Page 272 - Aerodynamics for Engineering Students

Finite wing theoly 255

Equating like terms:
peal sin 0 = A1 (1 + po) sin 0
psin20=~~(1+2h)sin20
2v
0 = A3( 1 + 3p0) sin 30 etc.

Thus the spanwise distribution for this case is
r =4sV[A1sin0+Aasin28]

and the coefficients are

and

A2 = ( ” ) ws
2(1 + 2po) 7

5.6.3 Load distribution for minimum drag
Minimum induced drag for a given lift will occur if C, is a minimum and this will be
so only if S is zero, since S is always a positive quantity. Since S 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 = A, = 0. Such a distri-
bution is I? = 4sVA1 sin8 and substituting z = -scos8

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 A, = 0, n # 1 in Eqn
(5.60) gives
p(a - a0) = A1 sin0 1 +-
( ste)

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 - 00 - 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 I = pv~ CL - pv2c (5.62)
=
2