# Unsteady Aerodynamic Design of a Flapping Wing Combined with a Bionic Wavy Leading Edge
## 2 Aerodynamic Characteristics Analysis of Baseline Flapping Wing
### Grid and Time Step Independence Verification  
Overset grid method is used in this paper, where the component grid is fluid domain grid including flapping wing. The boundary layer is set on the surface of the flapping wing to ensure y+ \(<\) 1. The background grid is a cylindrical area with a radius of 25c and a height of 10c, and it is locally refined to ensure that the mesh size at the junction of the background grid and the component grid is close. The component and background grid are divided by polyhedral unstructured grids. The computational domain grids and boundary conditions are shown in Figure 5. And Figure 6 shows the y+ contours of baseline case.  
Figure 4: Comparison of average thrust coefficient.  
Figure 3: Comparison of average lift coefficient.  
In order to ensure the accuracy of numerical calculation and save computational cost, the independence of computational mesh and time step is verified by the straight flapping wing model. The fifth flapping cycle is selected for comparison, and the calculation results are stable. The lift and trust coefficient time history of the fifth and sixth cycles is compared in Figure 7. Firstly, three groups of overset grids with different number of grids are selected as shown in Table 3 to calculate. The calculation results are shown in Figure 8. As can be seen, the grid solution results of Case2 and Case3 tend to coincide, which can be considered that the grid has reached the grid independence requirements. Therefore, the grid division scheme of Case2 is selected for subsequent calculation. Secondly, to verify the independence of the time step, three different time steps for calculation are taken; the unit is s, and the result is shown in Figure 9. When the time step is 0.0025 s and 0.001 s, the relative error between the time average lift coefficient, and the time average trust coefficient is less than 2%, which meets the requirement of independence. Therefore, in the subsequent calculation, the time step is set to 0.0025 s.  
Figure 5: Grid and boundary conditions. (**a**) Background grid; (**b**) Component grid; (**c**) Boundary layer and surface mesh.  
Figure 6: The y+ contours of baseline case.  
Figure 8: Grid independence verification. (**a**) Lift coefficient; (**b**) Trust coefficient.  
Figure 7: The comparison of time history of lift and thrust coefficient. (**a**) Lift coefficient; (**b**) Trust coefficient.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \begin{tabular}{c} **Number of** \\ **Background** \\ **Grid (_uv_)** \\ \end{tabular} & \
\begin{tabular}{c} **Number of** \\ **Component** \\ **Grids (_uv_)** \\ \end{tabular} &
\begin{tabular}{c} **Total Number** \\ **of Grids (_uv_)** \\ \end{tabular} & \multirow{2}{*}{**Total Faces (_uv_)**} & \multirow{2}{*}{**Total Nodes**} \\  & & & & & \\ \hline Case1 & 70 & 562 & 632 & 3612 & 2565 \\ \hline Case2 & 135 & 706 & 841 & 4554 & 3150 \\ \hline Case3 & 289 & 836 & 1125 & 6908 & 4816 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Grid independence verification.  
Figure 9: Time step independence verification. (**a**) Lift coefficient; (**b**) Trust coefficient.