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The lock-on of the frequency of acoustic oscillations to the frequency of the vortex shedding in a reacting flow in a backward-facing step combustor is demonstrated using a reduced-order model. Individual models for the flow, flame and acoustic models are developed. A two-dimensional flow field with growth and shedding of vortices is modeled using a potential flow model derived via a conformal mapping of the step geometry. Combustion and acoustic models are derived using thermo-diffusive equations and the one-dimensional Galerkin method, respectively. A modulation of heat release rate fluctuations is thus enabled by vortices advecting past the flame, overcoming the time and space-localized assumptions in vortex kicked oscillator models. Comparisons with results from kicked oscillator models show a better prediction of the dominant frequencies during stable and unstable operations. A Reynolds number sweep is performed to reveal the transition from stable to unstable operation and thus the onset of instability. A shift in the dominant frequency of pressure fluctuations from the natural duct acoustic mode to the natural vortex shedding mode is observed, indicative of a lock-on. The results compare well with past experimental and computational data for similar geometry and flow conditions.

A Reduced-order Model for Lock-on via Vortex-combustion-acoustic Closed-loop Coupling in A Step Combustor