Abstract
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 C1C2 methane-air mechanism and a mixture-averaged transport model indicate that speedup factors of about 15 are achieved over the starting split-stiff scheme
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References
A. Majda J. Sethian (1985) Comb. Sci. and Technology 42 185–205
S. Turek (1997) Comput. Methods Appl. Mech. Eng. 143 271–288 Occurrence Handle0898.76069 Occurrence Handle1445153 Occurrence Handle10.1016/S0045-7825(96)01155-3
P.M. Gresho (1990) Int. J. Numer Meth. Fluids 11 587–620 Occurrence Handle0712.76035 Occurrence Handle1074825
P.M. Gresho (1990) Int. J. Numer. Meth. Fluids 11 621–659 Occurrence Handle0712.76036 Occurrence Handle1074826
Patankar S.V. (1980). Numerical Heat Transfer and Fluid Flow chapter 6. Hemisphere Pub. Corp., McGraw-Hill Co., New York
A.J. Chorin (1968) Math. Comput. 22 745–762 Occurrence Handle0198.50103 Occurrence Handle242392
A.J. Chorin (1969) Math. Comput 23 341–353 Occurrence Handle0184.20103 Occurrence Handle242393
R. Temam (1969) Arch. Rat. Mech. Anal. 33 377–385 Occurrence Handle0207.16904 Occurrence Handle244654
J. Kim P. Moin (1985) J. Comput. Phys 59 308–323 Occurrence Handle796611 Occurrence Handle10.1016/0021-9991(85)90148-2 Occurrence Handle0582.76038
D.L. Brown R. Cortez M.L. Minion (2001) J. Comput. Phys. 168 464–499 Occurrence Handle1826523 Occurrence Handle10.1006/jcph.2001.6715 Occurrence Handle1153.76339
M.O. Henriksen J. Holmen (2002) J. Comput. Phys 175 438–453 Occurrence Handle1880114 Occurrence Handle10.1006/jcph.2001.6907 Occurrence Handle1059.76045
P.A. McMurtry W.-H. Jou J.J. Riley R.W. Metcalfe (1986) AIAA J 24 IssueID6 962–970 Occurrence Handle10.2514/3.9371
Rutland C., Ferziger J.H., Cantwell B.J. (1989). Report TF-44, Thermosciences Div., Mech. Eng., Stanford University, Stanford, CA.
C.J. Rutland J.H. Ferziger (1991) Combustion and Flame 84 343–360 Occurrence Handle10.1016/0010-2180(91)90011-Y
Najm H.N. (1996). in Transport Phenomena in Combustion. Chan, S. (ed.), Vol. 2, Taylor and Francis, Wash. DC, pp. 921–932
H.N. Najm P.S. Wyckoff (1997) Combustion and Flame 110 IssueID1–2 92–112
A.S. Almgren J.B. Bell P. Colella L.H. Howell M. Welcome (1998) J. Comput. Phys. 142 1–46 Occurrence Handle1618033 Occurrence Handle10.1006/jcph.1998.5890 Occurrence Handle0933.76055
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.
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.
G.E. Karniadakis M. Israeli S.A. Orzag (1991) J. Comput. Phys. 97 414 Occurrence Handle1137607 Occurrence Handle10.1016/0021-9991(91)90007-8 Occurrence Handle0738.76050
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).
D.A. Anderson J.C. Tannehill R.H. Pletcher (1984) Computational Fluid Mechanics and Heat Transfer Hemisphere Pub. Co. New York Occurrence Handle0569.76001
H.N. Najm P.S. Wyckoff O.M. Knio (1998) J. Comp. Phys 143 IssueID2 381–402 Occurrence Handle1631172 Occurrence Handle0936.76064
H.N. Najm O.M. Knio P.H. Paul P.S. Wyckoff (1998) Comb. Sci. Tech 140 IssueID1–6 369–403
O.M. Knio H.N. Najm P.S. Wyckoff (1999) J. Comp. Phys 154 428–467 Occurrence Handle1712580 Occurrence Handle0958.76061
Hundsdorfer, W. H. (1996). Report NM-N9603, CWI, Amsterdam, http://info4u.cwi.nl
Spee, E. J. (1995). in Air Pollution III H. P. et al(ed.), Vol. 1, Comput. Mech. Publ., Southampton-Boston, pp. 319–326.
J.G. Verwer J.G. Blom M. Loon ParticleVan E.J. Spee (1995) Atmos. Eviron. 30 49–58
W. Hundsdorfer J.G. Verwer (1995) App. Num. Math 18 191–199 Occurrence Handle1357916 Occurrence Handle0833.65099
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
J.G. Verwer E.J. Spee J.G. Blom W.H. Hundsdorfer (1999) SIAM. J. Sci. Comput. 20 1456–1480 Occurrence Handle1691050 Occurrence Handle0928.65116
E.J. Spee J.G. Verwer P.M. Zeeuw Particlede J.G. Blom W. Hundsdorfer (1998) Math. Comp. Simulation 48 177–204
L.A. Khan P.L.-F. Liu (1995) Comput. Methods Appl. Mech. Engrg. 127 181–201 Occurrence Handle1365374 Occurrence Handle10.1016/0045-7825(95)00839-5 Occurrence Handle0862.76060
G. Strang (1968) SIAM. J. Numer. Anal 5 IssueID3 506–517 Occurrence Handle0184.38503 Occurrence Handle235754 Occurrence Handle10.1137/0705041
S.Z. Burstein A.A. Mirin (1970) J. Comp. Phys 5 547–571 Occurrence Handle282545 Occurrence Handle0223.65053
H. Yoshida (1990) Physics Letters A 150 IssueID5–7 262–268 Occurrence Handle1078768
Q. Sheng (1989) IMA. J. Numer Anal. 9 199–212 Occurrence Handle0676.65116 Occurrence Handle1000457
J.P. Wright (1998) J. Comp. Phys 140 421–431 Occurrence Handle0920.65057
M.S. Day J.B. Bell (2000) Combust Theory Modelling 4 535–556 Occurrence Handle10.1088/1364-7830/4/4/309 Occurrence Handle0970.76065
Kee R.J., Rupley F.M., Miller J.A. (1993). Sandia Report SAND89-8009B, Sandia National Labs., Livermore, CA
J.G. Verwer (1996) App. Num. Math. 22 359–379 Occurrence Handle0868.65064 Occurrence Handle1424308
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.
Paul, Phillip H. (1997). Sandia Report SAND98-8203, Sandia National Laboratories, Albuquerque, New Mexico
Paul, P., and Warnatz, J. (1998). Twenty-Seventh Symposium (International) on Combustion:TheCombustion Institute, pp. 495–504.
P.J. Vander Houwen B.P. Sommeijer (1980) ZAMM 60 479–485 Occurrence Handle614285
J.G. Verwer (1982) ZAMM 62 561–563 Occurrence Handle0531.65040 Occurrence Handle682569
J.G. Verwer W.H. Hundsdorfer B.P. Sommeijer (1990) Numer Math. 57 157–178 Occurrence Handle1048310 Occurrence Handle10.1007/BF01386405 Occurrence Handle0697.65072
B.P. Sommeijer L.F. Shampine J.G. Verwer (1997) J. Comput. Appl. Math. 88 315–326 Occurrence Handle1613246
P.J. Vander Houwen (1972) Numer Math. 20 149–164 Occurrence Handle317547
P.J. Vander Houwen (1977) Construction of integration formulas for initial value problems North-Holland Amsterdam-New York
J.G. Verwer (1977) J. Comp. App. Math. 3 IssueID3 155–166 Occurrence Handle0373.65030 Occurrence Handle488771 Occurrence Handle10.1016/S0377-0427(77)80002-2
J.G. Verwer (1980) ACM Trans on Math Software 6 IssueID2 188–205 Occurrence Handle0431.65069
Medovikov, A. A. (1996) In Numerical 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.).
V.I. Lebedev (1998) Russ. J. Numer. Ansl. Math. Modelling 13 IssueID2 107–116 Occurrence Handle0914.65087
A.A. Medovikov (1998) BIT 38 IssueID2 372–390 Occurrence Handle0909.65060 Occurrence Handle1638136
M.I. Golushko E.A. Novikov (1999) Russ. J. Numer. Anal. Math. Modelling 14 IssueID1 71–85 Occurrence Handle1678999 Occurrence Handle0923.65046
A. Abdulle (2000) BIT 40 IssueID1 177–182 Occurrence Handle0956.65068 Occurrence Handle1759041 Occurrence Handle10.1023/A:1022378621048
Bakker M. (1971). Technical Note TN 62, Mathematical Center, Amsterdam, (in Dutch)
P.J. Vander Houwen (1994) CWI Report NM-R9420 CWI Amsterdam
Guillou, A., and Lago, B. (1961).Recherche de formules a grand rayon de stabilitéIer Congr. Assoc. Fran. Calcul, AFCAL, Grenoble, Sept. 1960, pp. 43–56
H. Schlichting (1979) Boundary-Layer Theory EditionNumber7 McGraw-Hill New York Occurrence Handle0434.76027
F.A. Williams (1985) Combustion Theory EditionNumber2 Addison-Wesley New York
Najm, H. N., Knio, O. M., and Paul, P. H. (2003). Sandia Report SAND2003-8412, Sandia National Laboratories.
P.N. Brown G.D. Byrne A.C. Hindmarsh (1989) SIAM. J. Sci. Stat. Comput. 10 1038–1051 Occurrence Handle1009555 Occurrence Handle10.1137/0910062 Occurrence Handle0677.65075
S. Mahalingam B.J. Cantwell J.H. Ferziger (1990) Phys Fluids A 2 720–728 Occurrence Handle10.1063/1.857725
B. Sportisse (2000) J. Comp. Phys. 161 140–168 Occurrence Handle0953.65062 Occurrence Handle1762076
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Najm, H.N., Knio, O.M. Modeling Low Mach Number Reacting Flow with Detailed Chemistry and Transport. J Sci Comput 25, 263–287 (2005). https://doi.org/10.1007/s10915-003-4643-x
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DOI: https://doi.org/10.1007/s10915-003-4643-x