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 Leeff and the effective burning speed ratio ST /SL , which are influenced by the flow through flow-enhanced diffusion. The resultant effective diffusivity matrix is, in general, neither a scalar nor a diagonal matrix and therefore induces anisotropic effects on the propagation of multi-dimensional flames. As the flow Peclet number Pe becomes large, the flow-enhanced fuel diffusion coefficient and the thermal diffusivity behave respectively like (PeLe )σ and Peσ , where Le is the Lewis number and σ ≤ 2 is a constant which depends on the flow and the direction of flame propagation. The maximal value σ = 2 is achieved for steady, unidirectional, spatially periodic shear flows, while for steady two-dimensional square vortices, we have σ = 1/2. In general, the constant σ 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 Leeff Le1−σ and ST /SL ∼ (Pe/Le )σ /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/Leσ for Le < 1 and diminished by the same factor for Le > 1, due to the flow-enhanced diffusion. The potential implications of the results to better understand turbulent combustion are discussed. A special emphasis is placed on the dependence of the flame on Le in the presence of high-intensity, small-scale flows. In particular, it is shown that this dependence is intimately linked to the flow through Taylor-dispersion like enhanced diffusion, rather than through the traditional molecular diffusion coupled with curvature effects. The flow-dependent effective Lewis number identified may also provide an explanation to the peculiar experimental observation that turbulence appears to facilitate ignition in Le > 1 mixtures and to inhibit it in Le < 1 mixtures.