## 2 Methodology
### Computational Setups  
As for constructing computational grids of the generated channels in Figure 1, the grid resolution is determined based on [PERSON] and [PERSON]'s study [16], who performed numerical simulations using [PERSON]'s channel. Please note that the meandering channels generated in this study have the aspect ratio (\(W\)/\(h\)), \(Re\), and \(Fr\) identical to the experimental conditions of [PERSON]'s meandering flume [52], as aforementioned. [PERSON] and [PERSON] [16], and [PERSON] and [PERSON] [17] reported that the grid resolution is optimized to 122, 24 and 14 cells or 122, 26 and 12 cells in the longitudinal, transverse and vertical directions, improving the computational efficiency with adequate accuracy. The present study also simulates [PERSON]'s channel for the model validation using the coarse grid resolution (\(122\times 24\times 14\)) of [PERSON] and [PERSON]'s study and a fine grid resolution (\(320\times 52\times 20\)). The boundary conditions used in the flow simulations are as follows: (a) At the inlet, the boundary conditions for the velocity and turbulence components are implemented, specifying the inlet velocity and turbulence intensity, respectively; (b) at the outlet, zero-gradient boundary condition is assigned for the velocity and turbulence properties, and the pressure outlet boundary condition is imposed; (c) at the solid walls, the non-slip boundary condition and wall functions are used for the velocity and turbulence variables; respectively; (d) at the water surface, this study employs the symmetric boundary condition, assuming the free surface as a frictionless plane [12; 13; 57]. The Pressure-Implicit with Splitting of Operators (PISO) algorithm is adopted for the pressure-velocity coupling. The time step is set to maintain the Courant number, \(Cr=U_{i}\Delta t/\Delta x\), less than 0.4, and this study runs the flow simulations until all variables of \(U\), \(k\), \(\varepsilon\), and \(\omega\) converge to the steady-state condition. Figure 2 shows results simulated with the \(k-\varepsilon\), \(k-\omega\) and SST \(k-\omega\) models. The model validation with [PERSON]'s observation is implemented at a cross-section of the second bend apex, which can be found in [56]. Herein, the difference in the grid resolution does not result in the noticeable difference in the simulated velocity fields since the simulations produce the relative difference around 1% between the coarse grid resolution and fine grid resolution, which is consistent of results reported in other numerical studies of [PERSON]'s channel [16; 17]. According to Figure 2, all the turbulence models adequately reproduce general flow patterns with respect to streamwise and spanwise velocity distributions. Additionally, no significant difference between the turbulence models is found in the simulated velocity profiles. For the spanwise velocity, all the turbulence models successfully predict the emergence of vortical secondary flows, which can be explained by the enhanced spanwise velocities near the free surface and channel bed if the relatively large discrepancies are observed near the free surface. [PERSON] and [PERSON] [17] reported that these near-free surface errors may be ascribed to a small \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & \(L_{s}\) **(m)** & \(\theta\) (\({}^{\circ}\)) & \(\lambda\) **(m)** & \(R_{c}\) **(m)** & \(R_{d}\)/\(W\) & \(\mathbf{L/L_{s}}\) \\ \hline S136 & 150 & 150 & 21.70 & 5.62 & 2.40 & 1.36 \\ S157 & 131 & 180 & 18.72 & 4.68 & 2.00 & 1.57 \\ S190 & 108 & 210 & 15.50 & 4.00 & 1.71 & 1.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of meandering channels. counter-rotating cell of the secondary flow adjacent to the sidewalls and free surface. Nevertheless, the influence of this near-wall secondary flow cell on the primary flow structures would be less significant owing to the large aspect ratio of the channel [58].