Taylor dispersion in premixed combustion: Questions from turbulent combustion answered for laminar flames

Abstract

We present a study of Taylor dispersion in premixed combustion and use it to clarify fundamental issues related to flame propagation in a flow field. In particular, simple analytical formulas are derived for variable density laminar flames with arbitrary Lewis number Le providing clear answers to important questions arising in turbulent combustion, when these questions are posed for the case of one-scale laminar parallel flows. Exploiting, in the context of a laminar Poiseuille flow model, a thick flame distinguished asymptotic limit for which the flow amplitude is large with the Reynolds number Re fixed, three main contributions are made. First, a link is established between Taylor dispersion [G. Taylor, Proc. R. Soc. London Ser. A 219, 186 (1953)] and Damköhler's second hypothesis [G. Damköhler, Ber. Bunsen. Phys. Chem. 46, 601 (1940)] by describing analytically the enhancement of the effective propagation speed U T due to small flow scales. More precisely, it is shown that Damköhler's hypothesis is only partially correct for one-scale parallel laminar flows. Specifically, while the increase in U T due to the flow is shown to be directly associated with the increase in the effective diffusivity as suggested by Damköhler, our results imply that U T ∼ Re (for Re 1) rather than U T ∼ √ Re, as implied by Damköhler's hypothesis. Second, it is demonstrated analytically and confirmed numerically that, when U T is plotted versus the flow amplitude for fixed values of Re, the curve levels off to a constant value depending on Re. We may refer to this effect as the laminar bending effect as it mimics a similar bending effect known in turbulent combustion. Third, somewhat surprising implications associated with the dependence of U T and of the effective Lewis number Le eff on the flow are reported. For example, Le eff is found to vary from Le to Le −1 as Re varies from small to large values. Also, U T is found to be a monotonically increasing function of Re if Le < √ 2 and a nonmonotonic function if Le > √ 2.

Effective Lewis number and burning speed for flames propagating in small-scale spatio-temporal periodic flows

“…The behaviours of other values of δ, lie between the above two limiting cases δ = 0 and δ = 1 just considered. It should be noted that the 1 4 th power dependence of S T /S L on Pe given in (29) for the case of square vortices was first identified by Audoly et.al. [4] and later confirmed in [8,36,37]; see also [38, pp.186-187].…”

“…The solution of problem ( 20)-( 22) for large β, as done e.g. in [29], provides the following formulas for the effective burning speed and the effective Lewis number…”

“…This upper bound indicates that maximal enhancement of diffusion is achieved when σ = 2. The maximal enhancement in fact occurs for steady, unidirectional periodic (or confined) shear flows and is associated with the Taylor's dispersion mechanism [29]. It is instructive to consider two classes of flow fields, namely, unsteady unidirectional flows for which the exponent σ can be determined explicitly and the two-dimensional so-called Childress-Soward flows for which σ will be computed numerically.…”

Effects of thermal expansion on Taylor dispersion-controlled diffusion flames

“…We will show that the reduced parametric dependence of the solution on the "effective" Lewis number, characteristic of the thermodiffusive approximation, no longer holds. It should be noted here that effects of thermal expansion have been studied previously [2,3] in the context of steadily propagating premixed flames in narrow pipes and channels. There, asymptotic solutions for premixed flames were derived in the limit that the flame thickness becomes large compared with the radius of the pipe.…”

“…Here D and D F /Le may respectively be considered as the Taylor diffusivities for the oxidizer (or, temperature) and fuel, measured in units of D O∞ . The diffusivities have been previously identified for premixed flames [2,3] where ρ 0 D and ρ 0 D F /Le (i.e. including ρ 0 ) were termed as the Taylor diffusion coefficients.…”