## 3 Results  
### Effects of Junction Angle on Flow Characteristics  
In this study, the reproducibility of the flow structures associated with the junction angle of the designed confluence was evaluated based on numerical simulation results. Figure 6 presents the depth-averaged velocity and streamlines in the recirculation zone developed by the tributary inflow. The inflow of the tributary causes flow separation and the maximum velocity is generated by flow contraction at the right bank. The stagnation zone appears in the upstream junction corner and the velocity at this point decreases. The size of the recirculation zone increases as the junction angle increases, which has been reported in previous studies [12, 23, 38, 53]. Figure 7 compares the \(u\) and \(v\)-\(w\) velocity vectors at the cross-section in the middle of the recirculation zone depicted in Figure 6. The negative sign of the velocity contour in Figure 7 indicates flow heading in the downstream direction. The channel-scale helical cell is developed under the simulation conditions of \(M_{r}>1\)[54]. As the junction angle increases, the center of the clockwise helical cell is positioned near the bottom of the channel. Figure 8 presents more detailed features of the transversal velocity associated with the clockwise helical motion shown in Figure 7. The depth-averaged vertical deviation of the \(v\) velocity is given as \(I=\sqrt{\int_{0}^{h}v^{\prime 2}dz/h}\), where \(v^{\prime}\) is the vertical deviation from the depth-averaged velocity (\(\overline{v}\)) and indicates the strength of the circular motion (Figure 8a). The maximum value of \(I\) occurs in the C90 case and decreases as the junction angle decreases. At each of the maximum values of \(I\), the vertical distributions of vorticity in the \(x\)-direction (\(\omega_{x}\)) are plotted according to the junction angle (Figure 8b). The results also reveal that helical motion develops strongly as the junction angle increases. Additionally, the strength of the helical motion increases near the left bank (\(0<y<2\) m), where the recirculation zone is developed as the junction angle decreases. This indicates that the transverse transport of the solute in the recirculation zone may be weakened as the junction angle increases. In Figure 9, the length (\(L_{s}/W\)) and width (\(b_{s}/W\)) of the recirculation zone, based on the depth-averaged velocity fields shown in Figure 6, are compared to the results obtained using Equations (2) and (3), as suggested by [PERSON] et al. [38]. The values of \(L_{s}/W\) and \(b_{s}/W\) are significantly reduced in the range of \(45^{\circ}\)-\(75^{\circ}\) in the simulation results. These \(L_{s}/W\) changes are similar to those obtained using Equation (2), but in the C30 case, the recirculation zone length is overestimated in the simulation results. The results of \(b_{s}/W\) exhibit some underestimation compared to the calculations using Equation (3). These discrepancies in the simulation results could have occurred as a result of the hydrostatic assumption in the EFDC model. However, despite these limitations of the model, the overall results of the numerical simulations only exhibit minor differences compared to the empirical equations based on experiments, and the trend of the recirculation zone area changes with changes in the junction angle was reproduced reasonably accurately. Figure 6: Depth-averaged velocity contours and streamlines near the recirculation zone: (**a**) C90; (**b**) C75; (**c**) C60; (**d**) C45; (**e**) C30. Figure 8: Properties of clock-wise helical cell by junction angle: (**a**) Lateral variations of the depth-averaged transversal velocity deviations; (**b**) Vertical distributions of the \(x\)-directional vorticity at the center of clockwise helical cell. Figure 7: Comparisons of \(u\)-velocity magnitude and \(v\)-\(w\) velocity vectors at the center of the recirculation zone: (**a**) C90 (\(x^{*}\) = 6.7 m); (**b**) C75 (\(x^{*}\) = 6.6 m); (**c**) C60 (\(x^{*}\) = 5.1 m); (**d**) C45 (\(x^{*}\) = 2.7 m).