We describe the theoretical foundations and associated numerical algorithm of a new method for the solution of compressible flows, using Cartesian grids (CGs). A unique feature that distinguishes this method from other CG approaches is its treatment of solid boundaries and their corresponding boundary conditions. Starting with a standard finite–element–method formulation based on the method of weighted residuals, we show that it is possible to treat the boundary in a manner that makes it unnecessary to determine its intersection with the surrounding CG. This feature is of great significance both algorithmically and numerically, for it eliminates complex topological operations hitherto considered essential in CG methods.
In this new method, we require only a determination of the grid cells that envelop a series of known integration points on the body surface, which is a trivial problem for a CG. The resulting pseudo–finite–element formulation leads to a very robust numerical algorithm and computer code (EULAIR).
We show the results of calculations performed on two–dimensional bodies and aerofoils to demonstrate the power of this new technique. The Euler equations have been solved in all these cases. However, the method is applicable to any partial differential equation that may be analysed by a CG approach. A computer code applying the three–dimensional formulation to solve flows around full aircraft configurations is currently under intensive development.