Numerical computation of macroscopic turbulence quantities in representative elementary volumes of the porous medium

In this study, fully developed macroscopic turbulence quantities-based on their definitions in some existing turbulence models for porous media as well as those based on definitions introduced in a recently developed model [F.E. Teruel, Rizwan-uddin, A new turbulence model for porous media flows. Part I: Constitutive equations and model closure, Int. J. Heat Mass Transfer (2009)]-are computed and analyzed for different Reynolds numbers as well as for different porosity levels. When computed based on the definition introduced in the new model, these numerically computed, pore-level turbulent quantities provide closure to the formulation. A large set of microscopic turbulent flow simulations of the REV of a porous medium, formed by staggered square cylinders, is carried out to achieve these tasks. For each Reynolds number selected, ten different porosities are simulated in the 5-95% range. The Reynolds number is varied from Re = 103 to Re = 105, covering four different cases of the turbulence flow regime. Numerical results obtained for the macroscopic turbulent kinetic energy based on the new definition show that the spatial dispersion of the mean flow is the main contributor to this quantity at low porosities. Additionally, it is found that for high porosities, the spatial average of the turbulent kinetic energy is the main contributor but the spatial dispersion of the mean flow cannot be neglected. The new definition of the macroscopic dissipation rate is found to asymptotically approach the volume average of this quantity at high Reynolds numbers. It is confirmed that microscopic numerical simulations are consistent with the macroscopic law that states that the macroscopic dissipation rate is determined by the pressure-drop through the REV.