A Force Correction Scheme for Volume Filtered Large Eddy Simulation of Flow Over a Permeable Bed of Particles*

A volume filtered formulation (Whitaker 1996) for large eddy simulation of flow over a bed of randomly packed particles, with size on the order of 10 wall units, is investigated. Volume filtering across the fluid-particle interface results in bed-normal porosity variation with unit porosity in the freestream that decreases sharply in the interface region to a closed-pack value within the bed. The streamwise velocity decreases near the fluid-particle interface and reaches Darcy velocity deep inside the bed. Drag force closure model based on the Ergun equation with correlations for permeability and Forchheimer tensor are evaluated using explicitly filtered, pore-resolved direct numerical simulation data at low Reynolds numbers. The closure model based on the Ergun equation works well deep inside the bed in the homogeneous porosity region, but overpredicts the drag in the fluid-particle interface region with inhomogeneous porosity variations. A uniform filter kernel width proportional to the particle size within the bed is shown to incorrectly sample large values from the free stream resulting in significantly higher drag in the transition region. A variable filter width bound by a unit Lipschitz constant that is proportional to the grid size in the freestream, rapidly increases in the interface region, and reaches a constant value proportional to the particle size inside the bed is devised to obtain a better representation of the velocity in the bed while avoiding non-physical oscillations in porosity. The variable filter width together with the assumption of an exponential Brinkman velocity solution in the interface region are used to develop a correction scheme that minimizes the error in the drag force compared to the filtered pore-resolved, direct numerical solution. The proposed correction and the model parameters are tested on a range of randomly packed bed porosities to show good predictive capability.