In this paper we perform direct numerical simulation (DNS) on the problem of fluid flow through a filter plate backed by a packed bed of spheres, resembling a cake layer on top of a membrane. For both the complete problem, and its single components (the filter plate and a bed of spheres of finite height) we have observed three flow regimes, depending on the Reynolds number. In each regime the flow resistance is showing a different scaling with the Reynolds number. In the Stokes flow regime the total flow resistance can be decomposed in linear independent components. The interior flows inside the filter holes and inside the packed bed follow the same correlations as hold for the single component. However, at the transition zone between filter plate and packed bed, there is a diverging flow in the first row of the packed bed, whose contribution in the flow resistance scales with the fractional hole to the power −1.5. Similar scaling exponent has been found earlier for the viscous-inertial regime with Reynolds numbers larger than 10, which has been modelled using the porous medium approach. Our findings suggest that also in the Stokes flow and the weakly flow regime the problem can also be solved with the same porous medium approach using the Navier-Stokes equation having Darcy–Brinkman terms incorporated. This investigation provides a good basis for developing more accurate analytical models for the flow resistance of membrane filters with a cake layer on top.
- Fluid flow