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

### Abstract

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, using an approximate
projection, 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 with
a redistribution procedure to eliminate time-step restrictions due to
small cells where the boundary intersects the mesh.
The approximate projection, which is based on a Cartesian grid
method for potential flow, 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.