## 2 Numerical Analysis
### Implementation of the Boundary Condition and Inflow Length Scale  
In this study, a hexahedral grid system was used for the numerical simulation of making fully developed turbulent flow. To improve the convergence efficiency in the solution of the governing equations, a stretch mesh was used so that the grid was concentrated on the wall surface, and the resolution was sufficiently high close to the wall, but coarse at the far field from the wall. The Reynolds number (\(Re_{\tau}\)) on the smooth wall-bounded channel flow was 150 based on the friction velocity (\(u_{\tau}\)) and the half height (\(d\)) of the channel. The size of the whole computational domain was \(60d\times 2d\times 3.5d\) in the streamwise (\(x\)), wall normal (\(y\)), and spanwise (\(z\)) directions, respectively. The number of cells in the whole domain was approximately 2.5 million, as shown in Figure 1. To improve the convergence of the analysis, a stretch mesh satisfying \(y^{+}<1\) in the first cell center was considered for the wall-normal direction. For the \(x\) and \(z\) directions, uniform meshes were used, and the resolutions were \(\triangle x^{+}=15,\triangle z^{+}=7.5\). The number of cells in each direction was \(600\times 60\times 70\) in the \(x\), \(y\), and \(z\) directions, respectively. All statistics were averaged by \(40t_{*}\), where \(t_{*}=tu_{\tau}/d\) is a non-dimensional time. Averaging for the statistical analysis was performed after the initialization period of \(400t_{*}\). The Smagorinsky subgrid-scale model was adopted with a constant \(c_{s}=0.065\) ([PERSON] and [PERSON] [35]). The time step satisfies the condition that the CFL number should be smaller than one. The corresponding \(\triangle t_{*}=\triangle t\times u_{\tau}/\delta=0.001\). A second-order implicit scheme was used for time discretization, and a second-order central difference scheme was used for spatial discretization. Furthermore, the transient incompressible flow solver was used, and the PISO algorithm was adopted as the velocity-pressure coupling scheme. The PRESTO algorithm was used as a discretization technique for pressure ([PERSON] et al. [38]). Figure 1: Schematic diagram of the computational domain (total number of cells: \(600\times 60\times 70\)). The boundary conditions of this channel flow were: a no-slip condition on the top and bottom walls, a periodic boundary condition in the lateral direction, and a zero-gradient condition with no diffusion of all the flow variables on the outlet. The synthetic turbulence obtained by the synthetic inflow generator was interpolated into the cell center of the inlet plane. The statistical information of turbulent flow, including the first and second moment statistics and the integral length scales used as input parameters to generate the synthetic inflow, were obtained from the DNS database that analyzed the plane channel flow ([PERSON] and [PERSON] (2018)). The integral length scale \(L_{i}\) was obtained using Equation (15) with a two-point correlation and \(R(x_{i})\) in the DNS database. \[L_{i}=\int_{0}^{\infty}R(x_{i})dx_{i},\;\;i=1,2,3 \tag{15}\]  
Figure 2 shows the length scale obtained from the DNS database and the simplified length scale applied to the synthetic inflow generation. Figure 2a shows the distribution of the streamwise direction length scale, \(L_{x}\), along the wall-normal direction. If the relative turbulence intensities are sufficiently low, according to [PERSON]'s hypothesis, the length scale can be determined by the local mean velocity \(\overline{u}\) and the Eulerian time scale \(T\). Therefore, \(L_{x}(y)=T\times\overline{u}(y)\) is applicable ([PERSON] (1970)). In the equation, \(T\) is a constant, and the local mean velocity in the streamwise direction increases with the distance from the wall, while the streamwise length scale also becomes large. In this study, the turbulence intensity of the flow simulation was smaller than 0.3, and [PERSON]'s hypothesis can therefore be applied across the entire domain.