Upwind schemes for the Navier-Stokes equations

A discretization method is presented for the full, steady, compressible Navier-Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discnetization of the diffusive part. In the present paper, the emphasis lies on the discretization of the convective part. The applied solution method directly solves the steady equations by means of a Newton method, which requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme}, results of a quantitative error analysis are presented. Osher's scheme appaars to be more and more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of convection is chosen. Based on this higher-order scheme, a new limner is constructed. Further, for van Leer's scheme, a solid wall - boundary condition treatment is proposed, which ensures a continuous transition from the Navier-Stokes flow regime to the Euler flow regime. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave - boundary layer interaction. The resu Its obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters.