Pulsating instability of detonations with a two-step chain-branching reaction model: theory and numerics

The nonlinear dynamics of Chapman–Jouguet pulsating detonations are studied both numerically and asymptotically for a two-step reaction model having separate induction and main heat release layers. For a sufficiently long main heat release layer, relative to the length of the induction zone, stable one-dimensional detonations are shown to be possible. As the extent of the main reaction layer is decreased, the detonation becomes unstable, illustrating a range of dynamical states including limit-cycle oscillations, period-doubled and four-period solutions. Keeping all other parameters fixed, it is also shown that detonations may be stabilized by increasing the reaction order in the main heat release layer. A comparison of these numerical results with a recently derived nonlinear evolution equation, obtained in the asymptotic limit of a long main reaction zone, is also conducted. In particular, the numerical solutions support the finding from the analytical analysis that a bifurcation boundary between stable and unstable detonations may be found when the ratio of the length of the main heat release layer to that of the induction zone layer is O(1/ε), where ε (?1) is the inverse activation energy in the induction zone.