Finite element solutions of the Navier-Stokes equations for compressible internal flows

A finite element method for the solution of the compressible Navier-Stokes equations, in primitive variables form, is presented in this paper. To provide stability and the necessary coupling, a simple pressure dissipation, in the form of a Laplacian, is introduced into the continuity equation. The equations are discretized by a weak-Galerkin weighted residual method and equal interpolation for all the unknowns is used. Newton linearization is utilized, and at each iteration, the linear algebraic system is solved directly for the velocity components and pressure in a fully coupled manner. Convergence of the algorithm is linear because of the lagging of the solution of the energy equation. Results are presented for two- and three-dimensional subsonic flows in various internal flow geometries.