WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 12, 2013
TVD and ENO Applications to Supersonic Flows in 3D – Part I
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Abstract: In this work, first part of this study, the high resolution numerical schemes of Lax and Wendroff, 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 three-dimensions. With the exception of the Lax and Wendroff 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 flows along a compression corner and along a ramp are solved, in the inviscid case. For the viscous case, the transonic flow along a convergent-divergent nozzle is solved. In the inviscid 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 Harten and Osher algorithm, in its ENO version, presents the best solutions in the inviscid compression corner and ramp problems; whereas the Lax and Wendroff algorithm has presented the best solution to the nozzle problem.
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Keywords: Lax and Wendroff algorithm, Yee, Warming and Harten algorithm, Yee algorithm, Harten and Osher algorithm, TVD and ENO flux splitting, Euler and Navier-Stokes equations, Finite volume, Three-dimensions