On the fast-time oscillatory instabilities of Liñán's diffusion-flame regime

Abstract

Fast-time instability for diffusion flames, with Lewis numbers set equal for fuel and oxidizer but greater than unity, is numerically analysed by the activation energy asymptotics and Evans function method. The time and length scales being chosen to be those of the inner reactive-diffusive layer, the problem corresponds to the instability problem for the Liñán's diffusion-flame regime. The instability is primarily oscillatory and emerges prior to reaching the turning point of the characteristic C-curve, usually known as the Liñán's extinction condition. A critical Lewis number, L c , is also found, across which the instability changes its qualitative character. Below L c , the instability possesses primarily a pulsating nature in that the two real branches of the dispersion relation existing for small wave numbers merge at a finite wave number, from which a pair of complex conjugate branches bifurcate. The maximum growth rate is found at the zero wave number. For Lewis numbers greater than L c , the eigensolution branch for small reactant leakages is found to be purely complex with the maximum growth rate found at a finite wave number, thereby exhibiting a travelling nature. As the reactant-leakage parameter is further increased, the instability characteristics turns into a pulsating type, similar to that for 1 < L < L c . The switching between different instability characters is found to correspond to the Bogdanov-Takens bifurcation.

Period doubling and chaotic transient in a model of chain-branching combustion wave propagation

“…On the other hand, for Lewis numbers below the critical value corresponding to Bogdanov-Takens bifurcation no pulsating behaviour occurred. A similar scenario has been observed recently in diffusion flames (Gubernov & Kim 2006). This result correlates with the analysis in Aldushin et al (1973).…”

Pulsating Instabilities of Combustion Waves in a Chain-Branching Reaction Model

“…It should be noted that similar scenario is reported in Gubernov et al [2004] (see Fig. 7) and Gubernov & Kim [2006] for the case of one-step premixed and diffusion flames respectively.…”

“…It is worthwhile noting that the cellular instabilities are known to originate from the extinction point at L A = 1 for the case of one-step reaction models [Gubernov & Kim, 2006]. Therefore we can expect that for our current model the cellular instabilities may also appear from the Bogdanov-Takens bifurcation point and it is the bifurcation responsible for the onset of instabilities of all types.…”

“…It is surprising that the Bogdanov-Takens bifurcation point is located on the line L A = 1. Previously, we have shown the existence of this bifurcation for the case of both one-step premixed [Gubernov et al, 2004] and diffusion [Gubernov & Kim, 2006] flames. In these investigations the Bogdanov-Takens bifurcation is demonstrated to play an important role in the onset of pulsating instabilities.…”

“…In particular we study the properties of the Hopf bifurcation leading to the onset of pulsations in detail and investigate the pulsating solutions emerging as a result of this bifurcation. We also investigate the existence of the Bogdanov-Takens bifurcation which is shown to play a significant role in the emergence of oscillations in both premixed [Gubernov et al, 2004] and diffusion flames [Gubernov & Kim, 2006] with single-step kinetics. The paper is organized as follows.…”