## 2 Numerical Simulation  
### Research Object  
The research object selected in this paper is a horizontal end-suction centrifugal pump with six blades, designed for transporting clear water medium at room temperature. This centrifugal pump is manufactured by Hunan Credo Pump Co., Ltd. Xiangtan, China. Its main structural and performance parameters are listed in Table 1. The structural design and performance configuration of this pump are carefully optimized to meet the requirements of practical applications, ensuring its reliability and efficiency in the process of transporting clear water medium.  
The hydraulic model is extracted based on the three-dimensional structural assembly of this centrifugal pump, as shown in Figure 1. The fluid domain inside the centrifugal pump is mainly divided into three parts: the impeller region, the suction chamber, and the discharge chamber.  
In order to ensure that the fluid flowing into and out of the computational domain is fully developed for sufficient computational accuracy, the inlet and outlet parts of the pump are extended to form the inlet pipe and the outlet pipe. The lengths of the inlet and outlet pipes are each five times their respective diameters. This pipe extension design can effectively mimic the actual flow conditions at the pump's inlet and outlet, minimizing the impact of boundary conditions on the calculation results and enhancing the reliability and accuracy of the computational analysis of the centrifugal pump's internal flow field.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Rated flow & \(Q_{d}\) & \(12.5\,\mathrm{m}^{3}/\mathrm{h}\) \\ Rated head & \(H_{d}\) & \(22.7\,\mathrm{m}\) \\ Rated speed & \(n_{d}\) & \(2900\,\mathrm{r}/\mathrm{min}\) \\ Specific speed & \(N_{s}\) & \(2.97\) \\ NPSH required & NPSHr & \(4.0\,\mathrm{m}\) \\ Impeller inlet diameter & \(D_{1}\) & \(0.061\,\mathrm{m}\) \\ Impeller outlet diameter & \(D_{2}\) & \(0.139\,\mathrm{m}\) \\ Impeller outlet width & \(b_{2}\) & \(0.007\,\mathrm{m}\) \\ Number of blades & \(N\) & \(6\) \\ Volute base diameter & \(D_{3}\) & \(0.15\,\mathrm{m}\) \\ Pump inlet diameter & \(D_{4}\) & \(0.05\,\mathrm{m}\) \\ Pump outlet diameter & \(D_{5}\) & \(0.032\,\mathrm{m}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main structure and performance parameters of the research object.  
Figure 1: Three-dimensional diagram of the whole hydraulic model. **1**: Impeller region; **2**: Discharge chamber; **3**: Section region; **4**: Inlet pipe; **5**: Outlet pipe; **6**: Inlet; **7**: Outlet; **8**: Blade.  
To more clearly illustrate the impeller structure of the research object and its relative relationship with the volute, the mid-section of the impeller and volute regions in Figure 1 is intercepted, as shown in Figure 2. In Figure 2, a Cartesian rectangular coordinate system is established with the central position of the impeller as the coordinate origin. The axial direction of the impeller is defined as the Z direction (perpendicular to the paper plane and not drawn in the figure), while the X and Y directions are defined as the two radial directions of the impeller.  
Meanwhile, the rotational position of the impeller--when a straight line is formed by connecting the root of a certain blade (Blade 5 in the figure), the center point of the impeller, and the protruding point at the end of the tongue--is defined as its initial phase. This initial phase also serves as the initial moment for the calculation of the unsteady flow field within the centrifugal pump.