Theory of unstable one-dimensional detonations

Abstract

The stability of one-dimensional piston-supported gaseous detonations is examined both numerically and analytically. Numerical calculations are presented for a one-step, first-order, irreversible reaction, obeying an Arrhenius rate expression. An approximate linearized stability theory is then developed for the case of high activation-energy reactions, and the mechanism of instability is identified. The analysis has demonstrated that the interaction between the irreversible temperature fluctuations and the finite reaction zone induces an oscillating energy-source field, which then leads to shock perturbations and thereby the temperature fluctuations. The low- and the high-frequency modes previously reported by Fickett and his coworkers are but two of the unstable resonance modes of this oscillatory system. However, the theory is also capable of predicting all the succeeding modes. Both oscillation periods and amplification (or attenuation) rates are obtained, and the results agree well with the findings of both the numerical calculations and the exact linearized stability analysis. The predicted periods also shown well with those observed in blunt-body flow experiments. It is also shown that at very high frequencies, the flow is stable—a result which invalidates the predictions of Zaidel on the basis of a square-wave model.