Multidimensional stability analysis of overdriven gaseous detonations

Abstract

Multidimensional instabilities of planar detonations that lead to cellular structures are addressed by use of a distinguished limit in which the propagation Mach number is large and the difference between the specific heats at constant pressure and at constant volume is small. In this limit, the Neumann-state Mach number is small, and the fractional variations of the pressure change after the Neumann state also are small for the overdriven conditions that are considered, under which the heat release is comparable in magnitude with the thermal enthalpy at the Neumann state. The resulting post-shock flow is quasi-isobaric in the first approximation. For all linear modes the analysis provides a dispersion relation expressing the frequency and the linear growth rate in terms of the transverse wavelength. The analysis serves to demonstrate how the interactions among the entropy waves, the varying rate of heat release and the transverse flow induced by the large density change across the wrinkled shock result in the multidimensional instability. The instabilities have a large transverse length but oscillate and evolve on a short time, comparable with the transit time of a fluid particle through the detonation. The coupling with the acoustic waves is a stabilizing factor, dominant at short wavelengths for assuring a suitably large ratio of transverse wavelength to detonation thickness for instability. Even when the detonation is stable to planar disturbances, so that there is a range of stability at long wavelengths, it is shown that there always exists an intermediate range of wavelengths for which the detonation is unstable. This is true even when heat-release rates are entirely independent of temperature, corresponding to detonations that are very stable to planar disturbances. A sufficiently large temperature sensitivity modifies the multidimensional instability. The general chemical kinetics adopted, having different temperature sensitivities for induction and for heat release, the former possibly large, is of the same character as that previously developed by the first authors for describing the planar stability and nonlinear oscillations of galloping detonations but differs from one-step activation-energy asymptotics, which produces nonphysical, spurious instability under all conditions. From the present extension to multiple dimensions with moderate overdrive, inferences are drawn concerning differences that arise for strong overdrive and when Chapman–Jouguet conditions are approached.