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Theory of Weakly Unstable Multi-Dimensional Detonation

An asymptotic analysis of the multi-dimensional detonation instability has been developed in the limit that the ratio q of the reaction heat release over the enthalpy at the leading shock is much smaller than unity. The leading order solution corresponds to a wave equation describing the dynamics of the inert shock wave. By considering the next order destabilizing effects related to the heat release fluctuation and the non-uniform distribution of the stationary solution, an expression of the dispersion relation. which is valid in the vicinity of the bifurcation limits, has been obtained. The present work extends our previous analysis (He I996a) in the Newtonian limit to general situation for q≪1. It is found that the dispersion relation so determined is identical to that presented in (He 1996a), which was obtained in the Newtonian limit and q≪1 by carrying out an asymptotic analysis to the second order. The theoretical results for the growth rate and the bifurcation limits agree well with the numerical solutions of the exact linear instability problem. The physical mechanisms involved in the cellular detonations have been discussed.