# Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge
## 4 Numerical Results
### Effects of Different Wavelengths on Aerodynamic Performance of Flapping Wing  
Figures 18 and 19, respectively, show the time history comparison of lift coefficient and thrust coefficient of straight flapping wing and wavy leading edge flapping wing with different wavelengths in one flapping cycle. It can be seen that the lift coefficient of the flapping wing with the wavy leading edge is higher than that of the straight wing in the process of downward flapping (t/T = 0-0.5) and in the last quarter stage of upward flapping (t/T = 0.875-1). In the first three quarters of upward flapping (t/T = 0.5-0.875), the lift coefficient is similar to or slightly lower than that of the straight wing. The thrust coefficient is close to that of the straight wing in the whole process of the downward flapping (t/T = 0-0.5) and the first half of the upward flapping (t/T = 0.5-0.75), and it  
Figure 16: The time history of lift and thrust coefficient (Configuration 11). (**a**) Lift coefficient; (**b**) Trust coefficient. Figure 17: The time history of lift and thrust coefficient (Configuration 3-4). (**a**) Lift coefficient; (**b**) Trust coefficient. decreases obviously in the latter half (t/T = 0.75-1), which has strong regularity. That is, the smaller the wavelength of the wavy leading edge flapping wing, the more the thrust coefficient decreases. Figure 20 compares the time averaged lift and thrust coefficients of the straight wing and wavy leading edge wings. The wavy leading edge inevitably reduces the thrust coefficient while increasing the lift coefficient, and the smaller the wavelength, the larger the time averaged lift coefficient and the smaller the time averaged thrust coefficient. Figures 21 and 22, respectively, show the time-averaged lift coefficient and thrust coefficient during downstroke (t/T = 0-0.5) and upstroke (t/T = 0.5-1). The wavy leading edge wings mainly increases the lift in the downstroke, while the thrust decreases in both the up-and-down stroke of the flapping. Table 5 shows the average growth rate of time averaged lift coefficient relative to straight wing and the average reduction rate of time averaged thrust coefficient for flapping wing with different wavelengths. When the wavelength is \(\overline{W}~{}=~{}0.01\), the average lift coefficient increases the most, which is 41.04% of the straight wing, but at the same time, the average thrust coefficient decreases by 25.97%. Therefore, it is necessary to weigh the lift and thrust of the flapping wing during design, or to increase the thrust by increasing the flapping frequency. Wavy leading edge flapping wing can withstand greater take-off weight, and it has better low-speed flight ability. Figure 21: Comparison of average lift coefficient of downstroke and upstroke. (**a**) Time averaged lift coefficient of downstroke; (**b**) Time averaged lift coefficient of upstroke. Figure 22: Comparison of average thrust coefficient of downstroke and upstroke. (**a**) Time averaged thrust coefficient of downstroke; (**b**) Time averaged trust coefficient of upstroke. Figure 20: Comparison of average lift coefficient and thrust coefficient. (**a**) Time averaged lift coefficient; (**b**) Time averaged trust coefficient. With the purpose of further clarifying the mechanism of wavy leading edge changing the characteristics of flow field and the influence of different wavelengths, four typical moments (t/T = 0, 0.2, 0.5 and 0.8) in a flapping cycle are compared and analyzed by taking the straight wing (\(\overline{W}~{}=~{}0\)) and the wavy leading edge flapping wing with \(\overline{A}~{}=~{}0.1\) as an example. Figure 23 shows the pressure contours of the upper surface of the flapping wing, and the unit of pressure is Pa. In the downward stage (t/T = 0 and 0.2), the pressure on the upper surface of the wavy leading edge flapping wing decreases, while in the upward stage (t/T = 0.5 and 0.8), the pressure on the upper surface increases.