# Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge
## 3 Design of Wavy Leading Edge Flapping Wing
### Characteristic Parameters of Wavy Leading Edge  
Based on the above analysis, the aerodynamic performance of the wavy leading edge flapping wing with NACA0014 as the base section in the unsteady flapping state is studied. The structure of the wavy leading edge wing is shown in Figure 13. The sinusoidal wavy is defined by the two parameters of amplitude \(A\) and wavelength \(W\) and varies along the wing wingspan according to Equation (5). In addition, \(x\) is the spanwise length from any section to the wing root. The average chord length and projection area of the wavy leading edge flapping wing are the same as those of the reference flapping wing.  
\[Y(x)\ =\ \frac{A}{2}\sin(\frac{2\pi}{W}x) \tag{5}\]  
Figure 11: Flipper’s front structure of humpback whale [41].  
Figure 12: Upper surface limit streamline of wavy leading edge wing.  
Figure 13: Wavy leading edge wing.  
A sinusoidal wave is generated by five sections along the spanwise direction, as shown in Figure 14a. The airfoil data for each section are defined by Equation (6), and the airfoil at the middle is the same as the airfoil of the baseline wing.  
\[\left\{\begin{array}{rcl}x_{uaxy}&=&\left\{\begin{array}{rcl}\frac{x_{ baseline}}{x_{\max}}\left[x_{\max}+\left(Y(x)-c\right)\right]-\left[Y(x)-c\right] \right.x_{uaxy}<x_{\max}\\ x_{baseline}\left.x_{uavy}\geq x_{\max}\end{array}\right.\\ z_{uaxy}&=&z_{baseline}\end{array}\right. \tag{6}\]  
The \(x_{baseline}\) and \(x_{uavy}\) represent the abscissas of the baseline wing and the wavy leading edge wing, respectively, and \(x_{\max}\) represents the abscissa at the maximum thickness of the baseline wing. The \(z_{baseline}\) and \(z_{uavy}\) represent the ordinates of the baseline wing and the wavy leading edge wing, respectively. When generating spanwise section airfoil data, only the geometry from the leading edge to the maximum thickness is changed, while the geometry from the maximum thickness to the trailing edge is kept unchanged. The baseline airfoil is stretched or compressed according to the chord length at different spanwise positions, as shown in Figure 14b.  
For purpose of further clarifying the influence of design parameters on the aerodynamic performance and flow control effect of the bionic wavy leading edge flapping wing, dimensionless processing was carried out for amplitude and wavelength, as shown in Equation (7). Twelve different flapping wing configurations composed of three amplitudes and four wavelengths are selected for calculation and comparison. The specific parameters of the configurations are shown in Table 4, and the values of these parameters are within the range of protuberances found on the humpback whale's pectoral fin in nature [42]. Figure 15 shows the schematic diagrams of all twelve configurations. The comparison of different design parameters of the wavy leading edge wing configurations can be seen through Figure 15.  
\[\left\{\begin{array}{rcl}\overline{A}&=&\frac{A}{\bar{W}}\\ \overline{W}&=&\frac{\bar{W}}{L}\end{array}\right. \tag{7}\]  
Figure 14: Construction method of wavy leading edge wing. (**a**) Wing spanwise section of wavy leading edge wing; (**b**) Airfoil of wavy leading edge wing.