WING OF FINITE THICKNESS AT ZERO LIFT

3-6-1 Displacement Problem of the Wing

The theory of the wing of infinite span as discussed in the previous sections of this chapter was based on the assumption of a very thin profile (skeleton). For the theory of the wing of finite span, the extension from the skeleton theory to the theory of the inclined wing of finite thickness (profile teardrop) has long been available (Sec. 244). A similar extension for the wing of finite span and finite thickness is still lacking. However, for the wing of finite span and finite thickness of the wing profiles (symmetric profiles), there does exist a computational method that allows the determination of the displacement effect of the wing and thus of the pressure distribution on the surface of such wings, provided that the lift is zero. It represents, therefore, a teardrop theory for wings of finite span; note the publications of Keune [42] and Neumark [65] and compare also [43]. The method of singularities is used in which the body within the flow field is replaced by a system of sources and sinks. The fundamentals of this method have been established in Sec. 2-4-3 for two-dimensional flow and applied to the problem of an airfoil in plane flow.

For the assessment of the effect of compressibility in both two-dimensional and, in particular, three-dimensional flow, it is important to know the maximum perturbation velocity on the wing. The computational procedures, treated in the following sections, for the velocity disturbance on wings of finite span and finite thickness are of significance, therefore, for the aerodynamics of the wing of high subsonic velocities.

3-6-2 Method of Source-Sink Distribution

Source system of the wing of finite span For the computation of the three – dimensional flow field about a slender body resembling a wing of finite span and finite thickness, a distribution of three-dimensional sources and sinks is established in the plane of the surface A (wing planform plane). An area element dx dy carries the source strength

у) = q(x, у) dxdy (3-171)

when q(x, y) designates the source strength per unit area. The source strength q(x, y) must satisfy the so-called closure condition

fj q{x, y)dxdy = 0 (3-172)

U)

in order to form a closed body shape. Compare also the corresponding expressions for the plane case, Eq. (2-92).

Velocity distribution on the wing contour Superposition of the velocity field, produced by the source distribution, with a translational flow of velocity £/«, the direction of which lies in the source plane (Fig. 3-70), produces a closed stream surface that can be interpreted as the contour of the wing of finite thickness; compare again Sec. 24-3. Let u, v, and w be the velocities induced by the source distribution (perturbation velocities) and let z^x, y) = z(x, y) be the shape of the wing contour, symmetric to the xy plane. Then the condition for tangency of the velocity resultant on the entire contour is

» = + «)-J*-+ »(З-ПЗ*)

= A (3-1736)

ax

This is the kinematic flow condition. Since, for slender bodies, the velocities и and v are small compared to the incident flow velocity except for the immediate

vicinity of the leading edge and the wing tips, it is sufficient to work with the simplified form, Eq. (3-173Z?). This is, formally, the same relationship as in the plane case. Both for the kinematic flow condition and for the computation of the pressure distribution on the contour, the velocity components u, v, w are required on the contour. For slender bodies, it is sufficient, however, to compute the velocity components in the wing plane, as in the teardrop theory of plane flow. This simplifies the problem, considerably.

The source distribution of Eq. (3-171) constitutes a three-dimensional source. Thus, the velocity potential of the source distribution q(x’,y’) at a point x, y, z is obtained as

(3-174)

where the integration is performed over the wing area A covered by sources. The corresponding velocity components are found from Eq. (3-45). At a point x, у of

ji,

the wing plane z — 0, they become

u{x, y, 0) v{x, y, 0) ■W (z, y, 0)

The upper sign is valid for z >0, the lower for z <0. Hence, the induced velocities normal to the xy plane are discontinuous across the source layer (wing plane).

Introduction of Eq. (3-175c) into the kinematic flow condition Eq. (3-1736) yields

q(x, y)^2V^~- (3-176)

dX

Consequently, the source strength is proportional to the slope of the contour in the zx plane [see also Eq. (2-906)]. The formulas obtained by properly introducing Eq. (3-176) into Eqs. (3-175a) and (3-175Z?) describe the velocities added at the location of the wing by flow displacement (profile teardrop) of the wing. Presentation of these formulas is omitted. For the wing of infinite span (plane problem), Eq. (2-94) yields

^Because of the singular points of the integrands in Eqs. (3-175a)-(3-175c), integration of Eq. (3-175д) must be conducted first over x! and then over y’. For Eq. (3-175Z?) the reverse order of integration is necessary. If the integration is to be performed in a different order, however, the Cauchy principal value must be taken for the second integration in either equation.

Also, in this case ypl = 0.

It should be stated here that the velocity differentials (perturbation velocities) of wings of finite span in Eqs. (3-175a), (3-1756), and (3-176) are proportional to the profile thickness ratio 5 = tjc, in analogy to the two-dimensional profile theory [Eq. (3-177)]. The above linear theory is sufficiently accurate for all practical purposes up to about 5 = .

The resultant velocity on the contour is

(3-178)

when quadratic terms in и and v are neglected.