Analysis of limit cycles sustained by two modes in the flame describing function framework

Abstract

The Flame Describing Function (FDF) framework, developed for the nonlinear instability analysis of combustors, has been validated more recently in a generic configuration comprising an upstream manifold, an injection unit and a flame tube. This system featuring a wide variety of dynamical phenomena, is used here to explore a new range of self-sustained flame oscillations. Depending on the geometry, the system exhibits stable or variable amplitude limit cycles. In the first case, oscillations have an essentially constant amplitude and are well retrieved with the FDF framework, whereas in the second case, limit cycles feature different types of amplitude unsteadiness and require some further consideration. The present article is concerned with one type of unstable oscillation in which a regular period modification occurs in the presence of two modes, leading to frequency heterodyning. It is found from the FDF analysis that such oscillations sustained by two modes may occur when there is an overlap between modes corresponding to super and subcritical bifurcations. An additional condition which has to be fulfilled to obtain this behavior is inferred from experiments.