Finite rotating and translating vortex sheets

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 uniqueness and stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we demonstrate that, within the class of rotating configurations involving a single open vortex sheet with endpoints, the straight segment revolving around its midpoint is a unique equilibrium. Second, 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. Third, we obtain and analyze equations describing the time evolution of a straight vortex sheet in linear external fields. Finally, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite-aspect 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 limits of the analogous solutions known for vortex patches.