# Albatross-Like Utilization of Wind Gradient for Unpowered Flight of Fixed-Wing Aircraft
## 4 Rayleigh Cycle Method
### Energy Perspective  
Here, \(E\) denotes the mechanical energy, including the gravitational potential energy and kinetic energy, and the _O-xyz_ coordinate system is as shown in Figure 2. According to the kinetic energy theorem, the change in the mechanical energy is equal to the work done against the drag and inertial force. Because the lift is perpendicular to the movement, it does no work. The following expression can thus be derived:  
\[dE=-Ddl+mG_{w}v_{z}dx \tag{33}\]  
Here, \(dl\) is defined as  
\[dl=vdt \tag{34}\]  
This implies that  
\[\begin{array}{ll}dE&=-P_{D}v^{3}dt+mG_{w}v_{z}dx\\ &=m(-P_{Dm}v^{3}+G_{w}v_{z}\frac{dx}{dt})dt\\ &=m(-P_{Dm}v^{3}+G_{w}v_{z}v_{x})dt\end{array} \tag{35}\]  
Hence, the normalized rate of the mechanical energy acceleration is equal to  
\[\frac{d}{dt}(E/m)=-P_{Dm}v^{3}+G_{w}v_{z}v_{x} \tag{36}\]  
It should be noted that the temporal variation of the mechanical energy is independent of the control input \(\varphi\). The coordinates can thus be transformed as follows:  
\[\begin{array}{ll}v_{x}=r\cos\theta\cos\psi\\ v_{y}=r\cos\theta\sin\psi\\ v_{z}=r\sin\theta\end{array} \tag{37}\]  
Figure 4: Illustration of a typical Rayleigh cycle.  
where \(r\) is the radial velocity, \(\theta\) is the pitch angle, and \(\psi\) is the heading angle. Equation (36) can therefore be rewritten as follows:  
\[\frac{d}{dt}(E/m)=-P_{Dm}r^{3}+G_{w}r^{2}\sin\theta\cos\theta\cos\psi \tag{38}\]  
Figure 5 is a sectional drawing of a contour plot of the normalized rate of the mechanical energy acceleration \(d(E/m)/dt\) for \(P_{Dm}=0.002\) and \(G_{w}=0.4\). It clearly shows that Equation (38) is symmetrical about the \(vxOv_{z}\) plane. The equation of the surface on which the energy is invariant (i.e., \(d(E/m)/dt=0\)) is as follows:  
\[r=\frac{G_{w}}{P_{Dm}}\sin\theta\cos\theta\cos\psi \tag{39}\]  
This corresponds to the green surface in Figure 5. It exists where \(v_{xxz}>0\) and comprises two leaves joined at the origin, having a spindle-like shape. Energy is only harvested inside the spindle, which is thus called an energy-harvest spindle. In practice, the Rayleigh cycle is a loop across the two leaves of the spindle. The upwind climb (\(v_{x}>0\), \(v_{z}>0\)) and downwind dive (\(v_{x}<0\), \(v_{z}<0\)) obviously occur in the area where \(v_{xxz}>0\). Energy would obviously be harvested if the aircraft was positioned within the two leaves.  
There is a maximum rate of the mechanical energy acceleration in the spindle. In addition, Equation (38) can be rewritten as  
\[\frac{d}{dt}(E/m)=-P_{Dm}r^{3}+\frac{1}{2}G_{w}r^{2}\sin 2\theta\cos\psi \tag{40}\]  
To maximize Equation (40), the term sin2\(\theta\)cos\(\psi\) should be 1, because 1 is the maximum of sin2\(\theta\)cos\(\psi\) when \(\theta=\pi/4\), \(\psi=0\) or \(\theta=-\pi/4\), \(\psi=\pi\). This implies that the maximum point is on the  
Figure 5: Sectional drawing of the contour plot of \(d(E/m)/dt\).  
straight line defined by \(v_{x}=v_{z}\) and \(v_{y}=0\). The maximum point is obtained by setting the derivative of Equation (40) to zero as  
\[\frac{d}{dr}\left(\frac{d}{dt}(E/m)\right)=\frac{d}{dr}\left(-P_{Dm}r^{3}+\frac{ 1}{2}G_{w}r^{2}\right)=-3P_{Dm}r^{2}+G_{w}r=0\]  
from which the following is obtained:  
\[r=\frac{G_{w}}{3P_{Dm}} \tag{41}\]  
Equation (41) can be expressed in the Cartesian coordinate frame as  
\[v_{x}=v_{z}=\pm\frac{\sqrt{2}G_{w}}{6P_{Dm}},v_{y}=0 \tag{42}\]  
The maximum power is thus given by  
\[\left(\frac{d}{dt}E/m\right)_{\max}=\frac{{G_{w}}^{3}}{54{P_{Dm}}^{2}} \tag{43}\]  
This leads to the conclusion that a larger gradient, smaller drag coefficient, lower air density, and larger wing loading enhance energy harvesting for unpowered flight. This agrees with the results of the proposed criterion presented below in Section 5.1.