Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T08:39:13.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability Analysis of a Fully Coupled Implicit Scheme for Inviscid Chemical Non-Equilibrium Flows

Part of: Hypersonic flows Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical linear algebra

Published online by Cambridge University Press:  19 September 2016

Yu Wang ,
Jinsheng Cai  and
Kun Qu
Yu Wang*
Affiliation:
National Key Laboratory of Science and Technology on Aerodynamic Design and Research, Northwestern Polytechnical University, Xi'an 710072, China
Jinsheng Cai*
Affiliation:
National Key Laboratory of Science and Technology on Aerodynamic Design and Research, Northwestern Polytechnical University, Xi'an 710072, China
Kun Qu*
Affiliation:
National Key Laboratory of Science and Technology on Aerodynamic Design and Research, Northwestern Polytechnical University, Xi'an 710072, China
*
*Corresponding author. Email:[email protected] (Y.Wang), [email protected] (K. Qu), [email protected] (J. S. Cai)
*Corresponding author. Email:[email protected] (Y.Wang), [email protected] (K. Qu), [email protected] (J. S. Cai)
*Corresponding author. Email:[email protected] (Y.Wang), [email protected] (K. Qu), [email protected] (J. S. Cai)
Get access

Abstract

Von Neumann stability theory is applied to analyze the stability of a fully coupled implicit (FCI) scheme based on the lower-upper symmetric Gauss-Seidel (LU-SGS) method for inviscid chemical non-equilibrium flows. The FCI scheme shows excellent stability except the case of the flows involving strong recombination reactions, and can weaken or even eliminate the instability resulting from the stiffness problem, which occurs in the subsonic high-temperature region of the hypersonic flow field. In addition, when the full Jacobian of chemical source term is diagonalized, the stability of the FCI scheme relies heavily on the flow conditions. Especially in the case of high temperature and subsonic state, the CFL number satisfying the stability is very small. Moreover, we also consider the effect of the space step, and demonstrate that the stability of the FCI scheme with the diagonalized Jacobian can be improved by reducing the space step. Therefore, we propose an improved method on the grid distribution according to the flow conditions. Numerical tests validate sufficiently the foregoing analyses. Based on the improved grid, the CFL number can be quickly ramped up to large values for convergence acceleration.

Keywords

StabilityLU-SGSnon-equilibrium flowsEuler equationsflux Jacobiangrid refinement

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65F10: Iterative methods for linear systems 76K05: Hypersonic flows
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 6 , December 2016 , pp. 953 - 970
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Anderson, J. D., Hypersonic and High Temperature Gas Dynamics, McGraw-Hill Book Company, New York, 1989.Google Scholar
[2] Chiang, T. L. and Hoffmann, K., Determination of computational time step for chemically reacting flows, AIAA paper 89-1855, 1989.CrossRefGoogle Scholar
[3] Li, C. P., Chemistry split techniques for viscous reactive blunt body flow computations, AIAA paper 87-0282, 1987.Google Scholar
[4] Li, C. P., Computational aspects of chemically reacting flows, AIAA-91-1574-CP, 1991.CrossRefGoogle Scholar
[5] Olsen, M. E., Prabhu, D. K. and Olsen, T., Implementation of finite rate chemistry capability in overflow, AIAA paper 04-2372, 2004.CrossRefGoogle Scholar
[6] Bussing, T. R. and Murman, E. M., Finite volume method for the calculation of compressible chemically reacting flows, AIAA J., 26 (1988), pp. 1070–1078.CrossRefGoogle Scholar
[7] Eberhardt, S. and Imlay, S., Diagonal implicit scheme for computing flows with finite rate chemistry, J. Thermophys. Heat Transfer, 6 (1992), pp. 208–216.Google Scholar
[8] Candler, G. and MacCormack, R. W., The computation of hypersonic ionized flow in chemical and thermal non-equilibrium, AIAA paper 88-0511, 1988.CrossRefGoogle Scholar
[9] Spiegel, S. C., Stefanski, D. L., Luo, H. and Edwards, J. R., A cell-centered finite volume method for chemically reacting flows on hybrid grids, AIAA paper 10-1083, 2010.Google Scholar
[10] Spiegel, S. C., Stefanski, D. L., Luo, H. and Edwards, J. R., A regionally structured/ unstructured finite volume method for chemically reacting flows, AIAA paper 11-3048, 2011.Google Scholar
[11] Lambert, J. D., Computational Methods in Ordinary Differential Equations, John Wiley & Sons, 1973.Google Scholar
[12] Fassbender, J. K., Improved Robustness for Numerical Simulation of Turbulent Flows around Civil Transport Aircraft at Flight Reynolds Numbers, Ph.D. Dissertation, Institute of Aerodynamics and Flow Technology, DLR, Brauschweig, 2003.Google Scholar
[13] Dwight, R. P., Efficiency Improvements of RANS-Based Analysis and Optimization Using Implicit and Adjoint Methods on Unstructured Grids, Ph.D. Dissertation, School of Mathematics, University of Manchester, 2006.Google Scholar
[14] Venkateswaran, S. and Michael, O., Stability analysis of fully coupled and loosely coupled schemes for combustion CFD, AIAA paper 03-3543, 2003.Google Scholar
[15] Kim, S. L., Jeung, I. S., Park, Y. H. and Choi, J. Y., Approximate Jacobian methods for efficient calculation of reactive flows, AIAA paper 00-3384, 2000.Google Scholar
[16] Ju, Y., Lower-upper scheme for chemically reacting flow with finite rate chemistry, AIAA J., 33 (1995), pp. 1418–1425.Google Scholar
[17] Glaz, H. and Colella, P., Non-equilibrium effects in oblique shock wave reflection, AIAA J., 26 (1985), pp. 46–53.Google Scholar
[18] Gollan, R. J., Yet another finite-rate chemistry module for compressible flow codes, Division Report, 9 (2003).Google Scholar
[19] Beam, R. M. and Warming, R. F., An implicit finite difference algorithm for hyperbolic system in conservation law form, J. Comput. Phys., 22 (1976), pp. 87–109.Google Scholar
[20] Yoon, S. and Jameson, A., Lower-upper symmetric-gauss-sediel method for the Euler and Navier-Stoker equations, AIAA J., 26 (1988), pp. 1025–1026.Google Scholar
[21] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43 (1981), pp. 357–372.Google Scholar
[22] Gordon, S. and McBride, B. J., Computer program for calculation of complex chemical equilibrium compositions and alpplications: I Analysis, NASA Reference Publication 1311, NASA, 1994.Google Scholar
[23] Hassan, B., Candler, G. V. and Olynick, D. R., Thermo-chemical nonequilibrium effects on the aerothermodynamics of aerobraking vehicles, J. Spacecraft Rockets, 30 (1993), pp. 647–655.Google Scholar
[24] Muylaert, J., Walpot, L. and Hauser, J., Standard model testing in the european high enthalpy facility F4 and extrapolation to flight, AIAA paper 92-3905, 1992.Google Scholar
[25] Chen, B., Wang, L. and Xu, X., An implicit upwind parabolized Navier-Stokes code for chemically non-equilibrium flows, Acta Mech. Sinica, 29 (2013), pp. 36–47.CrossRefGoogle Scholar
[26] Bian, Y. G. and Xu, L. G., Aerothermodynamics (Second Edition), Press of University of Science and Technology of China, He Fei, 2011.Google Scholar