WSEAS Transactions on Fluid Mechanics
Print ISSN: 1790-5087, E-ISSN: 2224-347X
Volume 7, 2012
A Review of Some Numerical Methods to the Euler Equations in Two-Dimensions
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Abstract: This work aims to describe the numerical implementation of the Lax and Friedrichs, Lax and Wendroff TVD, Boris and Book, Beam and Warming and MacCormack, on a finite volume and structured spatial discretization contexts, to solve the Euler equations in two-dimensions. The Lax and Wendroff algorithm was implemented according to the TVD formulation of Yee. The Beam and Warming scheme was implemented only in its explicit version. Hence, it is possible to distinguish four categories of algorithms studied in this work: symmetrical (Lax and Friedrichs and Beam and Warming), FCT (Boris and Book), Predictor/Corrector (MacCormack) and TVD (Lax and Wendroff). They are applied to the solution of the steady state problem of the moderate supersonic flow along a compression corner. A spatially variable time step procedure is employed to accelerate the convergence of the numerical methods to the steady state condition. This procedure has demonstrated a meaningful gain in terms of convergence ratio, as reported by Maciel. The results have demonstrated that the Beam and Warming scheme, using the nonlinear dissipation operator, provides the best results in terms of quality (good capture of shock wave thickness and wall pressure profile) and quantity (good prediction of the oblique shock wave angle).
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Keywords: Symmetrical schemes, FCT scheme, Predictor/Corrector scheme, TVD scheme, Finite volumes, Euler equations, Two-dimensions