LOW-FREQUENCY 2-DIMENSIONAL LINEAR INSTABILITY OF PLANE DETONATION

An analytical dispersion relation describing the linear stability of a plane detonation wave to low-frequency two-dimensional disturbances w ith arbitrary wavenumbers is derived using a normal mode approach and a combination of high activation energy and Newtonian limit asymptotic s, where the ratio of specific heats gamma --> 1. The reaction chemist ry is characterized by one-step Arrhenius kinetics. The analysis assum es a large activation energy in the plane steady-state detonation wave and a characteristic linear disturbance wavelength which is longer th an the fire-zone thickness. Newtonian limit asymptotics are employed t o obtain a complete analytical description of the disturbance behaviou r in the induction zone of the detonation wave. The analytical dispers ion relation that is derived depends On the activation energy and exhi bits favourable agreement with numerical solutions of the full linear stability problem for low-frequency one- and two-dimensional disturban ces, even when the activation energy is only moderate. Moreover, the d ispersion relation retains vitally important characteristics of the fu ll problem such as the one-dimensional stability of the detonation wav e to low-frequency disturbances for decreasing activation energies or increasing overdrives. When two-dimensional oscillatory disturbances a re considered, the analytical dispersion relation predicts a monotonic increase in the disturbance growth rate with increasing wavenumber, u ntil a maximum growth rate is reached at a finite wavenumber. Subseque ntly the growth rate decays with further increases in wavenumber until the detonation becomes stable to the two-dimensional disturbance. In addition, through a new detailed analysis of the behaviour of the pert urbations near the fire front, the present analysis is found to be equ ally valid for detonation waves travelling at the Chapman-Jouguet velo city and for detonation waves which are overdriven. It is found that i n contrast to the standard imposition of a radiation or piston conditi on on acoustic disturbances in the equilibrium zone for overdriven wav es, a compatibility condition on the perturbation jump conditions acro ss the fire zone must be satisfied for detonation waves propagating at the Chapman-Jouguet detonation velocity. An insight into the physical mechanisms of the one- and two-dimensional linear instability is also gained, and is found to involve an intricate coupling of acoustic and entropy wave propagation within the detonation wave.