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Flow, Turbulence and Combustion

, Volume 75, Issue 1–4, pp 173–190 | Cite as

Numerical Solvers for Radiation and Conduction in High Temperature Gas Flows

  • Mohammed SeaïdEmail author
  • Axel Klar
  • René Pinnau
Article

Abstract

In this paper, the authors introduce a robust numerical technique for radiation–conduction heat transfer in the high temperature fields of gas turbine combustors. The conduction and radiation effects are analyzed by a differential and an integral equation, respectively. Using discrete ordinates for the angular discretization of the integral equation for the radiation effects and a Galerkin discretization for the heat equation, the authors propose a fast multilevel algorithm to solve the fully discretized problem. The algorithm uses the same mesh hierarchy for both radiation and conduction effects, but with two different smoothing operators. Numerical results are shown for test problems in three space dimensions, and comparisons to other methods are also given.

Key Words

radiation–conduction heat transfer discrete ordinates Galerkin method multilevel algorithms 

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References

  1. 1.
    Jamaluddin, A. and Smith, P., Predicting radiative transfer in axisymmetric cylindrical enclosures using the discrete ordinates method. Combust. Sci. Technol. 62 (1988) 173–186.Google Scholar
  2. 2.
    Selçuk, N., Evaluation for radiative transfer in rectangular furnaces. Int. J. Heat Mass Transf. 31 (1988) 1477–1482.Google Scholar
  3. 3.
    Selçuk, N. and Kayakol, N., Evaluation of discrete ordinates method for radiative transfer in rectangular furnaces. Int. J. Heat Mass Transf. 40 (1997) 213–222.Google Scholar
  4. 4.
    Liu, F., Becker, H. and Bindar, Y., A comparative study of radiative heat transfer modelling in gas-fired furnaces using the simple grey gas and the weighted-sum-of grey-gases models. Int. J. Heat Mass Transf. 41 (1998) 3357–3371.Google Scholar
  5. 5.
    Hottel, H. and Sarofim, A., Radiative Transfer. McGraw-Hill, New York (1967).Google Scholar
  6. 6.
    Steward, F. and Cannom, P., The calculation of radiative heat flux in a cylindrical furnace using the Monte Carlo method. Int. J. Heat Mass Transf. 14 (1971) 245–262.CrossRefGoogle Scholar
  7. 7.
    Chandrasekhar, S., Radiative Transfer. Oxford University Press, London (1950).Google Scholar
  8. 8.
    Fiveland, W., The selection of discrete ordinate quadrature sets for anisotropic scattering. ASME HTD. Fundam. Radiat. Heat Transf. 160 (1991) 89–96.Google Scholar
  9. 9.
    Briggs, W., Henson, V. and McCormick, S., A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia, PA (1999).Google Scholar
  10. 10.
    McCormick, S., Multilevel Adaptive Methods for Partial Differential Equations. SIAM (1989).Google Scholar
  11. 11.
    Hackbusch, W., Multi-Grid Methods and Applications: Vol. 4, Springer Series in Computational Mathematics. Springer-Verlag, New York (1985).Google Scholar
  12. 12.
    Mihalas, D. and Mihalas, B., Foundations of Radiation Hydrodynamics. Oxford University Press, New York (1984).Google Scholar
  13. 13.
    Lewis, E. and Miller, W., Computational Methods of Neutron Transport. Wiley, New York (1984).Google Scholar
  14. 14.
    Adams, M. and Larsen, E., Fast iterative methods for deterministic particle transport computations. Preprint (2000).Google Scholar
  15. 15.
    Dinshaw, B., Fast and accurate discrete ordinate methods for multidimensional radiative transfer: part I, basics methods. JQSRT 69 (2001) 671–707.Google Scholar
  16. 16.
    Brown, P., A linear algebraic development of diffusion synthetic acceleration for three-dimensional transport equations. SIAM. J. Numer. Anal 32 (1995) 179–214.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Turek, S., An efficient solution technique for the radiative transfer equation. IMPACT, Comput. Sci. Eng. 5 (1993) 201–214.zbMATHMathSciNetGoogle Scholar
  18. 18.
    Turek, S., A generalized mean intensity approach for the numerical solution of the radiative transfer equation. Computing 54 (1995) 27–38.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Seaïd, M. and Klar, A., Efficient preconditioning of linear systems arising from the discretization of radiative transfer equation. Lect. Notes Computat. Sci. Eng. 35 (2003) 211–236.Google Scholar
  20. 20.
    Kelley, C., Multilevel source iteration accelerators for the linear transport equation in slab geometry. Transp. Theor. Stat. Phys. 24 (1995) 679–707.zbMATHMathSciNetGoogle Scholar
  21. 21.
    Saad, Y. and Schultz, M., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM. J. Sci. Statist. Comput. 7 (1986) 856–869.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Kelley, C., Iterative Methods for Linear and Nonlinear Equations. SIAM. Philadelphia, PA (1995).Google Scholar
  23. 23.
    Banoczi, J. and Kelley, C., A fast multilevel algorithm for the solution of nonlinear systems of conductive-radiative heat transfer equation. SIAM J. Sci. Comput. 19 (1998) 266–279.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Seaïd, M., Frank, M., Klar, A., Pinnau, R. and Thömmes, G., Efficient numerical methods for radiation in gas turbines. J. Comp. Appl. Math 170 (2004) 217–239.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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