## 3 Multi-Physical Coupled Fields Simulation Analyses
### Aerodynamic Performance
The 6\({}^{\circ}\) installation angle configuration generates higher \begin{table}
\begin{tabular}{c c c c c c} \hline
**Mesh** & **Elements** & **A,rms (m)** & **Change Rate** & **Cl,rms** & **Change Rate** \\ \hline M1 & 57,344 & 0.0164 & — & 0.5445 & — \\ M2 & 81,097 & 0.0167 & 1.80\% & 0.5594 & 2.66\% \\ M3 & 114,689 & 0.0168 & 0.63\% & 0.5628 & 0.61\% \\ \hline \end{tabular}
\end{table} 
Table 3: Mesh independence validation. energy output compared to the 3\({}^{\circ}\) and 0\({}^{\circ}\) configurations at wind speeds below 8.5 m/s, but its performance converges with that of the 0\({}^{\circ}\) configuration at higher wind speeds. The 3\({}^{\circ}\) installation angle configuration shows energy output characteristics comparable to the 0\({}^{\circ}\) configuration at wind speeds below 9 m/s; notably, it achieves the highest energy output capacity among all configurations when wind speeds surpass 9 m/s. These experimental results clearly indicate that configurations with smaller installation angles enhance the energy harvesting efficiency of the airfoil-based flutter energy harvesting system under high wind speed conditions. Figure 6 illustrates the phase diagrams and time histories of displacement and pitch angle for the airfoil-based flutter piezoelectric energy harvesting system at 8 m/s wind speed under varying installation angles. The results demonstrate that increasing installation angles leads to a progressive shift in neutral points away from zero for both displacement and pitch angle measurements. Notably, the system exhibits significantly larger  
Figure 5: Variation in output characteristics with the installation angle: (**a**) root mean square displacement; (**b**) root pitch angle; (**c**) root mean square output lift; (**d**) root mean square moment; (**e**) root mean square output voltage; (**f**) root mean square output power. displacement and pitch angle amplitudes at 6\({}^{\circ}\) and 9\({}^{\circ}\) installation angles compared to those at 0\({}^{\circ}\) and 3\({}^{\circ}\). At the 3\({}^{\circ}\) installation angle, the maximum positive displacement shows an increment of 0.0007 m relative to the 0\({}^{\circ}\) configuration, while the maximum negative displacement decreases by 0.0018 m, resulting in a larger overall displacement amplitude at 0\({}^{\circ}\) installation angle. Interestingly, at the 9\({}^{\circ}\) installation angle, while the displacement magnitude decreases compared to the 6\({}^{\circ}\) configuration, the pitch angle demonstrates a larger amplitude. This observation aligns with the trends presented in Figure 5, where the displacement at the 9\({}^{\circ}\) installation angle becomes progressively smaller than that at 6\({}^{\circ}\) as wind speed increases. This phenomenon can be attributed to stall occurrence at this specific wind speed under a 9\({}^{\circ}\) installation angle, effectively limiting the displacement. Stall, characterized by a sudden reduction in lift force despite increasing pitch angle beyond the critical stall angle, fundamentally alters the system's aerodynamic behavior. Figure 7 illustrates the phase diagrams and time histories of lift and moment magnitudes for the airfoil-based flutter piezoelectric energy harvesting system operating at a wind speed of 8 m/s under varying installation angles. The analysis reveals that the lift and moment magnitudes at the installation angles of 6\({}^{\circ}\) and 9\({}^{\circ}\) substantially exceed those observed at 0\({}^{\circ}\) and 3\({}^{\circ}\). Within each oscillation cycle, the systems with 6\({}^{\circ}\) and 9\({}^{\circ}\) installation angles demonstrate distinct characteristics, manifesting in sharp fluctuations of both lift and moment. These abrupt variations effectively constrain the pitch angle increment, which explains the observed phenomenon in Figure 5b,d, where despite experiencing larger moments, the systems with 6\({}^{\circ}\) and 9\({}^{\circ}\) installation angles did not achieve correspondingly higher pitch angles.