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A theoretical foundation for two widely used statistical representations of multiphase flows, namely the Eulerian-Eulerian (EE) and Lagrangian-Eulerian (LE) representations, is established in the framework of the probability density function (p.d.f.) formalism. Consistency relationships between fundamental statistical quantities in the EE and LE representations are rigorously established. It is shown that fundamental quantities in the two statistical representations bear an exact relationship to each other only under conditions of spatial homogeneity. Transport equations for the probability densities in each statistical representation are derived. Exact governing equations for the mean mass, mean momentum and second moment of velocity corresponding to the two statistical representations are derived from these transport equations. In particular, for the EE representation, the p.d.f. formalism is shown to naturally lead to the widely used ensemble-averaged equations for two-phase flows. Galilean-invariant combinations of unclosed terms in the governing equations that need to be modelled are clearly identified. The correspondence between unclosed terms in each statistical representation is established. Hybrid EE-LE computations can benefit from this correspondence, which serves in consistently transferring information from one representation to the other. Advantages and limitations of each statistical representation are identified. The results of this work can also serve as a guiding framework for direct numerical simulations of two-phase flows, which can now be exploited to precisely quantify unclosed terms in the governing equations in the two statistical representations. A theoretical foundation for two widely used statistical representations of multiphase flows, namely the Eulerian-Eulerian (EE) and Lagrangian-Eulerian (LE) representations, is established in the framework of the probability density function (p.d.f.) formalism. Consistency relationships between fundamental statistical quantities in the EE and LE representations are rigorously established. It is shown that fundamental quantities in the two statistical representations bear an exact relationship to each other only under conditions of spatial homogeneity. Transport equations for the probability densities in each statistical representation are derived. Exact governing equations for the mean mass, mean momentum and second moment of velocity corresponding to the two statistical representations are derived from these transport equations. In particular, for the EE representation, the p.d.f. formalism is shown to naturally lead to the widely used ensemble-averaged equations for two-phase flows. Galilean-invariant combinations of unclosed terms in the governing equations that need to be modelled are clearly identified. The correspondence between unclosed terms in each statistical representation is established. Hybrid EE-LE computations can benefit from this correspondence, which serves in consistently transferring information from one representati...

Modeling turbulent multiphase flows, such as sprays, is a major challenge owing to droplet (or solid-particle) interactions with a wide range of turbulence length and time scales. In a broad class of Lagrangian-Eulerian models, the instantaneous Lagrangian dispersedphase velocity evolves on a time scale that is proportional to the particle response time 2 ()/(18) p d p f f d. Numerical simulations of a model from this class reveal a nonmonotonic and unphysical increase of the turbulent kinetic energy (TKE) in the dispersed phase k d that is not seen in direct numerical simulations (DNS) of decaying, homogeneous turbulence laden with solid particles. Analysis of this class of models shows that for a linear drag law corresponding to the Stokes regime, the entire class of models will predict an anomalous increase in k d for decaying turbulent flow laden with solid particles or droplets. Even though the particle response time is the appropriate time scale to characterize momentum transfer between sub-Kolmogorov-size dispersed-phase particles and the smallest turbulent eddies (for droplet/particle Reynolds number of < 1), it is incapable of capturing the range of time-and length-scale interactions that are reflected in the evolution of k d. A new model that employs a time scale based on a multiscale analysis is proposed. This model succeeds in capturing the dispersed-phase TKE and fluid-phase TKE evolution observed in DNS. The model also correctly predicts the trends of TKE evolution in both phases for different Stokes numbers.

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