The Collapse of an Axi-symmetric, Swirling Vortex Sheet

An axi-symmetric and swirling vortex sheet is investigated as the simplest ow in which there is non-trivial vortex stretching and as a possible setting for studying vortex cancellation and singularity formation. Rayleigh's criterion indicates linear stability of a single sheet but instability for other conngurations of sheets. Due to the simplicity of vortex sheet problems, the linear modes and growth rates (or frequencies) can be explicitly expressed. Subsequent nonlinear evolution is numerically simulated using a vortex method. The numerical results 1 for an axi-symmetric swirling sheet with a vortex line along the axis of symmetry show detachment of a vortex ring from the sheet into the outer uid, and collapse of the sheet onto the vortex line at some points. Vortex cancellation, which in the presence of viscosity would likely lead to vortex line reconnection, seems to occur in both of these phenomena. The evolution of two co-axial, axi-symmetric, swirling vortex sheets is similar.