# Time-Domain Aeroelasticity Analysis by a Tightly Coupled Fluid-Structure Interaction Methodology
## 6 Results and Discussion
### Flutter Boundary Analysis of AGARD 445.6 Wing  
AGARD 445.6 wing is a standard aeroelastic configuration and its flutter responses are extensively simulated to evaluate and assess the FSI methodology. The geometry and experimental data are provided by the NASA Langley Research Center. The flutter response of the weakened structure model mounting on the wall is analyzed. To evaluate the uncertainties of the CFD results brought by the mesh resolution, grid independence studies are conducted on five multiblock structural meshes, named by Mesh1 with about 7000 cells, Mesh2 with about 15,030 cells, Mesh3 with about 30,000 cells, Mesh4 with about 42,000 cells and Mesh5 with 78,340 cells. In the test case, the free-stream Mach number is 0.96 and the angle of attack is 2\({}^{\circ}\). The drag coefficients corresponding to the five meshes are compared in Figure 6. It can be seen that at first the drag coefficient shows a decreasing trend with grid number increasing, then it tends towards stability even the grid number increases further. Hence, Mesh4 is employed as the final grid. A multiblock structural fluid mesh with 42,000 cells and 121 structure nodes are adopted in the simulation, the surface mesh and the structure nodes are plotted in Figure 7. The first four vibration modes, first bending, first torsion, second bending and second torsion, are utilized in the structure dynamic model. Further, the frequencies of these four modes are 9.6 Hz, 38.17 Hz, 48.35 Hz and 91.54 Hz, respectively. The modal shapes of the four modes are shown in Figure 8.  
The cross-section of the AGARD 445.6 wing is the NACA 65A004 symmetric airfoil, and the responses are simulated at the angle of attack of 0\({}^{\circ}\) with a small disturbance imposed to motivate the model at the beginning of the simulation. The dynamic aeroelastic responses at six different Mach numbers ranging from 0.499 to 1.141 are analyzed and corresponding generalized displacement responses on the flutter boundary are shown in Figure 9. On the flutter boundary of each case, all the generalized displacements perform the equi-amplitude oscillation with the same frequency, the aeroelastic system is neutral stable at the corresponding speed index. Figure 5: Phase portraits of pitching and plunging coordinates with different initial movements. Figure 8: Modal shapes of the first four modes. Figure 6: Analysis of grid independence. Figure 7: Surface mesh and structure nodes of AGARD 445.6 wing. The predicted flutter speed indexes and flutter frequency ratios are shown in Tables 1 and 2, respectively, together with the wing tunnel results. The results computed by the method proposed in the paper are in excellent agreement with the experimental data for low speed cases, M = 0.499 and M = 0.678, and the relative error of the simulation is subtle. Figure 9: Response histories of the generalized displacement on the flutter boundary. In transonic regime, M = 0.901 and M = 0.960, the results are reasonable with the relative error under 7%. For the supersonic cases, M = 1.072 and M = 1.141, however, coupled method overpredicts the flutter velocities by about 30% higher than the experimental data. The simulations predict the flutter frequency ratio in better accuracy compared with the flutter speed indexes, and the largest relative error is less than 10%. The simulation results are also compared with those predicted by other researchers. The comparisons of flutter speed indexes and the flutter frequency ratios are shown in Figures 10 and 11. As is noted that [20; 21] also overpredicted the flutter speed indexes in the supersonic region, and the larger differences are observed in those predicted results. Compared to their results, our results show some improvement by performing preconditioned Navier-Stokes calculation. The amount of improvement is relatively large, especially in the prediction of flutter frequency compared to the differences between other computational results and the experiment.