# Numerical Simulation of Flapping Airfoil Aerodynamic Characteristics
## 3 Results and Analysis
### Effects of Motion Parameters on the Aerodynamic Characteristics of Flapping Airfoils
#### 3.1.3 Impact of Motion Trajectory on Flapping Airfoil Aerodynamic Characteristics
When flapping to the left and down, the larger lift component of the flapping airfoil along the y direction is caused by the airfoil attitude angle, and the smaller drag component along the positive x direction. When flapping the airfoil to the upper right, the lift component along the y-direction is small or negative, which is caused by the attitude angle of the airfoil, but the drag component along the x-negative direction is larger. It can be concluded that the aircraft to flying the left is caused by this motion trajectory. The cycle-averaged lift coefficients and cycle-averaged drag coefficients of different trajectory flapping airfoils are revealed in Table 2. It can be observed that these coefficients vary among the different trajectories. Specifically, the cycle-averaged lift coefficients of the flapping airfoils with the horizontal straight trajectory are the largest, while the cycle-averaged lift coefficients of the trajectory with the 45\({}^{\circ}\) diagonal line are the smallest. The cycle-averaged lift coefficient of the \"\(\infty\)\" shape is reduced by about 3% relative to the straight trajectory, the elliptical trajectory is reduced by about 4.9% in relation to the straight trajectory, and the 45\({}^{\circ}\) trajectory is reduced by about 14.4% compared to the straight trajectory. It can be concluded that the horizontal straight trajectory, elliptical trajectory, and \"\(\infty\)\" shape have very small differences in lift characteristics, while the difference in lift characteristics compared with the 45\({}^{\circ}\) trajectory is more obvious. Based on the research conditions in this paper, it can be hypothesized that the differences in the lift characteristics of the horizontal straight trajectory, elliptical trajectory, and \"\(\infty\)\" trajectory may be related to the existence of upward and downward motions of the elliptical trajectory and \"\(\infty\)\" trajectory or the airfoil attitude angle in the case of the same motion parameters when there is no incoming flow. The differences in the lift characteristics of the three \"\(\infty\)\" shapes may be related to the existence of upward and downward motions in the elliptical trajectories and \"\(\infty\)\" shapes, or to the attitude angle of the airfoil. Speculation 2: Under the same motion amplitude and rate in the absence of incoming flow, the lift characteristics are significantly affected by the different symmetry axes of the motion trajectories. From Table 2, it can be established that there are differences in the average drag coefficients of the flapping airfoil cycle for these four trajectories. The periodic average drag coefficient of a flapping airfoil with a 45\({}^{\circ}\) tilt trajectory is the highest, while the periodic average drag coefficient of an elliptical trajectory is the lowest. The periodic average drag coefficients of the \"\(\infty\)\" shaped trajectory and the 45\({}^{\circ}\) inclined trajectory are both negative, indicating that the left thrust will be generated by the flapping airfoil period, and the flight direction of the aircraft is right. The periodic average drag coefficient of the elliptical trajectory and the horizontal straight trajectory is positive, indicating that the thrust to the right of the period is generated by the flapping airfoil, and the flight direction of the aircraft is to the right. The flow field cloud maps of flapping airfoils with different trajectories are presented in Figures 17 and 18. It can be identified that the leading edge vortex delay stall mecha \begin{table}
\begin{tabular}{c c c c} \hline
**Trajectory Type** & **T-Cycle Average Lift Coefficient** & **Periodic Average Drag Coefficient** & **Lift-Drag Ratio** \\ \hline elliptical trajectory & 0.541513372 & 0.007599804 & — \\ “\(\infty\)” shaped trajectory & 0.552286478 & \(-0.024008642\) & \(-23.00365366\) \\
45\({}^{\circ}\) tilt trajectory & 0.487343095 & \(-0.315957838\) & \(-1.54243078\) \\ horizontal straight trajectory & 0.569575793 & 0.123585176 & 4.608771136 \\ \hline \end{tabular}
\end{table}