Hydrodynamics: Strouhal number and propulsive efficiency

The biological optimum, 0.25 to 0.35

The Strouhal number (St = f × A / U) compares the frequency of tail beats with swimming speed. Aeronautical engineers found that all the high-performing swimmers and fliers of the animal kingdom, from the dolphin to the tuna, from the hummingbird to the shark, converge on the optimal range 0.25 to 0.35. A table for 18 cetaceans and reference animals.

Going further

Hydrodynamics connects to swimming and to metabolism: swimming performance, morphometrics, energy & metabolism and diving physiology.

The Strouhal number: why 0.25-0.35?

The Strouhal number (St) is a dimensionless number defined as St = f × A / U, where f is the beat frequency of the fluke, A the amplitude and U the swimming speed. Triantafyllou et al. (1991, MIT) showed that undulatory propulsion is energetically optimal in the window St = 0.25 to 0.35, where the wake forms a train of counter-rotating vortices (a “vortex street”) that pushes efficiently.

Cetaceans, sharks and tunas, seabirds and even insects in flight all converge on this same Strouhal band: a spectacular case of convergent evolution under a universal hydrodynamic constraint. That rule now inspires the biomimetics applied to underwater drones (MIT’s RoboTuna) and to turbines.