## 2 Numerical Simulation and Tests  
### Description of Geometry Models  
The model of the pump-turbine with splitter blades in this study was of a 1:7.47 scale for a pump-turbine in a domestic pumped storage power station. Basic parameters of the pump-turbine model are shown in Table 1.  
This paper studies the characteristics of the pump-turbine in the working condition of the pump. The working condition of the pump is as follows: First, the runner rotates anticlockwise, and water enters the draft tube. After passing through the runner, it is guided into the volute by the double-row cascade with a stay guide vane. Then, the casing converts the kinetic energy of the fluid to static pressure \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Data** \\ \hline Number of stay vanes & 16 \\ Number of guide vanes & 16 \\ Number of impeller vane & 10 \\ Impeller outlet minimum diameter D1 (mm) & 300 \\ Impeller outlet maximum diameter D2 (mm) & 584 \\ runner outlet diameter (mm) & 55 \\ volute outlet diameter (mm) & 320 \\ Draft tube outlet diameter (mm) & 600 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic parameters of the pump-turbine model. geometric characteristics of the pump-turbine model and calculation accuracy requirements, ICEM CFD software was used to divide the full flow passage into five parts (the solute, stay vanes, guide vanes, draft tube and runner), then converted into hexahedral meshes. The initial block was cut and O-type subdivided to form a decagonal shape, which is a better mapping of the curved section. Then, the node was combined with the array block. The quality of the generated structured grids was higher than 0.5, and the angle was greater than 25\({}^{\circ}\). Meshes of each part are shown in Figure 3.  
Since the number of grids has a great impact on the calculation results, the grid number independence analysis was performed for five sets of grids. According to the rated operating condition, with guide vanes set to openings of 17.5\({}^{\circ}\), eight groups of grid number schemes were set at uniform intervals between 4 million and 10 million. It can be seen from Table 2 that with the increase of the total number of grids, the simulation results tended to be stable. When the total number of grids reached 7.2 million or more, the simulated relative error rate on the head changes very little, so it can be considered that the number of grids has little influence on the calculation results. Considering the computing resources and solution accuracy, Case4 was selected, and the total number of grids was 7,213,000. The numerical calculations for different schemes were performed with the fluid simulation code ANSYS CFX, using the unsteady Reynolds-averaged Navier-Stokes equation, which is based on the finite volume method 14. The turbulence inside the pump was simulated by means of the shear stress transport (SST) model, and the reference pressure was set to 101,325 Pa. The constant total pressure and mass flow rate were given at the pump inlet and outlet as boundary conditions, respectively, since \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **Value** & **Stay** & **Guide** & **Draft** & **Runner** & **Total** & **\(\Delta\)H** \\  & **(10\({}^{6}\))** & **Vanes** & **Vanes** & **Tube** & **(10\({}^{6}\))** & **(10\({}^{6}\))** & **(10\({}^{6}\))** & **(\%)** \\ \hline Case1 & 0.486 & 0.883 & 0.982 & 0.606 & 1.309 & 4.266 & 2.732 \\ Case2 & 0.651 & 1.102 & 1.197 & 0.775 & 1.459 & 5.184 & 2.093 \\ Case3 & 0.845 & 1.297 & 1.364 & 1.009 & 1.638 & 6.153 & 1.285 \\ Case4 & 1.076 & 1.528 & 1.589 & 1.176 & 1.844 & 7.213 & 0.828 \\ Case5 & 1.260 & 1.732 & 1.771 & 1.353 & 2.065 & 8.181 & 0.826 \\ Case6 & 1.492 & 1.916 & 1.965 & 1.548 & 2.281 & 9.202 & 0.825 \\ Case7 & 1.707 & 2.097 & 2.166 & 1.733 & 2.467 & 10.170 & 0.824 \\ \hline \end{tabular}
\end{table} 
Table 2: Influence of number of grid cells on accuracy. Figure 3: Comparison of performance between simulation and experiment. the water enters radially and leaves axially.