# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 3 Results and Discussion  
### Roughness Step Change  
Figure 2 displays 3D iso-surfaces of \(\lambda_{ci}\), where \(\lambda_{ci}\) is a measure of swirling strength [49], used to identify vortical motions, focusing on the region around the step change at \(x/\delta_{0}\) = 700. The iso-surfaces illustrate 5% of the maximum value of \(\lambda_{ci}\), with color depths representing wall-normal locations. The cubic roughness elements become visible in the region \(x/\delta_{0}>700\). At \(x/\delta_{0}=700\), horizontal vortices are observed near the bottom of the upstream face of the elements. Vortex sheets emerge on the top and sides of the elements and are then ejected toward the downstream direction, leading to the generation of hairpin-like structures [48]. After a few rows of elements, hairpin-like structures originate from a strong shear layer over elements with less contribution from vortex sheets [28; 32; 33; 50].  
Figure 1: A schematic view of cubic roughness elements. The height of each element is \(k\), and \(l_{x}\) and \(l_{z}\) are streamwise and spanwise sizes of each element, respectively. The distance between adjacent elements is \(p_{x}\) and \(p_{z}\) in the streamwise and spanwise directions, respectively. Virtual origin is \(\varepsilon\), and \(y^{\prime}\) is defined as \(y-\varepsilon\). P\({}_{0}\), P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\) denote data locations.  
Figure 3a-c show the streamwise variations of \(\delta\), displacement thickness (\(\delta^{\star}\)), and \(Re_{\theta}\), respectively. In Figure 3, all quantities are spatially averaged, and each circle represents the averaged result over \(p_{x}/k=2\). The boundary layer thickness is defined as the wall-normal location where \(U=0.99U_{\infty}\), and \(\delta^{\star}\) gradually increases with an increase in \(x/\delta_{0}\). The magnitude of \(Re_{\theta}\) increases up to 1500 near the exit boundary, representing an increase in \(\theta\) as \(x/\delta_{0}\) increases. Figure 3d represents the streamwise variations of the virtual origin normalized by the height of cubic roughness (\(\varepsilon/k\)). The magnitude of \(\varepsilon/k\) is 0.81, which is lower than one, similar to that of [PERSON] et al. [32]. A larger \(\varepsilon/k\) implies that the contribution of the form drag to the total drag is dominant compared to the frictional drag, which can be negligible [51]. The friction velocity over a rough wall can be obtained from the total drag, the sum of frictional and form drag, as \(u_{\tau}/{U_{\infty}}^{2}=0.5(C_{f}+C_{p})\)[7]. The streamwise variations of \(u_{\tau}\) are displayed in Figure 3e, where the magnitude of \(u_{\tau}\) decreases with an increase in \(\delta_{0}\). Based on the streamwise variations of \(\delta\) and \(u_{\tau}\), the friction Reynolds number can be obtained (Figure 3f). The magnitude of \(Re_{\tau}\) gradually increases over the step change and then converges. In the present study, a DNS dataset is analyzed in the range \(x/\delta_{0}=839\)-\(857\), where the averaged \(Re_{\tau}\) is 816 with \(\delta/\delta_{0}=13.14\).