WSEAS Transactions on Fluid Mechanics
Print ISSN: 1790-5087, E-ISSN: 2224-347X
Volume 7, 2012
Frink, Parikh and Pirzadeh and Liou and Steffen Jr. TVD Algorithms and Implicit Formulations Applied to the Euler Equations in 2D
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Abstract: In this work, the Frink, Parikh and Pirzadeh and the Liou and Steffen Jr. schemes are implemented following a MUSCL approach, aiming to guarantee second order accuracy and to achieve TVD properties, and employing an implicit formulation to solve the Euler equations in the two-dimensional space. These schemes are implemented according to a finite volume formulation and using a structured spatial discretization. The former scheme is a flux difference splitting one, whereas the latter is a flux vector splitting scheme. The MUSCL approach employs five different types of nonlinear limiters, which assure TVD properties, namely: Van Leer limiter, Van Albada limiter, minmod limiter, Super Bee limiter and -limiter. All variants of the MUSCL approach are second order accurate in space. The implicit schemes employ an ADI approximate factorization to solve implicitly the Euler equations. Explicit and implicit results are compared, as also the computational costs, trying to emphasize the advantages and disadvantages of each formulation. The schemes are accelerated to the steady state solution using a spatially variable time step, which has demonstrated effective gains in terms of convergence rate according to Maciel. The algorithms are applied to the solution of the physical problem of the moderate supersonic flow along a compression corner. The results have demonstrated that the most accurate solutions are obtained with the Frink, Parikh and Pirzadeh TVD scheme using the Van Leer and Super Bee nonlinear limiters, when implemented in its explicit version.
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Keywords: Frink, Parikh and Pirzadeh algorithm, Liou and Steffen Jr. algorithm, MUSCL procedure, Implicit formulation, Flux difference splitting, Flux vector splitting, Euler equations, Two-dimensions