Direct numerical simulations of turbulent flow over a permeable wall using a direct and a continuum approach

A direct numerical simulation (DNS) has been performed of turbulent channel flow over a three-dimensional Cartesian grid of 30×20×9 cubes in, respectively, the streamwise, spanwise, and wall-normal direction. The grid of cubes mimics a permeable wall with a porosity of 0.875. The flow field is resolved with 600×400×400 mesh points. To enforce the no-slip and no-penetration conditions on the cubes, an immersed boundary method is used. The results of the DNS are compared with a second DNS in which a continuum approach is used to model the flow through the grid of cubes. The continuum approach is based on the volume-averaged Navier–Stokes (VANS) equations [ S. Whitaker, “The Forchheimer equation: a theoretical development,” Transp. Porous Media 25, 27 (1996) ] for the volume-averaged flow field. This method has the advantage that it requires less computational power than the direct simulation of the flow through the grid of cubes. More in general, for complex porous media one is usually forced to use the VANS equations, because a direct simulation would not be possible with present-day computer facilities. A disadvantage of the continuum approach is that in order to solve the VANS equations, closures are needed for the drag force and the subfilter-scale stress. For porous media, the latter can often be neglected. In the present work, a relation for the drag force is adopted based on the Irmay [ “Modèles théoriques d’écoulement dans les corps poreux,” Bulletin Rilem 29, 37 (1965) ] and the Burke–Plummer model [ R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena (Wiley, New York, 2002) ], with the model coefficients determined from simulations reported by W. P. Breugem, B. J. Boersma, and R. E. Uittenbogaard [“Direct numerical simulation of plane channel flow over a 3D Cartesian grid of cubes,” Proceedings of the Second International Conference on Applications of Porous Media, edited by A. H. Reis and A. F. Miguel (Évora Geophysics Center, Évora, 2004), p. 27 ]. The results of the DNS with the grid of cubes and the second DNS in which the continuum approach is used, agree very well