Analysis of the flow in inhomogeneous particle beds using the spatially averaged two-fluid equations

The drag force term appearing in two-fluid models for fluid–particle flows is commonly closed by expressing it as a function of the local quantities, such as the local particle volume fraction, the local slip velocity between the particle and fluid phases, and the local mean-squared fluctuating velocity of the particles. The adequacy of such closures for inhomogeneous suspensions has been debated in the literature and some researchers have suggested the need for additional terms involving spatial gradients in these quantities. To test this proposition, simulations of flow in inhomogeneous steady beds of particles have been performed using the lattice-Boltzmann method. The particle beds consisted of disordered assemblies with a density profile on a scale much larger than the particle radius. Inhomogeneous beds with a controlled density profile were generated in three different ways, (i) by inhomogeneous stretching of the particle bed in one direction, (ii) by applying an inhomogeneous force to the particle phase during random motion of the particles, and (iii) by taking snapshots of a direct simulation of a traveling wave in a fluidization simulation. The global structure of the three beds was comparable, while assessment of the radial distribution functions showed that the three beds exhibited clearly different microscopic structures.The force profiles along the inhomogeneous direction of the particle bed were obtained from the flow simulations. These were analyzed by applying spatial averaging in a manner identical to the averaging procedure that is used to derive the two-fluid model equations. The force profiles were compared to predictions based on flow simulations of homogeneous disordered particle beds over a range of volume fractions. To assess the role of the microstructure of the particle bed also simulations were executed where the homogeneous disordered bed was modified by either applying homogeneous stretching or by applying a lubrication force during generation of the particle bed.This study demonstrated that the microscopic structure of the particle bed has a severe impact on the closure of the drag force. Our computations did not reveal any evidence supporting the need for terms involving gradients in particle volume fraction in the drag force closure.