Page 274 - Aerodynamics for Engineering Students

Finite wing theory 257

and

for an elliptic chord distribution, so that on substituting in Eqn (5.63) and rearran-
ging
(5.64)

This equation gives the lift-curve slope a for a given aspect ratio (AR) 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
a
am =
1 - [a/744R)I
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 I1 fighter. It would be pleasing to report that the
wing shape was chosen with due regard being paid to aerodynamic theory. Unfortu-
nately 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.*

5.7 Swept and delta wings

Owing to the dictates of modern flight many modern aircraft have sweptback or
slender delta wings. Such wings are used for the benefits they confer in high-
speed flight - see Section 6.8.2. Nevertheless, aircraft have to land and take off.
Accordingly, a text on aerodynamics should contain at least a brief discussion of
the low-speed aerodynamics of such wings.

5:7.1 Yawed wings of infinite span

For a sweptback wing of fairly high aspect ratio it is reasonable to expect that away
from the wing-tips the flow would be similar to that over a yawed (or sheared) wing
of infinite span (Fig. 5.36). In order to understand the fundamentals of such flows it
is helpful to use the coordinate system (x’, y, z’), see Fig. 5.36. In this coordinate
system the free stream has two components, namely U, cos A and U, sin A, per-
pendicular and parallel respectively to the leading edge of the wing. As the flow

*L. Deighton (1977) Fighter Jonathan Cape Ltd