On the rules for aquatic locomotion

We present unifying rules governing the efficient locomotion of swimming fish and marine mammals. Using scaling and dimensional analysis, supported by new experimental data, we show that efficient locomotion occurs when the values of the Strouhal (St) number $\mathrm{St}(=fA/U)$ and $A^{*}(=A/L)$, two nondimensional numbers that relate forward speed $U$, tail-beat amplitude $A$, tail-beat frequency $f$, and the length of the swimmer $L$ are bound to the tight ranges of 0.2–0.4 and 0.1–0.3, respectively. The tight range of 0.2–0.4 for the St number has previously been associated with optimal thrust generation. We show that the St number alone is insufficient to achieve optimal aquatic locomotion, and an additional condition on $A^{*}$ is needed. More importantly, we show that when swimming at minimal power consumption, the Strouhal number of a cruising swimmer is predetermined solely by the shape and drag characteristics of the swimmer. We show that diverse species of fish and cetaceans cruise indeed with the St number and $A^{*}$ predicted by our theory. Our findings provide a physical explanation as to why fast aquatic swimmers cruise with a relatively constant tail-beat amplitude of approximately 20% of the body length, and their swimming speed is nearly proportional to their tail-beat frequency.

Figure 1

A snapshot of a live freely swimming trout.

Figure 2

(a) Variation of the Strouhal number with swimming speed per body length for trout (new data), cetaceans (new data), cetaceans [15] (various species of dolphin and killer whale), dace [14], mackerel [35], saithe [35], blacktip shark [36], and rainbow trout [37]. (b) (Square root of) The drag coefficient of streamlined bodies with thickness ratios of $6\%,12\%$, and $25\%$ as a function of the Reynolds number [38].

Figure 3

Schematic of the robotic flapper to generate fishlike motion. (a) Perspective view and (b) top view.

Figure 4

Measured self-propelled speed and input power coefficients for the pitching foil under study: (a) Re number (dimensionless speed) contours (solid lines), and their corresponding St number contours (dashed line; $\mathrm{St}=f^{*}{A}^{*}$) and (b) power coefficient for each Re number contour as a function of nondimensional amplitude $A^{*}. A_c^{*}$ is denoted by a solid red disk.

Figure 5

Gait data for trout (new data), cetaceans (new data), cetaceans [15] (various species of dolphin and killer whale), dace [14], mackerel [35], saithe [35], blacktip shark [36], and rainbow trout [37] on the nondimensional plane of amplitude versus frequency. Only the data with sufficiently high speed [i.e., $U/L>1\left(1/\mathrm{s}\right)$] are reported to be consistent with the assumption of high-speed cruising. The dashed and dashed-dotted lines correspond to $\mathrm{St}=0.2$ and 0.4 and $A^{*}=0.1$ and $0.3,$ respectively.