A Framework for Linear Stability Analysis of Finite-Area Vortices

TL;DR

This paper develops a comprehensive framework for analyzing the linear stability of 2D steady Euler flows with finite-area vortices, combining shape calculus, analytical validation, and a spectral numerical method.

Contribution

It provides a systematic derivation of the linearized stability equations for finite-area vortices, including new methods for contour integral linearization and a spectrally-accurate numerical approach.

Findings

Validated the stability equations with classical vortex cases

Developed a spectral numerical method for general vortex shapes

Provided analytical and numerical tools for vortex stability analysis

Abstract

In this investigation we revisit the question of the linear stability analysis of 2D steady Euler flows characterized by the presence of compact regions with constant vorticity embedded in a potential flow. We give a complete derivation of the linearized perturbation equation which, recognizing that the underlying equilibrium problem is of the free-boundary type, is done systematically using methods of the shape-differential calculus. Particular attention is given to the proper linearization of the contour integrals describing vortex induction. The thus obtained perturbation equation is validated by analytically deducing from it the stability analyses of the circular vortex, originally due to Kelvin (1880), and of the elliptic vortex, originally due to Love (1893), as special cases. We also propose and validate a spectrally-accurate numerical approach to the solution of the stability problem for vortices of general shape in which all singular integrals are evaluated analytically.