Green’s function analysis of nonlinear thermoacoustic effects under the influence of noise in a combustion chamber

This paper presents an analytical study to predict the effect of noise in a thermoacoustic system. The starting point of our analysis is the acoustic analogy equation with two source terms: one represents the fluctuating heat release rate, and the other one represents the noise. The heat release rate is nonlinear and modelled by a generalised-law with amplitude-dependent time-lag and coupling coefficients. A Green’s function approach is used to convert the acoustic analogy equation (a PDE) into an integral equation. An essential element in this approach is the tailored Green’s function of the combustion chamber. We calculate this analytically for a 1-D combustion chamber with general end conditions and a non-uniform mean temperature. The integral equation is then used for predictions in the time-domain and frequency-domain. We focus on the following three phenomena: transient oscillations, noise-induced triggering, and hysteresis. Our predictions are in line with experimental observations reported in earlier studies: (1) Noise speeds up the time it takes a transient oscillation with growing amplitude to reach its limit cycle. (2) Noise can launch a linearly stable system into an unstable state. (3) Noise reduces the width of a hysteresis zone. Both white noise and pink noise are considered; pink noise is found to be more effective.