Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field

Raphaël C Assier,Xuesong Wu

doi:10.1017/jfm.2014.525

Abstract

AbstractThe stability of premixed flames in a duct is investigated using an asymptotic formulation, which is derived from first principles and based on high-activation-energy and low-Mach-number assumptions (Wu et al., J. Fluid Mech., vol. 497, 2003, pp. 23–53). The present approach takes into account the dynamic coupling between the flame and its spontaneous acoustic field, as well as the interactions between the hydrodynamic field and the flame. The focus is on the fundamental mechanisms of combustion instability. To this end, a linear stability analysis of some steady curved flames is undertaken. These steady flames are known to be stable when the spontaneous acoustic perturbations are ignored. However, we demonstrate that they are actually unstable when the latter effect is included. In order to corroborate this result, and also to provide a relatively simple model guiding active control, we derived an extended Michelson–Sivashinsky equation, which governs the linear and weakly nonlinear evolution of a perturbed flame under the influence of its spontaneous sound. Numerical solutions to the initial-value problem confirm the linear instability result, and show how the flame evolves nonlinearly with time. They also indicate that in certain parameter regimes the spontaneous sound can induce a strong secondary subharmonic parametric instability. This behaviour is explained and justified mathematically by resorting to Floquet theory. Finally we compare our theoretical results with experimental observations, showing that our model captures some of the observed behaviour of propagating flames.

Highlights

Combustion instability, referred to as thermo-acoustic instability, arises due to a strong interaction between the heat released by a flame and the acoustic fluctuations of a combustion chamber
Concluding remarks As an effort to shed further light on the fundamental mechanisms of combustion instability, a flame–flow–acoustic interaction model was derived from the asymptotic theory of WWMP by making a weak nonlinearity assumption, under which the hydrodynamic field is linearised while the geometric nonlinearity is retained
As the flame and the spontaneous sound are both allowed to evolve in this model, the two crucial mechanisms, the parametric instability and the radiation of spontaneous sound when a flame wrinkles, operate simultaneously and are coupled dynamically as they are in experiments

Summary

Introduction

Combustion instability, referred to as thermo-acoustic instability, arises due to a strong interaction between the heat released by a flame and the acoustic fluctuations of a combustion chamber. When the unsteady heat release rate and acoustic fluctuations. Instability of a curved flame under the influence of its spontaneous sound 181 are in phase, a small perturbation to the system will amplify according to the criterion of Rayleigh (1878). Combustion instability may occur in numerous real-life situations such as ramjet engines (Yu, Trouvé & Daily 1991), rocket engines (Harrje & Reardon 1972) and, more generally, any type of gas turbine engine (Lieuwen & Yang 2005). The two-way interaction between the flame and the acoustics can lead to strong self-sustained fluctuations, which may have disastrous effects on the components of an engine, for example by causing vibrations and structural fatigue. A significant amount of research has been undertaken, both theoretical (e.g. Bloxsidge, Dowling & Langhorne 1988; Dowling 1995) and experimental (e.g. Poinsot et al 1987; Durox et al 2009; Steinberg et al 2010)

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