APPLICATION OF MULTIPOLE EXPANSIONS TO SOUND GENERATION FROM DUCTED UNSTEADY COMBUSTION PROCESSES

Predicting and controlling the behavior of oscillatory combustion systems requires an understanding of the interactions between the periodic combustion and acoustic processes. Due to the complexity of these interactions, a number of past analyses of these problems have assumed a “concentrated”, infinitely thin, combustion region. Although such treatments of the combustion process as a lumped element are attractive because of their simplicity, this paper shows that they may produce significant errors. It is further shown that such errors can be minimized by use of multipole expansion techniques similar to those used in acoustic radiation problems. Specifically, this paper develops a formalism for describing the combustion region as a series of lumped parameters whose magnitudes are proportional to ascending powers of the ratio L/λ, where L and λ are the length of the combustion region and the acoustic wavelength respectively. In the limit as L/λ goes to zero, only the first term of this expansion is significant and the “concentrated” combustion approximation is recovered. As L/λ increases, additional parameters (e.g., higher order terms) are needed to accurately describe the effect of the combustion process on the acoustic oscillations and must be included in the analysis. The paper closes with example calculations showing that accuracy in modelling combustion–acoustics interactions can be significantly increased by implementation of the developed technique.