Do flame describing functions suitably represent combustion dynamics under self-sustained oscillations?

Abstract

Transfer function concepts that appear in many areas and most notably in control systems have been extensively used to represent the flame response in low-order models of combustion instability. Much of the theoretical work is based on flame transfer functions (FTFs). In recent years, its nonlinear extension, namely the flame describing function (FDF), has been used to get a more accurate representation of the flame response when the level of oscillation becomes large and the system reaches a limit cycle. Despite their wide and reasonably successful use in predicting instabilities, the direct validity of using FTF/FDFs to represent the flame response still remains to be experimentally substantiated. This article is aimed at providing a direct assessment of the capacity of the FDF to suitably describe the flame behavior under self-sustained oscillations (SSOs) for a spray-swirl flame anchored by an injector that is weakly-transparent to acoustic waves. This is accomplished by making use of an experimental combustion configuration that not only exhibits unstable oscillations but also features a set of driver units to modulate the flame (namely stable flame modulation or SFM). The flame is modulated at the frequency of SSO, and the amplitude of incident velocity modulations is then progressively varied until it coincides with that found under SSO. The injector dynamics is shown to be different between SSO and SFM for an injector that is weakly-transparent to acoustic waves and imposes a certain degree of decoupling between plenum and chamber. For such injectors, the FDF built with the upstream velocity would not suitably represent SSO, as this lumps the injector and flame dynamics together. It is then important to use velocity measurement at the injector outlet, at a point where the relative velocity fluctuation matches the relative volumetric flow rate fluctuation. The describing function with velocity reference at the injector outlet is measured for various input levels and found to approximately match those measured under SSO. The best match is obtained when the amplitude of external modulation induces a level of velocity oscillations that comes closest to that prevailing under SSO. This demonstrates that the FDF may suitably capture the nonlinearity of the flame response, at least in the configuration investigated in this research.