Topological fluid mechanics of the formation of the Kármán-vortex street

Abstract

We explore the two-dimensional flow around a circular cylinder with the aim of elucidating the changes in the topology of the vorticity field that lead to the formation of the Kármán vortex street. Specifically, we analyse the formation and disappearance of extremal points of vorticity, which we consider to be feature points for vortices. The basic vortex creation mechanism is shown to be a topological cusp bifurcation in the vorticity field, where a saddle and an extremum of the vorticity are created simultaneously. We demonstrate that vortices are first created approximately 100 diameters downstream of the cylinder, at a Reynolds number, $Re_{K}$, which is slightly larger than the critical Reynolds number, $Re_{crit}\approx 46$, at which the flow becomes time periodic. For $Re$ slightly above $Re_{K}$, the newly created vortices disappear again a short distance further downstream. As $Re$ is further increased, the points of creation and disappearance move rapidly upstream and downstream, respectively, and the Kármán vortex street persists over increasingly large streamwise distances.