Stationary solutions and Neumann boundary conditions in the Sivashinsky equation

Abstract

New stationary solutions of the (Michelson) Sivashinsky equation of premixed flames are obtained numerically in this paper. Some of these solutions, of the bicoalescent type recently described by Guidi and Marchetti, are stable with Neumann boundary conditions. With these boundary conditions, the time evolution of the Sivashinsky equation in the presence of a moderate white noise is controlled by jumps between stationary solutions.

Self-Turbulent Flame Speeds

“…In our experiments we find larger flame velocities, in the range from 1.5U L to 3U L see fig. 4, but it is generally recognized [24] that the velocity increase relative to the laminar velocity is higher in 3-D than in 2-D. Another important result is that the inclined flame velocity is larger than the slightly asymmetric one, a property which is not found in the Sivashinsky equation [22], where these two solutions have the same velocity, but which is found in a potential model [23]. Increased velocity of inclined flames has been also demonstrated in direct numerical simulations of flame dynamics [12].…”

Premixed flames propagating freely in tubes

“…increases, so that as the ratio domain width/ cut-off length scale is increased, the solution becomes more and more complicated. Among the many solutions that have been found in [29], some solutions are linearly stable, see particularly figure 1 of [29], where slanted or almost symmetrical solutions (see the experimental results of the next section for similar solutions in the cylindrical tube) have been obtained. One interesting property of these solutions found in this article is that a moderate noise (the term u(x,t) of the equation or the residual turbulence in the experiment) causes jumps between the different stationary solutions.…”

“…It has been found in [9] that simple solutions of this equation exist, described by a certain number of poles in the complex plane. The typical number of poles of the solution (called optimal number of poles in [29]) increases when the number of unstable modes 1/! increases, so that as the ratio domain width/ cut-off length scale is increased, the solution becomes more and more complicated.…”

“…The fractal exponent was close to 1/3. However, slanted flames were also observed with a much larger propagation speed, a result not expected from Sivashinsky simulations where all solutions have the same speed and for which the more slanted flame is less stable [29]. The present work aims at a better description of these two regimes of propagation by measuring simultaneously the instantaneous position of the leading flame bump and its radius of curvature.…”

“…The first interrogation was on the existence of slanted flames: do they evolve with a well defined velocity of propagation slightly larger than the symmetrical ones as suggested by theoretical approaches [2,27], or is there a large number of discrete solutions as those obtained from the Sivashinsky equation [29]? The flame trajectories in the tubes appear relatively rectilinear with two distinct velocities of propagation well correlated to the symmetry of the flame (Fig.3), even if the flame can transit from one mode of propagation to the other one during the experiment.…”