An analytical model for azimuthal thermoacoustic modes in an annular chamber fed by an annular plenum

This study describes an analytical method for computing azimuthal modes due to flame/acoustics coupling in annular combustors. It is based on a quasi-one-dimensional zero-Mach-number formulation where N burners are connected to an upstream annular plenum and a downstream chamber. Flames are assumed to be compact and are modeled using identical flame transfer function for all burners, characterized by an amplitude and a phase shift. Manipulation of the corresponding acoustic equations leads to a simple methodology called ANR (annular network reduction). It makes it possible to retain only the useful information related to the azimuthal modes of the annular cavities. It yields a simple dispersion relation that can be solved numerically and makes it possible to construct coupling factors between the different cavities of the combustor. A fully analytical resolution can be performed in specific situations where coupling factors are small (weak coupling). A bifurcation appears at high coupling factors, leading to a frequency lock-in of the two annular cavities (strong coupling). This tool is applied to an academic case where four burners connect an annular plenum to a chamber. For this configuration, analytical results are compared with a full three-dimensional Helmholtz solver to validate the analytical model in both weak and strong coupling regimes. Results show that this simple analytical tool can predict modes in annular combustors and investigate strategies for controlling them.