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.
Thermal-Diffusive Instability of Premixed Flames for a Simple Chain-Branching Chemistry Model with Finite Activation Energy