Evolution of a cylindrical and a spherical vortex sheet

Institute for Mathematics and its Applications University of Minnesota Minneapolis, MN 55455, USA

Abstract

Point vortex and vortex blob computations are used to investigate the evolution of the planar and the axisymmetric vortex sheet which initially induce flow past a cylinder and past a sphere respectively. In both cases the sheet develops a singularity at two points in the symmetry plane at a finite time. It rolls up at these points forming a vortex pair in the planar case and a vortex ring in the axisymmetric case. The computations show differences in the shape of the sheet before singularity formation as well as in the time and location at which the singularity appears. The ensuing roll-up is faster, smaller and less symmetric in the axisymmetric case than in the planar case. Furthermore, unlike the planar pair, the axisymmetric ring sheds approximately 25% of the total vorticity into its wake which then in turn rolls up into a secondary ring. At large times, irregular particle motion appears. This is believed to result from the onset of chaos in a perturbed dynamical system, and to be caused by oscillations in the downstream motion of the vortices.