Inclusion of higher harmonics in the flame describing function for predicting limit cycles of self-excited combustion instabilities

The Flame Describing Function (FDF) is widely used to model non-linear thermo-acoustic phenomena, e.g. limit cycle oscillations in a combustor. The FDF is a weakly non-linear approach, because it accounts for the amplitude dependence of the flame response, but besides that relies on quasi-linear assumptions. In particular, it neglects the excitation of higher harmonics - a typical non-linear feature, which may play a major role in certain thermo-acoustic systems. Consequently, the FDF may provide inaccurate or incomplete results in such cases. In this study, we propose an efficient way to include higher harmonics of the flame response by an extended FDF, which includes additional transfer functions that relate higher harmonics of the heat release rate to the forcing velocity. The extended FDF is also a weakly non-linear approach, and requires the same effort for determination as the standard FDF. This paper shows how to determine the extended FDF and how to employ it for the prediction of limit cycle oscillations. The proposed concept is applied to predict and analyse the limit cycle of a laminar premixed burner for which the standard FDF delivered inaccurate results. Results obtained with the extended FDF show good agreement with fully compressible numerical simulation of the same configuration. Thus, the extended FDF proves its ability to provide accurate predictions in situations where higher harmonics play an important role in thermo-acoustic limit cycles.