High-resolution TVD schemes for the analysis of: I. Inviscid supersonic and transonic flows. II. Viscous flows with shock-induced separation and heat transfer

Application of Total Variation Diminishing (TVD) schemes to both inviscid and viscous flows is considered. The mathematical and physical basis of TVD schemes is discussed. First and second-order accurate TVD schemes, and a second-order accurate Lax-Wendroff scheme, are used to compute solutions to the Riemann problem in order to investigate the capability of each to resolve shocks, rarefactions, and contact surfaces. Second-order finite-volume and finite-difference TVD schemes are used to obtain solutions to inviscid supersonic and transonic cascade flow problems. TVD schemes are shown to be superior to the Lax-Wendroff family of schemes for both transient and steady-state computations.; TVD methodology is extended to the solution of viscous flow problems. A first-order time accurate, second-order space accurate algorithm is contrasted against a second-order time and space accurate algorithm for the solution of the viscous Burgers' equation. Necessity of using the fully second-order accurate algorithm at low Reynolds numbers is shown. Solutions are computed to the problems of laminar shock-boundary-layer interaction and unsteady, laminar, shock-induced heat transfer using the new algorithms. These algorithms provide the capability, for the first time, to accurately predict separation, reattachment, and pressure and skin friction profiles for shock-boundary-layer interaction. Additionally, extremely accurate comparison with theory and experiment is evident for the unsteady, shock-induced heat transfer problem. These solutions are contrasted against solutions computed with the Beam-Warming algorithm, and the TVD solutions are shown to be vastly superior.