# Numerical Simulation of the Tip Leakage Vortex Characteristics in a Semi-Open Centrifugal Pump
## 3 Results and Discussions
### Vortex Analysis of Tip Leakage Vortex  
The relative vorticity transport equation was used to analyze the initiation, development, and dissipation of the tip leakage vortex in the present study, which facilitated further analysis to receive a comprehensive understanding of the motion mechanism of vortices [34; 35; 36]. The equation is shown as,  
\[\frac{D\overset{\rightarrow}{\Omega}}{Dt}=\left(\overset{\rightarrow}{\Omega} \cdot\
abla\right)\overset{\rightarrow}{W}-\overset{\rightarrow}{\Omega} \left(\
abla\cdot\overset{\rightarrow}{W}\right)-2\
abla\times\left(\omega \times\overset{\rightarrow}{W}\right)+\frac{\
abla p\times\
abla p}{\rho^{2}}+ \
u\
abla^{2}\overset{\rightarrow}{\Omega} \tag{11}\]  
where \(\omega\) and \(\
u\) denote the rotating angular velocity and effective kinematic viscosity, respectively. The left term indicates the variation ratio of vorticity, and there are four terms on the right hand. The first term is the relative vortex stretching term (RVS), which is related to the velocity gradient. The second term is the relative vorticity dilation term, which reflects the change in vorticity due to fluid compressibility. The third term is the Coriolis force term (CORF), which is related to the rotary movement. The fourth term is the baroclinic torque term, which represents the effect of pressure gradient and density gradient on vorticity. The last term is the viscous diffusion term (VISD), which is caused by the viscosity of the fluid. The water used as a working fluid is considered incompressible, so the second term and fourth terms could be ignored [37]. Six circumferential sections corresponding to spans 0.7, 0.8, 0.9, 0.95, 0.97, and 0.99 were set up to study the aforementioned term distribution rules under the designed flow rate and comprehensively get the flow field around the tip clearance region. Figure 14 shows the RVS distribution. When the span was 0.95 and 0.97, the high RVS region began at the leading edge and moved downstream to 50% M along the suction surface. At 40% M, this region developed and extended to the leading edge of the adjacent blade with zonal distribution in the flow passage. The high RVS region coincided with the distribution of core and breakdown region of TLV A and TLV B by comparisons of Figures 11 and 12. This finding indicated that high RVS was the main reason for the formation and development of tip leakage vortex. RVS decreased with the span; therefore, the tip leakage vortex gradually dissipated during the movement along the blade span direction. The stretching and bending of the leakage vortex during the movement was the main reason for the vorticity change. Figure 15 demonstrates the distribution of CORF. The CORF of the tip region at the suction surface of the blade was found to be the largest among all span planes. CORF considerably decreased and exhibited ladder-like distribution as the span decreased from 0.99 to 0.7. The Coriolis force was generated by the rotation of the impeller, which induced the pressure gradient on the pressure surface and suction surface. The tip leakage flow was formed under the action of the pressure difference. Figure 14: Distribution of relative vortex stretching term on different span planes. Therefore, CORF was an important factor in driving the tip leakage flow. Furthermore, CORF could reflect changes in vorticity caused by tip leakage flow. The distribution of VISD is presented in Figure 16. The micro-jet was formed in the tip clearance region under the action of the pressure difference between the blade pressure surface and suction surface. The motion direction of the micro-jet was opposite to that of the blade rotation. Therefore, the relative motion with different velocities was formed between micro-jet and rotating impeller and fixed shroud. The relative motion generated vorticity on the wall, and VISD became a vorticity source. When the span was 0.99, high VISD almost occupied the entire blade passage.