## 3 Results and Discussion  
### Channel Flow Simulation Using the Synthetic Inflow Generator  
Given that the synthetic inflow generator was available, the channel flow simulation on the LES platform was initially conducted with the aim of observing the development of the turbulent flow in the whole domain. The channel flow started with the data from the synthetic inflow generator, adopting it in the inlet section, and developed the flow downstream to the smooth wall-bounded channel. The development of the axial and shear stress profiles are very important for identifying the fully developed flow, ([PERSON] et al. [37], [PERSON] et al. [41]), as well as obtaining an appropriate turbulent flow downstream. Figure 4 shows the development of dimensionless axial and shear stress distributions. Figure 4a indicates the variation in the wall shear stress close to the surface from the inlet to downstream of the channel; the ordinate axis indicates the non-dimensional wall shear stress, defined as \(\langle\tau_{w}\rangle^{+}=\tau_{w}/(\rho u_{\tau}^{2})\), and the abscissa represents the streamwise distance (\(x\)) from the inlet normalized by the half height (\(d\)) of the channel. To observe the effect of the longitudinal domain size, two different domains (\(x/d=12\) and \(60\)) were designed and calculated. As shown in the figure, the wall shear stress decreased near the inlet of the channel and slowly recovered downstream, becoming almost constant at around \(x/d=30\) (see solid line). Moreover, the short domain case (\(x/d=12\)) did not recover (see dashed line), and the difference between the highest and lowest magnitudes of \(\langle\tau_{w}\rangle^{+}\) was much  
Figure 3: Statistical profiles from DNS and the synthetic inflow generator. (**a**) mean velocity profiles (**b**) streamwise stress profiles (**c**) vertical and transverse stress profiles (**d**) Reynolds stress profiles. larger than that of the domain size \(x/d=60\). Therefore, if the domain size was set to be less than \(x/d=30\), the wall shear stresses would be very low. Figure 4b shows the Reynolds stress profiles at several equi-spaced positions, \(x/d=0\), 10, 20, 30, 40, and 50, and the abscissa indicates the wall-normal distance from the wall surface normalized by \(d\). Figure 4c-e indicates the variation of axial stress profiles along the whole domain. Note here that the solid lines in the figure represent the existing DNS data. The axial (longitudinal) stress profiles began initially with the data from the synthetic inflow generator; however, they gradually developed and reached the fully developed profile downstream, which corresponded to the existing DNS data. The vertical and spanwise stress profiles were also not properly developed upstream, but they almost fully developed downstream. Figure 5 shows the gradient of the dimensionless shear stress, \(<\tau_{w}>^{+}/\triangle x\), in the streamwise direction with the aim of identifying more precisely the critical point indicating a fully developed boundary layer. As shown in Figure 5, there was a steep gradient near the inlet, and the values \(<\tau_{w}>^{+}/\triangle x\) gradually recovered and converged to an almost constant value (i.e., around zero) in the region where \(x/d\geq 40\). Figure 6 shows the mean and turbulent statistics in the channel flow and compares them with the existing DNS data ([PERSON] et al. [42]). In the figure, the fully developed profile was obtained at \(x/d=50\), and the inlet data from the inflow generator and the DNS results for the reference and existing data were also compared. As shown in the figure, most of the profiles agreed perfectly well with each other, which was very satisfactory. The turbulent statistics for both cases, that is the inlet and \(x/d=50\), also showed a similar pattern. In particular, the non-dimensional mean velocity, \(u^{+}\), was almost identical. The axial longitudinal stress profile, \(\langle u^{\prime}u^{\prime}\rangle^{+}\) at \(x/d=50\) also showed a similar distribution near the wall surface compared to that at the inlet.