The Kármán gait: novel body kinematics of rainbow trout swimming in a vortex street
1 Department of Organismic and Evolutionary Biology, Harvard University,
Cambridge, MA 02138, USA
2 Department of Ocean Engineering, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
3 Department of Organismic and Evolutionary Biology, Harvard University,
Cambridge, MA 02138, USA
4 Department of Ocean Engineering, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
* Author for correspondence (e-mail: jliao{at}oeb.harvard.edu)
Accepted 23 December 2002
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Summary |
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Key words: Kármán street, vortex street, drag wake, vortex, cylinder, hydrodynamic perturbation, swimming kinematics, rainbow trout, Oncorhynchus mykiss, Kármán gait, drafting, tuning, tacking, entraining
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Introduction |
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Fishes swimming in flowing water often prefer to exploit turbulence
associated with physical structures to reduce locomotory costs
(Heggenes, 1988;
Fausch, 1993
;
Webb, 1993
;
Gerstner, 1998
;
Gerstner and Webb, 1998
;
Webb, 1998a
). Numerous field
studies (Puckett and Dill,
1985
; Heggenes,
1988
,
2002
;
McMahon and Gordon, 1989
;
Shuler et al., 1994
;
McLaughlin and Noakes, 1998
;
Pavlov et al., 2000
) attest to
the strong ecological and commercial importance of understanding how fish
interact with unsteady flow regimes. Fisheries scientists need to assess the
impact of hydrodynamic perturbations on migratory species in order to make
management decisions about stream design, habitat management and fish
passageways (Fausch, 1993
;
McLaughlin and Noakes, 1998
).
For engineers and biomechanists, recognizing how fish utilize heterogeneous
flows could provide valuable insight into the governing principles of aquatic
propulsion and stability.
Despite these applied and theoretical benefits, relatively few laboratory
studies (but see Sutterlin and Waddy,
1975; Gerstner,
1998
; Webb, 1998a
)
have quantitatively described the effects of hydrodynamic perturbations at the
organismal level, in part because of the difficulty in generating and
interpreting repeatable hydrodynamic perturbations. The wake behind simple
geometric objects in a moving fluid, a subject well characterized by fluid
mechanists, presents an attractive opportunity to generate consistent flow
perturbations. By altering the dimensions of a given object, it is possible to
manipulate vortical flows systematically to investigate their effect on
swimming fish. In this study, we take advantage of the periodicity of vortex
shedding behind a cylinder to generate repeatable hydrodynamic
perturbations.
For Reynolds numbers between 300 and 150,000, flow past a stationary
cylinder generates a staggered array of discrete, periodically shed, columnar
vortices of alternating sign, collectively known as a vortex of
Kármán street (Blevins,
1990). By changing flow velocity and cylinder diameter, it is
possible to change the frequency of the Kármán vortices (vortex
shedding frequency) and the wavelength that describes their spacing and size
(wake wavelength).
Our goal is to use a controlled experimental system to contribute to an
understanding of how swimming fish interact with hydrodynamic perturbations.
We describe axial body kinematics of rainbow trout swimming behind different
diameter cylinders at two different flow speeds and compare them to trout
swimming in the free stream flow, and also describe the kinematics of trout
swimming in the bow wake in front of a cylinder. Relative to the free stream,
trout behind a cylinder invariably encounter lower flow velocity. We define
the ability of a fish to use this region of reduced flow as `drafting', and
distinguish this from `entraining', which describes the use of upstream flow
in the suction region directly behind a cylinder to hold station
(Webb, 1998a). We employ the
term `tuning' to describe the voluntary alteration of body kinematics to match
a cyclical, external, hydrodynamic stimulus. Furthermore, we make the
distinction that the mechanism for drafting and tuning may or may not be
propulsive in nature. Although we could alter cylinder wake parameters, we
could not separate the presence of the shed vortices from the phenomenon of
reduced flow velocity in this study. Therefore, our goal is to determine if
trout behind cylinders are tuning in addition to drafting. We test the
hypothesis that trout are drafting with propulsive locomotion versus
an alternative hypothesis in which trout are drafting using a non-propulsive
mechanism of locomotion. If trout are only using the drag wake of the cylinder
for its reduced flow, then tail-beat frequency and amplitude should be similar
to trout swimming at free stream flows equivalent to the velocity deficit
behind the cylinder. However, if trout are actually altering their body
kinematics to reflect interaction with the vortices in the cylinder wake, then
we expect that tail-beat frequency and body wavelength will match the
corresponding vortex shedding frequency and wake wavelength of the cylinder,
respectively, and that the general swimming kinematics will differ
substantially from propulsive swimming kinematics in unobstructed flows.
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Materials and methods |
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Experimental procedures
Fish swam in an aerated, 600 liter recirculating flow tank (working section
28 cmx28 cmx80 cm) maintained at 15±1°C. We used 5.0 cm
(large) and 2.5 cm (small) diameter, solid polyvinyl chloride cylinders cut in
half lengthwise (D-section cylinders) to promote more discrete vortex shedding
(Blevins, 1990). The projected
areas of the large and small D-section cylinder (herein referred to simply as
cylinder) were 18% and 9% of the cross-sectional area of the flow tank,
respectively. Cylinders were secured in place from above with the lower end
resting against the bottom of the flow tank to avoid self-oscillation
(Fig. 1A). A RedLake high-speed
digital video camera (250 frames s-1, 1/500th s shutter
speed) recorded the ventral view of the trout against a lighted background,
which was accomplished by aiming the camera at a 45° front-surface mirror
placed below the flow tank (Fig.
1). To minimize turbulence in the flow tank, water was directed
through a baffle consisting of 6 mm diameter flow straighteners located
approximately 50 cm upstream from the working section of the flow tank.
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We specifically designed three `downstream' cylinder treatments to identify
the hydrodynamic variables that might be responsible for the kinematic
behaviors observed (Fig. 2).
Using the Strouhal number (0.2) appropriate for the Reynolds numbers of our
experiment (5600-20,000 and 5000-40,000 using cylinder diameter and fish
length, respectively; Blevins,
1990), we determined the expected vortex shedding frequency
(f) according to the equation:
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Four of the five treatments were conducted at a free stream flow velocity of 4.5 L s-1 (high flow): downstream of the small cylinder (smD4.5), downstream of the large cylinder (laD4.5), free stream (FS4.5), and bow wake in front of the large cylinder (BW4.5). The remaining treatment (smD2.5) consisted of a small cylinder at a velocity of 2.5 L s-1 (low flow), and was designed to replicate the shedding frequency of the large cylinder at high flow, despite halving its wavelength and therefore decreasing the vortex spacing in the x direction (Fig. 2). For a given Strouhal number, vortex shedding frequency can be altered by cylinder diameter or flow speed (Equation 1). In contrast, by solving for f in Equation 1 and substituting into Equation 3, it is apparent that wake wavelength depends only on cylinder diameter. We were interested in testing the effect of vortex shedding frequency and wake wavelength on fish swimming, so we designed flow velocity and cylinder diameter combinations for our treatments to hold one wake parameter constant while varying the other. For example, subjecting trout to two different speeds using the same cylinder allowed us to observe differences in swimming kinematics caused by changing the vortex shedding frequency without changing the wake wavelength (Fig. 2A,B).
The high and low flow velocities were chosen because they allowed
manipulation of cylinder wake variables, elicited recognizable, stereotypical
body kinematics, and were within the range of swimming speeds attained by
trout swimming behind boulders and woody debris in the field
(Heggenes, 1988;
McMahon and Gordon, 1989
;
Shuler et al., 1994
). For all
treatments, we analyzed only those sequences in which fish were swimming more
than 0.3 L above the bottom of the flow tank. Because cylinders were
positioned in the middle of the flow tank, all swimming sequences by necessity
were obtained with the fish at least 1 L away from the side
walls.
We used digital particle image velocimetry (DPIV) to verify the presence of
a Kármán street behind the cylinders, confirm calculated
shedding frequencies and wake wavelengths, and make measurements of the
reduced flow (velocity deficit) in the center of the wake. Silver-coated glass
spheres (12 µm diameter) were seeded in the flow tank and illuminated by a
horizontal light sheet (15 cmx28 cmx0.1 cm) generated by an 8 W
argon-ion Coherent laser. Two-frame cross-correlation analysis of particle
images recorded 4 ms apart yielded a 31x36 matrix of 1116 velocity
vectors, from which a plot of vorticity was generated and overlaid
(Fig. 3A; Insight version 3.0
software, TSI Inc., St. Paul, MN, USA). DPIV procedures follow those used
previously to study locomotor hydrodynamics in freely swimming fishes
(Drucker and Lauder, 1999;
Liao and Lauder, 2000
;
Wilga and Lauder, 2000
;
Nauen and Lauder, 2002
). At
approximately the same distance (20 cm) and height (5 cm) that fish were
observed holding station downstream from the large cylinder, a single row of
36 velocity vectors (Fig. 3A,
white arrows) along the z direction was averaged for 103 continuous
video frames (in the absence of a fish), representing a random, 8 s sequence
of the wake (Fig. 3B).
Predicted values of vortex shedding frequency and wavelength (Equations 1-3)
were verified by statistical comparison to empirically derived values based on
visual selection of vortex centers.
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From the video sequences, we measured tailbeat frequency, wavelength, wave
velocity, magnitude and location of maximum body curvature, maximum amplitude
of four body locations (snout, center of mass, 50% L and tail),
maximum head angle and downstream distance from snout to cylinder. Tail-beat
frequency was determined by averaging the number of tail-beat oscillations
over a known time (at least four oscillations per individual). Body wavelength
was calculated according to Equation 3 by substituting the length-specific
body wave speed (V, obtained by measuring the average speed of the
maxima moving down the midline) for the nominal flow velocity
(Uf) and the tail-beat frequency for the vortex shedding
frequency (f). The four body locations were selected to provide a
general indication of the effects of different cylinder wakes on longitudinal
body kinematics. Amplitudes at these four points were measured as the maximum
lateral excursion from the midline. We experimentally determined the center of
mass (COM) post-mortem for straight-stretched fish by iteratively
balancing the body between right and left side pins. Head angle was calculated
as the maximum angle of the head relative to the axis of the free stream flow.
In addition, we calculated slip as Uf/V, where
Uf is taken to be the length-specific swimming velocity of
the fish. Strouhal numbers for swimming trout were calculated according to
Equation 2 by substituting tail-beat frequency for vortex shedding frequency
(f) and tail-beat amplitude (using peak-to-peak amplitude; as in
Triantafyllou et al., 2000)
for cylinder diameter (d).
Image analysis
For all treatments, at least 20 video frames (separated in time by 12-20
ms, depending on the swimming speed of the fish) were captured for each fish
for each of four tail-beat trials. We used an automated digitizing program
written in Matlab v6.1 to digitize 20 points on each side of the axial body
silhouette, thresholding out interferences with fin profiles, for a total of
40 points per image. After a series of cubic spline functions were fitted to
these outline points, a midline spline was constructed and then divided into
20 segments, the longitudinal positions of each which were recorded as
relative body lengths (as in Jayne and
Lauder, 1995).
Statistical tests
Means and standard errors were calculated for all kinematic variables
associated with swimming. Two-sample unpaired t-tests were conducted
to see if tail-beat frequency differed from vortex shedding frequency and if
body wavelength differed from wake wavelength. All statistical tests were
performed at an level of 0.05.
A principal-components analysis determined which of the kinematic variables contributed to most of the variation in the entire data set. Data from the bow wake treatment were excluded since we were primarily interested in detecting differences in swimming kinematics between downstream cylinder treatments and free stream swimming. A multivariate analysis of variance (MANOVA) was performed on principal component 1 (PC1) and principal component 2 (PC2) to determine if there were significant differences among treatment means when all kinematic variables were considered together.
A MANOVA and subsequent post-hoc tests on all treatments except
the bow wake (to make maximum use of the data set, 7 individuals) determined
if there were differences between the downstream cylinder treatments and the
free stream treatment as well as within cylinder treatments. We performed a
MANOVA on all five treatments (4 individuals) to analyze bow wake data in the
context of other treatments. Two-way, mixed-model analyses of variance (ANOVA)
were used to determine the effects of individual (random) and cylinder
treatment (fixed) on the kinematic variables. The F value of the
fixed effect was calculated as the mean square of the fixed effect over the
interaction term of the random effect and the fixed effect. To account for
multiple simultaneous ANOVAs, the level of significance was adjusted within
columns using the sequential Bonferroni technique
(Rice, 1989). Where
appropriate, we note when P values are less than 0.05 but do not meet
the Bonferroni adjusted significance value. For all kinematic variables, we
recorded 3-4 tail-beat cycles for each fish for each treatment. Statistical
tests were performed by Systat (version 4.5 for the PC), Statview (version
5.0.1 for the PC), or calculated from Zar
(1999
).
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Results |
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Holding station behind cylinders versus free stream
swimming
Trout holding station behind a cylinder display unique axial body
kinematics that are readily distinguishable from steadily swimming trout using
active undulatory propulsion in the absence of a cylinder
(Fig. 4). In general, trout
holding station behind a cylinder appear passive, resembling the motion of a
flag flapping slowly in the wind. Compared to trout swimming in the free
stream at 4.5 L s-1, trout holding station behind the
large cylinder at the same speed show a 303% decrease in tail-beat frequency
and a 354% increase in body wavelength
(Fig. 5). In addition, body
amplitudes increase 300-800% (depending on the body point), owing to the
lateral translation of the entire body in the wake (Figs
4,
6). All body amplitudes
increase significantly (P<0.001) for trout behind the large
cylinder as compared to trout swimming in the free stream. For trout behind
small cylinders, the increase in the snout and COM amplitude is greater
relative to the posterior half of the body compared to trout swimming in the
free stream (Table 1).
Irrespective of flow velocity, trout behind a small cylinder display larger
midbody and tail amplitudes (P<0.002,
Table 1) than those of trout in
the free stream, but have similar snout and COM amplitudes (P=0.68,
P=0.59, respectively). Maximum body curvature for trout behind the
large cylinder is not statistically different from that of trout swimming in
the free stream (P=0.054, Fig.
6), though the absolute radius of curvature is larger. Compared to
trout swimming with active undulatory propulsion in the free stream, the
location of maximum curvature is closer to the head for trout holding station
behind cylinders (P<0.05, N=108). Trout behind the large
cylinder at high flow and the small cylinder at low flow have larger head
angles than free stream swimming trout (P<0.001,
Fig. 7).
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Traditional metrics of fish swimming efficiency further illustrate how swimming behind a cylinder differs from swimming in the free stream. Trout behind the large cylinder have a lower slip value, resulting from a higher body wave speed, than trout swimming in the free stream at 4.5 L s-1 (P=0.001, N=108, Fig. 8A). By contrast, the body wave speeds for trout behind the small cylinders are lower than that for trout swimming in the free stream, resulting in higher slip values (P=0.001, N=108, Fig. 8A). The Strouhal number for trout holding station behind a large cylinder is not statistically different from that of a trout swimming in the free stream. In comparison, trout holding station behind small cylinders have much lower Strouhal numbers (Fig. 8B).
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Locomotor kinematics behind different cylinder treatments
Trout alter their axial swimming kinematics to reflect the hydrodynamic
characteristics of the cylinder wake. We found that trout generally match
their tail-beat frequency to the expected vortex shedding frequency for all
three downstream cylinder treatments (see
Fig. 5 and
Table 1). Trout behind
cylinders have a body wavelength that is always longer than the corresponding
wake wavelength (P<0.05, N=108;
Table 1). For example, behind
the large cylinder the body wavelength (41 cm) is more than twice as long as
the wake wavelength (20 cm; Figs
2,
5). Body wavelength differs
among trout behind the large and small cylinders (P<0.001,
N=108) but not between the two small cylinder treatments
(P=0.19, N=108, Fig.
2), reflecting the relationships in wake wavelength. There is a
larger difference between the body and wake wavelengths for trout behind the
large cylinder treatment than for trout behind the small cylinder treatments.
Comparisons among the three downstream cylinder treatments reveal that trout
do not match their body wave speed to the speed at which they encounter shed
vortices.
Regardless of flow speed, trout behind the small cylinder have body
amplitudes that are not statistically different from each other
(P0.37, N=108, Fig.
6A), indicating that shedding frequency does not affect body
amplitude. These body amplitudes behind the small cylinders are, in turn, much
smaller than that of trout behind the large cylinder (P<0.001,
N=108). The amplitude of the snout, COM and midbody are similar for
trout behind the large cylinder, while the tail-beat amplitude is about twice
this magnitude. Trout subjected to the small cylinder treatments have snout
amplitudes that are similar to the COM amplitude, but much larger midbody and
the tail-tip amplitudes (3x and 10x, respectively), resulting in a
steeper amplitude envelope than seen in trout swimming behind the large
cylinder.
The degree and location of body bending depends on the size and strength of the shed vortices and not on the frequency of vortex shedding. Maximum curvature along the body of the trout does not track cylinder diameter or flow speed individually; only a combination of the two causes larger body curvatures. For example, the maximum body curvature for trout behind the large cylinder is not significantly different from that of trout behind the small cylinder at high flow (Fig. 6B), but is higher than for trout behind the small cylinder at low flow (P<0.05, N=108, Fig. 6B,C; Table 2). Likewise, the maximum body curvatures for trout behind the small cylinder treatments at two different flow speeds are not statistically different (P<0.05, N=108, Fig. 6B,C; Table 2). Thus, when flow speed is held constant, trout behind the large cylinder do not bend their body to a greater degree than trout behind the small cylinder; the only difference is that trout behind the large cylinder use more of the body to participate in the curvature. Of the three downstream cylinder treatments, maximum body curvature is closest to the tail for trout behind the small cylinder, high flow treatment, and closest to the head for trout behind the large cylinder (P<0.05, N=108, Fig. 6C). Longitudinal position of maximum body curvature does not vary with flow speed, as shown by comparing the values from the two small cylinder treatments. Thus, the degree of maximum body curvature depends on the size and strength of the vortices generated by the combination of cylinder diameter and flow speed, while the location of maximum curvature depends only on the cylinder diameter.
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Additional locomotor variables illustrate that trout behind cylinders are
reacting to wake parameters. Trout hold station further downstream from the
large cylinder treatment than the small cylinder treatment at low flow
(P<0.005, N=81); however, downstream distances are not
different between other treatments (P0.1157, N=81,
Fig. 7A). The two treatments
that generate the lowest vortex shedding frequency (smD2.5 and laD4.5,
Fig. 7B) also elicit the
largest head angles (which are not statistically different from each other).
Body wave speed increases with the combination of increased cylinder diameter
and flow velocity, and is not affected by either variable individually
(P=0.001, N=108). Specifically, trout behind the large
cylinder have a faster body wave than trout behind the small cylinder at high
flow, which in turn have a faster body wave than the small cylinder at low
flow (P=0.001, N=108). This generates a low slip value for
trout behind the large cylinder (Fig.
8). Strouhal numbers for all high flow treatments, with the
exception of the bow wake, fall within the range 0.25-0.35.
Locomotor kinematics in the bow wake versus free
stream
Only certain kinematic parameters differ between trout swimming in front of
a cylinder and trout swimming in the free stream after Bonferroni adjustment.
Trout in the bow wake have a much lower tail-beat frequency than trout
swimming in the free stream (P<0.0001, N=79,
Fig. 5A). Although trout in the
bow wake (Figs 4,
6) have body amplitudes that
are not different than trout swimming in the free stream after Bonferroni
adjustment, they have lower tail-beat amplitudes (P<0.05,
N=79) prior to Bonferroni adjustment. Likewise, maximum body
curvature is significantly lower (P<0.05, N=79) for fish
in the bow wake only prior to Bonferroni adjustment. Maximum body curvature is
located further posteriorly along the body for bow wake fish than for free
stream fish (P<0.005, N=79,
Fig. 6). Fish swimming in the
bow wake do not have a smaller maximum head angle after Bonferroni adjustment.
Although body wavelength is not statistically different between the two
treatments, the speed at which this wave passes down the body is much lower
(P<0.0001, N=79) and more variable for fish swimming in
the bow wake, thus generating a higher slip value
(Fig. 7). Strouhal number for
the bow wake is 0.17, the lowest of all treatments.
Principal components analysis
A principal components analysis on 13 kinematic variables for all
treatments except the bow wake indicated that PC1 and PC2 accounted for 50%
and 18% of the total variation and in the dataset, respectively
(Fig. 9). The variables most
responsible for separating out the treatments along PC1 were body wavelength
and body amplitude, while the variables most responsible for separating out
the treatments along PC2 were tail-beat frequency, body wave speed, and
location of minimum amplitude. Performing a MANOVA on these scores indicated a
highly significant difference among the treatment means (Wilks' lambda
F=23.94, P=0.004). Post-hoc tests indicate that for
PC1 all treatment means were different (P<0.05, N=108)
except between the two small cylinder treatments
(Fig. 9). For PC2, all
treatment means were different (P<0.005, N=108) except
between the small cylinder and large cylinder at high flow speed.
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Discussion |
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Previous studies using smaller cylinders
(Sutterlin and Waddy, 1975;
Webb, 1998a
) found that fish
entrain (hold station using upstream flow) with their head just behind and to
one side of a cylinder. In addition to several biotic factors
(Webb, 1998a
), entraining may
depend on the ratio of cylinder diameter to fish length, which at 1:12
(Sutterlin and Waddy, 1975
) is
much smaller than for the large cylinder treatment in this study (1:2). The
proximity of the body to the cylinder in these earlier studies
(Sutterlin and Waddy, 1975
;
Webb, 1998a
) indicates that
fish were taking advantage of actual upstream flow in the low-pressure region
directly behind the cylinder (Vogel,
1994
; Zdravkovich,
1997
). In addition, the proximity of the fish to the cylinder can
disrupt the vortex shedding frequency and may be responsible for the aperiodic
body movements observed. Station holding strategies may also differ with
species. Trout in this study position themselves further downstream from a
cylinder compared to river chub and smallmouth bass, despite a similar
cylinder diameter to fish length ratio (1:4) and swimming speed (3.5-4.5
Ls-1; Webb,
1998a
).
Given our experimental conditions, trout do not entrain in the suction
region behind cylinders. Instead, they hold station far enough away from the
cylinder that the net flow they experience is always directed downstream.
Trout choose not to position themselves close to the cylinder (<1
L) because the average low-pressure region, which extends less than
two diameters downstream from the cylinder
(Gerrard, 1966), is strong
enough to draw them into contact with the cylinder and thereby disrupt their
station holding behavior. The distance where trout hold station behind the
large cylinder coincides with the region in which the Kármán
vortices are fully formed; depending on the Reynolds number, this is 3-5
cylinder diameters or 1.5-2.5 L downstream of the cylinder
(Weihs, 1973
;
Zdravkovich, 1997
;
Fish, 1999
). Our data show a
tendency for trout to hold station further downstream as cylinder diameter and
flow speed increase (Fig.
7A).
We propose that the onset of the Kármán gait coincides with
the hydrodynamic stability and strength of the cylinder wake. By using long,
fine-diameter flow straighteners and large D-cylinders, we able to generate a
relatively stable vortex street (verified by DPIV) at these high Reynolds
numbers to observe the Kármán gait. Only within a certain range
of flow velocities and cylinder sizes do vortices shed periodically and
discretely enough to provoke a regular kinematic response. During low flow,
fish avoid swimming behind cylinders, and at high flow (6
Ls-1), turbulence displaces fish from the wake, as
revealed by sudden, high amplitude lateral excursions of the whole body.
Although we confirmed the periodicity of the shed vortices with DPIV, at the
Reynolds numbers of our experiment the three-dimensional cylinder wake is
still subject to turbulence, axial flow and braid vortices
(Blevins, 1990). We attribute
interruptions during the otherwise steady Kármán gait to
corrective, stabilizing movements used to compensate for these flow
irregularities. Preliminary results from a cleaner, more stable wake generated
by an automated, heaving cylinder verify that trout adopt a
Kármán gait for longer periods with fewer corrective
motions.
Kármán gait kinematics: evidence and implications of
tuning
The hypothesis that fish are swimming behind cylinders only to take
advantage of the reduced flow velocity (i.e. drafting) is not supported. On
the contrary, the distinctive and consistent changes in axial body kinematics
among different cylinder treatments supports the hypothesis that trout can
tune their swimming kinematics to capture energy from vortices shed from
cylinders. The body wave speed of trout behind small cylinders is lower than
the body wave speed for similar sized trout swimming at the equivalent free
stream speed (Webb et al.,
1984), indicating that trout are not producing thrust in the same
manner as when they swim in the free stream. The dramatic differences in the
Kármán gait as compared to trout swimming in the free stream at
4.5 Ls-1 probably reflect different mechanisms of
locomotion. Since the flow velocity behind the large cylinder is much lower
than the free stream velocity, it can be argued that a more legitimate
comparison would be between the kinematics of the Kármán gait
behind the large cylinder and the kinematics of trout swimming at an
unobstructed flow equivalent to the reduced flow behind the large cylinder.
Though we did not measure the tail-beat frequency and body wavelength of trout
swimming at a free stream velocity of 2.7 Ls-1, values
taken from similarly sized trout at a similar flow velocity (approx. 4.5 Hz
and 0.97 L, respectively; Webb et
al., 1984
) are significantly higher and shorter than trout
exhibiting the Kármán gait behind the large cylinder.
As mentioned above, the ratio of fish length to cylinder diameter is probably a key factor in determining the ability of fish to use the Kármán gait. Certain cylinder diameters will prevent fish of a given length from conforming their bodies to the size and spacing of vortices in the Kármán street. Contrary to one of our predictions, fish do not alter their body wavelength to match the wake wavelength prescribed by the cylinder. Why do trout consistently adopt a longer body wavelength than the wavelength dictated by the spacing of the vortices, and why does this relationship change with cylinder size? One explanation is that during the Kármán gait, trout position their head differently relative to a vortex than their tail. For example, by positioning the head to intercept a vortex and the tail to slalom around it, a trout will have a longer body wavelength than the cylinder wavelength despite interacting with each vortex in a consistent manner. Therefore, even though the wavelength along the body does not exactly match the wake wavelength for each of the three respective downstream cylinder treatments, the body wavelengths for trout subjected to the small cylinder treatments are still more similar to each other than either is to the wake wavelength behind the large cylinder. Since it is likely that trout utilize vortices by maintaining the body at an angle of attack relative to the localized region of vortical flow, then as vortex diameter increases, body wavelength may increase to insure this condition. In contrast to the wake wavelength, the shedding frequency, or the speed at which trout encounter vortices, has less of an effect on how trout shape their body into waves. What shedding frequency does dictate is the location of maximum body curvature, revealing that fish are not only conforming their bodies to interact with the shed vortices, but that they do so in a consistent way despite encountering differences in vortex size and strength.
Wake parameters differ in every respect between the small and large cylinder at high flow and yet trout show similar maximum body curvatures when swimming behind them. While body curvature may reflect the degree of body bending around large vortices shed from the large cylinder, the same magnitude of curvature for fish behind the small cylinder treatment may reflect a shift in station holding strategy to include active propulsion in addition to tuning. In support of this, maximum body curvature in trout swimming behind a small cylinder at high flow is no different from that of trout swimming with undulatory propulsion in the free stream at the same speed. For trout swimming behind small cylinders, more of the body moves laterally and by a greater factor relative to the center of mass than trout behind the large cylinder. The steeper amplitude envelope and larger relative midbody amplitude for trout behind the small cylinder treatments suggest that either body amplitude reflects a hydrodynamic relationship particular to shorter wake wavelengths, or that there is a degree of active undulation that is not present for trout swimming behind the large cylinder.
Head angles track vortex shedding frequency, further supporting the idea that body kinematics reflect interaction with vortices and not just undulatory propulsion through the reduced flow behind a cylinder (Fig. 7B). Fish subjected to treatments with the same shedding frequency adopt similar head angles, suggesting that buffeting of the head by the vortices, combined with active contraction of anterior axial muscles, may be what determines the position of the head in the current. Alternatively, head angle may result from an entirely passive motion generated by the buffeting effect of the vortices.
Our calculations of slip (Lighthill,
1975) for trout behind cylinders are likely to be an
overestimation because we considered swimming velocity equivalent to the
nominal velocity (Uf), rather than the reduced velocity
behind the cylinder. If slip is calculated by substituting the appropriate
reduced velocity value behind a cylinder for fish swimming speed, then slip
values for fish behind all cylinders are about half those of free stream
swimming fish. Slip values may not be a useful metric of efficiency for fish
swimming in perturbed environments since the proportion of the body wave that
generates thrust propulsively is unknown. Thus, the faster body wave in trout
behind cylinders may indicate vortex energy capture and wake buffeting rather
than axial propulsion. If trout are holding station behind the large cylinder
via some mechanism of vorticity control, they may be experiencing
enhanced locomotory performance compared to trout swimming in the free stream
(Anderson, 1996
;
Triantafyllou et al.,
2000
).
Hypothesized mechanism of the Kármán gait
Our results, which clarify the relationship between body kinematics and
hydrodynamic parameters, lead us to propose that axial muscles may be largely
inactive during the Kármán gait, especially behind the large
cylinder. By what mechanism can a fish hold station or move upstream entirely
passively if there is no net flow upstream? Drawing on a nautical analogy, we
hypothesize that trout use their body like a sail to tack upstream. By
changing the camber and angle of attack of the body to establish a
differential pressure gradient (using the low pressure region of each oncoming
vortex), it is possible to generate a net circulation around the body that, on
average, keeps a component of the lift force directed upstream
(Fig. 10). Thus, a major
component of the Kármán gait may consist of a passive,
lift-based mechanism that relies on the interaction of a body with the high
velocity regions of shed vortices, similar to the mechanism of thrust
production in foils oscillating in a vortex street
(Streitlien and Triantafyllou,
1996).
|
Although not quantified in this study, it is possible that by positioning
the body at a high angle of attack relative to the incident flow the lift
force generated will direct the fish upstream as well as laterally, thereby
causing the fish to hold position relative to the cylinder instead of drifting
downstream (Fig. 10A).
However, at times the incident flow may be parallel to the body
(Fig. 10B). At this moment,
the resultant lift vector can only be resolved into a lateral force, and the
fish translates sideways but does not generate thrust. Over the course of one
Kármán gait cycle, on average, the angle of attack of the body
should generate enough thrust to overcome drag, allowing the fish to hold
station while exhibiting large lateral oscillations of the body. Postural
changes of the body such as these probably generate `trimming' forces that
stabilize the body in the presence of hydrodynamic perturbations
(Webb, 2002).
An additional mechanism may contribute to the ability of trout to hold
station using the Kármán gait. Since vorticity decays abruptly
with distance from its source in turbulent systems
(Glezer and Coles, 1990), the
low-pressure region of a vortex returns to ambient as it drifts downstream. A
trout positioning its head near an oncoming low-pressure vortex and its tail
near a passing vortex (which has a relatively higher pressure) would be
continuously drawn upstream relative to the flow, but hold station relative to
the cylinder.
Bow wake
When trout swim in the low velocity, high-pressure region in front of a
cylinder, their low tail-beat frequency, body wave speed and posterior body
curvature suggest they are using the least amount of energy of all the
treatments (Figs 5,
6). At the behavioral level,
once fish discover the bow wake they do not leave, choosing to return
immediately to the front rather than behind the cylinder if displaced. A
similar phenomenon is seen in dolphins riding in the bow wake of ships
(Scholander, 1959;
Fejer, 1960
;
Bose and Lien, 1990
). Slip
values for bow wake fish indicate the highest mechanical efficiencies of all
other treatments; however, since the flow velocity near the tail is not the
same as along the anterior region of the body, calculating slip overestimates
the forward swimming velocity of the fish. In addition, we do not know what
portion of the tail-beat amplitude and frequency reflects propulsive thrust
generation. Strouhal number is lowest for fish swimming in the bow wake,
indicating a suboptimal swimming efficiency for what seems to be the most
economical kinematic behavior of all the treatments. Since highly efficient
swimmers have Strouhal numbers between 0.25-0.30
(Triantafyllou et al., 2000
),
the low value for bow wake fish (0.17) illustrates another unique mechanism of
station-holding separate from the Kármán gait.
Conclusions and future directions
Our understanding of fish locomotion is dominated by detailed studies of
steady swimming in homogenous, laboratory flow environments. The ubiquity of
turbulence in nature provides ample justification for exploring the effects of
hydrodynamic perturbations on fish swimming. Quantitatively assessing the
reactions of fish to unpredictable hydrodynamic perturbations found in the
natural environment is not currently within our grasp. Examining how fish
alter their swimming behavior in response to periodic perturbations in the
laboratory, however, is a productive intermediate step that promises to shed
light on biomechanical questions regarding stability and control of aquatic
animals, as well as stimulate practical applications for fisheries management.
While the general tendency of salmonids to reduce locomotory costs by
exploiting flow refugia has been shown in the field
(Puckett and Dill, 1985;
Heggenes, 1988
;
McMahon and Gordon, 1989
;
Fausch, 1993
), the results of
this paper detail the ability of trout to hold station by (1) adopting a novel
gait to utilize the vortex street behind a cylinder and (2) swimming in the
bow wake in front of a cylinder.
Ongoing investigations using electromyography, DPIV and oscillating hydrofoils should clarify the neuromuscular and hydrodynamic basis of the Kármán gait, enabling us to better address the mechanisms by which fish can make use of hydrodynamic perturbations created by inanimate objects in moving flow or by schooling fish. Current projects and key future questions include the following.
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Acknowledgments |
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Footnotes |
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References |
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Alexander, R. M. (1989). Optimization and gaits
in the locomotion of vertebrates. Physiol. Rev.
69,1199
-1227.
Anderson, J. M. (1996). Vorticity control for efficient propulsion. PhD thesis, Joint Program MIT/Woods Hole Oceanographic Institution.
Blake, R. W. (1983). Fish Locomotion. Cambridge: Cambridge University Press.
Blevins, R. D. (1990). Flow Induced Vibration, 2nd edition. Malabar, Florida: Krieger Publishing Company.
Bose, N. and Lien, J. (1990). Energy absorption from ocean waves: a free ride for cetaceans. Proc. R. Soc. Lond. B 240,591 -605.[Medline]
Drucker, E. G. and Lauder, G. V. (1999).
Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics
quantified using digital particle image velocimetry. J. Exp.
Biol. 202,2393
-2412.
Fausch, K. D. (1993). Experimental analysis of microhabitat selection by juvenile steelhead (Oncorhynchus mykiss) and coho salmon (Oncorhynchus kisutch) in a British Columbia stream. Can. J. Fish. Aquat. Sci. 50,1198 -1207.
Fejer, A. A. (1960). Porpoises and the bow-riding of ships under way. Nature 188,700 -703.
Fish, F. E. (1999). Energetics of swimming and flying in formation. Comments Theor. Biol. 5, 283-304.
Gerrard, J. H. (1966). Formation region of vortices behind bluff bodies. J. Fluid Mech. 25,401 -413.
Gerstner, C. L. (1998). Use of substratum ripples for flow refuging by Atlantic cod, Gadus morhua. Environ. Biol. Fishes 55,455 -460.
Gerstner, C. L. and Webb, P. W. (1998). The station-holding performance of plaice, Pleuronectes platessa, on artifical substratum ripples. Can. J. Zool. 76,260 -268.[CrossRef]
Glezer, A. and Coles, D. (1990). An experimental study of a turbulent vortex ring. J. Fluid Mech. 211,243 -283.
Heggenes, J. (1988). Effects of short-term flow fluctuations on displacement of, and habitat use by, brown trout in a small stream. Trans. Am. Fish. Soc. 117,336 -344.[CrossRef]
Heggenes, J. (2002). Flexible summer habitat selection by wild, allopatric brown trout in lotic environments. Trans. Am. Fish. Soc. 131,287 -298.[CrossRef]
Jayne, B. C. and Lauder, G. V. (1995). Speed
effects on midline kinematics during steady undulatory swimming of largemouth
bass, Micropterus salmoides. J. Exp. Biol.
198,585
-602.
Liao, J. and Lauder, G. V. (2000). Function of
the heterocercal tail in white sturgeon: flow visualization during steady
swimmming and vertical maneuvering. J. Exp. Biol.
203,3585
-3594.
Lighthill, J. (1975). Mathematical Biofluiddynamics. Philadelphia: Society for Industrial and Applied Mathematics.
McLaughlin, R. L. and Noakes, D. L. G. (1998). Going against the flow: an examination of the propulsive movements made by young brook trout in streams. Can. J. Fish. Aquat. Sci. 55,853 -860.[CrossRef]
McMahon, T. E. and Gordon, F. H. (1989). Influence of cover complexity and current velocity on winter habitat use by juvenile coho salmon (Oncorhynchus kisutch). Can. J. Fish. Aquat. Sci. 46,1551 -1557.
Nauen, J. C. and Lauder, G. V. (2002).
Hydrodynamics of caudal fin locomotion by chub mackerel, Scomber
japonicus (Scombridae). J. Exp. Biol.
205,1709
-1724.
Pavlov, D. S., Lupandin, A. I. and Skorobogatov, M. A. (2000). The effects of flow turbulence on the behavior and distribution of fish. J. Ichthyol. 40,S232 -S261.
Puckett, K. J. and Dill, L. M. (1985). The energetics of feeding territoriality in juvenile coho salmon (Oncorhynchus kisutch). Behaviour 92, 97-110.
Rayner, J. M. V. (1995). Dynamics of the vortex wakes of flying and swimming vertebrates. In Biological Fluid Dynamics (ed. C. P. Ellington and T. J. Pedley), pp.131 -155. Cambridge: Company of Biologists Limited.
Rice, W. R. (1989). Analyzing tables of statistical tests. Evolution 43,223 -225.
Scholander, P. F. (1959). Wave-riding dolphins, how do they do it? Science 129,1085 -1087.
Shuler, S. W., Nehring, R. B. and Fausch, K. D. (1994). Diel habitat selection by brown trout in the Rio Grande river, Colorado, after placement of boulder structures. N. Am. J. Fish. Manage. 14,99 -111.
Streitlien, K. and Triantafyllou, G. S. (1996). Efficient foil propulsion through vortex control. Amer. Inst. Aeronaut. Astronaut. J. 34,2315 -2319.
Sutterlin, A. M. and Waddy, S. (1975). Possible role of the posterior lateral line in obstacle entrainment by brook trout (Salvelinus fontinalis). J. Fish. Res. Board Can. 32,2441 -2446.
Tobalske, B. W. (2000). Biomechanics and physiology of gait selection in flying birds. Physiol. Biochem. Zool. 73,736 -750.[CrossRef][Medline]
Triantafyllou, M. S., Triantafyllou, G. S. and Yue, D. K. P. (2000). Hydrodynamics of fishlike swimming. Ann. Rev. Fluid Mech. 32,33 -53.[CrossRef]
Vogel, S. (1994). Life in Moving Fluids: The Physical Biology of Flow, 2nd edition. Princeton: Princeton University Press.
Webb, P. W. (1991). Composition and mechanics of routine swimming of rainbow trout, Oncorhynchus mykiss. Can. J. Fish. Aquat. Sci. 48,583 -590.
Webb, P. W. (1993). The effect of solid and
porous channel walls on steady swimming of steelhead trout Oncorhynchus
mykiss. J. Exp. Biol. 178,97
-108.
Webb, P. W. (1998a). Entrainment by river chub
Nocomis micropogon and smallmouth bass Micropterus dolomieu
on cylinders. J. Exp. Biol.
201,2403
-2412.
Webb, P. W. (1998b). Swimming. In The Physiology of Fishes (ed. D. H. Evans), pp.3 -21. Boca Raton, Florida: CRC Press.
Webb, P. W. (2002). Control of posture, depth, and swimming trajectories of fishes. Integ. Comp. Biol. 42,94 -101.
Webb, P. W., Kostecki, P. T. and Stevens, E. D. (1984). The effect of size and swimming speed on the locomotor kinematics of rainbow trout. J. Exp. Biol. 109, 77-95.
Weihs, D. (1973). Hydromechanics of fish schooling. Nature 241,290 -291.
Wilga, C. D. and Lauder, G. V. (2000).
Three-dimensional kinematics and wake structure of the pectoral fins during
locomotion in leopard sharks Triakis semifasciata. J. Exp.
Biol. 203,2261
-2278.
Zar, J. H. (1999). Biostatistical Analysis, 4th edition. Englewood Cliffs: Prentice-Hall, Inc.
Zdravkovich, M. M. (1997). Flow Around Circular Cylinders: A Comprehensive Guide Through Flow Phenomena, Experiments, Applications, Mathematical Models, and Computer Simulations. Oxford: Oxford University Press.