One-dimensional overdriven detonations with branched-chain kinetics

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The dynamics of time-dependent, planar propagation of gaseous detonations is addressed on the basis of a three-step chemistry model that describes branched-chain processes. Relevant nondimensional parameters are the ratio of the heat release to the thermal enthalpy at the Neumann state, the nondimensional activation energies for the initiation and branching steps, the ratio of the branching time to the initiation time and the ratio of the branching time to the recombination time. The limit of strong overdrive is considered, in which pressure remains constant downstream from the leading shock in the first approximation, and the ratio of specific heats γ is taken to be near unity. A two-term expansion in the strong overdrive factor is introduced, and an integral equation is derived describing the nonlinear dynamics and exhibiting a bifurcation parameter, the reciprocal of the product of (γ−1), the nondimensional heat release and the nondimensional branching activation energy, with an acoustic correction. A stability analysis shows that, depending on values of the parameters, either the mode of lowest frequency or a mode of higher frequency may be most unstable. Numerical integrations exhibit different conditions under which oscillations die, low-frequency oscillations prevail, high-frequency oscillations prevail, highly nonlinear oscillations persist, or detonation failure occurs. This type of parametric analysis is feasible because of the relative simplicity of the model, which still is more realistic than a one-step, Arrhenius chemical approximation. In particular, by addressing the limit of slow radical recombination compared with branching, explicit results are derived for the critical value of the bifurcation parameter, involving the ratio of the recombination time to the induction time. The results help to clarify the general nature of one-dimensional detonation instability and provide simplifications that can be employed in efficiently relating gaseous detonation behavior to the true underlying chemistry.