Relaxation of overdriven detonation waves with finite reaction rate

A numerical study is made of the interaction of a detonation wave having finite reaction velocity with a rarefaction wave of different intensity which approaches it from the rear, for the Zeldovich-Neumann-Doring (ZND) model with a single irreversible reaction A B. It is found that, for a fixed value of the parameter characterizing the initial supercompression (depending on the activation energy and the heating value of the mixture), the considered interaction leads either to a gradual relaxation of the detonation wave and its transition to the Chapman-Jouguet (CJ) regime, or to the development of undamped oscillations.Interest in the problems of detonation and supersonic combustion has increased in recent years. This is associated with the appearance and development of new experimental and theoretical techniques; it is also associated with the further development of air-breathing reaction engines, and other practical requirements. The present state of detonation theory is reflected in the survey [1].It has been established [2] that the detonation wave in gases nearly always has a complex nonuniform structure. Transverse disturbances are observed under a wide range of conditions and differ both in amplitude and wavelength. At the same time, behind the detonation leading front there is a region of uncompletely burned gas corresponding to the effective ignition induction period [3]. In spinning detonation the induction period is significantly longer than the heat release period and transverse detonation waves traveling in the induction zone of the head wave appear [3, 4]. Such a secondary detonation wave is free of transverse disturbances. The same is true of the detonation waves observed in the wake behind a body moving at high speed in a combustible medium [5] or in a gas which has been preheated by a shock wave [6].Although it is possible, under favorable conditions, to study in detail the system of discontinuities accompanying detonation, information on the extensive zones in which heat release takes place is scarce, the mechanism of detonation wave autonomy (in particular, the role of the rarefaction zone behind the wave) is not entirely clear, and the fact that, in spite of the complex structure, an autonomous detonation propagates with the CJ velocity calculated on the basis of one-dimensional theory has not yet been explained.In studying the nonlinear phenomena associated with the finite reaction rate it is quite acceptable to investigate only the simple one-dimensional detonation model, with which it is convenient to restrict ourselves to a single effective chemical reaction. This model is particularly reasonable since, in certain cases, the real detonation is virtually one-dimensional.The question of the stability of the one-dimensional detonation wave to disturbances of its structure has been examined by several authors [7–13]. The use of computers makes possible the direct computation of flows with heat release and the study of their properties. This method has been used in [11–13] to study the stability problem for a detonation wave with respect to finite disturbances.In the present paper we present a numerical study of the interaction of a detonation wave having finite chemical reaction rate with a rarefaction wave of different intensity approaching it from the rear for the ZND model with a single irreversible reaction A B. It is found that for a fixed value of the parameter characterizing the difference between detonation and the CJ waves, depending on the activation energy E and the mixture heating value Q_m, the interaction in question leads either to a gradual relaxation of the detonation wave and its transition to the CJ regime (this relaxation may be accompanied by decaying oscillations) or to the appearance of undamped oscillations (the unstable regime). The parameters E and Q_m affect the wave stability differently: with increase of Q_m, the wave is stabilized; with increase of E, it is destabilized. The boundary between the stable and unstable detonation wave propagation regimes is found. This boundary has a weak dependence on the rarefaction wave intensity. Estimates and calculated examples show that the amplitude of the unstable wave oscillations is finite and that the average detonation propagation velocity is close to the CJ velocity computed for the given heating value Qm.The author wishes to thank G. G. Chernyi for his guidance and L. A. Chudov for advice on computational questions.