A Finite Element Method for Computing Transonic Potential Flow

Abstract

The finite element method has gained widespread acceptance as a technique for solving elliptic partial differential equations, where its ability to handle quite general geometries has proved an advantage over finite differences. For the steady flow of an inviscid fluid, the governing equations are elliptic as long as the flow speed is everywhere subsonic. In such cases the effects of compressibility are generally small, and results of adequate accuracy for engineering purposes can be obtained by applying approximate corrections to the solution obtained for incompressible flow. However, incompressible flow can be calculated using the so-called panel method or boundary integral equation approach. Because it reduces the problem from one of solving for an entire field to that of solving for quantities on the boundary, the latter approach is more efficient than either finite elements or finite differences. This fact has prevented the widespread use of finite elements for purely subsonic flow. Attention has therefore recently focused on the possibility of using the finite element method to treat flows which are not purely subsonic, and which thus contain regions in which the governing equation is hyperbolic. This paper describes one such attempt which makes use of the ‘artificial compressibility’ concept developed in ref. 1.