# Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge
## 4 Numerical Results
### Effects of Different Amplitudes on Aerodynamic Performance of Flapping Wing  
Figures 26 and 27, respectively, show the time history comparison of lift coefficient and thrust coefficient of straight flapping wing and wavy leading edge flapping wing with different amplitudes in one flapping cycle. When the wavelength is large, the change of amplitude has little effect on the lift coefficient, and at the same wavelength, the larger the amplitude, the more the thrust coefficient decreases.  
Figure 23: Pressure contours of the upper surface of flapping wing (Pa). (**a**) t/T = 0; (**b**) t/T = 0.2; (**c**) t/T = 0.5; (**d**) t/T = 0.8.  
Figure 24: Pressure coefficient of 80% wingspan section. (**a**) t/T = 1; (**b**) t/T = 0.2; (**c**) t/T = 0.5; (**d**) t/T = 0.8.  
Figure 28 compares the time averaged lift and thrust coefficients of the straight wing and wavy leading edge wings with different amplitudes, and Figures 29 and 30, respectively, show the time-averaged lift coefficient and thrust coefficient during downstroke (t/T = 0-0.5) and upstroke (t/T = 0.5-1). Table 6 shows the average growth rate of time averaged lift coefficient and the average reduction rate of time averaged thrust coefficient of them. As can be seen, at the same wavelength, the larger the amplitude, the higher the average lift coefficient and the more the time-averaged thrust coefficient decreases.  
Figure 28: Comparison of average lift coefficient and thrust coefficient. (**a**) Time averaged lift coefficient; (**b**) Time averaged trust coefficient.  
Taking the straight wing (\(\overline{A}~{}=~{}0\)) and the wavy leading edge flapping wing with different amplitudes at wavelength \(\overline{W}~{}=~{}0.02\) as an example, the four typical moments in a flapping cycle (t/T = 0, 0.2, 0.5, and 0.8) are compared and analyzed. The pressure contours of the upper surface of the flapping wing are shown in Figure 31; the unit of pressure is Pa. It can be seen that the wavy leading edge changes the pressure distribution on the flapping wing. In the downward flapping stage, the larger the amplitude, the smaller the pressure behind the trough, the smaller the pressure on the upper surface of the flapping wing, the larger the pressure difference between the upper and lower surfaces, and the higher the lift.  
\begin{table}
\begin{tabular}{c c c c} \hline \(\overline{A}\) & **0.05** & **0.1** & **0.2** \\ \hline Time averaged lift coefficient \(\overline{C_{L}}\) & 24.34\%\(\uparrow\) & 29.33\%\(\uparrow\) & 36.52\%\(\uparrow\) \\ \hline Time averaged trust coefficient \(\overline{C_{T}}\) & 8.75\%\(\downarrow\) & 15.53\%\(\downarrow\) & 29.03\%\(\downarrow\) \\ \hline \end{tabular}
\end{table}
Table 6: Average change rates of flapping wings at different amplitudes.  
Figure 30: Comparison of average thrust coefficient of downstroke and upstroke. (**a**) Time averaged trust coefficient of downstroke; (**b**) Time averaged trust coefficient of upstroke.  
Figure 29: Comparison of average lift coefficient of downstroke and upstroke. (**a**)Time averaged lift coefficient of downstroke; (**b**) Time averaged lift coefficient of upstroke.  
Figure 32 shows the Mach number contours at 50% spanwise section of the straight wing and the wavy leading edge flapping wing with different amplitudes when t/T = 0, where is the trough position. When the gas flows through the wavy leading edge, the airflow on the upper surface of the wing accelerates, and the airflow on the lower surface decelerates. Moreover, the larger the amplitude, the more obvious the change in velocity, so the greater the lift increase.  
Figure 31: Pressure contours of the upper surface of flapping wing. (**a**) t/T=0; (**b**) t/T = 0.2; (**c**) t/T = 0.5; (**d**) t/T = 0.8.