## 3 Numerical Validation
### FSI with Free Motion
#### 3.2.2 Flutter of the AGARG 445.6 Wing
The displacements in the x- and y-direction can be ignored compared with those in the z-direction, though all of them are in divergence. At the same time, the distinction of z-direction amplitudes between the leading edge and the trailing edge reflects that the torsional motion exists. Judging from the dynamic motion of z-direction displacement, its frequency is 14.5 Hz, computed by the fast Fourier transform, which approaches the first-order bending. In other words, current flutter, dominated by bending deformation, is caused by the first-order bending mode. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Nodes** & **Experiment** & **Numerical** & **Wing Entity** & **Shell** & **Plate with Thickness** \\ \hline
1 & 9.6 & 9.5992 & 9.6224 & 9.5733 & 9.6 \\
2 & 38.1 & 38.1650 & 40.925 & 35.736 & 35.778 \\
3 & 50.7 & 48.3482 & 49.863 & 58.446 & 58.515 \\
4 & 98.5 & 91.5448 & 98.796 & 97.663 & 97.724 \\
5 & / & 118.1132 & 124.37 & 159.65 & 159.42 In order to depict the flutter boundary along a Mach number, it is necessary to run several simulations for the specified Mach number until the neutral response is achieved. The practice of identifying critical onset flutter is similar to the studies performed by [PERSON] where a detailed flutter analysis is conducted with various wind velocities [8; 9]. Here, flutter velocity \(V_{f}^{*}\) varies through the fixed value of density and manual adjustment of pressure. After repeating the aforementioned process for different Mach numbers, the flutter boundary is obtained, as shown in Figure 25. It compares the present result with the work of others [62; 64; 65; 66], including experimental and numerical data. It is found that a good agreement with [PERSON] experiment is obtained. In the subsonic flow, it is considered that the classical bending-torsion coupling flutter takes place due to the inertia coupling between the bending and torsion of the elastic wing. The additional aerodynamic forces yielding from flexural and torsional deformation will excite the wing. The wing becomes an energy conversion switch, which converts the energy of the incoming flow into energy with complex vibration properties, resulting in the occurrence of wing flutter. In the initial stage shown in Figure 26, there exists a slight  
Figure 24: Displacement in x-, y-, z-directions at the probe of (**a**) the leading edge and (**b**) the trailing edge for Ma = 0.96 and \(V_{f}^{*}=0.292\). Figure 25: Flutter speed index \(V_{f}^{*}\) as a function of a free-stream Mach number for AGARD 445.6 wing. phase difference in the z-direction displacement between the leading edge and the trailing edge. It indicates that the torsional deformation invites the flutter. Then, the phases are synchronized by aerodynamic instability in a hurry. The mode of flutter is transited into bending-torsion coupling vibration. The transonic dip occurs in the region of Ma = 0.9-1.1. For a Mach number \(Ma=0.96\), the flutter velocity \(V_{f}^{*}\) is equal to 0.283. In fact, with the Mach number falling into the transonic region, unstable shock waves occur and move back and forth on the surface of the wing, further inducing a changeable aerodynamic force. Therefore, it is inferred that the complex flow contributes to the dip of the flutter boundary as depicted in Figure 25. Meanwhile, for supersonic situations, the predicted flutter velocity is prone to the experiment results obtained by [PERSON]. It is clarified that the obvious discrepancy from the numerical results obtained by others is due to the viscous effect.