Self-similarly developing, premixed, turbulent flames: A theoretical study

Abstract

The development of a premixed turbulent flame in a statistically stationary and uniform, planar, one-dimensional flow of a homogeneous mixture is theoretically studied in the case of frozen turbulence. A balance equation for the mean combustion progress variable is proposed to generalize a number of currently available models of premixed turbulent combustion. The equation invokes two unspecified scales and three arbitrary functions to model turbulent diffusion, countergradient transport, and the mean rate of product creation, the rate being assumed to depend straightforwardly on the flame development time t. The equation supplemented with the averaged mass balance equation is analyzed by invoking a single assumption of the self-similarity of the mean flame structure. The assumption, well supported by numerous experiments, allows us to split the aforementioned partial differential balance equations into two ordinary differential equations, which separately model spatial variations of the mean combustion progress variable c¯ and time development of flame thickness Δt. The complementary error function profile of c¯, documented in numerous experiments, is predicted. The profile is independent of mixture properties and turbulence characteristics, as indicated by currently available experimental data. Closures of the countergradient transport term and of the mean rate of product creation are obtained. They substantiate the so-called Zimont equation and generalize it. They also imply that gradient diffusion dominates during the initial stage of flame development, followed by the transition to countergradient turbulent scalar flux ρu″c″¯ in a sufficiently developed flame. A criterion of the transition, obtained in the paper, shows that the transition is promoted by the density ratio. An analytical dependence of a fully developed flame thickness Δt,∞=Δt(t→∞) on the scale of countergradient transport and density ratio is found. Analytical expressions for the development of flame thickness are obtained in the simplified case of a time-independent mean rate of product creation.