Stability of hypersonic reacting stagnation flow of a detonatable gas mixture by dynamical systems analysis

The stability characteristics of the reacting hypersonic flow of the fuel/oxidizer mixture in the stagnation region of a blunt body are studied. The conditions for oscillations of the combustion front are assumed to be determined mainly by the flow conditions at the stagnation region. The density at the stagnation region is assumed to be constant at hypersonic flow conditions. By assuming a simplified flow model, the time dependent flow equations, including the heat addition due to the chemical reactions, are reduced to a second-order nonlinear differential equation for the instantaneous temperature. The solutions are analyzed assuming a one-step chemical reaction with zero-order and first-order processes using dynamical systems methods. These methods are used to determine the stability boundaries in terms of the flow and chemical reaction parameters. It is shown that the zero-order reaction has nonperiodic solutions that may lead to explosion whereas the first-order and higher-order reactions may have periodic solutions indicating oscillations. The zero-order analysis also reaffirms the requirements for a minimum size blunt body for the establishment of a detonation (in agreement with classical detonation theory) and the first-order analysis indicates a minimum body size for establishment of oscillations. The oscillation frequencies are calculated using the small perturbation approximation for the temperature oscillations. These frequencies are compared with results from published data on spheres and hemisphere cylindrical bodies fired into hydrogen-oxygen and acetylene oxygen mixtures. Very good agreement is found between the measured and calculated results.