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 [Formula: see text] 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 [Formula: see text], 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.

Keywords: biolocomotion; drag; flight; swimming.

Fig. 1.

A typical efficiency curve showing efficiency $\eta$ as a function of $St$. Data are for a heaving and pitching NACA0012 foil (5) ($A/L=0.19$, and heaving leads pitching by $90^{°}$).

Fig. 2.

Swimmers and fliers can be decomposed into thrust-producing (orange) and drag-producing (blue) parts, with the propulsor aptly represented by an oscillating airfoil.

Fig. 3.

Efficiency $\eta$ as a function of $St$. Data are as given in Fig. 1 for a heaving and pitching NACA0012 foil (5). Solid lines are given by Eq. 6 with a fixed proportionality constant of 0.155. The drag constant, $b_1$, is set to 0.5, 0.35, 0.23, 0.15, 0.1, and 0.05 as the colors vary from dark to light, and we have set $g\left(\mathrm{\Theta }\right)=\mathrm{\Theta }$. The proportionality constant and the value of $b_1$ corresponding to the experimental data were calculated by a total least squares fit to the data.