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

1 Abstract Many problems in uid dynamics have domains with complicated internal or external boundaries of the ow. Here we present a method for calculating time-dependent incompressible inviscid ow using a \Cartesian grid" approach for representing geometry. In this approach, the body is represented as an interface embedded in a regular Cartesian mesh. The basic algorithm is a fractional-step projection method based on an approximate projection. The advection step is based on a Cartesian grid algorithm for compress-ible ow, in which the discretization of the body near the ow uses a volume-of-uid representation with a redistribution procedure to eliminate time-step restrictions due to small cells where the boundary intersects the mesh. The approximate projection incorporates knowledge of the body through volume and area fractions. The method is demonstrated on ow past a half-cylinder with vortex shedding. 2 Introduction Modeling of low Mach number ows in complex geometries is often required in engineering applications. In this paper we present a Cartesian grid algorithm for the unsteady incompressible Euler equations. in which the problem geometry is represented as a \tracked front" embedded in a uniform Cartesian grid. The incompressible Eu-ler equations provide a prototype for more general low Mach number ows such as low speed combustion ((7, 9, 10]). The basic integration scheme uses a fractional step approach in which the nonlinear convection equations are approximated to construct a velocity eld without enforcing the divergence constraint. In the second step of the algorithm a discrete projection is applied to the intermediate velocity eld computed in the rst step to enforce the incompressibility