Evolution of an unstable horizontal stratified vortex

A trailing vortex behind a wing moving through a stratified fluid will twist surfaces of constant density such that the density overturns along the axis of the vortex. Recent results indicate that this flow is unstable to a wide range of modes. This conclusion was limited by the presence of a rigid cylinder on the outer boundary. The base flow also was assumed to have constant axial vorticity. Evolution of the unstable flow in a more realistic vortex is treated here with numerical simulations. The initial velocity field matches a Rankine vortex, with constant axial vorticity in the core, and zero vorticity outside the core. The initial density field is constructed by rotating a statically stable density profile uniformly along the vortex axis such that the density field is axially periodic. Up to ten periods of this density field are included in each simulation. The results depend on the Froude number, the pitch of the density field, and the Reynolds number. Results with large Froude numbers experience strong mixing, resulting in the interior of the vortex becoming fully mixed, similar to previous simulations of a stratified vortex with uniform overturning. Results with smaller Froude numbers show strong dependence on the pitch of the density field, in agreement with the previous stability theory.