Analysis of unsteady forces in ordered arrays of monodisperse spheres

Abstract

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 ($0.1\,{<}\,\phi\,{<}\,0.4$), background flow Reynolds numbers ($0.25 \,{\le}\, \hbox{\it Re}_p\,{\le}\, 60$, where $\hbox{\it Re}_p\,{=}\,2 a u_f / \nu$) and oscillatory flow Reynolds numbers ($0.9 \,{\leq}\, {Re_\omega} \,{\leq}\, 420$ with ${Re_\omega}\,{=}\,2a^2 \omega / \nu$). Here $u_f$ is the superficial velocity of the fluid through the bed, $a$ is the particle radius, $\nu$ is the kinematic viscosity of the fluid, and $\omega$ is the oscillation frequency.

In the limit of ${Re_\omega} \,{\to}\, 0$ the quasi-steady-state drag force was obtained. At low $\hbox{\it Re}_p$ this force approached the steady-state drag force, while its increase with $\hbox{\it Re}_p$ 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 ${Re_\omega} \,{\to}\, \infty$ 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 ${\sim} 0.5$ at high ${Re_\omega}$, which is the well-known limit for isolated spheres in inviscid flows, to ${\sim} 1.0$ at low ${Re_\omega}$. This change is clearly a consequence of viscous effects. The ${Re_\omega}$ at which the transition between these limits occurs increases with $\phi$. The history force exhibits a strong decay towards lower values of ${Re_\omega}$ in accordance with the results of Mei et al. (1991) for isolated spheres; however, the ${Re_\omega}$ value at which this decay sets in increases appreciably with $\phi$. This $\phi$-dependence is associated with the limited separation between the particles available for the Stokes boundary layer.

It was found that the unsteady drag coefficient $\beta'$ varies with ${Re_\omega}$. At low $Re_p$, the drag coefficient initially decreases with increasing ${Re_\omega}$, passes through a minimum and then increases strongly. With increasing ${Re_\omega}$ the relative contribution of pressure and friction forces to the unsteady drag force changes.