Load distribution for minimum drag

Minimum induced drag for a given lift will occur if CD is a minimum and this will be so only if 8 is zero, since 8 is always a positive quantity. Since 8 involves squares of all the coefficients other than the first, it follows that the minimum drag condition coincides with the distribution that provides A3 = A5 — A7 = An = 0. Such a distri­bution is Г = 4sVAi sin# and substituting z = —s cos в

which is an elliptic spanwise distribution. These findings are in accordance with those of Section 5.5.3. This elliptic distribution can be pursued in an analysis involving the general Eqn (5.60) to give a far-reaching expression. Putting An — 0, n ф 1 in Eqn

(5.60)

gives

and rearranging

(5.61)

Now consider an untwisted wing producing an elliptic load distribution, and hence minimum induced drag. By Section 5.5.3 the downwash is constant along the span and hence the equivalent incidence (a — ao — w/V) anywhere along the span is constant. This means that the lift coefficient is constant. Therefore in the equation

1 ,

lift per unit span / = pVT = c

as / and Г vary elliptically so must c, since on the right-hand side C^pV2 is a constant along the span. Thus

and the general inference emerges that for a spanwise elliptic distribution an untwisted wing will have an elliptic chord distribution, though the planform may not be a true ellipse, e. g. the one-third chord line may be straight, whereas for a true ellipse, the mid-chord line would be straight (see Fig. 5.35).

It should be noted that an elliptic spanwise variation can be produced by varying the other parameters in Eqn (5.62), e. g. Eqn (5.62) can be rearranged as

V

T = CL-c

and putting

Cl = Яоо[(<* — <*o) – є] from Eqn (5.57) Г ос cax(a – ao) – є]

Thus to make Г vary elliptically, geometric twist (varying (a – ao)) or change in aerofoil section (varying ax and/or ao) may be employed in addition to, or instead of, changing the planform.

Returning to an untwisted elliptic planform, the important expression can be obtained by including c = cq sin в in p to give

p = po sin в where po =

Then Eqn (5.61) gives

Al

l + Mo

But

Ai = – fi-v from Eqn (5.47) TT{AK)

Now

Fig. 5.35 Three different wing planforms with the same elliptic chord distribution

and

for an elliptic chord distribution, so that on substituting in Eqn (5.63) and rearran­ging

1 + [а^/ж{АК)

This equation gives the lift-curve slope a for a given aspect ratio (A R) in terms of the two-dimensional slope of the aerofoil section used in the aerofoil. It has been derived with regard to the particular case of an elliptic planform producing minimum drag conditions and is strictly true only for this case. However, most practical aerofoils diverge so little from the elliptic in this respect that Eqn (5.64) and its inverse

1 – [a/ir{AR)

can be used with confidence in performance predictions, forecasting of wind-tunnel results and like problems.

Probably the most famous elliptically shaped wing belongs to the Supermarine Spitfire – the British World War II fighter. It would be pleasing to report that the wing shape was chosen with due regard being paid to aerodynamic theory. Unfortu­nately it is extremely doubtful whether the Spitfire’s chief designer, R. D. Mitchell, was even aware of Prandtl’s theory. In fact, the elliptic wing was a logical way to meet the structural demands arising from the requirement that four big machine guns be housed in the wings. The elliptic shape allowed the wings to be as thin as possible. Thus the true aerodynamic benefits were rather more indirect than wing theory would suggest. Also the elliptic shape gave rise to considerable manufacturing problems, greatly reducing the rate at which the aircraft could be made. For this reason, the Spitfire’s elliptic wing was probably not a good engineering solution when all the relevant factors were taken into account.[26]