## 3 Results and Discussion
### Effect of Inlet Length Scale on Wall Shear Stress
It can be observed that the change in the spanwise length scale in the range \(0.1<y/d<0.2\) had a significant influence on the development of the turbulent boundary layer. Figure 8 shows the averaged turbulent boundary layer profile, and \(y/d=0.1,0.2\) corresponds to \(y^{+}=15.2,30.3\), respectively, i.e., \(y/d=0.1\) was placed in the buffer layer and \(y/d=0.2\) in the logarithmic layer. With respect to the turbulence characteristics near the wall, some studies ([PERSON] et al. [3], [PERSON] [43], [PERSON] [44]) reported that the region close to the wall has a highly active turbulence movement, which is associated with the behavior of low-speed streaks and horizontal shear stress in the spanwise direction. In addition, there was a breakup in the region \(10<y^{+}<30\), which is called \"bursting\", contributing as much as 70 % of the turbulence. [PERSON] et al. [45] and [PERSON] [46] identified counter-rotating streamwise vortices in the region \(14\leq y^{+}\leq 25\), which contained the most energetic vortices. According to existing research, the \(10<y^{+}<30\) region contributed greatly to the generation of turbulence. As shown in Figure 7, Cases 01 and 06 had the spanwise length scale of \(1L_{z}^{*}\), which was a relatively large scale in \(15<y^{+}<30\), whereas Case 04 had \(0.3L_{z}^{*}\), which means that it had a lower length scale in this region, which was insufficient  
Figure 7: Development of dimensionless wall shear stress with different length scales. (**a**) Cases 01–06 (**b**) Cases 01, 07 and 08.  
to generate turbulence statistics. In addition, the lower length scale reduced the energy production such that the minimum distance to recover the shear stress close to the wall was longer. It can be seen here that the change in the length scale in the region \(y^{+}<15\) did not significantly impact the recovery of the shear stress. Therefore, the results of Case 01 and Case 06 were very similar. The change in the length scale in the approximate region of \(15\) (\(0.1\))\(<y^{+}(y/d)<30\) (\(0.2\)) contributed significantly to the generation of turbulent energy near the wall surface. Figure 9 shows the streamwise development of the wall shear stress \(<\tau_{w}>^{+}\) in the domain with the deviation, which indicates the magnitude of data variability as error bars. As the length scale near the wall became smaller (Case 05), the deviation became larger. Interestingly, as shown in the figure, Cases 01 and 06 were very close to each other, whereas the shear stress deviation of Case 06 was relatively large. In this case, the only difference was that Case 06 had a length scale of \(0.3L_{z}^{*}\) in the region \(y/d<0.1\) (i.e., \(y^{+}<15\)). Therefore, if the spanwise length scale was small, the spatial correlation between two different points became lower such that this case required a relatively long longitudinal length and time to fully develop in the domain. As the length scale increased, the wall shear stress tended to recover quickly. However, when the length scale was \(2L_{z}^{*}\) or more, the distance required for the full development of the boundary layer did not change significantly. Figure 10a shows the change in the wall shear stress in the channel domain. In particular, Figure 10b-e shows the distribution of Reynolds stress profiles, \(\langle u^{\prime}u^{\prime}\rangle^{+},\langle v^{\prime}v^{\prime}\rangle^{ +},\langle w^{\prime}w^{\prime}\rangle^{+}\) at each longitudinal position of \(x/d\). As depicted in the figure, Case 01 revealed that the downstream region had fully recovered the shear stress; however, Case 03 had a poor distribution in the domain. This implies that the size of the inlet length scale close to the wall greatly influenced the development of the shear stress. In addition, if the wall shear stress was fully recovered, the axial stresses \(\langle u^{\prime}u^{\prime}\rangle^{+},\langle v^{\prime}v^{\prime}\rangle^{ +},\langle w^{\prime}w^{\prime}\rangle^{+}\) remained constant in the whole domain. Figure 8: Mean streamwise velocity profiles.