# zbMATH — the first resource for mathematics

A robust numerical model for premixed flames with high density ratios based on new pressure correction and IMEX schemes. (English) Zbl 1305.76078
Summary: We present a model for the interaction of premixed flames with obstacles in a channel flow. Although the flow equations are solved with Direct Numerical Simulation using a low Mach number approximation, the resolution used in the computation is limited ($$\sim$$1 mm) hence the inner structure of the flame and the chemical scales are not solved. The species equations are substituted with a source term in the energy equation that simulates a one-step global reaction. A level set method is applied to track the position of the flame and its zero level is used to activate the source term in the energy equation only at the flame front. An immersed boundary method reproduces the geometry of the obstacles. The main contribution of the paper is represented by the proposed numerical approach: an IMEX (implicit-explicit) Runge-Kutta scheme is used for the time integration of the energy equation and a new pressure correction algorithm is introduced for the time integration of the momentum equations. The approach presented here allows to calculate flames which produce high density ratios between burnt and unburnt regions. The model is verified by simulating first simple solutions for one- and two-dimensional flames. At last, the experiments performed by Masri and Ibrahim with square and rectangular bodies are calculated.
##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 80M25 Other numerical methods (thermodynamics) (MSC2010) 80A25 Combustion 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
Algorithm 547
Full Text:
##### References:
 [1] Masri, A.R.; Ibrahim, S.S.; Nehzat, N.; Green, A.R., Experimental study of premixed flame propagation over various solid obstructions, Experimental thermal and fluid science, 21, 109-116, (2000) [2] Ibrahim, S.S.; Hargrave, G.K.; Williams, T.C., Experimental investigation of flame/solid interactions in turbulent premixed combustion, Experimental thermal and fluid science, 24, 99-106, (2001) [3] Hargrave, G.K.; Jarvis, S.; Williams, T.C., Experimental investigation of flame/solid interactions in turbulent premixed combustion, Measurement science and technology, 13, 1036-1042, (2002) [4] Peters, N., Turbulent combustion, (2000), Cambridge University Press · Zbl 0955.76002 [5] Rehm, Ronald G.; Baum, Howard R., The equations of motion for thermally driven. buoyant flows, Journal of research of national bureau of standards, 83, (1978) · Zbl 0433.76072 [6] McMurtry, P.A.; Jou, W.-H.; Riley, J.J.; Metcalfe, R.W., Direct numerical simulation of a reactive mixing layer with chemical heat release, AIAA journal, 24, 962-970, (1986) [7] Majda, A.; Lamb, K.G., Simplified equations for low Mach number combustion with strong heat release, Dynamical issues in combustion theory, (1991), Springer-Verlag · Zbl 0751.76068 [8] Mller, B., Low Mach number asymptotics of the navier – stokes equations, Journal of engineering mathematics, 34, 97-109, (1998) [9] Pelce, P.; Calvin, P., Influences of hydrodynamics and diffusion on the stability limits of laminar premixed flames, Journal of fluid mechanics, 124, 219, (1982) [10] Piana, J.; Veynante, D.; Candel, S.; Poinsot, T., Direct numerical simulation analysis of the G-equation in premixed combustion, (), 321-330 [11] Poinsot, T.; Veynante, D., Theoretical and numerical combustion, (2001), R.T. Edwards [12] Clavin, P.; Joulin, G., Premixed flames in large scale and high intensity turbulent flow, Journal de physique lettres, 44, L1-L12, (1983) [13] Blint, R.J., The relationship of the laminar flame width to flame speed, Combustion science and technology, 49, 79-92, (1986) [14] Poinsot, T.; Echekki, T.; Mungal, M.G., A study of the laminar flame tip and implications for premixed turbulent combustion, Combustion science and technology, 81, 45-73, (1992) [15] Breugem, W.P.; Boersma, B.J., Direct numerical simulations of turbulent flow over a permeable wall using a direct and a continuum approach, Physics of fluids, 17, 2, 2005, (2004) · Zbl 1187.76067 [16] Paravento, F.; Pourquie, M.J.; Boersma, B.J., An immersed boundary method for complex geometry and heat transfer, Flow, turbulence and combustion, (2007) · Zbl 1257.76063 [17] T. Treurniet, Direct Numerical Simulation of Premixed Turbulent Combustion, Ph.D. Thesis, TU Delft, 2002. [18] E. Lindblad, D.M. Valiev, B. Mller, J. Rantakokko, P. Lotstedt, M.A. Liberman. Implicit – explicit Runge-Kutta method for combustion simulation, in: European Conference on Computational Fluid Dynamics, ECCOMAS CFD, 2006. [19] R.B. Lowrie, J.E. Morel, Discontinuous Galerkin for Hyperbolic Systems with Stiff Relaxation, Los Alamos National Laboratory, Applied Theoretical and Computational Physics Division, Report LA-UR-99-2517, May, 1999. [20] Pareschi, L., Central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms, SIAM journal on numerical analysis, 39, 4, 1395-1417, (2001) · Zbl 1020.65048 [21] Najm, H.N.; Wyckoff, P.S.; Knio, O.M., A semi-implicit numerical scheme for reacting flow, I stiff chemistry, Journal of computational physics, 143, 381-402, (1998) · Zbl 0936.76064 [22] Lambert, J.D., Numerical methods for ordinary differential systems, (1991), John Wiley & Sons · Zbl 0745.65049 [23] Hirsch, C., Numerical computation of internal and external flows, (1988), John Wiley & Sons [24] L. Pareschi, G. Russo, Stability analysis of implicit – explicit Runge-Kutta schemes for balance laws, 2007, Private communication. [25] Treurniet, T.C.; Nieuwstadt, F.T.M.; Boersma, B.J., Direct numerical simulation of homogeneous turbulence in combination with premixed combustion at low Mach number modelled by the G-equation, Journal of fluid mechanics, 565, 25-62, (2006) · Zbl 1177.76167 [26] Chorin, A.J., A numerical method for solving incompressible viscous flow problems, Journal of computational physics, 2, 12-26, (1976) · Zbl 0149.44802 [27] P. Rauwoens, K. Nerinckx, J. Vierendeels, E. Dick, B. Merci, A stable pressure-correction algorithm for low-speed turbulent combustion simulations, in: ECCOMAS CFD, 2006. [28] Jiang, Guang-Shan; Peng, Danping, Weighted ENO schemes for hamilton – jacobi equations, SIAM journal on scientific computing, 21, 6, 2126-2143, (2000) · Zbl 0957.35014 [29] Rauwoens, P.; Vierendeels, J.; Merci, B., A solution for the odd – even decoupling problem in pressure-correction algorithms for variable density flows, Journal of computational physics, 227, 79-99, (2007) · Zbl 1126.76039 [30] S.P. Van der Pijl, Computation of Bubbly Flows with a Mass-Conserving Level-set Method, Ph.D. Thesis, TU Delft, 2005. [31] Sussman, M.; Fatemi, E., An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow, SIAM journal on scientific computing, 20, 4, 1165-1191, (1999) · Zbl 0958.76070 [32] E.R.A. Coyajee, M. Herrmann, B.J. Boersma, Simulation of dispersed two-phase flow with a coupled volume-of-fluid/level-set method, in: Proceedings of the Summer Program 2004, Center of Turbulence Research, 2004. [33] F. Paravento, Numerical Simulation of Premixed Flames interacting with Obstacles, Ph.D. Thesis. TU Delft, 2009. [34] Duris, C.S., Algorithm 547 Fortran routines for discrete cubic spline interpolation and smoothing [E1], [E3], ACM transactions on mathematical software, 6, 1, 92-103, (1980) [35] Chunming Li, Chenyang Xu, Changfeng Gui, Martin D. Fox, Level set evolution without re-initialization: a new variational formulation, in: IEEE Proceedings of CVPR, 2005. · Zbl 1371.94226 [36] Patel, S.N.D.H.; Ibrahim, S.S.; Yehia, M.A., Flamelet surface density modelling of turbulent deflagrating flames in vented explosions, Journal of loss prevention in the process industries, 16, 451-455, (2003) [37] Veynante, D.; Vervisch, L., Turbulent combustion modeling, Progress in energy and combustion science, 28, 193-266, (2002) [38] Ibrahim, S.S.; Masri, A.R., The effects of obstructions on overpressure resulting from premixed flame deflagration, Journal of loss prevention in the process industries, 14, 3, 213-221, (2001) [39] Osher, S.; Shu, C.-W., High-order essentially non-oscillatory schemes for hamilton – jacobi equations, Journal of numerical analysis, 28, 907-922, (1991) · Zbl 0736.65066 [40] Peng, D.; Merriman, B.; Osher, S.; Zhaoand, H.; Kang, M., A PDE-based fast local level set method, Journal of computational physics, 155, 410-438, (1999) · Zbl 0964.76069 [41] J.D. Buckmaster, G.S.S Ludford, Lectures on Mathematical Combustion, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, 1983. · Zbl 0574.76001 [42] Di Sarli, V.; Di Benedetto, A.; Russo, G.; Jarvis, S.; Long, E.J.; Hargrave, G.K., Large eddy simulation and PIV measurements of unsteady premixed flames accelerated by obstacles, Flow, turbulence and combustion, 83, 227-250, (2009) · Zbl 1405.80008 [43] Matalon, M.; Matkowsky, B.J., Flames as gasdynamic discontinuities, Journal of fluid mechanics, 124, November, 239-259, (1982) · Zbl 0545.76133 [44] Pitsch, H., A consistent level set formulation for large-eddy simulation of premixed turbulent combustion, Combustion and flame, 143, 587-598, (2005) [45] H.G. Im, T.S. Lund, J.H. Ferziger, Study of turbulent premixed flame propagation using a laminar flamelet model. Annual Research Briefs 1995, 347-360. Center for Turbulence Research, Stanford, 1995. [46] Cambray, P.; Joulin, G., On moderately-forced premixed flames, (), 61-67
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.