Effects of multiplicative and additive colored noises on the stability of a simplified thermoacoustic combustor

Thermoacoustic oscillations are favorable in thermoacoustic engines, but unwanted in many propulsion systems, due to the induced detrimental outcomes to the combustors. To prevent or eliminate the onset of large-amplitude thermoacoustic oscillations, a better understanding of its mechanism is necessary. In the present work, the effects of two different types of background colored noises on the stability of an open-ended standing wave thermoacoustic system are studied theoretically and numerically. A multiplicative and an additive colored noises are considered, which act as a frequency disturbance and a combustion-related disturbance to acoustic waves, respectively. First, a noisy nonlinear thermoacoustic system with a confined premixed flame is modeled. The unsteady heat release rate of the premixed flame to acoustic pressure is described by a fifth-order polynomial. Then an expansion of Galerkin series is applied to decouple the acoustic waves in the spatial and time domains. Stochastic averaging is utilized, and analytical stationary probability density functions (PDFs) of acoustic pressure amplitudes are obtained. Through qualitative changes of PDFs, the stochastic P-bifurcations are discussed. To validate the theoretical predictions of this thermoacoustic system driven by the multiplicative and additive colored noises, a Monte Carlo simulation is conducted. By comparison, the analytical PDFs agree well with the numerical PDFs. Several important results have been obtained: (1) for the two types of colored noises, the noise intensity D and the correlation time τ have significant but opposite effects on the bimodal width and stability range. Furthermore, as the correlation time τ increases, the width of the bimodal region for the noise intensity is increased, and a larger noise intensity is needed to destabilize the thermoacoustic system; (2) compared with the multiplicative colored noise, the additive colored noise impacts the stability of the system in a limited range of operating conditions; (3) by varying the parameters of colored noise, two different types of stochastic P-bifurcations have been observed.