Efficient cruising for swimming and flying animals is dictated by fluid drag

Abstract

Many swimming and flying animals are observed to cruise in a narrow range of Strouhal numbers, where the Strouhal number ${St = 2fA/U}$ is a dimensionless parameter that relates stroke frequency $f$, amplitude $A$, and forward speed $U$. Dolphins, sharks, bony fish, birds, bats, and insects typically cruise in the range $0.2 < St < 0.4$, which coincides with the Strouhal number range for maximum efficiency as found by experiments on heaving and pitching airfoils. It has therefore been postulated that natural selection has tuned animals to use this range of Strouhal numbers because it confers high efficiency, but the reason why this is so is still unclear. Here, by using simple scaling arguments, we argue that the Strouhal number for peak efficiency is largely determined by fluid drag on the fins and wings.