## 3 CFD Analysis of the Francis Turbine Unit  
The hydropower plant under study is equipped with four hydro-generator units, each with a rated power of 46 MW (Table 1). The runner diameter is 3.5 m with 13 three-dimensional twisted blades. Each hydro-generator unit has 24 stay vanes and 24 guide vanes. The unit has a rated speed of 166.7 rpm, a rated head of 54 m, and a rated flow of 91.8 m\({}^{3}\)s\({}^{-1}\), respectively. Figure 2 presents the three-dimensional flow domain of the prototype Francis turbine unit. The flow domain of the turbine unit consists of the flow passages of the spiral case, stay and guide vanes, runner and draft tube flow passage, and the chambers of the crown and the band. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Rated head & 54 & m \\ Maximum head & 57 & m \\ Rated power & 46 & MW \\ Rated flow & 91.8 & m\({}^{3}\)s\({}^{-1}\) \\ Rated speed & 166.7 & rpm \\ Stay vane number & 24 & - \\ Guide vane number & 24 & - \\ Runner blade number & 13 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the hydro-generator unit. The finite-volume computational model of the turbine flow passages is established using high-quality tetrahedral and hexahedral meshes generated by the Ansys Meshing tool (Figure 3), and the meshes near the surfaces of the runner, stay vanes, and guide vanes, etc., are refined with five layers with a layer ratio of 1.2 to achieve accurate and reliable predictions of turbulent flow in the turbine fluid passage. Three groups of meshes of the fluid regions for the Francis turbine unit were generated with 27.8 million, 11.2 million, and 6.7 million elements, respectively. The first two grid sets yielded similar results for the turbine torque compared to the measurements at 100% load. However, with the same number of 48 CPU cores, the computation time for the first mesh group was nine times longer than for the second mesh group. The third group of mesh with fewer elements had a large deviation of around 3%. In order to ensure the simulation  
Figure 3: CFD calculation model of the Francis turbine unit. Figure 2: Flow domain of the CFD analysis of the Francis turbine unit. accuracy and save the calculation time, the second mesh group with 11.2 million elements was used for the CFD simulation of the Francis turbine (Table 2). The flow domain of the runner was located in the rotating frame, and the rotating speed was 166.7 rpm. The flow passages of the spiral case, stay vane, guide vane, draft tube, and the chambers of the crown and band were located in the stationary frame. Transient rotor-stator interfaces were specified between the rotating runner flow domain and the adjacent stationary flow domains. The inlet of the spiral case and the outlet of the draft tube were set up as the flow discharge and static pressure according to the site measurement data, respectively. All surfaces were set to non-slip wall conditions except for the domain interface, inlet, and outlet boundary conditions. To ensure convergence of the numerical calculations, the total CFD simulation time was set to 3.6 s, which is equivalent to 10 revolutions for the Francis turbine. Each revolution of the Francis turbine was uniformly divided into 720 timesteps. The high-resolution solver was used for the CFD analysis with a residual threshold of \(10^{-6}\) to improve the accuracy and detail capture of simulation solutions. In order to satisfy the load regulation requirements of the power grid, the turbine-generator units in this power plant had to operate in a wide range from 25% to 100% of the rated power. Four representative load conditions at 25%, 50%, 75% and 100% of the rated power were selected for CFD analysis and corresponding flow-induced dynamic stress analysis.