The multi-dimensional stability of weak-heat-release detonations

The stability of an overdriven planar detonation wave is examined for a one -step Arrhenius reaction model with an order-one post-shock temperature-sca led activation energy theta in the limit of a small post-shock temperature- scaled heat release beta. The ratio of specific heats, gamma, is taken such that (gamma - 1) = O(1). Under these assumptions, which cover a wide range of realistic physical situations, the steady detonation structure can be e valuated explicitly, with the reactant mass fraction described by an expone ntially decaying function. The analytical representation of the steady stru cture allows a normal-mode description of the stability behaviour to be obt ained via a two-term asymptotic expansion in beta. The resulting dispersion relation predicts that for a finite overdrive f, the detonation is always stable to two-dimensional disturbances. For large overdrives, the identific ation of regimes of stability or instability is found to depend on a choice of distinguished limit between the heat release beta and the detonation pr opagation Mach number D*. Regimes of instability are found to be characteri zed by the presence of a single unstable oscillatory mode over a finite ran ge of wavenumbers.