Horizontal locomotion of a vertically flapping oblate spheroid

Abstract

We consider the self-induced motions of three-dimensional oblate spheroids of density $\unicode[STIX]{x1D70C}_{s}$

with varying aspect ratios

$AR=b/c\leqslant 1$

, where

$b$

and

$c$

are the spheroids’ centre-pole radius and centre-equator radius, respectively. Vertical motion is imposed on the spheroids such that

$y_{s}(t)=A\sin (2\unicode[STIX]{x03C0}ft)$

in a fluid of density

$\unicode[STIX]{x1D70C}$

and kinematic viscosity

$\unicode[STIX]{x1D708}$

. As in strictly two-dimensional flows, above a critical value

$Re_{C}$

of the flapping Reynolds number

$Re_{A}=2Afc/\unicode[STIX]{x1D708}$

, 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

$St(AR)=2Af/U$

, 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

$Re_{A}\gtrsim Re_{C}$

, 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 structures above and below an oscillating spheroid, recently calculated using Floquet analysis by Deng et al. ( Phys. Rev. E, vol. 94, 2016, 033107). Such ‘in-phase’ instabilities arise in a relatively narrow band of

$Re_{A}\gtrsim Re_{C}$

, which band shifts to higher Reynolds number as the aspect ratio increases. (Indeed, for horizontally fixed spheroids with aspect ratio

$AR=0.2$

, Floquet analysis actually predicts stability at

$Re_{A}=45$

.) For such a spheroid (

$AR=0.2$

,

$Re_{A}=45$

, with sufficiently small mass ratio

$m_{s}/m_{f}=\unicode[STIX]{x1D70C}_{s}V_{s}/(\unicode[STIX]{x1D70C}V_{s})$

, where

$V_{s}$

is the volume of the spheroid) which is free to move horizontally, the second type of three-dimensional motion is observed, initially taking the form of a ‘snaking’ trajectory with long quasi-periodic sweeping oscillations before locking into an approximately elliptical ‘orbit’, apparently manifesting a three-dimensional generalisation of the

$QP_{H}$

quasi-periodic symmetry breaking discussed for sufficiently high aspect ratio two-dimensional elliptical foils in Deng & Caulfield ( J. Fluid Mech. , vol. 787, 2016, pp. 16–49).