A New Analytic pdf for Simulations of Premixed Turbulent Combustion

Abstract

A new reaction rate source term ωm(c) for modelling of premixed combustion with a single progress variable c is proposed. ωm(c) mimics closely the Arrhenius source term ωA(c) for a large range of activation energies and density ratios while admitting analytic evaluation of many quantities of interest. The analytic flame profile cm(ξ) very closely approximates the numerically integrated Arrhenius flame profiles cA(ξ). An important feature of cm(ξ) is that it is analytically invertible into a ξm(c). Analytic estimates of the laminar flame Eigenvalue Λ and of the Le dependence of the laminar flame speed sL are derived, which are more accurate than classic results based on asymptotic analyses. The flamelet pdf p(c) = 1 / (Δ ∗ c∗ (1 - cm)) for a flat laminar flame front in a LES cell of width Δ is derived. The exact mean of the reaction rate ω(c) ¯ is compared to a beta pdf result, which is shown to be inaccurate for large ratios of filter width to flame thickness Δ / δf and particularly for high activation energy flames. For multidimensional flame wrinkling we derive the exact relationship p(c) = p1D(c) I(c) Ξ (c) between the 3D pdf p(c), the 1D flat flame pdf p1D(c) , a correction factor I(c) for change of inner flame structure and a geometrical wrinkling factor Ξ (c). We show that the c dependence of these quantities cannot be neglected for small Δ / δf. A simple model of a sinusoidally wrinkled flame front qualitatively demonstrates the effect of flame wrinkling on p(c). We show that for large Δ / δf, a wrinkling of the reaction zone almost constantly increases p(c) in the reaction zone by a wrinkling factor Ξ ∗ (defined for the surface of the isosurface of maximum heat release) while reducing it near c= 0 , 1 as required for normalisation of p(c). The 1D p(c) evaluated using a reduced filter width Δ ′= Δ / Ξ ∗ may be a good approximation of the wrinkled flame pdf for evaluation of ω(c) ¯ for such cases.