# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 3 Results and Discussion
### Data Validation  
In this section, turbulence statistics are compared with previous results for validation. Figure 4a shows the streamwise mean velocity normalized by \(U_{\infty}\) with respect to \(y^{\prime}/\delta\). The grey line represents \(U/U_{\infty}\) for smooth, and the blue squares depict the results of [PERSON] et al. [27]. Since \(\varepsilon\) is zero in a smooth wall, \(y^{\prime}\) for smooth is the same as \(y\). The profile of \(U/U_{\infty}\) deviates from the smooth result near the wall, but both results converge well in the region \(y^{\prime}/\delta>0.6\). Although the 3D cubic elements extend up to  
Figure 3: Streamwise variations of spatially averaged (**a**) boundary layer thickness (\(\delta/\delta_{0}\)), (**b**) displacement thickness (\(\delta^{\star}/\delta_{0}\)), (**c**) momentum thickness Reynolds number (\(Re_{\theta}\)), (**d**) virtual origin (\(\varepsilon/k\)), (**e**) friction velocity (\(u_{\tau}\)), and (**f**) friction Reynolds number (\(Re_{\tau}\)). Figure 2: Three-dimensional iso-surfaces of 5% of the maximum of \(\Lambda_{ci}\). The color depths of iso-surfaces illustrate wall-normal locations. The grey represents cubic roughness elements. \(y^{\prime}/\delta=0.023\), where the profile of \(U/U_{\infty}\) has an inflection point with a strong shear layer, the streamwise velocity is recovered. As shown in Figure 4a, \(U/U_{\infty}\) is similar to the result of [PERSON] et al. [27]. Figure 4b represents the streamwise mean velocity in wall units with respect to \(y^{\prime}/y_{0}\). Here, \(y_{0}\) is the roughness length, which can be evaluated from the fitting of the logarithmic law of \(U^{+}=1/\kappa\ln(y^{\prime}/y_{0})\), where \(\kappa\) is the von Karman constant of 0.41. In the present study, \(y_{0}/\delta\) is 0.010, and \(y_{0}{}^{+}\) is 8.26. The dashed grey line in Figure 4b indicates the logarithmic law of \(U^{+}=1/\kappa\ln(y^{\prime}/y_{0})\). The magnitude of \(U^{+}\) varies logarithmically in the range \(y^{\prime}/y_{0}=\) 10-26. The squares in Figure 4b represent the results of [PERSON] et al. [30] and [PERSON] et al. [28], and they align with the logarithmic law in the range \(y^{\prime}/y_{0}=\) 6-70. A wider range for the logarithmic law is observed at a higher \(Re_{\tau}\) compared to the present result. The streamwise Reynolds stress in wall units (\(\langle uu\rangle^{+}\)) is compared to the experimental results of [PERSON] et al. [30] and [PERSON] et al. [28]. The bracket \(\langle\cdot\rangle\) indicates the time- and ensemble-averaged quantity. As shown in Figure 5, the magnitude of \(\langle uu\rangle^{+}\) is lower than that of the experimental results, which is attributed to the relatively lower \(Re_{\tau}\)[35]. [PERSON] and [PERSON] [35] reported that long-wavelength energy in the outer region penetrates the near-wall region, called footprints, leading to an enhancement of streamwise Reynolds stress at the inner peak. The dashed grey line represents the logarithmic law of \(\langle uu\rangle^{+}=B_{1}-A_{1}\ln(y^{\prime}/\delta)\). Here, \(B_{1}\) is a constant that depends on the flow geometry and wake parameter, and \(A_{1}=\) 1.26 is the slope constant proposed by [PERSON] et al. [52]. The profile of \(\langle uu\rangle^{+}\) follows the logarithmic law in the range \(y^{\prime}/\delta=\) 0.36-0.48 and overlaps with the experimental results of [PERSON] et al. [28] over \(y^{\prime}/\delta=\) 0.36. Figure 4: Wall-normal profiles of streamwise mean velocity (\(U\)) (**a**) normalized by free-stream velocity (\(U_{\infty}\)) and (**b**) in wall units. The dashed grey line in (**b**) represents the logarithmic law of \(U^{+}=1/\kappa\ln(y^{\prime}/y_{0})\). Figure 5: Wall-normal profile of streamwise Reynolds stress (\(\langle uu\rangle^{+}\)). The dashed grey line corresponds to the logarithmic law of \(\langle uu\rangle^{+}=B_{1}-A_{1}\ln(y^{\prime}/\delta)\). [PERSON] and [PERSON] [53] proposed a diagnostic function to plot the streamwise mean velocity and standard deviation of \(u\).