Simple scaling law predicts peak efficiency in oscillatory propulsion

Abstract

Oscillatory propulsion is ubiquitous among swimming and flying animals, and may some day be practical as a replacement for rotary propulsion in watercraft and small air vehicles. The strength and efficiency of flapping thrust production closely depend on a dimensionless parameter called the Strouhal number ( St ), representing the ratio of the transverse oscillation speed to forward speed of the propulsor ( St = 2 fA / U , where f is the oscillation frequency, A is its amplitude at the trailing edge, and U is the forward speed; Fig. 1 A ). Crucially, the propulsive efficiency, defined as the ratio of propulsive power output to mechanical power input, usually peaks at a Strouhal number intermediate between the value associated with the onset of thrust production and the value associated with maximal thrust production. While the exact Strouhal number giving peak propulsive efficiency depends on the relative amplitude and phase of the rotational pitch and translational heave components of flapping, its optimum typically falls in the range 0.2 < St < 0.4 for most natural combinations of pitch and heave. Remarkably, nature’s swimmers and fliers have been found to operate in the same narrow range of Strouhal number when cruising (1, 2). What determines the Strouhal number giving peak propulsive efficiency? Writing in PNAS, Floryan et al. (3) think they have the answer. Fig. 1. Significance of Strouhal number ( St ) in oscillatory propulsion. ( A ) St = 2 fA / U represents the ratio of transverse oscillation speed to forward speed of the propulsor, given by the ratio of trailing-edge excursion (2 A ) to distance traveled per oscillation ( U / f ), where f is frequency, A is amplitude, and U is speed. ( B ) At low St , the wing produces a net drag and typically forms a wake structure called a Karman vortex street. ( C ) At … [↵][1]1Email: graham.taylor{at}zoo.ox.ac.uk. [1]: #xref-corresp-1-1