This study describes an analytical method to compute azimuthal modes due to flame/acoustics coupling in annular combustors. It is based on a quasione-dimensional zero-Mach number formulation where N burners are connected to an upstream annular plenum and a downstream chamber. Flames are supposed 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 allows to retain only the useful information related to the azimuthal modes of the annular cavities. It yields a simple dispersion relation which can be solved numerically and allows 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 to 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 allows to predict modes in annular combustors and investigate strategies to control them.

International audienceThis study describes a simple analytical method to compute the azimuthal modes appearing in annular combustion chambers and help analyzing experimental, acoustic and large eddy simulation (LES) data obtained in these combustion chambers. It is based on a one-dimensional zero Mach number formulation where N burners are connected to a single annular chamber. A manipulation of the corresponding acoustic equations in this configuration leads to a simple dispersion relation which can be solved by hand when the interaction indices of the flame transfer function are small and numerically when they are not. This simple tool is applied to multiple cases: (1) a single burner connected to an annular chamber (N = 1), (2) two burners connected to the chamber (N = 2), and (3) four burners (N = 4). In this case, the tool also allows to study passive control methods where two different types of burners are mixed to control the azimuthal mode. Finally, a complete helicopter chamber (N = 15) is studied. For all cases, the analytical results are compared to the predictions of a full three-dimensional Helmholtz solver and a very good agreement is found. These results show that building very simple analytical tools to study azimuthal modes in annular chambers is an interesting path to control them