# Flow-Induced Fatigue Damage of Large Francis Turbines Under Multiple Operating Loads
## 2 Flow Dynamics Analysis on the Francis Turbine Unit
The power output of the simulation using the first set mesh deviates from the measured value by about 4%, while the power output simulated with the second and third set meshes is similar to the measured value. However, the computation time for the third set of mesh is much longer than that of the second set. Considering the balance between simulation accuracy and computation time, the second set mesh was selected for CFD analysis in this study (Table 2). With the increasing flexibility of grid load scheduling, the power plant's turbine units must operate within a wide range of 30 to 100 percent of their rated power. Three representative operating conditions--low partial load, high partial load, and full load--are selected for CFD calculations of the turbine unit. LPL and HPL in this study, corresponding to 1/3 and 2/3 of the full load, are typical loads that are prone to vortex and backflow phenomena, which may lead to flow instability and non-uniform pressure distribution. According to the statistics of the power station operation log, the turbine unit often operates at about 74 m head. The operating parameters of the selected three load conditions are listed in Table 3. \begin{table}
\begin{tabular}{c c c} \hline
**Fluid Domain** & **Element Number** & **Element Type** \\ \hline Spiral case and stay vane & 2.0 \(\times\) 10\({}^{5}\) & Hexahedral \\ Guide vane & 1.0 \(\times\) 10\({}^{5}\) & Hexahedral \\ Francis runner & 2.34 \(\times\) 10\({}^{7}\) & Hexahedral and tetrahedral \\ Crown and band chamber & 2.3 \(\times\) 10\({}^{5}\) & Hexahedral and tetrahedral \\ Draft tube & 2.8 \(\times\) 10\({}^{5}\) & Hexahedral \\ Total & 2.42 \(\times\) 10\({}^{7}\) & Hexahedral and tetrahedral \\ \hline \end{tabular}
\end{table} 
Table 2: Mesh of the fluid domains in the Francis turbine unit. Figure 3: Flow domain mesh of the Francis turbine unit. The flow domain of the runner is located in the rotating frame, and the rotating speed is 100 rpm. The flow passages of the spiral case, stay vane, guide vane, draft tube, and the chambers of the crown and band are located in the stationary frame. Transient rotor-stator interfaces are 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 are set up as flow discharge and static pressure according to the site measurement data, respectively. All surfaces are 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 6 s, which is equivalent to 10 revolutions for the Francis turbine. Each revolution of the Francis turbine was uniformly divided into 720 time-steps. A high-resolution solver is used for the CFD analysis with a residual threshold of \(10^{-6}\) to improve the accuracy and detail capture of simulation solutions. Figure 4: Mesh of the flow passages of the spiral case and stay vanes, guide vanes, and draft tube of the Francis turbine unit. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Operating Condition** & \begin{tabular}{c} **Power Output** \\ **(MW)** \\ \end{tabular} & \
\begin{tabular}{c} **Guide Vane** \\ **Opening (\%)** \\ \end{tabular} &
\begin{tabular}{c} **Flow Discharge** \\ **(\%)** \\ \end{tabular} \\ \hline Low partial load (LPL) & 60 & 52 & 43.9 \\ High partial load (HPL) & 120 & 76 & 71.4 \\ Full load (FL) & 180 & 100 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The analyzed load cases of the prototype unit. The pressure distributions of the Francis turbine under three different loads show that the pressure decreases gradually from the inlet of the spiral case to the draft tube outlet. For the LPL operating condition, the guide vane opening is relatively small, and the pressure difference between the upstream and downstream of the guide vane is obvious. The high-pressure zone of the flow channel is mainly located in the spiral case flow domain.