Stokes number effects in Lagrangian stochastic models of dispersed two-phase flows

The statistical properties of fluid velocities along particle trajectories in turbulent flows have a conditional dependency upon particle velocity. It is shown that the formulation of Lagrangian stochastic (LS) models for particle trajectories in terms of the well-mixed condition for these conditional velocity statistics is exactly analogous to the formulation of second-order LS models for fluid-particle trajectories. The particle aerodynamic response time is shown to be incorporated at second order, which together with the Lagrangian timescale introduced at first order, defines the Stokes number. Reynolds-number effects can be incorporated at third order. The corresponding Fokker-Planck equation is shown to be identical to that advocated by Pozorski and Minier [Phys. Rev. E 59 (1999) 855], who included the fluid velocities "seen" by a particle in the probability density function (pdf) formalism of Reeks and co-workers as a means of circumventing the closure problem (prescribing a closure for the particle flux induced by the fluid) associated with that approach. It is demonstrated that the neglect of Stokes-number effects accounts, in part, for the tendency of first-order LS models to underpredict particle deposition velocities in the diffusion-impaction regime.