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 St(= f A/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.