| 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. |
