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On the dynamics of multi-dimensional detonation

Published online by Cambridge University Press:  26 April 2006

Jin Yao  and
D. Scott Stewart
Jin Yao
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, USA
D. Scott Stewart
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, USA
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Abstract

We present an asymptotic theory for the dynamics of detonation when the radius of curvature of the detonation shock is large compared to the one-dimensional, steady, Chapman-Jouguet (CJ) detonation reaction-zone thickness. The analysis considers additional time-dependence in the slowly varying reaction zone to that considered ill previous works. The detonation is assumed to have a sonic point in the reaction- zone structure behind the shock, and is referred to as an eigenvalue detonation. A new, iterative method is used to calculate the eigenvalue relation, which ultimately is expressed as an intrinsic, partial differential equation (PDE) for the motion of the shock surface. Two cases are considered for an ideal equation of state. The first corresponds to a model of a condensed-phase explosive, with modest reaction rate sensitivity, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, Dn, the first normal time derivative of the normal shock velocity, Dn, and the shock curvature, K. The second case corresponds to a gaseous explosive mixture, with the large reaction rate sensitivity of Arrhenius kinetics, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, Dn, its first and second normal time derivatives of the normal shock velocity, Dn, Dn, and the shock curvature, K, and its first normal time derivative of the curvature, k. For the second case, one obtains a one-dimensional theory of pulsations of plane CJ detonation and a theory that predicts the evolution of self-sustained cellular detonation. Versions of the theory include the limits of near-CJ detonation, and when the normal detonation velocity is significantly below its CJ value. The curvature of the detonation can also be of either sign, corresponding to both diverging and converging geometries.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 309 , 25 February 1996 , pp. 225 - 275
Copyright
© 1996 Cambridge University Press

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References

Aslam, T. D., Bdzil, J. B. & Stewart, D. S. 1995 Level-set techniques applied to modeling detonation shock dynamics. J. Comput. Phys. (submited). Also TAM Rep. 773, UIUC-ENG- 94–6029, University of Illinois, June 1995.Google Scholar
Bdzil, J. B. 1981 Steady state two-dimensional detonation. J. Fluid Mech. 108, 185–226.Google Scholar
Bdzil, J. B. & Stewart, D. S. 1989 Modeling of two-dimensional detonation with detonation shock dynamics. Phys. Fluids A 1, 1261–1267.Google Scholar
Buckmaster, J. D. 1988 Pressure transients and the genesis of transverse shock in unstable detonation. Combust. Sci. Tech. 61, 1–20.Google Scholar
Erpenbeck, J. J. 1964 The stability of idealized one dimensional detonations. Phys. Fluids 7, 684–696.Google Scholar
Fickett, W. & Wood, W. W. 1966 Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 1, 528–534.Google Scholar
Klein, R., Kroc, J. C. & Shepherd, J. E. 1995 Critical conditions for quasi-steady curved detonations. Preprint
Klein, R. & Stewart, D. S. 1993 The relation between curvature and rate state-dependent detonation velocity. SIAM J. Appl. Maths 53, 1401–1435.Google Scholar
Lee, H. I. & Stewart, D. S. 1990 Calculation of linear detonation instability: one-dimension instability of plane detonation. J. Fluid Mech. 216, 103–132.Google Scholar
Lee, J. 1990 Detonation shock dynamics of composite energetic materials. PhD thesis, New Mexico Institute of Mining and Technology, Soccoro, NM.
Osher, S. & Sethian, J. 1988 Fronts propagating with curvature-dependent speed: Algorithms based on Hamiliton-Jacobi formulations. J. Comput. Phys. 79, 12–49Google Scholar
Stewart, D. Scott. 1993 Lectures on detonation physics: Introduction to the theory of detonation shock dynamics. TAM Rep. 721. UILU-ENG-93–6019.Google Scholar
Stewart, D. S., Aslam, T. D., Yao, J. & Bdzil, J. B. 1995 Level-set techniques applied to unsteady detonation propagation. Modeling in Combustion Science (ed . J. Buckmaster & T. Takeno). Lecture Notes in Physics, vol. 449, pp. 352–369. Springer. (Also TAM Rep. 773, UIUC-ENG- 94–6029, University of Illinois, Oct. 1994.)
Stewart, D. S. & Bdzil, J. B. 1988 The shock dynamics of stable multi-dimensional detonation. Combust. Flame 72, 311–323.Google Scholar
Stewart, D. S. & Bdzil, J. B. 1993 Asymptotics and multi-scale simulation in a numerical combustion laboratory. Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters. NATO ASI Series C: Vol 34, pp. 163–187.
Strehlow, R. A., Liaugminas, R., Watson, R. H. & Eyman, J. R. 1967 Transverse wave structures in detonation. Eleventh Symp. (Intl) on Combustion, pp. 683–691. Pittsburgh: The Combustion Institute.
Wood, W. W. & Kirkwood, J. G. 1954 Diameter effects in condensed explosives: The relation between velocity and radius of curvature. J. Chem. Phys. 22, 1920–1924.Google Scholar
Yao, J. 1996 The dynamics of multi-dimensional detonation. PhD thesis, Theoretical and Applied Mechanics, University of Illinois.
Yao, J. & Stewart, D. S. 1995 On the normal detonation shock velocity curvature relationship for materials with large-activation energy. Combust. Flame 100, 519–528.Google Scholar