Upscaling the Navier-Stokes Equation for Turbulent Flows in Porous Media Using the Method of Volume Averaging

The closure problem for turbulent flows through packed beds has drawn much attention in the recent decade, because of the extensive applications in both natural and engineering systems. Different from the conventional method of directly modifying the Darcy-Forchheimer correlation, using RANS models or the double-averaging approach, the method of volume averaging (Whitaker 1996) is applied to develop the volume averaged Navier-Stokes equation (VANS). An algebraic model is proposed to close the VANS equation by upscaling the data obtained from direct numerical simulations (DNS). The DNS is performed in a face-centered cubic porous unit cell, where flows at different Reynolds numbers (10, 100, 200, 300, 500 and 1000) are simulated. The simple algebraic model is then assessed by comparing with a conventional, Forchheimer-correction-based model. The proposed model shows a better prediction of the macroscopic velocity especially at high Reynolds number flow regime. Finally, DNS of turbulent flows through a randomly packed porous bed is conducted to test the proposed model.