Evans Function Stability of Combustion Waves

Abstract

In this paper we investigate the linear stability and properties, such as speed, of the planar travelling combustion front. The speed of the front is estimated both analytically, using the matched asymptotic expansion, and numerically, by means of the shooting and relaxation methods. The Evans function approach extended by the compound matrix method is employed to numerically solve the linear stability problem for the travelling wave solution.stability of the travelling wave and therefore allows us to continue the solution branch over a broader parameter range.As we vary the parameters of the problem, a steady propagating planar front can lose stability, giving rise to either pulsating or cellular flames [5,9]. Analytical investigation of the stability using the MAE leads to the so-called closure problem. In contrast to steady travelling waves, in this case the leading order equations depend on first order corrections, first order equations include second order terms of the asymptotic expansion, etc. In order to find the solution to the leading order problem, an infinite number of equations have to be investigated. One of the ways of overcoming this obstacle is just to truncate the expansion [1,2,3,4,5,9]. This yields a closed problem with a replacement of the Arrhenius reaction rate by a delta-function source depending on the temperature at the reaction front. The truncated model has been used extensively for the stability analysis of combustion waves [1,2,3,4]. However, the model with the delta-function source suffers from inconsistencies, as was noted in [5,9]. The inverse of the small parameter of the expansion appears explicitly in the exponential terms describing the strength of the source. In other words, temperature variations behind the front are considered to be small in the exponential terms and of leading order elsewhere.An asymptotically consistent approach was proposed in [5] for the system with the Lewis number of the order of unity. In this case the enthalpy does not change at the leading order. A closed problem was derived for the leading order temperature and the first order enthalpy. However, until recently [9], there has not been a consistent approach that treats the model with arbitrary Lewis number.In [9] a generalization of the MAE method was introduced. The coefficients in the expansions are allowed to depend on the expansion parameter. This enables the correct scaling of the temperature variations ahead of and behind the reaction zone; namely, a restriction is imposed connecting the leading order temperature in the preheat zone and the first two terms of the asymptotic expansion in the product zone. The constraint reflects the fact that small temperature variations behind the front change the leading order terms in the preheat zone. The resulting model includes equations for the leading order variables ahead of the reaction zone and for the first order temperature variations in the product zone, together with matching and boundary conditions. There is no restriction on the range of the Le...

Spectral Behavior of Combustion Fronts with High Exothermicity

“…Several papers have also examined the dependence of the wave speed on the exothermicity; see, for example, [5,43,49,48]. Several similar approximate formulas have been obtained, each describing exponential convergence c → 0 as β → ∞.…”

“…System (1) originated as a result of a nontraditional nondimensionalization procedure which led to physical values of temperature varying between 0 and 1 β . Investigation of (1) in a series of papers [49,5,46,19,21,43] significantly complemented the information obtained for the traditional system (see [41,7,33,38]).…”

“…In the case of two-dimensional perturbations, the instability is expected to manifest itself as transversally propagating wavetrains [3,40]. The onset of this instability was studied in [5] using Evans function methods [1], which determine the unstable eigenvalues of the linearized operator about the combustion profile; see also [49,43,46]. The goal of this paper is to examine the spectral behavior of one-dimensional combustion fronts for values of high exothermicity.…”

Stability of combustion waves in a simplified gas–solid combustion model in porous media

“…In this section, we study numerically the discrete spectrum of fast combustion waves using the Evans function [8,13,14]. More precisely, we study the discrete spectrum of the operator L α defined in the previous section, where α = (α − , α + ) ∈ R 2 has been chosen to stabilize the essential spectrum; thus 0 < α ± < c f − a.…”

“…We continue the linear stability analysis for fast combustion waves by performing a numerical computation of the Evans function to find the discrete spectrum [8]. Some of the waves have no unstable discrete spectrum; others have an unstable eigenvalue because of a saddle-node bifurcation of travelling waves.…”