# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 3 Results and Discussion
### Turbulent Kinetic Energy Budgets
Figure 13a shows the pressure transport of streamwise (\(Pv_{11}\)), wall-normal (\(Pv_{22}\)\({}^{+}\)), and spanwise (\(Pv_{33}\)\({}^{+}\)) Reynolds stresses in wall units. The summation of \(Pv_{i}\)\({}^{+}\) is the same as \(2 Pv^{+}\) in Figure 12a. Wall-normal profiles of \(Pv_{11}\)\({}^{+}\), \(Pv_{22}\)\({}^{+}\), and \(Pv_{33}\)\({}^{+}\) smooth are displayed in Figure 13b. The magnitude of \(Pv_{11}\)\({}^{+}\) is negative at all \(y^{\prime+}\), whereas that of \(Pv_{22}\)\({}^{+}\) and \(Pv_{33}\)\({}^{+}\) is positive, indicating that energy is transferred from the streamwise component to the wall-normal and spanwise components [72]. As shown in Figure 13a, the dominant contributions of \(Pv_{11}\)\({}^{+}\) to \(Pv_{0}\)\({}^{+}\) near \(y^{\prime+}=k^{+}\) result in negative \(Pv^{+}\). The profile \(Pv_{33}\)\({}^{+}\) has a peak at \(y^{\prime+}=25\), which is higher than the peak location (\(y^{\prime+}=k^{\prime+}\)) of \(Pv_{22}\)\({}^{+}\). For smooth, the peak location of \(Pv_{33}\)\({}^{+}\) is closer to the wall than that of \(Pv_{22}\)\({}^{+}\). This discrepancy in  
Figure 12: (**a**) Wall-normal profiles of turbulent kinetic energy budgets: \(Pd\) (black), \(Ds\) (orange), \(Td\) (blue), \(Pv\) (green), and \(Vd\) (red). Vertical dashed line shows \(y^{\prime+}=k^{\prime+}\). (**b**) Wall-normal profiles of those of smooth. peak locations is attributed to cubic roughness elements with additional walls at \(y^{\prime+}=k^{\prime+}\). The magnitude of \(Pv_{33}{}^{+}\) is larger than that of \(Pv_{22}{}^{+}\) in the region \(y^{\prime+}>21\), in accordance with the result of smooth near the wall. The pressure transport of Reynolds stress budgets is conditionally averaged at four locations: P\({}_{0}\), P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\). Figure 14a represents the conditionally averaged pressure transport of streamwise Reynolds stress (\(Pv_{11,i}{}^{+}\)), which is negative at all \(y^{\prime+}\) except \(Pv_{11,1}{}^{+}\) (green) at \(y^{\prime+}<k^{\prime+}\). The magnitude of \(Pv_{11,2}{}^{+}\) (blue) is more than twice as low as that of \(Pv_{11,3}{}^{+}\) (red) near \(y^{\prime+}=k^{\prime+}\), representing that most of the energy in the streamwise direction is reduced at the point of P\({}_{2}\). Note that the cubic elements are located immediately downstream of P\({}_{2}\). The energy transfer by Reynolds stress budgets is in accordance with the results of flow visualization near the elements [33; 37]. The profiles of \(Pv_{11,0}{}^{+}\) (black), \(Pv_{11,1}{}^{+}\), \(Pv_{11,2}{}^{+}\), and \(Pv_{11,3}{}^{+}\) are thoroughly overlapped in the region \(y^{\prime+}>150\). Figure 14b,c shows wall-normal profiles of \(Pv_{22,i}{}^{+}\) and \(Pv_{33,i}{}^{+}\), respectively. The magnitude of \(Pv_{22,1}{}^{+}\) is negative at \(y^{\prime+}<10\), where the energy in the wall-normal direction is reduced. However, the other profiles have positive contributions, indicating a gain of energy from other directions. In the region \(y^{\prime+}<k^{\prime+}\), the energy in the streamwise direction is mainly reduced at P\({}_{2}\), whereas the energy in the wall-normal and spanwise directions is dominantly enhanced at P\({}_{3}\) and P\({}_{2}\), respectively. In addition, the spanwise energy increases at P\({}_{0}\) in the region \(y^{\prime+}>k^{\prime+}\). Similar to \(Pv_{11,i}{}^{+}\), each profile set of \(Pv_{22,i}{}^{+}\) and \(Pv_{33,i}{}^{+}\) coincides well at \(y^{\prime+}>150\). Figure 14: Conditionally averaged pressure transport of (**a**) streamwise, (**b**) wall-normal, and (**c**) spanwise Reynolds stresses at P\({}_{0}\) (black), P\({}_{1}\) (green), P\({}_{2}\) (blue), and P\({}_{3}\) (red). Vertical dashed line indicates \(y^{\prime+}=k^{\prime+}\). Figure 13: (**a**) Wall-normal profiles of pressure transport of Reynolds stresses: streamwise, \(Pv_{11}{}^{+}\) (black); wall-normal, \(Pv_{22}{}^{+}\) (blue); and spanwise, \(Pv_{33}{}^{+}\) (red) components. Vertical dashed line represents \(y^{\prime+}=k^{\prime+}\).