Acta Mechanica Sinica

, Volume 33, Issue 3, pp 486–499 | Cite as

High-order discontinuous Galerkin method for applications to multicomponent and chemically reacting flows

Review Paper

Abstract

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.

Keywords

Discontinuous Galerkin method High-order schemes Reacting flows Multicomponent flows 

References

  1. 1.
    Zeldovich, Y.A., Raizer, Y.P.: Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Dover, Mineola (2002)Google Scholar
  2. 2.
    Liñán, A., Williams, F.A.: Fundamental Aspects of Combustion. Oxford University Press, Oxford (1993)Google Scholar
  3. 3.
    Hinze, J.O.: Turbulence, 2nd edn. McGraw-Hill, New York (1975)Google Scholar
  4. 4.
    Lu, T., Law, C.K.: Toward accommodating realistic fuel chemistry in large-scale computations. Prog. Energy Combust. Sci. 35, 192–215 (2009)CrossRefGoogle Scholar
  5. 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)Google Scholar
  6. 6.
    Abgrall, R., Karni, S.: Computations of compressible multifluids. J. Comput. Phys. 169, 594–623 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lee, J.H.S.: The Detonation Phenomenon. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  8. 8.
    Shepherd, J.E.: Detonation in gases. Proc. Combust. Inst. 32, 83–98 (2009)CrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 11.
    Gamezo, V.N., Desbordes, D., Oran, E.S.: Formation and evolution of two-dimensional cellular detonations. Combust. Flame 116, 154–165 (1999)CrossRefGoogle Scholar
  12. 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)CrossRefMATHGoogle Scholar
  13. 13.
    Lv, Y., Ihme, M.: Discontinuous Galerkin method for multicomponent chemically reacting flows and combustion. J. Comput. Phys. 270, 105–137 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Klöckner, A., Warburton, T., Bridge, J., et al.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228, 7863–7882 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Report LA-UR-73-479, America (1973)Google Scholar
  16. 16.
    Johnson, C., Pitkäranta, J.: An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comput. 46, 1–26 (1986)MathSciNetCrossRefMATHGoogle Scholar
  17. 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)MathSciNetCrossRefMATHGoogle Scholar
  18. 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)MathSciNetMATHGoogle Scholar
  19. 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)MathSciNetCrossRefMATHGoogle Scholar
  20. 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)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 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)MathSciNetCrossRefMATHGoogle Scholar
  23. 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)MathSciNetCrossRefMATHGoogle Scholar
  24. 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)MathSciNetCrossRefMATHGoogle Scholar
  25. 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)Google Scholar
  26. 26.
    Peraire, J., Persson, P.-O.: The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J. Sci. Comput. 30, 1806–1824 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 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)MathSciNetCrossRefMATHGoogle Scholar
  28. 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)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2013)Google Scholar
  30. 30.
    Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994)CrossRefMATHGoogle Scholar
  32. 32.
    Rusanov, V.V.: Calculation of intersection of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961)Google Scholar
  33. 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)Google Scholar
  34. 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)MathSciNetCrossRefGoogle Scholar
  35. 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)MathSciNetCrossRefMATHGoogle Scholar
  36. 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)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zhang, X., Shu, C.-W.: A minimum entropy principle of high order schemes for gas dynamics equations. Numer. Math. 121, 545–563 (2012)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lv, Y., Ihme, M.: Entropy-bounded discontinuous Galerkin scheme for Euler equations. J. Comput. Phys. 295, 715–739 (2015)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. AIAA 2006-112 (2006)Google Scholar
  40. 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)MathSciNetCrossRefMATHGoogle Scholar
  41. 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)MathSciNetCrossRefMATHGoogle Scholar
  42. 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)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Tadmor, E.: A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2, 211–219 (1986)MathSciNetCrossRefMATHGoogle Scholar
  44. 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)MathSciNetCrossRefMATHGoogle Scholar
  45. 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)Google Scholar
  46. 46.
    Krivodonova, L.: Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys. 226, 879–896 (2007)MathSciNetCrossRefMATHGoogle Scholar
  47. 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)MathSciNetCrossRefMATHGoogle Scholar
  48. 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)MathSciNetCrossRefMATHGoogle Scholar
  49. 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)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Nguyen, N.C., Persson, P.-O., Peraire, J.: RANS solutions using high order discontinuous Galerkin methods. AIAA 2007-914 (2007)Google Scholar
  51. 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)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Haas, J.F., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech. 181, 41–76 (1987)CrossRefGoogle Scholar
  53. 53.
    Quirk, J.J., Karni, S.: On the dynamics of a shock–bubble interaction. J. Fluid Mech. 381, 129–163 (1996)Google Scholar
  54. 54.
    Johnsen, E., Colonius, T.: Implementation of WENO schemes in compressible multicomponent flows. J. Comput. Phys. 219, 715–732 (2006)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Ranjan, D., Oakley, J., Bonazza, R.: Shock–bubble interactions. Annu. Rev. Fluid Mech. 43, 117–140 (2011)MathSciNetCrossRefMATHGoogle Scholar
  56. 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)Google Scholar
  57. 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)Google Scholar
  58. 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)CrossRefGoogle Scholar
  59. 59.
    Gamezo, V.N., Desbords, D., Oran, E.S.: Two-dimensional reactive flow dynamics in cellular detonation. Shock Waves 9, 11–17 (1999)CrossRefGoogle Scholar
  60. 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)CrossRefGoogle Scholar
  61. 61.
    Burke, M.P., Chaos, M., Ju, Y., et al.: Comprehensive H\(_2\)/O\(_2\) kinetic model for high-pressure combustion. Int. J. Chem. Kinet. 44, 444–474 (2012)CrossRefGoogle Scholar
  62. 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)CrossRefGoogle Scholar
  63. 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)Google Scholar
  64. 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)CrossRefGoogle Scholar
  65. 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)MathSciNetCrossRefGoogle Scholar
  66. 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)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringStanford UniversityStanfordUSA

Personalised recommendations