Stability and Instability of Detonation Waves

  • Tong Li
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 130)

Abstract

We study nonlinear stability and instability of detonation waves through a dynamic model which includes nonlinear convection, chemical reaction, weak curvature effect and induction kinetics. The stability of the plane and the divergent detonation waves were established rigorously when there is no induction-zone. Furthermore, when the wave front has a small positive curvature and when there is an induction-zone behind the shock front, we captured dynamically unstable solutions through numerical simulations. The unstable solutions are accompanied by velocity fluctuations.

Keywords

Shock Front Detonation Wave Travel Wave Solution Nonlinear Stability Unstable Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. [1]
    G.E. Abouseif and T.Y. Toong, Theory of unstable one-dimensional detonations, Comb. and Flame, 45 (1982), 67–94.CrossRefGoogle Scholar
  2. [2]
    R.L. Alpert and T.Y. Toong, Astronautica Acta, 15 (1972), 539–560.Google Scholar
  3. [3]
    J.B. Bdzil, Perturbation methods applied to problems in detonation physics,in Proceedings of the Sixth International Symposium on Detonation, (1976) 352–369.Google Scholar
  4. [4]
    P. Clavin and L. He, Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases, J. Fluid Mech., 306 (1996), 353–378.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    J.J. Erpenbeck, Stability of steady-state equilibrium detonations, Phys. Fluid, 5 (1961), 604–614.CrossRefGoogle Scholar
  6. [6]
    E. Engquist and B. Sjogreen, Robust difference approximations of stiff inviscid detonation waves, Technical Report CAM Report 91-03, Department of Mathematics, University of California, Los Angeles, 1991.Google Scholar
  7. [7]
    W. Fickett and W.W. Wood, Flow calculations for pulsating one-dimensional detonations, Phys. Fluids, 9 (1966), 903–916.CrossRefGoogle Scholar
  8. [8]
    W. Fickett, Decay of small planar perturbations on a strong steady detonation: A differential-difference equation for shock, Phys. Fluids, 30 (1987), 1299–1309.MATHCrossRefGoogle Scholar
  9. [9]
    W. Fickett, Approach to the steady solution for a plane Chapman-Jouguet detonation, Phys. Fluids A, 1 (1989), 371–379.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    W.E. Gordon, A.J. Mooradian and S.A. Harper, Limit and spin effects in hydrogen-oxygen detonations, in the Seventh Symposium (International) on Combustion, Academic Press, New York, 1959, 752–759.Google Scholar
  11. [11]
    S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solution to system for one dimensional gas motion, Comm. Math. Phys., 101 (1985), 97–127.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    H.I. Lee and D.S. Stewart, Calculation of linear detonation instability: one-dimensional instability of plane detonations, J. Fluid Mech., 216 (1990), 103–132.MATHCrossRefGoogle Scholar
  13. [13]
    H.F. Lehr, Experiments on shock-induced combustion, Astronaut. Acta, 17 (1972), 589–597.Google Scholar
  14. [14]
    T. Li, Time-asymptotic limit of solutions of a combustion problem,J. of Dynamics and Differential Equations, 10 (1998), 577–605, in press.MATHCrossRefGoogle Scholar
  15. [15]
    T. Li, Stability of strong detonation waves and rates of convergence,Electronic Journal of Differential Equations, 1998 (1998), 1–17.Google Scholar
  16. [16]
    T. Li, Rigorous asymptotic stability of a CJ detonation wave in the limit of small resolved heat release, Combustion Theory and Modelling, 1 (1997), 259–270.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    T. Li, Stability of a Transonic Profile Arising From Divergent Detonations,submitted to Comm. in PDE, February, 1998.Google Scholar
  18. [18]
    T. Li, On a Dynamic Model for Detonation Instability, preprint.Google Scholar
  19. [19]
    T.-P. Liu and L.A. Ying, Nonlinear stability of strong detonations for a viscous combustion model, SIAM J. Math. Anal., 26 (1995), 519–528.MathSciNetMATHGoogle Scholar
  20. [20]
    N. Manson, C. Brochet, J. Brossard and Y. Pujol, Vibratory phenomena and instability of self-sustained detonations in gases, in the Ninth Symposium (International) on Combustion, Academic Press, New York, 1963, 461–469.Google Scholar
  21. [21]
    A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys., 165 (1994), 83–96.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    G. Mundy, F.R.S. Ubbelhode and I.F. Wood, Fluctuating detonation in gases, Proc. Roy. Soc. A, 306 (1968), 171–178.CrossRefGoogle Scholar
  23. [23]
    R. Rosales and A. Majda, Weakly nonlinear detonation waves, SIAM J. Appl. Math., 43 (1983), 1086–1118.MathSciNetMATHGoogle Scholar
  24. [24]
    M. Short, Multidimensional linear stability of a detonation wave at high activation energy, SIAM J. Appl. Math., 57 (1997), 307–326.MathSciNetMATHGoogle Scholar
  25. [25]
    D.S. Stewart and J.B. Bdzil, The shock dynamics of stable multidimensional detonation, Comb. and Flame, 72 (1988), 311–323.CrossRefGoogle Scholar
  26. [26]
    G. Strang, On the construction and comparison of finite-difference schemes,SIAM J. Numer. Anal., 5 (1968), 506–517.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. Google Scholar
  28. [28]
    R.M. Zaidel, The stability of detonation waves in gaseous mixtures, Dokl. Akad. Nauk. SSSR, 136 (1961), 1142–1145.Google Scholar
  29. [29]
    Ya. B. Zeldovich, On the theory of detonation propagation in gaseous systems, (1940), in Selected Works of Yakov Borisovich Zeldovich, Princeton University Press, 1 (1992), 411–451.Google Scholar

Copyright information

© Springer Basel AG 1999

Authors and Affiliations

  • Tong Li
    • 1
  1. 1.Department of MathematicsUniversity of IowaIowaUSA

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