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 \(\xi=P(\varphi)\eta\), in which both \(\varphi\) (the level set function) and \(\eta\) (the vorticity strength vector) are smooth. We derive coupled equations for \(\varphi\) 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. The method also has the advantage of automatically allowing topological changes such as merging of surfaces. Numerical exmaples, including two- and three-dimensional vortex sheets, two-dimensional vortex dipole sheets, and point vortices, are given.
two- and three-dimensional vortex sheets
scientific article; zbMATH DE number 938005