Analysis of unsteady forces in ordered arrays of monodisperse spheres

Time-dependent flows of a Newtonian fluid through periodic arrays of spheres were simulated using the lattice-Boltzmann scheme. By applying a constant body force per unit mass to the fluid, a steady background fluid flow through the array of stationary spheres was first established. Subsequently, a small-amplitude perturbation to the body force, which varied periodically in time, was added and the long-time behaviour of the unsteady flow fields and the forces on the particles were determined. From the simulations, the pressure and friction (shear) forces acting on the particles were determined for a range of conditions. Results on simple cubic lattices are presented. Computations spanned a range of particle volume fractions (0.1<φ< 0.4), background flow Reynolds numbers (0.25 Rep 60, where Rep =2auf /ν) and oscillatory flow Reynolds numbers (0.9 Reω 420 with Reω =2aω/ν). Here uf is the superficial velocity of the fluid through the bed, a is the particle radius, ν is the kinematic viscosity of the fluid, and ω is the oscillation frequency. In the limit of Reω → 0 the quasi-steady-state drag force was obtained. At low Rep this force approached the steady-state drag force, while its increase with Rep was stronger than the steady-state drag force, similar to that for isolated spheres given by Mei et al. (J. Fluid Mech., vol. 233, 1991, p. 613). The unsteady force was decomposed into pressure and friction components. The phase angles of these components in the limit Reω →∞ indicate that the virtual mass force contributes to the unsteady pressure force while the history force contributes to the friction force. The remainder of the unsteady friction and pressure forces is attributed to unsteady drag force. The apparent virtual mass coefficient was found to vary from ∼0.5 at high Reω, which is the well-known limit for isolated spheres in inviscid flows, to ∼1.0 at low Reω. This change is clearly a consequence of viscous effects. The Reω at which the transition between these limits occurs increases with φ. The history force exhibits a strong decay towards lower values of Reω in accordance with the results of Mei et al. (1991) for isolated spheres; however, the Reω value at which this decay sets in increases appreciably with φ. This φ-dependence is associated with the limited separation between the particles available for the Stokes boundary layer. It was found that the unsteady drag coefficient β ′ varies with Reω. At low Rep , the drag coefficient initially decreases with increasing Reω, passes through a minimum and then increases strongly. With increasing Reω the relative contribution of pressure and friction forces to the unsteady drag force changes.