The aerodynamic shape optimization of transonic wings requires Reynolds-averaged Navier–Stokes (RANS) modeling due to the strong nonlinear coupling between airfoil shape, wave drag, and viscous effects. While there has been some research dedicated to RANS-based aerodynamic shape optimization, there has not been an benchmark case for researchers to compare their results. In this investigations, a series of aerodynamic shape optimizations of the Common Research Model wing defined for the Aerodynamic Design Optimization Workshop are presented. The computational fluid dynamics solves Reynolds-averaged Navier–Stokes equations with a Spalart–Allmaras turbulence model. A gradient-based optimization algorithm is used in conjunction with a discrete adjoint method that computes the derivatives of the aerodynamic forces. The drag coefficient at the nominal flight condition is minimized subject to lift, pitching moment and geometric constraints. A multilevel acceleration technique is used to reduce the computational cost. A total of 768 shape design variables are considered, together with a grid with 28.8 million cells. The drag coefficient of the optimized wing is reduced by 8.5% relative to the baseline. The single-point design has a sharp leading edge that is prone to flow separation at off-design conditions. A more robust design is achieved through a multi- point optimization, which achieves more reliable performance when lift coefficient and Mach number are varied about the nominal flight condition.
We minimize the drag coefficient by varying the shape design variables subject to a lift constraint (CL = 0.5). In addition, the pitching moment is constrained to be CMy <= 0.17. The shape design variables are z-coordinates of 768 control points on the FFD volume, and angle-of-attack. There are 750 thickness constraints imposed in a grid with 25 chordwise and 30 spanwise stations. The thickness is set to by greater than 25% of the initial thickness at each location. Finally, the internal volume is constrained to be greater than or equal to the initial volume.
In this video, the baseline wing results are shown in red and the optimized wing results are shown in blue. At the optimum, the lift coefficient target is met and the pitching moment is reduced to the lowest allowed value. The lift distribution of the optimized wing is much closer to the elliptical distribution, indicating an induced drag that is close to the theoretical minimum. This is achieved by fine-tuning the twist distribution and airfoil shapes. The baseline wing has a near linear twist distribution. The optimized design has more twist at the root and at the tip, and less twist near mid wing. The overall twist angle only changed slightly from 8.06 degrees to 7.43 degrees.
The optimized thickness distribution is significantly different from that of the baseline. Due to the volume constant, the overall volume has to be conserved. Therefore, the optimizer chooses to increase the thickness at the root and decrease the thickness at the tip. The root t/c is over 20%. The low thickness near the tip would in practice incur structural weight penalty. To obtain a more realistic design, we also performed additional optimization with a more strict thickness constraint.
The baseline wing exhibits a front of very closely spaced pressure contour lines spanning a significant portion of the wing, indicating a shock. The optimized wing shows parallel pressure contour lines with roughly equal spacing, indicating a nearly shock-free solution at the nominal flight condition. This is confirmed by the shock surface plots: we can see that the baseline wing has a shock on the upper surface, while the optimized wing does not show shocks at the design condition. The shock elimination can also be seen on the airfoil Cp distributions. The sharp increase in local pressure due to the shock becomes a gradual change from the leading edge to the trailing edge.
Another noticeable feature in the optimized wing is the sharp leading edge. The optimizer explores the weak- ness in the problem formulation. With a single-point optimization, there is no penalty for thinning out the leading edge. However, sharp leading edge airfoils experience adverse performance at off-design conditions, since the flow is prone to separation at off-design angles-of-attack. We explore these issues in more detail and perform a multi-point optimization in shown below.