# Flow-Induced Strength Analysis of Large Francis Turbine Under Extended Load Range
## 5 Conclusions  
In the study, the geometrical model CFD calculation models of the entire fluid domain including the spiral case, stay vanes, guide vanes, turbine runner, crow and band chambers, and draft tube of the turbine unit were constructed. CFD simulations and the flow-induced strength analysis under three operating conditions in the extended load range from the 33.3% rated power load to the full load have been performed. The Francis turbine flow domain pressure drops from the spiral case to the draft tube under 33.3%, 66.6%, and 100% loads of rated power, according to the CFD calculation results. Nevertheless, the guide vane region's local pressure distributions change with various loads. The pressure differential between the guide vanes' upstream and downstream is evident for the 33.3% and 66.6% loads of rated power, with the upstream pressure being higher than the downstream pressure. The pressure gradient in the guide vane region is lower at the 100% rated power load when the guide vanes are fully opened than when it is at 33.3% and 66.6% loads of rated power. The flow velocity distributions of the turbine under three operating conditions vary from one another. The flow in the spiral case is relatively stable for all loads. However, the flows in the guide vane region and the vaneless zone are strongly related to the guide vane openings. The flow forms a high-speed ring in the vaneless region at the 33.3% rated power load due to the small guide vane opening. The fluid velocity in the draft tube at the 33.3% and 66.6% rated power loads is asymmetric and disordered with the vortex and backflow. At the 100% rated power load, the flow of the whole turbine flow domain is more regular and stable. The high-speed ring in the vaneless region and the backflow and vortex in the draft tube disappear. According to the Francis runner's flow-induced strength analysis, the runner's maximum stress progressively rises as the power increases from 33.3% to 100% of the rated power load. Therefore, the maximum stress on the runner blades increases with the Francis runner's load. Under 100% of the rated power load, the analyzed runner's maximum stress is 2.2 and 1.4 times that under 33.3% and 66.6% of the rated power load, respectively, but it is only about one-third of the runner material's yield stress. This runner is safe to operate throughout the extended range of 33.3% to 100% of the rated power load. Because of the Francis runner's distinctive design, the flow-induced structural stress on the runner blades varies nonlinearly with the runner load. The study provides an in-depth analysis of flow-induced stresses of a large Francis turbine over an extended load range. By integrating CFD and FEA approaches, this study fills a critical gap in understanding the operating limits of large Francis turbines to ensure their safe operation under varying load conditions. The analysis approach proposed in this study can be applied to other hydraulic machinery including Francis turbines, pumps, and pump-turbines. **Author Contributions:** Conceptualization, X.H. ; methodology, X.H. ; investigation, X.L. and X.H. ; validation, W.C.; writing--original draft preparation, X.H. and X.L. ; writing--review and editing, X.H. and W.C.; supervision, Z.W. All authors have read and agreed to the published version of the manuscript. **Funding:** 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. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. **Conflicts of Interest:** The authors [PERSON] and [PERSON] are employed by the company Power Station Management Bureau, Y.R. Wanjiazhai Water Multi-Purpose Dam Project Co., Ltd.