Critical Slow Dynamics of Detonation in a Gas with Non-Uniform Initial Temperature and Composition: A Large-Activation-Energy Analysis

Abstract

We study the dynamic conditions of existence of the self-sustained detonation regime in gases with non-uniform initial temperature and dilution in the case where the gradients of initial state are parallel to the propagation direction. We analyze the slow dynamics of the detonation supposing that the initial variations have the same order of smallness as the unsteadiness and curvature of the leading shock of the detonation front. We obtain a hyperbolic evolution law for the detonation shock front as a solution to an eigenvalue problem solved in the asymptotic limit of large activation energies for which the induction time of the chemical decomposition process is considered much larger than the heat-release time. We study the effect of the shock dynamics of this self-sustained detonation front on the induction time and we integrate the evolution law for some examples of distinct or combined, dispersive or localized, initial variations in the temperature and diluent mass fraction. We show the existence of a bifurcation set of initial conditions that separates the self-sustained detonations that will continuously achieve the CJ regime from those whose dynamics eventually fail to initiate the chemical decomposition because the dynamic induction length becomes too long. We find that the characteristic critical lengths associated with the initial or boundary conditions are larger than the reference CJ characteristic chemical length by several orders of magnitude, a well-established experimental feature of gaseous detonations. We identify subcritical and supercritical limits which, respectively, give necessary and sufficient conditions for detonation. Below the subcritical limit, the combustion front cannot be coupled to the shock front because the volumetric expansion rate behind the shock is too large. Above the supercritical limit, the relaxation from an overdriven detonation regime to the self-sustained regime is a continuous process, without decoupling and recoupling of the combustion and shock fronts. We interpret the critical domain between these limits as the parametric zone in which experiments show successive decouplings and transverse recouplings of the combustion and shock fronts before the final self-sustained CJ regime.