# Flow-Induced Fatigue Damage of Large Francis Turbines Under Multiple Operating Loads
## 2 Flow Dynamics Analysis on the Francis Turbine Unit  
The hydropower plant under study in this paper is equipped with six large Francis turbine generator units, and the rated power of each unit is 180 MW. The design parameters of the large Francis turbine units of the investigated power plant are presented in Table 1.  
A completed geometrical model of the flow passage for the turbine unit is presented in Figure 2. The entire flow passage is constructed with the flow passages of the spiral case, stay vane, guide vane, turbine runner, draft tube, and the chambe \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline Rated power & MW & 180 \\ Rated head & m & 68 \\ Rated speed & rpm & 100 \\ Stay vane number & - & 24 \\ Guide vane number & - & 24 \\ Runner blade number & - & 13 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the prototype unit. Computational fluid dynamics (CFD) is a powerful tool used to simulate the complex fluid flow phenomena in a hydro turbine unit. The turbulent flow in the turbine flow passages follows the law of conservation of momentum (Equation (1)) and the law of conservation of mass (Equation (2)). \[\frac{\partial(\rho_{f}\mathbf{v})}{\partial t}+\
abla\cdot(\rho_{f}\mathbf{v} \,\mathbf{v})=-\
abla p+\
abla\cdot\tau+\rho_{f}\mathbf{f_{f}} \tag{1}\]  
\[\frac{\partial\rho_{f}}{\partial t}+\
abla\cdot(\rho_{f}\mathbf{v})=0 \tag{2}\]  
where \(t\) is time; \(\rho_{f}\) and \(p\) are the density and pressure of the fluid; \(\mathbf{v}\), \(\tau\), and \(\mathbf{f_{f}}\) are the velocity, deviatoric stress tensor, and accelerations of the fluid, respectively. The [PERSON]'s shear stress transport (SST) turbulence model, developed by [PERSON] in the 1990s, is used to analyze turbulent flows in the turbine units. The [PERSON]'s SST model combines the advantages of the \(k-e\) and \(k-\omega\) turbulence models to provide accurate predictions of flow characteristics, particularly in regions with adverse pressure gradients and separation [29,30]. The flow of the turbine unit enters the spiral case at high pressure and velocity, and then, the stay vanes help direct the flow smoothly towards the guide vanes. The guide vane can be adjusted to control the direction and flow rate of the water entering the runner under different working conditions. By changing the guide vane opening, the flow direction in the guide vane passage can be better aligned with the flow direction of the runner to minimize energy loss and improve the efficiency of the hydraulic turbine. The runner is the key rotating component of the Francis turbine for energy conversion. As water flows drive the runner to rotate, the kinetic and potential energy of the water is converted into the mechanical energy of the rotating shaft system, and then, the rotating generator cuts magnetic lines and generates electricity. The draft tube is located at the outlet of the runner and ensures a smooth discharge of water downstream. It also helps to recover the kinetic energy of the water by gradually enlarging its cross-sectional area, reducing the flow velocity, and increasing the pressure of the outgoing water. The finite volume computational model of the turbine flow passages is established using structured and unstructured high-quality meshes (Figures 3 and 4), and the meshes near the surfaces of the runner, stay vanes, and guide vanes, etc., are refined to achieve accurate and reliable predictions of turbulent flow in the turbine fluid passage. Figure 2: Flow domain model of the Francis turbine unit. To ensure the accuracy and reliability of the simulation results, three sets of meshes of the fluid domains in the Francis turbine are created using elements of different sizes for mesh independence analysis. The total element numbers of the three sets of meshes are 1.62 \(\times\) 10\({}^{7}\), 2.42 \(\times\) 10\({}^{7}\), and 2.94 \(\times\) 10\({}^{7}\), respectively.