## 5 Analysis of Simulation Results
### Transverse Circulation  
The distribution of circulation strength along the flow is very complex, and there are many ways to express it. In this paper, the circulation strength proposed by [PERSON] [34] is used to analyze the hydraulic characteristics of bend flow. The expression is as follows:  
\[S_{xy}=\left(\frac{V_{xy}^{2}}{2g}\right)/\left(\frac{V^{2}}{2g}\right) \tag{10}\]  
\(V_{xy}=(v^{2}+w^{2})^{0.5}\), \(V=(u^{2}+v^{2}+w^{2})^{0.5}\), \(u\), \(v\), \(w\) are the component velocities in three directions: streamwise, transverse and vertical, and \(g\) is the gravitational acceleration. The distribution of circulation strength is shown in Figure 10. After the flow enters the bend, the circulation intensity increases rapidly with the increase of centrifugal stress and flow velocity and reaches the maximum near section D60. After crossing the curved top section, the circulation strength gradually weakens. The circulation intensity has a strong correlation with the change of the main streamline of the bend flow. The main streamline of the water flow near section D60 began to shift to the concave bank, and the main streamline shifted slowly after passing the D90 section so that the secondary flow intensity began to weaken. The lateral circulation begins to form a clockwise secondary circulation near the surface of the concave bank on the D60 section, which makes the secondary flow stronger. Figure 10: The distributions of circulation strength along the curve. Figure 9: Velocity distribution and streamlines at different water depths: (**a**) \(z/H\) = 0.9; (**b**) \(z/H\) = 0.5; (**c**) \(z/H\) = 0.2. From the maximum position of circulation strength corresponding to different parameters, as shown in Table 3, it can be found that the position of the maximum circulation intensity has nothing to do with the parameters, such as the diameter-to-width ratio, the width-to-depth ratio, and the Froude number but has a strong correlation with the center angle of the curve. In the test with the center angle of the curve being 180\({}^{\circ}\), the maximum value of the circulation intensity appears near 60\({}^{\circ}\). The transverse circulation distribution of the section is shown in Figure 11. The circulation structure of each section shows that the upper part flows to the concave bank and the lower part flows to the convex bank, forming a main circulation the center of which is located below the convex bank. The position is continuously shifted downward toward the ledge. After passing through section D30, the circulation intensity of the convex bank is greater than that of the concave bank. Starting from the D60 section, a secondary circulation with the direction opposite to the main circulation forms on the surface layer of the concave bank. The secondary circulation continues to develop along the way with increasing intensity and increasing scope. When water enters the bend, the open channel water flow is subjected to the joint action of water pressure and centrifugal stress. At the cross section of the bend, the surface water flow moves to the concave bank under the action of centrifugal stress, and the bottom water flow moves from the concave bank to the convex bank under the action of pressure gradient, forming a counterclockwise bend circulation. Figure 12 illustrates the transverse circulations of the bend sections. The vortex structure obtained with LES is clearer. No circulation structure is formed in the straight sections of the curve. At section D30, a double-vortex structure is formed. A counterclockwise  
Figure 11: The distributions of transverse circulation for sections: (**a**) D0, (**b**) D30, (**c**) D60, (**d**) D90, (**e**) D150,(**f**) D180.