Lagrangian Stochastic Modeling of Subfilter Motion for LES of Particle- Laden Flow

In this chapter, the adoption and development of a Langevin-type Lagrangian stochastic model for the transport of inertial particles by subfilter motion in LES is detailed. The different assumptions and numerical requirements needed for the implementation of the model are discussed.

The theoretical and numerical formulations of the Langevin model have been extensively discussed in the framework of particle-laden RANS. Its use is extended herein with the necessary modifications for the modeling of the fluid velocity seen by particles in LES framework. We introduce below the two formulations of Langevin equation used to model the time increment of the fluid velocity seen by inertial particles. For the first formulation, called standard formulation, closure of both drift and diffusion terms are similar to the one used for the fluid particle case. However, the SGS time scale with which inertial particles see the turbulence is modified to account for inertia and cross trajectories effects. For the second formulation, referred to as complete formulation, closure forms for the drift and the diffusion terms are described.

Numerical issues linked to the solution of the resulting stochastic differential equations as well as some important properties of the Langevin-type equations are discussed.

Keywords: Large eddy simulation, multiphase flows, turbulent flows, Lagrangian modeling, stochastic modeling, Langevin equation, drift term, diffusion matrix, spurious drift, fluid velocity seen