Modern gas turbine combustors often have annular geometries. These are able to sustain thermoacoustic modes which vary in both the longitudinal and circumferential directions. Effects such as nonlinearity of the flame response to perturbations and differing burners around the annulus lead to the coupling of acoustic modes with different circumferential mode numbers. Such coupling renders differing spatial patterns of instability possible — for example purely longitudinal modes, circumferential standing modes, circumferential spinning modes, mixed modes and slanted modes. Accurately predicting the spatial pattern of limit cycle oscillations resulting from thermoacoustic instability remains an open challenge. This work develops a frequency domain low-order thermoacoustic network model for annular combustors which is notable in (i) accounting for both longitudinal and circumferential modes and (ii) allowing for generic acoustic boundary conditions at either end of the network. Linear acoustic waves are considered, with the different circumferential wavenumbers decoupled for sections both before and after the flames. Modal coupling occurs only at the flames, and is accounted for by summing all modal contributions prior to application of the flame models, and decomposing back into circumferential modes after application of flow conservation equations across the flames. By applying acoustic boundary conditions at either end of the network, an eigenvalue system is established which allows the thermoacoustic modes of the whole combustion system to be analysed. This low order modelling approach is applied to a simplified annular combustor set-up and is demonstrated to be able to capture limit cycles exhibiting longitudinal modes, circumferential spinning modes, circumferential standing modes and even the recently identified slanted modes.

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