On the rules for aquatic locomotion

Abstract

We present unifying rules governing the efficient locomotion of swimming fish and marine mammals. Using scaling and dimensional analysis, supported by new experimental data, we show that efficient locomotion occurs when the values of the Strouhal (St) number St(=f A/U) and A * (=A/L), two nondimensional numbers that relate forward speed U , tail-beat amplitude A, tail-beat frequency f , and the length of the swimmer L are bound to the tight ranges of 0.2-0.4 and 0.1-0.3, respectively. The tight range of 0.2-0.4 for the St number has previously been associated with optimal thrust generation. We show that the St number alone is insufficient to achieve optimal aquatic locomotion, and an additional condition on A * is needed. More importantly, we show that when swimming at minimal power consumption, the Strouhal number of a cruising swimmer is predetermined solely by the shape and drag characteristics of the swimmer. We show that diverse species of fish and cetaceans cruise indeed with the St number and A * predicted by our theory. Our findings provide a physical explanation as to why fast aquatic swimmers cruise with a relatively constant tail-beat amplitude of approximately 20% of the body length, and their swimming speed is nearly proportional to their tail-beat frequency.

Study of the thrust–drag balance with a swimming robotic fish

“…At small amplitudes, the model based on the independent analyses of thrust and drag is rather good, given that there is no free parameter. These results provide evidence that the Strouhal number of a self-propelled swimmer is associated with the thrust-drag balance 22,27 and that performing independent analyses is relevant to characterize the swimming properties to the leading order.…”

“…For elastic plates forced by an oscillating torque distribution, the highest velocities are also observed near the resonant frequency forcing. 21 Data collected from natural swimmers show that the swimming characteristics in the turbulent regime, i.e., a Reynolds number Re larger than 3000, 22 are governed by the relative uniformity of two dimensionless numbers, the Strouhal number St = Af /U 0 ∼ 0.3 [22][23][24][25][26][27] and the ratio A/L ∼ 0.2, [26][27][28][29] with swimming velocity U 0 , fish length L, tailbeat amplitude A, and frequency f. To understand these data in terms of swimming efficiency, numerous Refs. 4-7, 10, and 30-34 have assumed that the mean thrust F T scales with the dynamic pressure and that the thrust coefficient should be defined as F T / 1 2 ρS T U 2 0 , where ρ is the fluid density and a) Author to whom correspondence should be addressed: mederic.argentina@ unice.fr S T is the effective propulsive area of the fish.…”

Flow interactions of two- and three-dimensional networked bio-inspired control elements in an in-line arrangement

“…The Strouhal number of St = 0.25 is chosen since this leads to the peak propulsive efficiency for the isolated three-dimensional pitching wing with θ 0 = 7.5 • (see section 3.1). This Strouhal number also falls within a range that is typical for high efficiency, bio-inspired propulsion [30][31][32]. Since the amplitude and St are fixed, the reduced frequency is also fixed by the definition, k = St/(A/c).…”

Scaling laws for the propulsive performance of three-dimensional pitching propulsors

“…Many aquatic animals efficiently propel themselves by oscillating their bodies and caudal fins in unsteady motions. During self-propelled locomotion, swimmers reach a cruising condition where there is a balance between the time-averaged thrust, generated mainly by their caudal fins, and the time-averaged drag incurred by their bodies (Saadat et al 2017). By following this idea, numerous studies (Chopra 1976;Read et al 2003;Dong et al 2005) have investigated the performance of heaving and pitching propulsors in isolation instead of the combined body and propulsor of other studies (Lighthill 1960).…”