The nonlinear stability of a pulsating detonation wave driven by a three-step chain-branching reaction is studied. The reaction model consists sequentially of a chain-initiation step and a chain-branching step, both governed by Arrhenius kinetics, followed by a temperature-independent chain-termination step. The model mimics the essential dynamics of a real chain-branching chemical system, but is sufficiently idealized that a theoretical analysis of the instability is possible. We introduce as a bifurcation parameter the chain-branching cross-over temperature (TB), which is the temperature at which the chain-branching and chain-termination rates are equal. In the steady detonation structure, this parameter controls the ratio of the chain-branching induction length to the length of the recombination zone. When TB is at the lower end of the range studied, the steady detonation structure, which is dominated by the temperature-independent recombination zone, is found to be stable. Increasing TB increases the length of the chain-branching induction region relative to the length of the recombination zone, and a critical value of TB is reached where the detonation becomes unstable, with the detonation shock pressure evolving as a single-mode low-frequency pulsating oscillation. This single-mode nonlinear oscillation becomes progressively less stable as TB is increased further, persisting as the long-term dynamical behaviour for a significant range of TB before eventually undergoing a period-doubling bifurcation to a two-mode oscillation. Further increases in TB lead to a chaotic behaviour, where the detonation shock pressure history consists of a sequence of substantive discontinuous jumps, followed by lower-amplitude continuous oscillations. Finally, for further increases in TB a detonability limit is reached, where during the early onset of the detonation instability, the detonation shock temperature drops below the chain-branching cross-over temperature causing the wave to quench.

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A detailed investigation of the hydrodynamic stability to transverse linear disturbances of a steady, one-dimensional detonation in an ideal gas undergoing an irreversible, unimolecular reaction with an Arrhenius rate constant is conducted via a normal-mode analysis. The method of solution is an iterative shooting technique which integrates between the detonation shock and the reaction equilibrium point. Variations in the disturbance growth rates and frequencies with transverse wavenumber, together with two-dimensional neutral stability curves and boundaries for all unstable low- and higher frequency modes, are obtained for varying detonation bifurcation parameters. These include the detonation overdrive, chemical heat release and reaction activation energy. Spatial perturbation eigenfunction behaviour and phase and group velocities are also obtained for selected sets of unstable modes. Results are presented for both Chapman–Jouguet and overdriven detonation velocities. Comparisons between the earlier pointwise determination of stability and interpolated neutral stability boundaries obtained by Erpenbeck are made. Possible physical mechanisms which govern the wavenumber selection underlying the initial onset of either regular or irregular cell patterns are also discussed.

The one-and two-dimensional linear stability of a plane detonation wave characterized by a one-step Arrhenius chemical reaction is studied for large activation energies using a normal mode analysis based on the approach of Lee and Stewart [J. Fluid Mech., 216 (1990), p. 103]. It is shown that for one-dimensional disturbances, a low-frequency oscillatory mode present for moderate activation energies bifurcates into a slowly evolving nonoscillatory mode and a fasterevolving nonoscillatory mode as the activation energy is increased. It is also shown that for large activation energies, the stability spectrum consists of a large number of unstable one-dimensional modes, as predicted by the asymptotic analysis of Buckmaster and Neves [Phys. Fluids, 31 (1988), p. 3571], possessing a maximum growth rate at very high frequencies. For nonplanar disturbances, it is found that as the wavenumber increases, the two nonoscillatory modes present for zero wavenumber collapse into a single oscillatory unstable mode before stabilizing at a short wavelength. the activation energy is increased. Equivalently, for a fixed activation energy the detonation becomes more unstable as the degree of overdrive is decreased. In addition, for a fixed activation energy and degree of overdrive, high-frequency roots are in general found to be stable. However, Lee and Stewart [17] did not show results for large enough activation energies to ascertain if the latter trend continued. Namah et al. [19] have also conducted an exact stability analysis based on a normal modes approach for one-dimensional disturbances but using a pseudospectral collocation technique.Numerical simulations of unstable detonation waves have until recently been hampered by the vastly disparate spatial scales in the detonation structure. Consequently, early simulations had been able to resolve only the early stages of the onset of the instability and not the long-term behavior of the fully nonlinear instability. For example, Fickett and Wood [15], using a characteristic method, and Abouseif and Toong [1], using a finite-difference solution, have investigated the one-dimensional pulsating instability of overdriven detonation waves. Their results were found to agree well with the predictions of the onset of instability predicted by Erpenbeck [10,11]. Recently, however, Bourlioux, Majda, and Roytburd [4] have conducted a series of numerical studies into one-dimensional unstable detonation propagation using high-order finitevolume techniques and a numerical mesh refinement strategy specifically implemented for following detonation flows. They have produced high resolution calculations which agree remarkably well with the linear stability predictions of Lee and Stewart [17] and their own weakly nonlinear analysis describing the transition of a stable detonation to an unstable detonation through a Hopf bifurcation [4]. Bourlioux and Majda [5] have also obtained two-dimensional unstable detonation wave simulations through this method. It appears that regular cellular structures are ...