# Flow-Induced Strength Analysis of Large Francis Turbine Under Extended Load Range
## 4 Results and Discussions of Flow-Induced Strength Analysis  
### Flow Pressure Distributions and Velocity Distributions of the Francis Turbine Unit  
Under three different operating loads of 33.3%, 66.6%, and 100% of the rated power, the simulation results of the power output of the Francis turbine approach the measured ones, which verifies the reliability of the CFD calculation results. In order to better compare the calculation results of the Francis turbine under different operating conditions, the pressure and velocity of the turbine in this study are normalized according to the flowing formulae:  
\[p^{*}=\frac{p}{\rho_{f}gH} \tag{21}\]  
\[\mathbf{u}^{*}=\frac{\mathbf{u}}{u_{max.}} \tag{22}\]  
where \(g\) is the acceleration of gravity, \(H\) is the operating head of the Francis turbine unit, and \(u_{max. }\) is the maximum velocity value of the three load cases. The normalized pressure distribution in the flow domain of the Francis turbine unit at three different operating conditions, 33.3%, 66.6%, and 100% of the rated power, are demonstrated in Figures 6, 7 and 8, respectively. The results reveal that at 33.3%, 66.6%, and 100% loads of the rated power, the pressure of the Francis turbine flow domains decreases from the spiral case to the draft tube in general. The pressure in the spiral case is the highest in the flow domain, where the hydraulic energy is dominated by pressure energy. In the runner flow domain, the pressure drops significantly across the runner due to the pressure energy being transferred from the fluid to the rotating blades as kinetic energy. After the runner runner outlet, the fluid flows into the draft tube, which acts as a diffuser to recover part of the kinetic energy into pressure energy. The pressure in the draft tube is the lowest in the entire flow passages of the Francis turbine unit. Under various loads, the flow rate and flow direction of the water entering the runner can be controlled by adjusting the guide vane openings. Since the guide vane opening angles are significantly different at the three loads, the local pressure distributions in the guide vane domains also vary with each other. At a 33.3% rated power load, the guide vane opening is relatively small, and the water flow is ejected to the runner blade fluid channels from the narrow gaps between adjacent guide vanes. There is a significant difference in pressure between the upstream and downstream of the guide vanes. The pressure upstream of the guide vane is higher, and the pressure downstream of the guide vane is lower. With the guide vane opening increases at a 66.6% rated power load, the discharge increases accordingly to generate more power. And the pressure level of the flow between the guide vanes increases. At a 100% rated power load, the guide vanes are fully opened, and the pressure gradually decreases from the spiral case toward the guide vane outlet. The pressure gradient in the guide vane region is less than that at 33.3% and 66.6% loads of rated power. The normalized flow velocity distributions of the Francis turbine flow domain at 33.3%, 66.6%, and 100% loads of rated power are shown in Figures 9-11, respectively. Figure 8: Normalized pressure distribution of the Francis turbine unit (OP3, 100% rated power load). Figure 9: Normalized flow velocity distribution of the Francis turbine unit (OP1, 33.3% load of rated power). The flow streamlines of the Francis turbine under different operating conditions from 33.3% to 100% of the rated power load have obvious differences. At the 33.3% rated power load, the flow in the spiral case is relatively stable, but the flow between guide vanes and in the vaneless region is obviously irregular. Due to the small opening of the guide vanes and the narrow water flow passages in between, the fluid velocity increases when passing through the gap between adjacent guide vanes, forming a high shear zone with high speed locally. The fluid streamline distribution in the draft tube is disordered and asymmetric with obvious vortex and backflow.