Solid-liquid flows span a large parameter space, with dimensionless coordinates such as Stokes numbers, the solids volume fraction, the density ratio between the phases, and Reynolds numbers (e.g. associated with the continuous phase flow). We are interested in systems with appreciable inertia effects—i.e. non-zero Stokes and Reynolds numbers—having density ratios of the order of one and solids volume fractions of at least 0.1. In such flows, direct numerical simulations are desired to reveal the relevant interactions. The resolution required for DNS limits the size of the systems that we are able to simulate to the meso-scale. In this paper, examples of direct simulations based on the lattice-Boltzmann method of dense solid-liquid flows are presented, along with suggestions as to how to use their results at the macro-scale.

In a fluidized bed, the drag force acts to oppose the downward force of gravity on a particle, and thus provides the main mechanism for fluidization. Drag models that are employed in large-scale simulations of fluidized beds are typically based on either fixed-particle beds or the sedimentation of particles in liquids. In low-Reynolds-number (Re) systems, these two types of fluidized beds represent the limits of high Stokes number (St) and low St, respectively. In this work, the fluid-particle drag behaviour of these two regimes is bridged by investigating the effect of St on the drag force in low-Re systems. This study is conducted using fully resolved lattice Boltzmann simulations of a system composed of fluid and monodisperse spherical particles. In these simulations, the particles are free to translate and rotate based on the effects of the surrounding fluid. Through this work, three distinct regimes in the characteristics of the fluid-particle drag force are observed: low, intermediate and high St. It is found that, in the low-Re regime, a decrease in St results in a reduction in the fluid-particle drag. Based on the simulation results, a new drag relation is proposed, which is, unlike previous models, dependent on St.