  Direct numerical simulation (DNS) of flow in a turbulent channel with a
random rough wall is performed at $Re_{\tau}=400$ and $600$. The roughness
geometry corresponds to the experiments of Flack and Schultz (personal
communication). The rough surface is used on the bottom wall of the channel.
DNS of a smooth channel flow and a rod-roughened channel flow are also
performed at $Re_{\tau}=400$ for validation and comparison. The skin friction
coefficient of the random-rough channel shows good agreement with the
experimental results of Flack and Schultz. Due to the roughness, the slip
velocity on the rough wall increases while the mean velocity decreases in the
log-law region. The streamwise and spanwise velocity fluctuations are enhanced
near the rough wall. The pressure fluctuations show a significant increase in
the roughness layer and exhibit a good collapse with the smooth wall in the
outer layer. The streamwise mean momentum balance shows that pressure and
viscous stress gradients are induced in the roughness layer, and the gradients
are amplified at higher $Re_{\tau}$. The statistics of wall-shear stress
fluctuations in the peak (above the mean height location) and valley (below the
mean height location) regions are examined. The results indicate that reverse
flow mainly occurs in the valley regions of the random roughness, and is
enhanced at higher $Re_{\tau}$, but is not as strong as the recirculation
within the cavities in the rod-roughened channel. The probability distribution
function of wall-shear stress shows a better collapse after subtracting the
mean and normalizing by the root-mean-squared value. The distribution tail is
widened by the random roughness, implying that the probability of extreme
events is increased. The probability of extreme events in the random-rough
channel increases with increasing $Re_{\tau}$, in accordance with previous
studies on smooth-wall flows.
