Analytic structure of vortex sheet dynamics. Part 1. Kelvin–Helmholtz instability

Abstract

The instability of an initially flat vortex sheet to a sinusoidal perturbation of the vorticity is studied by means of high-order Taylor series in time t. All finite-amplitude corrections are retained at each order in t. Our analysis indicates that the sheet develops a curvature singularity at t = tc < ∞. The variation of tc with the amplitude a of the perturbation vorticity is in good agreement with the asymptotic results of Moore. When a is O(1), the Fourier coefficient of order n decays slightly faster than predicted by Moore. Extensions of the present prototype of Kelvin-Helmholtz instability to other layered flows, such as Rayleigh-Taylor instability, are indicated.