Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-06-27T23:35:52.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Multidimensional stability analysis of gaseous detonations near Chapman–Jouguet conditions for small heat release

Published online by Cambridge University Press:  10 April 2009

PAUL CLAVIN  and
FORMAN A. WILLIAMS
PAUL CLAVIN*
Affiliation:
Institut de Recherche sur les Phénomènes Hors Equilibre, Universités d'Aix Marseille et CNRS, 49 rue Joliot Curie, BP 146, 13384 Marseille cedex 13, France
FORMAN A. WILLIAMS
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093-0411, USA
*
†Email address for correspondence: clavin@irphe.univ-mrs.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Multidimensional instability of planar detonations, leading to cellular structures, is studied analytically near Chapman–Jouguet conditions, in the limit of small heat release, with small (Newtonian) differences between heat capacities, by using an expansion in a small parameter representing the ratio of the heat release to the thermal enthalpy of the fresh mixture. In this limit, the dynamics of detonations is governed by the interaction between the acoustic waves and the heat-release rate inside the inner detonation structure, the entropy–vorticity wave playing a negligible role at leading order. This situation is just opposite from that considered in our 1997 study of strongly overdriven detonations. The present analysis offers a step towards improving our understanding of the cellular structures of ordinary detonations, for which both the entropy–vorticity waves and the acoustic waves are involved in the instability mechanism. The relevant bifurcation parameter is identified, involving the degree of overdrive and the sensitivity of the rate of heat release to temperature at the Neumann state, and the onset of the instability is studied analytically for a realistic model of the inner structure of gaseous detonations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 624 , 10 April 2009 , pp. 125 - 150
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Bourlioux, A. & Majda, A. J. 1992 Theroretical and numerical structure for unstable two-dimensional detonation. Combust. Flame 90, 211–29.CrossRefGoogle Scholar
Buckmaster, J. D. 1989 A theory of triple point spacing in overdriven waves. Combust. Flame 77, 219–228.CrossRefGoogle Scholar
Buckmaster, J. D. & Ludford, G. S. S. 1988 The effect of structure on stability of detonations. I. Role of the induction zone. 21st Symp. (Intl) Combust. 28, 1669.CrossRefGoogle Scholar
Clavin, P. 2002 a Instabilities and nonlinear patterns of overdriven detonations in gases. In Nonlinear PDE's in condensed matter and reactive flows (ed. Berestycki, H. & Pomeau, Y.), NATO ASI Series B: Physics, vol. 569, pp. 49–97. Kluwer.CrossRefGoogle Scholar
Clavin, P. 2002 b Self-sustained mean streaming motion in diamond patterns of a gaseous detonation. Intl J. Bifur. Chaos 12 (11), 2535–2546.CrossRefGoogle Scholar
Clavin, P. & Denet, B. 2002 Diamond patterns in the cellular front of an overdriven detonation. Phy. Rev. Lett. 88 (4), 1–4.CrossRefGoogle ScholarPubMed
Clavin, P. & He, L. 1996 a Acoustic effects in the nonlinear oscillations of planar detonations. Phys. Rev. E 53, 4778–4784.Google ScholarPubMed
Clavin, P. & He, L. 1996 b Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases. J. Fluid Mech. 306, 353–378.CrossRefGoogle Scholar
Clavin, P., He, L. & Williams, F. A. 1997 Multidimensional stability analysis of overdriven gaseous detonations. Phys. Fluids 9 (12), 3764–3785.CrossRefGoogle Scholar
Clavin, P. & Williams, F. A. 2002 Dynamics of planar gaseous detonations near Chapman–Jouguet conditions for small heat release. Combust. Theory Model. 6, 127–129.CrossRefGoogle Scholar
Daou, R. & Clavin, P. 2003 Instability threshold of gaseous detonations. J. Fluid Mech. 482, 181–206.CrossRefGoogle Scholar
D'Yakov, S. P. 1954 The stabiliy of shockwaves: investigation of the problem of stability of shock waves in arbritary media. Zh. Eksp. Teor. Fiz 27, 288.Google Scholar
Erpenbeck, J. J. 1962 Stability of steady-state equilibrium detonations. Phys. Fluids 5, 604–614.CrossRefGoogle Scholar
Erpenbeck, J. J. 1964 Stability of idealized one-reaction detonation. Phys. Fluids 7, 684.CrossRefGoogle Scholar
Erpenbeck, J. J. 1965 Stability of idealized one-reaction detonation: zero activation energy. Phys. Fluids 8, 1192–3.CrossRefGoogle Scholar
Erpenbeck, J. J. 1966 Detonation stability for disturbances of small transverse wavelength. Phys. Fluids 9, 1293–1306.CrossRefGoogle Scholar
Kontorovich, V. M. 1957 Concerning the stability of shock waves. Zh. Eksp. Teor. Fiz 33, 1525.Google Scholar
Oran, E. S. & Boris, J. P. 1987 Numerical Simulation of Reactive Flows. Elsevier.Google Scholar
Short, M. & Stewart, D. S. 1998 Cellular detonation stability. Part 1. A normal-mode linear analysis. J. Fluid Mech. 368, 229–262.CrossRefGoogle Scholar
Short, M. & Stewart, D. S. 1999 The multidimensional stability of weak-heat-release detonations. J. Fluid Mech. 382, 109–135.CrossRefGoogle Scholar
Yao, J. & Stewart, D. S. 1996 On the dynamics of multidimensional detonation waves. J. Fluid Mech. 309, 1142–1145.CrossRefGoogle Scholar