# Airborne Quantum Key Distribution Performance Analysis under Supersonic Boundary Layer
## 3 Evaluation Result and Discussion  
Here, the typical airfoil that named \"NACA0015\" is chosen for the performance analysis of our specified air-to-ground QKD system. The specific parameter settings of the aircraft, source, ground station and protocols are shown in Table 1. The boundary layer will be generated around the airfoil and its density field distribution can be simulated by the computational fluid dynamics software (Ansys Fluent, Canonsburg, PA, USA) as shown in Figure 3a, with the parameters shown in Table 1. Usually, the boundary layer thickness based on the velocity boundary layer concept that the region in which flow adjusts from zero velocity at the wall to a maximum in the main stream of the flow is termed the boundary layer. According to the simulation results, the airfoil velocity boundary layer thickness is about 10 mm. Here, in order to ensure that the path lengths in the boundary layer are approximately equal and the results are more accurate under different incident angles, we expend the concept of the boundary layer as the field that caused by the motion of aircraft. So, the generalized boundary layer thickness is 400 mm. Literature [34] used the nano-particle-based planar laser scattering technique to measure the density distribution of the supersonic (Ma = 3.0) turbulent boundary layers, as shown in Figure 3b. The velocity boundary layer thickness is about 10 mm [33], and the generalized boundary layer thickness is about 13 mm. Although this kind of boundary layer is not the actual aircraft boundary layer, the turbulence structure is similar to the actual situation. Due to the limited amount of data, we only analyze the results at a certain moment, without considering an average over enough runs. It is also meaningful to research the influence of this kind of boundary layer on QKD performance. When the photon trajectories of different incident angles pass through the boundary layer and reached the ground station, the evaluated deflection angle and the drifted offset of the beam are shown in Figure 2. The deflection angle can be calculated as \(|\theta_{1}-\theta_{n}|\) which is absolute value of the angle difference between the incident angle \(\theta_{1}\) and the last refracted angle \(\theta_{n}\) when the photon propagated via the boundary layer. In both cases, the deflection angle and correspondingly drifted offset of the beam were varied obviously in Figures 4 and 5. Because of the complex small-scale turbulence structures in the supersonic boundary layer, the spatial distribution of the density field is anisotropic and random, and the results vary randomly as shown in Figure 5. When the azimuth angle is near zero, the incident photon will pass through the region with a large refractive index gradient, as shown in Figure 3b, which could influence the deflection angle obviously. In a practical airborne quantum communication system, owing to the ATP technique, the center axis of the beam can always be aligned with the center of the receiver telescope. Therefore, the  
Figure 3: The sectional drawing of refractive index field of (**a**) the airfoil boundary layer and (**b**) the supersonic boundary layer. The coordinate axis represents the size of the boundary layer, in millimeters. The incident points are respectively (x = 340 mm, y = 0) and (x = 9 mm, y = 0). And the color represents the refractive index distribution. deflection angle and the drifted offset were not analyzed emphatically in this paper, which could been ignored. Afterward, the photons distribution probability that eventually reaches the receiver telescope at azimuth angle is 10\({}^{\circ}\),was calculated as shown in Figure 6, when the photon via the supersonic BL or not. It shows that the photons distribution probability is redistributed when the scattered photon passes through the supersonic BL. But the photons distribution probability is fundamentally invariant when the photon via the BL when flight speed is 0.7 Ma, which is not displayed. Figure 4: (**a**) The deflection angle of transmitted photons.