5.1. Optimization Problem
We define a simple trapezoidal wing, as shown in
Figure 20, as the initial configuration referring to the design scheme of Aerion AS2, and the geometric parameters are shown in
Table 7.
The double-circular-arc supersonic airfoil NACA 2S-(40)(1.5)-(40)(1.5) is used, and its parameters are defined as shown in
Figure 21. The existence of a sharp leading edge of a typical supersonic airfoil has an adverse effect on the geometric deformation. The displacement of the FFD control point can make the smooth arc sharp, but because the initial surface is not continuous by
order, the reverse process cannot be accomplished. Therefore, we use a small arc instead of the ideal sharp leading edge. The diameter of the small arc is 0.4 mm. The blunted leading edge of the airfoil is shown in
Figure 22.
ANSYS-ICEM is used to generate the structural grid of the initial configuration, as shown in
Figure 23. Likewise, a grid convergence study is performed on the supersonic leading edge configuration.
Figure 24 shows three levels of grids, where the grid is refined from L2 to L0.5. The different mesh amounts, related drag coefficients, and GCIs based on Equations (
17) and (
19) are shown in
Table 8 to show a converging tendency, where the refinement ratio
r is 1.29. The solutions should fall within the estimated solution’s asymptotic range of convergence for each grid level. This can be confirmed by using two GCI values calculated over three grid levels, as shown in Equation (
20).
Considering the solving efficiency, L1 mesh is utilized in our optimization design, which can provide the solution with sufficient accuracy. The amount of L1 mesh is 6.3 million, and the height of the first layer of the boundary layer is
. A conical far field whose size is twice the length of the fuselage is used, as shown in
Figure 23.
The FFD control frame is shown in
Figure 25, where the red control points are chosen as design variables. The fuselage and wing root remain unchanged, and the leading and trailing edges of the wing are also restricted to remain unchanged (green points). The parameterization is mainly conducted for the shape deformation of the wing. For the wing profile deformation, there are a total of 120 FFD control points, of which four sections are distributed along the span direction, 15 positions are distributed along the chord direction, and two layers are in the normal direction. We select three stations in the span direction except the wing root to control the twist of the wing section and achieve the deflection of each station by rotating the 30 FFD control points at the same span location around a given rotation axis, which is always the leading edge.
The design condition is , , . The optimization objective is to minimize the total drag coefficient, and the design variables are the angle of attack, the displacement of FFD control points (a total of 120 design variables) on the wing and three deflection angles of FFD control profile.
Considering engineering application, the thickness and volume of the wing are also constrained. Thickness constraints are shown in
Figure 26, distributed in four spanwise positions of the semi-span. There are 25 parts uniformly distributed in each section within the range of 5–90% chord length. To compare the optimized configurations under different thickness constraints, the optimal design is carried out under the condition that the thickness constraints are, respectively,
and
. The volume constraints are distributed in the same way as that of the subsonic leading edge configuration. In order to avoid the adverse affect, the pitching moment coefficient is constrained within a range not less than 0. The optimization problem statement for supersonic leading edge configuration is shown in
Table 9.
5.2. Optimization Results
The comparison of aerodynamic force coefficients between the optimized configuration and the initial configuration is shown in
Table 10, where Opt_0.7t represents the optimized configuration with a thickness constraint of
, and Opt_0.9t represents the optimized configuration with a thickness constraint of
.
Compared with the initial configuration, the drag coefficient of the Opt_0.7t optimized configuration is reduced by 13.04 counts, a decrease of 4.53%. The drag coefficient of the Opt_0.9t optimized configuration is reduced by 10.1 counts, a decrease of 3.51%. In the optimization process, the angle of attack is also used as a design variable. Compared with the initial configuration, the angle of attack of the optimized configuration of Opt_0.7t and Opt_0.9t increases by 0.03 and 0.01, respectively, which are small. Meanwhile, the pitching moment coefficient changes slightly before and after optimization, which satisfies the pitching moment constraint.
The comparison of normalized lift and twist angle distribution along the span direction before and after optimization is shown in
Figure 27. Since the initial configuration is the spanwise chord length distribution of the trapezoidal plane, the initial lift load is close to the elliptical distribution, which is an ideal distribution to reduce the induced drag. However, the lift distribution of the two optimized configurations shows a tendency to deviate from the elliptical lift distribution. Therefore, unlike the subsonic leading edge configuration, the supersonic leading edge configuration takes reducing shock wave drag as the optimization dominant direction. Compared with the initial configuration, the lift in the optimized configuration increases in the inner wing section. It decreases nonlinearly in the middle wing section, which is close to an elliptical lift distribution. In the outer wing section, the lift gradually increases, and the overall lift load is transferred to the wing root and wing tip.
Corresponding to the changing trend of lift, while the middle wing section is negatively twisted, both the inner wing section and the outer wing section have positive twist angles, and the lift is influenced by the local angle of attack. Moreover, the changing trend of the twist angle under the two thickness constraints is the same compared with the initial configuration, and there is only a slight difference in the amount of change.
Figure 28 shows the distribution of normalized drag along the span direction before and after optimization. Compared with the initial configuration, the drag change trend of Opt_0.7t and Opt_0.9t is the same. The drag decreases in the range of 40–90% semi-span and increases in the range of 10% semi-spans of the wingtip. For a supersonic airliner, the volume wave drag is closely related to the thickness of the wing. Therefore, compared with the Opt_0.9t optimized configuration, the Opt_0.7t optimized configuration further reduces the thickness of the outer wing section, which further reduces the volume wave drag, and finally obtains greater drag reduction.
The contour of the pressure coefficient on the upper surface of the wing with the optimized configuration and the initial configuration is shown in
Figure 29. Corresponding to
Figure 30, a–d represents the slices in spanwise direction, respectively
. The pressure distribution and airfoil comparison of a typical section is shown in
Figure 30.
According to
Figure 30a,b, in the inner wing section, compared with the initial configuration, the optimized configuration reduces the thickness of the leading and trailing edges, which is conducive to weakening the shock wave strength at the leading edge and reducing wave drag. At the same time, the thickness of the airfoil increases at
, and the position of the maximum thickness is delayed.
Comparing the airfoil before and after optimization in
Figure 30c,d, it can be found that in the outer wing section, the thickness of the optimized airfoil is smaller than that of the initial airfoil in the entire chord length range. Corresponding to the change in the spanwise distribution of normalized drag in
Figure 28, the reduction in the airfoil thickness of the middle and outer wing sections reduces the drag. Although the drag of the inner wing may increase because of the larger thickness to satisfy the volume constraint, it is beneficial to the improvement of the overall drag performance.
The spatial distribution of the pressure coefficient and Mach number at the
50% and
90% of the wing span before and after optimization is shown in
Figure 31. The results show that there are obvious differences, mainly on the lower surface of the leading edge on the wing, and each section presents the same change trend. After optimization, the pressure coefficient of the leading edge of each section after the shock wave is significantly reduced, and the Mach number after the wave is obviously increased. Therefore, the optimized configuration weakens the shock wave intensity at the leading edge of the wing and effectively reduces the shock wave drag.
For the purpose of qualitatively analyzing the trade-offs between lift and drag, as well as airfoil deformation and constraints in aerodynamic optimization design, sensitivity analysis is performed using gradient information obtained from the discrete adjoint equation for the initial configuration.
The derivative distribution of the lift/drag coefficient on the upper and lower surfaces of the wing with respect to the
z displacement of the surface grid points in the initial configuration is shown in
Figure 32, where the displacement direction is positive along the positive direction of the coordinate axis. For the supersonic leading edge configuration, when the lower surface deforms along the
z axis, it has a greater influence on the lift/drag coefficient than the upper surface. Compared with the lift coefficient, the drag coefficient is more sensitive to the
z displacement of the upper and lower surface.
The comparison of the airfoils before and after optimization (
Figure 30) shows that, compared with the initial configuration, the upper and lower surfaces of the optimized configuration both move in a large range along the positive direction of the
z axis. Combined with the sensitivity analysis results, when the lower surface moves in the positive direction of the
z axis, the drag coefficient decreases accordingly, and the drag reduction effect is more significant. On the one hand, this deformation trend increases the lift coefficient in the dark red area of the sensitivity contour figure. On the other hand, reducing drag is the dominant direction of optimization since the lift can be maintained to meet the constraint through the deformation of the local profile. According to the results of sensitivity analysis, the movement of the upper surface of the wing along the negative direction of the
z axis reduces the drag coefficient, but the actual optimization results show a contrary deformation. The reason is that the drag coefficient is more sensitive to the deformation of the lower surface than to the deformation of the upper surface. On balance, the optimizer mainly focuses on the lower surface deformation, and the upper surface moves accordingly to make sure that the final optimization configuration satisfies the thickness and volume constraints.
The convergence process of the optimized supersonic leading edge configuration is shown in
Figure 33. After 35 main iterations, the constraints and the convergence conditions of the objective function are satisfied. From the convergence process of the main iteration, we can obtain the same conclusion as in subsonic leading edge optimization that the drag coefficient is reduced quickly in the first 10 main iterations of the gradient-based optimization.
The intermediate process of the airfoil at the typical sections of the inner wing 40% section and the outer wing 90% section from the initial configuration to the final optimized configuration is given, respectively. First, the comparison chart of P1 and P0 shows that in the beginning stage of the optimization, the changing trend of the inner and outer airfoils is the same, and the shock wave strength is weakened by reducing the thickness of the leading and trailing edges, while the camber of the airfoil increases, and the position of the maximum thickness moves backward. Compared with P1, the airfoil thickness of the inner and outer wing sections of P2 is reduced in a larger range near the leading and trailing edges, and the position of the maximum thickness is further moved back. Furthermore, the thickness of the airfoil of the outer wing section of P3 is reduced in the entire chord length range, while the thickness of the airfoil of the inner wing section is mainly increased in the middle section. The main reason for this phenomenon is that, considering the combined effect of the drag reduction objective and constraints, the optimizer increases the thickness of the inner wing section in exchange for a reduction in the thickness of the outer wing section and an overall greater drag reduction benefit. In the final stage of optimization, P4 has minor changes compared to P3, mainly to make small adjustments to the airfoil shape to make it smoother.