High-order discontinuous Galerkin method for applications to multicomponent and chemically reacting flows. (English) Zbl 1372.76068
Summary: This article focuses on the development of a discontinuous Galerkin (DG) method for simulations of multicomponent and chemically reacting flows. Compared to aerodynamic flow applications, in which DG methods have been successfully employed, DG simulations of chemically reacting flows introduce challenges that arise from flow unsteadiness, combustion, heat release, compressibility effects, shocks, and variations in thermodynamic properties. To address these challenges, algorithms are developed, including an entropy-bounded DG method, an entropy-residual shock indicator, and a new formulation of artificial viscosity. The performance and capabilities of the resulting DG method are demonstrated in several relevant applications, including shock/bubble interaction, turbulent combustion, and detonation. It is concluded that the developed DG method shows promising performance in application to multicomponent reacting flows. The paper concludes with a discussion of further research needs to enable the application of DG methods to more complex reacting flows.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 92-08 | Computational methods for problems pertaining to biology |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 92E20 | Classical flows, reactions, etc. in chemistry |
| 76V05 | Reaction effects in flows |
| 35Q35 | PDEs in connection with fluid mechanics |
| 80A32 | Chemically reacting flows |
| 74F25 | Chemical and reactive effects in solid mechanics |
Software:
HE-E1GODFReferences:
| [1] | Zeldovich, Y.A., Raizer, Y.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover, Mineola (2002) |
| [2] | Liñán, A., Williams, F.A.: Fundamental Aspects of Combustion. Oxford University Press, Oxford (1993) |
| [3] | Hinze, J.O.: Turbulence, 2nd edn. McGraw-Hill, New York (1975) |
| [4] | Lu, T., Law, C.K.: Toward accommodating realistic fuel chemistry in large-scale computations. Prog. Energy Combust. Sci. 35, 192-215 (2009) · doi:10.1016/j.pecs.2008.10.002 |
| [5] | Ma, P.C., Lv, Y., Ihme, M.: An entropy-stable hybrid scheme for simulations of transcritical real-fluid flows. J. Comput. Phys. 340, 330-357 (2017) · Zbl 1376.76033 |
| [6] | Abgrall, R., Karni, S.: Computations of compressible multifluids. J. Comput. Phys. 169, 594-623 (2001) · Zbl 1033.76029 · doi:10.1006/jcph.2000.6685 |
| [7] | Lee, J.H.S.: The Detonation Phenomenon. Cambridge University Press, Cambridge (2008) · doi:10.1017/CBO9780511754708 |
| [8] | Shepherd, J.E.: Detonation in gases. Proc. Combust. Inst. 32, 83-98 (2009) · doi:10.1016/j.proci.2008.08.006 |
| [9] | Pintgen, F., Eckett, C.A., Austin, J.M., et al.: Direct observations of reaction zone structure in propagating detonations. Combust. Flame 133, 211-229 (2003) · doi:10.1016/S0010-2180(02)00458-3 |
| [10] | Maley, L., Bhattacharjee, R., Lau-Chapdelaine, S.M., et al.: Influence of hydrodynamic instabilities on the propagation mechanism of fast flames. Proc. Combust. Inst. 35, 2117-2126 (2015) · doi:10.1016/j.proci.2014.06.134 |
| [11] | Gamezo, V.N., Desbordes, D., Oran, E.S.: Formation and evolution of two-dimensional cellular detonations. Combust. Flame 116, 154-165 (1999) · doi:10.1016/S0010-2180(98)00031-5 |
| [12] | Hu, F.Q., Hussaini, M.Y., Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151, 921-946 (1999) · Zbl 0933.65113 · doi:10.1006/jcph.1999.6227 |
| [13] | Lv, Y., Ihme, M.: Discontinuous Galerkin method for multicomponent chemically reacting flows and combustion. J. Comput. Phys. 270, 105-137 (2014) · Zbl 1349.76239 · doi:10.1016/j.jcp.2014.03.029 |
| [14] | Klöckner, A., Warburton, T., Bridge, J., et al.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228, 7863-7882 (2009) · Zbl 1175.65111 · doi:10.1016/j.jcp.2009.06.041 |
| [15] | Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Report LA-UR-73-479, America (1973) · Zbl 1125.65091 |
| [16] | Johnson, C., Pitkäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46, 1-26 (1986) · Zbl 0618.65105 · doi:10.1090/S0025-5718-1986-0815828-4 |
| [17] | Peterson, T.E.: A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM J. Numer. Anal. 28, 133-140 (1991) · Zbl 0729.65085 · doi:10.1137/0728006 |
| [18] | Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. J. Sci. Comp. 52, 411-435 (1989) · Zbl 0662.65083 |
| [19] | Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J. Comput. Phys. 84, 90-113 (1989) · Zbl 0677.65093 · doi:10.1016/0021-9991(89)90183-6 |
| [20] | Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199-224 (1998) · Zbl 0920.65059 · doi:10.1006/jcph.1998.5892 |
| [21] | Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173-261 (2001) · Zbl 1065.76135 · doi:10.1023/A:1012873910884 |
| [22] | Arnold, D.N., Brezzi, F., Cockburn, B., et al.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749-1779 (2002) · Zbl 1008.65080 · doi:10.1137/S0036142901384162 |
| [23] | Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440-2463 (1998) · Zbl 0927.65118 · doi:10.1137/S0036142997316712 |
| [24] | Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267-279 (1997) · Zbl 0871.76040 · doi:10.1006/jcph.1996.5572 |
| [25] | Bassi, F., Rebay, S.: GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations. In: Cockburn, B., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, Springer, Berlin, 197-208 (2000) · Zbl 0989.76040 |
| [26] | Peraire, J., Persson, P.-O.: The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput. 30, 1806-1824 (2008) · Zbl 1167.65436 · doi:10.1137/070685518 |
| [27] | Hartmann, R., Houston, P.: An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations. J. Comput. Phys. 227, 9670-9685 (2008) · Zbl 1359.76220 · doi:10.1016/j.jcp.2008.07.015 |
| [28] | Gassner, G., Lörcher, F., Munz, C.-D.: A discontinuous Galerkin scheme based on a space-time expansion II. Viscous flow equations in multi dimensions. J. Sci. Comput. 34, 260-286 (2008) · Zbl 1218.76027 · doi:10.1007/s10915-007-9169-1 |
| [29] | Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2013) |
| [30] | Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357-372 (1981) · Zbl 0474.65066 · doi:10.1016/0021-9991(81)90128-5 |
| [31] | Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25-34 (1994) · Zbl 0811.76053 · doi:10.1007/BF01414629 |
| [32] | Rusanov, V.V.: Calculation of intersection of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267-279 (1961) |
| [33] | Ma, P.C., Lv, Y., Ihme, M.: Discontinuous Galerkin scheme for turbulent flow simulations. Annual Research Briefs, Center for Turbulence Research, 225-236 (2015) · Zbl 0927.65118 |
| [34] | Wang, Z.J., Fidkowski, K., Abgrall, R., et al.: High-order CFD methods: current status and perspective. Int. J. Numer. Methods Fluids 72, 811-845 (2013) · Zbl 1455.76007 · doi:10.1002/fld.3767 |
| [35] | Zhang, X., Shu, C.-W.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918-8934 (2010) · Zbl 1282.76128 · doi:10.1016/j.jcp.2010.08.016 |
| [36] | Wang, C., Zhang, X., Shu, C.-W., et al.: Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations. J. Comput. Phys. 231, 653-665 (2012) · Zbl 1243.80011 · doi:10.1016/j.jcp.2011.10.002 |
| [37] | Zhang, X., Shu, C.-W.: A minimum entropy principle of high order schemes for gas dynamics equations. Numer. Math. 121, 545-563 (2012) · Zbl 1426.76444 · doi:10.1007/s00211-011-0443-7 |
| [38] | Lv, Y., Ihme, M.: Entropy-bounded discontinuous Galerkin scheme for Euler equations. J. Comput. Phys. 295, 715-739 (2015) · Zbl 1349.65465 · doi:10.1016/j.jcp.2015.04.026 |
| [39] | Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. AIAA 2006-112 (2006) · Zbl 0662.65083 |
| [40] | Krivodonova, L., Xin, J., Remacle, J.-F., et al.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48, 323-338 (2004) · Zbl 1038.65096 · doi:10.1016/j.apnum.2003.11.002 |
| [41] | Vuik, M.J., Ryan, J.K.: Multiwavelet troubled-cell indicator for discontinuity detection of discontinuous Galerkin schemes. J. Comput. Phys. 270, 138-160 (2014) · Zbl 1349.65490 · doi:10.1016/j.jcp.2014.03.047 |
| [42] | Lv, Y., See, Y.C., Ihme, M.: An entropy-residual shock detector for solving conservation laws using high-order discontinuous Galerkin methods. J. Comput. Phys. 322, 448-472 (2016) · Zbl 1351.76068 · doi:10.1016/j.jcp.2016.06.052 |
| [43] | Tadmor, E.: A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2, 211-219 (1986) · Zbl 0625.76084 · doi:10.1016/0168-9274(86)90029-2 |
| [44] | Guermond, J.-L., Pasquetti, R.: Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. C. R. Acad. Sci. Paris, Ser. I 346, 801-806 (2008) · Zbl 1145.65079 · doi:10.1016/j.crma.2008.05.013 |
| [45] | Lax, P.D.: Shock waves and entropy. In: Zarantonello E.H. (ed.) Contributions to Nonlinear Functional Analysis. Academic Press, New York and London, 603-634 (1971) · Zbl 0268.35014 |
| [46] | Krivodonova, L.: Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys. 226, 879-896 (2007) · Zbl 1125.65091 · doi:10.1016/j.jcp.2007.05.011 |
| [47] | Luo, H., Baum, J.D., Löhner, R.: A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J. Comput. Phys. 225, 686-713 (2007) · Zbl 1122.65089 · doi:10.1016/j.jcp.2006.12.017 |
| [48] | Zhu, J., Zhong, X., Shu, C.-W., et al.: Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes. J. Comput. Phys. 248, 200-220 (2013) · Zbl 1349.65501 · doi:10.1016/j.jcp.2013.04.012 |
| [49] | Hartmann, R.: Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 51, 1131-1156 (2006) · Zbl 1106.76041 · doi:10.1002/fld.1134 |
| [50] | Nguyen, N.C., Persson, P.-O., Peraire, J.: RANS solutions using high order discontinuous Galerkin methods. AIAA 2007-914 (2007) · Zbl 0933.65113 |
| [51] | Barter, G.E., Darmofal, D.L.: Shock capturing with PDE-based artificial viscosity for DGFEM: part I formulation. J. Comput. Phys. 229, 1810-1827 (2010) · Zbl 1329.76153 · doi:10.1016/j.jcp.2009.11.010 |
| [52] | Haas, J.F., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech. 181, 41-76 (1987) · doi:10.1017/S0022112087002003 |
| [53] | Quirk, J.J., Karni, S.: On the dynamics of a shock-bubble interaction. J. Fluid Mech. 381, 129-163 (1996) · Zbl 0877.76046 |
| [54] | Johnsen, E., Colonius, T.: Implementation of WENO schemes in compressible multicomponent flows. J. Comput. Phys. 219, 715-732 (2006) · Zbl 1189.76351 · doi:10.1016/j.jcp.2006.04.018 |
| [55] | Ranjan, D., Oakley, J., Bonazza, R.: Shock-bubble interactions. Annu. Rev. Fluid Mech. 43, 117-140 (2011) · Zbl 1299.76125 · doi:10.1146/annurev-fluid-122109-160744 |
| [56] | Sjunnesson, A., Nelsson, C., Max, E.: LDA measurements of velocities and turbulence in a bluff body stabilized flame. Laser Anemometry 3, 83-90 (1991) |
| [57] | Sjunnesson, A., Olovsson, S., Sjoblom, B.: Validation rig—a tool for flame studies. In: 10th International Symposium on Air Breathing Engines. Nottingham, England, 385-393 (1991) |
| [58] | Ghani, A., Poinsot, T., Gicquel, L., et al.: LES of longitudinal and transverse self-excited combustion instabilities in a bluff-body stabilized turbulent premixed flame. Combust. Flame 162, 4075-4083 (2015) · doi:10.1016/j.combustflame.2015.08.024 |
| [59] | Gamezo, V.N., Desbords, D., Oran, E.S.: Two-dimensional reactive flow dynamics in cellular detonation. Shock Waves 9, 11-17 (1999) · doi:10.1007/s001930050134 |
| [60] | Ohyagi, S., Obara, T., Hoshi, S., et al.: Diffraction and re-initiation of detonations behind a backward-facing step. Shock Waves 12, 221-226 (2002) · doi:10.1007/s00193-002-0156-z |
| [61] | Burke, M.P., Chaos, M., Ju, Y., et al.: Comprehensive \[H_22/O_22\] kinetic model for high-pressure combustion. Int. J. Chem. Kinet. 44, 444-474 (2012) · doi:10.1002/kin.20603 |
| [62] | Lv, Y., Ihme, M.: Computational analysis of re-ignition and re-initiation mechanisms of quenched detonation waves behind a backward facing step. Proc. Combust. Inst. 35, 1963-1972 (2015) · doi:10.1016/j.proci.2014.07.041 |
| [63] | de Wiart, Carton C., Hillewaert, K., et al.: Implicit LES of free and wall-bounded turbulent flows based on the discontinuous Galerkin/symmetric interior penalty method. Int. J. Numer. Methods Fluids 78, 335-354 (2015) |
| [64] | Kanner, S., Persson, P.-O.: Validation of a high-order large-eddy simulation solver using a vertical-axis wind turbine. AIAA J. 54, 101-112 (2015) · doi:10.2514/1.J054138 |
| [65] | Beck, A.D., Bolemann, T., Flad, D., et al.: High-order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations. Int. J. Numer. Methods Fluids 76, 522-548 (2014) · doi:10.1002/fld.3943 |
| [66] | Gassner, G.J., Beck, A.D.: On the accuracy of high-order discretizations for underresolved turbulence simulations. Theor. Comput. Fluid Dyn. 27, 221-237 (2013) · doi:10.1007/s00162-011-0253-7 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
