Proceedings of International Conference on Applications and Design in Mechanical Engineering (iCADME) 11 – 13 October 2009, Batu Ferringhi, Penang, MALAYSIA 1F-1 Accurate Compact Flowfield-Dependent Variation Method for Compressible Euler Equations Abdulhafid M. Elfaghi * , Waqar Asrar and Ashraf A. Omar Department of Mechanical Engineering, Kulliyah of Engineering, International Islamic University Malaysia P. O. Box 10, 50728, Kuala Lumpur, Malaysia * Emails: hafied@yahoo.com Fax: +603-6196-4455 Telephone: +603-6196-4591 Abstract- Numerical simulations of compressible inviscid flows using fully implicit high order compact flowfield- dependent variation (HOC-FDV) method has been developed. The third-order accurate in time flowfield-dependent variation (FDV) scheme is used for time discritization and the fourth- order compact Pade scheme is used to approximate the first and second spatial derivatives. The solution procedure consists of a number of tri-diagonal matrix operations with considerable saving in computing time, and produces an efficient solver. The method has been tested and verified for two numerical examples, a flow over a channel flow with compression/expansion and a flow past NACA0012 airfoil. Good agreement with the analytical and other numerical solutions has been obtained in both cases. Numerical results show the validity and accuracy of the proposed method. Keywords— Flowfield-dependent variation (FDV), Higher- order compact (HOC), Inviscid flows, Euler equations, Supersonic flow I. INTRODUCTION Numerical predictions of the compressible supersonic flows play very important role in computational fluid dynamics (CFD) and are very important. The FDV method was originally introduced by T. J. Chung et al. [1]. The FDV idea began from the need to address the transitions from one type of flow to another and interactions between two distinctly different flows. The approach begins by obtaining the implicitness FDV parameters from the current flowfield variables at each time step and every nodal point. These parameters are used to adjust governing equations at each regime according to the current flowfield situation. The physical interpretation of the FDV first-order parameter, s 1 , and the second-order parameter, s 2 , is the foundation of the FDV method. Large values of these parameters reflect large changes in the conservation variables. These changes may occur between adjacent nodal points within the special nodes as well as between adjacent time steps. The second-order FDV parameter, s 2 , is chosen to be exponentially proportional to the first-order FDV parameter, s 1 . This choice is based on the fact that the first-order FDV parameter tends to assure accuracy of the solution, whereas the second-order FDV parameter provides numerical stability [1-3]. A fourth-order compact Pade scheme was used to approximate the first and second order spatial derivatives. Together with the FDV technique this results in a higher- order compact flow field dependent variable (HOC-FDV) scheme. This technique was successfully applied to solve the unsteady non-linear viscous Burgers equation [6]. In this paper, the HOC-FDV scheme is implemented to solve the two-dimensional Euler equations. Two steps; along i and j directions are used for each time level. The spatial derivatives are calculated using higher-order compact approximation Pade Scheme. Each step is solved using block tri-diagonal algorithm. The problems associated with multidimensionality are solved by using the alternating direction implicit (ADI) factored algorithm which reduces the formidable matrix inversion problem to a series of small band width matrix inversion problems that have efficient solution algorithms. Cross-derivatives terms arising from the FDV derivation are excluded from the implicit side and are evaluated explicitly at the known time step. II. GOVERNING EQUATIONS The two dimensional Euler form of the FDV equations proposed by Chung [1] can be written in the compact factorized ADI form as: RHS U x D x D I = Δ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + * 2 2 11 1 (1) * 1 2 2 22 2 U U y D y D I n Δ = Δ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + + (2) Where ( ) n A s D 1 1 t Δ = (3) ( ) n B s D 1 2 t Δ = (4) n n A s D ⎩ ⎨ ⎧ ⎭ ⎬ ⎫ Δ − = 2 2 2 11 2 t (5)