To get a better understanding of the formation of thermoacoustic oscillations in an annular gas turbine combustor, an analysis of the acoustic eigenmodes has been conducted using the finite element method. The influence of different boundary conditions and a space-dependent velocity of sound has been investigated. The boundary conditions actually define the eigenfrequency spectrum. Hence, it is crucial to know, e.g., the burner impedance. In case of the combustion system without significant mixing air addition considered in this paper, the space-dependence of the velocity of sound is of minor importance for the eigenfrequency spectrum leading to a maximum deviation of only five percent in the eigenvalues. It is demonstrated that the efficiency of the numerical eigenvalue analysis can be improved by making use of symmetry, by splitting the problem into several steps with alternate boundary conditions, and by choosing the shift frequency $ωs$ in the range of frequencies one is interested in.

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