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-scaled activation energy θ in the limit of a small post-shock temperature-scaled heat release β. The ratio of specific heats, γ, is taken such that (γ - 1) = O(1). Under these assumptions, which cover a wide range of realistic physical situations, the steady detonation structure can be evaluated explicitly, with the reactant mass fraction described by an exponentially decaying function. The analytical representation of the steady structure allows a normal-mode description of the stability behaviour to be obtained via a two-term asymptotic expansion in β. The resulting dispersion relation predicts that for a finite overdrive f, the detonation is always stable to two-dimensional disturbances. For large overdrives, the identification of regimes of stability or instability is found to depend on a choice of distinguished limit between the heat release β and the detonation propagation Mach number D*. Regimes of instability are found to be characterized by the presence of a single unstable oscillatory mode over a finite range of wavenumbers.