## 2 Methods and Background
### Model Settings and Validations  
Numerical simulations were conducted to analyze changes in solute mixing with flow and mixing changes stemming from junction angle modifications. The channel used for numerical simulation was designed based on the laboratory experimental channel considered by [PERSON] [51], and the length scale was increased to 10 times the original size. Through simulations using the scaled-up channel, flow and mixing patterns in small-scale streams, which are often found in urban areas covered with concrete (\(n=0.013\)) with a small width-to-depth ratio, were reproduced. According to the Froude similitude, the channel length, width, and discharge values used in our experiments are listed in Table 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{\(Q_{r}\)} & \multirow{3}{*}{\(W/dx\)} & \multicolumn{6}{c}{**MAPE (\%)**} \\ \cline{3-6}  & & \multicolumn{2}{c}{**Recirculation Zone**} & \multicolumn{2}{c}{**Water Depth**} \\ \cline{3-6}  & & \(L_{\text{s}}\) & \(b_{\text{s}}\) & \(y/W\) = 0.17 & \(y/W\) = 0.50 & \(y/W\) = 0.82 \\ \hline \multirow{3}{*}{0.25} & 10 & 100.0 & 100.0 & 6.4 & 5.5 & 5.6 \\ \cline{2-6}  & 15 & 28.5 & 32.3 & 3.7 & 2.8 & 3.2 \\ \cline{1-1} \cline{2-6}  & 20 & 6.6 & 15.8 & 2.6 & 1.8 & 2.0 \\ \cline{1-1} \cline{2-6}  & 25 & 4.9 & 5.3 & 2.6 & 1.8 & 2.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Grid sensitivity analysis conditions and error analysis results. Figure 2: Grid sensitivity analysis results for a 90\({}^{\circ}\) confluence angle: (**a**) Mean percentage error for the recirculation zone; (**b**) Mean percentage error for the water depth. The confluence channel used for simulation was set to have the same width of \(W=9.14\) m for the main and tributary channels in all cases. The downstream boundary water depth (\(l_{0}\)) for the numerical simulations was 2.96 m, which conforms to the geometric ratio. The junction angle was changed in 15\({}^{\circ}\) intervals with five cases, from 30\({}^{\circ}\) to 90\({}^{\circ}\). Based on the results of grid sensitivity analysis, the grid size was set to be \(W/dx>20\), and the number of vertical layers was set to 10 for the flow and mixing simulations by applying quasi-three-dimensional characteristics. The computational grids are presented in Figure 3.  
For our simulations, the Smagorinsky model was adopted for RANS simulations, and the Smagorinsky constant varied from 0.15 to 0.40 to calculate the horizontal eddy viscosity and diffusion coefficient. The background vertical eddy viscosity was set to 2.09 \(\times\) 10\({}^{-2}\) m\({}^{2}\)/s according to the depth-averaged value of parabolic distribution, where \(\epsilon_{z}=\frac{1}{6}\kappa hu^{*}\), \(\kappa=0.41\) is the von Karman constant, \(u^{*}=\sqrt{gR_{h}S_{f}}\) is the shear velocity, \(R_{h}\) is the hydraulic radius, and \(S_{f}=\left(nU/R_{h}^{2./3}\right)^{2}\). Time step (\(dt\)) was set to 0.005 s to satisfy the Courant-Friedrichs-Lewy (CFL) condition. The upstream boundary conditions for the main channel and tributary were set to a constant discharge, and the downstream boundary condition was set to a constant water depth, as listed in Table 2. The discharge of the main and tributary channels was set to \(\text{Fr}_{d}=0.37\) for a discharge ratio (\(Q_{r}\)) of 0.25, such that the numerical simulations would have a separation zone downstream of the junction. Based on these hydraulic conditions, the momentum flux ratio (\(M_{r}\)) in this study was 1.62, which is included in the range for previous studies (0.29-2.52) [16; 30]. The free-slip condition was employed for the wall boundary. The kinematic boundary condition was applied for the free surface and bottom boundary, and \(w\) becomes 0 at the bottom and water surface.