Integral Equation for Circulation Distribution from Wing Theory

Vortex system of the lifting surface To simplify the problem, it was assumed in Sec. 3-2-1 that the circulation representing the wing was concentrated on one line (lifting-line theory); see Fig. 3-7. This concept is a fairly good approximation for a real wing only when its chord is much smaller than its span (wing of large aspect ratio). When the chord is no longer much smaller than the span, it is necessary to replace the concept of a lifting line by that of a distribution of lifting vortices over the wing chord. Such a continuous vortex distribution over the wing chord was the basis for the skeleton theory (Sec. 24-2). In the preceding section, the free vortices were assumed to be distributed on the surface. By applying this concept of a continuous circulation distribution logically to the wing of finite span, a vortex distribution on the surface results that varies in chord and span direction (lifting surface). An outline of this lifting-surface theory will now be derived. This theory is of practical importance particularly for wings of small aspect ratio, for swept-back and delta wings, and for yawed wings. This vortex distribution on the surface can be taken to be a distribution of singularities in the sense of Sec. 24-2. During the further development of wing theory, instead of vortex distributions, dipole distributions will be used occasionally; see, for example, Prandtl [69 (1936)].

After the fundamental publication of Prandtl on wing theory using vortex distributions, Blenk [69] further developed this theory by extending the two – dimensional Bimbaum-Ackermann theory, Chap. 2 [8], to three dimensions.

The distribution of vortex strength over a given surface can be accomplished in various ways. Let the wing surface have an arbitrary shape, and let a rectangular wing-fixed coordinate system be chosen whose у axis is normal to the incident flow direction.

A first possible approach to the replacement of the wing by a vortex distribution is to cover this surface with two areal vortex distributions kx(x, у) and ky(x, y), as in Fig. 3-15. The former distribution consists of vortex lines parallel to the x axis, the latter of those parallel to the у axis. The ky vortices are of the kind that was previously applied to the two-dimensional wing theory (see Fig. 2-20); the kx vortices, however, resemble the free vortices in the vortex sheet behind the wing (see Fig.- 3-9). Only the ky vortices contribute to the lift of the wing when the incident flow is in the x direction. The vortex distributions kx(x, y) and ky(x, y)

Direction of incident flow

Figure 3-15 Wing with areal vortex distribution. kx — vortex density of vortex lines in the x direction, ky — vortex density of vortex lines in the у direction.

cannot be chosen arbitrarily; rather, they must produce velocities induced by the vortex sheet that satisfy the condition of irrotationality dufdy — dvfdx = 0.

According to Eq. (2-46л), in the vicinity of the vortex sheet (z -> 0) the perturbation velocities are

(3-37)

where the upper sign is valid above, the lower sign below the vortex sheet. Hence

This relationship is called the condition of source-free vortex distribution.

The connection between circulation and rotation (Stokes’s theorem) yields kx ~ cox and ку~Ыу, which is another formulation of the spatial vortex conservation law.

A second possible way to represent a wing by a vortex distribution consists, as suggested by Glauert [23], of replacing the wing by so-called elementary wings of infinitesimal span dy and of chord c(y) (Fig. 3-16). Each elementary wing occupies its special location within the wing boundaries as defined by the wing geometry. The vortex system of each elementary wing consists of a number of vortex lines, one behind the other, parallel to the у axis, which is equivalent to a series arrangement of horseshoe vortices as introduced in Sec. 3-2-1. This representation was given for arbitrary wing planforms by Truckenbrodt [84], among others. In Fig. 3-17, this concept is again demonstrated by the example of a yawed swept-back wing. Note that the free vortices of the individual horseshoe vortices have been drawn separately in this picture, but only for clarity; actually, all of them are located on two parallel lines of distance dy. The circulation distribution density of

Figure 3-16 Substitution of a lifting wing by an elementary wing of span dy and chord c(y).

the elementary wing in direction of the chord (x direction) is k(x) per unit length. In the terminology of Fig. 3-15, к corresponds to ky of this figure. From Eq. (244), it follows that the circulation of a surface element of the elementary wing with span dy and chord dx is

tІГ{х, у) = Jc{x, у) сіх (3-38)

and the total circulation of the bound vortex of the elementary wing at the wing section у becomes

xr

Г(у) = J k(x, y) dx (3-39)

xf

where Xf(y) and xr(y) designate the x coordinates of the front and rear edges of the section, respectively. The same circulation is found in the two free vortices originating at the trailing edge of the elementary wing.

Figuie 3-17 Vortex system of a yawed wing, from [84].

Within the framework of linear wing theory, that is, limitation to small profile camber of the individual wing sections and to small angles of attack, it can be assumed that bound and free vortices of all elementary wings lie in the same plane (,xy plane). This assumption was also made for the profile theory in Sec. 2-4-2.

Equation for the determination of the circulation distribution To establish an equation for the computation of the circulation distribution, an expression first must be developed for the requirement that the lifting surface carrying the vortices is a stream surface, that is, that the normal component of the resultant velocity is equal to zero on this surface. This is the so-called kinematic flow condition. In Fig.

3- 18 a wing cross section у of the lifting surface (skeleton surface) z^(x, y) = z(x, y) is sketched. It is located in a flow field of incident flow velocity [/„ that forms the geometric angle of attack ag(y) = clf + z(y) with the chord.* Here aF is the angle of attack, measured from the x axis, and e(y) is the twist angle.

The kinematic flow condition becomes, in analogy to Eq. (249),

(340)

where w(x, y) is the velocity in the z direction induced by the total vortex system at the point x, у of the xy plane (w > 0 in the direction of the positive z axis). The brackets contain the term describing the angle between the incident flow direction and the skeleton tangent. Equation (340) must be satisfied in all points x, у of the lifting surface.

Furthermore, as a next step, the induced velocity w(x, y) on the lifting surface must be determined from the given vortex distribution k(x, y). To simplify the problem, the induced velocity is computed, however, at the projection of the lifting surface on the xy plane that is identical with the vortex sheet. The induced velocity w(x, y) at an arbitrary point of the xy plane is obtained by first determining the contribution of one horseshoe vortex of one elementary wing (Fig. 3-19). The total induced velocity w(x, y) is then the result of integrating first over one elementary wing in the x direction and consecutively in the у direction over the total number

^Contrary to Sec. 3-2-1, the incident flow velocity is designated now by U„ instead of V.

Figure 3-18 Illustration of the kinematic flow condition of wing theory.

Free vortex pair.

Control point

Figure 3-19 Explanation of the determination of the induced velocity w(x, y) of the horseshoe vortex of an elementary wing.

– <7{x, у; у) – =£ G'<*• V-V"> dy’ £ J ІУ — У )“

(3-41)*

of elementary wings. Execution of this integration yields the following result, as shown in detail in [84]:

with the kernel function

xr(y’)

G{x, у: у’) = Г к(х’, у’) (1 + —=^=) dx’ (ЗА2а)

J У(® — О2 + (у – уп

х/(у’)

X

У, У) — 2 jk(x’, у) dp’ (3-42Ъ)

xf(y)

For the derivation of Eqs. (3-41) and (342), the Biot-Savart theorem must be applied in such a way that the point in which the induced velocity w{x, y) is to be computed is first positioned outside of the vortex sheet (z Ф 0). It is then shifted into the vortex sheet (z-* 0). On the right-hand side of Eq. (341), the first term represents the self-induction of the elementary wing in the section у =y (downwash), whereas the second term represents the external induction of all other elementary wings in the sections у Фу (upwash). Finally, introducing Eq. (341) into the kinematic flow condition Eq. (340) yields

G(x, yy’) is related to k(x, y) through Eq. (342). Equation (343) is an integral equation for the circulation distribution k(x, y) of the lifting surface in which the angle of attack aF and the wing shape zix, y) are given quantities. To satisfy the

Kutta condition for the wing, the vortex density k(x, у) at the trailing edge x = xr(y) must disappear [see Eq. (2-51)]. After having determined the vortex density k(x, y) from Eq. (343), the resultant of the pressure distribution of lower and upper surface at the point x, y, from Eq. (2-53), takes the form

(344)

Here, <7oo = gU’Lfl is the dynamic pressure of the incident flow.

As in the case of the Prandtl wing theory (Sec. 3-2-1), the wing geometry (twist and camber) can be established with Eq. (343) when the wing area and vortex distribution k(x, y) are given quantities. The indirect problem requires quadratures as in Eqs. (342) and (343). When the wing geometry (planform and angle of attack) is given, Eq. (343) produces the vortex distribution on the wing surface. This direct problem leads to an integral equation for the vortex distribution k(x, y), the solution of which poses considerable mathematical difficulties. Approximation methods need to be applied, therefore, which can be laid out in various ways.

A first possibility for obtaining an approximate solution is given by imposing beforehand the vortex distribution k(x, y) in the direction of the wing span y. By selecting for k(y) an expression of m terms, the first of which may, for instance, represent the elliptic distribution, the integral equation Eq. (343) can no longer be satisfied on the whole lifting surface, but only on m sections in the chord direction.

A second possibility for the establishment of approximate solutions consists of imposing beforehand the vortex distribution k(x, y) in the direction of the wing chord x, for example, using the Birnbaum normal distribution of Eq. (2-61). If one selects for k(x) an expression of n terms, then the integral equation can be satisfied only on n lines along the span. Such procedures have been established for n = 1 (first normal distribution) by Weissinger [95], for n — 2 (first and second normal distributions) by Multhopp [62] and Truckenbrodt [84], and for n = 5 by Wagner [91] and also by Kulakowski and Haskell [12].

A third possibility consists of imposing beforehand distributions with m terms over the span and simultaneously distributions with n terms over the chord. In this case the integral equation can be satisfied at (m • n) points suitably distributed over span and chord. Such a procedure was applied by Blenk [69]. More recently, the so-called panel procedure was developed [46] (see Sec. 6-3-1).

Previously, Falkner [14] presented a procedure in which discrete vortices were arranged in both the chord and span directions. Also, the work of Jones [39] and Lan [51] must be mentioned.

Velocity potential The induced velocity field of the vortex system of a wing can also be defined by means of a spatial velocity potential Ф(х, у, z). Here the velocity components induced by the vortex system are

According to Truckenbrodt [84], the potential is

with Cr{x, y, z; y’) = f k(x’, y’) (1 4 -……………… ж ~ – dx’ (347)

J 1 {z – *’)2 -f (y — y’f 4 з2 /

xf(y’)

This expression for the velocity potentials of a lifting surface had been presented earlier in similar form by von Karman [89] and Burgers [69]; see also [84].

The potential is discontinuous at the lifting vortex sheet and in the free vortex sheet behind the wing. Closer investigation shows that it changes abruptly when crossing the. vortex sheet from the upper to the lower surface. This step of the potential above (index u) and below (index Ї) the vortex sheet is given at the lifting surface [xf(y) <x <л:г(у)] by

Фы(4 у) — Ф/ (4 у) = J к(х’, у) dx’ (348а)

Xf(.V)

and the free vortex sheet [x >xr(y)] by

xr(y)

фи(х> У) — фі (4 у) — S к(х’> У) dx’ — г(у) (348Ь)

х/(у)

Very far upstream and very far downstream of the wing, the function Ф, in terms of Г from Eq. (3-39) becomes

Ф ( — oo, y, z) = 0 (349a)

S

(349b)

—s

Equation (349b) represents the two-dimensional potential of the induced velocity field in the yz plane far behind the wing (potential in the Trefftz plane [69]).

Acceleration potential For the treatment of the problem of the lifting surface by means of the Laplace potential equation there is available, besides the method of the velocity potential just discussed, the method of the acceleration potential. This was first published by Prandtl [69 (1936)].

The method of the acceleration potential has been applied to the circular plate by Kinner [44] and to the elliptic plate by Krienes [47].