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Comparative Study
, 5 (20), 329-38

Speed Limits on Swimming of Fishes and Cetaceans

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Comparative Study

Speed Limits on Swimming of Fishes and Cetaceans

G Iosilevskii et al. J R Soc Interface.

Abstract

Physical limits on swimming speed of lunate tail propelled aquatic animals are proposed. A hydrodynamic analysis, applying experimental data wherever possible, is used to show that small swimmers (roughly less than a metre long) are limited by the available power, while larger swimmers at a few metres below the water surface are limited by cavitation. Depending on the caudal fin cross-section, 10-15 m s(-1) is shown to be the maximum cavitation-free velocity for all swimmers at a shallow depth.

Figures

Figure 1
Figure 1
Swimming model. With fish body moving along the solid horizontal line, the tail is assumed to move along the dotted sawtooth trajectory. Its wave-length, x, is the stride length and its depth is 2h. The ratio of 4h and x is also the ratio of the (presumably constant) lateral tail velocity v to the (constant) swimming velocity u.
Figure 2
Figure 2
Minimal power required for (a) swimming and (b) the associated lift coefficient. CD,0 varies between 0.008 and 0.02 (about twice its maximal expected value), and k varies between 0.05 and 0.1. (a,b) All feasible combinations of these parameters result in practically indistinguishable differences.
Figure 3
Figure 3
Estimated cavitation-free envelopes on (a) the lateral tail velocity and (b) the stride length; cavitation is avoided inside the envelopes. Right extending envelopes correspond to a typical dolphin-like swimmer with 0.2 chord thick fin having CD,b/τ=0.2, CD,0/τ=0.01 and τk=0.05; inner envelopes correspond to a typical tuna-like swimmer with 0.09 chord thick fin having CD,b/τ=0.8, CD,b/τ=0.04 and τk=0.0125. Broken lines mark the approximation (3.8), circles mark the maximal speed estimate of equations (3.5) and (3.6) and squares mark cavitation appearing at CL,1=1.2.
Figure 4
Figure 4
Maximal swimming velocity possible with (a) no cavitation near the sea surface and (b) the caudal fin lift coefficient at that velocity, plotted against caudal fin thickness-to-chord ratio. Solid lines mark exact numerical solution for the maximum of equation (3.4) and broken lines mark the estimate based on equations (3.5) and (3.7). (a) The dotted line marks the maximal swimming velocity possible with no cavitation when the fin is at CL=1.2. CD,b equals 0.1 (line 1), 0.2 (line 2) and 0.3 (line 3); CD,0=0.01 and k changes between 0.05 and 0.1 (the difference is imperceptible in the figure).
Figure 5
Figure 5
(a) Minimal lateral tail velocity and (b) maximal stride length. Broken lines mark approximation (3.11). The values of CD/CL in the figure range between 0.03 and 0.3; they are marked to the right of the respective lines.
Figure 6
Figure 6
Maximal swimming velocity with h¯=0.2, CD,0=0.01, k=0.05, CL,max=1.2, (a) CD,b=0.2 and (b) CD,b=0.4. Curved solid lines mark power limit on the swimming velocity—the respective power per unit mass of the fish is marked to the right of each line. The horizontal lines mark cavitation limits for 9% (solid lines) and 20% (dashed lines) thick foils near the surface (lower sets) and at the depth of 50 m (higher sets).
Figure 7
Figure 7
Minimal pressure coefficient on the surface of a 0.2 chord thick dolphin caudal fin. Circles mark the data extracted from fig. 2 of Lang (1966), dashed line marks the approximation (A 12) with t=0.2c and solid line marks the approximation (A 14) with rLE=0.042c.

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