## 3 Computational-Fluid-Dynamics Analysis
### Simulation Results of Non-Planar Quadrotor  
Figure 8 shows the pressure and the velocity contours of the planar quadrotor on the vertical plane with rotor spacing of 1.05 \(d\). It can be seen that with small spacing, the inter-rotor interference is obvious, and the high pressure areas are overlapped and reduced. Obviously, the wakes between the rotors are attracted to each other, which reduce the flow velocity and cause the pressure difference between the upper and lower surfaces on the adjacent rotors. These results are consistent with the analysis of the twin-rotor interference wake [10]. Figure 9 shows the velocity distribution on the vertical plane of non-planar quadrotors. It can be seen that the rotor is affected by the left-facing rotor, and the tilt angle causes a large change in the flow field below the rotor. The flow field distributions of 10 and 50 deg at l/d = 1.1 are different from each other. Obviously, the inflow is stronger at 50 deg which is mainly affected by the left-facing rotor. Therefore, the high-speed airflow distribution area above the adjacent blades tip is relatively large, which not only reduces the pressure in the negative pressure region but also increases the inflow, thereby increasing the thrust. The outflow of the back-to-back rotors considerably extends beyond the range of this rotor and leaves the blade in an upward, then formed the unsteady flow field. According to the momentum blade element theory under the generalized motion of the rotor, the wake moves in the opposite direction with respect to the airflow, this may lead the incensement of thrust. The thrust is increased again because \(\chi_{mn}\) is less than 0, and finally the actual induced speed is increased. Moreover, the flow field distributions of 10 and 50 deg with \(l\) = 2 \(d\) show the same trend, but the interference is much weaker than that with \(l\) = 1.1 \(d\). However, a large spacing may still increase the rotor thrust. In order to further study the overlapping flow field distribution caused by the rotor inclination angle, Figure 10 shows the different velocity vector distribution of the traditional planar rotors and non-planar rotors. Figure 8: The pressure and velocity contours of the planar quadrotor with rotor spacing of 1.05 \(d\). As showed in the figure, the downwash of the back-to-back rotors are flushed together and the inflow is limited, while the outflow of both rotor pairs is increased. Since the two vortices are diffused and reformed under the blade, the interaction of the lower blade surface is much more intense with the airflow. Figure 11 illustrates 816 s\({}^{-2}\) small vortex distribution defined by the Q criterion. It can be seen from the figure that the vortex at \(l=1.2\ d\) are clearly attracted to each other and deformed near the fuselage. There is no significant change at \(l=1.6\ d\). Hence, the interaction of the low vortex system decreases with an increasing spacing. Figure 12 shows the tip vortex distribution at 30-1.1, 30-1.2 and 50-1.2. The vortex of both the 30-1.1 and 30-1.2 is about 73,000 s\({}^{-2}\), while the vorticity of 50-1.2 is 13,200 s\({}^{-2}\). It indicates that the spacing has a minor effect on the strength of the tip vortex of the blade surface. However, it is more obvious changed with the tilt angle. Thus, larger tilt angle may result in higher power for non-planar quadrotors. Figure 13 shows the pressure and velocity distribution of the upper surface of the rotor 4 (see Figure 1) with different configurations. Figure 10: Velocity vector distribution of planar rotors and non-planar rotors: (**a**) traditional planar rotors; (**b**) back-to-back non-planar rotors; (**c**) face-to-face non-planar rotors. Figure 9: The velocity distribution contours of typical non-planar configuration: (**a**) 10-1.1; (**b**) 50-1.1; (**c**) 10-2; (**d**) 50-2. Figure 11: Vorticity distribution of 816 s\({}^{-2}\) defined by the Q criterion: (**a**) 50-1.2 (**b**) 50-1.6.