# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 4 Extended Quasi-Steady Aerodynamic Model
### Aerodynamic Forces
Recently, [PERSON] and [PERSON] give out 3D lift curve slope based on Extended Lifting Line Theory and Prandtl lifting line theory, respectively [30; 44]. Here, the 3D lift curve slope given by [PERSON] is adopted because it takes care of some correction for unsteady aerodynamic mechanism through edge correction except aspect ratio. According to the Prandtl lifting line theory [30], the 3D wing lift curve slope (\(C_{La}\)) for aspect ratio above 3 can be given by \[C_{La}=\frac{C_{La,2d}}{E_{\text{vari},cc}+kC_{La,2d}/\pi AR_{\text{vari}}}, \tag{17}\]  
where \(C_{La,2d}\) is 2D aerofoil lift curve for a flat plate. The value of 0.09 deg\({}^{-1}\) is adopted for rigid thin wing planform at typical insect wing's \(Re\)[30]. The variable aspect ratio, \(AR_{\text{vari}}\), is equal to \(R_{\text{eff,vari}}/C_{\text{aver,vari}}\) for a single wing. The parameter \(k\), which can be evaluated by induced power factor according to a hovering actuator disc model [30], is termed '\(k\)-factor'. The \(k\)-factor for the normal hovering flight of a fruit fly can take the value of 1.51 in terms of the estimating result of [PERSON] and Crowther through including three contributors of the non-uniform downwash velocity distribution, tip losses and effective flapping disc area [30]. The parameter, \(E_{\text{vari,ec}}\), is edge correction based on lifting line theory considering the 3D effect of variation of wing shape [30]. Accounting for simplification in the integral of wing planform semi-perimeter, we assume that the edge correction (\(E_{\text{vari,ec}}\)) for dynamically scaled wing planform can be estimated as  
\[E_{\text{vari,ec}}=\lambda_{1}R_{\text{ratio}}C_{\text{ratio}}E_{\text{orig,ec}}, \tag{18}\]  
where a factor of shape, \(\lambda_{1}\), is introduced to match 3D lift coefficient of original wing planform to 2D quasi-static lift coefficient fitted from experimental data for dynamically scaled fruit fly wing model [34] (Figure 5). A value of 0.62 is used with root-mean-squared error (RMSE) of 0.04 relative to 2D quasi-static data. The edge correction for the original wing planform, \(E_{\text{orig,ec}}\), is calculated by the quotient of the wing semi-perimeter to its length [33], as given by  
\[E_{\text{orig,ec}}=\frac{C_{\text{semi,perim}}}{R_{\text{eff,vari}}}, \tag{19}\]  
here, \(C_{\text{semi,perim}}=\frac{S_{\text{le,orig}}+S_{\text{tr,orig}}}{2}\) is semi-perimeter of original fruit fly wing planform. The arc length of leading-edge and trailing edge profiles can be expressed as  
\[S_{i,\text{orig}}=\int_{x_{r,\text{orig}}}^{R_{\text{eff,orig}}}\left(1+\left( \frac{dz_{i,\text{orig}}\left(r\right)}{dr}\right)^{2}\right)^{\frac{1}{2}}dr, \tag{20}\]  
where \(i\) (\(i=\text{le, tr}\)) denotes leading-edge and trailing edge profiles. Once the 3D translational lift coefficient is obtained, the drag coefficient can be obtained by using trigonometry:  
\[C_{\text{D}}\left(\alpha\right)=\lambda_{2}C_{\text{L}}\left(\alpha\right) \tan\left(\alpha\right)=\lambda_{2}C_{\text{L}}\left(\alpha\right)\sin^{2} \left(\alpha\right), \tag{21}\]  
Figure 5: Translational circulations lift and drag coefficients. Similarly, a factor of shape, \(\lambda_{2}\), is introduced to match 3D drag coefficient of original wing planform to 2D quasi-static drag coefficient (Figure 5). A value of 0.93 is used with RMSE of 0.11 relative to 2D quasi-static data. As shown in Figure 5, the 3D lift and drag coefficients of (16) and (21) have a good approximation with the fitting formulas based on experimental data with little RMSE after introduced two factors of shape. Thus these formulas of translational lift and drag coefficients constructed by 3D lift curve slope based on Prandtl lifting line theory [30] are also adopted for comparing the possible effect of wing aerodynamic performance arising from variation of wing shape on optimization involving WGP.