# Design and Verification of Large-Scaled Flapping Wings for High Altitude Environment
## 3 Results
### Numerical Simulation of Large-Scaled Rigid Wings at High Altitudes  
Before the flight test, it is necessary to numerically simulate the flapping wing at different altitudes and adjust the flight state at high altitudes. The numerical method is mainly used to predict the effect of altitude on the trim state and power consumption. This section aims to give guidance for subsequent flight tests. The states of the flight parameters shown in Table 6 were selected for numerical calculation. The effects of altitude, flight speed, angle of attack, and flapping frequency on lift and drag were studied by intercomparison. Table 7 shows the time-averaged aerodynamic performance results for the calculated cases presented in Table 6. In order to visually compare the differences between the eight cases, the results of the second to eighth cases are expressed as percentages of the corresponding parameters compared to the first case, i.e., the plain environment. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **Altitudes (m)** & **Speeds (m/s)** & **Flapping Frequencies (Hz)** & **Angle of Attack (\({}^{\circ}\))** & \(Re\) & \(St\) \\ \hline
1 & 0 & 7 & 4 & 15 & 127,470 & 0.1749 \\
2 & 1000 & 7 & 4 & 15 & 117,785 & 0.1749 \\
3 & 2000 & 7 & 4 & 15 & 108,527 & 0.1749 \\
4 & 3000 & 7 & 4 & 15 & 99,933 & 0.1749 \\
5 & 4000 & 7 & 4 & 15 & 91,805 & 0.1749 \\
6 & 4000 & 10 & 4 & 15 & 131,150 & 0.1224 \\
7 & 4000 & 7 & 4 & 10 & 91,805 & 0.1749 \\
8 & 4000 & 7 & 5 & 10 & 91,805 & 0.2186 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Numerically calculated flight parameter states. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Wind Speed** & **Attack Angle** & **Flapping Frequency** & \(Re\) & \(St\) \\ \hline
9 th & 7.5 & 14.3 & 3.6 & 137,296 & 0.1469 \\
10 th & 8.2 & 10.4 & 3.7 & 150,111 & 0.1381 \\
11 th & 9 & 12.9 & 4.5 & 164,756 & 0.1530 \\
12 th & 8.8 & 11.7 & 4.5 & 161,094 & 0.1565 \\ \hline \hline \end{tabular}.  
\end{table} 
Table 5: The value of the independent variables that satisfy the trim state. Figure 16a,b indicate that altitude plays a significant role in the transient force during downstrokes. As the altitude increases, the peak positive lift in the middle of the downstroke decreases considerably, while the peak negative lift in the upstroke becomes slightly larger. The drag force shows the same trend as altitude. It is also known from the periodic-averaged aerodynamic performance in Table 7 that the increment of altitude leads to a slight decrease in the lift coefficient. The absolute values of the time-averaged lift, drag, and pitching moment all decrease remarkably with increasing altitude. Specifically, when the altitude increased from 0 to 4000 m, the \(Re\) was subsequently reduced by 27.98%, and the time-averaged lift, drag, and pitching moment decreased by 33.31%, 33.08%, and 33.33%, respectively. However, the lift coefficient decreased by only 0.25%, the drag coefficient increased by only 0.10%, and the pitching moment decreased by only 0.28%. The lack of lift will be one of the most serious problems for the vehicle at high altitudes if the flight state is kept constant. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case** & \(\mathbf{C_{L}}\) & \(\mathbf{C_{D}}\) & \(\mathbf{C_{M_{X}}}\) & \(\mathbf{L}\) & \(\mathbf{D}\) & \(\mathbf{M_{X}}\) & \(\mathbf{P}\) \\ \hline
1 & 1.35075 & 0.24894024 & \(-\)0.80316 & 1726.98 & 318.278 & \(-\)4.1075 & 103.2327 \\
2 & 99.94\% & 100.001\% & 99.94\% & 90.72\% & 90.77\% & 90.72\% & 90.72\% \\
3 & 99.85\% & 99.98\% & 99.85\% & 82.00\% & 82.11\% & 82.00\% & 82.00\% \\
4 & 99.83\% & 100.08\% & 99.81\% & 74.08\% & 74.26\% & 74.06\% & 74.06\% \\
5 & 99.75\% & 100.10\% & 99.72\% & 66.69\% & 66.92\% & 66.67\% & 66.67\% \\
6 & 82.26\% & 89.13\% & 81.19\% & 112.24\% & 121.62\% & 110.78\% & 110.78\% \\
7 & 72.63\% & 37.70\% & 73.24\% & 48.56\% & 25.20\% & 48.96\% & 48.96\% \\
8 & 80.90\% & 34.10\% & 82.25\% & 54.09\% & 22.80\% & 54.99\% & 68.74\% \\ \hline \hline \end{tabular}
\end{table}