## 3 Results and Discussion
### Internal Flow Field  
The absolute flow angle can reflect the flow pattern of the fluid. Therefore, the absolute flow angle on the blade leading edge was investigated. Figure 9 shows the absolute flow angle distribution along the circumferential direction of 0.93 span. The absolute flow angle presents evident periodicity. Under the high-flow condition, the absolute flow angle increases to 95\({}^{\circ}\) in the circumferential middle position and then decreases to 85\({}^{\circ}\) on the blade leading edge. The distribution range of absolute flow angle is between 85\({}^{\circ}\) and 100\({}^{\circ}\). The initial position of the leakage vortex is 0.33 M, and the leakage vortex flows to the downstream with the mainstream. As a result, the leading edge of the blade is almost unaffected by the leakage vortex, which indicates that the water can successfully flow to the impeller. Under the BEP condition, the absolute flow angle increases remarkably. The distribution range of absolute flow angle is between 150\({}^{\circ}\) and 170\({}^{\circ}\). There are two peaks in one period. The locations of the maximum value and minimum value are circumferential middle position and blade leading edge, respectively. Under the low-discharge condition, the absolute flow angle is further increased. The distribution range of the absolute flow angle is between 160\({}^{\circ}\) and 180\({}^{\circ}\). The absolute flow angle on the leading edge of every blade is close to 180\({}^{\circ}\). The inlet of the blade is considerably affected by the leakage vortex because the initial position of the TLV moves forward to the upstream. A low-speed region is formed, and an obvious reverse flow appears. For fluid machinery, hydraulic loss caused by unstable flows is inevitable. Entropy production represents the irreversibility of a system. According to entropy theory, the location and strength of the energy dissipation can be ensured quantitatively. Thus, entropy theory was adopted to determine the internal flow-loss mechanism in rotating machinery, such as centrifugal pumps [30], centrifugal fans [31], and pump turbines [32]. To analyze the three-dimensional structure of the flow at the tip clearance, different sections were made from the inlet to the outlet of the blade, and the entropy production rate (EPR) distribution was adopted to reflect the action range of the TLV. As shown in Figure 10, the high-EPR region is caused by the TLV, and the low-entropy production area represents the unaffected area. Figure 10 shows that the high EPR is mainly distributed in two regions: the blade tip clearance and the inside of the flow channel. On one hand, tip leakage flow is  
Figure 9: Absolute flow-angle distribution along the circumferential direction at 0.93 span. formed in the blade tip region. The high EPR is caused by the high-velocity gradient in the blade tip. On the other hand, the tip leakage flow mixes and rolls up with the main flow in the channel, which leads to the appearance of TLV. The high EPR distributes around the TLV. To quantitatively analyze the change in EPR under different flow conditions, the distribution of circumferential average EPR in the impeller is shown in Figure 11. There are two different peaks in the curves. The peak of EPR on the trailing edge is affected by the wake vortex. Regardless of the influence of the wake vortex, the distribution line of the EPR can be approximated as a \"bell curve\". This curve first increases and then decreases. Therefore, it can be divided into two parts: the dominant part of the main flow with a positive slope and the dominant part of leakage flow with a negative slope. The position of the peak depends on the balance between the main flow momentum and tip leakage flow momentum. Figure 11: EPR distribution along the streamwise direction. Figure 10: Tip streamline distribution along the streamwise direction: (**a**) 1.6 \(Q_{d}\); (**b**) 1.0 \(Q_{d}\); (**c**) 0.7 \(Q_{d}\). The peak of the \"bell curve\" moves upstream from the high- to low-discharge condition.