A model for the approximation of vortex rings by almost rigid bodies

TL;DR

This paper models vortex rings as almost rigid bodies in 3D Euler equations, deriving an ODE system that converges to point vortex dynamics, and demonstrates leapfrogging motion in certain configurations.

Contribution

It introduces a novel approximation of vortex rings using almost rigid bodies and derives the limiting dynamics as these bodies shrink to filaments.

Findings

System described by an ODE in body positions

Convergence to point vortex system in the limit

Existence of leapfrogging motion in the model

Abstract

We consider a model that approximates vortex rings in the axisymmetric 3D Euler equation by the movement of almost rigid bodies described by Newtonian mechanics. We assume that the bodies have a circular cross-section and that the fluid is irrotational and interacts with the bodies through the pressure exerted at the boundary. We show that this kind of system can be described through an ODE in the positions of the bodies and that in the limit, where the bodies shrink to massless filaments, the system converges to an ODE system similar to the point vortex system. In particular, we can show that in a suitable set-up, the bodies perform a leapfrogging motion.