SEPARATION OF THE THICKNESS AND THE LIFTING PROBLEMS

At this point of the discussion, the boundary condition (Eq. (4.17)) is defined for a thin wing and is linear. The shape of the wing is then defined by the contours of the upper r]u and lower r), surfaces as shown in Fig. 4.2,

z = t)u{x, y) (4.18a)

z = ri,(x, y) (4.18b)

This wing shape can also be expressed by using a thickness function r/„ and a camber function ijc, such that

r}c = 2(,nu + til) (4.19a)

Vt = 2(riu-Vi) (4.19b)

Therefore, the upper and the lower surfaces of the wing can be specified alternatively by using the local wing thickness and camberline (Fig. 4.2):

Ли = гіс + r, (4.20a)

Vi = Tlc- V, (4.20b)

Now, the linear boundary condition (Eq. (4.17)) should be specified for both the upper and lower wing surfaces,

(4.21&)

The boundary condition at infinity (Eq. (4.2)), for the perturbation potential Ф, now becomes

ІітУФ = 0 (4.21c)

Г-* oo

Since the continuity equation (Eq. (4.11)) as well as the boundary conditions (Eqs. (4.21a-c)) are linear, it is possible to solve three simpler

FIGURE 4.3

Decomposition of the thick cambered wing at an angle of attack problem into three simpler problems.

problems and superimpose the three separate solutions according to Eqs. (4.21a), and (4.21b), as shown schematically in Fig. 4.3. Note that this decomposition of the solution is valid only if the small-disturbance approxima­tion is applied to the wake model as well. These three subproblems are:

1. Symmetric wing with nonzero thickness at zero angle of attack (effect of thickness) о II 4 *> (4.22) with the boundary condition: (4.23)

where + is for the upper and – is for the lower surfaces.

2. Zero thickness, uncambered wing at angle of attack (effect of angle-of – attack)

V2<&2 = 0 (4.24)

0±) = – Q~a (4-25)

3. Zero thickness, cambered wing at zero angle of attack (effect of camber)

V2<&3 = 0 (4.26)

^W,0 ±)=~C – (4-27)

The complete solution for the cambered wing with nonzero thickness at an angle of attack is then

Ф = Ф, + Ф2 + Ф3 (4.28)

Of course, for Eq. (4.28) to be valid all three linear boundary conditions have to be fulfilled at the wing’s projected area on the z = 0 plane.