Stochastic dynamics of vortex-acoustic lock-in: stochastic bifurcation and resonance

We perform a theoretical study to understand vortex-acoustic lock-in in the presence of an additive Gaussian white noise using a lower-order model. The acoustic field is realized as an external harmonic excitation. Frequency, harmonic excitation and noise amplitudes are varied over ranges encountered in practical and lab-scale combustors. Probability density functions (PDFs) for shedding time periods and phase instants are obtained in the Fokker-Planck framework. Unlike the general scenario, where stochastic bifurcation is identified by a qualitative change in the stationary PDFs, in this case, stochastic lock-in is identified by a qualitative change in the spectrum of the transition probability matrix. The effect of the noise is to reduce the extent of lock-in while preserving the underlying deterministic dynamical features. Although various orders of lock-in are identified, 1 :1 stochastic lock-in with the excitation frequency close to the natural vortex shedding frequency is found to be the most favourable condition for combustion instability to occur. We show that lock-in can be accompanied by both instability and amplitude suppression. Stochastic resonance is also observed; however, its contribution to instability is marginal.