Self Similarity, Critical Points, and Hill’s Spherical Vortex
Hill’s spherical vortex, a classical solution dating from 1894, is examined in a new perspective. The axisymmetric vorticity transport equation is taken to its low Reynolds number limit (no convective term) and cast into a form that is self similar in time. The subsequent equation is solved by separation of variables and shown to have two linearly independent solutions. One of these two solutions satisfies the Navier-Stokes equation exactly for all Reynolds numbers. However, to satisfy boundary conditions for the unbounded problem, both of these solutions as well as irrotational components are required. The results are represented by self similar particle paths. The flow topology is examined in the context of critical points of the self similar particle paths. It is shown that this result undergoes two transitions in topological structure with changing Reynolds number. Also shown is that the creeping flow (low Reynolds number) solution has three possible topological states in the axisymmetric case.
KeywordsVortex Vorticity Tate Geophysics Topo
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