EFFECT OF TAPER AND SWEEPBACK

The aspect ratio of a wing is the primary factor in determining the three-dimensional characteristics of the ordinary wing and its drag due to lift. However, certain local effects take place throughout the span of the wing and these effects are due to the distribution of area throughout the span. The distribution of lift along the span of a wing cannot have sharp discontinuities. (Nature just doesn’t arrange natural forces with sharp discontinuities.) The typical lift distribution is arranged in some elliptical fashion. A representative dis­tribution of the lift per foot of span along the span of a wing is shown in figure 1.32.

The natural distribution of lift along the span of a wing provides a basis for appreciating the effect of area distribution and taper along the span. If the elliptical lift distribution is

matched with a planform whose chord is dis­tributed in an elliptical fashion (the elliptical wing), each square foot of area along the span produces exactly the same lift pressure. The elliptical wing planform then has each section of the wing working at exactly the same local Lift coefficient and the induced downflow at the wing is uniform throughout the span. In the aerodynamic sense, the elliptical wing is the most efficient planform because the uni­formity of lift coefficient and downwash incurs the least induced drag for a given aspect ratio. The merit of any wing planform is then meas­ured by the closeness with which the distribu­tion of lift coefficient and downwash approach that of the elliptical planform.

The effect of the elliptical planform is illus­trated in figure 1.32 by the plot of local lift

coefficient to wing lift coefficient, 4r, versus

Oz,

semispan distance. The elliptical wing pro­duces a constant value of4r = 1.0 throughout

the span from root to tip. Thus, the local section angle of attack, a0, and local induced angle of attack, are constant throughout the span. If the planform area distribution is anything other than elliptical, it may be ex­pected that the local section and induced angles of attack will not be constant along the span.

A planform previously considered is the simple rectangular wing which has a taper ratio of 1.0. A characteristic of the rectangular wing is a strong vortex at the tip with local downwash behind the wing which is high at the tip and low at the root. This large non­uniformity in downwash causes similar varia­tion in the local induced angles of attack along the span. At the tip. where high downwash exists, the local induced angle of attack is greater than the average for the wing. Since the wing angle of attack is composed of the sum of cr( and a0, a large local a, reduces the local ao creating low local lift coefficients at the tip. The reverse is true at the root of the rectangular wing where low local downwash exists. This situation creates an induced angle of attack at the root which is less than the average for the wing and a local section angle of attack higher than the average for the wing. The result is shown by the graph of figure 1.32 which depicts a local lift coefficient at the root almost 20 percent greater than the wing lift coefficient.

The effect of the rectangular planform may be appreciated by matching a near elliptical lift distribution with a planform with a constant chord. The chords near the tip develop less lift pressure than the root and consequently have lower section lift coeffi­cients. The great nonuniformity of local lift coefficient along the span implies that some sections carry more than their share of the load while others carry less than their share of the load. Hence, for a given aspect ratio, the rectangular planform will be less efficient than the elliptical wing. For example, a rectangular wing of AR=6 would have 16 percent higher induced angle of attack for the wing and 5 percent higher induced drag than an elliptical wing of the same aspect ratio.

At the other extreme of taper is the pointed wing which has a taper ratio of zero. The extremely small parcel of area at the pointed tip is not capable of holding the main tip vortex at the tip and a drastic change in down – wash distribution results. The pointed wing has greatest downwash at the root and this downwash decreases toward the tip. In the immediate vicinity of the pointed tip, an upwash is encountered which indicates that negative induced angles of attack exist in this area. The resulting variation of local lift coefficient shows low a at the root and very high ct at the tip. This effect may be appre­ciated by realizing that the wide chords at the root produce low lift pressures while the very narrow chords toward the tip are sub­ject to very high lift pressures. The varia­tion of U – throughout the span of the wing of taper ratio=0 is shown on the graph of figure

1.32. As with the rectangular wing, the non­uniformity of downwash and lift distribution result in inefficiency of this planform. For example, a pointed wing of AR=6 would have 17 percent higher induced angle of attack for the wing and 13 percent higher induced drag than an elliptical wing of the same aspect ratio.

Between the two extremes of taper will exist pianforms of more tolerable efficiency.

The variations of p for a wing of taper ratio

=0.5 closely approximates the lift distribution of the elliptical wing and the drag due to lift characteristics are nearly identical. A wing of AR=6 and taper ratio = 0.5 has only 3 percent higher а і and 1 percent greater CDi than an elliptical wing of the same aspect ratio.

A separate effect on the spanwise lift dis­tribution is contributed by wing sweepback. Sweepback of the planform tends to alter the lift distribution similar to decreasing the taper ratio. Also, large sweepback tends to increase induced drag.

The elliptical wing is the ideal of the sub­sonic aerodynamic planform since it provides a minimum of induced drag for a given aspect ratio. However, the major objection to the elliptical planform is the extreme difficulty of mechanical layout and construction. A highly tapered planform is desirable from the stand­point of structural weight and stiffness and the usual wing planform may have a taper ratio from 0.45 to 0.20. Since structural con­siderations are quite important in the develop­ment of an airplane configuration, the tapered planform is a necessity for an efficient configu­ration. In order to preserve the aerodynamic efficiency, the resulting planform is tailored by wing twist and section variation to obtain as near as possible the elliptic lift distribution.