Results of the Teardrop Theory. for Wings of Finite Span

Rectangular wing of finite span In the simple case of a rectangular wing c(y) — c of constant profile over the span, the contour is represented by z(x, у) = z{x) for —s<y <s. Hence, introduction into Eqs. (3-176) and (3-175д) and integration over у yield

V(* — x’f + (e – F yf

V(® — a0* + {a — yf

with

Uoo % J 8x ^ У (a: — a;/)3 + *s x

It is easily verified that the quantity Ли is negative in general. This means that the perturbation velocities on the contour of a finite wing are reduced in comparison with those on the infinitely long wing. For the wing middle (root) section, у — 0, the perturbation velocity becomes

(3-181)

For Л — 0 this reduces to и = 0.

On the parabola profile Z — 25X(l —X) of thickness ratio 5 = tjc, the

maximum (perturbation) velocity at station x = c/2 of the middle section у = 0 is, from Eqs. (3-179) and (3-180),

^ = -5/lsmh-‘ () (3-182)[23]

Uoo 77 J

Thus, for the plane case, Д-*°°, it follows that umzx pl/£/oo =48/л, in agreement with Eq. (2-97).

Results for the rectangular and the parabolic profiles are given in Fig. 3-71. Figure 3-7lor shows the maximum perturbation velocity, which lies at X= 0.5, as a function of the aspect ratio for the two sections 77= 0 and 77= 0.5. In Fig. 3-7lb the maximum perturbation velocities are depicted for various aspect ratios over the span coordinate. In conclusion, it should be stated that the maximum perturbation velocity on the wing of finite span becomes noticeably smaller than on the wing of infinite span only for aspect ratios A < 2.

Elliptic wing The theory of Sec. 3-6-2 for the computation of the perturbation velocities and the above example for the rectangular wing were based on approximate methods valid for small thickness ratios 5. One example of an exact solution will now be given.

A wing of elliptic planform and elliptic profile as in Fig. 3-72 is a general ellipsoid of which the two axial ratios are very different. Let au b, сг be the three semiaxes of the ellipsoid. Then

б = – r = г – (3-183Д)

Cr О і

Л = ¥- = -— (3-1836)

А л a. x

Figure 3-71 Maximum perturbation velocities on rectangular wings with parabolic profile at zero lift. For infinite spantrmaXp] = (4/-7г)5 £/«, at station x/c = X = 0.5. (c) Dependency on aspect ratio.1. (b) Dependency on span coordinate ц =yjs.

Figure 3-72 The geometry of the elliptic wing with elliptic profile (general ellipsoid).

For a general ellipsoid, the velocity distribution on the contour can be determined in closed form. The pressure distribution on the surface of the ellipsoid in a flow parallel to the x axis is, from Maruhn, Chap. 5 [40],

where cp = (p — p «>)/(р/2)£/£ is the dimensionless pressure coefficient and A — Афіїаі, Ci(bi), a quantity that depends on the two axis ratios of the ellipsoid.

Equation (3-184) demonstrates that the pressure minimum and thus the velocity maximum lie at x = 0. This velocity maximum is constant along the у axis. From Eq. (3-184), Cpmjn = 1 A2 = 1 > with Umzx — Um "fwmax

being the maximum velocity on the contour. Hence, the maximum perturbation velocity becomes

^=A(8,A)-i (3-185)

h oc

In Fig. 3-73, the ratio wmax/wmaxpl is plotted against the aspect ratio A for various thickness ratios 5, where umaxpl =5(7*. The curve 5 ->0 represents linear

theory. The curves for the other values of 5 show the deviations of the exact solutions from linear theory.

Swept-back wing Another example is the swept-back wing of constant chord. The wing of infinite span (see Fig. 3-74) is considered first. Its sweep-back angle is ip and the profile z(x, у) = z(xr) is constant over the span, where xr is the x coordinate of the middle (root) section. The wing sections at large distance from the plane of symmetry are always in quasi-two-dimensional flow. Its velocity distribution can be determined by assuming an incident flow normal to the leading edge of the magnitude СЛ» cos </>. The result is a perturbation velocity in the x direction that, for the swept-back wing, is smaller by the factor cos than that for the unswept wing (plane problem):

Uy(y °°) = Mpi cos y

Now the velocity distribution on the middle section is to be computed. From

The integration requires special caution [see footnote to Eq. (3-175)]. The result for the middle section is

(3-187)

This relationship was first published by Neumark [65]. The first term represents the velocity distribution on a wing section far away from the wing plane of symmetry

Figure 3-74 Geometry of the swept-back wing of infinite scan.

as in Eq. (3-186). The second term represents the change in velocity distribution caused by wing folding. In the case of backward sweepback (tp > 0), the perturbation velocity in the front part of the middle section is reduced, and in the rear part of the middle section it is increased.

The above equation has been extended to generalized parabolic profiles. The result is presented in Fig. 3-75 for profiles with relative thickness positions Xt = xtjc = 0.2, 0.3, and 0.5. The curves for the sweepback angles v? = —45°, 0°, and +45° show a very considerable influence of sweepback on the velocity distribution over the middle section. The maximum perturbation velocities are shown once more separately in Fig. 3-76 over the sweepback angle.

Figure 3-76 Maximum perturbation velocity at mid­dle (root) section of swept-back wings of constant chord and infinite span vs. sweepback angle <p; see Fig. 3-75.

Figure 3-77 Velocity distribution of swept-back wing of constant chord, with aspect ratio -1 = 2.0 and sweepback angle y? = 53° at zero lift for several sections along the span, according to Neuxnark. Wing profile: parabolic profile Xt = 0.5. ы^тахСV = “) = (4/tt)(6 cos = maximum perturbation velocity of swept-back wing of infinite span at section у — «.

For a swept-back wing of constant chord and finite span, corresponding computations have been made by Neumark [65]. The velocity distribution м of a wing of aspect ratio /1= 2 and sweepback angle у = 53° is illustrated in Fig. 3-77 for various sections along the span. It is related to the maximum perturbation velocity of the swept-back wing of infinite span at a section far away from the wing root [Eq. (3-186)]. For the same wing, the lines of constant velocity (isobars) are drawn on the wing planform in Fig. 3-78. This figure demonstrates particularly well that, as a result of the sweepback, the maximum perturbation velocity increases

Figure 3-78 Isobars of a swept-back wing of aspect ratio л = 2, with sweepback angle |*э= 530 at zero lift, from [65]. Curves u{x, y)/u<Pttaax(y= «) = const; see Fig. 3-77.

considerably in the vicinity of the wing middle (root) section, and that the velocity maximum at this middle section has shifted far back.

Investigations on the pressure distribution over the middle section of a lifting swept-back wing of infinite span have been conducted by Kiichemann and Weber [48], and those on a swept-back wing with an arbitrary symmetric profile by Weber [923.