Structure and stability of weak–heat–release detonations for finite Mach numbers

Abstract

The stability of an overdriven detonation wave for a one–step Arrhenius reaction in an ideal gas is examined in the limits of a finite detonation Mach number, small modified heat release γ−1)β and large activation energy θ, limits that are relevant to the generation of the regular cellular detonation structures observed experimentally in highly diluted mixtures. Here, γ is the specific heat ratio and β is the formation energy. The limiting case, β = 0, corresponds to a finite Mach number reactionless shock, which in an ideal gas is always stable. Nevertheless, for small but finite values of (γ−1)β, we will show that detonation instabilities can arise when perturbations in the reaction rate generated by a small disturbance at the shock front are sufficient in magnitude to balance the corresponding acoustic fluctuations that are also generated. These acoustic disturbances are also those that govern the stability of a reactionless shock (β = 0). Such reaction–rate perturbations are only generated for sufficiently large activation energies, and it is the precise magnitude of the activation energy that leads to detonation instability that we identify here for various choices of β and γ. In particular, subject to the restriction on the modified heat release (γ−1)β ≪ 1, three new weakly exothermic ordered limits are identified that are relevant to detonation instability: (1) θ(γ−1) = O (1) and (γ−1) = O (1); (2) θ(γ−1) = O (1) and (γ−1) = o (1); and (3) θ(γ−1) ≫ 1 and (γ−1) = o (1).