Nonlinear thermoacoustic mode synchronization in annular combustors

Abstract

Nonlinear coupling between azimuthal and axisymmetric modes in annular combustors is studied analytically. Based on the thermoacoustic wave equation, a model featuring three nonlinearly coupled oscillators is derived. Two oscillators represent the dynamics of an azimuthal mode, and the third accounts for the axisymmetric mode. A slow-time system for the evolution of the mode amplitudes and phases is obtained through the application of the method of averaging. The averaged system is shown to accurately reproduce the solutions of the full oscillator model. Analysis of this five-dimensional dynamical system shows that a standing azimuthal mode may synchronize with an axisymmetric mode, provided that their individual resonance frequencies and growth rates are similar. This phase-coupled two-mode oscillation corresponds to the so-called slanted mode, observed in recent experiments involving an annular model combustion chamber. Quantitative conditions for the occurrence of mode synchronization are derived in terms of the growth rate ratio and a frequency detuning parameter. The analysis results are found to be consistent with experimental observations of the slanted mode.