Desingularization of periodic vortex sheet roll-up

Abstract

The equations governing periodic vortex sheet roll-up from analytic initial data are desingularized. Linear stability analysis shows that this diminishes the vortex sheet model's short wavelength instability, yielding a numerically more tractable set of equations. Computational evidence is presented which indicates that this approximation converges, beyond the critical time of singularity formation in the vortex sheet, if the mesh is refined and the smoothing parameter is reduced in the proper order. The results suggest that the vortex sheet rolls up into a double branched spiral past the critical time. It is demonstrated that either higher machine precision or a spectral filter can be used to maintain computational accuracy as the smoothing parameter is decreased. Some conjectures on the model's long time asymptotic state are given.