Numerical study of Kelvin-Helmholtz instability by the point vortex method

Rosenhead's classical point vortex numerical method for studying the evolution of a vortex sheet from analytic initial data (Kelvin-Helmholtz instability) is examined by the discrete Fourier analysis techniques. One cause for the chaotic motion previously observed in computations using a large number of vortices is that short wavelength perturbations are introduced spuriously by finite precision arithmetic and become amplified by the model's dynamics. The formation of a singularity in a finite time which was previously found by Moore and Meiron, Baker and Orszag using different techniques of analysis was confirmed. The point vortex approximation remains consistent at the critical time. The singularity is interpreted physically as a discontinuity in the strain rate along the vortex sheet and also as the start of roll up on a small scale. Some aspects of the dependence of the solution on the initial condition were studied and agreement with Moore's asymptotic relation between the initial amplitude and the critical time was found. For large initial amplitudes, two cusps form in the sheet strength, corresponding to double roll up. It is explained why the Poincare recurrence theorem does not imply that the sheet will eventually unroll. It is suggested that beyond the critical time, the vortex sheet becomes a spiral with infinite arc length.