On two-way coupling in gas-solid turbulent flows

An analysis of kinetic energy transfer in particle-laden turbulent flows is presented. The present study focuses on the subset in which dispersed-phase motion is restricted to particles in translation, particle diameters are smaller than the smallest lengthscales in the turbulent carrier flow, and the dispersed phase is present at negligible volume fraction. An analysis of the separate and exact two-fluid mean and turbulent kinetic energy transport equations shows that momentum exchange between the phases results in a transfer of kinetic energy from the mean to the fluctuating motion of the two-phase mixture. The source term accounting for fluid-particle coupling in the fluid turbulent kinetic energy equation is written as the sum of three parts, the first part representing the production of velocity fluctuations in the particle wake ("pseudo turbulence"), the second and third contributions - which act primarily on the larger scales of the fluid turbulent motion - representing a damping effect due to the turbulent fluctuation of the drag force and the effect of the transport of the particles by the fluid turbulence against their mean relative motion. A schematic representation of the energy transfers in particle-laden mixtures is also presented for the simplified systems under consideration, consistent with the separation of scales between perturbations introduced at the scale of the particle and the large, energy-containing scales of fluid turbulent motion. Implications of the energy transfers for ensemble-averaged modeling approaches are discussed, along with computational techniques that account for the back-effect of the particles on the flow using the point-force approximation. It is shown that the point-force approximation as typically implemented only accounts for the modulation of the large eddies, the contribution to wake production is not included, being implicitly assumed to be in local equilibrium with the corresponding viscous dissipation.