Horizontal locomotion of a vertically flapping oblate spheroid

摘要

We consider the self-induced motions of three-dimensional oblate spheroids of density rho(s) with varying aspect ratios AR = b/c = 1, where b and c are the spheroids' centre-pole radius and centre-equator radius, respectively. Vertical motion is imposed on the spheroids such that y(s)(t) = A sin(2 pi ft) in a fluid of density rho and kinematic viscosity nu. As in strictly two-dimensional flows, above a critical value Re-C of the flapping Reynolds number Re-A = 2Afc/nu, the spheroid ultimately propels itself horizontally as a result of fluid-body interactions. For Re-A sufficiently above Re-C, the spheroid rapidly settles into a terminal state of constant, unidirectional velocity, consistent with the prediction of Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107) that, at sufficiently high Re-A, such oscillating spheroids manifest m = 1 asymmetric flow, with characteristic vortical structures conducive to providing unidirectional thrust if the spheroid is free to move horizontally. The speed U of propagation increases linearly with the flapping frequency, resulting in a constant Strouhal number St(A) = 2Af/U, characterising the locomotive performance of the oblate spheroid, somewhat larger than the equivalent St for two-dimensional spheroids, demonstrating that the three-dimensional flow is less efficient at driving locomotion. St decreases with increasing aspect ratio for both two-dimensional and three-dimensional flows, although the relative disparity (and hence relative inefficiency of three-dimensional motion) decreases. For flows with Re-A greater than or similar to Re-C, we observe two distinct types of inherently three-dimensional motion for different aspect ratios. The first, associated with a disk of aspect ratio AR = 0.1 at Re-A = 45, consists of a 'stair-step' trajectory. This trajectory can be understood through consideration of relatively high azimuthal wavenumber instabilities of interacting vortex rings, characterised by in-phase vortical structure