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Abstract
A family of Probability Density Functions (PDF's) generated by Johnson-Edgeworth Translation (JET) is used for statistical modeling of the mixing of an initially binary scalar in isotropic turbulence. The frequencies obtained by this translation are shown to satisfy some of the characteristics of the PDF's generated by the Amplitude Mapping Closure (AMC) (Kraichnan, 1989; Chen et al., 1989). In fact, the solution obtained by one of the members of this family is shown to be identical to that developed by the AMC (Pope, 1991). Due to this similarity and due to the demonstrated capabilities of the AMC, a justification is provided for the use of other members of JET frequencies for the modeling of the binary mixing problem. This similarity also furnishes the reasoning for the applicability of the Pearson Family (PF) of frequencies for modeling of the same phenomena. The mathematical requirements associated with the applications of JET in the modeling of the binary mixing problem are provided, and all the results are compared with data generated by Direct Numerical Simulations (DNS). These comparisons indicate that the Logit-Normal frequency portrays some subtle features of the mixing problem better than the other closures. However, none of the models considered (JET, AMC, and PF) are capable of predicting the evolution of the conditional expected dissipation and/or the conditional expected diffusion of the scalar field in accordance with DNS. It is demonstrated that this is due to the incapability of the models to account for the variations of the scalar bounds as the mixing proceeds. A remedy is suggested for overcoming this problem which can be useful in probability modeling of turbulent mixing, especially when accompanied by chemical reactions. While in the context of a single-point description the evolution of the scalar bounds cannot be predicted, the qualitative analytical-computational results portray a physically plausible behavior.
