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Dimensionless numbers are important in biomechanics because their constancy can imply dynamic similarity between systems, despite possible differences in medium or scale1. A dimensionless parameter that describes the tail or wing kinematics of swimming and flying animals is the Strouhal number1, St = fA/U, which divides stroke frequency (f) and amplitude (A) by forward speed (U)2-8. St is known to govern a well-defined series of vortex growth and shedding regimes for airfoils undergoing pitching and heaving motions6,8. Propulsive efficiency is high over a narrow range of St and usually peaks within the interval 0.2 < St < 0.4 (refs 3-8). Because natural selection is likely to tune animals for high propulsive efficiency, we expect it to constrain the range of St that animals use. This seems to be true for dolphins2-5, sharks3-5 and bony fish3-5, which swim at 0.2 < St < 0.4. Here we show that birds, bats and insects also converge on the same narrow range of St, but only when cruising. Tuning cruise kinematics to optimize St therefore seems to be a general principle of oscillatory lift-based propulsion.
Experiments with isolated pitching or heaving foils have measured extremely high peak propulsive efficiencies within the interval 0.2 < St < 0.4 (modal peak at St [approximate] 0.3)3-7. In this range, the propulsive efficiency (defined as the ratio of aerodynamic power output to mechanical power input) can be as high as 70% (ref. 7) or even 80% (ref. 6). Optimal St depends subtly on kinematic parameters including geometric angle of attack, amplitude-to-chord ratio, airfoil section and phase of motion6-8 but, for any given motion, efficiency is usually high (>60%) over a range narrower than 0.2 < St < 0.4 (refs 3-7). For example, measured efficiency can plummet from 80% at St = 0.27 to 10% at St = 0.09 (ref. 6) and also drops off at higher St, albeit more gently7,8. Measured propulsive efficiency usually peaks when the kinematics result in maximum amplification of the shed vortices in the wake and an average velocity profile equivalent to a jet3,4,6.
Theoretical treatments of flapping wings3,4,6,8 further confirm the empirical result that St tightly constrains propulsive efficiency. In fact, St is bound to affect aerodynamic force coefficients and propulsive efficiency, because it defines the maximum aerodynamic...