Induced-Drag Minimization of Nonplanar Geometries Based on the Euler Equations

The induced drag of several nonplanar configurations is minimized using an aerodynamic shape optimization algorithm based on the Euler equations. The algorithm is first validated using twist optimization to recover an elliptical lift distribution. Planform optimization reveals that an elliptical planform is not optimal when side-edge separation is present. Optimized winglet and box-wing geometries are found to have span efficiencies that agree well with lifting-line analysis, provided the bound constraints on the entire geometry are accounted for in the linear analyses. For the same spanwise and vertical bound constraints, a nonplanar split-tip geometry outperforms both the winglet and box-wing geometries, because it can more easily maximize the vertical extent at the tip. The performance of all the optimized geometries is verified using refined grids consisting of 88-152 million nodes.