Nonlinear thermoacoustics: flames on the edge of chaos

Abstract

Thermoacoustic oscillations occur when a flame is confined within an acoustic cavity, such as a combustion chamber. Their amplitude grows if the oscillating flame releases more heat at times of higher pressure and less heat at times of lower pressure. The phase between the pressure and the additional heat release is critical. It can vary from cycle to cycle, resulting in quasiperiodic, multi-periodic, or chaotic oscillations, as observed in experiments and numerical simulations.

Simulations also reveal a multitude of periodic and quasiperiodic unstable attractors, which attract the system in many directions in phase space and repel in one direction. The system's state can pass within the vicinity of several unstable attractors before arriving at a stable attractor, which has similar features to bypass transition to turbulence in hydrodynamics. Sometimes small differences in initial states lead to diverging paths in phase space and different final states.

In some linearly stable thermoacoustic systems, thermoacoustic oscillations can be triggered by a small pulse. A simple thermoacoustic system containing a stable fixed point, an unstable periodic solution and a stable periodic solution is examined. The `minimal seed' is found with nonlinear adjoint looping. Growth to the stable periodic solution is shown to exploit non-normal transient growth around the unstable periodic solution, rather than non-normal transient growth away from the fixed point.