## 4 Flow-Induced Structural Stress Analysis of the Francis Turbine Unit  
Fluid-structure interaction (FSI) is a critical step during the design phase to evaluate the flow-induced structural stress of the Francis turbine, allowing a comprehensive analysis of the force characteristics and dynamic response of the turbine under different operating conditions. Due to the complex geometry of the Francis runner blades, especially at the rounded corners where the curvature varies greatly, these regions tend to be high-stress concentration points. In order to capture the accurate stress distribution and dynamic behavior in these regions, the mesh at the rounded corners of the runner blades must be refined (Figure 9). The mesh of the Francis runner uses high-quality tetrahedral elements with a total number of 0.66 million elements. The boundary conditions including the rotational speed, gravity acceleration, and fixed constraints are applied to the Francis runner. The density, Young's modulus, and Poisson's ratio of the Francis runner are 7700 kg m\({}^{-3}\), 210 GPa, and 0.3, respectively. From the previously described CFD analysis, the pressure results of the whole flow channel of the turbine under different operating conditions can be exported, and then the  
Figure 8: Flow streamlines of the Francis turbine under different loads: (**a**) 25% load, (**b**) 50% load, (**c**) 75% load, (**d**) 100% load. Figure 9: Finite element model of the Francis turbine for the flow-induced stress analysis. pressure data obtained from the CFD analysis can be accurately mapped to the nodes of the finite element model for the following flow-induced structural stress analysis (Figure 10). The analysis of the flow-induced runner stress under four different operating conditions from 25% to full load shows that the locations of the maximum stress of the runner under four different operating conditions vary with the power output. Figure 11 shows the maximum stress of the runner at 25% load, which appears at the chamfered surface connecting the blade outlet edge and the upper crown. From 50% load to 100% load, the maximum stress of the runner gradually increases, and the maximum stress of the runner appears at the chamfered surface connecting the blade inlet edge and the band (Figure 12). The maximum stress of the runner gradually increases with the increase in the power output of the Francis turbine unit (Figure 13), and the maximum stress of the runner at 100% is almost twice the stress at 50% load. From 50% load to 100% load, the maximum stress value of the runner and the unit output are linearly related. Since the unit is a low-head unit, the results of the stress analysis of four typical operating conditions, 100% (full load), 75% load, 50% load, and 25% load, show that the stress of the unit is 20% of the yield stress of the runner material, so the runner will not suffer damage during normal service. Figure 11: Normalized stress distribution of the Francis runner at 25% load. Figure 10: Transfer the pressure data to the finite element model of the Francis turbine. In order to prove that this investigation is of practical reference significance, Figure 14 shows that the Francis turbines from other power plants subjected to high flow-induced stresses under different loads eventually crack. Thus, the poor design of the Francis turbine can indeed lead to severe damage to the runner blades. After several years of operation, the investigated Francis turbine will be aged, and the runner stress level may increase. As a next step, it is planned to measure the stresses induced by the flow of the Francis runner under 30 to 50 different operating conditions between 0 and 100% loads, as well as transient processes such as start-up, stop-down, and load rejections. CFD and FEM simulations will also be performed to fully understand the flow patterns and flow-induced stress characteristics under these operating conditions. Figure 12: Normalized stress distribution of the Francis runner from 50% to 100% load.