The effects of different types of noise on thermoacoustic systems using a Green's function approach

In this paper, we take advantage of using a Green's function approach as an analytical tool to study linear and nonlinear aspects of thermoacoustic systems. Green's function is defined as the impulse response of a system, it has a clear physical meaning in combustion systems, and it provides a fast and flexible tool to predict thermoacoustic instabilities in both time and frequency domains. In this study, we consider a Rijke tube, which houses two sources: a heat source and an external noise source. The heat source is modelled by an amplitude-dependent nτ-law, the noise source is assumed to emit two types of noise (pink noise or white noise). We use the Green's function approach to derive an integral equation for the acoustic field in the Rijke tube, and we also derive an algebraic equation for the thermoacoustic eigenfrequency. Both equations are validated. The results that we found, show that the presence of noise (whether pink or white noise) results in "triggering" an instability and in accelerating the growth of the amplitude. These effects become more pronounced as the level of noise increases. The influence of pink noise is stronger than that of white noise. We also studied how the hysteresis behavior (a nonlinear effect in dynamical systems) is affected by the noise, using the heat source position and heater power as bifurcation parameters. Our study reveals that the width of the bistable region decreases as the strength of noise increases.