## 2 Structure Design and Numerical Method
### CFD Model and Validation  
In order to calculate the flow field of the flapping wing rotor and make a comparison between the lift obtained by CFD, a CFD model was built. The CFD solver, used in this paper, is fluent 19.0 and the mesh is plotted by Hypermesh software, as well as the meshing mode of fluent as given in Figure 7. The whole fluid field is defined as a sphere and can be mainly divided into two parts: Inner fluid and outer fluid. There is also a region around the wing surface (\"wall\" boundary condition) in which structured mesh is employed. The outer boundary of this region is selected to be a 1 time chord from the wing surface. Then, the outer bound radius of inner fluid (dark blue domain) is about 5 times the wing chord length while the outer bound radius of outer fluid (light blue) accounts for 10 times the wing chord length, in order to meet the requirements of far field boundary condition. A user-defined function is coded in C language to make the mesh in dark blue rotate about the center of sphere. These rotation motions can be decomposed into three mutually orthogonal components to simulate the flapping, twisting and rotary motion of our FWR model. The overset boundary condition is applied to the interface between inner fluid and outer fluid. Since the Reynolds number of this single wing is 7000-23,000 when its flapping frequency ranges from 1 Hz to 3 Hz, the \(k\)-\(\omega\) shear stress transfer (SST) model is utilized in this paper as that used in [28].  
In order to validate the CFD model built-up in this paper, a case comparison is made between the lift and moment results in our CFD simulation and the results obtained by [PERSON] et al. [23]. The variation function of flapping angle (\(\varphi\)), twisting angle (\(\theta\)) and rotary angle (\(\psi\)) as shown in Figure 8 are set the same as those in the literature. According to [PERSON], the shape of the wing is simplified as a flat plane with 3% relative thickness and the aspect ratio \(\lambda=5\). The reference length (\(L_{ref}\)) and reference  
Figure 6: Trace line of the flapping wing rotor.  
Figure 7: Mesh used in the CFD calculation.  
velocity (\(U_{ref}\)) are defined as wing chord \(L_{ref}=c\) and the average flapping velocity at the wing tip \(U_{ref}=4c\lambda f\Delta\varphi\). Based on the above definition, the lift and rotational moment coefficients \(C_{L}\) and \(C_{m}\) are calculated as follows,  
\[C_{L}=\frac{2L}{\rho\big{(}U_{ref}\big{)}^{2}S} \tag{4}\]  
\[C_{m}=\frac{2m_{rotary}}{\rho\big{(}U_{ref}\big{)}^{2}Sc} \tag{5}\]  
where \(L\) (lift) and \(m_{rotary}\) (rotational moment) are the components of the aerodynamic force and moment in rotary direction. Besides, \(\rho\) denotes the air density. After calculation, the simulation results are shown in Figure 9.  
According to Figure 9, the trend of lift coefficients' and rotational moment coefficients' curves by fluent software in this paper is the same as those by [PERSON] et al. (2019). The average lift coefficients and rotational moment coefficients are \(\overline{C_{L}}=0.92\) and \(\overline{C_{m}}=0.13\) in [PERSON]'s results. Meanwhile, in this paper, the \(\overline{C_{L}}\) equals to \(0.922\) and \(\overline{C_{m}}\) equals to \(0.123\) which is quite similar to those given by [PERSON] et al. (2019). Overall, the comparisons above show that the CFD model, in this paper, is suitable for the aerodynamic study of the FWR.