Direct numerical simulations have been carried out for a fully developed turbulent channel flow with a smooth upper wall and a lower wall consisting of square bars separated by a rectangular cavity. A wide range of $w/k$, the cavity width to roughness height ratio, was considered. For $w/k\,{\ge}\,7$, recirculation zones occur immediately upstream and downstream of each element while mean streamlines and spatial distributions of the skin frictional drag indicate that each element is virtually isolated. The maximum form drag occurs at $w/k\,{=}\,7$ and coincides with the minimum skin frictional drag. The dependence on $w/k$ of the Clauser roughness function reflects that of the form drag.

Computations of channel flow with rough walls comprising staggered arrays of cubes having various plan area densities are presented and discussed. The cube height h is 12.5% of the channel half-depth and Reynolds numbers (u_τh/ν) are typically around 700 – well into the fully rough regime. A direct numerical simulation technique, using an immersed boundary method for the obstacles, was employed with typically 35 million cells. It is shown that the surface drag is predominantly form drag, which is greatest at an area coverage around 15%. The height variation of the axial pressure force across the obstacles weakens significantly as the area coverage decreases, but is always largest near the top of the obstacles. Mean flow velocity and pressure data allow precise determination of the zero-plane displacement (defined as the height at which the axial surface drag force acts) and this leads to noticeably better fits to the log-law region than can be obtained by using the zero-plane displacement merely as a fitting parameter. There are consequent implications for the value of von Kármán's constant. As the effective roughness of the surface increases, it is also shown that there are significant changes to the structure of the turbulence field around the bottom boundary of the inertial sublayer. In distinct contrast to two-dimensional roughness (longitudinal or transverse bars), increasing the area density of this three-dimensional roughness leads to a monotonic decrease in normalized vertical stress around the top of the roughness elements. Normalized turbulence stresses in the outer part of the flows are nonetheless very similar to those in smooth-wall flows.

DNS results for turbulent channel flow with two- and three-dimensional roughness elements are discussed. The focus is on surfaces where the separation between consecutive roughness elements is small. In these cases, the roughness function is strongly influenced by the shape of the roughness elements. Usually, the roughness function is plotted against geometrical parameters and a satisfactory correlation does not emerge. Alternatively, a rather good collapse of the data is achieved when the roughness function is plotted against the rms of the normal velocity on the plane of the crests. This good relationship was observed previously for two-dimensional geometrical disturbances (Orlandi, P., Leonardi, S. and Antonia, R. A., 2006, Turbulent channel flow with either transverse or longitudinal roughness elements on one wall. Journal of Fluid Mechanics, to appear), here it is extended to three-dimensional geometries that are more often found in practical applications. Three-dimensional roughness has a strong influence on the overlying flow than two-dimensional roughness. This is shown through visualizations of the instantaneous velocity and vorticity, and through the comparison among the turbulent intensities and Reynolds shear stress distributions.

Roughness is classified by the so-called roughness function, which represents the downward shift of the velocity profile relative to a smooth wall. The dependence of the roughness function on the Reynolds number is discussed with the aim of clarifying the difference between d-type and k-type behaviors. This difference has been traditionally associated with the stability of the flow within the roughness elements. The present direct numerical simulation results indicate that the difference more correctly reflects the different contributions from the frictional drag and pressure drag to the total stress.