On the dynamics of pulsating detonations

Abstract

To understand the circumstances in which a pulsatingdetonation wave may prefer a low-frequency mode ofoscillation, and the implications this has formathematical modelling, the dynamics of a detonation waveare studied when the underlying linear stability spectrumconsists of at most two unstable modes. One mode α1 has a period much larger than the time-scaleof particle passage through the half-reaction length inthe steady one-dimensional detonation; the second α2 has a smaller period than α1. Thequestion addressed in this paper concerns the long-timebehaviour of the pulsating detonation in the presence ofthe unstable modes α1 and α2. Westudy two general scenarios related to a crossing of theneutral stability boundaries traced out by α1 and α2. In all cases where the mode α1 is unstable, the pulsating detonation emerges with alow-frequency, large-amplitude oscillation, regardlessof the relative growth rates of α1 and α2. Only in one case, where thehigher-frequency mode α2 is alone unstable, isthe long-time nonlinear pulsation of high frequency. Inthis case, the amplitude of the oscillation issignificantly smaller than that observed in thelow-frequency oscillations. In all cases, the finalperiod of the nonlinear oscillation is closely related tothat of the relevant underlying linear mode.