## 3 Results and Discussion
### Second Invariant Flow Structure near the Wall  
A second invariant of the velocity gradient tensor can be used to visualize the flow structure near the wall. The related Q-criterion is defined as follows:  
\[Q=\frac{1}{2}\big{(}\Omega_{ij}\Omega_{ij}-S_{ij}S_{ij}\big{)} \tag{18}\]  
where \(\Omega_{ij}=\frac{1}{2}\Big{(}\frac{\partial u_{i}}{\partial x_{j}}-\frac{ \partial u_{j}}{\partial x_{i}}\Big{)}\) and \(S_{ij}=\frac{1}{2}\Big{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}\Big{)}\). \(\Omega_{ij}\) denotes the rotation rate, and \(S_{ij}\) denotes the rate of the strain. From the equation, when \(Q\) is positive, it means that the intensity of the rotation rate is higher than the deformation rate.  
Figure 14 shows the iso-surface of the Q-criterion in the range \(y/d<0.25\) near the wall. The Q distribution showed a remarkably similar tendency to that of the wall shear stress \(\langle\tau_{w}\rangle^{+}\) distribution in Figure 7. In fact, Case 03, as shown in Figure 7, tended to have a continuous reduction in the shear stress \(\langle\tau_{w}\rangle^{+}\) from the inlet to \(x/d<5\), and this was also similar in the Q distribution. In Case 04, as shown in Figure 14b, the Q structure was not observed in the region of \(1<x/d<35\). This means that when the shear stress was lower than 0.8, the vorticity strength was weak and therefore not enough to form the vortex. Thus, the vortex did not form. In addition, as the shear stress recovered, the vortex reformed when the shear stress was higher than 0.8.  
In Figure 14d, that is, in Case 05 and the cases having a length scale higher than 0.5 near the wall, the shear stress was higher than 0.8 in the entire region, and the vortex appeared in the whole area. The vortex structure showed a slight discrepancy depending  
Figure 13: Lateral integral length scale distribution at the inlet and downstream sections. In the figure, (**a**) denotes all the lateral integral length scales at the inlet section (\(\circ\): Case01, \(\square\): Case03, \(\diamond\): Case06, \(\times\): Case07, \(\triangle\): Case08, \(\bullet\): DNS database), (**b**) the lateral integral length scale of streamwise velocity components at \(x/d=50\) (\(\bullet\): DNS database, \(\xrightarrow{}\): Averaged \(L_{z}(u)\)), and (**c**) the lateral integral length scale of spanwise velocity components at \(x/d=50\) (\(\bullet\): DNS database, \(\xrightarrow{}\): averaged \(L_{z}(w)\)).  
on the shear stress and the size of the length scale. Interestingly, in the region where the boundary layer was fully developed, the Q structures in all cases (Case 01, Case 07, Case 08) were similar.  
Figure 15 shows the energy spectrum distribution at the inlet and for \(y/d=50\) compared to the existing DNS data. Figure 15a,b includes the results of (c) Case 01 and (d) Case 09. As mentioned earlier, Case 09 had length scales that were two times larger in the streamwise and spanwise directions compared with those of Case 01. In the figure, the symbol k is the wave number, which is defined as \(2\pi f/U_{m}\). Figure 15a,c shows that the energy spectrum at the inlet had an appropriate level at the low wave number region; however, in the higher wave number region, it did not agree with the DNS data. In the figure, the dashed line has a slope of -5/3 in the spectrum. Regardless of the integral length scale at the inlet, the energy spectrum in the fully developed region agreed well with the DNS data.  
Figure 14: Iso-surface of the Q-criterion at \(y/d<0.25\) and the value of \(Q>1\).