# Albatross-Like Utilization of Wind Gradient for Unpowered Flight of Fixed-Wing Aircraft
## 5 Simulation Results and Discussion  
### Discussion of Criterion for Unpowered Climbing  
#### 5.1.1 Optimum Lift Coefficient  
The drag coefficient is generally expressed as [43]  
\[C_{D}=C_{D0}+nC_{L}^{2} \tag{51}\]  
where \(n\) and \(C_{D0}\) are parameters that depend on the airfoil itself. When an aircraft is in flight, its angle of attack can be adjusted by the elevators to vary the lift coefficient \(C_{L}\). The drag coefficient \(C_{D}\) varies with \(C_{L}\) according to Equation (51), and \(C_{L}\) can thus be considered as a control input. Substituting Equations (51) and (11) into Equation (20) yields  
\[\Pi_{A}=h\left(\frac{C_{L}}{C_{D0}+nC_{L}^{2}}\right)\left[C_{L}^{2}+\left(C_{ D0}+nC_{L}^{2}\right)^{2}\right]^{\frac{1}{4}} \tag{52}\]  
Equation (52) expresses \(\Pi_{A}\) as a function of \(C_{L}\). Minimizing \(\Pi_{A}\) optimizes \(C_{L}\) and thus minimizes the climbing criterion in Equation (17). In the case of an albatross, \(C_{D0}\) and \(n\) are respectively 0.033 and 0.019 [22]. The function \(\Pi_{A}(C_{L})\) for this case is plotted in Figure 7.  
Figure 7 indicates that the optimal lift coefficient \(C_{L}\) for an albatross is approximately 0.1. The lift coefficient corresponding to the maximum lift-to-drag ratio can be calculated using [21] as  
\[\sqrt{C_{D0}/n}=1.32\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**State**} & \multicolumn{3}{c}{\(\boldsymbol{v_{x}}\)} \\ \cline{3-4} \multicolumn{1}{c}{} & \multicolumn{1}{c}{**>0**} & \multicolumn{1}{c}{**<0**} \\ \hline \(v_{z}\) & \begin{tabular}{c} \textgreater{}0 \\ \textless{}0 \\ \end{tabular} & \
\begin{tabular}{c} Upwind Climb \\ High Altitude Turn \\ \end{tabular} &
\begin{tabular}{c} Low Altitude Turn \\ Downwind Dive \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Initial entry logic.  
As can be observed, these two \(C_{L}\) values significantly differ. To understand the reason for the small optimal \(C_{L}\), we will consider a simple calculation. For \(C_{L}=1.32\), we have \(L/D=20\), \(h=1.08\), and  
\[\left[C_{L}^{2}+\left(C_{D0}+nC_{L}^{2}\right)^{2}\right]^{\frac{1}{4}}\Bigg{|} _{C_{L}=1.32}=1.15\]  
Thus, from Equation (20), \(\Pi_{A}=1.24\).  
Further, for \(C_{L}=0.1\), we have \(L/D=3\), \(h=1.8\), and  
\[\left[C_{L}^{2}+\left(C_{D0}+nC_{L}^{2}\right)^{2}\right]^{\frac{1}{4}}\Bigg{|} _{C_{L}=0.1}=0.32\]  
From Equation (20), \(\Pi_{A}=0.58\).  
Although the maximum \(L/D\) minimizes \(h\), the other item in Equation (20) has a much larger effect on the calculated value of \(\Pi_{A}\). In the general soaring problem without a wind gradient, the maximum lift-to-drag ratio is optimal for the soaring range [44]. The situation is, however, different in the presence of a wind gradient. An intuitive explanation of the difference can be presented from an energy viewpoint. In the absence of a wind gradient, maintaining the maximum lift-to-drag ratio enables energy saving, based on traditional flight mechanics. When there is a wind gradient, the aircraft can harvest extra energy from it, and a smaller lift coefficient would cause the velocity to increase, because of the higher velocity needed to generate the same lift to offset the weight. Thus, the aircraft crosses more wind layers during the same period, acquiring energy more efficiently, although it also saves less energy compared to that at the maximum lift-to-drag ratio. It is thus necessary to make a compromise between harvesting and saving energy. Therefore, the calculated optimal lift coefficient in Figure 4 is smaller than the corresponding maximum lift-to-drag ratio.