## 5 Numerical Method and Verification  
### Analysis of Flow Field in the Hump Section  
At rotational speed n = 2200 r/min, the flow pattern in the hump section of the siphon outlet channel under the three characteristic conditions is shown in Figure 13. Under the low flow condition (\(R_{\text{Q}}\) = 0.5), the flow pattern at the bottom of the hump section is smooth, and the red filament lines are affixed to the wall along the flow direction. The red filament lines on the right side of the wall along the flow direction are inclined to the top of the hump at an average angle of 38 degrees, while the red filament lines on the left side are inclined to the bottom of the hump at an average angle of 33 degrees, and the red filament lines on both sides are crossed. It shows that the flow inside the hump section is spiral and staggered under the low flow condition, the residual circulation at the outlet of these guide vane is large under the low flow condition and the control of the transverse flow rate on the flow pattern is the main factor. Under the optimum condition (\(R_{\text{Q}}\) = 1.0), the red wire line at the top and bottom of the hump section is consistent with the flow direction as a whole. In the local area at the top of the hump section, there is the intersection of the red wire line deflection direction, left-side wall and right side. The red silk line on the side wall deviates from the bottom of the hump section at an average deflection angle of about 38 degrees. Under the high flow condition (\(R_{\text{Q}}\) =1.20), the red silk line on the top and bottom of the hump section is consistent with the direction of the water flow movement as a whole; the red silk line on the left and right sides deviated from the bottom of the hump section at an average deflection angle of about 21\({}^{\circ}\). At a certain speed, with the decrease of the flow rate, the spiral intensity of the flow pattern in the hump section increases, and the more complex the flow pattern is, the greater  
Figure 12: Model test and numerical simulation comparisons of the pump unit. the proportion of the transverse flow velocity at the outlet of the guide vane to the flow pattern. The flow phenomena of cross-section A-A (Figure 14) at the top of the hump section of the siphon outlet conduit are analyzed by using three-dimensional helicity. The three-dimensional helicity \(H_{y}\) calculation formula is as follows:  
\[H_{\text{y}}=\left(\
abla\times\overset{\rightarrow}{v}\right)\cdot\overset{ \rightarrow}{v} \tag{17}\]  
where \(\
abla\) is the gradient operator, and \(\overset{\rightarrow}{v}\) is the velocity vector. As the flow decreases, the A-A vortices in the cross-section of the hump section of the siphon outlet conduit gradually increase; under different flow conditions, there are reverse vortex pairs formed on the left- and right-side walls of cross-section A-A, and the flow distribution in cross-section A-A does not show good symmetry, which is affected by the circumferential velocity at the outlet of the guide blade and the restriction of the solid wall of the conduit. The uniformity of the axial velocity distribution and the average velocity circulation [27] are introduced to further analyze cross-section A-A of the hump section. The calculation methods are shown in Equations (18) and (19), and the calculation results are shown in Figure 15. In the range of the flow ratio 0.4-0.7, with the increase of the flow, the axial velocity distribution of cross-section A-A increases linearly; in the range of flow ratio 0.7-1.2, with the increase of the flow, the axial velocity distribution uniformity of cross-section A-A increases from 55.4% to 75.4% and tends to be stable. As the flow rate increases, the average velocity circulation of cross-section A-A decreases from 0.056 m\({}^{2}\)/s to 0.008 m\({}^{2}\)/s, the circumferential velocity of the water flow gradually decreases with the gradual expansion of the geometrical solid side wall of the siphon outlet conduit as a whole.