zbMATH — the first resource for mathematics

Semi-implicit iterative methods for low Mach number turbulent reacting flows: operator splitting versus approximate factorization. (English) Zbl 1373.76336
Summary: Two formally second-order accurate, semi-implicit, iterative methods for the solution of scalar transport-reaction equations are developed for direct numerical simulation (DNS) of low Mach number turbulent reacting flows. The first is a monolithic scheme based on a linearly implicit midpoint method utilizing an approximately factorized exact Jacobian of the transport and reaction operators. The second is an operator splitting scheme based on the Strang splitting approach. The accuracy properties of these schemes, as well as their stability, cost, and the effect of chemical mechanism size on relative performance, are assessed in two one-dimensional test configurations comprising an unsteady premixed flame and an unsteady nonpremixed ignition, which have substantially different Damköhler numbers and relative stiffness of transport to chemistry. All schemes demonstrate their formal order of accuracy in the fully-coupled convergence tests. Compared to a (non-)factorized scheme with a diagonal approximation to the chemical Jacobian, the monolithic, factorized scheme using the exact chemical Jacobian is shown to be both more stable and more economical. This is due to an improved convergence rate of the iterative procedure, and the difference between the two schemes in convergence rate grows as the time step increases. The stability properties of the Strang splitting scheme are demonstrated to outpace those of Lie splitting and monolithic schemes in simulations at high Damköhler number; however, in this regime, the monolithic scheme using the approximately factorized exact Jacobian is found to be the most economical at practical CFL numbers. The performance of the schemes is further evaluated in a simulation of a three-dimensional, spatially evolving, turbulent nonpremixed planar jet flame.
Reviewer: Reviewer (Berlin)

76V05 Reaction effects in flows
76F65 Direct numerical and large eddy simulation of turbulence
Full Text: DOI
[1] Yoo, C. S.; Sankaran, R.; Chen, J. H., Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: flame stabilization and structure, J. Fluid Mech., 640, 453, (2009) · Zbl 1183.76925
[2] Aspden, A. J.; Day, M. S.; Bell, J. B., Turbulence-flame interactions in Lean premixed hydrogen: transition to the distributed burning regime, J. Fluid Mech., 680, 287-320, (2011) · Zbl 1241.76435
[3] Chen, J. H., Petascale direct numerical simulation of turbulent combustion - fundamental insights towards predictive models, Proc. Combust. Inst., 33, 1, 99-123, (2011)
[4] Attili, A.; Bisetti, F.; Mueller, M. E.; Pitsch, H., Formation, growth, and transport of soot in a three-dimensional turbulent non-premixed jet flame, Combust. Flame, 161, 7, 1849-1865, (2014)
[5] Kolla, H.; Hawkes, E. R.; Kerstein, A. R.; Swaminathan, N.; Chen, J. H., On velocity and reactive scalar spectra in turbulent premixed flames, J. Fluid Mech., 754, 456-487, (2014) · Zbl 1329.76266
[6] Lecoustre, V. R.; Arias, P. G.; Roy, S. P.; Luo, Z.; Haworth, D. C.; Im, H. G.; Lu, T. F.; Trouvé, A., Direct numerical simulations of non-premixed ethylene-air flames: local flame extinction criterion, Combust. Flame, 161, 11, 2933-2950, (2014)
[7] Attili, A.; Bisetti, F.; Mueller, M. E.; Pitsch, H., Damköhler number effects on soot formation and growth in turbulent nonpremixed flames, Proc. Combust. Inst., 35, 2, 1215-1223, (2015)
[8] Selle, L.; Lartigue, G.; Poinsot, T.; Koch, R.; Schildmacher, K.-U.; Krebs, W.; Prade, B.; Kaufmann, P.; Veynante, D., Compressible large eddy simulation of turbulent combustion in complex geometry on unstructured meshes, Combust. Flame, 137, 4, 489-505, (2004)
[9] Wolf, P.; Staffelbach, G.; Gicquel, L. Y.; Müller, J.-D.; Poinsot, T., Acoustic and large eddy simulation studies of azimuthal modes in annular combustion chambers, Combust. Flame, 159, 11, 3398-3413, (2012)
[10] Mueller, M. E.; Pitsch, H., LES model for sooting turbulent nonpremixed flames, Combust. Flame, 159, 6, 2166-2180, (2012)
[11] Najm, H. N.; Wyckoff, P. S.; Knio, O. K., A semi-implicit numerical scheme for reacting flow. I. stiff chemistry, J. Comput. Phys., 143, 2, 381-402, (1998) · Zbl 0936.76064
[12] Bisetti, F., Integration of large chemical kinetic mechanisms via exponential methods with Krylov approximations to Jacobian matrix functions, Combust. Theory Model., 16, 5, 940, (2012) · Zbl 1262.80089
[13] Savard, B.; Xuan, Y.; Bobbitt, B.; Blanquart, G., A computationally-efficient, semi-implicit, iterative method for the time-integration of reacting flows with stiff chemistry, J. Comput. Phys., 295, 740-769, (2015) · Zbl 1349.80041
[14] Peters, N., Turbulent combustion, (2000), Cambridge University Press · Zbl 0955.76002
[15] Law, C. K., Combustion physics, (2006), Cambridge University Press
[16] Ju, Y., Lower-upper scheme for chemically reacting flow with finite rate chemistry, AIAA J., 33, 8, 1418-1425, (1995) · Zbl 0845.76058
[17] Desjardins, O.; Blanquart, G.; Balarac, G.; Pitsch, H., High order conservative finite difference scheme for variable density low Mach number turbulent flows, J. Comput. Phys., 227, 15, 7125-7159, (2008) · Zbl 1201.76139
[18] Hawkes, E. R.; Sankaran, R.; Sutherland, J. C.; Chen, J. H., Direct numerical simulation of turbulent combustion: fundamental insights towards predictive models, J. Phys. Conf. Ser., 16, 65-79, (2005)
[19] Chen, J. H.; Choudhary, A.; de Supinski, B.; DeVries, M.; Hawkes, E. R.; Klasky, S.; Liao, W. K.; Ma, K. L.; Mellor-Crummey, J.; Podhorszki, N.; Sankaran, R.; Shende, S.; Yoo, C. S., Terascale direct numerical simulations of turbulent combustion using S3D, Comput. Sci. Discov., 2, 1, (2009)
[20] Sankaran, R.; Hawkes, E. R.; Chen, J. H.; Lu, T.; Law, C. K., Structure of a spatially developing turbulent Lean methane-air bunsen flame, Proc. Combust. Inst., 31, 1, 1291-1298, (2007)
[21] Hawkes, E. R.; Chatakonda, O.; Kolla, H.; Kerstein, A. R.; Chen, J. H., A petascale direct numerical simulation study of the modelling of flame wrinkling for large-eddy simulations in intense turbulence, Combust. Flame, 159, 8, 2690-2703, (2012)
[22] Gou, X.; Sun, W.; Chen, Z.; Ju, Y., A dynamic multi-timescale method for combustion modeling with detailed and reduced chemical kinetic mechanisms, Combust. Flame, 157, 6, 1111-1121, (2010)
[23] Pepiot-Desjardins, P.; Pitsch, H., An efficient error-propagation-based reduction method for large chemical kinetic mechanisms, Combust. Flame, 154, 1-2, 67-81, (2008) · Zbl 1158.80325
[24] Lu, T.; Law, C. K.; Yoo, C. S.; Chen, J. H., Dynamic stiffness removal for direct numerical simulations, Combust. Flame, 156, 8, 1542-1551, (2009)
[25] Majda, A.; Sethian, J., The derivation and numerical solution of the equations for zero Mach number combustion, Combust. Sci. Technol., 42, 3-4, 185-205, (1985)
[26] Nicoud, F., Conservative high-order finite-difference schemes for low-Mach number flows, J. Comput. Phys., 158, 1, 71-97, (2000) · Zbl 0973.76068
[27] Moin, P.; Reynolds, W. C.; Ferziger, J. H., Large eddy simulation of incompressible turbulent channel flow, (1978), Stanford Univ., Tech. Rep. Report TF-12
[28] Kim, J.; Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 2, 308-323, (1985) · Zbl 0582.76038
[29] Chorin, A. J., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2, 1, 12-26, (1967) · Zbl 0149.44802
[30] Chorin, A. J., On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comput., 23, 106, 341, (1969) · Zbl 0184.20103
[31] Moin, P.; Kim, J., Numerical investigation of turbulent channel flow, J. Fluid Mech., 118, 1, 341, (1982) · Zbl 0491.76058
[32] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166, (1987) · Zbl 0616.76071
[33] Akselvoll, K.; Moin, P., Large-eddy simulation of turbulent confined coannular jets, J. Fluid Mech., 315, 387-411, (1996) · Zbl 0875.76444
[34] Pierce, Charles D., Progress-variable approach for large-eddy simulation of turbulent combustion, (2001), Stanford University, Ph.D. thesis
[35] Morinishi, Y.; Lund, T.; Vasilyev, O.; Moin, P., Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys., 143, 90-124, (1998) · Zbl 0932.76054
[36] Eberhardt, S.; Imlay, S., Diagonal implicit scheme for computing flows with finite rate chemistry, J. Thermophys. Heat Transf., 6, 2, 208-216, (1992)
[37] Marchuk, G. I., Some application of splitting-up methods to the solution of mathematical physics problems, Appl. Math., 10, 2, 2-6, (1968)
[38] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 3, 506-517, (1968) · Zbl 0184.38503
[39] LeVeque, R.; Yee, H., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys., 86, 1, 187-210, (1990) · Zbl 0682.76053
[40] Cohen, S. D.; Hindmarsh, A. C.; Dubois, P. F., CVODE, a stiff/nonstiff ODE solver in C, Comput. Phys., 10, 2, 138, (1996)
[41] Singer, M. A.; Pope, S. B.; Najm, H. N., Operator-splitting with ISAT to model reacting flow with detailed chemistry, Combust. Theory Model., 10, 2, 199-217, (2006) · Zbl 1121.80333
[42] Ren, Z.; Xu, C.; Lu, T.; Singer, M. A., Dynamic adaptive chemistry with operator splitting schemes for reactive flow simulations, J. Comput. Phys., 263, 19-36, (2014) · Zbl 1349.76881
[43] 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) · Zbl 0958.76061
[44] Ropp, D. L.; Shadid, J. N., Stability of operator splitting methods for systems with indefinite operators: reaction-diffusion systems, J. Comput. Phys., 203, 2, 449-466, (2005) · Zbl 1073.65088
[45] Ren, Z.; Pope, S. B., Second-order splitting schemes for a class of reactive systems, J. Comput. Phys., 227, 17, 8165-8176, (2008) · Zbl 1147.65056
[46] Descombes, S.; Duarte, M.; Dumont, T.; Laurent, F.; Louvet, V.; Massot, M., Analysis of operator splitting in the nonasymptotic regime for nonlinear reaction-diffusion equations. application to the dynamics of premixed flames, SIAM J. Numer. Anal., 52, 3, 1311-1334, (2014) · Zbl 1320.65080
[47] Yang, B.; Pope, S. B., An investigation of the accuracy of manifold methods and splitting schemes in the computational implementation of combustion chemistry, Combust. Flame, 112, 1-2, 16-32, (1998)
[48] Day, M. S.; Bell, J. B., Numerical simulation of laminar reacting flows with complex chemistry, Combust. Theory Model., 4, 4, 535-556, (2000) · Zbl 0970.76065
[49] Duarte, M.; Descombes, S.; Tenaud, C.; Candel, S.; Massot, M., Time-space adaptive numerical methods for the simulation of combustion fronts, Combust. Flame, 160, 6, 1083-1101, (2013)
[50] Sportisse, B., An analysis of operator splitting techniques in the stiff case, J. Comput. Phys., 161, 1, 140-168, (2000) · Zbl 0953.65062
[51] Williams, F. A., Combustion theory, (1985), Addison-Wesley
[52] Shunn, L.; Ham, F.; Moin, P., Verification of variable-density flow solvers using manufactured solutions, J. Comput. Phys., 231, 9, 3801-3827, (2012) · Zbl 1402.76107
[53] Wilke, C. R., A viscosity equation for gas mixtures, J. Chem. Phys., 18, 4, 517, (1950)
[54] Kee, R. J.; Rupley, F. M.; Miller, J. A.; Coltrin, M. E.; Grcar, J. F.; Meeks, E.; Moffat, H. K.; Lutz, A. E.; Dixon-Lewis, G.; Smooke, M. D.; Warnatz, J.; Evans, G. H.; Larson, R. S.; Mitchell, R. E.; Petzold, L. R.; Reynolds, W. C.; Caracotsios, M.; Stewart, W. E.; Glarborg, P.; Wang, C.; Adigun, O.; Houf, W. G.; Chou, C. P.; Miller, S. F.; Ho, P.; Young, D. J., CHEMKIN release 4.0, (2004), Reaction Design, Inc.
[55] Burali, N.; Lapointe, S.; Bobbitt, B.; Blanquart, G.; Xuan, Y., Assessment of the constant non-unity Lewis number assumption in chemically-reacting flows, Combust. Theory Model., 7830, 1-26, (July 2016)
[56] Tosatto, L.; Bennett, B. A.; Smooke, M. D., Parallelization strategies for an implicit Newton-based reactive flow solver, Combust. Theory Model., 15, 4, 455-486, (2011) · Zbl 1253.80022
[57] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 1, 202-228, (1996) · Zbl 0877.65065
[58] Leonard, B., A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comput. Methods Appl. Mech. Eng., 19, 1, 59-98, (1979) · Zbl 0423.76070
[59] Chan, T. F.; Van der Vorst, H. A., Approximate and incomplete factorizations, (Parallel Numer. Algorithms, (1997), Kluwer), 167-202 · Zbl 0865.65015
[60] Roquemore, W. M.; Litzinger, T. A.; Colket, M.; Katta, V.; McNesby, K.; Sidhu, S.; Santoro, R., Combustion science to reduce PM emissions for military platforms, (2012), Tech. Rep. WP-1577 (SERDP)
[61] Davis, S. G.; Joshi, A. V.; Wang, H.; Egolfopoulos, F., An optimized kinetic model of H_2/CO combustion, Proc. Combust. Inst., 30, 1, 1283-1292, (2005)
[62] G.P. Smith, D.M. Golden, M. Frenklach, N.W. Moriarty, B. Eiteneer, M. Goldenberg, C.T. Bowman, R.K. Hanson, S. Song, W.C. Gardiner Jr., V.V. Lissianski, Z. Qin, GRI-Mech 3.0.
[63] Blanquart, G.; Pepiot-Desjardins, P.; Pitsch, H., Chemical mechanism for high temperature combustion of engine relevant fuels with emphasis on soot precursors, Combust. Flame, 156, 3, 588-607, (2009)
[64] Narayanaswamy, K.; Blanquart, G.; Pitsch, H., A consistent chemical mechanism for oxidation of substituted aromatic species, Combust. Flame, 157, 10, 1879-1898, (2010)
[65] Narayanaswamy, K.; Pepiot, P.; Pitsch, H., A chemical mechanism for low to high temperature oxidation of n-dodecane as a component of transportation fuel surrogates, Combust. Flame, 161, 4, 866-884, (2014)
[66] Gao, Y.; Liu, Y.; Ren, Z.; Lu, T., A dynamic adaptive method for hybrid integration of stiff chemistry, Combust. Flame, 162, 2, 287-295, (2015)
[67] Pope, S. B., Turbulent flows, (2000), Cambridge University Press · Zbl 0966.76002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.