A kinematic model of a ducted flame

A premixed ducted flame, burning in the wake of a bluff-body flame-holder, is considered. For such a flame, interaction between acoustic waves and unsteady combustion can lead to self-excited oscillations. The concept of a time-invariant turbulent flame speed is used to develop a kinematic model of the response of the flame to flow disturbances. Variations in the oncoming flow velocity at the flame-holder drive perturbations in the flame initiation surface and hence in the instantaneous rate of heat release. For linear fluctuations, the transfer function between heat release and velocity can be determined analytically from the model and is in good agreement with experiment across a wide frequency range. For nonlinear fluctuations, the model reproduces the flame surface distortions seen in schlieren films. Coupling this kinematic flame model with an analysis of the acoustic waves generated in the duct by the unsteady combustion enables the time evolution of disturbances to be calculated. Self-excited oscillations occur above a critical fuel-air ratio. The frequency and amplitude of the resulting limit cycles are in satisfactory agreement with experiment. Flow reversal is predicted to occur during part of the limit-cycle oscillation and the flame then moves upstream of the flame-holder, just as in experimental visualizations. The main nonlinearity is identified in the rate of heat release, which essentially 'saturates' once the amplitude of the velocity fluctuation exceeds its mean. We show that, for this type of nonlinearity, describing function analysis can be used to give a good estimate of the limit-cycle frequency and amplitude from a quasi-nonlinear theory.