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

Abstract

This article revisits the instability of sharp shear interfaces, also called vortex sheets, in incompressible fluid flows. 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 circular vortex sheets: such a configuration has a delicate linear stability, 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 prove that a common regularized kernel used in numerical simulation and analysis destabilizes the circular vortex sheet. Absent a destabilizing kernel regularization, our work suggests that the nonlinear dynamics are critical for understanding circular vortex sheet instability, and so the essential mechanism of the Kelvin-Helmholtz instability is dependent on global vortex sheet geometry. As expected, nonlinear numerical simulations utilizing the regularized kernel exhibit unstable behavior. Finally, we show experimental results which qualitatively match the types of instabilities that are observed numerically, demonstrating the persistence of the Kelvin-Helmholtz instability in real circular shear flows.