Solution of the parabolized Navier-Stokes equations using Osher's upwind scheme

Abstract

A new, explicit, finite volume algorithm based on Osher's upwind method is applied to the two-dimensional parabolized Navier-Stokes equations to model hypersonic flows. The algorithm is second-order accurate and employs flux limiters to make the scheme total variation diminishing (TVD). The pressure gradient in the subsonic region is limited in the streamwise direction to maintain a hyperbolic inviscid equation set. Secondorder central differencing is applied to the viscous terms and upwind differencing is applied to the inviscid terms in both the subsonic and supersonic portions of the flowfield. The new algorithm is demonstrated by computing four laminar-flow cases; supersonic flow over a flat plate, supersonic flow in a diffuser, hypersonic flow over a 15-deg ramp, and hypersonic flow in a converging inlet. Extensive comparisons of heat transfer, skin friction, and pressure coefficients are made between Osher's and Roe's upwind schemes. For this class of flows, Osher and Roe upwinding yield very similar results. I. Introduction D ESIGN and analysis of flow about supersonic vehicles is now being accomplished with the aid of the parabolized Navier-Stokes equations (PNS). By using the PNS equations the computational burden can be reduced by an order of magnitude. In recent years, upwinding has been applied to the inviscid flux terms in the PNS equations. Upwinding improved the quality of the solutions and eliminated the necessity of adding damping terms associated with the conventional central-difference algorithms. In computational fluid dynamics four major upwind schemes are used: Engquist and Osher,1 Roe,2 Steger and Warming,3 and van Leer.4 StegerWarming splitting is not continuously differentiable across the sonic points where the eigenvalues change sign and, consequently, it has a glitch at the sonic point.5 Van Leer splitting is known to yield smeared contact discontinuities, thus requiring more grid points for resolving viscous shear layers. Roe's scheme has gained wide acceptance for the solution of the PNS and Navier-Stokes (NS) equations. The algorithm is fairly easy to implement and the results, for most flows, are very good. Unfortunately, Roe's scheme does not strictly enforce the entropy condition and expansion shocks are possible.68 PNS schemes912 which have employed Roe upwinding require an entropy correction model.9-13-14 While entropy correction models do eliminate nonphysical features, they are dissipative and they generally introduce another point into the shock-transition process and increase error in the flowfield. However, Osher's15 scheme rules out expansion shocks and actually enforces the entropy condition. It is continuously differentiable which provides smoothness in transition points and sharp contact discontinuities/shocks. Thus, in cases where the flow experiences large expansions, Osher's method should prove superior to existing upwind methods which have been applied to the PNS equations. Therefore, this current work involves the application of Osher's scheme to the PNS equations.