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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM 1586
REMARK ON THE THEORY OF LIFTING SURFACES*
By Aldo Muggia
First of all, a brief synopsis of the Weissinger method, as it applies
to a rectangular wing, is set forth, in order to show how lifting surface
theory is applied in this simple case and to show that his idealization
of the vortex system is justifiable in this particular instance. By
building on this framework and merely adding a few approximations and
unrestrictive understandings, it is demonstrated how the same sort of
vortex system can be devised, and can find sanction, for the treatment
of the lift problem presented by any thin wing of arbitrary plan form.
1. To begin with, let attention be directed to the aerodynamics of
a lifting surface (which is a suitable idealization of an actual wing)
having the simple physical property that it departs but slightly from the
flat surface S, which is the projection of the lifting surface on the
xy-plane. Furthermore, this surface is to be considered immersed in an
incompressible fluid of density p and to have a free-stream pressure and
velocity denoted by pm and V,, respectively. This impinging stream
is assumed to be directed along and have the same sense as the positive
x-axis. In the right-handed Cartesian coordinate system employed here
(see fig. 1) it will be assumed that the z-axis is in the direction of
the vertically downward pointing vector, as illustrated.
On the basis of the above-stated hypotheses, it is legitimate to
assume that the perturbations to the free-stream uniform velocity V,
that are produced by the presence of the lifting surface, will be small.
Consequently, it follows that the local pressure p will be an harmonic
function (a solution to Laplace's equation) of the position-coordinates x,
y, z, and, in addition, the overpressure at such a point will be dependent
upon the potential (p describing the behavior of the local incremental
velocities, through means of the relation
p p = -pV -
*"Sulla teoria delle superfici portanti." Atti della Accademia delle
Scienze di Torino, vol. 87, 1952-19553. Introduced by Carlo Ferrari, Active
National Member, at the Session of 15 May 19535.
NACA TM 1586
The pressure jump occasioned by passage from the underside to the
topside of the surface will be a certain unknown function of the points
of S; call it f(x,y). The value of this function must become zero along
the edge of the surface S, and thus one may write that the overpressure
is given by
p(xyz) p 1 f(x',y')z dx'dy'
r = ((x x')2 + (y y')2 + z2
Upon invoking the stipulation that one must have Cq = 0 at an
infinite distance upstream from the surface, it is seen that the sought
potential will have the form
qp(x,y,z) = -/4npV f dx
^ilpVoo -m <
Now let it be assumed, for convenience' sake, that the equation
denoting the leading edge of the lifting surface is to be taken as
x = xl(y) and that for the trailing edge as x = x2(y), while, likewise,
as is customary, the semispan is to be denoted by b. Then, upon
carrying through a few obvious transformations it is possible to rewrite
the expression giving the perturbation potential as
4 n P V. f-
f oC+m dx x2 Y')
2x(y') r' x (y')
Furthermore, let the following conventions be hypothesized:
Take S' to be the wake region of the xy-plane; ie., the region
which lies downstream of the S region and lying in between the two
straight boundaries y = tb.
NACA TM 1586 5
Then make the definitions that on the wing region S
m(x,y) = lf f(x,y)dx
while on the wake region S'
m(y) = 1 f(x,y)dx
Consequently, it may be seen that the perturbation potential is now
T (., Y; z m dx'dy'
p(x,y,z) = -
which merely says that the sought value of qp is the potential which
describes the behavior of a distribution of doublets of strength m
per unit area in the S + S' region.
This system of doublets may be replaced by a system of vortices,
spread over the same S + S' region, by having recourse to the equiva-
lency principle between doublets and vortices. The local strength of
this equivalent vortex distribution, per unit distance, will have x- and
y-components given by the following expressions:
In the S region,
x = = L
y pPV xl(y) a
= -m =f(xy)
Yy ax PV'
while in the S' region,
=, k x2(y)
7x =y P V" ) fa
NACA TM 1386
The above-described system of vortices will induce, at any arbitrary
general point of the S region, a certain velocity, the vertical component
of which will be given by the integral
v= =41 1 f +-L -+- dxdy+
S4xpVo f s by' y yI x x'
Now this vertical component of the induced velocity must be of such a
magnitude that = holds true, in order that the velocity vector
representing the total flow at the point in question shall be tangent
to the wing surface. Thus one now has obtained an integrodifferential
equation to work with for the determination of the unknown function,
2. The exact solution of this equation will, however, entail rather
formidable difficulties, and for this reason it is best to fall back
upon a much simpler approximate procedure for attaining the desired
result. To this end, let the situation in regard to the simple rectan-
gular wing be examined first of all refss. 1 and 2). In this case one
xi(y) = -2 and x2(y) +'
The angle of attack for any profile section situated at a spanwise dis-
tance y from the center line of the wing may be denoted by a(y), where
this symbol is meant to denote the true aerodynamic angle of attack of
the profile in question, measured from the angle of attack for zero lift.
Thus it follows that the governing integrodifferential equation may be
1/2 1+ x1/2 + x
a(y)V. = 0 dx= vz dx
Z J-a/2 Ix x x J- /2 1- x
r2/ x F2
= 1 -2 CX 1 (1 + -r dx'dy'
a 2pVo 1 11/2 (2 x -
Now make the approximation that the value of Ix x'l, that enters
into the expression for r, is to be replaced by its average, which is
simply 1/2 in this case. Making this substitution, and changing the
NACA TM 1586 5
order of integration will result in the following simplified form for
the integrodifferential equation:
c.(y)V. = 1 b d .AL[ d (12 +26y
a2t -b dy' y-/' Q Y y' 2
r(y') = f(x',y')x' = 2 ly7 (x',y')dx'
is the circulation function for the velocity distribution around the
profile that is situated at the spanwise location denoted by y'.
The above-derived approximate equation equates the vertical velocity
component a(y)Vg to the induced velocity produced at the point (,y,o)
by action of a special system of vortices, which may be considered as
derived from the actual distribution of vortices by concentration of all
the bound vortices along the quarter-chord line.
5. The line of reasoning followed in section 2 is only valid for
the case of rectangular wings, but the result obtained is known to hold
true even for other cases, as has already been pointed out by Weissinger.
The wider generality of this result may be established upon the basis
of the following considerations:
Let it be assumed that the bound vortices are concentrated along a
line denoted by x = xo(y), as illustrated in figure 2. The precise way
in which this line is to be selected will be explained in full later on.
It may be remarked here, however, that the acceptance of this represen-
tation for the bound vortices is equivalent to replacement of the actual
distribution of doublets in the S region by means of a special kind of
doublet system, consisting of null doublets everywhere upstream of the
x = xg(y) line and a distribution of doublets of strength (per unit
area) described by means of the m(y) function throughout the S1 region
(and in the S' region as well), where the Si region is that area of
the wing which lies downstream of the x = xO(y) dividing line. The
potential which describes this new redistribution of doublets is express-
qp*(x,y,z) =- -- m(y)dx'dy'
6 NACA TM 1386
and thus the error in this approximate expression for the potential of
the flow is given by the difference in the two potentials, i.e., this
fx.y' zI -Z.Iq-1xy, 4 f / C4 r-3 K frx(,y' dx +
No-//' let2^'1 ttentio e s u
-nJ'.',, J x ly'l
veta odxay')t ae(7 Xhic e friIY dx' +
iJIIL.V-. X YJ I
4 .t o. io I F. f m 1.1c
t- Y 2- + w X, In fx'
Sr .*X' I' ly z [ X ,'- I
[( y') J
Now let attention be focused upon that region of space which is
composed of all points which have quite small absolute values for the
vertical coordinate |zi and which when projected upon the xy-plane
fall within the S region; let this portion of space be labeled the
S control volume. For points within E, therefore, one may replace
the cp function with the cp* approximation land thus it will be per-
missible to substitute C-- for 2 on the surface of the wing) pro-
kz 6z /
vided the quantity standing within the curlicue brackets is of small
Further, it is to be noted that for wings with sufficiently large
spans, the value of j(x x')2 + (y y')2 + does not vary to any
marked degree as one ranges over the values of x' of interest, provided
the distances V(y y') + z remain large enough, while on the other
hand, if the distances V(y y')2 + z are small, then the value
that x x'| takes on in the E control zone can be represented to
good approximation by use of its average value I I(y) where the chord
distribution function I(y) is defined as
1(y) = x2y) x1(Y)
NACA TM 1386
Thus, in analogy to what was done in the case of the rectangular
wing, it will be legitimate to make the approximation that
V(x x)2 + (y ')2 + z2 V xo 2 (y j2+ )2 z2
provided the point with coordinates (x,y,z) is so situated that it makes
ix xO(y) = 1(y)
If this is true, it follows, in consequence, that
X2(y') x2 (y1) f7y
x2(y') (x',y')(x x')dx' x xO(y'] f(x',y')dx
x (y') (y')
from which one obtains the desired definition for x0(y') as
1x2 x2f(x',y')dx' 7 (x',y')dx'
The interpretation of the relationship just deduced is as follows:
The proper xo(y') abscissa coordinate to choose at each profile section
through the wing is the one which corresponds to the location of the
barycenter of the circulation for that section. In other words, it is
the barycenter of the moments of the vector forces fi taken about the
points P(x',y'), where i is the unit vector in the direction of the
x-axis and where P is the radius vector out to any arbitrary general
point in the S region at which the circulation-function value is
Thus, to close approximation, one may select the x0 abscissae
values according to the rule
xo(y') x (y') + (y')
8 NACA TM 1586
while, when applying the boundary condition, it will be necessary, in
addition, to make use of the potential values (or the induced velocity
values) which appertain to the locations (x,y,O) for which it is true
x(y) = x0(y) + 1. 1(y) = x1(y) + l(y)
This result, which has been deduced by aid of the above-mentioned
list of specific observations and series of approximations, may be arrived
at by examination of the general equations applying to lifting surfaces.
This result is important, for example, in those cases where one wishes
to obtain the distribution of circulation (and thus of the lift) which
exists out along the span of swept wings refss. 5 and 4).
Translated by R. H. Cramer
Cornell Aeronautical Laboratory, Inc.
NACA TM 1386 9
1. Reissner, E.: Note on the Theory of Lifting Surfaces. Proc. Nat. Acad.
Sci., vol. 55, no. 4, Apr. 1949.
2. Lawrence, H. R.: The Lift Distribution on Low Aspect Ratio Wings at
Subsonic Speeds. Jour. Aero. Sci., vol. 18, no. 10, Oct. 1951,
3. DeYoung, John, and Harper, Charles W.: Theoretical Symmetric Span
Loading at Subsonic Speeds for Wings Having Arbitrary Plan Form.
NACA Rep. 921, 1948.
4. DeYoung, John: Theoretical Antisymmetric Span Loading for Wings of
Arbitrary Plan Form at Subsonic Speeds. NACA Rep. 1056, 1951.
(Supersedes NACA TN 2140.)
NACA TM 1586
Figure 1.- Orientation of coordinate axes and definitions of integration
NACA TM 1386 11
x x (y) X X2 (y)
x= x0 V) x
Figure 2.- Location of the bound vortex line and areas of integration.
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