| Two major categories of Upwind Differencing (UD) algorithms, namely flux vector splitting and flux difference splitting are studied and modified during the course of this research. These schemes are the van Leer scheme from the FVS family and the Roe scheme of the FDS type. To check and assess the accuracy of the modified algorithms the inviscid Burgers equations, Riemann problem (shock tube problem) and Enter equations in two-dimensions are taken as model equations. The method is later extended to the Navier-Stokes equations.; For the spatial discretization of inviscid terms (convective and pressure terms) three different cases are studied: (1) second order upwind scheme, in which the numerical oscillations are damped by minmod Total Variation Diminishing (TVD), (2) third order upwind biased scheme with the van Albada flux limiter and (3) mixed second and third order scheme with no flux limiter. In cases (1) and (2) the primitive variables are extrapolated to the left and right faces of the cell by the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) idea, while in case (3) the inviscid fluxes are directly extrapolated to the cell faces. For the spatial discretization of the viscous terms, central difference is used. For the temporal discretization, either Euler's first order forward, or second order two step predictor-corrector from the Lax-Wendroff family is used. The governing equations are recast into generalized coordinates and are solved in these coordinates.; The method is applied to a variety of test cases ranging from the low subsonic regime (Mach of 0.5) to supersonic flows (Mach 4.0). Turbulence is modeled by the Baldwin and Lomax zero-equation model. The meshes, which are structured, are developed by an algebraic or by an orthogonal grid generator in a separate module and then transferred to the main solver. |
