On the Critical Conditions for the Initiation of a Detonation in a Nonuniformly Perturbed Reactive Fluid

Abstract

The critical conditions for the initiation of a strong Zeldovich--von Neumann--Döring (ZND) detonation in a nonuniformly perturbed reactive fluid are derived using a high activation-energy asymptotic analysis. The initial disturbances are taken to have amplitudes of the order of the small inverse activation energy, with a characteristic wavelength of the order of an acoustic scale, and consist of initial temperature, concentration, pressure, velocity, or density perturbations. The detonation initiation mechanism is based on the work of Dold and Kapila, which describes how ignition of a reactive fluid leads to the generation of a supersonic, shockless, weak detonation. A transition from the weak detonation to a ZND detonation occurs if the initial disturbance is sufficiently strong to induce a gradient of thermal ignition times that cause the weak detonation to slow to the Chapman--Jouguet detonation velocity. Underpinning the whole process is the ability to calculate the path of the weak detonation. In Dold and Kapila's theory this has to be done numerically. Here, the path of the weak detonation, and thus the precise location and time of a transition to strong detonation, is derived analytically through a long-wavelength analysis of the induction zone of the explosion by assuming the initial disturbances vary slowly on the characteristic acoustic scale. Comparisons of the analytically derived results with exact numerical solutions of the induction zone evolution, the thermal runaway, and the weak detonation path, arising from initial disturbances which vary explicitly on the acoustic scale, are shown to be excellent. Moreover, this analysis provides a theoretically based understanding of how each type of initial disturbance, be it an entropy disturbance involving initial temperature nonuniformities or an acoustic disturbance involving initial velocity and pressure fluctuations, influence both the location and the time of the initial point of thermal runaway in the fluid, and the subsequent generation and the speed of the weak detonation. In combination with the analysis of Dold and Kapila, the theory contained herein gives a completely analytical description of how a ZND detonation is generated from an initially perturbed reactive fluid. An extension of the analysis to determine the path of a weak detonation arising from an initial nonuniformity in a three-dimensional geometry is also presented. Finally, as a by-product of the present analysis, a rational derivation of Zeldovich's notion of a constant volume spontaneous flame is given; moreover, it results in a significant improvement on Zeldovich's results.