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Decay of plane detonation waves to the self-propagating Chapman–Jouguet regime

  • Paul Clavin (a1) and Bruno Denet (a1)


A theoretical study of the decay of plane gaseous detonations is presented. The analysis concerns the relaxation of weakly overdriven detonations toward the Chapman–Jouguet (CJ) regime when the supporting piston is suddenly arrested. The initial condition concerns propagation velocities ${\mathcal{D}}$ that are not far from that of the CJ wave ${\mathcal{D}}_{CJ}$ , $0<({\mathcal{D}}/{\mathcal{D}}_{CJ}-1)\ll 1$ . The unsteady inner structure of the detonation wave is taken into account analytically for small heat release, i.e. when the propagation Mach number of the CJ wave $M_{u_{CJ}}$ is small, $0<(M_{u_{CJ}}-1)\ll 1$ . Under such conditions the flow is transonic across the inner structure. Then, with small differences between heat capacities (Newtonian limit), the problem reduces to an integral equation for the velocity of the lead shock. This equation governs the detonation dynamics resulting from the coupling of the unsteady inner structure with the self-similar dynamics of the centred rarefaction wave in the burnt gas. The key point of the asymptotic analysis is that the response time of the inner structure is larger than the reaction time. How, and to what extent, the result is relevant for real detonations is discussed in the text. In a preliminary step the steady-state approximation is revisited with particular attention paid to the location of the sonic condition.


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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, 126.
Chandrasekhar, S.1943 On the decay of plane shock wave. Rep. 423. Ballistic Research Laboratory.
Clavin, P. 2017 Nonlinear dynamics of shock and detonation waves in gases. Combust. Sci. Technol. 189 (5), 129.
Clavin, P. & He, L. 1996 Stability and nonlinear dynamics of one-dimensional overdriven detonations in gases. J. Fluid Mech. 306, 353378.
Clavin, P., He, L. & Williams, F. A. 1997 Multidimensional stability analyses of overdriven gaseous detonations. Phys. Fluids 9 (12), 37643785.
Clavin, P. & Searby, G. 2016 Combustion Waves and Fronts in Flows. Cambridge University Press.
Clavin, P. & Williams, F. A. 2002 Dynamics of planar gaseous detonations near Chapman–Jouguet conditions for small heat release. Combust. Theor. Model. 6, 127129.
Clavin, P. & Williams, F. A. 2009 Multidimensional stability analysis of gaseous detonations near Chapman–Jouguet conditions for small heat release. J. Fluid Mech. 624, 125150.
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Interscience Publishers.
Faria, L., Kasimov, A. & Rosales, R. 2015 Theory of weakly nonlinear self-sustained detonations. J. Fluid Mech. 784, 163198.
Friedrichs, K. O. 1948 Formation and decay of shock waves. Commun. Appl. Maths 1 (3), 211245.
He, L. & Clavin, P. 1994 On the direct initiation of gaseous detonations by an energy source. J. Fluid Mech. 277, 227248.
Larin, O. B. & Levin, V. A. 1971 Study of the attenuation of a detonation wave with a two-front structure by the boundary (shock) layer method. Mekh. Zhidk. Gaza (3), 5965; translated in Fluid Dyn. 6 (3) 413–418 (1971).
Levin, V. A. & Chernyi, G. G 1967 Asymptotic laws of behavior of detonation waves. Prikl. Mat. Mekh. 31, 393405.
Liñan, A., Kurdyumov, V. & Sanchez, A. L. 2012 Initiation of reactive blast waves by external energy source. C. R. Méc. 340, 829844.
Medvedev, S. A. 1969 Relaxation of overdriven detonation waves with finite reaction rate. Mekh. Zhidk. Gaza 4 (3), 2230.
Shchelkin, K. I. & Troshin, Ya. K 1965 Gasdynamics of Combustion. Mono Book Corp.
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Decay of plane detonation waves to the self-propagating Chapman–Jouguet regime

  • Paul Clavin (a1) and Bruno Denet (a1)


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