# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 3 Results and Discussion
### Turbulent Kinetic Energy Budgets  
The transport equation of Reynolds stress can be expressed as follows:  
\[\frac{\partial\langle u_{i}u_{j}\rangle}{\partial t}+U_{k}\frac{\partial \langle u_{i}u_{j}\rangle}{\partial x_{k}}=Pd_{ij}-Ds_{ij}+Td_{ij}+Pv_{ij}+Vd_ {ij}, \tag{3}\]  
Figure 11: (**a**) Relative contributions of each quadrant to [PERSON] shear stress \((\langle-uv\rangle_{i}^{*})\): Q1 (green), Q2 (black), Q3 (blue), and Q4 (red). (**b**) Wall-normal profiles of Reynolds shear stress of quadrant events \((\langle-uv\rangle_{i}^{+})\). The dashed lines denote [PERSON] shear stress \((\langle-uv\rangle^{+})\). Each squares in (**a**) and each circles in (**b**) represent \(AF_{i}\) of [PERSON] et al. [33] and smooth, respectively. Figure 10: Area fraction (\(AF_{i}\)) of quadrant events of Reynolds shear stress with respect to (**a**) \(y/k\) and (**b**) \(y/\delta\): Q1 (green), Q2 (black), Q3 (blue), and Q4 (red). Each squares in (**a**) and each circles in (**b**) represent \(AF_{i}\) of [PERSON] et al. [33] and smooth, respectively. where \(Pd_{ij}\), \(Ds_{ij}\), \(Td_{ij}\), \(Po_{ij}\), and \(Vd_{ij}\) denote the production, viscous dissipation, turbulent transport, pressure transport, and viscous diffusion, respectively. Each term is defined as  
\[\left\{ \begin{array}{l}Pd_{ij}=\langle-u_{i}u_{k}\rangle\frac{\partial U_{i}}{ \partial x_{k}}+\left\langle-u_{j}u_{k}\right\rangle\frac{\partial U_{i}}{ \partial x_{k}},\ Ds_{ij}=2v\Big{\langle}\frac{\partial u_{i}}{\partial x_{k}} \frac{\partial u_{j}}{\partial x_{k}}\Big{\rangle},\ Td_{ij}=-\frac{\partial \left\langle u_{i}u_{i}u_{k}\right\rangle}{\partial x_{k}},\\ Pv_{ij}=-\frac{1}{p}\Big{(}\Big{(}\frac{\partial p}{\partial x_{i}}u_{j} \Big{)}+\Big{\langle}\frac{\partial p}{\partial x_{j}}u_{i}\Big{)}\Big{)},\ Vd_{ij}=v^{\frac{ \partial}{2}\left\langle u_{i}u_{j}\right\rangle}.\end{array} \right. \tag{4}\]  
Here, \(\rho\) is the density. The turbulent kinetic energy (TKE) is defined as \(0.5\) (\(\langle u_{1}u_{1}\rangle+\langle u_{2}u_{2}\rangle+\langle u_{3}u_{3}\rangle\)), and the budget of the TKE equation can be obtained using Equation (3). Here, \(u_{1}\), \(u_{2}\), and \(u_{3}\) are equivalent to \(u\), \(v\), and \(w\), respectively. Figure 12a shows wall-normal profiles of the TKE budgets, among which the production (\(Pd^{+}\)), viscous dissipation (\(D\)s\({}^{+}\)), turbulent transport (\(Td^{+}\)), pressure transport (\(Pv^{+}\)), and viscous diffusion (\(Vd^{+}\)) are introduced in wall units. The magnitude of each budget is significantly changed near the top of the elements due to a strong shear layer over elements [25; 71]. In addition, the TKE budgets of smooth are displayed in Figure 12b. Positive budget magnitudes are interpreted as a gain of TKE, whereas a negative one represents a loss of TKE. The magnitude of each TKE budget is lower than that of smooth. The profile of \(Pd^{+}\) has a positive peak, similar to that of smooth. This could imply that the production process of TKE is not sensitive to elements, while other profiles of TKE budgets exhibit different trends, especially near \(y^{\prime+}=k^{+}\). A positive peak of \(Vd^{+}\) and a negative peak of \(Ds^{+}\) are observed near \(y^{\prime+}=k^{+}\), where viscosity predominantly influences the flows from the top of elements. The re is a positive plateau of \(Td^{+}\) in the region \(y^{\prime}+<k^{+}\), while a positive peak can be found at \(y^{\prime+}=5\) for smooth. The location of a negative peak of \(Td^{+}\) is similar to that of \(Pd^{+}\), representing that generated TKE is transferred to the near-wall region around elements through turbulent transport. Convex peaks with negative \(Pv^{+}\) emerge on both sides of a concave peak at \(y^{\prime+}=k^{+}\), whereas the magnitude of \(Pv^{+}\) of smooth is almost zero at \(y^{\prime+}>15\). Further analysis of the pressure transport of each Reynolds stress is necessary.