The Kármán gait: novel body kinematics of rainbow trout swimming in a vortex street
James C. Liao1,*,
David N. Beal2,
George V. Lauder3 and
Michael S. Triantafyllou4
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
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Fig. 1. Diagram of the experimental setup (A) showing a D-cylinder (not to scale)
in the center of the flow tank with gray arrows representing the direction of
water flow. Images of swimming fish were obtained with a high-speed video
camera aimed at a 45° front-surface mirror positioned below the flow tank.
An image of the ventral view of the fish (B) silhouetted against a lighted
background provided a high-contrast image that could be digitized.
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Fig. 2. Diagram of the experimental design showing the effect of cylinder diameter
and flow speed on vortex shedding frequency and wake wavelength. Objects are
not drawn to scale. (A) The small, 2.5 cm diameter (d) D-cylinder in low flow
(ambient flow speed is set at 2.5 L s-1 prior to solid
blocking effects) has a low shedding frequency f (2.2 Hz) and a short
wavelength  (11 cm). (B) Using the same cylinder and increasing the
ambient flow velocity to 4.5 L s-1, the shedding frequency
almost doubles (4.0 Hz) but the wavelength remains the same. (C) Using the
large, 5 cm diameter D-cylinder at high flow results in a shedding frequency
of 2.2 Hz, which is the same as in A, except that for the large cylinder the
wake wavelength almost doubles (20 cm), representing a substantial difference
in downstream—upstream vortex spacing. Vortex shedding frequency can be
changed by altering cylinder size or flow speed, while wavelength depends only
on cylinder diameter. Shedding frequency and wavelength values reported are
calculated from constricted flow velocities (U; see Materials and
methods). Uf, nominal flow velocity — see text.
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Fig. 3. (A) Superimposed vorticity and velocity vector plot of the wake in the
region that trout were observed holding station downstream behind the 5 cm
cylinder at 4.5 L s-1. The color plot represents vorticity
and the length and orientation of the arrows represents velocity magnitude and
direction. The single row of white arrows represents the region of the wake in
which vectors were selected for statistical analysis of the velocity in (B).
Flow is from bottom to top. One counterclockwise vortex (red) has been shed
into the middle of the frame and a second, clockwise vortex (blue) has just
entered the field of view. Two centers of vorticity are shed per period to
form the Kármán street, with a wavelength of 20.30±0.43
cm (mean ± S.E.M., N=29). (B) Time-averaged velocity profile
of the downstream flow component (x direction) reveals that the
minimum velocity in the wake (2.7±0.2 L s-1; mean
± S.E.M., N=103) is over half the free stream velocity (4.5
L s-1).
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Fig. 4. Superimposed body outlines (A) and body midlines (B) for three treatment
conditions, from left to right; free stream or `no cylinder' treatment, bow
wake treatment in front of a 5 cm D-cylinder, and downstream treatment behind
the 5 cm D-cylinder, illustrating the Kármán gait. The ten body
outlines for each treatment (A) showing approximately one tail-beat cycle were
recorded at intervals of 24 ms, 48 ms and 48 ms, respectively. Spacing along
the x direction between successive outlines reflects relative
swimming velocities. Superimposed midlines (B) show the motion of the body at
higher time resolution for the three treatments. The distance of the fish
relative to the downstream edge of the cylinder (where the downstream edge of
the cylinder is zero, the region upstream of the cylinder is negative, and
downstream of the cylinder is positive) is given on the x-axis, while
the lateral position of the fish relative to the center of the cylinder is
given on the z-axis. For the free stream treatment, values on the
x-axis illustrate the position of the trout relative to a fixed point
upstream (approximately 2.5 L downstream of where the cylinder would
be located for other treatments). Therefore, during the free stream treatment
fish hold station about a half body length further downstream in the flow tank
than during cylinder treatments. The tip of the caudal fin for a fish swimming
in the bow wake is positioned 5 cm upstream from the downstream edge of the
cylinder, or 2.5 cm in front of the cylinder.
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Fig. 5. Mean tail-beat frequency (A) and normalized body wavelength (B) for each of
the five treatments (smD2.5 is the small D-cylinder at 2.5 L
s-1, smD4.5 is the small D-cylinder at 4.5 L
s-1, laD4.5 is the large D-cylinder at 4.5 L
s-1, FS4.5 is the free stream at 4.5 L s-1, and
BW4.5 is the bow wake in front of the large cylinder at 4.5 L
s-1, where L is the total length of the fish). Values are
means ± S.E.M., but in most instances the error bars are small enough
to be obscured by the data symbol. Circles represent fish data (A) and squares
represent cylinder data (see text for calculation). For all three downstream
cylinder treatments, the tail-beat frequency is similar to the vortex shedding
frequency and the fish adopts a body wavelength that is longer than the
cylinder wake wavelength.
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Fig. 6. (A) Lateral body amplitudes taken relative to the midline at four
locations. Circles represent the snout; squares, the center of mass (COM);
diamonds, a point 50% down the body; triangles, the tail tip. (B) Maximum
curvatures and (C) their corresponding positions along the body. Trout behind
the large cylinder have the largest and most anteriorly located body
curvatures, while trout in the bow wake have the smallest and most posteriorly
located curvatures.
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Fig. 7. (A) Distance from the tip of the snout to the downstream edge of the
D-cylinder for the three downstream cylinder treatments. Fish are located
furthest downstream from the laD4.5 treatment, followed by the smD4.5 and
smD2.5 treatments. (B) Mean maximum head angle relative to the axis of free
stream flow. Head angles are higher for trout behind cylinders than for trout
swimming in the free stream and the bow wake. Head angles are not
statistically different between the cylinder treatments that have the same
vortex shedding frequency but different wavelengths (smD2.5 and laD4.5), and
are different between the treatments that have different shedding frequencies
but the same wavelength (smD2.5 and smD4.5).
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Fig. 8. Slip (A) and Strouhal number (B) for all five experimental treatments. (A)
A low slip value, such as that for fish displaying the Kármán
gait behind the large cylinder, indicate that the body wave velocity is
relatively greater than the swimming velocity of the fish. For the slip values
shown here, swimming velocities were taken as the free stream velocity and not
the reduced velocity behind the cylinder. A high slip value for trout swimming
in the bow wake suggests a high mechanical swimming efficiency. (B) Strouhal
number for trout swimming behind the large cylinder is not statistically
different from the Strouhal number for free stream swimming fish. Strouhal
number is significantly lower for fish swimming in the bow wake. See
Discussion for the appropriateness of measuring slip and Strouhal number for
fish swimming behind and in front of cylinders.
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Fig. 9. Principal components analysis (PCA) on 13 kinematic variables for three
downstream cylinder treatments and the free stream treatment. The variables
that loaded high on PC1 were body amplitudes and wave speed, while the
variables that loaded high on PC2 were tail-beat frequency, body wave
velocity, and location of minimum amplitude. Results from a MANOVA show a
significant difference among treatment means (Wilks' lambda, P=0.004,
N=108). SmD4.5 is not statistically different (P>0.05)
from smD2.5 along PC1, or from laD4.5 along PC2.
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Fig. 10. Schematic of the hypothesized hydrodynamic mechanism of the
Kármán gait. A low-pressure, counterclockwise vortex (red
circle) is shed from the cylinder and approaches the head of the trout (A),
causing the incident flow (gray arrow) to be directed at an angle to the body
(simplified as the wide, dark gray line). The region of reduced flow behind
the cylinder is illustrated by the sinusoidal light gray lines. The relatively
large angle of attack of the body produces a lift force (light green arrow)
normal to the path of the incident flow and a drag force (olive arrow)
parallel to the flow. The resultant force (green arrow) can be decomposed into
a forward component (purple arrow) and a lateral component (lavender arrow).
At a small angle of attack (B), such as when a vortex is directly to the side
of the body, the lift force causes the fish to only move laterally, since the
thrust component of the lift force is zero (purple dot). A clockwise vortex
(blue circle) is forming on the opposite side of the cylinder. The shed
clockwise vortex is now just upstream of the trout in (C), and the
counterclockwise vortex has drifted past the body of the trout. Force vectors
are similar to that in A, only opposite in direction. Due to vorticity decay,
the upstream vortex has a lower pressure than the downstream vortex, which may
facilitate station holding.
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© The Company of Biologists Ltd 2003
