Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-19T08:52:20.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Calculation of linear detonation instability: one-dimensional instability of plane detonation

Published online by Cambridge University Press:  26 April 2006

H. I. Lee  and
D. S. Stewart
H. I. Lee
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, USA
D. S. Stewart
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The detonation stability problem is studied by a normal mode approach which greatly simplifies the calculation of linear instability of detonation in contrast to the Laplace transform procedure used by Erpenbeck. The method of solution, for an arbitrary parameter set, is a shooting method which can be automated to generate easily the required information about instability. The condition on the perturbations applied at the end of the reaction zone is shown to be interpreted as either a boundedness condition or an acoustic radiation condition. Continuous and numerically exact neutral stability curves and boundaries are given as well as growth rates and eigenfunctions which are calculated for the first time. Our calculations include the Chapman–Jouguet (CJ) case which presents no special difficulty. We give representative results for our detonation model and summarize the one-dimensional stability behaviour in parameter space. Comparison with previous results for the neutral stability boundaries and approximations to the unstable discrete spectrum are given. Parametric studies of the unstable, discrete spectrum's dependence on the activation energy and the overdrive factor are given with the implications for interpreting the physical mechanism of instability observed in experiments. This first paper is restricted to the case of one-dimensional linear instability. Extensions to transverse disturbances will be treated in a sequel.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 216 , July 1990 , pp. 103 - 132
Copyright
© 1990 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abouseif, G. & Toong, T. Y. 1982 Theory of unstable one-dimensional detonations. Combust. Flame 45, 67–94.Google Scholar
Alpert, R. L. & Toong, T. Y. 1972 Periodicity in exothermic hypersonic flows about blunt projectiles. Acta Astron. 17, 538–560.Google Scholar
Bdzil, J. B. & Stewart, D. S. 1986 Time-dependent two-dimensional detonation: the interaction of edge rarefactions with finite-length reaction zones. J. Fluid Mech. 171, 1–26.Google Scholar
Bourlioux, A., Majda, A. J. & Roytburd, V. 1989 Theoretical and numerical structure for unstable one-dimensional detonation. Preprint.
Buckmaster, J. D. & Ludford, G. S. S. 1988 The effect of structure on the stability of detonations I. Role of the induction zone. In Proc. Twenty-First Symp. (Intl) on Combustion, pp. 1669–1675. The Combustion Institute, Pittsburgh, PA.
Buckmaster, J. D. & Nevis, J. 1988 One-dimensional detonation stability — The spectrum for infinite activation energy. Phys. Fluids 31, 3571–3575.Google Scholar
Erpenbeck, J. J. 1962 Stability of steady-state equilibrium detonations. Phys. Fluids 5, 604–614.Google Scholar
Erpenbeck, J. J. 1963 Structure and stability of the square-wave detonation. In Ninth Symp. (Intl) on Combustion, pp. 442–453. Academic.
Erpenbeck, J. J. 1964 Stability of idealized one-reaction detonations. Phys. Fluids 7, 684–696.Google Scholar
Fickett, W. 1985 Stability of the square wave detonation in a model system. Physica 16D, 358–370.Google Scholar
Fickett, W. & Davis, W. C. 1979 Detonation. University of California Press.
Fickett, W., Jacobson, J. D. & Wood, W. W. 1970 The method of characteristics for one-dimensional flow with chemical reaction. Los Alamos Scientific Laboratory Rep. LA-4269.Google Scholar
Fickett, W. & Wood, W. W. 1966 Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 9, 903–916.Google Scholar
McVey, J. B. & Toong, T. Y. 1971 Mechanism of instabilities in exothermic blunt-body flows. Combust. Sci. Tech. 3, 63–76.Google Scholar
Majda, A. & Rosales, R. 1983 A theory for spontaneous Mach stem formation in reacting shock front, I. The basic perturbation analysis. SIAM J. Appl. Maths 43, 1310–1334.Google Scholar
Shepherd, J. E. 1986 Chemical kinetics of hydrogen—air—diluent detonations. Prog. Aeronaut. Astronaut. 106, 263–293.Google Scholar
Stewart, D. S. & Bdzil, J. B. 1988a The shock dynamics of stable multidimensional detonation. Combust. Flame 72, 311–323.Google Scholar
Stewart, D. S. & Bdzil, J. B. 1988b A lecture on detonation shock dynamics. In Mathematical Modelling in Combustion Science (ed. J. B. Buckmaster & T. Takeno). Springer.