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

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.