Skip to main content
SpringerLink
Log in

A Fully Implicit, Compact Finite Difference Method for the Numerical Solution of Unsteady Laminar Flames

  • Published:
Flow, Turbulence and Combustion Aims and scope Submit manuscript

Abstract

Forced laminar diffusion flames form an important class of problems that can help to bridge the significant gap between steady laminar flames in simple burner configurations and the turbulent flames found in many practical combustors. Such flames offer a much wider range of interactions between convection, diffusion, and chemical reaction than can be examined under steady-state conditions, and yet detailed simulations of them should be feasible without having to resort to “modeling” any of the relevant physics, above all without having prematurely to reduce the large kinetic mechanisms typical of hydrocarbon fuels. Nevertheless, the computation of time-dependent laminar diffusion flames with conventional numerical methods is hindered by technical challenges that, while not new, are more troublesome to surmount than in the calculation of otherwise similar, unforced flames. First, the intricate spatiotemporal coupling between fluid dynamics and combustion thermochemistry ensures that spurious numerical diffusion or spatial under-resolution of the mixing process at any stage of the computation can lead to inaccurate prediction of flame characteristics for the remainder thereof. Second, relatively long simulated flow times and extremely short chemical time scales make many standard time integration algorithms impractical on all but the largest parallel computer clusters. This paper introduces a new numerical approach for time-varying laminar flames that addresses these challenges through the use of high order compact finite difference schemes within a robust, fully implicit solver based on a Jacobian-free Newton–Krylov method. The capabilities of this implicit-compact solver are demonstrated on a periodically forced axisymmetric laminar jet diffusion flame with one-step Arrhenius chemistry, and the results are compared to those of a conventional low order finite difference solver.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics, Philadelphia (1998)

    MATH  Google Scholar 

  2. Baum, M., Poinsot, T.J., Haworth, D.C., Darabiha, N.: Direct numerical simulation of H 2/O 2/N 2 flames with complex chemistry in two-dimensional turbulent flows. J. Fluid Mech. 281, 1–32 (1994)

    Article  Google Scholar 

  3. Baum, M., Poinsot, T., Thévenin, D.: Accurate boundary conditions for multicomponent reactive flows. J. Comput. Phys. 116(2), 247–261 (1995)

    Article  MATH  Google Scholar 

  4. Bayliss, A., Matkowsky, B.J., Aldushin, A.P.: Dynamics of hot spots in solid fuel combustion. Phys. D: Nonlinear Phenom. 166(1–2), 104–130 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bennett, B.A.V., Smooke, M.D.: Local rectangular refinement with application to axisymmetric laminar flames. Combust. Theory Model. 2(3), 221–258 (1998)

    Article  MATH  Google Scholar 

  6. Bennett, B.A.V., Smooke, M.D.: Local rectangular refinement with application to nonreacting and reacting fluid flow problems. J. Comput. Phys. 151, 684–727 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bennett, B.A.V., Smooke, M.D.: Unsteady axisymmetric laminar diffusion flames: an application of local rectangular refinement. In: Eastern States Section of the Combustion Institute Biennial Technical Meeting, 2–5 December, Hilton Head, SC (2001)

  8. Bennett, B.A.V., Smooke, M.D.: Local rectangular refinement solution-adaptive gridding with application to oscillating laminar diffusion flames. In: Third Joint Meeting of the U.S. Sections of the Combustion Institute, 16–19 March, Chicago, IL (2003)

  9. Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182(2), 418–477 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Benzi, M., Haws, J.C., Tuma, M.: Preconditioning highly indefinite and nonsymmetric matrices. SIAM J. Sci. Comput. 22(4), 1333–1353 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bijl, H., Carpenter, M.H., Vatsa, V.N., Kennedy, C.A.: Implicit time integration schemes for the unsteady compressible Navier–Stokes equations: laminar flow. J. Comput. Phys. 179(1), 313–329 (2002)

    Article  MATH  Google Scholar 

  13. Bilger, R.W.: Future directions in turbulent combustion research. Combust. Sci. Technol. 98(4), 223–228 (1994)

    Article  Google Scholar 

  14. Björck, Å.: Numerics of Gram–Schmidt orthogonalization. Linear Algebra Appl. 197–198, 297–316 (1994)

    Article  Google Scholar 

  15. Bochev, P., Gunzburger, M.: An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations. SIAM J. Numer. Anal. 42(3), 1189–1207 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Braack, M., Ern, A.: Coupling multimodelling with local mesh refinement for the numerical computation of laminar flames. Combust. Theory Model. 8(4), 771–788 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Statistical Comput. 11(3), 450–481 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Carpenter, M.H., Kennedy, C.A., Bijl, H., Viken, S.A., Vatsa, V.N.: Fourth-order Runge–Kutta schemes for fluid mechanics applications. J. Sci. Comput. 25(1), 157–194 (2005)

    Article  MathSciNet  Google Scholar 

  19. Coffey, T.S., Kelley, C.T., Keyes, D.E.: Pseudotransient continuation and differential-algebraic equations. SIAM J. Sci. Comput. 25(2), 553–569 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Connelly, B.: Quantitative characterization of steady and time-varying, sooting, laminar diffusion flames using optical techniques. PhD thesis, Yale University (2009)

  21. Dahlquist, G., Björck, Å.: Numerical Methods. Dover, New York (2003)

    MATH  Google Scholar 

  22. Day, M.S., Bell, J.B.: Numerical simulation of laminar reacting flows with complex chemistry. Combust. Theory Model. 4(4), 535–556 (2000)

    Article  MATH  Google Scholar 

  23. Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  24. Deuflhard, P.: A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. Math. 22, 289–315 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  25. Dobbins, R.R.: Development of an implicit-compact finite difference method for the numerical simulation of unsteady laminar flames. PhD thesis, Yale University (2010)

  26. Don, W.S., Gottlieb, D.: Spectral simulation of supersonic reactive flows. SIAM J. Numer. Anal. 35(6), 2370–2384 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  27. Duff, I.S., Koster, J.: On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Anal. Appl. 22(4), 973–996 (2000)

    Article  MathSciNet  Google Scholar 

  28. Dworkin, S.B., Bennett, B.A.V., Smooke, M.D.: A mass-conserving vorticity-velocity formulation with application to nonreacting and reacting flows. J. Comput. Phys. 15(2), 430–447 (2006)

    Article  Google Scholar 

  29. Dworkin, S.B., Connelly, B.C., Schaffer, A.M., Bennett, B.A.V., Long, M.B., Smooke, M.D., Puccio, M.P., McAndrews, B., Miller, J.H.: Computational and experimental study of a forced, time-dependent, methane-air coflow diffusion flame. Proc. Combust. Inst. 31(1), 971–978 (2007)

    Article  Google Scholar 

  30. Ehrenstein, U., Guillard, H., Peyret, R.: Flame computations by a Chebyshev multidomain method. Int. J. Numer. Methods Fluids 9(5), 499–515 (1989)

    Article  MATH  Google Scholar 

  31. Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  32. Ern, A., Smooke, M.D.: Vorticity-velocity formulation for three-dimensional steady compressible flows. J. Comput. Phys. 105(1), 58–71 (1993)

    Article  MATH  Google Scholar 

  33. Ern, A., Giovangigli, V., Keyes, D.E., Smooke, M.D.: Towards polyalgorithmic linear system solvers for nonlinear elliptic problems. SIAM J. Sci. Comput. 15(3), 681–703 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  34. Fernández-Tarrazo, E., Sánchez, A.L., Liñán, A., Williams, F.A.: A simple one-step chemistry model for partially premixed hydrocarbon combustion. Combust. Flame 147(1–2), 32–38 (2006)

    Article  Google Scholar 

  35. Fuchs, L., Zhao, H.S.: Solution of three-dimensional viscous incompressible flows by a multi-grid method. Int. J. Numer. Methods Fluids 4(6), 539–555 (1984)

    Article  MATH  Google Scholar 

  36. Gamet, L., Ducros, F., Nicoud, F., Poinsot, T.: Compact finite difference schemes on non-uniform meshes. Application to direct numerical simulations of compressible flows. Int. J. Numer. Methods Fluids 29(2), 159–191 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  37. Gao, X., Groth, C.P.T.: A parallel adaptive mesh refinement algorithm for predicting turbulent non-premixed combusting flows. Int. J. Numer. Methods Fluids 20(5), 349–357 (2006)

    MATH  Google Scholar 

  38. Giovangigli, V., Darabiha, N.: Vector computers and complex chemistry combustion. In: Brauner, C.M., Schmidt-Laine, C. (eds.) Mathematical Modeling in Combustion and Related Topics, pp 491–503, Nijhoff, Dordrecht (1988)

    Google Scholar 

  39. Gottlieb, S., Mullen, J.S., Ruuth, S.J.: A fifth order flux implicit WENO method. J. Sci. Comput. 27(1–3), 271–287 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  40. Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuska–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59(1), 85–99 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  41. Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol 33. Springer, Berlin (2003)

    MATH  Google Scholar 

  42. Im, H.G., Chen, J.H.: Structure and propagation of triple flames in partially premixed hydrogen-air mixtures. Combust. Flame 119(4), 436–454 (1999)

    Article  Google Scholar 

  43. Isono, S., Zingg, D.W.: A Runge–Kutta–Newton–Krylov algorithm for fourth-order implicit time marching applied to unsteady flows. AIAA Paper 2004-0433 (2004)

  44. Jameson, L.: AMR vs high order schemes. J. Sci. Comput. 18(1), 1–24 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  45. Jiménez, C., Cuenot, B., Poinsot, T., Haworth, D.: Numerical simulation and modeling for lean stratified propane-air flames. Combust. Flame 128(1–2), 1–21 (2002)

    Article  Google Scholar 

  46. Kaplan, C.R., Baek, S.W., Oran, E.S., Ellzey, J.L.: Dynamics of a strongly radiating unsteady ethylene jet diffusion flame. Combust. Flame 96(1–2), 1–21 (1994)

    Article  Google Scholar 

  47. Kee, R.J., Dixon-Lewis, G., Warnatz, J., Coltrin, M.E., Miller, J.A.: A Fortran computer code package for the evaluation of gas-phase, multicomponent transport properties. Tech. Rep. SAND86-8246, Sandia National Laboratories (1986)

  48. Kee, R.J., Rupley, F.M., Miller, J.A.: The Chemkin thermodynamic database. Tech. Rep. SAND87-8215, Sandia National Laboratories (1987)

  49. Knio, O.M., Najm, H.N., Wyckoff, P.S.: A semi-implicit numerical scheme for reacting flow: II. Stiff, operator-split formulation. J. Comput. Phys. 154(2), 428–467 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  50. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  51. Knoll, D.A., McHugh, P.R., Keyes, D.E.: Newton–Krylov methods for low Mach number compressible combustion. AIAA J. 34, 961–967 (1996)

    Article  MATH  Google Scholar 

  52. Lacor, C., Smirnov, S., Baelmans, M.: A finite volume formulation of compact central schemes on arbitrary structured grids. J. Comput. Phys. 198(2), 535–566 (2004)

    Article  MATH  Google Scholar 

  53. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103(1), 16–42 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  54. Linden, J., Lonsdale, G., Steckel, B., Stüben, K.: Multigrid for the steady-state incompressible Navier–Stokes equations: a survey. In: Dwoyer, D.L., Hussaini, M.Y., Voight, R.G. (eds.) 11th International Conference on Numerical Methods in Fluid Dynamics. Lecture Notes in Physics, no. 323, pp 57–68. Springer, Berlin (1989)

    Chapter  Google Scholar 

  55. Majda, A., Sethian, J.: The derivation and numerical solution of the equations for zero Mach number combustion. Combust. Sci. Technol. 42, 185–205 (1985)

    Article  Google Scholar 

  56. Mohammed, R.K., Tanoff, M.A., Smooke, M.D., Schaffer, A.M., Long, M.B.: Computational and experimental study of a forced, time-varying, axisymmetric, laminar diffusion flame. Proc. Combust. Inst. 27(1), 693–702 (1998)

    Google Scholar 

  57. Müller, B.: Low Mach number asymptotics of the Navier–Stokes equations and numerical implications. 30th Computational Fluid Dynamics Lecture Series, 8–12 March. von Karman Institute for Fluid Dynamics (1999)

  58. Nagarajan, S., Lele, S.K., Ferziger, J.H.: A robust high-order compact method for large eddy simulation. J. Comput. Phys. 191(2), 392–419 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  59. Northrup, S.A., Groth, C.P.T.: Parallel implicit adaptive mesh refinement algorithm for unsteady laminar flames. In: Proceedings of the Combustion Institute/Canadian Section Spring Technical Meeting, 11–14 May, pp. 78–83, Toronto, Ontario, Canada (2008)

  60. Northrup, S.A., Groth, C.P.T.: Solution of laminar combusting flows using a parallel implicit adaptive mesh refinement algorithm. In: Deconinck, H., Dick, E. (eds.) Computational Fluid Dynamics 2006: Proceedings of the Fourth International Conference on Computational Fluid Dynamics, ICCFD, Ghent, Belgium, 10–14 July 2006, pp. 341–346. Springer, Berlin (2009)

    Google Scholar 

  61. Noskov, M., Smooke, M.D.: Primitive variable solver for modeling steady-state and time-dependent laminar flames with detailed chemistry and transport properties. In: Bathe, K.J. (ed.) Computational Fluid and Solid Mechanics: Proceedings, First MIT Conference on Computational Fluid and Solid Mechanics, 12–15 June 2001, pp. 1338–1341. Elsevier, Oxford (2001)

    Google Scholar 

  62. Noskov, M., Smooke, M.D.: An implicit compact scheme solver with application to chemically reacting flows. J. Comput. Phys. 203(2), 700–730 (2005)

    Article  MATH  Google Scholar 

  63. Noskov, M., Benzi, M., Smooke, M.D.: An implicit compact scheme solver for two-dimensional multicomponent flows. Comput. Fluids 36, 376–397 (2007)

    Article  MATH  Google Scholar 

  64. Orszag, S.A.: Spectral methods for problems in complex geometries. J. Comput. Phys. 37(1), 70–92 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  65. Pember, R.B., Howell, L.H., Bell, J.B., Colella, P., Crutchfield, W.Y., Fiveland, W.A., Jessee, J.P.: An adaptive projection method for unsteady, low-Mach number combustion. Combust. Sci. Technol. 140, 123–168 (1998)

    Article  Google Scholar 

  66. Poinsot, T., Veynante, D.: Theoretical and Numerical Combustion, 2nd edn. Edwards, Ann Arbor, MI (2005)

    Google Scholar 

  67. Ropp, D.L., Shadid, J.N., Ober, C.C.: Studies of the accuracy of time integration methods for reaction-diffusion equations. J. Comput. Phys. 194(2), 544–574 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  68. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2003)

    MATH  Google Scholar 

  69. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7(3), 856–869 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  70. Sánchez-Sanz, M., Bennett, B.A.V., Smooke, M.D., Liñán, A.: Influence of Strouhal number on pulsating methane-air coflow jet diffusion flames. Combust. Theory Model. 14(3), 453–478 (2010)

    Google Scholar 

  71. Sesterhenn, J., Müller, B., Thomann, H.: On the cancellation problem in calculating compressible low Mach number flows. J. Comput. Phys. 151(2), 597–615 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  72. Shin, D., Strikwerda, J.C.: Inf-sup conditions for finite-difference approximations of the Stokes equations. J. Aust. Math. Soc., Ser. B 39(1), 121–134 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  73. Smooke, M.D.: Solution of burner-stabilized premixed laminar flames by boundary value methods. J. Comput. Phys. 48(1), 72–105 (1982)

    Article  MATH  Google Scholar 

  74. Smooke, M.D.: Error estimate for the modified Newton method with applications to the solution of nonlinear, two-point boundary-value problems. J. Optim. Theory Appl. 39(4), 489–511 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  75. Smooke, M.D., Bennett, B.A.V.: Numerical modeling of multidimensional laminar flames. In: Baukal, C.E., Gershtein, V.Y., Li, X. (eds.) Computational Fluid Dynamics in Industrial Combustion, chap. 6, pp. 209–245. CRC Press, Boca Raton, FL (2001)

    Google Scholar 

  76. Smooke, M.D., Giovangigli, V.: Formulation of the premixed and nonpremixed test problems. In: Smooke, M.D. (ed.) Reduced Kinetic Mechanisms and Asymptotic Approximations for Methane-Air Flames. Lecture Notes in Physics, no. 384, chap. 1, pp. 1–28. Springer-Verlag, Berlin (1991)

    Chapter  Google Scholar 

  77. Smooke, M.D., Mitchell, R.E., Keyes, D.E.: Numerical solution of two-dimensional axisymmetric laminar diffusion flames. Combust. Sci. Technol. 67(4), 85–122 (1989)

    Article  Google Scholar 

  78. Sutherland, J.C., Kennedy, C.A.: Improved boundary conditions for viscous, reacting, compressible flows. J. Comput. Phys. 191(2), 502–524 (2003)

    Article  MATH  Google Scholar 

  79. Tomboulides, A.G., Lee, J.C.Y., Orszag, S.A.: Numerical simulation of low Mach number reactive flows. J. Sci. Comput. 12(2), 139–167 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  80. Tuminaro, R.S., Walker, H.F., Shadid, J.N.: On backtracking failure in Newton-GMRES methods with a demonstration for the Navier–Stokes equations. J. Comput. Phys. 180(2), 549–558 (2002)

    Article  MATH  Google Scholar 

  81. van der Hoeven, S., Boersma, B.J., Roekaerts, D.J.E.M.: Large eddy simulation of turbulent non-premixed jet flames with a high order numerical method. In: Koren, B., Vuik, K. (eds.) Advanced Computational Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 71, pp. 269–287. Springer, Berlin (2010)

    Chapter  Google Scholar 

  82. van der Vorst, H.A.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 13(2), 631–644 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  83. Wolfram Research: Mathematica, version 5.0, Champaign, IL (2003)

  84. Xu, Y., Smooke, M.D., Lin, P., Long, M.B.: Primitive variable modeling of multidimensional laminar flames. Combust. Sci. Technol. 90(5), 289–313 (1993)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Yale University, New Haven, CT, 06520-8284, USA

    Richard R. Dobbins & Mitchell D. Smooke

Authors
  1. Richard R. Dobbins
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mitchell D. Smooke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard R. Dobbins.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dobbins, R.R., Smooke, M.D. A Fully Implicit, Compact Finite Difference Method for the Numerical Solution of Unsteady Laminar Flames. Flow Turbulence Combust 85, 763–799 (2010). https://doi.org/10.1007/s10494-010-9278-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10494-010-9278-z

Keywords

Search

Navigation