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

Frédéric Boudy; Daniel Durox; Thierry Schuller; Sébastien Candel

doi:10.1016/j.crme.2012.10.014

Combustion, flow and spray dynamics for aerospace propulsion

Analysis of limit cycles sustained by two modes in the flame describing function framework
[Analyse de cycles limites impliquant deux modes dans le cadre de lʼéquivalent harmonique pour les flammes]

Frédéric Boudy ^1,² ; Daniel Durox ^1,² ; Thierry Schuller ^1,² ; Sébastien Candel ^1,²

¹ CNRS, UPR 288 – laboratoire EM2C, 92290 Châtenay-Malabry, France
² École centrale Paris, 92290 Châtenay-Malabry, France

Comptes Rendus. Mécanique, Volume 341 (2013) no. 1-2, pp. 181-190.

Résumé
Abstract

La méthode de lʼéquivalent harmonique (FDF), développée pour lʼanalyse non-linéaire des instabilités de combustion, a été récemment validée dans une configuration générique comportant un élément dʼalimentation, un dispositif dʼinjection et un tube à flamme. Ce système caractérisé par des comportements dynamiques très variés, est utilisé pour analyser une nouvelle gamme dʼoscillations de flamme. Suivant la géométrie, le système fait apparaître plusieurs types de cycles limites. Dans un premier cas, les oscillations ont une amplitude approximativement constante. Dans un second cas, lʼamplitude du cycle limite varie de façon instationnaire. Lʼétude traite dʼun régime dʼoscillation de ce dernier type impliquant deux modes et induisant une modification régulière de la période faisant aussi apparaître un hétérodynage de fréquence. Cette situation peut être envisagée dans le cadre de la méthode FDF lorsque les taux de croissance de deux modes, correspondant à des bifurcations supercritique et sous-critique, se chevauchent. Une deuxième condition nécessaire à ce comportement est déduite à partir des expériences.

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.

Publié le : 2013-01-01

DOI : 10.1016/j.crme.2012.10.014

Keywords: Combustion instabilities, Describing Function, Heterodyning, Limit cycles
Mot clés : Instabilités de combustion, Méthode de lʼéquivalent harmonique, Hétérodynage, Cycles limites

Affiliations des auteurs :

Frédéric Boudy ^{1, 2} ; Daniel Durox ^{1, 2} ; Thierry Schuller ^{1, 2} ; Sébastien Candel ^{1, 2}

¹ CNRS, UPR 288 – laboratoire EM2C, 92290 Châtenay-Malabry, France
² École centrale Paris, 92290 Châtenay-Malabry, France

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     title = {Analysis of limit cycles sustained by two modes in the flame describing function framework},
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Frédéric Boudy; Daniel Durox; Thierry Schuller; Sébastien Candel. Analysis of limit cycles sustained by two modes in the flame describing function framework. Comptes Rendus. Mécanique, Volume 341 (2013) no. 1-2, pp. 181-190. doi : 10.1016/j.crme.2012.10.014. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2012.10.014/

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