Two methods for the study of vortex patch evolution on locally refined grids

摘要

Two numerical methods for the solution of the two-dimensional Euler equations for incompressible flow on locally refined grids are presented. The first is a second order projection method adapted from the method of Bell, Colella, and Glaz. The second method is based on the vorticity-stream function form of the Euler equations and is designed to be free-stream preserving and conservative. Second order accuracy of both methods in time and space is established, and they are shown to agree on problems with a localized vorticity distribution. The filamentation of a perturbed patch of circular vorticity and the merger of two smooth vortex patches are studied. It is speculated that for nearly stable patches of vorticity, an arbitrarily small amount of viscosity is sufficient to effectively eliminate vortex filaments from the evolving patch and that the filamentation process affects the evolution of such patches very little. Solutions of the vortex merger problem show that filamentation is responsible for the creation of large gradients in the vorticity which, in the presence of an arbitrarily small viscosity, will lead to vortex merger. It is speculated that a small viscosity in this problem does not substantially affect the transition of the flow to a statistical equilibrium solution. The main contributions of this thesis concern the formulation and implementation of a projection for refined grids. A careful analysis of the adjointness relation between gradient and divergence operators for a refined grid MAC projection is presented, and a uniformly accurate, approximately stable projection is developed. An efficient multigrid method which exactly solves the projection is developed, and a method for casting certain approximate projections as MAC projections on refined grids is presented.