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

Abstract

The 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.

Nonlinear dynamics of premixed flames: from deterministic stages to stochastic influence

“…In this framework much effort has been made to obtain an equation for this interface in a closed form (i.e. in a form involving only quantities defined on the flame front) (Sivashinsky 1977; Sivashinsky & Clavin 1987; Joulin & Cambray 1992; Bychkov 1998; Bychkov & Kleev 1999; Bychkov, Kovalev & Liberman1999; Kazakov & Liberman 2002 a , b ; Kazakov 2005 a , b ; Joulin, El-Rabii & Kazakov 2008; El-Rabii, Joulin & Kazakov 2010; Assier & Wu 2014). However, to obtain such an equation is an extremely intricate mathematical problem because it requires solving the whole flow both ahead and behind the flame front.…”

Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors

“…the fluctuations of the heat release rate are present only if an external acoustic wave perturbs the flame. This means for example that we do not consider the flame response to its own acoustic field (Assier & Wu 2014) if it leads to a linearly unstable flame. We will study the problem both in the time and frequency domains.…”

Experimental and theoretical study of secondary acoustic instability of downward propagating flames: Higher modes and growth rates

“…Velocity coupling was analyzed theoretically, and transfer function was derived for weakly cellular flames [25] based on effect of hydrodynamics and diffusion on premixed flame stability [26]. Nonlinear coupled equations were later derived considering both these mechanisms [27] and later studied for weakly non-linear case [28]. Some aspects of experimental observation e.g.…”