A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries

Many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow. Here we present a method for calculating time-dependent incompressible inviscid flow which combines a projection method with a “Cartesian grid” approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The advection step is based on a Cartesian grid algorithm for compressible flow, in which the discretization of the body near the flow uses a volume-of-fluid representation. A redistribution procedure is used to eliminate time-step restrictions due to small cells where the boundary intersects the mesh. The projection step uses an approximate projection based on a Cartesian grid method for potential flow. The method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells. Convergence results are given for the projection itself and for the time-dependent algorithm in two dimensions. The method is also demonstrated on flow past a half-cylinder with vortex shedding.