# Time-Domain Aeroelasticity Analysis by a Tightly Coupled Fluid-Structure Interaction Methodology
## 6 Results and Discussion
### Flutter Boundary and Limit Cycle Oscillation of Isogai Wing  
Firstly, the tightly coupled FSI methodology is applied to analyze the Isogai wing. The model simulates the bending and torsional motion of a cross section of a sweptback wing. It consists of two DOF, plunging and pitching for a NACA 64A010 airfoil. Two aeroelastic phenomena with typically nonlinear characteristics, including the transonic flutter boundary and the LCO response, are simulated to validate the method.  
The test case A of Isogai wing is investigated, and the nondimensional structural parameters are: \(x_{a}=1.8\), \(r_{\omega}=1\), \(r_{a}^{2}=3.48\), \(r_{d}=60\) and the elastic axis located at the half-chord length. The responses to three different reduced velocities near the flutter boundary at the Mach number of 0.825 are calculated, and the corresponding displacement histories are plotted in Figure 2. The displacements in pitching and plunging coordinates oscillate with equi-amplitudes when \(V^{*}=0.63\), the system reaches a neutrally stable state known as flutter and the corresponding reduced velocity is the flutter speed index. At a lower reduced velocity, \(V^{*}=0.60\), the displacements decay with time and the system is stable for the condition. However, the displacements diverge when a higher reduced velocity is imposed, which may bring disastrous damage to the aeroelastic system. Similar simulations are implemented at different Mach numbers in transonic region, and the flutter boundary is obtained and shown in Figure 3. The speed index flutter boundary is S-shaped with a dip near the Mach number of 0.85, and the simulation result agrees well with other simulation results [18; 19; 20].  
Figure 2: Displacement responses history to different speed indexes.  
For the transonic flow, the shock wave appears in the flow field around the structure, which arouses nonlinear aerodynamic loads for the structure. As a response, the aeroelastic system performs a nonlinear behavior. At the Mach number of 0.85, LCOs are observed with \(V^{*}=0.80\), a reduced velocity higher than the flutter speed index. During the simulation, the model is forced to oscillate sinusoidally with a given amplitude for a cycle, the plunging coordinate is activated since the second cycle and the aeroelastic response begins in the meantime. Figure 4 describes amplitudes of LCO responses to different initial movements, \(\alpha_{0}=1^{\circ}\), \(\alpha_{0}=2^{\circ}\) and \(\alpha_{0}=8^{\circ}\). The pitching and plunging displacements diverge in the first several cycles, then come to equi-amplitude vibrations and the amplitudes are independent of the initial movements. It is found that the responses get to the LCO state more rapidly with the initial movement amplitude increasing. Figure 5 depicts the corresponding phase portraits of the two coordinates, and the responses to different initial movements in both Dofs reach the same orbit, respectively.  
Figure 4: LCO responses to different initial movements.  
Figure 3: Comparison of the flutter speed index in transonic region.  
For the two-dimensional test case, the flutter boundary in the transonic region is predicted by the tightly coupled FSI method and it is similar to those predicted by other researchers. The simulation results reveal the nonlinear response to the periodic nonlinear aerodynamic loads. The coupled CFD-CSD method can estimate the flutter boundary accurately.