Direct numerical simulation of high aspect ratio spanwise-aligned bars

We conduct minimal-channel direct numerical simulations of turbulent flow over two-dimensional rectangular bars aligned in the spanwise direction. This roughness has often been described as $d$-type, as the roughness function $\unicode[STIX]{x0394}U^{+}$ is thought to depend only on the outer-layer length scale (pipe diameter, channel half-height or boundary layer thickness). This is in contrast to conventional engineering rough surfaces, named $k$-type, for which $\unicode[STIX]{x0394}U^{+}$ depends on the roughness height, $k$. The minimal-span rough-wall channel is used to circumvent the high cost of simulating high Reynolds number flows, enabling a range of bars with varying aspect ratios to be investigated. The present results show that increasing the trough-to-crest height, $k$, of the roughness while keeping the width between roughness bars, ${\mathcal{W}}$, fixed in viscous units, results in non-$k$-type behaviour although this does not necessarily indicate $d$-type behaviour. Instead, for deep surfaces with $k/{\mathcal{W}}\gtrsim 3$, the roughness function appears to depend only on ${\mathcal{W}}$ in viscous units. In these situations, the flow no longer has any information about how deep the roughness is and instead can only ‘see’ the width of the fluid gap between the bars.