# Numerical Simulation of the Tip Leakage Vortex Characteristics in a Semi-Open Centrifugal Pump
## 3 Results and Discussions
### Internal Flow Field  
The streamline distributions of different blade spans are shown in Figures 8 and 9. When the span was 0.05 and 0.5, the velocity of TC was larger than that of NTC, and the low-speed region on the blade pressure surface was improved. However, the streamline was disturbed near the impeller outlet. When the span was 0.95, the flow field of NTC was relatively smooth. But a low-speed region was observed at the blade leading edge of TC, and it gathered in the middle of the passage that led to a blockage in the flow when it moved to downstream along the pressure surface because the plane near the tip clearance was considerably affected by the leakage vortex. Figure 6: Comparison of the head between experimental and numerical results. Figure 7: External characteristics of TC and NTC. The flow at the blade inlet was analyzed to explore the reasons for the above phenomena. Figure 10 presents the distribution of the relative flow angle along the circumferential direction of different spans at the blade inlet. The relative flow angle distribution showed evident periodicity when the span was equal to 0.05 and 0.5. The number of periods was equal to the number of blades. From the suction surface to the pressure surface of the adjacent blade, the relative flow angle increased first and then decreased. The maximum was obtained in the circumferential middle position of two blades. Compared with NTC, the relative flow angle of TC increased considerably, but the fluctuation range decreased, and the distribution law remained the same. The large inflow angle resulted in the improvement of the low-speed region of the blade pressure surface. When the span was 0.95, the relative flow angle considerably increased, and the distribution law evidently changed because closeness to the tip region remarkably influenced the tip clearance. Figure 8: Streamline distribution of NTC on different span planes. (a) Span = 0.05, (b) Span = 0.5, (c) Span = 0.95. Figure 9: Streamline distribution of TC on different span planes. (a) Span = 0.05, (b) Span = 0.5, (c) Span = 0.95. From the suction surface to the pressure surface, the relative flow angle increased first and then decreased to 68\({}^{\circ}\) in the circumferential middle position. Afterward, the relative flow angle increased again and then finally decreased to 15\({}^{\circ}\). Two maxima close to 90\({}^{\circ}\) and a minimum value close to 68\({}^{\circ}\) appeared in one flow passage. The change in relative flow angle distribution led to a large negative attack angle that appeared at the blade inlet. So, the flow separation occurred on the pressure surface of the blade, and the low-speed region was formed that blocked the flow passage. The pressure distribution on the blade surface could accurately reflect the turbocharging performance of the pump and was significantly affected by the structure of leakage vortex. The static pressure coefficient \(C_{p}\), which is expressed, as shown in the equation below, was introduced to illustrate the static pressure distribution law on the blade surface. \[C_{p}=\frac{P-\overline{P}}{0.5\rho{V_{in}}^{2}} \tag{8}\]  
where \(P\) is the static pressure on the blade, \(V_{in}\) and \(\overline{P}\) are the velocity and the static pressure of the inlet, respectively, and \(\rho\) stands for the fluid density. The \(C_{p}\) distribution on the blade surface is shown in Figure 11. \"M\" indicates the meridional distance from the leading edge to the trailing edge of the blade; \"0% M\" and \"100% M\" are the leading edge and trailing edge, respectively. The \(C_{p}\) of the blade without tip clearance smoothly increases with the meridional distance, but the intersection points appear near the trailing edge of the blade. The \(C_{p}\) of the pressure surface is smaller than that of the suction surface due to the wake vortex at the trailing edge of the blade. The tip clearance had a considerable effect on the distribution of \(C_{p}\) at any span of the blade.