On the relationship between the statistics of the resolved and true rate of dissipation of mixture fraction

The relationship between the one-point probability-density-function (PDF) of the dissipation rate of mixture fraction fluctuations and the corresponding resolved quantity available in large eddy simulation (LES) is analyzed. The investigation pursues two fronts: an a priori study using direct numerical simulation (DNS), and an analytic development that, using common turbulence physics simplifications, relates the one-point statistics of the resolved and true scalar dissipations. Particularly, the analysis reveals the connection between the multi-point correlations of the mixture fraction gradient and the one-point PDF of the resolved scalar dissipation. A DNS of a temporally evolving shear layer with and without heat release is used to quantify the accuracy of the analytical result. It is verified, both by filtering the DNS and from the theory, that increasing the filter cutoff width reduces the magnitude of the resolved scalar dissipation fluctuations, as expected and observed experimentally. Comparison with DNS indicates that the analytical relationship predicts the behavior of the resolved scalar dissipation PDF well at the center planes of the shear layer, where turbulence is locally more isotropic and homogeneous. Large-scale anisotropy and inhomogeneities in the DNS degrade the accuracy of the approximate analytical result close to the edges of the shear layer. These results may be improved with future investigations to account fully for the missing statistics in LES, which have the potential to allow a more accurate quantification of finite-rate chemistry effects in reacting flows.