# Albatross-Like Utilization of Wind Gradient for Unpowered Flight of Fixed-Wing Aircraft
## 5 Simulation Results and Discussion
### Discussion of Criterion for Unpowered Climbing
#### 5.1.3 Sensitivity Analysis of Aircraft Fraction  
To compare the sensitivity of the two aircraft fractions in Equation (19), we define the logarithm of the aircraft fractions as follows:  
\[\Sigma_{a}\triangleq\text{In}\Pi_{a}\qquad\Sigma_{A}\triangleq\text{In}\Pi_{ A}\qquad\Sigma_{5}\triangleq\text{In}\Pi_{5}\]  
The aircraft fraction can thus be obtained in the form of a summation from Equation (19), as follows:  
\[\Sigma_{a}=\Sigma_{A}+\Sigma_{5}\]  
Smaller values of these three parameters are also preferable, considering the multiplication criterion in Equation (17). Five example aircrafts were used to analyze the sensitivity of the fractions. The relevant parameters are presented in Table 2. The parameters for four of the aircrafts were obtained from references, while those for the National University of Defense Technology (NUDT) solar aircraft were measured and estimated.  
\(\Sigma_{A}\) was calculated with respect to \(C_{L}\) for each aircraft using Equation (52). The results are plotted in Figure 8. As can be observed, the plots for Avatar P2, SBXC, and Sky Sailor are very close, as well as those for the albatross and NUDT Solar Aircraft.  
It can also be seen from Figure 8 that all of the optimal lift coefficients are quite close to zero. The practical lift coefficient during flight is generally higher than the optimal value, and we therefore consider the operational \(\Sigma_{A}\) to range between the minimum value and the value corresponding to the maximum lift coefficient. On this basis, we calculated the values of \(\Sigma_{A}\), \(\Sigma_{S}\), and \(\Sigma_{a}\) for the different aircraft, as plotted in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Aircraft** & **Albatross D.S.** (1966) & **Avatar P2 D.S.** (1966) & **SBX C.S.** (1966) & **Sky Sailor Sailor (1966)** & **NUDT Solar Aircraft** \\ \hline \(S\) (m\({}^{2}\)) & 0.65 & 0.473 & 0.957 & 0.78 & 1.56 \\ \(m\) (kg) & 8.5 & 4.5 & 5.443 & 2.5 & 6.8 \\ \(C_{D0}\) & 0.033 & 0.0173 & 0.017 & 0.0191 & 0.03 \\ \(N\) & 0.019 & 0.0517 & 0.0192 & 0.0268 & 0.0223 \\ (\(C_{L}\))max & 1.6 & 1.1 & 1.0 & 1.3 & 1.0 \\ \(m/S\) & 13.08 & 9.51 & 5.69 & 4.49 & 4.36 \\ (kg/m\({}^{2}\)) & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Aircraft parameters.  
In Figure 9, the wide and narrow bars respectively indicate the maximum and minimum values. The lift coefficient can be adjusted to assign an aircraft fraction between the minimum and maximum values, and the aerodynamic configuration and wing loading can also be redesigned to modify the aircraft fraction. From Figure 9, it can be seen that the variation of \(\Sigma_{A}\) among the different aircrafts is as large as that of the wing loading fraction. This suggests that the aerodynamic fraction is generally as important as the wing loading fraction. Neither of these fractions can thus be ignored in a bid to enhance the sustainable climbing performance in a gradient wind.  
Figure 8: Logarithm of the aerodynamic fraction \(\Sigma_{A}\) with respect to \(C_{L}\) for different aircrafts.  
Figure 9: \(\Sigma_{A}\), \(\Sigma_{S}\), and \(\Sigma_{u}\) for different aircrafts (the wide and narrow bars respectively indicate the maximum and minimum values).