Phase-reduction analysis of periodic thermoacoustic oscillations in a Rijke tube

Abstract

Phase-reduction analysis captures the linear phase dynamics with respect to a limit cycle subjected to weak external forcing. We apply this technique to study the phase dynamics of the self-sustained oscillations produced by a Rijke tube undergoing thermoacoustic instability. Through the phase-reduction formulation, we are able to reduce these dynamics to a scalar equation for the phase, which allows us to efficiently determine the synchronisation properties of the system. For the thermoacoustic system, we find the conditions for which $m:n$ frequency locking occurs, which sheds light on the mechanisms behind asynchronous and synchronous quenching. We also reveal the optimal placement of pressure actuators that provide the most efficient route to synchronisation.