## 3 Results
### Solute Transport Results  
The effect of junction angle on solute mixing was evaluated. Figure 10 presents the depth-averaged concentration distributions of the simulation results for \(t=2.4\) min and \(4.1\) min. When the solute cloud crosses the recirculation zone, a relatively small amount of solute inflows into the recirculation zone compared to the flow zone. The boundary layer created between the recirculation and flow zones disturbs solute trapping by the recirculation zone. Once the solute is trapped in the recirculation zone, it persists there for a relatively long time. Comparisons of the concentration in the recirculation zone revealed that the concentration in the C30 case was the highest after \(2.4\) min as a result of a relatively weakened boundary layer effect as the junction angle decreases. In contrast, after \(4.1\) min, the solute in the C90 case is retained the longest, resulting in a concentration that is higher than in the other simulation cases. The C30 case exhibits different mixing characteristics than the C60 and C90 cases because the solute input from the tributary bounces off the right bank, resulting in higher concentrations. In Figure 11, the concentration distributions and streamlines of the \(v-w\) velocity, in the same cross-section shown in Figure 7, are depicted for \(t=2.4\) min. All cases reveal results in which the concentration at the bottom of the channel is relatively higher than that at the upper water surface. These results were obtained because the \(u\) velocity is lower at the bottom of the channel, resulting in less solute dispersion compared to the water surface, as shown in Figure 7. Additionally, the clockwise helical motion, which creates a strongly distorted mixing interface, causes rapid mixing with high discharge from the tributary. The distortion of the mixing interface results in an increase in the effects of \(\langle u^{\prime}w^{\prime}\rangle\) on turbulent diffusion [18]. As the junction angle increases, the vertical gradient of the mixing layer also increases as a result of the strongly developed helical cell. The solute cloud transported  
Figure 10: Depth-averaged concentration field with \(C\)-\(y\) curves: (**a**) C90 (\(t=2.4\) min); (**b**) C90 (\(t=4.1\) min); (**c**) C60 (\(t=2.4\) min); (**d**) C60 (\(t=4.1\) min); (**e**) C30 (\(t=2.4\) min); (**f**) C30 (\(t=4.1\) min). near the water surface by helical motion spreads to the outer bank. In contrast, near the channel bottom, the solute cloud moves to the inner bank where the recirculation zone is located. Then, the cloud is stored to accumulate the solute. The transverse mixing of solute inflows from the tributary was analyzed using dosage curves, which were calculated using Equation (5). Figure 12 presents the transverse change in the dimensionless dosage (\(\theta/\Theta\)) plotted versus the dimensionless cumulative discharge (\(\eta\)). The transverse distributions of \(\theta/\Theta\) gradually become flatter toward the downstream cross-section, except for the C30 case. In the C30 case in particular, \(\theta/\Theta\) increases at the right bank as a result of wall reflection. In the C45 case, wall reflection at the right bank causes an increase in \(\theta/\Theta\) from \(x=-3W\). In the C75 and C90 cases, the effects of wall reflection rarely appear, even as far as \(x=-5W\); in these cases, \(\theta/\Theta\) locally increases near the left bank at \(x=-W\) and \(x=-2W\) as a result of the recirculation zones. In the C90 case, the local increase in \(\theta/\Theta\) is maintained up to \(x=-3W\). The mixing metrics defined in Equation (4) can be utilized to evaluate transverse mixing. Figure 13 presents the changes in the dimensionless mixing metric (\(\sigma_{r}\)), which was normalized using the standard deviation of the dosage curve at \(x=0\) (\(\sigma_{us}\)), as shown in Figure 12a. Upstream of \(x=-2W\), the value of \(\sigma_{r}\) was not affected by the junction angle, except for the C30 case.