# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 3 Results and Discussion
### Turbulence Statistics
In addition, the profile of \(\langle ww\rangle^{+}\) logarithmically varies as \(\langle ww\rangle^{+}=-0.38\ln y^{\prime+}+3.38\) in the range \(y^{\prime}=130\)-\(220\)[56], and a plateau is observed in \(\langle-ww\rangle^{+}\). The se observations can be predicted by [PERSON] [55] and [PERSON] [65]. The logarithmic variation in \(\langle ww\rangle^{+}\) and the plateau in \(\langle-wv\rangle^{+}\) support [PERSON]'s attached eddy hypothesis over rough  
Figure 8: (a) Wall-normal profiles of streamwise Reynolds stress (\(\langle uu\rangle^{+}\)). The green dashed line in (a) represents the logarithmic law of \(\langle uu\rangle^{+}=-0.67\ln y^{\prime+}+5.75\). (b) Wall-normal profiles of wall-normal Reynolds stress (\(\langle vv\rangle^{+}\)), spanwise Reynolds stress (\(\langle vw\rangle^{+}\)), and Reynolds shear stress (\(\langle-wv\rangle^{+}\)). The red dashed line in (b) shows the logarithmic law of \(\langle ww\rangle^{+}=-0.38\ln y^{\prime+}+3.38\). Figure 7: Wall-normal profiles of (a) streamwise mean velocity (\(U^{+}\)). The grey line represents the result of smooth. The blue and green dashed lines show the logarithmic law of \(U^{+}=1/\kappa\ln y^{\prime+}+B\) and \(U^{+}=1/\kappa\ln y^{\prime+}+B-\Delta U^{+}\), respectively. (b) Wall-normal profiles of conditionally averaged streamwise velocity (\(U_{i}^{+}\)) at P\({}_{0}\) (black), P\({}_{1}\) (green), P\({}_{2}\) (blue), and P\({}_{3}\) (red). An inset shows the results in near-wall region. walls [66]. This could imply that wall-attached structures play an important role in the present flow. To further analyze the streamwise Reynolds stress, a pre-multiplied spanwise energy spectrum of \(u\) (\(k_{z}\phi_{uu}\)) is considered. Figure 9a illustrates a 2D contour of \(k_{z}\phi_{uu}\) normalized by the maximum of \(k_{z}\phi_{uu}\) (\(k_{z}\phi_{uu}/k_{z}\phi_{uu,max}\)) as functions of \(\lambda_{z}{}^{+}\) and \(y^{+}\). Here, \(k_{z}\) is the spanwise wavenumber, and \(\lambda_{z}\) (=2\(\pi\)/\(k_{z}\)) is the spanwise wavelength. In the present study, \(k_{z}\phi_{uu}\) is defined in the region \(y^{\prime+}>k^{\prime+}\) since there is non-homogeneity in the spanwise direction at \(y^{\prime+}<k^{\prime+}\). A strong peak is observed in \(k_{z}\phi_{uu}/k_{z}\phi_{uu,max}\) at \(y^{\prime+}=39\) and \(\lambda_{z}{}^{+}=383\), and a weak peak can be seen at \(y^{\prime+}=230\) and \(\lambda_{z}{}^{+}=550\). The spanwise wavelength of the strong peak is \(\lambda_{z}{}^{+}\approx 4k^{+}\), similar to spanwise sizes of low momentum regions over elements in instantaneous flow fields [31; 33]. On the other hand, the scales are clearly separated in smooth with an inner peak at \(y^{\prime+}=13\) and \(\lambda_{z}{}^{+}=115\) and an outer peak at \(y^{\prime+}=170\) and \(\lambda_{z}{}^{+}=660\), as shown in Figure 9b. Large and small scales can be decomposed based on \(k_{z}\phi_{uu}\) for wall-bounded turbulence at \(\lambda_{z}/\delta=0.5\) (\(\lambda_{z}{}^{+}\approx 400\)) [67; 68]. Although the scale separation is not clear at \(y^{\prime+}>k^{\prime+}\) due to the roughness elements, the energy at long wavelengths near \(y^{\prime+}=30\) is extended to \(\lambda_{z}{}^{+}=100\), and a weak peak arises in the outer region.