Page 268 - Aerodynamics for Engineering Students

Finite wing theory 251

and since
VE = w = -'/'Mdz from Eqn (5.32)
47r -3 z-21

(5.59)

This is Prandtl's integral equation for the circulation I? at any section along the span
in terms of all the aerofoil parameters. These will be discussed when Eqn (5.59) is
reduced to a form more amenable to numerical solution. To do this the general series
expression (5.45) for I' is taken:
r = 4s~C~,sinn~
The previous section gives Eqn (5.48):
VCnA, sin ne
W=
sin 8
which substituted in Eqn (5.59) gives together
4sVCAn sin ne V nA, sin ne
2 = V(a - ao) -
cam sin 6
Cancelling V and collecting caX/8s into the single parameter p this equation becomes:

(5.60)

The solution of this equation cannot in general be found analytically for all points
along the span, but only numerically at selected spanwise stations and at each end.

5.6.2 General solution of Prandtl's integral equation

This will be best understood if a particular value of 0, or position along the span, be
taken in Eqn (5.60). Take for example the position z = -0.5~~ which is midway
between the mid-span sections and the tip. From

Then if the value of the parameter p is p1 and the incidence from no lift is (a1 - ~01)
Eqn (5.60) becomes
[
+
pl(q - a01) = A1 sin60" [l + k] 1200 1 + -
sin
AZ
sin 60" s20"]
This is obviously an equation with AI, A2, A3, A4, etc. as the only unknowns.
Other equations in which Al, A2, A3, A4, etc., are the unknowns can be found by
considering other points z along the span, bearing in mind that the value of p and of
(a - ao) may also change from point to point. If it is desired to use, say, four terms in
the series, an equation of the above form must be obtained at each of four values of 6,
noting that normally the values 8 = 0 and T, i.e. the wing-tips, lead to the trivial