The use of larger aspect-ratios is a well known means to reduce the induced drag, that is the drag created by the spanwise distribution of circulation, ultimately due to the development of two strong tip vortices.
From three-dimensional small disturbance theory it is found that the optimum distribution of circulation is the one that creates a constant downwash velocity if the slipstream (wake). With this distribution of downwash velocity, the spanwise lift distribution is elliptic, and the induced drag is at a minimum. If the wing planform is also elliptic, there is a closed expression relating the lift to the induced drag.
Fig. 1 shows the behavior of the induced drag coefficient as a function of the wing aspect-ratio at two lift levels, from the small disturbance theory (Since this theory was developed for large aspect-ratio wings, values of CDi for aspect-ratios less than 5 are rather off.)
Figure 1: Induced drag vs. AR at different lift levels.
The actual aspect-ratio is a compromise between conflicting requirements. For example, for a transport aircraft the optimal aspect-ratio would be around 8 for minimum cost acquisition, and twice as much for minimum fuel consumption.
A fighter aircraft has a low aspect-ratio for manouvrability. Sail planes, human
powered planes, high altitude airplanes have a very large wing span, but have poor
manouvrability characteristics. The table below provides some typical values.
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