# Numerical Simulation of Flapping Airfoil Aerodynamic Characteristics
## 3 Results and Analysis
### Effects of Motion Parameters on the Aerodynamic Characteristics of Flapping Airfoils
#### 3.1.2 Impact of Flapping Frequency on the Aerodynamic Characteristics of Flapping Airfoils  
The flapping airfoil is an elliptical shape with a chord length of h and a maximum thickness of 0.12 h. The function of the flapping process is displayed in Formula (1), where \(\alpha_{0}=45^{\circ}\), \(h_{a}=\) h, \(\alpha_{a}=90^{\circ}\), \(\phi=90^{\circ}\), and f are taken as four operating conditions: 25 Hz, 50 Hz, 100 Hz, and 150 Hz. From the lift coefficient curve of flapping airfoils at different frequencies in Figure 8 and the average lift coefficient line graph at various frequency periods in Figure 9, it can be seen that during a period of flapping airfoil motion, the lift coefficient exhibits four peaks and four valleys. The variation law of the lift coefficient curve is described in Section 3.1.1, so it will not be repeated here. Additionally, the lift coefficient curves of flapping airfoils at different frequencies are mostly located directly above the \(x\)-axis, indicating that lift is generated most of the time. This is a common characteristic of the lift coefficient curves of flapping airfoils at various frequencies. However, negative lift is also generated when the direction of flapping airfoil travel changes, and it is increased with increasing frequency. Comparing the lift coefficient curves of flapping airfoils with distinct frequencies in the figure, it can be concluded that the lift coefficient curve with a flapping frequency of 150 Hz is at the top, followed by the lift coefficient curve with a flapping rate of 100 Hz. Below 100 Hz is the lift coefficient curve with a flapping frequency of 50 Hz, and the lift coefficient curve corresponding to a flapping frequency of 25 Hz is at the bottom. With the growth of frequency, the lift coefficient of the flapping airfoil and the average lift coefficient of the flapping airfoil are also raised. As indicated in Figure 10, the resistance coefficient curves of flapping airfoils at different frequencies are as follows: (1) the peak and valley values of the drag coefficient curve correspond to each other, as well as the peak and valley values of the lift coefficient curve. The absolute values of the lift coefficient and drag coefficient are positively correlated, meaning they increase or decrease simultaneously. This correlation arises because lift and drag are the components of the airfoil's combined force in the y and x directions, respectively, and changes in each component are consistent with changes in the combined force. (2) Additionally, the thrust direction in this article is positive to the right. When  
Figure 8: Lift coefficient curve at different frequencies. Figure 9: Average lift coefficient for different frequency cycles. flapping to the left, the force to the right is formed most of the time, as can be seen from the figure. That is, the flapping direction is to the left, and the aircraft flies to the right. During a short period towards the end of the journey, the rotation changes the pressure state on the airfoil surface. As a result, left drag is generated. This means that the flapping airfoil will experience resistance opposite to the flight direction before changing the flapping direction. When flapping to the right, it is similar. So, during the complete flapping cycle of a flapping airfoil, both forward thrust and forward resistance will be originated. (3) Comparing the resistance coefficient curves at various frequencies, it can be concluded that the peak and valley values corresponding to a rate of 150 Hz are larger than those of 100 Hz, and the peak and valley values of 100 Hz are larger than those of 50 Hz. Additionally, the peak and valley values of 25 Hz are the smallest. From the cycle-averaged line graphs of drag coefficients of fluttering airfoils with different frequencies in Figure 11, it can be concluded that the cycle-averaged thrust coefficients are increasing with the increase in frequency. When the flutter rate is 25 Hz, 50 Hz, 100 Hz, and 150 Hz, their cycle-averaged thrust coefficients are all positive.