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Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases

Published online by Cambridge University Press:  26 April 2006

Paul Clavin  and
Longting He
Paul Clavin
Affiliation:
Institut de Recherche sur les Phénomènes Hors Equilibre, Unite Mixte 138, CNRS – Universités d'Aix-Marseille, Centre Universitaire de Saint-Jérôme, Service 252, Avenue Escadrille Normandie Niemen, 13397 Marseille, Cedex 20, France
Longting He
Affiliation:
Institut de Recherche sur les Phénomènes Hors Equilibre, Unite Mixte 138, CNRS – Universités d'Aix-Marseille, Centre Universitaire de Saint-Jérôme, Service 252, Avenue Escadrille Normandie Niemen, 13397 Marseille, Cedex 20, France
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Abstract

The purpose of this analytical work is twofold: first, to clarify the physical mechanisms triggering the one-dimensional instabilities of plane detonations in gases; secondly to provide a nonlinear description of the longitudinal dynamics valid even far from the bifurcation. The fluctuations of the rate of heat release result from the temperature fluctuations of the shocked gas with a time delay introduced by the propagation of entropy waves. The motion of the shock is governed by a mass conservation resulting from the gas expansion across the reaction zone whose position fluctuates relative to the inert shock. The effects of longitudinal acoustic waves are quite negligible in pistonsupported detonations at high overdrives with a small difference of specific heats. This limit leads to a useful quasi-isobaric approximation for enlightening the basic mechanism of galloping detonations. Strong nonlinear effects, free from the spurious singularities of the square-wave model, are picked up by considering two different temperature sensitivities of the overall reaction rate: one governing the induction length, another one the thickness of the exothermic zone. A nonlinear integral equation for the longitudinal dynamics of overdriven detonations is obtained as an asymptotic solution of the reactive Euler equations. The analysis uses a distinguished limit based on an infinitely large temperature sensitivity of the induction kinetics and a small difference of specific heats. Comparisons with numerical calculations show a satisfactory agreement even outside the limits of validity of the asymptotic solution.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 306 , 10 January 1996 , pp. 353 - 378
Copyright
© 1996 Cambridge University Press

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References

Abouseif, G. E. & Toong, T. Y. 1982 Theory of unstable one-dimensional detonations. Combust. Flame 45, 67–94.Google Scholar
Abouseif, G. E. & Toong, T. Y. 1986 Theory of unstable two-dimensional detonations: Genesis of the transverse waves. Combust. Flame 63, 191–207.Google Scholar
Alpert, R. L. & Toong, T. Y. 1972 Periodicity in exothermic hypersonic flows about blunt projectiles. Astronaut. Acta 17, 539–560.Google Scholar
Blythe, P. A. & Crighton, D. G. 1989 Shock-generated ignition: the induction zone. Proc. R. Soc. Lond. A 426, 189–209.Google Scholar
Bourlioux, A. & Majda, A. J. 1992 Theoretical and numerical structure for unstable two-dimensional detonations. Combust. Flame 90, 211–229.Google Scholar
Bourlioux, A., Majda, A. J. & Roytburd, V. 1991 Nonlinear development of low frequency one-dimensional instabilities for reacting shock waves. In Dynamic Issues in Combustion (ed.P. Fife, A. Linan & F. A. Williams), vol. 35, pp. 63–83. Springer.
Clark, J. F. 1985 Finite amplitude waves in combustible gases. In The Mathematics of Combustion (ed. J. D. Buckmaster). SIAM, Philadelphia, 183.
Clavin, P. 1994 Premixed combustion and gasdynamics. Ann. Rev. Fluid Mech. 26, 321–352.Google Scholar
Clavin, P. & He, L. T. 1995 Acoustic effects in the nonlinear oscillations of planar detonations. Phys. Rev. E (submitted).Google Scholar
Clavin, P. & Sun, J. 1991 Theory of acoustic instability of plane flame propagating in sprays. Combust. Sci. Tech. 78, 265–288.Google Scholar
Clavin, P., Pelce, P. & He, L. T. 1990 One-dimensional vibratory instability of planar flames propagating in tubes. J. Fluid Mech. 216, 299–322.Google Scholar
Erpenbeck, J. J. 1963 Structure and stability of the square-wave detonation. 9th 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
Erpenbeck, J. J. 1967 Nonlinear theory of unstable one-dimensional detonation. Phys. Fluids 10 274–289.Google Scholar
Fickett, W. 1985 a Introduction to Detonation Theory.. University of California Press., Berkeley.
Fickett, W. 1985b Stability of the square-wave detonation in a model system. Physica D 16, 358–370.Google Scholar
Fickett, W. & Davis, W. C. 1979 Detonation.. University of California Press., Berkeley.
Fickett, W. & Wood, W. W. 1966 Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 9, 903–916.Google Scholar
He, L. T. 1995 Stability analysis of one-dimensional gaseous detonation. Europhys. Lett. (submitted).Google Scholar
He, L. T. & Clavin, P. 1992 Critical conditions for detonation initiation in cold gaseous mixtures by nonuniform hot pockets of reactive gases. 24th Symp. (Intl) on Combustion, pp. 1861–1867. The Combustion Institute.
He, L. T. & Clavin, P. 1993 Premixed hydrogen-oxygen flames. (Parts I and II) Flame structure and flammability limits. Combust. Flame 93, 391–420.Google Scholar
He, L. T. & Clavin, P. 1994 On the direct initiation of gaseous detonations by an energy source. J. Fluid Mech. 277, 227–248.Google Scholar
He, L. & Lee, J. H. 1995 On the dynamic limit of one-dimensional detonations. Phys. Fluids 7, 1151–1158.Google Scholar
Lee, H. I. & Stewart, D. S. 1990 Calculation of linear detonation instability: one-dimensional instability of plane detonations. J. Fluid Mech. 216, 103–132.Google Scholar
Lehr, H. F. 1972 Experiments on shock-induced combustion. Astronaut. Acta 17, 589–597.Google Scholar
Oran, E. S. & Boris J. P. 1987 Numerical Simulation of Reactive Flow. Prentice-Hall.
Searby, G. & Rochwerger, D. 1991 A parametric acoustic instability in premixed flames. J. Fluid Mech. 231, 529–543.Google Scholar
Shchelkin, K. I. 1959 Two cases of unstable burning. Sov. Phys., J. Exp. Theor. Phys. 36, 600–605.Google Scholar
Strehlow, R. A. 1985 Combustion Fundamentals. McGraw-Hill.
Williams, F. A. 1985 Combustion Theory. Benjamin/Cummings.
Zaidel, R. M. 1961 The stability of detonation waves in gaseous mixtures. Dokl. Akad. Nauk. SSSR 136, 1142–1145.Google Scholar
Zeldovich, Ya. B. 1940 On the theory of detonation propagation in gaseous systems. In Selected Works of Yakov Borisovich Zeldovich (ed.J. P. Ostriker, G. I. Barenblatt and R. A. Sunyaev), vol. 1, pp. 411–451. Princeton University Press. 1992.