One-dimensional overdriven detonations with branched-chain kinetics

The dynamics of time-dependent, planar propagation of gaseous detonations i s addressed on the basis of a three-step chemistry model that describes bra nched-chain processes. Relevant nondimensional parameters are the ratio of the heat release to the thermal enthalpy at the Neumann state, the nondimen sional activation energies for the initiation and branching steps, the rati o of the branching time to the initiation time and the ratio of the branchi ng time to the recombination time. The limit of strong overdrive is conside red, in which pressure remains constant downstream from the leading shock i n the first approximation, and the ratio of specific heats gamma is taken t o be near unity. A two-term expansion in the strong overdrive factor is int roduced, and an integral equation is derived describing the nonlinear dynam ics and exhibiting a bifurcation parameter, the reciprocal of the product o f (gamma -1), the nondimensional heat release and the nondimensional branch ing activation energy, with an acoustic correction. A stability analysis sh ows that, depending on values of the parameters, either the mode of lowest frequency or a mode of higher frequency may be most unstable. Numerical int egrations exhibit different conditions under which oscillations die, low-fr equency oscillations prevail, high-frequency oscillations prevail, highly n onlinear oscillations persist, or detonation failure occurs. This type of p arametric analysis is feasible because of the relative simplicity of the mo del, which still is more realistic than a one-step, Arrhenius chemical appr oximation. In particular, by addressing the limit of slow radical recombina tion compared with branching, explicit results are derived for the critical value of the bifurcation parameter, involving the ratio of the recombinati on time to the induction time. The results help to clarify the general natu re of one-dimensional detonation instability and provide simplifications th at can be employed in efficiently relating gaseous detonation behavior to t he true underlying chemistry. (C) 2001 American Institute of Physics.