Combustion instability due to the nonlinear interaction between sound and flame

It has been observed in experiments that significant levels of sound may be produced when a curved flame propagates downwards along a tube in a gravity field. In this paper, we present a mathematical description of this acoustic amplification process, which represents a simple form of combustion instability. First, based on the large-activation-energy and small-Mach-number assumptions, a general asymptotic formulation is derived, in which the nature of flame–sound coupling is brought out explicitly. This framework is then employed to study the weakly nonlinear coupling between a Darrieus–Landau (D-L) instability mode of the flame and an acoustic mode of the tube, which is the main mechanism for sound generation in the experiments. In order to provide a somewhat unified description, the linear coupling via the direct pressure effect has also been included in our analysis. A set of coupled equations which govern the evolution of the acoustic and D-L modes was derived. The solutions show that the nonlinear coupling leads to very rapid amplification of sound. After reaching an appreciable level, the sound inhibits the flame, causing the latter to flatten. The sound then saturates at an almost constant level, or continues to grow at a smaller rate owing to the pressure effect. The above theoretical predictions are in good qualitative agreement with experiments. The present study also considered the influence of weak vortical disturbances in the oncoming flow. It is shown that certain components in these perturbations may form resonant triads with the acoustic and D-L modes, thereby providing an additional coupling mechanism.