This study focuses on the numerical determination of thermo-acoustic instabilities using a combination of the Flame Describing Function (FDF) methodology and a numerical code solving the Helmholtz equation. In this framework, the FDF is defined by a set of Flame Transfer Functions (FTF) that depend on both the frequency and amplitude of acoustic perturbations. The FDF methodology has been recently used in combination with acoustic network methods to examine the nonlinear stability of generic configurations with simplified geometries. Its extension to complex 3D geometries requires the use of numerical tools such as a Helmholtz solver. In the present work, that combination is validated on a multiple injection combustor. The implementation of the FDF methodology in the Helmholtz solver is detailed before examining numerical predictions obtained by the use of an experimentally determined FDF in the Helmholtz solver. The instability frequencies and growth rates are determined for each perturbation level and different nonlinear behaviors are exhibited depending on the combustor geometry. The case of linearly unstable modes reaching limit cycles is first examined. A more complex case involving mode switching is then examined when two unstable modes are present. In this situation, the most unstable mode in the linear regime triggers another unstable mode at a higher perturbation level. These numerical calculations are compared with experimental data and exhibit a good match in terms of amplitude and frequency reached by the limit cycle.

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