# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 3 Results and Discussion
### Quadrant Analysis of Reynolds Shear Stress  
The quadrant analysis of Reynolds shear stress, as introduced by [PERSON] et al. [69], was conducted to provide a statistical interpretation of turbulence structures within turbulent shear flows. Quadrant events are defined based on the combination of \(u\) and \(v\), denoted as Q\({}_{1}\) (\(u>0\) and \(v>0\)), Q\({}_{2}\) (\(u<0\) and \(v>0\)), Q\({}_{3}\) (\(u<0\) and \(v<0\)), and Q\({}_{4}\) (\(u>0\) and \(v<0\)). Physically, Q\({}_{2}\) and Q\({}_{4}\) are associated with ejection and sweep motions, while Q\({}_{1}\) and Q\({}_{3}\) are interpreted as outward and inward motions.  
The area fraction of each quadrant event (\(AF_{i}\)) with respect to \(y/k\) is depicted in Figure 10a, where the results of [PERSON] et al. [33] are presented with square symbols for Q\({}_{1}\) (green), Q\({}_{2}\) (black), Q\({}_{3}\) (blue), and Q\({}_{4}\) (red). Across the entire region, Q\({}_{2}\) and Q\({}_{4}\) contribute dominantly to the occupied area, with \(AF_{4}\) (red) exhibiting prominence near the wall under roughness, particularly \(AF_{2}\) (black) at \(y/k>0.25\). The distributions of \(AF_{1}\) (green), \(AF_{2}\), \(AF_{3}\) (blue), and \(AF_{4}\) are found to be similar to those reported by [PERSON] et al. [33]. Figure 10b presents wall-normal profiles of \(AF_{i}\) as a function of \(y/\delta\), focusing on the outer region. Similar to the near-wall region, \(AF_{2}\) and \(AF_{4}\) emerge as dominant contributors in the outer region. However, the magnitude of \(AF_{4}\) increases with \(y/\delta\), while that of \(AF_{2}\) decreases. Notably, the magnitude of \(AF_{1}\) surpasses that of \(AF_{2}\) near the edge of the boundary layer. The results of smooth align well with the present results in the outer region, reminiscent of [PERSON]'s outer-layer similarity [55].  
Figure 11a illustrates the fractional contribution of each quadrant \((\langle-uv\rangle_{i}^{*})\) defined as \(\langle-uv\rangle_{i}/\sum_{k=1}^{4}\lvert\langle-uv\rangle_{k}\rvert\), which is compared with DNS data from [PERSON] et al. [33] denoted as square symbols for Q1 (green), Q2 (black), Q3 (blue), and Q4 (red). The magnitudes of \(\langle-uv\rangle_{1}^{*}\) (green) and \(\langle-uv\rangle_{3}^{*}\) (blue) are dominant near the wall under roughness but decrease with increasing \(y/k\). Additionally, the magnitude of \(\langle-uv\rangle_{4}^{*}\) (red) increases rapidly from the wall to the top of the elements and then decreases at \(y/k>1\). Ejections maintain their fractional contribution dominantly, especially in the outer region. As seen in Figure 11a, \(\langle-uv\rangle_{2}^{*}\) (black) and \(\langle-uv\rangle_{4}^{*}\) are dominant contributors in the range \(y/k>1.2\). All trends of \(\langle-uv\rangle_{i}^{*}\) are similar to the results of [PERSON] et al. [33]. Figure 11b presents wall-normal profiles of Reynolds shear stress for quadrant events \((\langle-uv\rangle_{i}^{+})\) with respect to \(y/\delta\), and they are compared to the results of smooth. The dashed grey lines indicate \(\langle-uv\rangle^{+}\), equivalent to the summation of \(\langle-uv\rangle_{i}^{+}\) at all quadrants. The magnitude of \(\langle-uv\rangle_{i}^{+}\) decreases toward the edge of the boundary layer. Ejections and sweeps dominate in the outer region, where \(\langle-uv\rangle_{2}^{+}\) is greater than \(\langle-uv\rangle_{4}^{+}\). Although \(\langle-uv\rangle_{i}^{+}\) is lower than that of smooth, the behavior of the profiles is similar, in agreement with the results of [PERSON] and [PERSON] [70].