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Modeling low Mach number reacting flow with detailed chemistry and transport. (English) Zbl 1203.80025
Summary: An efficient projection scheme is developed for the simulation of reacting flow with detailed kinetics and transport. The scheme is based on a zero-Mach-number formulation of the compressible conservation equations for an ideal gas mixture. It relies on Strang splitting of the discrete evolution equations, where diffusion is integrated in two half steps that are symmetrically distributed around a single stiff step for the reaction source terms. The diffusive half-step is integrated using an explicit single-step, multistage, Runge-Kutta-Chebyshev (RKC) method. The resulting construction is second-order convergent, and has superior efficiency due to the extended real-stability region of the RKC scheme. Two additional efficiency-enhancements are also explored, based on an extrapolation procedure for the transport coefficients and on the use of approximate Jacobian data evaluated on a coarse mesh. We demonstrate the construction in 1D and 2D flames, and examine consequences of splitting errors. By including the above enhancements, performance tests using 2D computations with a detailed \(C_{1}C_{2}\) methane-air mechanism and a mixture-averaged transport model indicate that speedup factors of about 15 are achieved over the starting split-stiff scheme.

80M25 Other numerical methods (thermodynamics) (MSC2010)
80A32 Chemically reacting flows
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76N15 Gas dynamics, general
76V05 Reaction effects in flows
Full Text: DOI
[1] Majda, A., and Sethian, J. (1985).Comb. Sci. and Technology 42, 185–205. · doi:10.1080/00102208508960376
[2] Turek, S. (1997).Comput. Methods Appl. Mech. Eng. 143, 271–288. · Zbl 0898.76069 · doi:10.1016/S0045-7825(96)01155-3
[3] Gresho, P. M. (1990).Int. J. Numer. Meth. Fluids 11, 587–620. · Zbl 0712.76035 · doi:10.1002/fld.1650110509
[4] Gresho, P. M. (1990).Int. J. Numer. Meth. Fluids 11, 621–659. · Zbl 0712.76036 · doi:10.1002/fld.1650110510
[5] Patankar, S. V. (1980).Numerical Heat Transfer and Fluid Flow chapter 6 Hemisphere Pub. Corp., McGraw-Hill Co., New York.
[6] Chorin, A. J. (1968).Math. Comput. 22, 745–762. · doi:10.1090/S0025-5718-1968-0242392-2
[7] Chorin, A. J. (1969).Math. Comput. 23, 341–353. · doi:10.1090/S0025-5718-1969-0242393-5
[8] Temam, R. (1969).Arch. Rat. Mech. Anal. 33, 377–385. · Zbl 0207.16904 · doi:10.1007/BF00247696
[9] Kim, J., and Moin, P. (1985).J. Comput. Phys. 59, 308–323. · Zbl 0582.76038 · doi:10.1016/0021-9991(85)90148-2
[10] Brown, D. L., Cortez, R. and Minion, M. L. (2001).J. Comput. Phys. 168, 464–499. · Zbl 1153.76339 · doi:10.1006/jcph.2001.6715
[11] Henriksen, M. O., and Holmen, J. (2002).J. Comput. Phys. 175, 438–453. · Zbl 1059.76045 · doi:10.1006/jcph.2001.6907
[12] McMurtry, P. A., Jou, W.-H., Riley, J. J., and Metcalfe, R. W. (1986).AIAA J. 24(6), 962–970. · doi:10.2514/3.9371
[13] Rutland, C., Ferziger, J. H., and Cantwell, B. J. (1989). Report TF-44, Thermosciences Div., Mech. Eng., Stanford University, Stanford, CA.
[14] Rutland, C. J., and Ferziger, J. H. (1991).Combustion and Flame 84, 343–360. · doi:10.1016/0010-2180(91)90011-Y
[15] Najm, H. N. (1996). inTransport Phenomena in Combustion. Chan, S. (ed.), Vol. 2, Taylor and Francis, Wash. DC, pp. 921–932.
[16] Najm, H. N., and Wyckoff, P. S. (1997).Combustion and Flame 110(1–2), 92–112. · doi:10.1016/S0010-2180(97)89577-6
[17] Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., and Welcome, M. (1998).J. Comput. Phys. 142, 1–46. · Zbl 0933.76055 · doi:10.1006/jcph.1998.5890
[18] Pember, R. B., Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H., and Lai, M. (1994). Preprint UCRL-JC-118634, Lawrence Livermore National Laboratory, Livermore, CA.
[19] Pember, R. B., Howell, L. H., Bell, J. B., Colella, P., Crutchfield, W. Y., Fiveland, W. A., and Jesse, J. P. (1997). Technical Report LBL-38551, Lawrence Berkeley National Laboratory, Berkeley, CA.
[20] Karniadakis, G. E., Israeli, M., and Orzag, S. A. (1991).J. Comput. Phys. 97, 414. · Zbl 0738.76050 · doi:10.1016/0021-9991(91)90007-8
[21] Courant, R., Friedrichs, K. O., and Lewy, H. (1928)Mathematische Annalen,100, 32–74 (Translated to: On the Partial Difference Equations of Mathematical Physics, IBM J. Res. Dev., vol. 11, pp. 215–234, 1967). · JFM 54.0486.01 · doi:10.1007/BF01448839
[22] Anderson, D. A., Tannehill, J. C., and Pletcher, R. H. (1984).Computational Fluid Mechanics and Heat Transfer, Hemisphere Pub. Co., New York. · Zbl 0569.76001
[23] Najm, H. N., Wyckoff, P. S., and Knio, O. M. (1998).J. Comp. Phys.,143(2), 381–402. · Zbl 0936.76064 · doi:10.1006/jcph.1997.5856
[24] Najm, H. N., Knio, O. M., Paul, P. H., and Wyckoff, P. S. (1998).Comb. Sci. Tech. 140(1–6), 369–403. · doi:10.1080/00102209808915779
[25] Knio, O. M., Najm, H. N., and Wyckoff, P. S. (1999).J. Comp. Phys. 154, 428–467. · Zbl 0958.76061 · doi:10.1006/jcph.1999.6322
[26] Hundsdorfer, W. H. (1996). Report NM-N9603, CWI, Amsterdam, http://info4u.cwi.nl.
[27] Spee, E. J. (1995). inAir Pollution III H. P.et al., (ed.), Vol. 1, Comput. Mech. Publ., Southampton-Boston, pp. 319–326.
[28] Verwer, J. G., Blom, J. G., Van Loon, M., and Spee, E. J. (1995).Atmos. Eviron. 30, 49–58. · doi:10.1016/1352-2310(95)00283-5
[29] Hundsdorfer, W., and Verwer, J. G. (1995).App. Num. Math. 18, 191–199. · Zbl 0833.65099 · doi:10.1016/0168-9274(95)00069-7
[30] Spee, E. J., de Zeeuw, P. M., Verwer, J. G., Blom, J. G., and Hundsdorfer, W. H. (1996). Report NM-R9620, CWI, Amsterdam, http://info4u.cwi.nl.
[31] Verwer, J. G., Spee, E. J., Blom, J. G., and Hundsdorfer, W. H. (1999).SIAM J. Sci. Comput. 20, 1456–1480. · Zbl 0922.65031 · doi:10.1137/S1064827597326651
[32] Spee, E. J., Verwer, J. G., de Zeeuw, P. M., Blom, J. G., and Hundsdorfer, W. (1998).Math. Comp. Simulation 48, 177–204. · doi:10.1016/S0378-4754(98)00155-4
[33] Khan, L. A., and Liu, P.L.-F. (1995).Comput. Methods Appl. Mech. Engrg. 127, 181–201. · Zbl 0862.76060 · doi:10.1016/0045-7825(95)00839-5
[34] Strang, G. (1968).SIAM J. Numer. Anal.,5(3), 506–517. · Zbl 0184.38503 · doi:10.1137/0705041
[35] Burstein, S. Z., and Mirin, A. A. (1970).J. Comp. Phys. 5, 547–571. · Zbl 0223.65053 · doi:10.1016/0021-9991(70)90080-X
[36] Yoshida, H. (1990).Physics Letters A 150(5–7), 262–268. · doi:10.1016/0375-9601(90)90092-3
[37] Sheng, Q. (1989).IMA J. Numer. Anal. 9, 199–212. · Zbl 0676.65116 · doi:10.1093/imanum/9.2.199
[38] Wright, J. P. (1998).J. Comp. Phys.,140, 421–431. · Zbl 0920.65057 · doi:10.1006/jcph.1998.5902
[39] Day, M. S., and Bell, J. B. (2000).Combust. Theory Modelling 4, 535–556. · Zbl 0970.76065 · doi:10.1088/1364-7830/4/4/309
[40] Kee, R. J., Rupley, F. M., and Miller, J. A. (1993). Sandia Report SAND89-8009B, Sandia National Labs., Livermore, CA.
[41] Verwer, J. G. (1996).App. Num. Math. 22, 359–379. · Zbl 0868.65064 · doi:10.1016/S0168-9274(96)00022-0
[42] Frenklach, M., Wang, H., Goldenberg, M., Smith, G. P., Golden, D. M., Bowman, C. T., Hanson, R. K., Gardiner, W. C., and Lissianski, V. (1995). Top. Rep. GRI-95/0058, GRI.
[43] Paul, Phillip H. (1997). Sandia Report SAND98-8203, Sandia National Laboratories, Albuquerque, New Mexico.
[44] Paul, P., and Warnatz, J. (1998).Twenty-Seventh Symposium (International) on Combustion, The Combustion Institute, pp. 495–504.
[45] Van der Houwen, P. J., and Sommeijer, B. P. (1980).ZAMM 60, 479–485. · Zbl 0455.65052 · doi:10.1002/zamm.19800601005
[46] Verwer, J. G. (1982).ZAMM 62, 561–563. · Zbl 0531.65040 · doi:10.1002/zamm.19820621008
[47] Verwer, J. G., Hundsdorfer, W. H., and Sommeijer, B. P. (1990).Numer. Math. 57, 157–178. · Zbl 0697.65072 · doi:10.1007/BF01386405
[48] Sommeijer, B. P., Shampine, L. F., and Verwer, J. G. (1997).J. Comput. Appl. Math. 88, 315–326. · Zbl 0910.65067 · doi:10.1016/S0377-0427(97)00219-7
[49] Van der Houwen, P. J. (1972).Numer. Math. 20, 149–164. · Zbl 0233.65039 · doi:10.1007/BF01404404
[50] Van der Houwen, P. J. (1977).Construction of integration formulas for initial value problems, North-Holland, Amsterdam-New York. · Zbl 0359.65057
[51] Verwer, J. G. (1977).J. Comp. App. Math. 3(3), 155–166. · Zbl 0373.65030 · doi:10.1016/S0377-0427(77)80002-2
[52] Verwer, J. G. (1980).ACM Trans. on Math. Software 6(2), 188–205. · Zbl 0431.65069 · doi:10.1145/355887.355892
[53] Medovikov, A. A. (1996) InNumerical Analysis and Its Applications, Vulkov, L., Waśniewski, J., and Yalamov, P., (eds.), Springer, Berlin, pp. 327–334, Lecture Notes in Computer Science 1196. Goos, G., Hartmanis, J., and van Leeuwen, J. (eds.).
[54] Lebedev, V. I. (1998).Russ. J. Numer. Ansl. Math. Modelling 13(2), 107–116. · Zbl 0914.65087 · doi:10.1515/rnam.1998.13.2.107
[55] Medovikov, A. A. (1998).BIT 38(2), 372–390. · Zbl 0909.65060 · doi:10.1007/BF02512373
[56] Golushko, M. I., and Novikov, E. A. (1999)Russ. J. Numer. Anal. Math. Modelling,14(1), 71–85. · Zbl 0923.65046 · doi:10.1515/rnam.1999.14.1.71
[57] Abdulle, A. (2000).BIT,40(1), 177–182. · Zbl 0956.65068 · doi:10.1023/A:1022378621048
[58] Bakker, M. (1971). Technical Note TN 62, Mathematical Center, Amsterdam, (in Dutch).
[59] Van der Houwen, P. J. (1994). CWI Report NM-R9420, CWI, Amsterdam.
[60] Guillou, A., and Lago, B. (1961).Recherche de formules a grand rayon de stabilité, ler Congr. Assoc. Fran. Calcul, AFCAL, Grenoble, Sept. 1960, pp. 43–56.
[61] Schlichting, H. (1979).Boundary-Layer Theory, 7th edn, McGraw-Hill, New York. · Zbl 0434.76027
[62] Williams, F. A. (1985).Combustion Theory, 2nd edn, Addison-Wesley, New York.
[63] Najm, H. N., Knio, O. M., and Paul, P. H. (2003). Sandia Report SAND2003-8412, Sandia National Laboratories.
[64] Brown, P. N., Byrne, G. D., and Hindmarsh, A. C. (1989).SIAM J. Sci. Stat. Comput. 10, 1038–1051. · Zbl 0677.65075 · doi:10.1137/0910062
[65] Mahalingam, S., Cantwell, B. J., and Ferziger, J. H. (1990).Phys. Fluids A,2, 720–728. · doi:10.1063/1.857725
[66] Sportisse, B. (2000).J. Comp. Phys. 161, 140–168. · Zbl 0953.65062 · doi:10.1006/jcph.2000.6495
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