Indeterminacy and stability of the detonation mode in a bounded medium

Abstract

It was shown in [1] that in the presence of nonmonotonic heat release the detonation mode in an unbounded medium can be indeterminate. Certain properties of the number of detonation modes and their stability with respect to transition from one mode to another were also investigated there.

The lateral scattering of products of detonation affects the latter's rate in a manner qualitatively similar to that of heat losses or of an endothermic reaction. Noting this similarity and the results cited in [1], we consider the case of monotonie heat release on the assumption that lateral scattering is the only source of losses.

The effect of lateral scattering on the rate and stability of detonation, as well as the existence of a critical diameter of the charge, was first confirmed experimentally by Khariton and Roaing [2,3]. The critical diameter and its dependence on specific properties of an explosive were subsequently investigated in numerous papers (see [4, 7] and other publications). Another interesting aspect of the link between the stability of detonation and the scattering of reaction products is the low detonation rate observed under certain conditions [4, 5, 8–12]. This phenomenon has not yet been fully explained. There is no doubt that the low detonation rate is, if not in all cases at least in the majority of them related to two or more heat-release stages taking place at substantially different rates [5, 10–13]. However, the question of the character and limits of stability as well as of the low sensitivity of the detonation rate to variation in external parameters [10,11] remains unanswered. Attempts at a theoretical explanation of the low detonation rate were made by Eyring et al [14], but their results contradict experimental data [11,12] (see also §3 below, and [15]). The problem of indeterminacy of the detonation rate has been partially considered by Schall [16]. A critical review of [16] appears in [1].

In connection with all this it is interesting to investigate the total number of detonation modes and their stability during transition from one mode to another. In view of the recently revealed instability of a plane (smooth) detonation front [17], we point out that in the following we consider the stability of any detonation front which is steady-state on the average. The detonation rate, expressed in terms of the heat of reaction, is the same as that of a smooth front. It is known that “turbulent pulsations” of an uneven front alter the detonation rate only very slightly [18, 19], and that such alterations are not always present [20]. Unevenness of the front also does not contradict the concept presented in [3] of a relation between the critical diameter and the width of the reaction zone [6, 7]. The quantitative expression of the criterion given in [3] depends on specific properties of the explosive. In the case of high activation energy the induction time for large diameters close to the critical one increases very greatly in the direction from the charge axis toward its periphery, owing to considerable curvature of the front. Under these conditions the effective induction time can be considerably longer than for a straight shock wave [21].

The problem is formulated in an approximation to a given streamtube shape in §1, which also deals with investigation of the general properties of detonation modes in a bounded medium. These properties are illustrated in §2 for a simple model of detonation with one or two irreversible chemical reactions. Section 3 is devoted to discussion of the results and to the conclusions.

Systematic mathematical analysis of detonation with lateral scattering is extremely difficult. This explains the present lack of a rigorous theory for non-one-dimension detonation, in spite of numerous investigations of this problem. Several existing approximate theories [14,22–25] give a qualitatively correct description of increase in the detonation rate with increasing diameter of a cylindrical charge or with decreasing front curvature [26].

However, knowledge of the exact pattern of expansion of the explosive in the reaction zone is not necessary for the analysis of the number of modes and their stability. The mode and stability of detonation depend on the relationship between the rate of expansion and that of the chemical reaction. Both of these rates are defined for a given charge diameter by the shock-wave intensity and by the complete range of gas-dynamic parameters behind the compression shock. The rate of a chemical reaction is usually much more affected by the shock-wave intensity than by the effect of lateral scattering. Consequently, in investigations of the number of detonation modes and of their stability, it is natural to consider the expansion law (the shape of the streamtubes) as given and independent of the wave intensity, except in the case of weak waves (see below). Any relatively minor dependence of the streamtube form on the wave intensity result only in a small variation of the detonation rate, while the functional relationships remain qualitatively unchanged.