NonPlanar Wing Systems
Summary
Nonplanar wing systems are optimally loaded wings with higher aerodynamic
efficiency than planar wings having the same aspectratios. Methods for the analysis
of such wings have evolved due to the requirements of minimum induced drag (or drag
due to lift) for powered aircraft, and later for endurance flight, flight altitude
flight and humanpowered flight.
The wing systems discussed in this section include wings with spanwise camber,
Cwings, ring wings, and various combinations of basic lifting surfaces. Wings with
endplates, winglets and other tip devices
belong to the same category of nonplanar systems.
Thick wings optimized for minimum liftinduced drag or other parameters (for
ex. aerodynamic center), are sometimes called warped wings, because they have
a distribution of camber and twist to fulfill the condition of equidistribution of
the wing loading.
The benefits of using large wing spans are well known from small perturbation theory:
The induced drag grows with the inverse of the aspectratio. However, there are
practical limitations to the span (weight, structural, operational, etc.).
An increase in aspectratio would produce an excessive increase in profile drag (because
of increased wetted area). The idea would be to find wing systems that are more
efficient (at given aspectratio) than a flat wing having elliptical loading (wing of
minimum theoretical induced drag).
By using linearized vortex theory, it is possible to prove that there exists an
infinite variety of nonplanar wings that have induced drag less than that of a flat
wing elliptically loaded.
This leads to the formulation of an effective
aspectratio kA, that allows to write the expression of the induced drag in the same
fashion as in the small perturbation theory for flat wings.
Values for the effectiveness factor k for selected configurations are
summarized in the following table (data compiled from Cone, 1962). The data are compared
at constant span.
Table 1: Effeciency factor of subsonic wings
Configuration  k 
Elliptically loaded flat wing  1.00 
Closed SemiCircle  1.50 
SemiCircle  1.50 
Full Circle  2.00 
Full Ellipse , a/b=0.5  1.50 
Wing w/ tip tanks, b=0.7, r=0.15  1.31 
Wing w/ upper endplates, h/b=0.15  1.22 
Other Systems
The closed lifting surfaces (circles, squares, ellipses) are in fact examples of
ring wings. The practical means for producing the optimal lift distributions
on such wings is not straightforward.
Span effectiveness for wings with endplates is also given by Schlichting and
Truckenbrodt, 1978. Other wings not covered here include: joined wings,
crescentshaped wings (SmithKroo, 1993), Cwings.
The use of linearized theory in supersonic flow is made difficult by the law on
forbidden signals (von Karman). Theories for subsonic flows are based on the fact
that perturbations car travel (instantaneously) downstream and upstream.
In supersonic flow perturbations travel downstream along characteristic lines at the
speed of sound. One way to approach the problem is through the aerodynamic influence
coefficients. The method is numerically involved, but in the end it is not much
different from lifting surface methods.
Although aerodynamic interference is a general subject, we will review here those
cases wherein two or more wings are combined together to provide a nonplanar lifting
system. Among the cases falling in this category there is the biplane and the
Ttail.
Linear interference problems at subsonic speeds can be reported directly to some
interference theorems (Munk’s stagger theorems, 1924). The solution for the biplane
is relatively simple (MilneThompson, 1966).
Related Material
Other Wing Configurations
Selected References
 Ashley H, Landahl M. Aerodynamics of Wings and Bodies,
AddisonWesley Publ. Company, Reading, Mass. 1965.
 Schlichting H, Truckenbrodt E. Aerodynamics of the Airplane,
McGrawHill, New York, 1979.
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