ON THE CRITICAL CONDITIONS FOR THE INITIATION OF A DETONATION IN A NONUNIFORMLY PERTURBED REACTIVE FLUID

The critical conditions for the initiation of a strong Zeldovich-von N eumann-Doring (ZND) detonation in a nonuniformly perturbed reactive fl uid are derived using a high activation-energy asymptotic analysis. Th e initial disturbances are taken to have amplitudes of the order of th e small inverse activation energy, with a characteristic wavelength of the order of an acoustic scale, and consist of initial temperature, c oncentration, pressure, velocity, or density perturbations. The detona tion initiation mechanism is based on the work of Dold and Kapila, whi ch describes how ignition of a reactive fluid leads to the generation of a supersonic, shockless, weak detonation. A transition from the wea k detonation to a ZND detonation occurs if the initial disturbance is sufficiently strong to induce a gradient of thermal ignition times tha t cause the weak detonation to slow to the Chapman-Jouguet detonation velocity. Underpinning the whole process is the ability to calculate t he path of the weak detonation. In Dold and Kapila's theory this has t o be done numerically. Here, the path of the weak detonation, and thus the precise location and time of a transition to strong detonation, i s derived analytically through a long-wavelength analysis of the induc tion zone of the explosion by assuming the initial disturbances vary s lowly on the characteristic acoustic scale. Comparisons of the analyti cally derived results with exact numerical solutions of the induction zone evolution, the thermal runaway, and the weak detonation path, ari sing from initial disturbances which vary explicitly on the acoustic s cale, are shown to be excellent. Moreover, this analysis provides a th eoretically based understanding of how each type of initial disturbanc e, be it an entropy disturbance involving initial temperature nonunifo rmities or an acoustic disturbance involving initial velocity and pres sure fluctuations, influence both the location and the time of the ini tial point of thermal runaway in the fluid, and the subsequent generat ion and the speed of the weak detonation. In combination with the anal ysis of Dold and Kapila, the theory contained herein gives a completel y analytical description of how a ZND detonation is generated from an initially perturbed reactive fluid. An extension of the analysis to de termine the path of a weak detonation arising from an initial nonunifo rmity in a three-dimensional geometry is also presented. Finally, as a by-product of the present analysis, a rational derivation of Zeldovic h's notion of a constant volume spontaneous flame is given; moreover, it results in a significant improvement on Zeldovich's results.