# 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  
The wing tip trajectories of birds and insects flying in various shapes such as an 8-shaped or elliptical shape have been researched. To investigate the impact of different wing tip trajectories on the aerodynamic characteristics of flapping airfoils, four various trajectories, namely the horizontal straight trajectory, the 45\({}^{\circ}\) inclined straight trajectory, the elliptical trajectory [32], and the \"\(\infty\)\" trajectory, were investigated in this section. The finite volume method was employed to numerically solve the flow field. The flapping airfoil is elliptical, with a chord length of h and a maximum thickness of 0.12 h, where the parameters in the flutter motion function are as follows: \(\alpha_{0}=45^{\circ}\), \(h_{a}=\) h, \(\alpha_{a}=90^{\circ}\), \(\phi=90^{\circ}\), and \(f=100\) Hz. The four types of flapping airfoil motion trajectories are demonstrated in Figure 14.  
The lift coefficient curves of different flapping airfoil trajectories obtained through numerical calculation are shown in Figure 15, and the drag coefficient curves of various flapping airfoil trajectories are demonstrated in Figure 16. Elliptic trajectories, \"\(\infty\)\" shaped trajectories, and horizontal straight line trajectories are all symmetrical about the plumb axis passing through the geometric center of the trajectory. Compared to Figures 15 and 16, their lift coefficient curves and drag coefficient curves are similar, with the peak times of the curves being basically the same. The trend of the curves is also consistent. Additionally, the lift coefficient curves of the three trajectories are generally above the \(x\)-axis, indicating lift generated by each flapping stroke of the flapping airfoil. The leading edge vortex delayed stall lift mechanism and the Trailing vortex capture lift mechanism both exist in three \begin{table}
\begin{tabular}{c c} \hline
**Frequency** & **Lifting Efficiency** \\ \hline
25 Hz & 53.67\% \\
50 Hz & 45.87\% \\
100 Hz & 34.41\% \\
150 Hz & 26.53\% \\ \hline \end{tabular}
\end{table} 
Table 1: Lifting efficiency of flapping airfoils with different flutter frequencies. trajectories. Under the action of the lift mechanism, the lift coefficient curve is enhanced, the lift mechanism ends, and the curve is reduced. When the flapping airfoil in the three types of flapping airfoil trajectories flaps to the left, it mostly receives a right thrust, and during the period from the end of the stroke to the beginning of the next stroke, it experiences resistance to the left. The force situation of flapping airfoil flapping to the right is similar, but the force direction is opposite. The 45\({}^{\circ}\) inclined trajectory is significantly different from other trajectories, with no plumb axis symmetry, but inclined axis symmetry. In the first three trajectories, higher lift and resistance are originated when the flapping airfoil is moved to the left or right. In the 45  
Figure 16: Drag coefficient curve of flapping airfoils with various trajectories. Figure 14: Schematic diagram of motion trajectory. (**a**) Horizontal linear motion of the center of mass; (**b**) elliptical motion of center of mass; (**c**) centroid “\(\infty\)” shaped motion; (**d**) centroid tilt 45\({}^{\circ}\) linear motion. Figure 15: Lift coefficient curve of flapping airfoils with different trajectories. trajectory, higher lift and less resistance are generated when the flapping airfoil is moved downwards to the left, while lower lift and greater resistance are experienced when the airfoil is moved upwards to the right. By combining the cloud map of flapping airfoil flow field and the lift coefficient and drag coefficient curve of flapping airfoil, it can be found that all lift mechanisms exist during flapping at a 45\({}^{\circ}\) trajectory.