(189d) Stochastic Drag Formulations for Particle-Laden Flows

Existing drag force closures for volume-filtered Eulerian-Lagrangian (EL) methods typically capture the mean hydrodynamic force experienced by an assembly of particles. As a result, sources to granular temperature and particle dispersion, arising from neighbor-induced drag force fluctuations, are not captured. This study provides a detailed account of stochastic approaches that may be utilized in EL simulations to account for a distribution of drag forces. 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 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 behavior, enabling control of both the granular temperature and dispersion. A Stokes number is defined that relates the Stokes response time to the integral time scale of the fluctuating force. Formal convergence of the FL and VL scheme is shown for large fluctuating force Stokes numbers -- i.e. rapid uncorrelated drag force fluctuations. In the opposite limit, the fluctuating drag forces are highly inertial and the FL scheme departs significantly from the VL scheme.