## 3 Multi-Physical Coupled Fields Simulation Analyses
### Aerodynamic Performance  
Figure 5 presents a comprehensive analysis of the dynamic characteristics and energy harvesting performance of the airfoil-based piezoelectric flutter energy harvester under various installation angles and wind speeds. The experimental results reveal that the installation angle plays a crucial role in determining the system's performance characteristics. Notably, the critical flutter speed demonstrates a consistent decreasing trend as the installation angle increases from 7.8 m/s at 0\({}^{\circ}\) to 7.6 m/s at 3\({}^{\circ}\), further reducing to 7.2 m/s at 6\({}^{\circ}\), and reaching 6.3 m/s at 9\({}^{\circ}\). This systematic reduction in critical flutter speed significantly broadens the effective operating range of the energy harvester. Figure 5a,b demonstrate the displacement and pitch angle characteristics of the airfoil-based flutter piezoelectric energy harvesting system at various installation angles (0\({}^{\circ}\), 3\({}^{\circ}\), 6\({}^{\circ}\), and 9\({}^{\circ}\)). The analysis reveals that at wind speeds below 7.5 m/s, the system installed at 9\({}^{\circ}\) exhibits the highest displacement and pitch angle among all configurations; however, these performance parameters progressively decrease as wind speeds exceed 7.5 m/s, with a significant reduction observed beyond 8.5 m/s. The system behavior at the 3\({}^{\circ}\) installation angle closely mirrors that of the 0\({}^{\circ}\) configuration, particularly in terms of displacement characteristics, while their pitch angles remain similar at wind speeds below 10 m/s, though the 3\({}^{\circ}\) configuration demonstrates higher pitch angles at speeds exceeding 10 m/s. Within the specific wind speed range of 7.5-8.5 m/s, the 6\({}^{\circ}\) installation angle configuration achieves maximum displacement and pitch angle values; as wind speeds surpass 8.5 m/s, the performance metrics of the 6\({}^{\circ}\) configuration converge with those of the 3\({}^{\circ}\) setup. Figure 5c presents the lift characteristics of the airfoil-based flutter piezoelectric energy harvesting system at various installation angles. The results demonstrate that as the installation angle increases, the wind speed required for the system to achieve greater lift decreases. However, as the wind speed continues to increase, the lift obtained by systems with different installation angles tends to converge. This phenomenon, when analyzed in conjunction with Figure 5a, explains why systems with larger installation angles can experience fluttering at lower wind speeds. Figure 5d illustrates the torque characteristics under different installation angles, showing that the torque generated by the system increases proportionally with the installation angle. The analysis reveals that the system with a 9\({}^{\circ}\) installation angle generates torque significantly higher than that of the 6\({}^{\circ}\) angle configuration, while the torque for the 6\({}^{\circ}\) installation angle is substantially higher than those observed at 3\({}^{\circ}\) and 0\({}^{\circ}\) angles. At wind speeds below 8.5 m/s, the torques generated by systems with 3\({}^{\circ}\) and 0\({}^{\circ}\) installation angles remain comparable; however, beyond this wind speed threshold, the torque for the 3\({}^{\circ}\) angle configuration becomes significantly higher than that of the 0\({}^{\circ}\) angle. Analysis of Figure 5b,c reveals that at higher wind speeds, the system with a 9\({}^{\circ}\) installation angle, despite generating higher torque, does not achieve larger pitch angles or lift. Figure 5e,f demonstrate the energy output characteristics of the airfoil-based flutter piezoelectric energy harvesting system at different installation angles under varying wind speeds. At wind speeds below 7.5 m/s, the configuration with a 9\({}^{\circ}\) installation angle exhibits superior energy output capacity. However, this configuration's performance significantly deteriorates when wind speeds exceed 8.5 m/s, resulting in the lowest energy output among all tested configurations.