The three-dimensional instabilities and destruction of the viscous Hill’s vortex

Abstract

The Hill vortex is a three-dimensional vortex structure solution of the Euler equations. For small amplitude axisymmetric disturbances on the external surface from the linear stability analysis by \citet{moff78} emerged the formation of a tail. Successive studies confirmed this fact, and in addition observed a different shape of the tail with azimuthal small amplitude disturbances. In this paper the Navier-Stokes equations are solved at high values of the Reynolds number imposing large amplitude axisymmetric and three-dimensional disturbances on the surface of the vortex. The conclusion is that the azimuthal disturbances produce a hierarchy of structures inside the vortex maintaining the shape of the vortex. On the other hand the axisymmetric disturbances are convected in the rear side, are dumped and form an axisymmetric wave longer higher the magnitude of the surface disturbance. Simulations in a moving frame allow to extend the evolution in time leading to the conclusion that the Hill vortex is unstable and produces a wide range of energy containing scales characteristic of three-dimensional flows.