Page 279 - Aerodynamics for Engineering Students

262 Aerodynamics for Engineering Students

Let the velocity potential associated with the perturbation velocities be denoted by
9'. For slender-wing theory cp' corresponds to the two-dimensional potential flow
around the spanwise wing-section, so that

(5.71)

Thus for an infinitely thin uncambered wing this is the flow around a two-dimen-
sional flat plate which is perpendicular to the oncoming flow component U, sin a.
The solution to this problem can be readily obtained by means of the potential flow
theory described above in Chapter 3. On the surface of the plate the velocity potential
is given by

cp' = &Urn ~ina.\/(b/2)~ - z2 (5.72)

where the plus and minus signs correspond to the upper and lower surfaces respect-
ively.
As previously with thin wing theory, see Eqn (4.103) for example, the coefficient of
pressure depends only on u' = aCp'/ax. x does not appear in Eqn (5.71), but it does
appear in parametric form in Eqn (5.72) through the variation of the wing-section
width b.

Example 5.5 Consider the slender delta wing shown in Fig. 5.39. Obtain expressions for the
coefficients of lift and drag using slender-wing theory.
From Eqn (5.72) assuming that b varies with x
I =-=*Umsina b db (5.73)
a(p'
ax 2 &zGdx
From the Bernoulli equation the surface pressure is given by
1
P = PO - 7 P(Um + u' + v' + wy N Po;, - pumu' + O(#)

So the pressure difference acting on the wing is given by
sina b db
Ap=pu:T&E-&Tdx

The lift is obtained by integrating Ap over the wing surface and resolving perpendicularly to
the freestream. Thus, changing variables to 5 = 2z/b, the lift is given by

(5.74)

Evaluating the inner integral first

Therefore Eqn (5.74) becomes

L=-sinacosapUil'b$dx (5.75)
K
2