Dynamical states of thermo-acoustic system with respect to frequency-phase relationship based on probabilistic oscillator model

We perform laboratory-scale experiments in a swirl combustor by reducing the equivalence ratio for two cases involving slightly different inlet air flow rates. Reduction in equivalence ratio at a constant air flow results in the combustor transiting from stable to unstable combustion, i.e., in exciting acoustics within the combustor. The dynamical nature of pressure oscillations traverses through stable, type 2 intermittency and beat oscillations. The beat oscillations are seen to fluctuate at a single frequency and hence suggestive of a process involving spatiotemporal variations in the phase of the driver (i.e., heat release rate fluctuations). This is understood by means of a reduced-order model, where the driver is considered to be an ensemble of phase oscillators with time delays and a probabilistic distribution of natural frequencies, similar to Kuramoto oscillator. The acoustic field is modeled as a van der Pol-Duffing system, with natural frequency equal to the duct acoustic mode. The coupling between the oscillators is varied based on the physical premise of Rayleigh criterion. The coupled system is observed to qualitatively and quantitatively match the pressure data obtained from experiments. Insights into various conditions illustrate the role of mean and fluctuating instantaneous frequency among the phase oscillators in determining the modeled pressure oscillations. By quantifying the extent of fluctuating frequency coupling among the phase oscillators, it is observed that beat oscillations have high correlation with lower deviation compared to intermittency and stable oscillations.