Roughness height effects on turbulent boundary layers over rod-and cuboid-roughened walls

Direct numerical simulations (DNSs) of spatially developing turbulent boundary layers (TBLs) over rod- and cuboid-roughened walls are conducted to investigate the effects of the roughness height on the flow characteristics in the outer layer. The rod and cuboid roughness height ($k)$ is varied in the range of 0.1$\le k$/$\theta_{in}\le $1.8 (13$\le \delta $/$k\le $285). As the roughness height increases, the roughness function ($\Delta U^{\mathrm{+}})$ increases and the magnitude of the Reynolds stresses in the outer layer also increases. The outer layer similarity between the flows over the rough and smooth-walls is established when $\delta $/$k \ge $ 250 and 100 for the 2D rod and 3D cuboid, respectively. The continuous increase of the Reynolds stresses in the outer layer with an increase of $k$/$\delta $is explained by a large population of very long structures over the rough-wall flows. Moreover, as $k$/$\delta $ increases, the wider characteristic width of the structure leads to frequent spanwise merging between adjacent structures. The active spanwise merging events between the spanwise-offset LSMs increase the appearance of meandering significantly as $k$/$\delta $ increases.