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On the Finite Differences Used in Reacting Flow Simulations

Part of: Basic methods in fluid mechanics

Published online by Cambridge University Press:  14 September 2015

Robert Prosser
Robert Prosser*
Affiliation:
Department of Mechanical, Aerospace and Civil Engineering, University of Manchester, PO Box 88 Manchester M13 9PL, UK
*
*Corresponding author. Email address: [email protected] (R. Prosser)
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Abstract

There exist many spatial discretization schemes that are well able to provide accurate and stable approximations for isothermal turbulent flows. Comparatively little analysis has been made of the performance of these schemes in the presence of temperature gradients driven by combustion. In this paper, the effects of temperature gradients on numerical stability are explored. A surprising result is that temperature gradients in the flow have a tendency to impinge on left half plane (LHP) stability of the spatial discretization scheme. Reasons for this tendency are explored and two remedies are proposed: one based on the particular class of finite difference schemes, and one based on an alternative method of boundary condition specification.

Keywords

Numerical methodscombustionstability

MSC classification

Secondary: 76M20: Finite difference methods
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015 , pp. 558 - 576
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Baum, M., Poinsot, T.J., and Thévenin, D.. Accurate boundary conditions for multicomponent reactive flows. J. Comput. Phys., 116:247–261, 1994.Google Scholar
[2]Cant, R.S. and Mastorakos, E.. An Introduction to Turbulent Reacting Flows. Imperial College Press, 2008.Google Scholar
[3]Carpenter, M.H., Gottlieb, D., and Abarbanel, S.. Stable and accurate boundary treatments for compact, high-order finite-difference schemes. Appl. Numer. Math., 12:55–87, 1993.Google Scholar
[4]Carpenter, M.H., Gottlieb, D., and Abarbanel, S.. The stability of numerical boundary treatments for compact high-order finite-difference schemes. J. Comput. Phys., 108:272–295,1993.Google Scholar
[5]Carpenter, M.H., Gottlieb, D., and Abarbanel, S.. Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes. J. Comput. Phys., 111:220–236, 1994.Google Scholar
[6]Chen, J.H., Choudhary, A., Supinsky, B. de, DeVries, M., Hawkes, E.R., Klasky, S., Liao, W.K., Ma, K.L., Mellor-Crummey, J., Podhorszki, N., Sankaran, R., Shende, S., and Yoo, C.S.. Terascale direct numerical simulations of turbulent combustion using S3D. Comput. Sci. Discov., 2:2–32, 2009.Google Scholar
[7]Colonius, T., Lele, S.K., and Moin, P.. Sound generation in a mixing layer. J. Fluid Mech., 330:375–409, 1997.Google Scholar
[8]Gradshteyn, I.S. and Ryzhik, I.M.. Table of Integrals, Series and Products. Academic Press, 6th edition, 2000.Google Scholar
[9]Gray, R.M.. Toeplitz and circulant matrices: A review. Foundations and Trends in Communication and Information Theory, 2:155–239, 2006.Google Scholar
[10]Gustafsson, B., Kreiss, H.-O., and Sundstr¨m, A.. Stability theory of difference approximations for mixed initial boundary value problems. II. Math. Comput., 26:649–686, 1972.Google Scholar
[11]Kennedy, C.A. and Carpenter, M.H.. Several new methods for compressible shear-layer simulations. Appl. Numer. Math., 14:397–433, 1994.Google Scholar
[12]Kreiss, H.O. and Scherer, G.. Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, 1977.Google Scholar
[13]Lele, S.K.. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys., 103:16–42, 1992.Google Scholar
[14]Majda, A. and Sethian, J.. The derivation and numerical solution of the equations for zero mach number combustion. Combust. Sci. and Tech., 42:185–205, 1985.Google Scholar
[15]Martins, E.A. and Silva, F.C.. Eigenvalues of matrix commutators. Linear and Multilinear Algebra, 39:375–389, 1995.Google Scholar
[16]Poinsot, T.J. and Lele, S.K.. Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys., 101:104–129, 1992.Google Scholar
[17]Prosser, R.. Towards improved boundary conditions for the DNS and LES of turbulent subsonic flows. J. Comput. Phys., 222:469–474, 2007.Google Scholar
[18]Prosser, R.. Improved boundary conditions for the DNS of reacting subsonic flows. Flow Turbul. Combust., 87:351–376, 2011.Google Scholar
[19]Reddi, S.S.. Eigenvector properties of Toeplitz matrices and their application to spectral analysis of time series. Signal Process., 7:45–56, 1984.Google Scholar
[20]Ryaben’kii, V.S. and Tsynkov, S.V., editors. A Theoretical Introduction to Numerical Analysis. Chapman & Hall/CRC, 2007.Google Scholar
[21]Strand, B.. Summation by parts for finite difference approximations for d/dx. J. Comput. Phys., 110:47–67, 1994.Google Scholar
[22]Sutherland, J.C. and Kennedy, C.A.. Improved boundary conditions for viscous, reacting, compressible flows. J. Comput. Phys., 191:502–24, 2003.CrossRefGoogle Scholar
[23]Thompson, K.W.. Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys., 68:1–24, 1987.Google Scholar
[24]Trefethen, L.N.. Group velocity in finite difference schemes. SIAM Review, 24:113–136, 1982.Google Scholar
[25]Trefethen, L.N.. Group velocity interpretation of the stability theory of Gustaffson, Kreiss and Sundström. J. Comput. Phys., 49:199–217, 1983.Google Scholar
[26]Wray, A.A.. Minimal Storage Time-Advancement Schemes for Spectral Methods. NASA Ames Research Center, 1990.Google Scholar
[27]Yoo, C.S., Wang, Y., Trouvé, A., and Im, H.G.. Characteristic boundary conditions for direct simulations of turbulent counterflow flames. Combust. Theory Modelling, 9:617–646, 2005.Google Scholar