Relationships for maximum flame surface density and brush thickness through conditional analysis in turbulent premixed combustion

Abstract

A new conditional averaging method is introduced in terms of the reaction progress variable, c, and its derivatives of successively higher orders, $\Sigma _f^{\prime} $Σf′ (or −∂c/∂n) and ∂2c/∂n2, in high Damköhler number (Da) turbulent premixed combustion. c is continuous and differentiable in space so that conditional averages may be defined for flamelets of finite thicknesses as well as for infinitesimally thin flamelets. A coupled set of transport equations are derived with corresponding conditional coefficient terms for mean reaction progress variable, $\bar c$c¯, and flame surface density (FSD), Σf. Statistics for all component terms are consistent with previous results and show the same qualitative trends irrespective of compressibility by the new averaging practice. It is shown that the maximum FSD, Σmax, occurs at the location where the relationship, $\nabla ^2 \bar c + \langle {\nabla \cdot {\bf n}} \rangle _f \Sigma _f = 0$∇2c¯+⟨∇·n⟩fΣf=0, holds in a flame brush. An analytical expression is derived for Σmax under the assumption of constant tangential strain rate and linear variation of mean curvature and mean displacement speed, ⟨Sd⟩f, in the $\bar c$c¯ space. Σmax becomes equal to one fourth ⟨|∇·n|⟩f or half the mean absolute curvature for fully turbulent flames with the magnitude of ⟨n⟩f close to zero. Once the turbulent burning velocity, ST, is determined by asymptotic behavior at the leading edge, the brush thickness, ℓF, may be obtained from local and integral relationships for the product, ℓFΣmax, in a steadily propagating one-dimensional flame. The newly proposed relationships reproduce proper trends of variation for ST, Σmax, and ℓF against measurement data in literature. They are validated to show good agreement with direct numerical simulation (DNS) results of the Lewis number equal to unity for freely propagating incompressible and stagnating compressible turbulent premixed flames.