An eulerian approach for vortex motion using a level set regularization procedure

We present an Eulerian, fixed grid, approach to solve the motion of an incompressible fluid, in two and three dimensions, in which the vorticity is concentrated on a lower dimensional set. Our approach uses a decomposition of the vorticity of the form xi = P(phi) eta, in which both phi (the level set function) and eta (the vorticity strength vector) are smooth. We derive coupled equations for phi and eta which give a regularization of the problem. The regularization is topological and is automatically accomplished through the use of numerical schemes whose viscosity shrinks to zero with grid size. There is no need for explicit filtering, even when singularities appear in the front, The method also has the advantage of automatically allowing topological changes such as merging of surfaces. Numerical examples, including two and three dimensional vortex sheets, two-dimensional vortex dipole sheets, and point vortices, are given. To our knowledge, this is the first three-dimensional vortex sheet calculation in which the sheet evolution feeds back to the calculation of the fluid velocity. Vortex in cell calculations for three-dimensional vortex sheets were done earlier by Trygvasson et al. (C) 1996 Academic Press, Inc.