# Design and Verification of Large-Scaled Flapping Wings for High Altitude Environment
## 2 Materials and Methods
### Numerical Simulation  
#### 2.2.1 Flow Solver  
The transient simulations are investigated using the CFD software Fluent (Fluent 2020R2) by solving the three-dimensional Navier-Stokes equations. The motions of the flapping wings are simulated based on an overset mesh technique. The overset mesh, consisting of a background mesh and a component mesh, is considered to be suitable for large-scale motion simulations. During the dynamic movement, the background mesh remains stationary, while the entire component mesh moves as a rigid body. The motion of the flapping wings is adjusted by modifying a user-defined function in the form of a coordinate transformation. While the wing is moving, the mesh around the wing in the deformable zone is updated using a dynamic mesh technique. The Reynolds number is calculated from the forward flight speed. The Reynolds number (\(Re\)) is:  
\[Re=\rho vd/\mu=1.27e^{5}, \tag{7}\]  
where \(\rho\) and \(\mu\) are the air density and dynamic viscosity corresponding to different altitudes, respectively, which are shown in Table 3. \(v\) is the forward flight speed equal to \(7\) m/s; \(d\) is the average chord equal to about \(0.266\) m. The Strouhal number (\(St\)) is a related dimensionless variable, defined as follows:  
\[St=fA/v, \tag{8}\]  
where \(A\) is the wingtip amplitude, defined as the vertical linear excursion of the wing tip above and below its position with wings level. \(f\) and \(v\) are the flapping frequency and the forward flight speed. \(St\) gives an indication of the angle with which the wing tip moves up and down, relative to the flight path, and this can be seen as an advantage when describing wake geometries.  
The model of the wing used for the calculation is the same airfoil and the planform of increasing the internal area of the wing as the real flapping wing. To simplify the model and improve numerical convergence, the wing was assumed to be a rigid body whose motion only consists of the rotation around the x-axis. The wing trajectory (i.e., flapping amplitude and frequency), which consists of a rotation, was implemented in the code using user-defined functions (UDFs).  
Wing rotation \(\Psi\) is defined as follows:  
\[\Psi(t)=\Psi_{0}\times\sin(2\pi ft), \tag{9}\]  
where \(\Psi_{0}\) indicates that the amplitude is equal to \(22.5^{\circ}\),\(f\) is the flapping frequency of 4 Hz, and the angle of attack of the wing is \(15^{\circ}\) (Figure 4).  
A k \(-\omega\) SST model is adopted to simulate the high-Reynolds-number flow situation. The boundary condition of the outlet is a pressure outlet. The component zone uses an overset boundary. A no-slip wall condition is applied to the wing. Momentum, turbulent kinematic energy, and the specific dissipation rate are discretized with a second-order upwind scheme. Second-order accuracy is applied to calculate the pressure. A coupled algorithm is employed for the pressure-velocity coupling. The mesh strategy used in this research is shown in Figure 5.  
Figure 4: Complete flapping cycle of rotation around the x-axis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Altitude (km)** & **Density (kg/m\({}^{3}\))** & **Dynamic Viscosity (N\(\times\)s/m\({}^{2}\))** & **Temperature (K)** & **Reynolds Number (Re)** \\ \hline
0 & 1.225 & \(1.7894\times 10^{-5}\) & 288.15 & \(1.27\times 10^{5}\) \\
1 & 1.112 & \(1.7579\times 10^{-5}\) & 281.65 & \(1.18\times 10^{5}\) \\
2 & 1.006 & \(1.7260\times 10^{-5}\) & 275.15 & \(1.09\times 10^{5}\) \\
3 & 0.909 & \(1.6937\times 10^{-5}\) & 268.65 & \(9.99\times 10^{4}\) \\
4 & 0.819 & \(1.6611\times 10^{-5}\) & 262.15 & \(9.18\times 10^{4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Air density, dynamic viscosity, temperature, and Reynolds number at different altitudes.