# Horizontal locomotion of a vertically flapping oblate spheroid

Article

We consider the self-induced motions of three-dimensional oblate spheroids of density $\rho_s$ with varying aspect ratios $AR=b/c \leq 1$, where $b$ and $c$ are the spheroids' centre-pole radius and centre-equator radius respectively. Vertical motion is imposed on the spheroids such that $y_s(t)=A \sin (2 \pi f t)$ 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 $Re_A=2A f c/\nu$ 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 \cite{deng2016pre} 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 $St(AR)=2 Af/U$, 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 \cite{deng2016pre}. 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 $m_s/m_f=\rho_s V_s/(\rho V_s)$ 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 generalization of the $QP_H$ quasi-periodic symmetry breaking discussed for sufficiently high aspect ratio two-dimensional elliptical foils in \cite{deng2016jfm}.