Page 259 - Aerodynamics for Engineering Students

242 Aerodynamics for Engineering Students
Writing the numerator as (z - zl) + z1:

=$[I
dz
+zl
sdz
Js
47rs -s&E-7 -sd?Zf(z-z1) 1
Evaluating the first integral which is standard and writing I for the second
TO
wz, =-[7r+z1l] (5.40)
47rs
Now as this is a symmetric flight case, the shed vorticity is the same from each side of
the wing and the value of the downwash at some point z1 is identical to that at the
corresponding point - z1 on the other wing.
So substituting for fzl in Eqn (5.40) and equating:

This identity is satisfied only if I = 0, so that for any point z - z1 along the span
r0
w=- (5.41)
4s
This important result shows that the downwash is constant along the span.

Induced drag (vortex drag) for elliptic distribution
From Eqn (5.36)

whence
A
2
D~ = -pro (5.42)
8
Introducing
1
Dv = Co,-pV2S
2
and from Eqn (5.39)
e, vs
ro =-
TS
Eqn (5.42) gives
eo, - P v2s = 5 P ( F)
1
CLVS
2
or

(5.43)