Page 252 - Aerodynamics for Engineering Students

Finite wing theoly 235

t=
2-11
x-xj-/4q
I
A /B

p1

C

Fig. 5.25 Geometric notation for L-shaped vortex element

Making use of Eqn (5.9) it can be seen that this induced velocity is perpendicular to
the y = 0 plane and can be written as
svi(xl,~~) = (svi),, + (6vi)Bc
-_ ksx [cosel -cos (5.26)
-
4n(x - XI)
From the geometry of Fig. 5.25 the various trigonometric expressions in Eqn (5.26)
can be written as
z - z1
cOsel =
&x - Xd2 + (2 -
x - x1
cose2 = -
J(x - + (z + sz - z1)2
(2 J(x - + (2 + sz -
COS e2 + - = - sin02 = z + sz - 21

The binomial expansion, i.e.

(a + b)" = d + nd-lb + *. . ;

can be used to expand some of the terms, for example

In
where r = d(x -XI)' + (z - ~1)~. this way, the trigonometric expressions given
above can be rewritten as