2019 - Nonlinear dynamics of closely spaced thermoacoustic modes in the presence of noise
Abstract: Practical combustion systems can have linearly unstable axial thermoacoustic modes that oscillate at close frequencies. These modes are nonlinearly coupled and their interaction can result in novel dynamics when the frequency spacing is much smaller compared to the mean frequency of oscillations. In this study, we consider frequency spacing of up to 10 percent of the mean frequency. In a recent paper, Acharya et al. (Combustion and Flame 2018) showed that a mode with larger growth rate could be suppressed depending on the frequency spacing between the modes. In this paper, we extend the work by Acharya et al. to include the effects of noise in the system and we explore how the deterministic dynamics change in the presence of noise. Noise can alter the behavior of the system significantly by changing the average amplitudes and the stability of the limit cycle oscillations. In order to study these noise-induced features, we derive the stochastic evolution equations for the amplitudes and phases of the two modes using an averaging procedure. Then, we analyze the dynamics of the system in its state space and numerically obtain the probability density functions of amplitudes and phases for a set of parameters. For sufficiently high noise intensity, we observe that both modes coexist and the system switches between them in an irregular manner when they evolve in time.