Inviscid coupling between point symmetric bodies and singular distributions of vorticity

Abstract

We study the inviscid coupled motion of a rigid body (of density ρb, in a fluid of density ρ) and singular distributions of vorticity in the absence of gravity, using for illustration a cylinder moving near a point vortex or dipolar vortex, and the axisymmetric interaction between a vortex ring and sphere.The coupled motion of a cylinder (radius a) and a point vortex, initially separated by a distance R and with zero total momentum, is governed by the parameter R4/(ρb/ρ+1)a4. When R4/(ρb/ρ+1)a4,≤,1, a (positive) point vortex moves anticlockwise around the cylinder which executes an oscillatory clockwise motion, with a mixture of two frequencies, centred around its initial position. When R4/(ρb/ρ+1)a4≫1, the initial velocity of the cylinder is sufficiently large that the dynamics become uncoupled, with the cylinder moving off to infinity. The final velocity of the cylinder is related to the permanent displacement of the point vortex.The interaction between a cylinder (initially at rest) and a dipolar vortex starting at infinity depends on the distance of the vortex from the centreline (h), the initial separation of the vortical elements (2d), and ρb/ρ. For a symmetric encounter (h=0) with a dense cylinder, the vortical elements pass around the cylinder and move off to infinity, with the cylinder being displaced a finite distance forward. However, when ρb/ρ<1, the cylinder is accelerated forward to such an extent that the vortex cannot overtake. Instead, the cylinder ‘extracts’ a proportion of the impulse from the dipolar vortex. An asymmetric interaction (h>0) leads to the cylinder moving off in the opposite direction to the dipolar vortex.To illustrate the difference between two- and three-dimensional flows, we consider the axisymmetric interaction between a vortex ring and a rigid sphere. The velocity perturbation decays so rapidly with distance that the interaction between the sphere and vortex ring is localized, but the underlying processes are similar to two-dimensional flows.We briefly discuss the general implications of these results for turbulent multiphase flows.