Simulation of vortex sheet roll-up: chaos, azimuthal waves, ring merger

This article reviews some recent simulations of vortex sheet roll-up using the vortex blob method. In planar and axisymmetric flow, the roll-up is initially smooth but irregular small-scale features develop later in time due to the onset of chaos. A numerically generated Poincaré section shows that the vortex sheet flow resembles a chaotic Hamiltonian system with resonance bands and a heteroclinic tangle. The chaos is induced by a self-sustained oscillation in the vortex core rather than external forcing. In three-dimensional flow, an adaptive treecode algorithm is applied to reduce the CPU time from O(N) to O(N log N), where N is the number of particles representing the sheet. Results are presented showing the growth of azimuthal waves on a vortex ring and the merger of two vortex rings. Vortex blob methods discrete, Applied to roll-up of a sheet, Will persuade any cynic That heteroclinic Tangles give insights quite neat.