*2.3.3 Simulation analysis*

In both the two considered altitudes (9.65 and 14 km), the numerical results agreed well with the experimental results of [13] with a relative error about 14%. This difference would be from the computation such as the quality of mesh and

For more details of the stability of the wing structure, the vibrated value of the

At a Mach number smaller than the flutter value, the damping coefficient was

At a Mach number near the flutter value, the damping coefficient was zero, and

At a Mach number greater than the flutter value, the damping coefficient was negative, and the vibration was divergent (**Figure 6c**). This divergent vibration

wing tip position at three Mach numbers were plotted as shown in **Figure 5**.

*Comparison of Cp distribution at M = 1.141, α = 0°. (a) 26 % semispan. (b) 75.5 % semispan.*

order of model in CFD and CSD.

*Aerodynamics*

**Figure 7.**

**62**

positive, and the wing was stable (**Figure 6a**).

the vibration was harmonic oscillation (**Figure 6b**).

Dynamic aeroelastic analysis was a problem related to fluid-structure interaction over a period of time. Therefore, the quality of aerodynamic grid and the time step strongly influenced the results of aeroelastic analysis. These parameters were also two of the most important problems in the dynamic aeroelastic analysis.

In order to evaluate the quality of aerodynamic grid, the coefficient of pressure of AGARD 445.6 wing was first estimated at 26% semispan and at 75.5% semispan and then was compared with Ref. [6] as shown in **Figure 7**. The presented results were in good agreement with the results in Ref. [6]. It could be concluded that these simulation settings were appropriate for solving the transonic flow.

To evaluate the time step size, three different time sizes were examined such as 0.001, 0.002, and 0.005 s. As it could be seen in **Figure 8**, the displacement of the wing tip was reduced with the reduction of time step size up to 0.002 s until the aeroelastic simulation did not change [6]. Therefore, the value 0.002 s of time step size in the numerical solution was chosen for both aerodynamic and structural analysis.

The limit of flutter was identified by using damping estimations for a large test point at each Mach number. At M = 0.499, the oscillation of the displacement of the wing tip was harmonic (**Figure 9**), and it was considered as a flutter point. At this limit of flutter, the air speed was calculated as 174.26 m/s, and the density of air was calculated as 0.432 kg/m3 . These values were very close to the experimental values: 172.46 m/s for flutter speed and 0.428 kg/m<sup>3</sup> for density of air (**Table 3**).

**Figure 8.** *Wing-tip oscillation depends on the selected time step.*

**Figure 9.** *Wing-tip oscillation of flutter point at M = 0.499.*


#### **Table 3.**

*Flutter characteristics at M = 0.499.*

This remark illustrated that the developed solution could be used to specify the transonic flutter characteristics with errors less than 10%.
