A Nonlinear Acoustic–Flame Synchronization Model for Thermoacoustic Instability in a Bluffbody-Stabilized Combustor

A nonlinear coupled-oscillator model is developed that can predict the various stages of synchronization between the acoustic and heat-release oscillations in the lead-up to thermoacoustic instability in a dump combustor with a bluffbody-anchored flame. The flame is perturbed by vortices shed from the shear layer formed at the combustor dump plane. The resulting heat-release-rate fluctuations generate acoustic waves that travel upstream and interact with the shear layer, thereby modulating the vortex shedding process and closing the feedback loop. Coupled nonlinear equations governing the temporal evolution of pressure and heat-release-rate fluctuations are derived and evolved along with an equation for the build-up of circulation at the dump plane, as well as an equation for vortex advection. The effects of vortex impingement on the flame are modeled as a localized, instantaneous source term in the flame oscillator equation. With the mean flow velocity $\Bar{u}$ as the control parameter, the nonlinear model is applied to investigate the acoustic--flame--vortex interactions. For smaller $\Bar{u}$, chaotic combustion noise is observed with essentially no synchronization between the pressure and heat-release-rate fluctuations, $p'$ and $\dot{q}'$, respectively. As $\Bar{u}$ is increased, intermittent bursts of periodic fluctuations are seen in $p'$ and $\dot{q}'$, which is followed by a weakly periodic state with only frequency synchronization. Eventually, when $\Bar{u}$ exceeds a critical value, limit-cycle oscillations are seen with both frequency and amplitude synchronization between $p'$ and $\dot{q}'$ (also known as generalized synchronization). Thus, the model qualitatively captures the successive stages of acoustic--flame coupling seen in the experiments of \citet{pawar2017}. Furthermore, the derived nonlinear equations accurately predict the natural frequencies of the pressure and heat-release oscillators, as well as the lock-on frequency of the two oscillators with vortex shedding.