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

TL;DR

This paper investigates how small-scale, rapidly-varying flows influence flame propagation, revealing new scaling laws for effective diffusivities and burning speeds, with implications for turbulent combustion and ignition processes.

Contribution

It introduces novel scaling laws for effective Lewis number and burning speed in small-scale flows using homogenization and asymptotics, highlighting flow-induced diffusion effects.

Findings

Effective diffusivities scale with flow Peclet number and Lewis number.

Flow-enhanced diffusion modifies flame quenching limits.

Scaling laws depend on flow type and direction, affecting combustion behavior.

Abstract

Propagation of premixed flames having thick reaction zones in rapidly-varying, small-scale, zero-mean, spatio-temporal periodic flows is considered. Techniques of large activation energy asymptotics and homogenization theory are used to determine the effective Lewis number $Le_{\mathrm{eff}}$ and the effective burning speed ratio $S_T/S_L$, which are influenced by the flow through flow-enhanced diffusion. As the flow Peclet number $Pe$ becomes large, the effective fuel and thermal diffusivities behave respectively like $(PeLe)^\sigma$ and $Pe^\sigma$, where $Le$ is the Lewis number and $\sigma\leq 2$ is a constant that depends on the flow and the flame-propagation direction. The maximal value $\sigma=2$ is achieved for steady, unidirectional, spatially periodic shear flows, while for steady 2D square vortices, we have $\sigma=1/2$. In general, the constant $\sigma$ is determined by solving a linear partial differential equation. The scaling laws for the diffusion coefficients lead to corresponding scaling laws for the effective Lewis number and the effective burning speed ratio of the form $Le_{\mathrm{eff}}\simeq Le^{1-\sigma}$ and $S_T/S_L\sim (Pe/Le)^{\sigma/2}$. Effects of thermal expansion and volumetric heat loss on the flame are also briefly discussed. In particular, it is shown that the quenching limit is enlarged by a factor $1/Le^\sigma$ for $Le<1$ and diminished by the same factor for $Le>1$, due to the flow-enhanced diffusion. Potential implications of the results for turbulent combustion are discussed. A special emphasis is placed on the dependence of the flame on $Le$ in high-intensity, small-scale flows. This dependence is intimately linked to flow-induced diffusion, rather than to the traditional molecular diffusion coupled with curvature effects. The effective Lewis number may also explain why turbulence aids ignition in $Le>1$ mixtures but hinders it in $Le<1$ mixtures.