Linear Stability of Inviscid Vortex Rings to Axisymmetric Perturbations

TL;DR

This paper analyzes the linear stability of inviscid vortex rings to axisymmetric perturbations using shape differentiation and spectral methods, revealing stability depends on vortex thickness.

Contribution

It introduces a novel stability analysis method for vortex rings via shape differentiation applied to contour dynamics in 3D axisymmetric geometry.

Findings

Thin vortex rings are neutrally stable.

Fat vortex rings become linearly unstable.

Eigenmodes resemble Rankine and Hill vortices in respective limits.

Abstract

We consider the linear stability to axisymmetric perturbations of the family of inviscid vortex rings discovered by Norbury (1973). Since these vortex rings are obtained as solutions to a free-boundary problem, their stability analysis is performed using recently-developed methods of shape differentiation applied to the contour-dynamics formulation of the problem in the 3D axisymmetric geometry. This approach allows us to systematically account for the effects of boundary deformations on the linearized evolution of the vortex ring. We investigate the instantaneous amplification of perturbations assumed to have the same the circulation as the vortex rings in their equilibrium configuration. These stability properties are then determined by the spectrum of a singular integro-differential operator defined on the vortex boundary in the meridional plane. The resulting generalized eigenvalue problem is solved numerically with a spectrally-accurate discretization. Our results reveal that while thin vortex rings remain neutrally stable to axisymmetric perturbations, they become linearly unstable to such perturbations when they are sufficiently ``fat''. Analysis of the structure of the eigenmodes demonstrates that they approach the corresponding eigenmodes of Rankine's vortex and Hill's vortex in the thin-vortex and fat-vortex limit, respectively. This study is a stepping stone on the way towards a complete stability analysis of inviscid vortex rings with respect to general perturbations.