TVD and ENO Applications to Supersonic Flows in 2D

In this work, first part of this study, the high resolution numerical schemes of Lax and Wend off, of Yee, Warming and Harten, of Yee, and of Harten and Osher are applied to the solution of the Euler and Navier-Stokes equations in two-dimensions. With the exception of the Lax and Wend off and of the Yee schemes, which are symmetrical ones, all others are flux difference splitting algorithms. All schemes are second order accurate in space and first order accurate in time. The Euler and Navier-Stokes equations, written in a conservative and integral form, are solved, according to a finite volume and structured formulations. A spatially variable time step procedure is employed aiming to accelerate the convergence of the numerical schemes to the steady state condition. It has proved excellent gains in terms of convergence acceleration as reported by Maciel. The physical problems of the supersonic shock reflection at the wall and the supersonic flow along a compression corner are solved, in the in viscid case. For the viscous case, the supersonic flow along a compression corner is solved. In the in viscid case, an implicit formulation is employed to marching in time, whereas in the viscous case, a time splitting approach is used. The results have demonstrated that the Yee, Warming and Harten algorithm has presented the best solution in the in viscid shock reflection problem, the Harten and Osher algorithm, in its ENO version, and the Lax and Wend off TVD algorithm, in its Van Leer variant, have yielded the best solutions in the in viscid compression corner problem, and the Lax and Wend off TVD algorithm, in its Minmod1 variant, has presented the best solution in the viscous compression corner problem.