ConstantDensity Flow; the Lifting Problem
For the purely antisymmetrical case we have
<p{x, y, z) = —<p{x, y, —z) (717)
with corresponding behavior in the velocity and pressure fields. The
principal boundary condition now reads
<Pz = — oc at z = 0± for (ж, y) on S. (718)
ox
We must make allowance for a discontinuity in <p not only on the planform projection S but over the entire wake surface, extending from the trailing edge and between the wingtips all the way to x = +oo on the x, yplane. For this reason, the simplest approach proves to be the use of (710) not as a means of expressing <p itself but the dimensionless жcomponent of the perturbation velocity
и = <pX! (719)
where и is essentially the pressure coefficient, in view of (531). Equation (181) shows us that we are also working with a quantity proportional to the smalldisturbance acceleration potential, and this is the starting point adopted by some authors for the development of subsonic lifting wing theory.
Obviously, и is a solution of Laplace’s equation, since the operation of differentiation with respect to x can be interchanged with V2 in (79). We may therefore write
(720)
We specify for the moment that S’ encompasses both wing and wake, since the derivation of (228) called for integration over all surfaces that are sources of disturbance and made no allowance for circulation around any closed curve in the flow external to the boundary. It is an easy matter to show, however, that the choice of и as dependent variable causes the first term in the (720) brackets to vanish except on S and the second term to vanish altogether. Because there can be no pressure jump except through a solid surface, и is continuous through z = 0 on the wake. But
are equal and opposite on top and bottom everywhere over S, so the firstterm contributions remain uncanceled only on S. There и jumps by an amount
У(хі, V) = uu — m. (722)
(By antisymmetry, щ = —uu.)
As regards the second bracketed term in (720), the condition of irrota – tionality reveals that
du _ du _ dw dn dz і dxi
on top and one finds that
on the bottom of S’. Both w and its derivatives are continuous through all of S’, and therefore the upper and lower integrations cancel throughout. One is left with
s
We have inserted (721) and (722) here, along with the fact that
A=+_*_____
dz d(z1 — 2)
d_ dz
when applied to a quantity which is a function of these two variables only in the combination (2 — 21).
The modification of (724) into a form suitable for solving liftingwing problems can be carried out in several ways. Perhaps the most direct is to observe that nearly always w(x, y, 0) is known over S, and (724) should therefore be manipulated into an expression for this quantity. This we do by noting that и = рх and w = so that
w(x, y, z) = u(x0, y, 2) dx0, (726)
where account has been taken that <p(— ж, у, z) — 0. Inserting (726) into (724) and interchanging orders of differentiation and integration, we get
w(x, y, z) = — ^.JjУ(хь Vi) s
_____________ dx0____________
V (x0 — Zl)2 + (у — У1)2 + 22′
When deriving (728b), an integration by parts* on yx is carried out at finite z. The singularity encountered as z —> 0 is then similar to the one in upwash calculation at a twodimensional vortex sheet and can be handled by the wellknown Cauchy principal value. The integral in (728a) containing 7(xj, уi) itself is, however, the more direct and directly useful form. In the process of arriving at it, we find ourselves confronted with the following steps:
dx0
V (x0 — Xl)2 + (y — Vi)2 + Z2
For г = 0 all terms here will vanish formally except the one arising from the 2derivative of the numerator which will give a nonintegrable singularity of the form 1 /{y — уi)2. It is precisely with such limits, however, that Mangler’s study of improper integrals [Mangier (1951)] is concerned. Indeed, if we examine Eqs. (33) and (34) of his paper, replacing his £ with our yi, we observe that our ^integral should be evaluated in accordance
with Mangler’s principalvalue technique and thereupon assumes a perfectly reasonable, finite value. The result implies, of course, that the selfinduced normal velocity on a vortex sheet should not be infinite if it is calculated properly.
, if an indefinite integral can be found for the integrand, the answer is obtained simply by inserting the limits y = a and yx = b, provided any logarithm of (y — yi) that appears is interpreted as In г/ — yx. The validity of Mangler’s principal value depends on the condition that the integrand, prior to letting z —* 0, be a solution of the twodimensional Laplace equation. It is clear that this is true, in the present case, of the function that causes the singularity in the ^integration of (729), since
Equation (732) provides confirmation for (535).
The question of exact or approximate solution of (728) is deferred to later sections. We note here that, when the angle of attack a and camber ordinates h(x, y) of the wing are given and the load distribution is required, (728) is a singular double integral equation for the unknown 7. Thus the problem is much more difficult mathematically than the corresponding thickness problem embodied in (714)(715). On the other hand, when the loading is given and the shape of the wing to support it desired, the potential and upwash distributions are available by fairly straightforward integrations from (731) and (728), respectively. Finally, the thickness
shape g(x, y) to generate a desired symmetrical pressure distribution must be determined by solving the rather complicated integral equation which results from гdifferentiation of (714).
We finish this section with some further discussion of the lift, drag, and nature of the wake. From (531), (719), and (722), we see that the difference in pressure coefficient across the wing is
. Cvi – CPu = 27. (733)
Using the definition of Cp,
Vi — Vu = PmUiy. (734)
Because the surface slopes are everywhere small, this is also essentially the load per unit plan area exerted on the wing in the positive zdirection. Since (734) is reminiscent of the Kutta formula (2157), we note that 7 can be interpreted as a circulation. As shown in Fig. 73, let the circulation about the positive ^/direction be computed around a small rectangular box C of length dx in a chordwise cross section of the wing. Since the contributions of the vertical sides cancel, except for terms of higher order in dx, we find that the circulation around C is
UK[ 1 + uu] dx — f7„[l + ui] dx = U«,(uu — щ) dx = f7„7 dx. (735)
Hence U„У(х, y) is the spanwise component, per unit chordwise distance, of the circulation bound to the wing in the vicinity of point (x, y).
U^oll + Uu] Fig. 73. Interpretation of 7 by determining circulation around a circuit of length dx along the wing chord. 
The lift per unit span at station у on the wing is l(y) = f [Pi — Pu] dx = pxUl ( У(х, y) dx = pxU^T(y), (736)
./chord./chord
Г being the total bound circulation. If the wing tips are placed at у — ±5/2, the total lift becomes
L = Ґ12 l(y) dy = pxt/o, fbl2 Г(у) dy. (737)
JЫ 2 Jbl 2
The total pitching moment, pitching moment per unit span about an
arbitrary axis, rolling moment, or any other desired quantity related to the loading may be constructed by an appropriate integration of (734).
Unlike a twodimensional airfoil, the finite lifting wing does experience a downstream force (drag due to lift or “induced drag,” Zh, sometimes also called “vortex drag, ” see Chapter 9) in a subsonic inviscid flow. An easy way to compute this resistance is by examining the wake at points remote behind the trailing edge. In fact, if we observe the wing moving at speed Ux through the fluid at rest, we note that an amount of mechanical work DiUx is done on the fluid per unit time. Since the fluid is nondissipative and can store energy in kinetic form only, this work must ultimately show up as the value of T (cf. 211) contained in a length Ux of the distant wake. The nature of this wake we determine by finding its disturbance velocity potential. Over the wing region S, the ^discontinuity is calculated, as in (726), to be
A<p(x, y) = <p(x, y, 0+)
(738)
The last line here follows from the definition of 7, (722), the lower limit — oo being replaced by the coordinate zle of the local leading edge since there is no wdiscontinuity ahead of this point. Beyond the trailing edge on the x, ?/plane, [uu — щ] = 0 in view of the condition of continuity of pressure. Hence
A<£>wake(2/) ~ / 7(^0, y) dx0 — ‘ (7“39)
J chord ^ qo
Since A^wake is independent of x, the wake must consist of a sheet of trailing vortices parallel to x and having a circulation per unit spanwise distance
A jp
Ux b(y) = Ux ± (AVwake) = J • (740)
The complete vortex sheet simulating the lifting surface, as seen from above, is sketched in Fig. 74. It is not difficult to show that the Kutta – Joukowsky condition of smooth flowoff is equivalent to the requirement that the vortex lines turn smoothly into the stream direction as they pass across the trailing edge. At points far downstream the motion produced by the trailing vortices becomes twodimensional in x, zplanes (the so – called “Trefftz plane”). Although the wake is assumed to remain flat in accordance with the smallperturbation hypothesis, some rolling up and downward displacement in fact occurs [Spreiter and Sacks (1951)]. This rolling up can be shown to have influences on the loading that are only of
third order in в and a. For the plane wake, we use (211) and the equality of work and kineticenergy increment to obtain
Di = — <p dS, (741)
*^wake
where Swake comprises the upper and lower surfaces of unit length of the wake, as seen by an observer at rest in the fluid. Clearly, dS = dy, while
rbl 2
д<Р = = 1 і jTP dyi
дП <Рг 2tU„ Jb/2 dy! (y — yi)
from the properties of infinite vortex lines. Using (742) and (739) in (741),

A more symmetrical form of (743) can be constructed by partial integration with respect to y. In the process, we make use of the Cauchy principal value operation at у = y and note the fact that Г(±Ь/2) = 0 in view of the continuous dropping of load to zero at the wing tips.
It is interesting that (737) and (744) imply the wellknown result that minimum induced drag for a given lift is achieved, independently of the
Г = r(0)Vl – yz/(b/2)2. 
(745a) 
Thus, if we represent the circulation as a Fourier sine series 

Г = UJ> ^ An sin пв, 
(745b) 
where 

b a 2 cos в = у, 
(745c) 
we find from (737) 

L = їряиї, Ь*Аі. 
(746) 
However, (743) or (744) yields 

n p. Ulb2 sr’ л 2 D г 7Г g 7іАП‘ 
(747) 
details of camber and planform shape, by elliptic spanwise load distribution 
n = l 
7 4 LiftingLine Theory
The first rational attempt at predicting loads on subsonic, threedimensional wings was a method due to Prandtl and his collaborators, which was especially adapted in an approximate way to the large aspectratio, unswept planforms prevalent during the early twentieth century. Although our approach is not the classical one, we wish to demonstrate here how the liftingline approximation follows naturally from an application of the method of matched asymptotic expansions. Let us consider a wing of the sort pictured in Fig. 74, and introduce a second small parameter e.4, which is inversely proportional to the aspect ratio. Then if we write the local chord as
c(y) = eAc(y) (749)
and examine the matched inner and outer solutions associated with the process tA —> 0 at fixed span b, we shall be generating a consistent high aspectratio theory. As far as the thickness ratio, angle of attack, and camber are concerned, we assert that they are small at the outset and
remain unchanged as eA —> 0. Our starting point is therefore the problem embodied in (529)(530), with M = 0; we study the consequences of superimposing a second limit on the situation which they describe.
Placing the wing as close as possible to the уaxis in the x, уplane, we define new independent variables
Fig. 75. Coefficients of lift plotted vs. angle of attack a and total drag coefficient Cd for several rectangular wings (values of A indicated on the figures). In parts (b) and (d), the same data are adjusted to a reference A = 5 by formulas based on liftingline theory and elliptic loading. [Adapted from Prandtl, Wiesels – berger, and Betz (1921).]