## 2 Methodology
### Solute Transport Model  
The transport of solute particles is simulated with the 3D LPT model. In general, the complex interplay between advection, molecular diffusion, and turbulent diffusion governs the solute transport. In turbulent open-channel flow systems, the molecular diffusion can be neglected because, in terms of length scale, the turbulent diffusion is usually dominant over the molecular diffusion. Hence, the solute particle motion is governed by 3D Langevin equations defined as:  
\[x_{i}(t+\Delta t)=x_{i}(t)+[U_{i}(x_{i})+u_{i}(x_{i},t)]\Delta t \tag{12}\]  
where \(x_{i}(t)\) is the particle position at \(t\); \(U_{i}(x_{i})\) is the mean velocity determining the advection process, as a function of space; and \(u_{i}(x_{i},t)\) is the instantaneous velocity fluctuation, which governs the turbulent diffusion process and varies spatiotemporally, depending on time and length scale of turbulent eddies.  
Since the RANS models are not capable of reproducing the fluctuating velocity, it is necessary to model \(u_{i}(x_{i},t)\) in order to consider the turbulent diffusion effects on particle dispersion. This study models the particle dispersion via the discrete random walk (DRW) method suggested by [PERSON] and [PERSON] [51]. This stochastic particle dispersion model characterizes the turbulent diffusion process using the RANS-simulated turbulence variables of \(k\), \(\varepsilon\), and \(\omega\), and it has successfully been applied to simulate sediment and solute dispersion in turbulent open-channel flows [44; 51]. In particular, [PERSON] et al. [52] adequately simulated the effects of the secondary flow on sediment transport and deposition in a meandering channel using the DRW model. However, the role of the meander-driven flow separation was not investigated in this study due to the channel topography with mildly curved bends. The DRW model calculates \(u_{i}(x_{i},t)\) from \(k\) using the relation below:  
\[u_{i}(x_{i},t)=\zeta\sqrt{2k/3} \tag{13}\]  
where \(\zeta\) is the normally distributed random number with a zero mean and a unit variance, which represent the arbitrary motion of the turbulent eddy. Here, \(\zeta\) is renewed after each lifetime (time scale) of the turbulent eddy, which is calculated as \(k/\varepsilon\) or \(1/\omega\). Therefore, the DRW model can consider both length scale and time scale of the turbulent eddy on particle dispersion using the RANS simulation results without instantaneous velocity fields only available with computationally intensive LES and DNS. References for detailed implementation of the DRW method are provided in [PERSON] and [PERSON] [51] and related application studies [53; 54].
Based on [PERSON]'s relations, this study generates meandering channels having three different channel curvatures, as shown in Figure 1. Here, a variation of \(\theta\) between 180 and 210\({}^{\circ}\) produces sinuosity ranging from 1.36 to 1.90, as summarized in Table 1. Please note that sinuosity is defined as \(L/L_{s}\), where \(L\) and \(L_{s}\) indicate the channel (curvilinear) length and linear distance between the inlet and outlet, respectively. The generated channels consist of 12 meander bends and have constant values of \(L\), \(W\), and \(h\) (flow depth), equal to about 205 m, 2.34 m, and 0.115 m, respectively.  
The mean inlet velocity, \(U_{0}\), is set to 0.366 m/s, which yields the Reynolds number, \(Re=U_{0}d_{h}/\
u=1.81\times 10^{5}\) and the Froude number, \(Fr=U_{0}/\sqrt{gh}=0.355\), where \(d_{h}\) denotes the hydraulic diameter. The channel width, flow depth, and inlet velocity adopted in this study follow conditions of [PERSON]'s experiments conducted in a meandering flume [56].