Simulation and scaling analysis of periodic surfaces with small-scale roughness in turbulent Ekman flow

Roughness of the surface underlying the atmospheric boundary layer causes departures of the near-surface scalar and momentum transport in comparison with aerodynamically smooth surfaces. Here, we investigate the effect of $56\times 56$ homogeneously distributed roughness elements on bulk properties of a turbulent Ekman flow. Direct numerical simulation in combination with an immersed boundary method is performed for fully resolved, three-dimensional roughness elements. The packing density is approximately $10\,\%$ and the roughness elements have a mean height in wall units of $10 \lesssim H^+ \lesssim 40$ . According to their roughness Reynolds numbers, the cases are transitionally rough, although the roughest case is on the verge of being fully rough. We derive the friction of velocity and of the passive scalar through vertical integration of the respective balances. Thereby, we quantify the enhancement of turbulent activity with increasing roughness height and find a scaling for the friction Reynolds number that is verified up to $Re_\tau \approx 2700$ . The higher level of turbulent activity results in a deeper logarithmic layer for the rough cases and an increase of the near-surface wind veer in spite of higher $Re_\tau$ . We estimate the von Kármán constant for the horizontal velocity $\kappa _{m}=0.42$ (offset $A=5.44$ ) and for the passive scalar $\kappa _{h}=0.35$ (offset $\mathbb {A}=4.2$ ). We find an accurate collapse of the data under the rough-wall scaling in the logarithmic layer, which also yields a scaling for the roughness parameters $z$ -nought for momentum ( $z_{0{m}}$ ) and the passive scalar ( $z_{0{h}}$ ).