# Numerical Simulation of the Tip Leakage Vortex Characteristics in a Semi-Open Centrifugal Pump
## 3 Results and Discussions
### Structure of the Tip Leakage Vortex  
The standard helicity \(H_{n}\)[27; 28] is defined as shown below to quantitatively identify the vortex structure. \[H_{n}=\frac{\overrightarrow{\Omega}\cdot\overrightarrow{W}}{\left| \overrightarrow{\Omega}\right|\left|\overrightarrow{W}\right|} \tag{9}\]  
where \(\overrightarrow{\Omega}\) and \(\left|\overrightarrow{\Omega}\right|\), respectively, denote the vector and modulus of relative vorticity. \(\overrightarrow{W}\) and \(\left|\overrightarrow{W}\right|\), respectively, denote the vector and module of relative velocity. \(H_{n}\) essentially represents the cosine value of the angle between the relative velocity vectors and vorticity vectors. \(H_{n}\) is equal to 1, \(-\)1, and 0 when the direction of the vorticity vector and relative velocity vector is the same, opposite, and orthogonal, respectively. Moreover, the \(H_{n}\) value of 1 or \(-\)1 represents the vortex core, which is characterized by strong vortex roll-up. The Q criterion can accurately characterize the structure of vortices. Therefore, this criterion is widely used in rotating machinery [29; 30; 31; 32; 33]. Q, which is also regarded as the vortex core, is the region where the rotation rate tensor exceeds the strain rate tensor. The vortex exists when Q \(>\) 0, and large Q signifies a strong vortex intensity. \[Q=\frac{1}{2}(\overrightarrow{\Omega}\overrightarrow{\Omega}-\overrightarrow {S}\overrightarrow{S}) \tag{10}\]  
where \(\overrightarrow{\Omega}\) and \(\overrightarrow{S}\) denote the vector of relative vorticity and strain rate tensor, respectively. In the present work, the tip leakage vortex was characterized by the Q criterion. Figure 12 shows the isosurface distribution when \(Q=1.0\times 10^{5}\) s\({}^{-2}\). The structure of the tip leakage vortex in the impeller mainly comprised tip leakage vortex A (TLV A), B (TLV B), and a passage vortex. Moreover, the tip leakage flow was observed near the trailing edge of the blade. The \(H_{n}\) of TLV A was close to 1, which indicated the formation of tip leakage vortex core, and the direction of the vortex vector was the same as that of relative velocity. By contrast, the TLV B had varying \(H_{n}\) from \(-\)1 to 1, which indicated the complexity of the vortex structure. The formation of the tip leakage vortex was accompanied by the breakdown of the vortex. The tip leakage vortex would also deteriorate the flow field. Therefore, the passage vortex was generated downstream of the tip leakage vortex. Figure 13 shows the streamline released from tip clearance for a detailed study of the tip leakage vortex structure. The red, black, and blue streamlines indicate release from 0-20% M, 30-50% M, and 75-85% M, respectively. All the streamlines were driven by the pressure gradient. TLV A formed at the leading edge of the blade, moved along the leakage core, and then broke at approximately 35% M. After mixing with the mainstream, TLV A flew downstream along the suction surface of the blade. Meanwhile, TLV B was formed near the middle of the blade. It moved to the leading edge of the pressure surface of adjacent blade B2. Most of the fluid was blocked by the wall. This condition changed the flow direction and made the fluid move in the opposite direction, and then TLV B broke at approximately 60% M. The broken fluid of TLV A and TLV B mixed and flew downstream along the suction surface of the blade. In addition, a small part of the fluid flows around the inlet of the adjacent blade B2 and moved along the suction surface of B2. The disturbance occurred in the adjacent flow passage. The flow was mainly the tip leakage flow rather than tip leakage vortex near the trailing edge of the blade. Under the action of pressure difference, the tip leakage flow moved circumferentially in the direction opposite to the impeller rotation. The entrainment effect was formed between the tip leakage flow and mainstream in the middle position of the B1 and B2, and the secondary leakage flow was formed at the trailing edge of B2.