## 1 Introduction  
Centrifugal pumps are important equipment in fluid-conveying systems and the main energy consumption component of industrial production [1]. The pressure difference causes the formation of tip leakage flow in a semi-open centrifugal pump. Thus, the energy performance and stability of the pump decrease [2; 3; 4]. Vortex, as a basic flow pattern, is defined as a region with concentrated vorticity. The location of a vortex can be determined by identifying the core of the vortex. The Q and \(\uplambda\) criteria are widely used in rotating machinery [5; 6; 7]. However, these two vortex identification methods have some limitations. The results are significantly influenced by the selection of critical values. Thus, a new omega method to identify the vortex was proposed in [8]. The coefficient R was raised by defining a ratio between the vortical vorticity and total vorticity. The vortex can be identified in different conditions when R is equal to 0.52. [PERSON] et al. [9; 10] compared and analyzed the Q and omega criteria and found that the omega criterion has striking advantages. The omega criterion can identify vortices with different intensities. This theory has been adopted to analyze the flow characteristics of pump turbines [11]. Tip leakage flow generated in the tip region is complex and includes separation vortex, leakage vortex, and secondary flows. It leads to an increase in the internal flow-field instabilities and a decline in pump performance. [PERSON] et al. [12] investigated the centrifugal impeller and reported that tip leakage flow caused by tip clearance could increase flow loss,thereby causing head and efficiency degradation. However, the hump effect was improved when the clearance was increased. [PERSON] et al. [13] and [PERSON] et al. [14] reached the same conclusion. The influence of tip clearance on the external characteristics of the pump is also related to the operating conditions. [PERSON] et al. [15] reported that the decline in pump head and efficiency under low-discharge conditions is remarkably smaller than that under high-flow conditions. However, [PERSON] et al. [16] studied a centrifugal compressor and drew different conclusions. The smaller the flow coefficient, the faster the decrease in pressure when the tip size is the same. TLV has two distinct characteristics, namely the unsteadiness of trajectory and the complexity of the motion mechanism. [PERSON] et al. [17] found that the static pressure at 90% span presents unsteady characteristics, which lead to a change in leakage vortex and a periodic shedding of the wake vortex. [PERSON] et al. [18] found that the period of unsteady flow is approximately half of the passage frequency of blades. [PERSON] et al. [19] investigated the connection between the stall and leakage vortex of an axial compressor. The leakage vortex trajectory flowed to the pressure surface with a decrease in discharge. A leading-edge overflow was formed, and tip stall was induced when the leakage vortex trajectory reached the leading edge of the adjacent blade. [PERSON] et al. [20] reported the same conclusion in their study of a centrifugal compressor. [PERSON] et al. [21] studied tip leakage flow in a centrifugal pump and argued that the low-pressure region near the blade tip changes periodically. As a result, a low-frequency vibration was developed. Furthermore, the TLV increased the amplitude of pressure fluctuation [22; 23; 24]. To better understand the TLV trajectory, [PERSON] et al. [25] first proposed the linear law of the leakage vortex trajectory of an axial compressor by decomposing the three-dimensional flow field. The angle between the leakage vortex trajectory and the blade skeleton line was assumed to be a function of the impeller inlet velocity and blade load. The coefficient k in the function was confirmed to be equal to 0.46 through experimentation. However, [PERSON] et al. [26] investigated a subsonic axial compressor and summarized that coefficient k was affected by the boundary condition of the clearance layer. [PERSON] et al.