# Flow-Induced Fatigue Damage of Large Francis Turbines Under Multiple Operating Loads
## 6 Conclusions  
In this investigation, the flow dynamics of large Francis turbine units are analyzed under three representative loading conditions--low partial load, high partial load, and full load--and the flow-induced stress of the Francis runner is analyzed under these loading conditions. The pressure of the Francis turbine fluid domains under three different loads decreases gradually from the spiral case inlet to the draft tube outlet. For low partial load, 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. The jet water flow between adjacent guide vanes forms a high-speed water ring in the vaneless space, and the flow pattern in the draft tube is extremely turbulent. As the guide vane opening and power output of the turbine increase, the high-speed water ring in the vaneless area gradually weakens, and the turbulence of the flow pattern in the draft tube decreases. When the guide vanes are fully opened at full load, the high-speed water ring disappears, and the overall flow pattern of the turbine flow channel tends to be smooth and stable. The pressure distribution in the entire flow domain of the turbine unit shows a gradually decreasing trend from the spiral case inlet to the draft tube outlet. The flow-induced stress analysis reveals that the maximum static and dynamic stresses of the runner at three representative loading conditions are located at the chamfered surface where the blade trailing edge connects to the runner crown. The maximum static stresses of the Francis runner are 284 MPa, 352 MPa, and 381 MPa at low partial load, high partial load, and full load, respectively, and above the allowable stress limits, defined as half of the yield stress of the runner material (550 MPa). The peak-to-peak values of runner dynamic stress at LPL, HPL, and FL are 15 MPa, 25 MPa, and 14.6 MPa, respectively. The high stress invoked by the unsteady flow under various loading conditions in this runner was the cause of the fatigue breakage of the runner blades. For the Francis runners in this power plant that have been in service, it is strongly recommended that different sensors be used to monitor the dynamic behavior of the runner in real-time and detect the early crack formation and crack propagation in the runner blades. It is recommended that regular inspection, overhaul, and maintenance be strengthened, non-destructive testing on the runner should be conducted regularly, and cracked blades should be repaired or replaced in time to ensure the integrity of the runner. Considering the long-term, safe, and stable operation of the power station, it is strongly recommendedthat the runner be redesigned and the blade design optimized to reduce flow unevenness and pressure pulsation intensity. It is necessary to increase the thickness of the blade trailing edge to reduce the stress concentration at the blade root. The entire runner can be manufactured using 3D printing technology to avoid the high residual stress caused by blade welding. High-strength stainless steel with better fatigue resistance and surface strengthening technology can be used to improve the fatigue strength of the blades. The conclusions of this investigation have important reference value for mitigating the fatigue damage of similar Francis runners and optimizing the unit operation. Conceptualization, X.H. ; methodology, X.H. ; investigation, P.L. and X.H. ; validation, T.Y. ; writing--original draft preparation, X.H. and P.L. ; writing--review and editing, X.H. and T.Y. ; supervision, Z.W. All authors have read and agreed to the published version of the manuscript. This research and the APC were funded by the project \"Study on Stable Operation of Units in Wanjiazhai Hydropower Station\" of Wanjiazhai Water Multi-purpose Dam Project Co., Ltd. Grant number: WJZ-ZB-2022-027. Not applicable. The original contributions presented in this study are included in the article.