## Results: What is Possible?

Each of these configurations provides particular advantages and disadvantages, although each benefits from some reduction in induced drag compared with the conventional monoplane. The reduction in vortex drag is shown below for biplanes, boxplanes, and winglets with varying ratios of height to span. These results were computed using an optimizing vortex lattice code, but agree with classical solutions from Prandtl, von Karman and Burgers, Cone, and Jones. Note that the boxplane achieves the lowest drag for a given span and height, although winglets are quite similar. Considerable savings in induced drag are possible for a fixed span if large vertical extents are permitted.

Of course, adding vertical surfaces such as winglets add wetted area and weight due to higher bending moments, while the weight of a cantilevered biplane is increased since for a fixed total area, the chords (and dimensional thickness) of each wing are halved. Jones showed that with fixed integrated bending moment (a rough indicator of wing weight) winglets produced about as much drag savings as planar tip extensions. More recent analyses using more realistic weight estimation methods have yielded similar results (but with much a less broad optimum).

For some applications, this discouraging result is not relevant since the aircraft must operate with a span constraint, or because the structural arrangement is not simply analyzed.

The figure below illustrates the effect of nonplanar wing shape on span efficiency. Each of the geometries, shown in front view below, is permitted a vertical extent of 20% of the wing span. Each design has the same projected span and total lift. The results were generated by specifying the geometry of the trailing vortex wake and solving for the circulation distribution with minimum drag. So, each of the designs is assumed to be optimally twisted. This was done by discretizing the vortex wake and solving a linear system of equations for minimum drag with a constraint on overall lift. Similar results for a variety of shapes have been described by Cone, Munk, Letcher, Jones, and others.

The results illustrate the variability in span efficiency among these designs. Note the relatively small gain for the diamond-shaped device and the wing with dihedral, while the C-wing shape achieves essentially the same drag as the boxplane.