A study of singularity formation in a vortex sheet by the point-vortex approximation

Abstract

The initial-value problem for perturbations of a flat, constant-strength vortex sheet is linearly ill posed in the sense of Hadamard, owing to Kelvin-Helmholtz instability. Previous numerical studies of this problem have experienced difficulty in converging when the mesh was refined. The present work examines Rosenhead's point-vortex approximation and seeks to understand better the source of this difficulty. Using discrete Fourier analysis, it is shown that perturbations introduced spuriously by computer roundoff error are responsible for the irregular point-vortex motion that occurs at a smaller time as the number of points is increased. This source of computational error is controlled here by using either higher precision arithmetic or a new filtering technique. Computations are presented which use a linear-theory growing eigenfunction of small amplitude/wavelength ratio as the initial perturbation. The results indicate the formation of a singularity in the vortex sheet at a finite time as previously found for other initial data by Moore and Meiron, Baker & Orszag using different techniques of analysis. Numerical evidence suggests that the point-vortex approximation converges up to but not beyond the time of singularity formation in the vortex sheet. For large enough initial amplitude, two singularities appear along the sheet at the critical time.