In this chapter, finite element analyses of both inviscid and viscous compressible flows are examined. Traditionally, computational schemes for compressible inviscid flow are developed separately from compressible viscous flows, governed by Euler equations and Navier-Stokes system of equations, respectively. However, it is our desire in this chapter to study numerical schemes capable of treating a compressible flow with or without the effect of viscosity or diffusion. Furthermore, it would be desirable to develop a scheme that can handle all speed regimes – not only the compressible flow, but the incompressible flow as well. To accomplish this goal, the most suitable governing equations to use are the Navier-Stokes system of equations written in conservation form in terms of conservation variables. Advantages of transforming the conservation variables into entropy variables and primitive variables will be explored. One of the most prominent features in compressible flow calculations is the ability of numerical schemes to resolve shock waves or discontinuities in high-speed flows. Furthermore, compressible viscous flows at high Mach numbers and high Reynolds numbers lead to significant numerical difficulties. We shall address these and other issues in this chapter.

To this end, we begin with the general description of the governing equations in Section 13.1, followed by the Taylor-Galerkin methods (TGM), generalized Galerkin methods (GGM), generalized Petrov-Galerkin (GPG) methods, characteristic Galerkin methods (CGM), and discontinuous Galerkin methods (DGM) in Sections 13.2 through 13.4. Finally, the flowfield-dependent variation (FDV) methods introduced in FDM and discussed earlier in Section 6.5 will be presented for FEM applications (Section 13.6). This subject will be treated again in Chapter 16, where many of the methods in both FDM and FEM can be shown to be the special cases of FDV methods.