Theory of premixed-flame propagation in large-scale turbulence

Abstract

A statistical theory is developed for the structure and propagation velocity of premixed flames in turbulent flows with scales large compared with the laminar flame thickness. The analysis, free of usual closure assumptions, involves a regular perturbation for small values of the ratio of laminar flame thickness to turbulence scale, termed the scale ratio ε, and a singular perturbation for large values of the non-dimensional activation temperature β. Any effects of the flame on the flow are considered to be given. In this initial study, molecular coefficients for diffusion of heat and reactants are set equal. The results identify convective-diffusive and reactive-diffusive zones in the flame and predict thickening of the flame by turbulence through streamwise displacement of the reactive-diffusive zone. Profiles for intensities of temperature fluctuations and for streamwise turbulent transport are obtained. A fundamental quantity occurring in the analysis is the longitudinal displacement of the reactive-diffusive zone in an Eulerian frame by turbulent fluctuations, and to first order in the scale ratio this equals the longitudinal displacement of fluid elements in an Eulerian frame by turbulent fluctuations, herein termed simply the Eulerian displacement. To first order in the scale ratio it is found that, if the Eulerian displacement experiences the same type of statistical non-stationarity as the corresponding Lagrangian displacement, then the diffusion approximation is valid for streamwise turbulent transport but the turbulent flame thickens as time increases, while if the Eulerian displacement is statistically stationary then the diffusion approximation necessitates a negative coefficient of diffusion in part of the flame but the flame thickness remains constant. By carrying the analysis to second order in the scale ratio it is shown that the turbulent-flame speed exceeds the laminar-flame speed by an amount proportional to the mean square of the transverse gradient of the Eulerian displacement. This result can be understood from the mechanistic viewpoint of a wrinkled laminar flame in terms of the increase in flame area produced by turbulence. Thus the theory provides a precise statistical quantification of the model of the wrinkled laminar flame for describing structures of turbulent flames.