Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency

Dimensionless numbers are important in biomechanics because their constancy can imply dynamic similarity between systems, despite possible differences in medium or scale(1). A dimensionless parameter that describes the tail or wing kinematics of swimming and flying animals is the Strouhal number(1), St = fA/U, which divides stroke frequency (f) and amplitude (A) by forward speed (U)(2-8). St is known to govern a well-defined series of vortex growth and shedding regimes for airfoils undergoing pitching and heaving motions(6,8). Propulsive efficiency is high over a narrow range of St and usually peaks within the interval 0.2 < St < 0.4 (refs 3-8). Because natural selection is likely to tune animals for high propulsive efficiency, we expect it to constrain the range of St that animals use. This seems to be true for dolphins(2-5), sharks(3-5) and bony fish(3-5), which swim at 0.2 < St < 0.4. Here we show that birds, bats and insects also converge on the same narrow range of St, but only when cruising. Tuning cruise kinematics to optimize St therefore seems to be a general principle of oscillatory lift-based propulsion.