Quantification of the wake of rainbow trout (Oncorhynchus mykiss) using three-dimensional stereoscopic digital particle image velocimetry
Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA
* Author for correspondence (e-mail: glauder{at}oeb.harvard.edu)
Accepted 7 August 2002
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Summary |
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Stereo-DPIV wake images showed three-dimensional views of oscillating jet flows high in velocity relative to free-stream values. These jet flows are consistent with the central momentum jet flows through the cores of shed vortex rings that have been previously viewed for caudal fin swimmers using two-dimensional DPIV. The magnitude and direction of U, V and W flows in these jets were determined over a time series of 6-8 consecutive strokes by each of four fish.
Although the fish swam at the same relative speed, the absolute magnitudes of U, V and W were dependent on individual because of body size variation. Vertical flows were small in magnitude (<1 cm s-1) and variable in direction, indicating limited and variable vertical force production during slow, steady, forward swimming. Thus, in contrast to previous data from sunfish (Lepomis macrochirus) and mackerel (Scomber japonicus), the trout homocercal caudal fin does not appear to generate consistent vertical forces during steady swimming. U flows were of the order of 3-6 cm s-1; lateral flows were typically strongest, with W magnitudes of 4-6 cm s-1. Such strong lateral flows have also been shown for more derived euteleosts with homocercal caudal fins.
The ratios of the magnitudes of wake flow,
U/(U+V+W), which is a flow equivalent to
mechanical performance, were also dependent on individual and ranged from 0.32
to 0.45, a range similar to the range of mechanical performance values
previously determined using standard two-dimensional DPIV methods for caudal
fin locomotion by more derived euteleosts. Strong lateral jet flow appears to
be a general feature of caudal fin locomotion by teleosts and may reflect the
nature of undulatory propulsion as a posteriorly propagated wave of bending.
Froude efficiency (p) was independent of individual; mean
p was 0.74, which is similar to previous findings for
trout.
Key words: locomotion, swimming, rainbow trout, efficiency, stereoscopic digital particle image velocimetry, Oncorhynchus mykiss
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Introduction |
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To date all studies of aquatic locomotion using DPIV have used
two-dimensional DPIV methods, in which flow is visualized separately in one
plane, typically the parasagittal (xy) or frontal (xz)
plane. A few studies have combined two-dimensional measurements of the three
flow velocity variables, downstream (U), vertical (V) and
lateral (W), calculated from separate kinematic events measured with
different planar light sheet orientations to estimate the three-dimensional
structure of wake flow patterns (e.g. Drucker and Lauder,
1999,
2000
;
Nauen and Lauder, 2002
). Such
three-dimensional reconstructions require study of a repetitive behavior that
shows little kinematic variation among repetitions, such as pectoral fin
locomotion by bluegill sunfish (Lepomis macrochirus) (Drucker and
Lauder, 1999
,
2000
) and caudal fin locomotion
by chub mackerel (Scomber japonicus)
(Nauen and Lauder, 2002
).
These limitations of two-dimensional analysis can be avoided by using
three-dimensional stereoscopic DPIV, in which the use of two cameras with two
different fields of view allows simultaneous visualization of all three
velocity components using only a single light sheet orientation. While
stereo-DPIV is well established in the engineering literature
(Gaydon et al., 1997;
Lawson and Wu, 1997
;
Willert, 1997
; Westerweel and
van Oord, 2000), this is the first application of this technique to the study
of biologically generated flow patterns. Hence, the first goal of this study
was to test the utility of stereo-DPIV to study wakes created by freely
swimming fishes.
Our second goal was to examine the vortex wake of a salmoniform fish. This
group represents a major phylogenetic grouping within ray-finned fishes for
which few experimental hydrodynamic data are available, although this clade
has been the subject of extensive previous research on locomotory performance.
For example, the steady swimming kinematics
(Stevens, 1979;
Webb et al., 1984
; Webb,
1988
,
1993
) and energetics (Webb,
1971a
,b
),
myotomal slow muscle locomotory function
(Hammond et al., 1998
) and
muscle power output (Coughlin,
2000
) of Oncorhynchus mykiss have been examined
previously, although the hydrodynamics of locomotion by O. mykiss
remain unexplored except for the preliminary report of Blickhan et al.
(1992
).
Wake structure reveals important information about the timing, magnitude
and direction of thrust production as well as potentially interesting data on
the mechanisms of thrust production. On the basis of previous studies of trout
steady swimming kinematics (Stevens,
1979; Webb et al.,
1984
; Webb, 1988
,
1993
) and wake structure
(Blickhan et al., 1992
), it is
likely that trout wakes are composed of linked vortex rings with oscillating
central jet flows that are high in lateral velocity (and are thus similar to
the wakes of other bodycaudal fin swimmers). The expectation that trout
wakes would be composed of linked vortex-ring wakes such as those previously
visualized using various two-dimensional techniques
(Anderson, 1996
;
Müller et al., 1997
;
Videler et al., 1999
;
Hanke et al., 2000
;
Drucker and Lauder, 2001b
;
Nauen and Lauder, 2002
)
offered an advantage in using trout rather than a completely uncharacterized
system as a test of the application of stereo-DPIV to aquatic locomotory
systems.
The third goal of this study was to examine two specific features of the
salmoniform vortex wake in the light of current data on the function of the
homocercal caudal fin in teleost fishes (for a review, see
Lauder, 2000). Previous
studies of more derived clades (Gibb et
al., 1999
; Lauder,
2000
; Nauen and Lauder,
2002
) have suggested that the homocercal caudal fin, which is
externally symmetrical in shape, can generate lift forces due to asymmetrical
movements that direct flow dorsally or ventrally. We quantified wake flow
patterns to determine whether the homocercal caudal fin of rainbow trout was
directing flow dorsally or ventrally, and thus producing lift, during steady
swimming. In addition, previous studies of more derived clades using
two-dimensional DPIV have demonstrated that lateral flow velocities are
typically higher than the downstream flow velocities, resulting in
surprisingly low caudal fin mechanical performance ratios of approximately
0.35 (out of a possible maximum value of 1) for taxa as varied as bluegill
sunfish Lepomis macrochirus
(Drucker and Lauder, 2001b
) and
chub mackerel Scomber japonicus
(Nauen and Lauder, 2002
). We
used the simultaneously measured magnitudes of U, V and W
produced by O. mykiss during steady swimming to determine whether the
ratio of downstream to total wake flow magnitude is similar to the previously
reported caudal fin mechanical performance measurements for more derived
species. Values of Froude propulsive efficiency, an index of overall
locomotory efficiency, were also calculated from the velocity measurements for
comparison both with values from other studies and with values of caudal fin
mechanical performance.
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Materials and methods |
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Stereoscopic DPIV flow visualization
Experiments were conducted using a 600 l recirculating flow tank. The flow
tank has a working area of 82 cmx28 cmx28 cm (length x width
x height). Water temperature was 15±1°C. Flow speed in the
tank was controlled using a variable-speed motor. Details of the flow tank and
flow speed calibration have been presented previously
(Jayne et al., 1996;
Gibb et al., 1999
). A flow
tank was used to control the speed of the fish and to provide a relatively
small field of view for the images (approximately 15 cmx7 cm, length
x height) for a close view of the wake (see Figs
2,
3).
|
|
|
|
As in previous two-dimensional DPIV studies conducted in this laboratory
(e.g. Drucker and Lauder, 2000;
Lauder, 2000
;
Wilga and Lauder, 2000
; Nauen
and Lauder, 2001
,
2002
), flow was visualized by
seeding the water with near-neutrally buoyant 12 µm diameter silver-coated
glass beads (density 1.3 g cm-3, Potters Industries, USA) that
reflected light from an 8 W argon-ion laser
(Fig. 1). The laser light sheet
was approximately 2 mmx7 cmx15 cm (depth x height x
width). It was positioned in the center of the working section of the flow
tank, and oriented in the parasagittal (xy) plane. Data were
collected when the fish was positioned so that the caudal fin beat through the
light sheet in mid-stroke.
A key characteristic of the stereo-DPIV arrangement is the use of two cameras (Fig. 1) to view the light sheet. For two-dimensional measurements (see studies cited above), a single camera is positioned perpendicular to the light sheet. Stereo-DPIV measurements require two synchronized cameras both positioned at a wide angle to the light sheet (Fig. 1), so that each camera records a different, but overlapping, perspective of the flow. In the present study, two Redlake Motionscope model PCI 500 digital video cameras were used. Images of 480x420 pixels resolution were recorded at a frequency of 250 Hz.
A second novel characteristic of the stereo-DPIV arrangement is the camera lens position and orientation. Lenses are uncoupled from the body housing the camera sensor and tilted relative to the camera to achieve the Scheimpflüg stereoscopic configuration (Fig. 1B) in which the object plane (in this case, the light sheet), the image plane and the lens principal plane intersect in a common line. Tilting the lens of each camera relative to the camera's CCD sensor (using a custom-designed mounting bracket) makes the plane of best focus the plane of the light sheet, rather than the plane perpendicular to the camera's optical axis. This results in images that are bright and in good focus across the field of view for accurate vector calculations.
The camera arrangement and Scheimpflüg lens orientation introduce
perspective distortion. For example, the cameras view a rectangular
calibration grid illuminated by the light sheet as a trapezoid
(Fig. 2A). This image
distortion is corrected by recording simultaneous views of a three-dimensional
calibration grid (Fig. 2A).
Calibration marker points in a square grid with a fiducial mark at grid center
(Fig. 2A) define the coordinate
system and match calibration image points to target locations in the fluid.
Alternating rows of calibration marks are offset in depth by 1 mm for
calibration in the z direction (this offset is not visible in the
Fig. 2A calibration images).
Images of the target from each camera are analyzed, providing a pair of
locations for each marker on the target: the x, y and z
location of the marker point on the target in the fluid and its x and
y pixel location in the image. A polynomial equation mapping function
is created based on calibration grid points using a least-squares method; this
equation is used to determine the spatial dimensions of the flow field. Use of
the calibration grid and image mapping functions had the added advantage of
aligning the images during processing so that the field of view of the cameras
did not have to be aligned pixel to pixel. Areas seen by each individual
camera that do not overlap the view from the other camera (approximately 15%
of the field of view, Fig. 2A)
are discarded prior to calculation of the three-dimensional vector field.
Stereo-DPIV calibration and the required calculations are discussed
extensively in the engineering literature
(Gaydon et al., 1997;
Lawson and Wu, 1997
;
Willert, 1997
; Westerweel and
van Oord, 2000) and will not be repeated here.
The first step in calculating the three-dimensional vector field is
two-dimensional analysis of the simultaneously collected left and right images
(Fig. 2B). Using Insight
software (v. 3.3, TSI, Inc., USA), sequential pairs of video images (4 ms
apart in time) were analyzed by subdividing the images into a series of
interrogation areas and comparing pixel intensity in these areas using
two-frame cross-correlation analysis. The areas of flow analyzed were
typically rectangular in shape. The cross-correlation analysis used a Hart
algorithm (Hart, 2000), a spot
size of 64 pixels and a peak:noise ratio of 1.5. DPIV calculation methods have
been previously described in detail
(Willert and Gharib, 1991
;
Anderson, 1996
;
Raffel et al., 1998
). The two
components of velocity nominally perpendicular to the camera optical axis (in
our case, U and V) are calculated from the two-dimensional
recorded images (Fig. 2B). Note
that the two-dimensional vectors are calculated in a symmetrical, aligned grid
(Fig. 2B) based on the
calibration grid (Fig. 2A).
The out-of-plane velocity component (in our case, W, Fig. 2C, D) is calculated using the two-dimensional vector files (Fig. 2B) and a set of four polynomial equations that determine the W component at each location in the two-dimensional vector fields. Interpolation of the right and left vector fields provides particle image displacements; fluid displacement is solved for using a least-squares method.
DPIV measurements
To visualize wake geometry, downstream free velocity
(Uo) was subtracted from each three-dimensional velocity
vector (see, for example, Fig.
3). Downstream rather than upstream velocity was subtracted
because the presence of the fish affects the flow field. The method for
subtracting Uo was based on the fact that the present
visualizations showed defined, oscillating, posteriolaterally oriented jet
flows in the trout wakes such as are associated with linked vortex ring wakes
produced by bodycaudal fin swimmers (e.g.
Müller et al., 1997;
Wolfgang et al., 1999
;
Drucker and Lauder, 2001b
;
Nauen and Lauder, 2002
). In
addition, previous studies of trout steady swimming kinematics
(Stevens, 1979
;
Webb et al., 1984
; Webb,
1988
,
1993
) and wake flow
(Blickhan et al., 1992
)
indicate that the jet flows seen here are associated with a linked vortex-ring
wake. The height of linked vortex-ring wakes produced during steady swimming
is typically equal to caudal fin span
(Blickhan et al., 1992
;
Müller et al., 2000
;
Nauen and Lauder, 2002
). Thus,
flow above and below the caudal fin represented free-stream velocity, and we
evaluated free-stream velocity magnitudes in regions of the stereo-DPIV images
above and below the caudal fin.
The magnitudes and directions of U, V and W were
determined for 10 vectors in the central jet flow of the wake (Figs
2,
3) created by each stroke of
the caudal fin. Six to eight consecutive strokes were analyzed for each
individual. The ratio of the magnitudes of
U/(U+V+W), which is referred to hereafter
as U/total wake flow, was calculated from the flow values. Froude
propulsive efficiency (p) was also calculated as
2Uo(U-Uo)/U2+V2+W2-Uo2)
based on Blake (1983
).
Single-factor analyses of variance (ANOVAs) were performed using SuperANOVA
(version 1.11) to determine the effect of individual on the measured
parameters, the ratio of U/total wake flow and
p. All
F-values were calculated as described by Zar
(1984
).
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Results |
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A series of jets was visible in the field of view of the cameras (e.g. Fig. 3B), which allows an evaluation of jet dynamics. The time-dependence of lateral (W) flows is depicted for an individual of 16.5 cm BL swimming at 1.2 BL s-1 (19.8 cm s-1). A wake flow of approximately 6.6 cm in length, which was created over 0.33 s, is visible in a single image. Lateral flows are strong and consistent and showed defined directional reversals that resulted from the change in movement direction of the caudal fin with each stroke.
Average vertical flows (V) were low in magnitude (typically less than 1 cm s-1, Fig. 4A) and variable in direction (Fig. 4A). The magnitude and direction of V were dependent on individual (ANOVA, d.f.=3,286, F=6.9, P=0.0002). One individual (no. 2) typically produced small vertical flows in the upward direction (or positive flows, Fig. 4A). The other three individuals produced small vertical flows that were positive or negative (Fig. 4A) depending on the particular tail beat.
The magnitude of downstream flow (U) was also dependent on individual (ANOVA, d.f.=3,286, F=51.0, P=0.0001). U flows, of the order of 3-6 cm s-1 above free-stream velocity (Fig. 4A), were relatively substantial. Lateral flows (W), were typically the largest in magnitude, with average values of 4-6 cm s-1 (Fig. 4A), and were dependent on individual (ANOVA, d.f.=3,286, F=6.7, P=0.0002).
The ratio of U/total wake flow (a flow index of mechanical
performance) was also dependent on individual (ANOVA, d.f.=3,286,
F=15.6, P=0.0001). Mean U/total wake flow ratios
for each individual (Fig. 4B)
varied from 0.32±0.012 to 0.45±0.014 (mean ± S.E.M.,
N=60-80). Froude efficiency (p) was independent of
individual (ANOVA, d.f.=3,286, F=1.67, P=0.17). On average,
p was 0.74±0.01 (mean ± S.E.M.,
N=290).
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Discussion |
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Previous studies have estimated three-dimensional wake geometry using
two-dimensional DPIV methods (see, for example,
Drucker and Lauder, 1999;
Nauen and Lauder, 2002
), which
requires combining two-dimensional measurements made in at least two different
planes (typically the parasagittal and frontal planes) measured during
independent kinematic events. This combination of measurements requires study
of a repetitive behavior with low kinematic variation. Such requirements are
unnecessary with stereo-DPIV, however, as simultaneous calculation of U,
V and W allows for three-dimensional visualization from
measurements made using a single light sheet orientation. The 1-2 mm thickness
of the light sheet used here limited the present W measurements to a
subset of the total wake volume. Nevertheless, stereo-DPIV has high potential
for three-dimensional flow visualization of non-repetitive, highly variable
behaviors for which combining separate measurements from orthogonally oriented
light sheets is not suitable.
Trout caudal fin locomotion
The swimming speed of 1.2 BL s-1 studied here has been
used in previous studies of wake structure produced by caudal fin locomotion
(Liao and Lauder, 2000;
Drucker and Lauder, 2001b
;
Nauen and Lauder, 2002
) and is
close to speeds examined in other two-dimensional DPIV studies of caudal fin
locomotion (Müller et al.,
1997
,
2001
), which allows for direct
comparison of the present wake structure data with wake structures described
for other species. The data of Webb et al.
(1984
) indicate that the speed
of 1.2 BL s-1 is well within the aerobic capacity limits
of O. mykiss.
Stereo-DIPV revealed a strong, oscillating jet with a large lateral flow
component in the wake of Oncorhynchus mykiss swimming steadily at 1.2
BL s-1. Such jets are typically associated with linked
vortex ring wakes produced by bodycaudal-fin locomotors (e.g.
Müller et al., 1997;
Wolfgang et al., 1999
;
Drucker and Lauder, 2001b
;
Nauen and Lauder, 2002
). The
magnitudes and directions of U, V and W flows in the wake
correspond to the magnitude of reaction forces to the jet flow that powers
locomotion. Of particular interest are the magnitude and direction of vertical
flows produced by the caudal fin that may act to rotate the fish about its
anteriorly positioned center of mass and the ratio of U/total wake
flow, which is a flow index of caudal fin mechanical performance.
Vertical flows measured here were very small in magnitude and were variable
in direction. O. mykiss, a member of the Salmoniformes, has a
symmetrically shaped, or homocercal, forked caudal fin. It is traditionally
assumed that caudal fins of this shape move symmetrically and generate forces
in the horizontal plane only (for a review, see
Lauder, 2000). The present
data, and study of more derived teleosts with forked homocercal caudal fins,
do not, however, support this assumption. During relatively slow, steady
swimming, the caudal fin of bluegill sunfish Lepomis macrochirus
tilts relative to the vertical (probably because of intrinsic tail muscle
activity; Lauder, 2000
) and
produces ventrally inclined flows (Lauder,
2000
). The homocercal forked caudal fin of chub mackerel
Scomber japonicus also tilts relative to vertical during steady
swimming (Gibb et al., 1999
)
and produces ventrally inclined flows and thus lift forces that tend to lower
the head (Nauen and Lauder,
2002
).
Vertical flows produced by homocercal caudal fins tend to be small in
magnitude. For example, vertical forces are less than 3% of the total force
production of slightly larger mackerel (23±1 cm fork length, mean
± S.E.M., N=4) swimming at 1.2 and 2.2 fork lengths
s-1 (Nauen and Lauder,
2002). The low-magnitude V flows measured in the present
study are thus similar to those for the more derived species
(Lauder, 2000
;
Nauen and Lauder, 2002
). The
present data indicate, however, that, unlike L. macrochirus
(Lauder, 2000
) and S.
japonicus (Nauen and Lauder,
2002
), O. mykiss shows variable caudal fin vertical
orientations during slow steady swimming. Indeed, on average, the flows
produced by three out of the four individuals studied here were directed
slightly upwards. This indicates the production of a lift force that would
tend to raise the head, which is the opposite of previous findings.
There is also typically a strong lateral component to jet flow produced by
caudal fin locomotion (Stamhuis and
Videler, 1995; Müller et
al., 1997
; Hanke et al.,
2000
; Lauder,
2000
; Liao and Lauder,
2000
; Müller et al.,
2000
,
2001
;
Drucker and Lauder, 2001b
;
Nauen and Lauder, 2002
), which
signifies relatively strong lateral force production. Studies of both bluegill
sunfish (Drucker and Lauder,
2001b
) and chub mackerel
(Nauen and Lauder, 2002
)
indicated that for animals swimming freely at a speed of approximately 1
BL s-1, lateral forces are approximately twice the
magnitude of thrust forces. W flows seen in the present study were
typically greater in magnitude than U, although by a factor of less
than two (Fig. 4).
High-magnitude W relative to U (and low-magnitude
V) resulted in U/total wake flow ratios in the range
0.32-0.45. This ratio is a flow-based estimate of true mechanical performance
(which is the ratio of thrust/total force). Mechanical performance ratios of
the order of 0.3-0.4 have been determined using two-dimensional DPIV for
caudal fin locomotion by slightly larger (approximately 23 cm BL)
bluegill sunfish (Drucker and Lauder,
2001b) and chub mackerel
(Nauen and Lauder, 2002
)
swimming at approximately 1 BL s-1. These values are
consistent with the present three-dimensional DPIV data for O.
mykiss. Thus, data collected using two methods, from three distantly
related euteleost taxa with a variety of body and fin shapes and a range of
swimming performance capacities, indicate a high lateral force component
during slow, steady caudal fin locomotion. It is possible that the relatively
low mechanical performance of these taxa reflects an inherent limitation to
caudal fin propulsion as a posteriorly propagated wave of lateral bending.
Supporting this hypothesis is the finding of Triantafyllou et al.
(1991
) of optimum mechanical
performance values of 0.2 for a foil oscillating transversely through uniform
incoming flow (which is a physical model of caudal fin locomotion).
In contrast to these caudal fin mechanical performance values, our
calculated mean Froude propulsive efficiency (p) of 0.74 was
relatively high. The
p value of 0.74 determined here for the
single relative speed of 1.2 BL s-1 is somewhat larger
than values determined previously using kinematic data. For example, Webb
(1971a
) calculated a
p value of 0.54 using larger trout (30 cm BL) than
those studied here swimming at speeds of approximately 1.25 BL
s-1. Values of
p approached 0.7 as swimming speed
increased to approximately 1.7-1.9 BL s-1 (see Table 11 in
Webb, 1971a
). In that study,
p was calculated as the ratio of fish swimming speed to the
speed of the body propulsive wave, on the basis that this is the ratio between
the momentum added to the water by the propulsive wave and the amount of
kinetic energy necessary to accelerate that water
(Webb, 1971b
). A later study
(Webb et al., 1984
) confirmed
the conclusion that
p increases with increasing speed for
O. mykiss. An efficiency value of 0.7 was, however, previously
determined for O. mykiss on the basis of power measurements
calculated from automatic particle tracking and laser-Doppler anemometry
techniques (Blickhan et al.,
1992
); however, few data and methodology details from that study
are available for direct comparison with the present study.
The above range of p values suggests a relatively high
propulsive efficiency for O. mykiss at slow cruising speeds, in
contrast to the caudal fin mechanical performance values of 0.32-0.45. The
p ratio reflects whole-body propulsive efficiency, including,
for example, the volume of water accelerated by the fish (see the discussion
in Blake, 1983
), whereas caudal
fin mechanical performance simply reflects the proportion of wake flow that is
thrust. The finding that
p is much greater than caudal fin
mechanical performance suggests the involvement of additional factors (such as
body shape and propulsive wave amplitude and frequency), which might result in
relatively high overall propulsive efficiencies even though the performance of
any given propulsor is relatively low.
The use of stereo-DPIV to determine simultaneously all three velocity
components allowed for calculations of both p and caudal fin
mechanical efficiency for single kinematic events in the present study (rather
than averaging over multiple events as is necessary with two-dimensional DPIV,
Drucker and Lauder, 1999
,
2000
;
Nauen and Lauder, 2002
). The
homocercal caudal fin shows flexibility in performance during steady forward
locomotion with increasing speed, particularly in regard to vertical force
production and increases in thrust/total force production as a function of
speed (Nauen and Lauder,
2002
). Future studies of the homocercal caudal fin of teleost
fishes under a wider range of hydrodynamic conditions (including unsteady and
maneuvering locomotion) are necessary to ascertain the extent of functional
versatility in the caudal fins of different taxa. Including calculations of
both
p and caudal fin mechanical performance will allow a
better understanding of the interplay between caudal fin performance and
locomotor efficiency.
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Acknowledgments |
---|
![]() |
Footnotes |
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References |
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