Nonlinear analytical flame models with amplitude-dependent time-lag distributions

In the present work, we formulate a new method to represent a given Flame Describing Function by analytical expressions. The underlying idea is motivated by the observation that different types of perturbations in a burner travel with different speeds and that the arrival of a perturbation at the flame is spread out over time. We develop an analytical model for the Flame Describing Function, which consists of a superposition of several Gaussians, each characterised by three amplitude-dependent quantities: central time-lag, peak value and standard deviation. These quantities are treated as fitting parameters, and they are deduced from the original Flame Describing Function by using error minimisation and nonlinear optimisation techniques. The amplitude-dependence of the fitting parameters is also represented analytically (by linear or quadratic functions). We test our method by using it to make stability predictions for a burner with well-documented stability behaviour (Noiray's matrix burner). This is done in the time-domain with a tailored Green's function approach.