A New Analytic pdf for Simulations of Premixed Turbulent Combustion

A new reaction rate source term $$\omega _m(c)$$ for modelling of premixed combustion with a single progress variable c is proposed. $$\omega _m(c)$$ mimics closely the Arrhenius source term $$\omega _A(c)$$ for a large range of activation energies and density ratios while admitting analytic evaluation of many quantities of interest. The analytic flame profile $$c_m(\xi )$$ very closely approximates the numerically integrated Arrhenius flame profiles $$c_A(\xi )$$ . An important feature of $$c_m(\xi )$$ is that it is analytically invertible into a $$\xi _m(c)$$ . Analytic estimates of the laminar flame Eigenvalue $$\Lambda$$ and of the Le dependence of the laminar flame speed $$s_L$$ are derived, which are more accurate than classic results based on asymptotic analyses. The flamelet pdf $$p(c)=1/(\Delta *c*(1-c^m))$$ for a flat laminar flame front in a LES cell of width $$\Delta$$ is derived. The exact mean of the reaction rate $$\overline{\omega (c)}$$ is compared to a beta pdf result, which is shown to be inaccurate for large ratios of filter width to flame thickness $$\Delta /\delta _f$$ and particularly for high activation energy flames. For multidimensional flame wrinkling we derive the exact relationship $$p(c)=p_{1D}(c)I(c)\Xi (c)$$ between the 3D pdf p(c), the 1D flat flame pdf $$p_{1D}(c)$$ , a correction factor I(c) for change of inner flame structure and a geometrical wrinkling factor $$\Xi (c)$$ . We show that the c dependence of these quantities cannot be neglected for small $$\Delta /\delta _f$$ . A simple model of a sinusoidally wrinkled flame front qualitatively demonstrates the effect of flame wrinkling on p(c). We show that for large $$\Delta /\delta _f$$ , a wrinkling of the reaction zone almost constantly increases p(c) in the reaction zone by a wrinkling factor $$\Xi ^*$$ (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 $$\Delta '=\Delta /\Xi ^*$$ may be a good approximation of the wrinkled flame pdf for evaluation of $$\overline{\omega (c)}$$ for such cases.