# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 3 Results and Discussion
### Turbulence Statistics  
Figure 7a displays wall-normal profiles of the streamwise mean velocity in wall units, with the result of smooth represented by the grey line. The green dashed line shows the logarithmic law of \(U^{+}=1/\kappa\ln y^{\prime^{+}}+B-\Delta U^{+}\) for a rough wall, and the blue dashed line represents \(U^{+}=1/\kappa\ln y^{\prime^{+}}+B\) for a smooth wall. Here, \(B\) is a constant of 5.1, and \(\Delta U^{+}\) is the roughness function. The magnitude of \(\Delta U^{+}\) is 10.15, estimated as the best fitting of \(U^{+}\) with logarithmic variation in the range \(y^{\prime+}\) = 30-200. The streamwise velocity is conditionally averaged at four locations, namely P\({}_{0}\), P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\), as illustrated in Figure 1. Figure 7b depicts the conditionally averaged streamwise velocity (\(U_{i}\)) with respect to \(y/\delta\). The four profiles of \(U_{i}\) overlap at \(y^{\prime}/\delta>0.1\), representing that the influences of roughness elements on the mean flow are restricted to the region \(y^{\prime}/\delta<0.1\). An inset in Figure 7b provides a magnified view of \(U_{i}{}^{+}\) near the roughness according to \(y^{+}\). The magnitude of \(U_{0}{}^{+}\) (black) is zero below \(y^{+}=k^{+}=98.6\), indicating that the no-slip boundary condition is well applied to the immersed boundary at surfaces of the cubic roughness elements. Reverse flows, as negative \(U_{1}{}^{+}\) (green) and \(U_{2}{}^{+}\) (blue), are observed in the region \(y/k<1\) at P\({}_{1}\) and P\({}_{2}\), consistent with the observations of [PERSON] et al. [25] and the flow patterns [33; 37]. Since roughness elements are located upstream of P\({}_{1}\) and downstream of P\({}_{2}\), the flow direction bypasses the elements toward the wall-normal and spanwise directions or is reversed. Reynolds stresses are averaged at all points of the wall-parallel plane and analyzed. Figure 8a presents the streamwise Reynolds stress in wall units \((\langle uu\rangle^{+})\) with respect to \(y^{\prime+}\). The magnitude of \(\langle uu\rangle^{+}\) rapidly increases at \(y^{\prime+}=k^{+}=19\) in accordance with the height of cubic roughness, and the profile of \(\langle uu\rangle^{+}\) has a peak at \(y^{\prime+}=37\). Here, \(k^{\prime}\) is defined as \(k-\varepsilon\)  
Figure 6: The profile of \(\langle uu\rangle^{0.5}/U\) with respect to \(U/U_{\infty}\). The dashed grey line shows the fully-rough regime from [PERSON] et al. [54]. The green dashed line represents the logarithmic law of \(\langle uu\rangle^{+}=-0.67\ln y^{\prime+}+5.75\) from the best fitting. The profile of \(\langle uu\rangle^{+}\) aligns with the logarithmic law in the range \(y^{\prime}=130\)-\(220\). The logarithmic variation in \(\langle uu\rangle^{+}\) is one of the statistical features of [PERSON]'s attached eddy hypothesis [55]. [PERSON] and [PERSON] [56] observed the logarithmic variation in the streamwise Reynolds stress reconstructed by wall-attached \(u\) structures of ZPG TBL in the range \(y^{+}=100\)-\(0.3\delta^{+}\), similar to a TBL subjected to adverse pressure gradient [57], pipe and channel flows [58; 59], and drag-reduced flow [60]. For high-Reynolds-number turbulence, logarithmic variations in \(\langle uu\rangle^{+}\) are reported in experiments [52; 61] and numerical simulations [62; 63; 64]. Although the present result is under relatively lower \(Re_{\tau}\), the statistical feature of wall-attached structures typical of high-Reynolds-number flows can be observed. Figure 8(b) displays three wall-normal profiles of wall-normal Reynolds stress (\(\langle vv\rangle^{+}\)), spanwise Reynolds stress (\(\langle ww\rangle^{+}\)), and Reynolds shear stress (\(\langle-wv\rangle^{+}\)). Each profile of \(\langle ww\rangle^{+}\) and \(\langle vv\rangle^{+}\) exhibits a peak at \(y^{\prime+}=84\) and \(140\), respectively.