Tuning of Strouhal number for high propulsive efficiency accurately predicts how wingbeat frequency and stroke amplitude relate and scale with size and flight speed in birds

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Abstract
The wing kinematics of birds vary systematically with body size, but we still, after several decades of research, lack a clear mechanistic understanding of the aerodynamic selection pressures that shape them. Swimming and flying animals have recently been shown to cruise at Strouhal numbers (St) corresponding to a regime of vortex growth and shedding in which the propulsive efficiency of flapping foils peaks (St ≈ fA/U, where f is wingbeat frequency, U is cruising speed and A ≈ bsin(θ/2) is stroke amplitude, in which b is wingspan and θ is stroke angle). We show that St is a simple and accurate predictor of wingbeat frequency in birds. The Strouhal numbers of cruising birds have converged on the lower end of the range 0.2 < St < 0.4 associated with high propulsive efficiency. Stroke angle scales as θ ≈ 67b−0.24, so wingbeat frequency can be predicted as f ≈ St. U/bsin(33.5b−0.24), with St = 0.21 and St = 0.25 for direct and intermittent fliers, respectively. This simple aerodynamic model predicts wingbeat frequency better than any other relationship proposed to date, explaining 90% of the observed variance in a sample of 60 bird species. Avian wing kinematics therefore appear to have been tuned by natural selection for high aerodynamic efficiency: physical and physiological constraints upon wing kinematics must be reconsidered in this light.