On a Model for the Response of Unstable Premixed Flames to Turbulence

Abstract

Using the Michelson-Sivashinsky (M.S.) non-linear equation as a starting point to study the dynamics of unstable premixed flames, we propose a way of incorporating an external noise intended to mimic an incoming turbulence. while preserving the possibility of pole-decompositions of the solutions.

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

“…2011). From the pole decomposition standpoint, as said before, the external perturbations act as a stochastic source of new poles (Joulin 1988; Kupervasser et al. 1996; Olami et al.…”

“…However, even if the TFH solution is the only stable one for both (3.1) and (3.4) (Vaynblat & Matalon 2000 a , b ), the number of equilibrium solutions is growing faster than linearly with , leading rapidly to a large number of solutions (Denet 2006). The external noise, which is responsible for the permanent addition of new poles (Joulin 1988; Kupervasser, Olami & Procaccia 1996; Olami et al. 1997), may cause the system to permanently jump from one solution to another if poles are introduced close enough to the real axis (Denet 2006).…”

“…However, even if the TFH solution is the only stable one for both (3.1) and (3.4) (Vaynblat & Matalon 2000a,b), the number of equilibrium solutions is growing faster than linearly with 1/ν, leading rapidly to a large number of solutions (Denet 2006). The external noise, which is responsible for the permanent addition of new poles (Joulin 1988;Kupervasser, Olami & Procaccia 1996;Olami et al 1997), may cause the system to permanently jump from one solution to another if poles are introduced close enough to the real axis (Denet 2006). Moreover, this unsteady behaviour can be explained without invoking nonlinear mechanisms, because of the existence of transient growth phenomena (Karlin 2002(Karlin , 2004) not taken into account in the modal analysis carried out by Vaynblat & Matalon (2000a) and making the TFH solution an unstable equilibrium for (3.1).…”

“…From the pole decomposition standpoint, as said before, the external perturbations act as a stochastic source of new poles (Joulin 1988;Kupervasser et al 1996;Olami et al 1997). But until now the new poles are introduced in a completely artificial manner, and how to determine the probability function of adding a new pole at a certain location in the complex plane remains an open question.…”

On the hydrodynamic stability of curved premixed flames

“…equation (2.1), in situations where u (X, t ) is a harmonic function of time and space : for several realizations of weak amplitude noises, a quantitative confirmation of the criterion (3.8) was obtained, at least in the cases of not too large cells, for which the numerical noise is indeed negligible and a steady-state, background profile F(X) can be obtained numerically. We also recently proposed a complementary model [27] in which u (X, t ) represents a special, spatially inhomogeneous shot-noise that is compatible with a pole-decomposition of (2.1) ; its influence is merely to implant new complex-conjugate pairs of poles, at random locations am ± ibM (m = 1, 2, ...) and times tm, without changing the pole-dynamics between the implants. When a spatially-periodic version of this noise is included in (2.1), u(X, t) reads as :…”