Theory of Detonation with an Embedded Sonic Locus

Abstract

Abstract.A steady planar self-sustained detonation has a sonic surface in the reaction zone that resides behind the lead shock. In this work we address the problem of generalizing sonic conditions for a three-dimensional unsteady self-sustained detonation wave. The conditions are proposed to be the characteristic compatibility conditions on the exceptional surface of the governing hyperbolic system of reactive Euler equations. Two equations are derived that are necessary to determine the motion of both the lead shock and the sonic surface. Detonation with an embedded sonic locus is thus treated as a two-front phenomenon: a reaction zone whose domain of influence is bounded by two surfaces, the lead shock surface and the trailing characteristic surface. The geometry of the two surfaces plays an important role in the underlying dynamics. We also discuss how the sonic conditions of detonation stability theory and detonation shock dynamics can be obtained as special cases of the general sonic conditions. 1. Introduction. A detonation wave is a shock wave that triggers exothermic reactions in an explosive as it propagates so that the energy released in the reactions sustains the shock propagation. Modern theories of detonation originate from the theory first developed independently by Zel'dovich, von Neumann, and Doering in the 1940s (ZND theory; see Fickett and Davis [5] for details) that describes the dynamics of a steady one-dimensional planar detonation in a gaseous explosive. The ZND theory is applicable to both self-sustained detonations, that is, autonomous waves whose motion is sustained entirely by the energy released in their reaction zone, and overdriven detonations which require an additional external support to maintain their motion at a nominal speed. In self-sustained steady one-dimensional planar detonations, which are also called Chapman−Jouguet (CJ) detonations, there exists an embedded sonic locus within or at the end of the reaction zone, such that at that point the flow speed is sonic relative to the shock. As a consequence, the leadshock dynamics is influenced only by the flow between the shock and the sonic locus. In contrast, the lead-shock dynamics of overdriven detonations is influenced by the entire region between the shock and the support (e.g., a piston); no sonic locus exists in such detonations. Without the condition of sonicity, the equations governing the CJ detonation (the mass, momentum, and energy equations) are not closed, since the detonation speed is unknown; the sonicity condition provides the necessary closure. Understanding the nature of the sonic conditions in detonations more general than

Asymptotic theory of evolution and failure of self-sustained detonations

“…The equations and boundary conditions form a closed system and allow a solution that describes the motion of the detonation shock, the evolution of the material states in the reaction zone and the motion of the sonic surface. The reader can find a detailed derivation of the conditions at the sonic locus in Kasimov (2004) and Stewart & Kasimov (2004). Here we present a concise derivation of a simplified version of the evolution equation that retains the leading-order curvature and shock-acceleration corrections to the quasi-steady planar solution.…”

“…As it turns out (for more details, see Kasimov 2004;Stewart & Kasimov 2004), most of the difficulties associated with approximating the structure of detonations with an embedded sonic locus concern this square root. An obvious difficulty is seen immediately by observing that for the steady detonation, the argument of the square root vanishes at the sonic point.…”

“…In Kasimov (2004) and Stewart & Kasimov (2004) we worked out the general three-dimensional formulation for this surface as a rear boundary condition. The sonic locus is coincident with a forward-propagating characteristic surface that remains at a finite distance behind the lead shock throughout the evolution.…”

“…In this work we derive a nonlinear evolution equation for a self-sustained detonation wave in the asymptotic limit of small curvature and slow time variation, which are measured in the scales of steady reaction-zone length and time in the same sense as the previous DSD theories. We assume that the detonation has an embedded sonic locus and employ the general characteristic conditions that we have developed in Kasimov (2004) and Stewart & Kasimov (2004). That the embedded sonic locus is assumed to exist initially implies that our analysis is restricted to the evolution of pre-existing detonation waves.…”

“…We have developed a general theory of detonation waves with an embedded sonic locus (Kasimov 2004;Stewart & Kasimov 2004) that applies to wide class of detonation waves in explosives with a general equation of state and complex chemistry, and recently illustrated the behaviour of the sonic locus by means of a numerical simulation in Kasimov & Stewart (2004b). The sonic locus in general is unambiguously defined to be a characteristic surface that serves as a separatrix and an information boundary for the reaction zone initiated by the lead shock.…”