Skip to main content
SpringerLink
Log in

High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we demonstrate the detailed numerical studies of three classical two dimensional detonation waves by solving the two dimensional reactive Euler equations with species with the fifth order WENO-Z finite difference scheme (Borges et al. in J. Comput. Phys. 227:3101–3211, 2008) with various grid resolutions. To reduce the computational cost and to avoid wave reflection from the artificial computational boundary of a truncated physical domain, we derive an efficient and easily implemented one dimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the two dimensional unsteady reactive Euler equation when one of the directions of domain is periodical and inflow/outflow in the other direction. The numerical comparison among characteristic, free stream, extrapolation and PML boundary conditions are conducted for the detonation wave simulations. The accuracy and efficiency of four mentioned boundary conditions are verified against the reference solutions which are obtained from using a large computational domain. Numerical schemes for solving the system of hyperbolic conversation laws with a single-mode sinusoidal perturbed ZND analytical solution as initial conditions are presented. Regular rectangular combustion cell, pockets of unburned gas and bubbles and spikes are generated and resolved in the simulations. It is shown that large amplitude of perturbation wave generates more fine scale structures within the detonation waves and the number of cell structures depends on the wave number of sinusoidal perturbation.

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

Similar content being viewed by others

References

  1. Shu, C.W.: High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542–567 (2005)

    Article  MATH  Google Scholar 

  5. Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3101–3211 (2008)

    Article  MathSciNet  Google Scholar 

  6. Jacobs, G.B., Don, W.S.: A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks. J. Comput. Phys. 228(5), 1365–1379 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jacobs, G.B., Don, W.S., Dittmann, T.: High-order resolution Eulerian-Lagrangian simulations of particle dispersion in the accelerated flow behind a moving shock. Theor. Comput. Fluid Dyn. (2010). doi:10.1007/s00162-010-0214-6

    Google Scholar 

  8. Zeldovich, Y.B.: On the theory of the propagation of detonation in gaseous system. Zh. Èksp. Teor. Fiz. 10, 542–568 (1940)

    Google Scholar 

  9. von Neumann, J.: Theory of detonation waves. In: John von Neumann, Collected Works, vol. 6. Macmillan, New York (1942)

    Google Scholar 

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

    Article  Google Scholar 

  11. Erpenbeck, J.J.: Stability of steady-state equilibrium detonations. Phys. Fluids 5, 604–614 (1962)

    Article  MathSciNet  Google Scholar 

  12. Oppenheim, A.K., Soloukhin, R.I.: Experiments in gasdynamics of explosions. Annu. Rev. Fluid Mech. 5, 31–58 (1973)

    Article  Google Scholar 

  13. Fickett, W., Davis, W.C.: Detonation. University of California Press, Berkeley (1979)

    Google Scholar 

  14. Lee, J.H.S., Moen, I.O.: The mechanism of transition from deflagration to detonation in vapor cloud explosion. Prog. Energy Combust. Sci. 6, 359–389 (1980)

    Article  Google Scholar 

  15. Sharpe, F., Sharpe, S.A.E.G.: Two-dimensional numerical simulations of idealized detonations. Proc. R. Soc. Lond. A 456, 2081–2100 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Azarenok, B.N., Tang, T.: Second-order Godunov-type scheme for reactive flow calculations on moving meshes. J. Comput. Phys. 206, 48–80 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, Z.C., Yu, S.T., He, H., Chang, S.C.: Direct calculation of two- and three-dimensional detonations by an extended CE/SE method. AIAA Paper 2001–0476 (2001)

  18. Papalexandris, M.V., Leonard, A., Dimotakis, P.E.: Unsplit algorithms for multidimensional system of hyperbolic conservation laws with source terms. Comput. Math. Appl. 44, 25–49 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. He, X., Karagozian, A.R.: Numerical simulation of pulse detonation engine phenomena. J. Sci. Comput. 19, 201–224 (2003)

    Article  MATH  Google Scholar 

  20. Wang, C., Ning, J., Lei, J.: Numerical study on propagation of explosion wave in H2-O2 mixtures. Adv. Nat. Comput. 4222, 816–819 (2006)

    Article  Google Scholar 

  21. Gao, Z., Don, W.S., Li, Z.: High order weighted essentially non-oscillation schemes for one-dimensional detonation wave simulations. J. Comput. Math. 29(6), 623–638 (2011)

    Article  MathSciNet  Google Scholar 

  22. Bourlioux, A., Majda, A.J., Roytburd, V.: Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math. 51, 303–343 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Papalexandris, M.V., Leonard, A., Dimotakis, P.E.: Unsplit schemes for hyperbolic conservation laws with source terms in one space detonation. J. Comput. Phys. 134, 31–61 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Quirk, J.J.: Godunov-type schemes applied to detonation flows. ICASE Report 93-15 (1993)

  25. Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hesthaven, J.S.: On the analysis and construction of perfectly matched layers for the linearized Euler equations. J. Comput. Phys. 142, 129–147 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hagstrom, T.: A new construction of perfectly matched layers for hyperbolic systems with applications to the linearized Euler equations. In: Cohen, G., Heikkola, E., Joly, P., Neittaanmäki, P. (eds.) Mathematical and Numerical Aspects of Wave Propagation Phenomena, pp. 125–129. Springer, Berlin (2003)

    Chapter  Google Scholar 

  28. Hu, F.Q.: A perfectly matched layer absorbing boundary condition for linearized Euler equations with a non-uniform mean-flow. J. Comput. Phys. 208, 469–492 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hu, F.Q.: On the construction of PML absorbing boundary condition for the non-linear Euler equations. AIAA Paper 2006-0798 (2006)

  30. Brachet, M.E., Meneguzzi, M., Sulem, P.L.: Small-scale dynamics of high-Reynolds-number two-dimensional turbulence. Phys. Rev. Lett. 57, 683–686 (1986)

    Article  Google Scholar 

  31. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230, 1766–1792 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dou, H.S., Tsai, H.M., Khoo, B.C., Qiu, J.: Simulations of detonation wave propagation in rectangular ducts using a three-dimensional WENO scheme. Combust. Flame 154, 644–659 (2008)

    Article  Google Scholar 

  33. Hu, F.Q.: On perfectly matched layer as an absorbing boundary condition. AIAA paper 96-1664 (1996)

  34. Bourlioux, A., Majda, A.J.: Theoretical and numerical structure for unstable two-dimensional detonations. Combust. Flame 90, 211–229 (1992)

    Article  Google Scholar 

  35. Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics, a practical introduction. Springer, Berlin (1999)

    MATH  Google Scholar 

  36. Erpenbeck, J.J.: Stability of idealized one-reaction detonations. Phys. Fluids 9, 684–696 (1964)

    Article  Google Scholar 

  37. Lee, H.I., Stewart, D.S.: Calculation of linear detonation instability, one-dimensional instability of plane detonation. J. Fluid Mech. 216, 103–132 (1990)

    Article  MATH  Google Scholar 

  38. Dou, H.S., Tsai, H.M., Khoo, B.C., Qiu, J.: Simulations of detonation wave propagation in rectangular ducts using a three-dimensional WENO scheme. Combust. Flame 154, 644–659 (2008)

    Article  Google Scholar 

  39. Don, W.S.: WENO pack, http://www.math.hkbu.edu.hk/~wsdon/

  40. Latini, M., Schilling, O., Don, W.S.: Effects of order of WENO flux reconstruction and spatial resolution on reshocked two-dimensional Richtmyer-Meshkov instability. J. Comput. Phys. 221, 805–836 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Marine Environment & Ecology, Ministry of Education, and School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, China

    Zhen Gao

  2. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China

    Wai Sun Don & Zhiqiu Li

Authors
  1. Zhen Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wai Sun Don
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhiqiu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wai Sun Don.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gao, Z., Don, W.S. & Li, Z. High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations. J Sci Comput 53, 80–101 (2012). https://doi.org/10.1007/s10915-011-9569-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-011-9569-0

Keywords

Search

Navigation