The role of particle-fluid velocity correlation in single-point statistical closures of dispersed turbulent two-phase flows

The evolution equation for the dispersed--phase turbulent kinetic energy in the standard continuum model for turbulent two--phase flow contains an unclosed term-- the Eulerian particle velocity--acceleration covariance---which is a two--point statistic. A widely-used Lagrangian linear slip-- velocity model for the particle acceleration implies a model for this quantity, which depends on the Eulerian {\em single--point} covariance of velocity between carrier fluid and dispersed-- phase particles. However, earlier Eulerian formulations have shown that the particle--fluid velocity covariance in a two--phase flow is a two--point statistic, which is always zero in the single--point limit, because the presence of one phase disallows the simultaneous presence of the other at the same space--time location. This paradox is resolved in this work, where it is shown that this Eulerian single--point covariance of carrier fluid velocity with dispersed--phase velocity is correctly interpreted as a single--point surrogate for the particle-- fluid velocity covariance. It is shown that in zero mean--slip homogeneous flows, this surrogate is nothing but the two-phase mixture velocity covariance, which can be expressed as a weighted sum of the velocity covariance in the fluid--phase and the velocity covariance in the dispersed-phase. Therefore, this single-- point surrogate for the particle--fluid velocity covariance should not form a part of the set of independent equations in {\em single-point} closures of turbulent two-phase flow.