Limiting and optimal Strouhal numbers or tip speed ratios for cruising propulsion by fins, flukes, wings and propellers.

Swimming and flying animals produce thrust with oscillating fins, flukes or wings. The relationship between frequency f, amplitude A and forward velocity U can be described with a Strouhal number St, where St = 2fA/U, where animals are observed to cruise with [Formula: see text]-0.4. Under these conditions, thrust is produced economically and a reverse von Kármán wake is observed. However, propeller-driven craft produce thrust with steadily revolving blades and a helical wake. Here, the simplified aerodynamic geometry of lift-based thrust production is described, applicable to both oscillating and revolving foils. The same geometric principles apply in both cases: if the foil moves too slowly, it cannot produce thrust; if it moves too fast, it produces thrust with excessive power demand. Effective, economic thrust production by animals is not the result of oscillating foils or cyclic vortex shedding; rather, the selection of amplitude and frequency, and wake vortex structure, are corollaries of driving an efficient foil velocity with finite amplitudes. Observed Strouhal numbers for cruising animals appear too low for optimal mechanical efficiency; however, the deviation from optimal efficiency may be small, and there are physical and physiological advantages to relatively low amplitudes and frequencies for swimming and flapping flight.