a 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 using the discrete Fourier analysis techniques of Sulem, Sulem & Frisch. 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. Methods for controlling this source of error are given and the results confirm the formation of a singularity in a finite time which was previously found by Moore and Meiron, Baker & Orszag using different techniques of analysis. A cusp forms in the vortex sheet strength at the critical time, explaining the onset of erratic particle motion in applications of the numerical methods of Van de Vooren and Fink & Soh to this problem. Unlike those methods, the point vortex approximation remains consistent at the critical time and we present the results of a long time calculation. 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. We numerically study some aspects of the dependence of the solution on the initial condition and find agreement with Moore's asymptotic relation between the initial amplitude and the critical time. For large initial amplitudes, two cusps form in the sheet strength, corresponding to double roll up. We explain why the Poincare recurrence theorem does not imply that the sheet will eventually unroll. Our results suggest that beyond the critical time, the vortex sheet becomes a spiral with infinite arclength although we have doubts about the approximation's accuracy in that regime.
- Pub Date:
- Physics: Fluid and Plasma