Nonplanar Wings and Winglets
One often begins the wing design problem by specifying a target Cp distribution and/or span loading and then modifying the wing geometry (either manually, by direct inverse, or by nonlinear optimization). In the case of planar wings, the elliptic loading results provide a useful benchmark in the creation of target loadings. (For high aspect ratio wings, 2D airfoil results may provide a useful reference for the chordwise loading.)
More complex methods for creating target Cp's are beyond the scope of this discussion, but we have little guidance at all when the wing is nonplanar.
This section deals with the problem of optimal loading for nonplanar lifting surfaces. It is easily generalized to multiple surfaces.
When the wing is not planar, many of the previous simple results are no longer valid. Elliptic loading does not lead to minimum drag and the span efficiency can be greater than 1.0.
Here we will describe a method for computing the minimum induced drag for planar and nonplanar wings. First, consider the distribution of downwash for minimum drag. This can be obtained by using the method of restricted variations as follows.
We consider an arbitrary variation in the circulation distribution represented by dG1 and dG2 which do not change the lift:
This implies: 
If the drag was minimized by the initial distribution:
So, 
That is, the downwash is constant behind a planar wing with minimum drag.
In the general case, with multiple surfaces or nonplanar wings, the same approach may be used. In this case, the condition for constant lift is:
where theta is the local dihedral angle of the lifting surface.
For minimum drag:
where Vn is the induced velocity in the Trefftz plane in a direction normal to the wake sheet (the normalwash).
In this case, 
so, Vn = k cos q.
The normalwash is proportional to the local dihedral angle. Thus, the sidewash on optimally-loaded winglets is 0, for example.
We may then solve for the distribution of circulation that produces this distribution of normalwash.
Alternatively, one may use a more direct optimization approach. With the circulation distribution represented as the row vector, {G} and the wake modeled as a collection of line vortices of strength {Gw}, we may write the wake vorticity in terms of the surface circulation, based on a discrete vortex model as shown below.
The drag is then given by: D = r/2 {Vn} · {G}
where Vn is the normal wash in the Trefftz plane computed using the Biot Savart law.
{Vn} is related to the circulation strengths by:
{Vn} = [VIC] {G}
where [VIC] is a function of the geometry.
So, D = r/2 [VIC] {G} · {G}
The lift is also a function of the circulations:
L = r U {G} · {cos q}
with theta the local dihedral angle.
Finally, the optimal values of {G} are given by setting
(D+l(L-Lref)) Gi = 0 where l is a Lagrange multiplier.
This problem is sometimes done as homework, but some results are summarized below:
· When the wing/winglet combination is optimized for minimum drag at fixed span, it achieves about the same drag as a planar wing with a span increased by about 45% of the winglet height.
· The wing lift distribution is as shown below with increased lift outboard compared with the no winglet case.
This increased tip loading along with the extra bending moment of the winglet leads to increased structural weight. When a bending moment constraint replaces the span constraint, wings with winglets are seen to have about the same minimum drag as the stretched-span planar wings. This is shown below.
Induced drag of wings with winglets and planar wings all with the same integrated bending moment (related to structural weight). Note that solutions to the left of the span ratio = 1.0 line are not meaningful.
The same approach may be taken for general nonplanar wake shapes. The figure below summarizes some of these results, showing the maximum span efficiency for nonplanar wings of various shapes with a height to span ration of 0.2.
Several points should be made about the preceding results.
1. The result that the sidewash on the winglet (in the Trefftz plane) is zero for minimum induced drag means that the self-induced drag of the winglet just cancels the winglet thrust associated with wing sidewash. Optimally-loaded winglets thus reduce induced drag by lowering the average downwash on the wing, not by providing a thrust component.
2. The results shown here deal with the inviscid flow over nonplanar wings. There is a slight difference in optimal loading in the viscous case due to lift-dependent viscous drag. Moreover, for planar wings, the ideal chord distribution is achieved with each section at its maximum Cl/Cd and the inviscid optimal lift distribution. For nonplanar wings this is no longer the case and the optimal chord and load distribution for minimum drag is a bit more complex.
3. Other considerations of primary importance include:
Stability and control
Structures
Other pragmatic issues
More details on the design of nonplanar wings may be found in a recent paper, "Highly Nonplanar Lifting Systems," accessible here.