 3D effects should no longer be neglected. There are two components to these effects: (1) the static 3D body shape and (2) the dynamic 3D motion of the body. As we begin to include these effects, we return to the roots of studies of fish swimming, by connecting hydrodynamics to functional morphology. With a 3D perspective, we can start to address questions of adaptation in a quantitative way, examining the relationship of swimming performance to 3D body shape and kinematics. Also, describing how maneuvers are accomplished requires a 3D analysis, due to the complex and often asymmetric positions and kinematics of fins during maneuvers.
In this paper, we first describe recent data on the 3D character of the
swimming motion, both in steady swimming and during maneuvering. We focus on
the dorsal and anal fins, which are often neglected entirely in the 2D
approximation (but not in all models) (see
Weihs, 1972) but, for many
fishes, constitute nearly as much fin area as the caudal fin and have their
own complex 3D motion. Then, we describe evidence that the 3D shape and motion
of the median fins is important to the overall flow structure around the fish.
The importance of 3D effects is seen in three ways: (1) the effect of the 3D
motion itself, which we illustrate by focusing on the caudal fin; (2) the
effect of the dorsal and anal fins, as an indication of the importance of 3D
body shape; and (3) the interaction between flows from different fins.
Problems with the 2D approximation
As an example of the problems with the 2D approximation, consider a classic
distinction in modes of fish swimming: the anguilliform mode, used by eels and
sharks primarily, and the carangiform mode, used by most other fishes
(Breder, 1926). Gray's early
studies of eels (Gray, 1933)
indicated erroneously that eels swim very differently from other fishes
(Lauder and Tytell, 2004). In
contrast to Gray's results, recent close examination of the midline kinematics
of these two modes has shown that the differences between them are quite
subtle (Fig. 1)
(Gillis, 1998;
Lauder and Tytell, 2006),
particularly at the low swimming speeds commonly used for long distance
swimming (van Ginneken et al.,
2005).
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; Nauen and Lauder,
2002a; Tytell and Lauder,
2004). This led Lauder and Tytell to conclude that differences in
3D body shape (Fig. 1A,C) must
account for the differences in wake structure
(Lauder and Tytell, 2006).
3D body shapes
To some extent, classic 2D theories can account for differences in body
shapes among fishes. Most theories, including Lighthill's elongated body
theory, include a term that expresses the dorsoventral height of the fish
(Lighthill, 1971). To satisfy
the assumptions in the theories, however, this term cannot vary too quickly
along the body length. In other words, fishes like tunas and mackerels with
pronounced narrow peduncles and broad tails cannot be approximated in this way
(Lighthill, 1971); instead,
Lighthill developed an alternative 2D theory that approximated just the tail
(Lighthill, 1970).
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), a
combination that cannot be treated well with a 2D approximation like Wu's
waving plate or elongated body theory
(Lighthill, 1971;
Wu, 1971). Computational
results supported the same conclusion for an approximation of a giant danio
morphology (Zhu et al., 2002).
Additionally, both elongated body theory
(Lighthill, 1971) and a
resistive model (after Taylor,
1952) were compared with PIV estimates of lateral forces in the
wake of an eel. Neither model individually, nor the combination of the two,
could fully account for the forces estimated experimentally
(Tytell, 2004).
3D kinematics
Not only are fish body shapes 3D, but the fins typically move in a 3D
manner. Median fins actively oscillate, bend, curve and flick in all
directions under the control of fin ray musculature. By erecting and
depressing their rays, fish can control both the shape of their fins and the
how much fin area is exposed to the flow.
Fig. 2 illustrates control
of fin shape. Bluegill sunfish move the dorsal and ventral edges of the caudal
fin further and faster than the middle
(Fig. 2A)
(Lauder, 2000;
Tytell, 2006), resulting in a
`cupping' motion into the flow. Bainbridge reported similar results for the
caudal fin of dace (Leuciscus leuciscus)
(Bainbridge, 1963). The dorsal
and anal fins also have a similar cupping motion
(Fig. 2B)
(Standen and Lauder, 2005;
Tytell, 2006).
Fig. 3 shows data on changes
in the exposed area of fins in bluegill sunfish. Bluegill sunfish, trout and
yellow perch all decrease dorsal and anal fin area with increasing swimming
speed (Fig. 3)
(Drucker and Lauder, 2001;
Drucker and Lauder, 2005;
Standen and Lauder, 2005).
Reducing surface area affects the function of the fin in two ways: (1) by
reducing the amount of drag on the fin, mostly from skin friction, and (2) by
reducing the fin area available to produce useful forces. Such changes in the
functional area of fins are not easily incorporated into 2D kinematic studies
of fish swimming.
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Importance of the 3D view
In the above sections, we have argued that both fishes and the swimming motion itself are inescapably 3D, particularly during unsteady behaviors like maneuvering. But how important is the third dimension? Is it really necessary to add the extra complexity that a 3D analysis brings? Below, we present three lines of argument that 3D body shape and kinematics must be considered to further the current understanding of fish swimming.
First, the kinematics themselves, even during steady swimming, have a 3D
character. We claim that the 3D motion probably affects the amount of force
produced and the efficiency by which the tail transfers momentum to the fluid.
In dace (Bainbridge, 1963) and
bluegill sunfish (Lauder,
2000; Tytell,
2006), the dorsal and ventral tips of the caudal fin actively cup
into the flow as the fin moves from side to side
(Fig. 2). The tips move about
twice as far as the horizontal midline in the same amount of time; therefore,
they move approximately twice as fast. In general, fluid dynamic forces scale
as velocity squared; thus, force could vary along the height of the caudal fin
by as much as four times. To correctly estimate force, one must therefore look
at the 3D motion of the fin, not the 2D motion of any individual horizontal
slice. Additionally, the cupping motion of the fin distorts the vortex ring
shed into the wake into a complex, non-circular form
(Tytell, 2006). A circular
loop, in contrast, is the most efficient way of transferring momentum to the
fluid (Lighthill, 1970). Thus,
the independent control of the dorsal and ventral lobes of the caudal fin,
which is important for maneuvering (e.g. as seen in
Fig. 4A), may reduce the
efficiency of steady swimming for bluegill sunfish.
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; Standen and
Lauder, 2007; Tytell,
2006) indicate that the dorsal and anal fins, which are typically
neglected in 2D approximations (but see
Weihs, 1972), actively produce
forces that are a sizeable contribution to the forces produced by the fish as
a whole. These studies use circulation, a measure of the strength of a vortex,
as an estimate of force production. Fig.
5 shows vortices shed by the dorsal and anal fins in trout and
bluegill sunfish. In bluegill sunfish, the dorsal and anal fins shed vortices
(Fig. 5A) with circulations
that are statistically indistinguishable from those produced by the caudal
fin, indicating that the total force from the anterior median fins is
comparable to that from the tail itself
(Tytell, 2006). Recordings
from the inclinator muscles that move the dorsal fin in sunfish
(Jayne et al., 1996) show that
bluegill sunfish actively move the soft dorsal fin: it is not oscillating
passively with the body. The dorsal fin wake has both lateral (side) and
thrust (posterior) components, with a ratio of side to posterior force of
approximately 2:1 for fish swimming at 1.0 lengths s–1
(L s–1) (Drucker
and Lauder, 2001). The side forces produced by the dorsal fin may
be important in maintaining stability, while the thrust component adds to
locomotor thrust forces generated by the tail and the anal fin. In trout,
also, the dorsal and anal fins produce substantial vortices
(Fig. 5B,C)
(Drucker and Lauder, 2005;
Standen and Lauder, 2007).
These vortices result in jets with a large lateral component, indicating that
their effect is more on roll stability of the fish, rather than propulsion
(Drucker and Lauder, 2005;
Standen and Lauder, 2007).
Because both fins are located above and below the rolling axis of trout, the
lateral forces from these fins result in rolling torque production during
swimming. In brook trout, these torques appear to balance
(Standen and Lauder, 2007),
helping to maintain the fish in an upright orientation. Similar dorsal fin
momentum jets are also observed in yellow perch (Figs
6,
7).
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Finally, the vortices shown diagrammatically in
Fig. 8 do not occur in
isolation; they influence each other and the flow around the body of the fish.
Drucker and Lauder showed that the dorsal fin of bluegill sunfish produces a
strong wake that moves posteriorly to encounter the tail as it beats from side
to side (Drucker and Lauder,
2001). The dorsal fin wake can directly alter thrust generated by
the tail as a result of changing the flow environment on the tail surface.
Akhtar and colleagues, using kinematic data from bluegill sunfish, showed
through computational fluid dynamic calculations that the dorsal fin wake can
substantially enhance thrust produced by the tail, if it is phased correctly
(Akhtar et al., 2007). As the
dorsal fin wake encounters the tail, it causes enhanced separation of the tail
leading edge vortex, increasing the low pressure region on the tail surface
during the time that the caudal fin is angled forward. This increases thrust.
In these simulations, the tail thrust coefficient nearly doubled and the
efficiency increased by 52% compared with the tail alone in the absence of a
dorsal fin wake (Akhtar et al.,
2007).
In rainbow trout, the dorsal fin generates little additional thrust, but
strong side forces: the side:posterior force ratio is almost 6:1 at a swimming
speed of 1.0 L s–1. The rainbow trout dorsal fin
wake consists of a chain of vortices that move toward the tail, and the caudal
fin oscillates through the centers of these shed vortices
(Drucker and Lauder,
2005).
In both sunfish and trout, the dorsal fin wake changes significantly with speed, but the changes differ considerably between the species. In sunfish, the dorsal fin oscillates with very low amplitude at slow swimming speeds, and increases both movement and force production as speed increases. In contrast, data on the rainbow trout dorsal fin show that amplitude and lateral force generation are maximal at low swimming speeds, and decrease as speed increases. The differences between species may reflect the different body shapes, and hence differences in the ability of each species to maintain stability. Further comparative hydrodynamic data will allow a quantitative, mechanistic examination of how body shape and median fin placement affect stability among species.
Many bony fishes possess more than one dorsal fin, and the major group of
teleost fishes known as the Acanthopterygii
(Lauder and Liem, 1983;
Nelson, 2006) possess a dorsal
fin with an anterior spiny portion and a posterior soft-rayed region. The
soft-rayed region of the fin possesses inclinator muscles which allow active
side-to-side motion (Jayne et al.,
1996) of the typical bony fish fin rays supporting this fin
(Alben et al., 2007), while the
spiny region of the fin possesses only erector and depressor muscles acting on
rigid spines, but no inclinators
(Winterbottom, 1974). The
spiny dorsal fin thus can move laterally only in concert with body motion. In
bluegill sunfish these two dorsal fin regions are attached (e.g.
Fig. 8), but in many species
the spiny region is separate from the soft-rayed region.
Yellow perch (Perca flavescens) possess distinct soft and spiny regions of the dorsal fin, and these two fin regions can function differently during locomotion (Fig. 9). During steady swimming, the height of both portions of the dorsal fin decreases as swimming speed increases, while during a c-start, the spiny region of the dorsal fin is erected while the soft portion of the fin changes little in height (Fig. 9). During steady swimming, the spiny dorsal fin sheds a drag wake that generates a shear layer that passes back over the soft dorsal fin (Fig. 7); no lateral momentum is evident in the wake of the spiny dorsal fin. The soft dorsal fin of perch also sheds a drag wake, but with significant lateral momentum that passes back and is encountered by the tail during steady locomotion (Fig. 7).
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Maneuvering
In steady swimming, the forces and moments from all of the fins must
balance over the course of a fin beat. When they do not balance, the velocity
of a fish changes, which is the most general definition of maneuvering. In
three dimensions, maneuvering can occur along three rotational axes: left or
right turns (yaw), rolling along the fish's long axis, and pitching the snout
upwards or downwards. Maneuvers often involve 3D movements of the fins and
body (Fig. 3)
(Standen and Lauder, 2005).
For example, during yaw turns, the dorsal and anal fins tend to oscillate
asymmetrically with a series of large excursions to one side of the body,
often pausing at maximum excursion between oscillations. This asymmetrical
behavior results in the formation of large lateral jets by dorsal and anal
fins and a lateral and often dorso-ventral displacement of the fish's body
(Fig. 10).
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The timing of maximum circulation and resultant jet peak velocity from
median fins do not always balance (Standen
and Lauder, 2007), indicating that fish can fine tune force
production around the body to control body position. Understanding the
mechanics of maneuvers will require examination of the contributions of all of
the fins and their 3D shapes and motions.
Conclusions and prospectus
Fish are 3D. Despite this fact, considerable progress has been made over
the past 30 years in understanding the basic dynamics of aquatic locomotion
using 2D models (e.g. Lighthill,
1975; Schultz and Webb,
2002). 2D simplifications have allowed progress in understanding
the fundamental mechanisms of thrust generation in fishes (reviewed in
Lauder and Tytell, 2006), and
have permitted the use of relatively less-complex technology to provide data
for testing models and understanding the first-order diversity of fish
movement patterns. Using only a single high-speed camera, one can readily
obtain body outlines for swimming fishes, quantify how patterns of body
bending change with speed, and relate body bending to muscle electrical
activity and strain (Shadwick and
Gemballa, 2006).
But quantifying the motion of the fish body, fins and water in three
dimensions is a more challenging proposition. Fortunately, technical
developments in recent years are increasingly making the move to three
dimensions feasible for experimentalists
(Lauder, 2006;
Lauder and Tytell, 2006), and
increases in computing power and the sophistication of computational
algorithms have made 3D computation of fluid flows more practical also
(Mittal, 2004;
Mittal et al., 2006;
Mittal and Iaccarino, 2005).
On the experimental side, the use of multiple high-speed cameras to provide a
variety of views of complexly deforming fish fins is now practical
(Lauder et al., 2006;
Standen and Lauder, 2005), and
the use of stereo particle image velocimetry, multiple simultaneous light
sheets, transverse light sheet orientations and scanning PIV all make 3D
reconstruction of fluid flows feasible
(Burgmann et al., 2006;
Nauen and Lauder, 2002b;
Standen and Lauder, 2007;
Tytell, 2006). The coming
marriage of 3D experimental and computational approaches promises exciting
progress toward understanding fish locomotor dynamics.
Acknowledgments
This work was supported by NIH grant 5 F32 NS054367 to E.D.T., by NSF grant IBN0316675 to G.V.L., and by ONR-MURI grant N00014-03-1-0897, monitored by Dr Thomas McKenna and initiated by Dr Promode Bandyopadhyay. We thank Drs Rajat Mittal and Promode Bandyopadhyay for many helpful discussions on bio-inspired propulsion. Video data presented herein on yellow perch fin function were collected in collaboration with Dr Eliot Drucker, and we are grateful for his permission to use those data here. Brooke Flammang kindly provided data on bluegill sunfish tail function during maneuvering.
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