On the relationship between stochastic Lagrangian models of turbulence and second-moment closures

A detailed examination is performed of the relationship between stochastic Lagrangian models—used in PDF methods—and second-moment closures. To every stochastic Lagrangian model there is a unique corresponding second-moment closure. In terms of the second-order tensor that defines a stochastic Lagrangian model, corresponding models are obtained for the pressure-rate-of-strain and the triple-velocity correlations (that appear in the Reynolds-stress equation), and for the pressure-scrambling term in the scalar flux equation. There is an advantage in obtaining second-moment closures via this route, because the resulting models automatically guarantee realizability. Some new stochastic Lagrangian models are presented that correspond (either exactly or approximately) to popular Reynolds-stress models.