Non-planar lifting-line theory for fixed and deformable geometries
In this thesis, the lifting-line approximation of a flat, unswept wing, originally attributed to Prandtl, is investigated. The original formulation for a flat wing is examined in detail. The governing integro-differential equation is developed from its components. The optimum and general solutions to the original formulation are presented and discussed. An expanded formulation is presented, which includes the effect of the wake of non-planar wings. The self-induced velocities of the bound vortex on the wing are assumed to be small for practical cases and not included in the model. The case of simple dihedral is considered and the general formulation is simplified to better illustrate the effect of the geometry on the governing equation. For the simplified dihedral case, the optimal solution remains the same as for a flat wing. A simplified finite element model is also included, which accounts for the bending due to the force generated by the bound vortex. This finite element model is combined with the non-planar lifting-line equation to create a static aeroelastic model for a wing. The solution of this problem is iterative, but converges quickly. Lift coefficient and span efficiency factor are provided for a set of wing geometries for cases of dihedral and wing bending, and the trends are examined compared to flat wings. Additionally, the resulting geometries after deformation of the wing are presented and the effect of circulation distribution on the resulting shape is discussed.