Stochastic models for capturing dispersion in particle-laden flows

This study provides a detailed account of stochastic approaches that may be utilized in Eulerian-Lagrangian simulations to account for neighbour-induced drag force fluctuations. The frameworks examined here correspond to Langevin equations for the particle position (PL), particle velocity (VL) and fluctuating drag force (FL). Rigorous derivations of the particle velocity variance (granular temperature) and dispersion resulting from each method are presented. The solutions derived herein provide a basis for comparison with particle-resolved direct numerical simulation. The FL method allows for the most complex behaviour, enabling control of both the granular temperature and dispersion. A Stokes number St(F) is defined for the fluctuating force that relates the integral time scale of the force to the Stokes response time. Formal convergence of the FL scheme to the VL scheme is shown for St(F) >> 1. In the opposite limit, St(F) << 1, the fluctuating drag forces are highly inertial and the FL scheme departs significantly from the VL scheme.