Skip to main content
SpringerLink
Log in

An efficient unstructured WENO method for supersonic reactive flows

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

An efficient high-order numerical method for supersonic reactive flows is proposed in this article. The reactive source term and convection term are solved separately by splitting scheme. In the reaction step, an adaptive time-step method is presented, which can improve the efficiency greatly. In the convection step, a third-order accurate weighted essentially non-oscillatory (WENO) method is adopted to reconstruct the solution in the unstructured grids. Numerical results show that our new method can capture the correct propagation speed of the detonation wave exactly even in coarse grids, while high order accuracy can be achieved in the smooth region. In addition, the proposed adaptive splitting method can reduce the computational cost greatly compared with the traditional splitting method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Roy, G.D.: Pulse detonation propulsion: challenges, current status, and future perspective. Prog. Energy Combust. 30, 545–672 (2004)

    Article  Google Scholar 

  2. Geßner, T.: Dynamic mesh adaption for supersonic combustion waves modeled with detailed reaction mechanisms. [Ph.D. Thesis], Technical University Dresden, German (2001)

  3. Fedkiw, R.P., Merriman, B., Osher, S.: High accuracy numerical methods for thermally perfect gas flows with chemistry. J. Comput. Phys. 132, 175–190 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Togashi, F., Löhner, R., Tsuboi, N.: Numerical simulation of H\(_2\)/air detonation using unstructured mesh. Shock Waves 192, 151–162 (2009)

    Article  MATH  Google Scholar 

  5. Ju, Y.: Recent progress and challenges in fundamental combustion research. Adv. Mech. 44, 1142–1157 (2014)

    Google Scholar 

  6. Chapman, D.L.V.I.: On the rate of explosion in gases. Lond. Edinb. Dublin Philos. Mag. J. Sci. 47, 90–104 (1899)

    Article  MATH  Google Scholar 

  7. Jouguet, E.: On the propagation of chemical reactions in gases. J. Math. Pureset Appl. 1, 347–425 (1905)

    MATH  Google Scholar 

  8. Zeldovich, Y.B.: On the theory of the propagation of detonation in gaseous systems. Tech. Memos. Nat. Adv. Comm. Aeronaut. 1950, 1261 (1950)

    MathSciNet  Google Scholar 

  9. Von Neuman, J.: Theory of Detonation Waves. Institute for Advanced Study, Princeton (1942)

    Google Scholar 

  10. Döring, W.: On detonation processes in gases. Ann. Phys. 43, 421–436 (1943)

    Article  Google Scholar 

  11. Bao, W., Jin, S.: The random projection method for hyperbolic conservation laws with stiff reaction terms. J. Comput. Phys. 163, 216–248 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bao, W., Jin, S.: The random projection method for stiff detonation capturing. SIAM J. Sci. Comput. 23, 1000–1026 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bao, W., Jin, S.: The random projection method for stiff multispecies detonation capturing. J. Comput. Phys. 178, 37–57 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Helzel, C., Leveque, R.J., Warnecke, G.: A modified fractional step method for the accurate approximation of detonation waves. SIAM J. Sci. Comput. 22, 1489–1510 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tosatto, L., Vigevano, L.: Numerical solution of under-resolved detonations. J. Comput. Phys. 227, 2317–2343 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dumbser, M., Enaux, C., Toro, E.F.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227, 3971–4001 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, W., Shu, C.W., Yee, H.C., et al.: High order finite difference methods with subcell resolution for advection equations with stiff source terms. J. Comput. Phys. 231, 190–214 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, B., Liu, H., Chen, F.: The equilibrium state method for hyperbolic conservation laws with stiff reaction terms. J. Comput. Phys. 263, 151–176 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Taki, S., Fujiwara, T.: Numerical analysis of two-dimensional non-steady detonations. AIAA J. 16, 73–77 (1978)

    Article  Google Scholar 

  20. Lefebvre, M.H., Oran, E.S., Kailasanath, K.: Computations of Detonation Structure: The Influence of Model Input Parameters. Naval Research Lab, Washington, DC (1992)

    Google Scholar 

  21. Sichel, M., Tonello, N.A., Oran, E.S., et al.: A two-step kinetics model for numerical simulation of explosions and detonations in H\(_2\)–O\(_2\) mixtures. R. Soc. 458, 49–82 (2002)

    Article  MATH  Google Scholar 

  22. Colella, P., Majda, A., Roytburd, V.: Theoretical and numerical structure for reacting shock waves. SIAM J. Sci. Comput. 7, 1059–1080 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. LeVeque, R.J., Yee, H.C.: A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 86, 187–210 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bussing, T.R.A., Murman, E.M.: Finite-volume method for the calculation of compressible chemically reacting flows. AIAA J. 26, 1070–1078 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gnoffo, P.A.: An upwind-biased, point-implicit relaxation algorithm for viscous, compressible perfect-gas flows. NASA Tech. Rep. 90, 17–42 (1990)

    Google Scholar 

  26. Zhong X.: New high-order semi-implicit Runge–Kutta schemes for computing transient nonequilibrium hypersonic flow. In: 30th Thermophysics Conference, New York, August 4–14 (1995)

  27. Oran, E.S., Weber, J.W., Stefaniw, E.I., et al.: A numerical study of a two-dimensional H\(_2\)–O\(_2\)–Ar detonation using a detailed chemical reaction model. Combust. Flame 113, 147–163 (1998)

    Article  Google Scholar 

  28. Xiao, X., Edwards, J.R., Hassan, H.A., et al.: Inflow boundary conditions for hybrid large eddy/Reynolds averaged Navier–Stokes simulations. AIAA J. 41, 1481–1489 (2003)

    Article  Google Scholar 

  29. Shen, Y., Zha, G.: Application of low diffusion E-CUSP scheme with high order WENO scheme for chemical reacting flows. In: The 40th Fluid Dynamics Conference and Exhibit, Chicago, June 28–30 (2010)

  30. Taylor, B.D., Kessler, D.A., Gamezo, V.N., et al.: Numerical simulations of hydrogen detonations with detailed chemical kinetics. Proc. Combust. Inst. 34, 2009–2016 (2013)

    Article  Google Scholar 

  31. Billet, G., Ryan, J., Borrel, M.: A Runge Kutta discontinuous Galerkin approach to solve reactive flows on conforming hybrid grids: the parabolic and source operators. Comput. Fluids 95, 98–115 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lv, Y., Ihme, M.: Discontinuous Galerkin method for multicomponent chemically reacting flows and combustion. J. Comput. Phys. 270, 105–137 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Dumbser, M., Castro, M., Parés, C., et al.: ADER schemes on unstructured meshes for non-conservative hyperbolic systems: applications to geophysical flows. Comput. Fluids 38, 1731–1748 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dumbser, M., Zanotti, O.: Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations. J. Comput. Phys. 228, 6991–7006 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Titarev, V.A., Tsoutsanis, P., Drikakis, D.: WENO schemes for mixed-element unstructured meshes. Commun. Comput. Phys. 8, 585–609 (2010)

    MathSciNet  MATH  Google Scholar 

  36. Zhang, Y.T., Shu, C.W.: Third order WENO scheme on three dimensional tetrahedral meshes. Commun. Comput. Phys. 5, 836–848 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Zhang, Y., Shu, C.: High-order WENO Sschemes for Hamilton–Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 24, 1005–1030 (2003)

    Article  MATH  Google Scholar 

  38. Togashi, F., Löhner, R., Tsuboi, N.: Numerical simulation of H\(_2\)/air detonation using unstructured mesh. Shock Waves 19, 151–162 (2009)

    Article  MATH  Google Scholar 

  39. Wang, Z.J.: High-order methods for the Euler and Navier–Stokes equations on unstructured grids. Prog. Aerosp. Sci. 43, 1–41 (2007)

    Article  Google Scholar 

  40. Ekaterinaris, J.A.: High-order accurate, low numerical diffusion methods for aerodynamics. Prog. Aerosp. Sci. 41, 192–300 (2005)

    Article  Google Scholar 

  41. Harten, A., Engquist, B., Osher, S., et al.: Uniformly high order accurate essentially non-oscillatory schemesIII. J. Comput. Phys. 71, 231–303 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  42. Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  43. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  44. Cockburn, B., Karniadakis, G.E., Shu, C.W.: The Development of Discontinuous Galerkin Methods. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  45. Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids basic formulation. J. Comput. Phys. 178, 210–251 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zheng, H., Zhao, W.G.: An improved unstructured WENO method for compressible multi-fluids flow. In. J. Numer. Methods Fluids 82, 113–129 (2016)

    Article  MathSciNet  Google Scholar 

  47. Stull, D., Prophet, H.: JANAF Thermo Chemical Tables, 2nd edn. NSRDS-NBS37, Washington (1971)

    Google Scholar 

  48. Kailasanath, K., Oran, E.S., Boris, J.P.: Determination of detonation cell size and the role of transverse waves in two-dimensional detonations. Combust. Flame 61, 199–209 (1985)

    Article  Google Scholar 

  49. Hu, Z., Mu, Q., Zhang, D., et al.: Numerical simulation of gaseous detonation wave propagation through bends with a detailed chemical reaction model. Chin. J. Comput. Phys. 21, 408–414 (2004)

    Google Scholar 

  50. Lv, Y., Ihme, M.: Development of discontinuous Galerkin method for detonation and supersonic combustion. In: 51st AIAA Aerospace Sciences Meeting, Grapevine, January 7–10 (2013)

  51. Deiterding, R., Bader, G.: High-Resolution Simulation of Detonations with Detailed Chemistry. In: Analysis and Numeric for Conservation Laws, Berlin Heidelberg, May 4–12 (2005)

  52. Eckett, C.A.: Numerical and Analytical Studies of the Dynamics of Gaseous Detonations [Ph.D. Thesis], California Institute of Technology, America (2000)

  53. Deiterding, R.: Parallel Adaptive Simulation of Multi-Dimensional Detonation Structures. [Ph.D. Thesis], Branderburg University of Technology Cottbus, German (2003)

  54. Li, J., Zhao, Z., Kazakov, A., et al.: An updated comprehensive kinetic model of hydrogen combustion. Int. J. Chem. Kinet. 36, 566–575 (2004)

    Article  Google Scholar 

  55. Stull, D.R., Prophet, H.: JANAF thermo chemical tables. Technical report, U. S. Department of Commerce (1971)

  56. Hu, C., Shu, C.W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  57. Dumbser, M., Käser, M., Titarev, V.A., et al.: Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J. Comput. Phys. 226, 204–243 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  58. Ivan, L.: High-order central ENO finite-volume scheme for hyperbolic conservation laws on three-dimensional cubed-sphere grids. J. Comput. Phys. 12, 157–182 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. Ivan, L., Clinton, P.T.: High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows. J. Comput. Phys. 11, 830–862 (2014)

  60. Billet, G., Ryan, J., Borrel, M.: A Runge Kutta discontinuous Galerkin approach to solve reactive flows on conforming hybrid grids: the parabolic and source operators. Comput. Fluids 1, 98–115 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  61. Billet, G., Ryan, J., Borrel, M.: A Runge Kutta discontinuous Galerkin approach to solve reactive flows on conforming hybrid grids: the parabolic and source operators. Comput. Fluids 23, 98–115 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  62. Ren, Z.: Dynamic adaptive chemistry with operator splitting schemes for reactive flow simulations. J. Comput. Phys. 263, 19–36 (2014)

    Article  MATH  Google Scholar 

  63. Cai, X., Liang, J., Lin, Z., et al.: Adaptive mesh refinement-based numerical simulation of detonation initiation in supersonic combustible mixtures using a hot jet. J. Aerosp. Eng. 28, 1–15 (2015)

    Google Scholar 

  64. Florian, E., Klaus, G., Sattelmayer, T.: Numerical simulation of the deflagration to detonation transition in inhomogeneous mixtures. J. Combust. 31, 1–15 (2014)

    Google Scholar 

  65. Zhou, R., Wu, D., Wang, J.: Progress of continuously rotating detonation engines. Chin. J. Aeronaut. 29, 15–29 (2016)

    Article  Google Scholar 

  66. Qing, Y., Zhang, X.: Discontinuous Galerkin immersed finite element methods for parabolic interface problems. J. Comput. Appl. Math. 1, 127–139 (2016)

    MathSciNet  MATH  Google Scholar 

  67. Assyr, A., Huber, M.E.: Discontinuous Galerkin finite element heterogeneous multiscale method for advection–diffusion problems with multiplescales. Numer. Math. 126, 589–633 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The project was supported by the National Natural Science Foundation of China (Grants 51476152, 11302213, and 11572336). The authors wish to thank Jun Meng, Li Liu, Jun Peng, Shengping Liu and Guangli Li for helpful discussions. Special thank goes to Prof. Yiqing Shen and Prof. Xinliang Li of the Institute of Mechanics CAS for their suggestions and help.

Author information

Authors and Affiliations

  1. China Academy of Aerospace and Aerodynamics, Beijing, 100074, China

    Wen-Geng Zhao, Xiao-Tian Shi, Jun Gao, Ning Hu, Meng Lv & Si-Cong Chen

  2. LHD, Institute of Mechanics, CAS, Beijing, 100190, China

    Hong-Wei Zheng

  3. School of Engineering Science, University of Chinese Academy of Sciences, Beijing, 100049, China

    Hong-Wei Zheng

  4. AECC Beijing Institute of Aeronautical Materials, Beijing, 100095, China

    Feng-Jun Liu

  5. Science and Technology on Scramjet Laboratory, Beijing Power Machinery Research Institute, Beijing, 100074, China

    Hong-Da Zhao

Authors
  1. Wen-Geng Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hong-Wei Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feng-Jun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiao-Tian Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jun Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ning Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Meng Lv
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Si-Cong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Hong-Da Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Wen-Geng Zhao or Xiao-Tian Shi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, WG., Zheng, HW., Liu, FJ. et al. An efficient unstructured WENO method for supersonic reactive flows. Acta Mech. Sin. 34, 623–631 (2018). https://doi.org/10.1007/s10409-018-0756-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-018-0756-1

Keywords

Search

Navigation