Flame Double Input Describing Function analysis

The Flame Describing Function (FDF) is a useful and relatively cheap approximation of a flame’s nonlinearity with respect to harmonic velocity fluctuations. When embedded into a linear acoustic network, it is able to predict the amplitude and stability of harmonic thermoacoustic oscillations through the harmonic balance procedure. However, situations exist in which these oscillations are not periodic, but their spectrum contains peaks at several incommensurate frequencies. If one assumes that two frequencies dominate the spectrum, these oscillations are quasiperiodic, and the FDF concept can be extended by forcing the flame with two amplitudes and two frequencies. The nonlinearity is then approximated by a Flame Double Input Describing Function (FDIDF), which is a more expensive object to calculate than the FDF, but contains more information about the nonlinear response.In this study, we present the calculation of a non-static flame’s FDIDF. We use a G-equation-based laminar conical flame model. We embed the FDIDF into a thermoacoustic network and we predict the nature and amplitude of thermoacoustic oscillations through the harmonic balance method. A criterion for the stability of these oscillations is outlined. We compare our results with a classical FDF analysis and self-excited time domain simulations of the same system. We show how the FDIDF improves the stability prediction provided by the FDF. At a numerical cost roughly equivalent to that of two FDFs, the FDIDF is capable to predict the onset of Neimark–Sacker bifurcations and to identify the frequency of oscillations around unstable limit cycles. At a higher cost, it can also saturate in amplitude these oscillations and predict the amplitude and stability of quasiperiodic oscillations.