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A Method for Estimating the Noise Level of Unstable Combustion Based on the Flame Describing Function

Practical combustion is often coupled by the system acoustics and under some conditions the process may become unstable giving rise to oscillations of finite amplitude and to discrete tone noise radiation. The objective of this investigation is to model the nonlinear features of unstable combustion to predict limit cycle amplitudes and thus get an estimation of the radiated noise level under such regimes of operation. The present study is based on an analysis derived in a companion paper (Noiray et al. (2008)) which contains a complete nonlinear description of the thermo-acoustic coupling encountered in an unconfined reaction zone layout. Theoretical predictions are successfully compared with a set of measurements. It is shown that nonlinear phenomena such as frequency shifts, triggering or hysteresis may be suitably predicted by an analytical model based on the flame describing function (FDF). The objective of the present paper is to make use of this method to investigate nonlinear acoustic-combustion dynamics in a confined geometry comprising an upstream manifold, an injector and a flame tube. The flame is identified as the main nonlinear element in the system and its response to perturbations is characterized in terms of a describing function which assumes that the gain and phase depend on the amplitude level of the input. A frequency domain analysis is then carried out by solving the dispersion relation for a range of amplitudes. This yields growth rates and eigenfrequencies, which depend on the amplitude level of perturbations impinging on the flame. The growth rates and eigenfrequencies are determined for a wide range of upstream manifold lengths while keeping equivalence ratio and mass flow rate constant. For certain values of this parameter one finds a positive growth rate for vanishingly small amplitude levels indicating that the system is linearly unstable. The growth rate then changes as the amplitude is increased and eventually vanishes for a finite amplitude indicating the existence of a limit cycle. For other values of the length, the growth rate is initially negative, becomes positive for a finite amplitude and drops to zero for a higher level. This indicates that the system is linearly stable but nonlinearly unstable. Using calculated growth rates it is effectively possible to predict amplitudes of acoustic pressure oscillations in the system when it operates on a limit cycle and thus determine the noise level generated under such conditions.