Copyright © A. Filippone (1999-2003). All Rights Reserved.

Wings for All Speeds

Non-Planar Wing Systems


Non-planar wing systems are optimally loaded wings with higher aerodynamic efficiency than planar wings having the same aspect-ratios. 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 human-powered flight.

The wing systems discussed in this section include wings with spanwise camber, C-wings, ring wings, and various combinations of basic lifting surfaces. Wings with endplates, winglets and other tip devices belong to the same category of non-planar systems.

Thick wings optimized for minimum lift-induced 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 equi-distribution of the wing loading.

Wings in Subsonic Flow

The benefits of using large wing spans are well known from small perturbation theory: The induced drag grows with the inverse of the aspect-ratio. However, there are practical limitations to the span (weight, structural, operational, etc.).

An increase in aspect-ratio 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 aspect-ratio) than a flat wing having elliptical loading (wing of minimum theoretical induced drag).

Linearized Vortex Theory

By using linearized vortex theory, it is possible to prove that there exists an infinite variety of non-planar wings that have induced drag less than that of a flat wing elliptically loaded.

This leads to the formulation of an effective aspect-ratio kA, that allows to write the expression of the induced drag in the same fashion as in the small perturbation theory for flat wings.

Wing Effectiveness

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 Semi-Circle 1.50
Semi-Circle 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, crescent-shaped wings (Smith-Kroo, 1993), C-wings.

Wings in Transonic and Supersonic Flow

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.

Some Interference Problems

Although aerodynamic interference is a general subject, we will review here those cases wherein two or more wings are combined together to provide a non-planar lifting system. Among the cases falling in this category there is the biplane and the T-tail.

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 (Milne-Thompson, 1966).

Related Material

Other Wing Configurations

Selected References

  • Ashley H, Landahl M. Aerodynamics of Wings and Bodies, Addison-Wesley Publ. Company, Reading, Mass. 1965.

  • Schlichting H, Truckenbrodt E. Aerodynamics of the Airplane, McGraw-Hill, New York, 1979.

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Copyright © A. Filippone (1999-2003). All Rights Reserved.