## 3 Results and Discussion
### Effect of Inlet Length Scale on Wall Shear Stress  
Figure 7 shows the distribution of the non-dimensional wall shear stress \(<\tau_{w}>^{+}\) for all the cases of different inlet length scales. Notably, the streamwise length scales were the same for all those cases. To observe the effect of the length scale close to the wall, the spanwise length scale was set from a minimum of \(0.15L_{z}^{*}\) to a maximum of \(3L_{z}^{*}\). It should be noted that the variation in the inlet length scale had a significant effect on the recovery of the wall shear stress downstream. As shown in Figure 7a, Case 01 revealed that the wall shear stress recovered fairly well at \(x/d=40\). The wall shear in other cases having a relatively smaller length scale than that in Case 01 was mostly unstable and did not properly recover downstream, except in Case 06. In Case 02, which was the worst case having no length scale implemented in the inflow, the wall shear stress continuously decreased downstream, and the wall shear stress substantially dropped to approximately 50 % at the outlet compared to that at the inlet. Case 03, in which the length scale was \(0.15L_{z}^{*}\), was used in the range \(y/d<0.2\), and was shown to be similar to Case 02, although it was slightly larger in terms of shear stress. This was similar to the case having a short domain size of \(x/d=12\) shown in Figure 4a, which implies that the domain needed to be at least longer than this size in the streamwise direction to obtain fully developed turbulent statistics. Case 04, which had a smaller length scale (i.e., \(0.3L_{z}^{*}\)) in the range \(y/d<0.2\), indicated that the shear stress had a local minimum turning point under this condition, and this meant that the shear stress decreased and gradually recovered at a specific point around \(x/d=20\). However, the shear stress continued to increase and finally became unstable at a point approximately 14 % higher than that of Case 01. For Case 05, in which the length scale was slightly smaller (i.e., \(0.5L_{z}^{*}\)) in the range \(y/d<0.2\), but larger (i.e., \(1.5L_{z}^{*}\)) for the rest of the range, the wall shear stress appeared to be quite close to that of Case 01 in the inlet region; however, it appeared scattered, wavy, and unstable downstream. Apart from Case 04, it did not tend to continue increasing continuously near the outlet, but it reached a value that was 10 % higher than that in Case 01. In this study, we tried to adjust the size of the length scale in certain zones, such as \(0.3L_{z}^{*}\) in \(L_{z}^{*}\) and \(L_{z}^{*}\) in \(y/d>0.1\), with the aim of understanding the effect of zone height on flow development. Interestingly, we found that the shear stress agreed fairly well with that in Case 01. This implies that depending on how the length scales were imposed in the zones, the wall shear stress would be adjusted in a target shape inside the domain. Figure 7b shows the wall shear stress distribution for the cases with a larger inlet length scale (approximately two and/or three times) than that of Case 01 (i.e., \(L_{z}^{*}\)). Interestingly, Cases 07 and 08 (i.e., \(2L_{z}^{*}\) and \(3L_{z}^{*}\)) showed a faster recovery of the wall shear stress compared with Case 01. As shown in the figure, Case 08 had a higher length scale than Case 07, but the distance required to recover in the shear stress would be almost the same, whereas the earlier drop in the shear stress close to the inlet in Case 08 was steeper than that in Case 07. In the case of the shear stress distribution of Case 01, Case 04, and Case 06 (see Figure 7), these three cases had the same spanwise length scale of \(1L_{z}^{*}\) at \(y/d>0.2\), but the conditions differed at \(y/d<0.2\). As shown in the figure, Cases 01 and 06 had very similar shear stress distributions, whereas Case 04 exhibited a markedly different tendency. The difference among these three cases was that the region with \(0.3L_{z}^{*}\) existed at \(y/d<0.2\) for Case 04 and at \(y/d<0.1\) for Case 06.