Impact of exothermicity on steady and linearized response of a premixed ducted flame

Thermoacoustic instabilities arise in power generation devices such as gas turbines and aero-engines when acoustic modes couple with unsteady heat released due to combustion in a positive feedback loop. This work focuses on the development of a reduced order model for understanding flame dynamics in the case of flameholder-stabilized premixed combustion in a duct-a situation typical in many of these applications. Similar to earlier studies in reduced order modeling of this flow, we employ a G-equation formulation to obtain kinematical representation of the premixed flame and ignore the impact of the unsteady (vortical) fluid dynamics downstream of the flameholder. Unlike those studies, however, we retain the impact of combustion exothermicity in the form of a density jump and associated volume generation at the flame front as well as the steady portion of the baroclinic vortical effect. The reduced order model yields analytical solutions for the flame location and for linear transfer functions between imposed (acoustic) perturbation and combustion heat release. We validate these solutions against numerical simulations and other results in literature. The role of expansion (dilatation) and baroclinic aspects of exothermic effects are discussed in detail. Results show that for realistic density ratios across the flame, the flow is accelerated in the streamwise direction on account of combustion exothermicity and the effects of confinement. This not only alters the flame location but also changes the linearized dynamics of the flame and brings into question conclusions drawn from similar analyses in which exothermicity effects were neglected. This is discussed in the context of modeling and controlling thermoacoustic instabilities.