The effect of flow speed and body size on Kármán gait kinematics in rainbow trout

Abstract

We have little understanding of how fish hold station in unsteady flows. Here, we investigated the effect of flow speed and body size on the kinematics of rainbow trout Kármán gaiting behind a 5 cm diameter cylinder. We established a set of criteria revealing that not all fish positioned in a vortex street are Kármán gaiting. By far the highest probability of Kármán gaiting occurred at intermediate flow speeds between 30 and 70 cm s(-1). We show that trout Kármán gait in a region of the cylinder wake where the velocity deficit is about 40% of the nominal flow. We observed that the relationships between certain kinematic and flow variables are largely preserved across flow speeds. Tail-beat frequency matched the measured vortex shedding frequency, which increased linearly with flow speed. Body wave speed was about 25% faster than the nominal flow velocity. At speeds where fish have a high probability of Kármán gaiting, body wavelength was about 25% longer than the cylinder wake wavelength. Likewise, the lateral (i.e. cross-stream) amplitude of the tail tip was about 50% greater than the expected lateral spacing of the cylinder vortices, while the body center amplitude was about 70% less. Lateral body center acceleration increased quadratically with speed. Head angle decreased with flow speed. While these values are different from those found in fish swimming in uniform flow, the strategy for locomotion is the same; fish adjust to increasing flow by increasing their tail-beat frequency. Body size also played a role in Kármán gaiting kinematics. Tail-beat amplitudes of Kármán gaiting increased with body size, as in freestream swimming, but were almost three times larger in magnitude. Larger fish had a shorter body wavelength and slower body wave speed than smaller fish, which is a surprising result compared with freestream swimming, where body wavelength and wave speed increased with size. In contrast to freestream swimming, tail-beat frequency for Kármán gaiting fish did not depend on body size and was a function of the vortex shedding frequency.

Keywords: fish swimming; turbulence; unsteady flow; vortex street.

Fig. 1.

Schematic diagram of the experimental setup. (A) Trout swam behind a stationary D-section cylinder positioned in a flow tank in which flow speed could be digitally controlled. A high-speed video camera was aimed at a front-surface mirror that was positioned under the flow tank at a 45 deg angle. (B) Ventral image of the fish in the flow tank. (C) Lateral view of the working section of the flow tank.

Fig. 2.

The probability of Kármán gaiting changes with flow speed. Fish Kármán gait the most at intermediate flow speeds (~30–70 cm s⁻¹) and the least at extreme speeds. Videos were binned into five flow speed categories, where each category consisted of a minimum of 50 videos from at least five different fish.

Fig. 3.

(A) Location of the body center (BC, black circles) of the body relative to the D-cylinder for all trials. The x- and y-axes show the downstream and cross-stream position, respectively, where 0 corresponds to the cylinder axis. (B) Downstream position of the BC relative to the cylinder as flow velocity increases. The BC positions are superimposed on a heat map illustrating the magnitude of the velocity deficit behind the cylinder as a percentage of the freestream velocity, where red represents the greatest relative flow reduction. The location of greatest flow reduction remains in a consistent region downstream of the cylinder across most flow speeds. Note that this plot does not distinguish the reversal in flow direction that is established in the suction region directly behind the cylinder. At the lowest speed, flow reduction can equate to no flow (100% reduction), whereas at higher speeds the largest flow reduction still results in some flow magnitude.

Fig. 4.

Relationships between flow speed, vortex shedding frequency, tail-beat frequency and body size. (A) Experimentally measured vortex shedding frequency increases linearly with flow velocity (r²=0.98, P<0.05). Reynolds number, using cylinder diameter as the length term, is shown at the top. (B) Tail-beat frequency for Kármán gaiting fish increases linearly and matches the vortex shedding frequency as flow velocity increases (r²=0.93, P<0.05, N=9 fish). (C) Tail-beat frequency normalized to flow speed (~50 cm s⁻¹), showing no differences across body size (P=0.11, N=15 fish). Values shown are means ± s.e.m. Note that error bars are partially obscured by the data symbols.

Fig. 5.

Body wave speed is the product of tail-beat frequency and body wavelength, and increases with flow speed. (A) Body wave speed increases linearly with, and is 25% faster than, the nominal flow speed (r²=0.85, P<0.05, N=9 fish). (B) At a flow speed of ~50 cm s⁻¹, small fish have a higher body wave speed than medium and large fish (*P<0.05, N=15 fish). Values shown are means ± s.e.m.

Fig. 6.

Body wavelength across speed and body size. (A) Body wavelength (solid line) starts lower than the cylinder wake wavelength (dashed line) and then rises above it as flow speed increases (r²=0.28, N=9 fish). At the lowest swimming speeds, the absence of a strong vortex street likely requires use of a shorter body wave similar to freestream swimming fish (see Discussion). (B) At a flow speed of ~50 cm s⁻¹, smaller fish have a longer body wavelength than larger fish (*P<0.05, N=15 fish). Values shown are means ± s.e.m.

Fig. 7.

Lateral body amplitude across speed and body size. (A) Tail tip amplitude (solid black line) is closest to the lateral vortex spacing defined by the cylinder diameter (dashed line) at the lowest speed and then rises above this as flow speed increases (r²=0.14, N=9 fish). Lateral BC amplitude (gray line) is much smaller than both the tail tip amplitude and the lateral vortex spacing (r²=0.07, N=9 fish). (B) At a flow speed of ~50 cm s⁻¹, tail tip amplitude increases with fish size except between medium and large fish (filled circles, *P<0.05, N=15 fish), unlike the BC amplitude (open circles, P=0.46, N=15 fish). Values shown are means ± s.e.m.

Fig. 8.

Lateral acceleration of the BC across speed and body size. (A) BC acceleration increases quadratically with speed (r²=0.69, N=9 fish). (B) There is no effect of body size on BC acceleration, although larger fish display more variation (P=0.51, N=15 fish). Values shown are means ± s.e.m.

Fig. 9.

Standard deviation of head angle from the axis of freestream flow across speed and body size. (A) Standard deviation of head angle decreases with flow velocity (r²=0.13, P<0.05, N=9 fish). (B) At a flow speed of ~50 cm s⁻¹, the head angle deviation for large fish is greater than that for small fish (P<0.05, N=15 fish). Values shown are means ± s.e.m.

Fig. 10.

Principal component analysis (PCA) of four Kármán gaiting kinematic variables on three body size treatments. PC1 and PC2 account for 85% and 11% of the variation, respectively. Small fish differ from medium and large fish only along PC1 (P<0.05, N=15 fish), in which the variable body wavelength loaded high. There is no difference between size groups along PC2 (P=0.08, N=15 fish).