1:_t i 7 +L AIAA 96-0764 High Resolution Euler Solvers Based on the Space-Time Conservation Element and Solution Element Method _X(..y. W&l'i_g, C.Y. Chow University of Color.do Boulder, CO and S.C. Chang NASA Lewis Cleveland, OH 34th Aerospace Sciences Meeting & Exhibit January 15-18, 1996 / Reno, NV '1 ' I II IIIn _ I ill I ' For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W.,-Washington, D.C. 20024 tlIGtt RESOLUTION EULER SOLVERS BASED ON THE SPACE-TIME CONSERVATION ELEMENT AND SOLUTION ELEMENT METHOD Xiao-Yen Wang,* Chuen-Yen Chowt Department of Aerospace Engineering, University of Colorado, Boulder, CO 80309 Sin-Chung Changl NASA Lewis Research Center, Cleveland, OH 44135 Abstract variables and their spatial derivatives are consid- ered as individual unknowns to be solved simultane- The I-D, quasi 1-D and 2-D Euler solvers based ously at each grid point. Third, no approximation on the method of space-time conservation element techniques other than Taylor's series expansion, no and solution element are used to simulate various monotonous constraints, and no characteristic-based flow phenomena including shock waves, Mach stem, techniques are used in the design of the scheme. De- contact surface, expansion waves, and their inter- tailed description of this method and the accompa- sections and reflections. Seven test problems are nied analysis are referred to Refs. 1and 2. solved to demonstrate the capability of this method Various efficient numerical schemes based on the for handling unsteady compressible flows in various CE/SE method have been developed for solving dif- configurations. Numerical results so obtained are ferent flow problems, especially the problems in the compared with exact solutions and/or numerical so- presence of shock waves with discontinuous flow lutions obtained by schemes based on other estab- properties. Of those schemes, the one- and two- lished computational techniques. Comparisons show dimensional time marching Euler solvers are em- that the present Euler solvers can generate highly ployed here to solve problems involving flow phe- accurate numerical solutions to complex flow prob- nomena that are more complex than those shown in lems in a straightforward manner without using any Refs. 1and 2. In addition, a quasi one-dimensional ad hoc techniques in the scheme. Euler solver is constructed in this work which is aimed at dealing with problems in axisymmetric con- 1. Introduction figurations. Comparisons of the computed results The method of space-time conservation element with published data are made to demonstrate the and solution element (to be abbreviated as the simplicity and accuracy of the present Euler solvers. CE/SE method) is a new numerical method devel- Detailed description of the governing equations oped by Chang for solving conservation laws. 1-4 It and CE/SE l-D, quasi l-D, 2-D Euler solvers used is different in both concept and methodology from in the following numerical tests is referred to Ref. 3, the well-established traditional methods such as the while numerical results and discussions are described finite difference, finite volume, finite element and in details as follows. spectral methods. It is designed from a physicist's perspective to overcome several key limitations of 2. Numerical Results the traditional methods. 2.1) 1-D shock-tube problems Simplicity, generality and accuracy are weighted in the development of the present method while satis- In the 1-D weighted-average Euler a-e scheme used fying the fundamental computational requirements. in the numerical tests for the following four shock- Its salient properties are summarized briefly as fol- tube problcms, the parameter e is set as 0.5, whilc lows. First, the concepts of conservation clement c_is l excel)t in the first problem where c_is 4. and solution element are introduced to enforce both local and global flux conservation in space and time a) The Lax problem instead of in space only. Second, all the dependent The initial conditions in the region [-8, 6] on the x axis are defined as *Graduate Research Assistant. tProfessor, AIAA Associate Fellow. p1=0.445, u1=0.6989, pt =3.5277 x<0 (1) !Senior Scientist. Copyright (_)1996 American Institute of Aeronautics and As- tronautics, Inc. All rights reserved. p,.--0.5, u_=0.0, p_ =0.571 x>0 (2) Thenumericaslolutionat I = 100At obtained by Pr = 1+ 0.2sin 5x, u_ = 0, p_ = 1 otherwise (6) using the present scheme, under the same compu- This problem does not have an exact solution. tational conditions (CFL=0.95 and Ax = 0.1) as Several upwind schemes have been used to solve this those used in Ref. 5, is shown with the exact solu- problem to compare their abilities in resolving the tion represented by solid lines in Fig. 1. Compar- peaks appeared in the solution. 9 The CE/SE solu- isons were made by Harten 5to appraise four numeri- tion at t=l.8 obtained by using 800 grid points with cal schemes, namely, the ROE, LW (Lax-Wendroff), ULT1, and ULTIC schemes, which were used to At = 0.0015 (CFL=0.582) is shown in Fig. 3. The present solution is comparable to those obtained in solve the same problem. The last two are the TVNI Ref. 9 by using the TVD1 and TVD2 schemes with finite-difference schemes designed by Harten. The the same number of grid points. numerical results plotted in Figs. 2(a)-(d) of Ref. 5 are not reproduced here. Among the four solutions, d) The Woodward-Colella problem the LW solution is the worst showing serious oscil- This problem, concerning the interaction of two lations near the shock discontinuity. In the ROE blast waves in a close-ended tube, was proposed by solution, the oscillations disappear but up to four Woodward and Colella without an exact solution.I° grid points are needed to resolve a shock wave, while The initial conditions are a significant smear is found either near the contact surface or within the expansion fan. Much improve- Pt = 1.0, ut=0, Pl= 1000 x<0.1 (7) ment is revealed in the ULT1 solution, which re- solves a shock discontinuity with only 2 grid points Pm = 1.0, u,,,=O, p,,=0.01 0.1

NASA Technical Reports Server (NTRS) 19980018847: High Resolution Euler Solvers Based on the Space-Time Conservation Element and Solution Element Method

Similar NASA Technical Reports Server (NTRS) 19980018847: High Resolution Euler Solvers Based on the Space-Time Conservation Element and Solution Element Method