Finite rotating and translating vortex sheets

TL;DR

This paper analyzes the stability of finite vortex sheets under rotation and translation, revealing linear instability with different growth behaviors, and connects these results to classical vortex and vortex patch solutions.

Contribution

It provides the first analytical stability analysis of finite vortex sheets under rotation and translation, extending classical results and including external field effects.

Findings

Rotating and translating finite vortex sheets are linearly unstable.

Unstable perturbations grow exponentially in rotation, algebraically in translation.

Results align with classical vortex and vortex patch stability analyses.

Abstract

We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyze equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite-aspect-ratio limit of the stability results known for Kirchhoff's ellipse (Love 1893; Mitchell & Rossi 2008) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite-aspect-ratio limits of the analogous solutions known for vortex patches.