Page 273 - Aerodynamics for Engineering Students

256 Aerodynamics for Engineering Students

as I and r vary elliptically so must c, since on the right-hand side c~$pV’ is
a constant along the span. Thus

c = cod1 - = cosine

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
r = cL-c
2
and putting

CL = a,[(a - QO) - E] from Eqn (5.57)
r 0: ca,[(a - ao) - 4

Thus to make I? vary elliptically, geometric twist (varying (a - ao)) or change in
aerofoil section (varying am 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 = co sin 8 in p to give

coam
p = po sin 8 where po = -
8s
Then Eqn (5.61) gives

(5.63)

But

A1 =- cL from Eqn (5.47)
4AR)
Now
CL
= a = three-dimensional lift slope
(a - a01

I I 1
I - /

I

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