  We investigate rough-wall turbulent flows through direct numerical
simulations of flow over three-dimensional transitionally rough sinusoidal
surfaces. The roughness Reynolds number is fixed at $k^+=10$, where $k$ is the
sinusoidal semi-amplitude, and the sinusoidal wavelength is varied, resulting
in the roughness solidity, $\Lambda$ (frontal area divided by plan area)
ranging from 0.05 to 0.54. The high cost of resolving both the flow around the
dense roughness elements and the bulk flow is circumvented by the use of the
minimal-span channel technique, recently demonstrated by Chung et al. (J. Fluid
Mech., vol. 773, 2015, pp. 418-431) to accurately determine the Hama roughness
function, $\Delta U^+$. Good agreement of the second-order statistics in the
near-wall roughness-affected region between minimal- and full-span rough-wall
channels is observed. In the sparse regime of roughness ($\Lambda \lesssim
0.15$) the roughness function increases with increasing solidity, while in the
dense regime the roughness function decreases with increasing solidity. It was
found that the dense regime begins when $\Lambda\gtrsim0.15-$0.18, in agreement
with the literature. A model is proposed for the limit of
$\Lambda\rightarrow\infty$, which is a smooth wall located at the crest of the
roughness elements. This model assists with interpreting the asymptotic
behaviour of the roughness, and the rough-wall data presented in this paper
show that the near-wall flow is tending towards this modelled limit. The peak
streamwise turbulence intensity, which is associated with the turbulent
near-wall cycle, is seen to move increasingly further away from the wall with
increasing solidity. In the sparse regime, increasing $\Lambda$ reduces
streamwise turbulent energy associated with the near-wall cycle, while
increasing $\Lambda$ in the dense regime increases turbulent energy
[continued]$\ldots$
