Linear instability of symmetric logarithmic spiral vortex sheets

Abstract

We consider Alexander spirals with $M\ge 3$ branches, that is symmetric logarithmic spiral vortex sheets. We show that such vortex sheets are linearly unstable in the $L^{\infty }$ (Kelvin-Helmholtz) sense, as solutions to the Birkhoff-Rott equation. To this end we consider Fourier modes in a logarithmic variable to identify unstable solutions with polynomial growth in time.

27 pages, 2 figures