## 3 Numerical Simulation Results
### Impact of the Internal Flow on the Flow Field  
A meridional velocity vector diagram of the impeller and the diffuser under design conditions is shown in Figure 4. When the influence of internal leakage is not considered, the main stream basically contains only the velocity vector along the flow direction. As can be seen in Figure 4a, tip leakage disturbs the direction of the flow of the fluid that is on the top of hydraulic components, leading to the appearance of backflow in the middle on the top of the vaned diffuser. As can be seen in Figure 4b, the backflow generated in the hub clearance damages the uniform flow field, resulting in a rapid increase in tip leakage and the upward velocity vector component of the fluid near the interface between the impeller and the vaned diffuser. At the same time, the upward velocity vector component of the main flow in the vaned diffuser increases, shortening the length of the top backflow section caused by tip leakage. As can be seen in Figure 4c, compared with hub leakage, the internal flow has a greater impact on the flow field of the pump. The static pressure of the main stream increases rapidly after it passes through the diffuser, then the partial \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Head/m** & \begin{tabular}{c} **Head Relative** \\ **Error (\%)** \\ \end{tabular} & **Efficiency (\%)** & \
\begin{tabular}{c} **Tip Leakage Flow** \\ **Relative Error (\%)** \\ \end{tabular} &
\begin{tabular}{c} **Tip Leakage Flow** \\ **(kg\(\cdot\)m\({}^{-3}\))** \\ \end{tabular} \\ \hline Tip leakage model & 28.28 & 2.5 & 38.5 & 16.0 & 0.0795 \\ Tip–hub leakage model & 28.42 & 3.0 & 35.2 & 6.0 & 0.1003 \\ Internal flow model & 27.61 & 0.1 & 34.5 & 3.9 & 0.1209 \\ Experimental values & 27.58 & & 33.2 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the results of different models. flow, namely the internal flow, flows back from the internal flow channel into the junction between the impeller and the vaned diffuser. As a consequence of the internal flow, both the upward velocity vector component of the fluid in the vaned diffuser and the tip leakage flow increase, and the flow disturbance exacerbates, resulting in the appearance of backflow. Generally, the existence of an internal flow complicates the flow field distribution in the impeller and the vaned diffuser, which further destroys the uniform flow field of the main stream. As a result, the tip leakage flow increases and both the position and degree of the backflow change. To convert the fluid from a radial flow to an axial flow and reduce the fluid diffusion problem caused by the excessive cross-sectional area in the subsequent flow channel, the vaneless diffuser is designed to have a bend, as shown in Figure 5. In the vaneless diffuser, due to the change in the channel's direction and the action of centripetal force, the velocity vector component of the fluid that points to the center of the curve increases. According to the law of flow at an elbow, the fluid at the top of the inlet of the vaneless diffuser diffuses while the fluid at the bottom shrinks. Moreover, there is a concave wall vortex zone at the top [18]. Such a flow field distribution is also affected by the flow field distribution of the former hydraulic component, which is the vaned diffuser. The longer the length of the backflow in the vaned diffuser, the smaller the disturbance at the top of the inlet of the vaneless diffuser. When there is some backflow at the top of the vaned diffuser, the upward velocity vector component of the nearby fluid is large, which cancels out the velocity vector component towards the center of the curve caused by the centripetal force so that the disturbance is reduced. The backflow and secondary flow at the outlet of the vaneless diffuser are mainly caused by the drastic change in the flowing cross-sectional area at the interface between the outlet and the motor flow channel.