## 3 Numerical Validation
### FSI with Free Motion
#### 3.2.2 Flutter of the AGARG 445.6 Wing  
AGARD 445.6 is a classical wing model for flutter analysis. [PERSON] et al. [62] has conducted mode analysis on AGARD 445.6 for two kinds of models. This section adapts the AGARD 445.6 model as the object for studying. Figure 20 depicts the geometry of the AGARD 445.6 wing and its airfoil. This wing takes NACA 65A004 as its airfoil and extends it with no twist or curvature along its quarter-chord sweep angle (45\({}^{\circ}\)). The corresponding tip and root chords are 0.559 m and 0.368 m, respectively, while the semi-span is 0.762 m. The material of the wing used in [PERSON] experiment is laminated mahogany, which has distinguished physical properties in different directions. Based on the experimental data in [PERSON] work and previous works by [PERSON] [63], the material properties of the wing model, such as elasticity modulus \(E\), Poisson's ratio \(\
u\), shear modulus \(G\) and density \(\rho\), are summarized and listed in Table 2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Material Properties** & **Value** \\ \hline E\({}_{11}\) & 3.2456144 GPa \\ E\({}_{22}\)E\({}_{33}\) & 0.416 GPa \\ \(v_{12}v_{23}v_{23}\) & 0.31 \\ \(G_{12}G_{23}G_{12}\) & 0.411942 GPa \\ \(\rho\) & 381.98 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The material properties in each direction for AGARD 445.6 wing. Figure 19: Pressure coefficient contours at four snapshots in a circle of LCO when Ma = 0.85. The red solid line represents the initial contour of the airfoil. Figure 20: Geometry of AGARD 445.6 and its NACA 65A004 airfoil. When it comes to solving the structural dynamic response, it is first necessary to extract its eigenvalues and eigenvectors as described in Equation (38). With the aid of finite element structural analysis software Patran/Nastran, mode analysis is conducted on models of the wing entity. Several simplifications of the structural model, shell model, and plate with constant thickness are analyzed as well. Their meshes are exhibited in Figure 21. Table 3 compares the first six mode frequencies with numerical and experimental results obtained by [PERSON]. It suggests that contrasted with the experiment, a good agreement for the wing entity is obtained. Due to the same principle assumption involved in the analysis of shell and thin plate, they perform similarly but have differences in the 3 rd, 5 th, and 6 th modes in comparison  
Then, tight coupling connects the fluid and structural solver. Within each physical time iteration, the fluid and structural solver interact with each other for twenty inner time iterations. As soon as the inner iterations are up to twenty or the residual is beneath the tolerance, the loop ends and marches physical time. Second-order dual time-stepping with 10 internal iterations is responsible for time discretization in the simulation of the flow field. In advance of flutter taking place, steady numerical simulation is solved with time step \(\Delta t=0.001\) and angle of attack \(\alpha=0\). Then, this perturbation of the steady-state solution is used as an initial condition for the unsteady state. In the definition of flutter velocity \(V_{f}^{*}\), b is the length of the semi-root chord and \(\omega_{a}\) is the first torsional frequency of the wing. The time history of generalized displacements is an indicator to determine the flutter boundary. The same as the flutter analysis in NACA64A010, only if \(V_{f}^{*}\) approaches the critical point, the generalized displacements are neutral. When \(V_{f}^{*}\) exceeds the critical point, the generalized displacements diverge. Otherwise, they behave with a damped response. Figure 23 shows the divergent responses of the first three generalized displacements corresponding to \(Ma=0.96\) and \(V_{f}^{*}=0.292\). Through Equation (39), its displacements in the x-, y-, and z-directions are easy to compute as shown in Figure 24. It describes the displacements at the probe of the leading edge and trailing edge, which are drawn in Figure 16b.