On multiphase turbulence models for collisional fluid–particle flows

Starting from a kinetic theory (KT) model for monodisperse granular flow, the exact Reynolds-averaged (RA) equations are derived for the particle phase in a collisional fluid–particle flow. The corresponding equations for a constant-density fluid phase are derived from a model that includes drag and buoyancy coupling with the particle phase. The fully coupled macroscale/hydrodynamic model, rigorously derived from a kinetic equation for the particles, is written in terms of the particle-phase volume fraction, the particle-phase velocity and the granular temperature (or total granular energy). As derived from the hydrodynamic model, the RA turbulence model solves for the RA particle-phase volume fraction, the phase-averaged (PA) particle-phase velocity, the PA granular temperature and the PA turbulent kinetic energy of the particle phase. Thus, unlike in most previous derivations of macroscale turbulence models for moderately dense granular flows, a clear distinction is made between the PA granular temperature $\Theta _\textit {p}$, which appears in the KT constitutive relations, and the particle-phase turbulent kinetic energy $k_\textit {p}$, which appears in the turbulent transport coefficients. The exact RA equations contain unclosed terms due to nonlinearities in the hydrodynamic model and we briefly discuss the available closures for these terms. Finally, we demonstrate by comparing model predictions with direct numerical simulation results that even for non-collisional fluid–particle flows it is necessary to provide separate models for $\Theta _\textit {p}$ and $k_\textit {p}$ in order to correctly account for the effect of the particle Stokes number and mass loading.