A new multi-scale mixing model for turbulent reacting flows

In the probability density function (PDF) description of a turbulent reacting flow, the local temperature and species concentration are replaced by a high-dimensional joint probability density function that describes the distribution of states in the fluid. The PDF has the great advantage of rendering the chemical reaction source terms closed, independent of their complexity. However, molecular mixing, which involves two-point information, must be modeled. Indeed, the qualitative shape of the PDF is sensitive to this modeling, hence the reliability of the model to predict even the closed chemical source terms rests heavily on the mixing model. Spectral models contain two-point statistical information of the velocity and scalar fields that can accurately capture all of the spectral dynamics of the scalars (enthalpy and species mass fractions). A new closure called `multi-scale mixing model' is presented for the mixing terms based on a spectral representation of the scalar field. Such a closure is developed in two levels. In level A, the eddy damped quasi normal Markovian (EDQNM) model, extended to multiple scalars by Ulitsky and Collins (Journal of Fluid Mechanics, 412, 2000), has been applied to the study of mixing of differential diffusing scalars and completely validated by direct numerical simulations (DNS). In level B, the `multi-scale model' of mixing is then developed based on this EDQNM closure. This model is implemented as an ensemble of stochastic particles, each carrying scalar concentrations distributed across different wavenumbers. Scalar exchanges within a given particle represent ``transfer' while scalar exchanges between particles represent ``mixing.' The model correctly predicts the evolution of an initial double delta function PDF (unmixed condition) into a self-similar Gaussian as seen in DNS by Eswaran and Pope (Physics of Fluids, 31, 1988). Comparisons of the model with DNS are in good agreement and extensions to multiple scalar mixing are also presented. To bring in the effect of chemical reactions, a new EDQNM model for a bimolecular reaction is developed (level A). This model is validated with DNS for a bimolecular reaction for unmixed initial conditions. A widely used PDF mixing model, Interaction by Exchange with Mean (Dopazo, Physics of Fluids,18, 1975) model fails even to qualitatively capture the product-reactant correlations, an essential statistic for multi-step chemical reactions, while EDQNM, due to the fact that it retains spectral information, is able to do a better job at predicting this statistic. Work has been initiated to include the effect of reaction into the multi-scale mixing model described earlier.