The Influence of Vortex Sheet Geometry on the Kelvin-Helmholtz Instability

Abstract

This article revisits the instability of sharp shear interfaces in incompressible fluids, which are also called vortex sheets. We study the Birkhoff-Rott equation, which describes the motion of vortex sheets according to the incompressible Euler equations in two dimensions. The classical Kelvin-Helmholtz instability demonstrates that an infinite, flat vortex sheet has a strong linear instability. We show that this is not the case for vortex sheets which are circular: such a configuration is linearly stable, and is the first example of a linearly stable solution to the Birkhoff-Rott equation. We subsequently derive a sufficient condition for linear instability of a circular vortex sheet for a family of generalized Birkhoff-Rott kernels, and show that a common regularized kernel used in numerics and analysis destabilizes the circular vortex sheet. Finally, we show experimental results which qualitatively match the types of instabilities that are observed numerically.

Existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations

“…On the other hand, a recent result of Murray and Wilcox [19] proves that the standard circular discontinuous shear flow does not admit Kelvin-Helmholtz instability, which shows that geometry of a vortex sheet could be related to its stability properties. Thus, considering logarithmic spirals (1.1), the question of their stability is interesting, as their geometry can be thought of as an interpolation between the straight and the circular vortex sheets.…”