The elliptic wing based on the potential theory


Material Information

The elliptic wing based on the potential theory
Series Title:
Physical Description:
45 p. : ill ; 27 cm.
Krienes, Klaus
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics   ( lcsh )
Potential theory (Mathematics)   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Summary: The present report deals with the elliptical wing in straight and angular flow on the basis of the potential theory. Conformably to the theory of first approximation upon which the calculation rests, the known requirements regarding the shape of the surface and its angle of attack must be met. A further condition is that the slope of the surface toward the stream lines must be a continuously differentiable function of the points of the surface. If this is not the case, in a given example (for instance, by aileron deflection or wing dihedral - the latter being of importance in sideslips), the discontinuities must be replaced by suitable rounding off. In general, the calculation of a given elliptic surface requires a series of infinitely many potential functions, the coefficients of which are afforded from linear infinite systems of equations. The expansion is stopped with a certain term, depending upon the degree of accuracy desired. Its effect on the integral quantities, lift and lift moment, is practically negligible. An immediate prediction of the induced drag is ruled out, since it would involve all the coefficients of the infinite number of potential functions. Otherwise, the lift distribution at the wing tips does not approach zero or the downwash becomes infinite, which is due to the fact that the load distribution of the lifting line is developed here by spherical functions (equation (80)) which do not approach zero at the wing tips as do the trigonometric functions employed elsewhere.
Includes bibliographic references (p. 43).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Klaus Krienes.
General Note:
"Report date March 1941."
General Note:
"Translation of "Die elliptische tragfläche auf potentialtheoretischer grundlage." Zietschrift für angewandte Mathematik und Mechanik Vol 20, No. 2, April 1940."

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University of Florida
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Full Text
kA-cW-m71 I

I /I '6




By Klaus Krienes


The present report deals with the elliptical wineg in
straight and angular flow on the basis of the potential
theory. Conformably. to the theory of first approximation
upon which the calculation rests, the k.:nown requirements
regarding the shape of the surface and', its angle of attack
must be met. A further condition is that the slope of the
surface toward the streamlines must be a continuously dif-
ferentiable function of tie points of the surface. If
this is not the case, in a given example (for instance,
by aileron deflection or wing dihedral the latter being
of importance in sideslips), the discontinuities must be
replaced by suitable rounding off. In general, the cal-
culation of a given elliptic surface requires a series of
.infinitely many potential functions, the coefficients of
which are afforded from liLear infinite systems of equa-
tions. The expansion is stopped with a certain term, de-
pending upon the degree of accuracy desired. Its effect
on ti-e integral quantities, lift and lift moment, is prac-
tically negligible. -.n immediate prediction of the in-
duced dra,. is ruled out, since it would involve all the
coefficients of the infinite number of potential functions.
Otherwise, the lift distribution at the wing tips does not
approach zero or the do'.-awash becomes infinite, which is
due to the fact tiiat the load distribution of the lifting
line is developed here by spherical functions (equation
(80)) which do not approach zero at the wing tips as do
the trigonometric functions employed elsewhere. Dn the
wing in sideslip, which can be suimarily replaced by a
lifting line, the so-called parasite drag (reference 2)
F = J /(Pu Pob) 7 d d y

*"Die elliptische Tragflgche auf potentialtheoretischer
Grundlage." L.f.aM.M., vol. 20, no. 2, April 1940,
pp. 65-88.

2 NACA Technical Memorandum No. 971

would have to be defined, first and the suction force on
the leading edge subtracted therefrom, where, however,
extrapolations are recommended because of the finite num-
ber of computed coefficients. Even the,resulting pres-
sure distribution is only conditionally valid by few ex-
pansion terms, especially near the wing tips.

It may be mentioned that the computed potential and
downwash functions change on transition to K2 -> 0
into Kinner's functions for the circular wing.

A large portion of the computations were made on the
calculating machine, the accuracy of the slide rule being
insufficient in the calculation of the elliptic integral
for higher n.


This article is intended as a contribution to the
theory of the lifting surface. The aerodynamics of the
elliptic wing in straight and oblique flow are explored
on the basis of the potential theory. The foundation of
the calculation is the linearized theory of the acceler-
ation potential (references 1 and 2) in which all small
quantities of high-er order are disregarded. This affords
the following. simplifications:

1. The z coordinate of every wing point is neglected,
i.e., the variati-on of the potential function
corresponding to-the pressure pump on the sur-
face is situated. on the base ellipsoid (fig. 1).

2. The streamlines, along which .the-convective inte-
gration of the acceleration is effected, are
straight lines parallel to the direction of the

In.the case of the elliptic boundary, solutions of
Laplace's differential equation A I= 0, as products of
Lame's function, are known.

The acceleration potential and Lame's functions.-
Suppose that the elliptic wing is in a stationary parallel
flow with the velocity V. The fluid is homogeneous, in-
compressible, frictionless,ahd not subjected to gravity
and non-vortical outside the lifting surface and the shed-
ding vortex band. Then:

.iJACA Technical Memorandum No. 971 3

= rad p (1)
dt p

The pressure p is expressed by the potential func-
tion \f

p p, = p V2 (2)

where p,._ is tne static pressure at infinity. Function
'j satisfies Laplace's differential equation

L = + + = 0 ( )
-2 2 + 2
ex :y3 oz

and may be visualized as being the result of a superpo-
sition of sources and sinks of intensity 'YXFp, LF, zp)
or every point of the surface.

u(xyz) = ,(xF,yF,z ) L 15 F (4)
o: \R,
where n denotes the direction of the normals of the sur-
face in ( ., yF,2 ) and
R =- (. x-.) + (y y. ) + (, z \)

the distance of the starting, point (x,y,z) from (x-,
yFsZ). The integral can be exchan-ed for one taken ever
the surface of ithe ground ellipse.

\j(T:,y,z) = ,' 3(X.r,y)-. d xx d 7-, (4a)

The mathemriaticl:. treatment of the present case be-
comes possible by the introduction of ellipsoidal-hyper-
boloidal coordinates (reference 3). The semi-axes of the
base ellipsoid being c in direction y and c il K2
in the direction of x (so that 2Kc becomes the distance
of the aerodynamic centers on axis y) the new system of
coordinates is:

X2 + y2 + c 2 C
p a p p 1

xa yE za
+ = c > > _L, > K >v >-K (5)
2 2 2 2 -

2 -2 z2 2
-+ C
K 2 2 -
K p 2 1 p

2TACA Technical Aemorandum No. 971

The solution of the equations leads to:

C- 2_ )2
x 2/= 1-p2 K _K2
46 K2 K

z =vp 1 21 (JI7-7
I K2

, y = cp-K

The surfaces p, 1, and v = const., respectively,
are confocal surfaces of the second degree; p = 1 yields
the elliptic base surface, t = 1 the plane z = 0 out-
side of the elliptic disk. The surface element of the
base elliosoid is

dx dy = c

2 2
S- v _


besides which

c ^1-9 1-^ = /c -y2 -
S- K21 -2 K

is valid for p = 1.

The introduction of new coordinates u, v, w


Y(u) = p2 1(1 + K2),

Y(v) = 1(1 + K2),

Y(w) = 1(1 + K)

for p, m, by means of -eierstrass' Y function
erence 4), appearing in equation (10)

d'(u) = 2 /(Y(u) ei)(Y(u) e2)(7(u) e3)




where the quantities

el = 1(2 K2); eg = (2 K 1); e3 = (1 + K2);

el + e2 + es = 0 (11)

i.e. ,

YNCA Technical Memoracum 7To. 971

..'e e3 = '/ K 2; /e2 e5 = ; ~/ e- = 1

ere posted., gives

S. u)
u = -- --
S2J/p e,/p e2,.p e3

and Lalplsce's equation reads

r. /dp
d p (1-2)
:I %/L47P- -

[Y(v) V(,I)]-. + [v(v) Y(u)] IL-+[Y(u)

Fcsting the solution in the form

'j(u,v,w) = 2(u)E(v)Z(w)

gives, for each of the three functions 3,
ential equation

-Y(v) ) 'V= 0
- Tv w--- 0



Laplace's differ-

d2(u) = [ + Yu)] E(u)
2 A+B' uu


with the separation constar.ts A and B. After posting

' = n(n + 1) and v = -
1 +


and again introducing o, u, o by means of equation (9),
Lame's differential equation (reference 3, vol. I, p. 359)

(p (.-) ( -i(" ) + 2 .2 1))an "

+ ~ +K cv u(n+)> (.., = 0 9


For B = n(n + 1) there are precisely two 2n + 1
values Vm, for which En () has the form (reference .3, p. 360)

S.V % l-v ib5 (a0 -1-'-C + a2 1 2- 3-+ .

CL .


NACA Technical .Memorandum No. 971

Onm(L) = ao0- 1l-E2-E3 + ... is an even polynomial in p
of degree n C 2 -EG. The values vm are dissimi-
lar solutions of an algebraic equation, resulting from
the condition that On (~) is a polynomial. With a view
to calculations later on, Enm(v) is defined as follows:

Enm() =1 41 U I i -) 0 3 m

= AP (-cos 9)2 (sin )On (m) (16a)

by putting

- = sin Cp; = -cos ( ; DA = 1 sin P (17)

The solution

1 nnm (0,p) -= nm ()Enm(V)Enm(p) (18)

achieved with equation (13) can also be represented in
different form (reference 5). For, on denoting the zero
places of the polynomial Onm(j) with Ps it is readily
seen from equations (5) and (6) that

Enm () nm (V )nm(p)

ScoXst y ( t2 y2 2 a
c t= ilyyz xyz -I -- +- c (19)
c- z zzx psakps- 2 K2 Ps Ps2- I

One of the factors contained in the parentheses
is selected and the product II is formed over all zero
places beonging to Onm( ). The zero places ps5 can
also be determined direct by applying Laplace's operator
a + + to the right side of equation (19) and

making the result equal zero. The system of equations for
ps is then as follows:

3 2 33 31 4 s = 1,2,...
+ + + 4 = 0 (20)
Ps 2Ps Ps -1 4 s ps -Pq m= n- E- 2- s3

ZACA Technical Memorandum No. 971

The potential functions in the respective form of
(1i) and (19) are so-called "inner" solutions of the po-
tential equation, not suitable for our purposes, since we
require potential functions which ordinarily disappear at
infinity. These are secured by taking the Lame function
of the second type in the variable p. This is the solu-
tion of Lamn's equation (15a), which, for p -> co as
cost -->0. It has the form

e integral can always be reduced to elliptic in-
tegrals of the first and second categories only. The
aspect of the outer solution of.the potential equation
is tnen as follows: "

-m m (E ()Enm(P) r do 22)
*Jp [Enm (p)32 P2-1 V/pp- K

and equation (19) yields

,m const1 x _xy ( x 2 y 2
"n (xy,z) = -- -ly yZ xyz X +
c z Zx Ps 3 ps

+ z c2' d (23)
2-- / r 2 n / P K 2
From the representation of :he potential function by
(4a) as source-sink superposition on the elliptic disk, it
is ?ppsrenit that the potential functions in th-e plane
z = 0 in the outside zone of the disk must be zero. Hence
1 containing factor z must be taken according to (23)
i.e., only such Lame functions as are of the form (cf. (6)
and (16)):

E m( ) = P2 Mnm(o) (24)

Lame's functions have orthogonality characteristics
similar to those of the spherical functions (reference 3,
vol. 1, pp. 369 and 379):

NACA Technical Memorandum No. 971

\ 'd 1 : .... ^ .O, "if m # t
S n() n- (25)
J- ./ -n if m = t
1 + m t --
P E nm(P.)Enm(iV)Es ()E t()V dpL du

0, if n s
or m t
= or, t (26)
I, if n = s
and m = t
Tl- connectionn with equation (4a) is established by
having recoul-se o the following representation of
1 (R =J(x XF)2 + (y y) + (z zF)2) (reference 3, v. .I,
p. 172)
1--- =. n+1m(,)E () Fm(p)Enm(PF) nmF)nlm() o >
R 2" c nFO m =i


B R/ z =0

) T, I
= 2cm-- nl nm Enm mP )nm(PF)Enm( l --
2c 1-p.y 1-p

leaving the summation over Enm(p) in the form (24) to
be effected. Then by assuming that the source-sink distri-
"bution on the elliptic disk approaches zero on the edge with
the root from the eice distance, i.e., with 1i-F ,
'(xF,yF) can be developed conformably to products of Lame's
"function of the type (24):
(T(x,yF).= E Ast Est .( F)Est(U-) (28)
s=1. t -

after which the foormulation of the integral (4a) gives,
based on the orthogonality of Lame-'s products (cf. (26)),

IACA Technical Memorandum l-o. 971 9

Srp)= A /MM Annmm(1)E m(,)E nm(.V)Fn (P) (29)
2 n=1 m

.cuation (29) proves the potential function (4a) to
be a certain linear combination of the functions Vn
which are analyzed next.

epDresentation of t.he potential function fnm as a
definite inteEral.- If

X = 1 Y(t )-e v'y(-)-en .'" (v)-e, vy 'V (w)-e.
ea.-e ) (ea..-e.y (30)

(eC defined by equation (11)) or if (6) and (3) are taken
into account

/- ','(t) eg x /, .t) -ex y e (t) -e z
,/e2-ev vea-e., c. /3 e-e. ve3 c Ze-e e1 -3 c

it can be shown that

M( ,) E; I (X) m() dt (d)
2n i
+i n 2 i T1

i, e e -e z -

x -:,o: nial [Y(t)] 1 (33)

if loop about tha points of the complex t plane cor-
respondin. to X = 1 is taken as integration pat.h from
Tn toward r 7. Qn(X) satisfies Leeenire's differential
eaua t i on

(1 X2) da x(x) 2X 'n) + n(n+ 1) -n(X) = 0 (34)
dX2 dX

and Enm(t) complies with Lamre's differential equation
(15). Then

[ACA TechniCal Memoraadum No. 971

^n(Wx.) 'Rn (x) n(n + 1)[Y(u) Y(t)] Qn(X) (35)
8 us 3tn

for any two variables each of t, u, v, w; i.e.,
Enm(t) i x) Am + n(n+ l)Y(u)) n(x)
L 8u2
Sm(t) a2 Qn(x) d,(x) Enm(t) (36)
a at2 0)tm (

-2 (nuvw)-[A[ + n(n+ 1)(u)] jm(u,v,w)

1 r m 8 n(X) d Enm(t) ](2
iE (t) (37)
2Tri Ln a t a -t JT

The same holds true if u is replaced by v or w.
The integration path is next so chosen that

EL nm(t 1 ) E tm(t) 1 = 0
at dat -Ti
i.e., '0nm in every one of the three variables u, v, w,
satisfies Lame's equation (15) and is accordingly a third
representation of the potential function defined by (22).
The points X = 1 in plane t are given by

t = v + w + u (X = -1); t = v + w u (X = +1) (38)

On the elliptic disk (p = 1), Y(u) = el; i.e.,

u = -=W (39)

The potential function 41 is cited as an example.
The'sole Lame function of the first type and first degree
equipped with the factor /p2 1 is:

El1() = p- l 1

NACA Technical Memorandum No. 971

According to (22) and (23), respectively, the poten-
tial function then reads, respectively:


: / (p 2 1) /.I 1 o 2 -

Lift and lift mom:ents.- The lift is given by

F = ell Pu Pob)dXdy = pV ff (V'ob 2u)dxdy (0o)


V nm n)Enn) n (1) = -=E m(u), n ( )
Sp= 1 ,: nm 1) /(

S- 1(41)

c 1 -(1) 2 41

Base on the orthogonality of Lame's functions, only
S1 contributes to the total lift

A R V2 F (42)
3 2 ell

-13 furnishes the pitching moment about the y axis

'A = i- c R/ FelI (43)

' rolling moment about the x axis

L c2 V ell (44)

(The negative index refers to the odd functions in y.)

The elliptic wing in straight flow.- Assume the el-
liptic wing in a stream in direction of the positive x
axis with velocity V, l'ow equation (1) enables t'he cal-

12 TACA*Technical Memorandum No. 971

culation of the velocities induced by the pressure poten-
tial \ in space and especially on the lifting surface.
The z component of equation (1) reads in the stationary

Ow aw 8w 1 p 3 8a (4\
(7 + u) -- + v + w = -- (45)
ax y p 8z Pz

Small quantities of higher order are disregarded,

-= V -- (46)
ox 8z

The z component w of the velocity vector w is
hereafter called "downwash" for short. The downwash on
the elliptic surface is obtained by integration of equa-
tion of equation (46) for z = 0 and y < c over x:
w = / d x (47)
V / oz

The calculation of the integral is readily secured by hav-
ing recourse to the representation (32) of the potential

function nm. After formulating n- by differentia-

tion below the integral sign, equation (47) gives
w m 1 / / (t) e x
V 2Tri I / I -
., J e'eg e1 .es e3

.+ Ve (t) e3 y v+ (t) el z m( dtd
/e ei -/e. eg2 W Ie e2 e e3 J (48)9


a3n(X) _d(n(Z) aX dn(X) 1'Y(t) e1 1
az dX z dX e e e~ c

_n(X) d %n(X) X dQn(X) h/7(t) e2n
8x dX 8x dX /Ze ee Ies e3 c

NIACA Technical i4ewdrandum 1To. 971

that is
-" _n(X) 2 (x) ,,Y(t) el (4
0 z a x ./Y(t) e2

which, when inserted in equation .(48) and followed by in-
tegration over x, affords since lim Qn(X) = 0 for
X --- -
wm i r e/"(t e
2 iK n(X).t)dt (50)
V ( t ) e2

4, (X) is given by Ieumann's representation

nY(x) = W, ,i (51)
2/ Z Y
Similarly X(t) is expressed with

(S) 'Y(s) X Y) a, y Y(s) e5 z
Y.s) = -___ c i~ .-+ (52)
.'e2-el,,'e2-e '. e3-el%,'e -e. 6'e1-e2A.'e -e.

whence (50) becomes

w I 1 ,- P (-(s)) dY /':(t)-e1 (5n)
-Wn n is -- t)dsedt (5,)
V 2ni 2 .(t)- (s) ds /Y(t)-e
The integrand has poles at t = s, because X(t) -
Y(s) = 0 and at t = + i'.: where (t) e2 has a
simple zero place. The behavior of the denominator near
the zero place is defined by Taylor expansion. It affords

./p(t) e2 = [t ('r. +iw3)] (- ./p(t)-e) .l+iw + ...

= [t (a + i iu3)] j /e e1 'e2- e + ... (54)

taking into account equation (l), as well as

(t) Y(s) = (t- s)' > +... = (t- s) --+ ... (55)
C st/t=s os

ITP.CA Technical Mermorandum N.o. 971

according to equation (52). Then the integration of
s = v + w + v. as far- a.s s :"v + w -u- corresponding to
Y = -1 to Y = +1 Cequation (38)) gives

1 P Pn(Y(s)) dY
-= Q(I(w + i W'2)) (56)
2 / X(W1 + i 23) Y(s) ads


/ P (Y(s)) l(s e nm(s) d s (57)

it is to be noted that the integrand for p = 1, i.e.,
u = -Wo has the period 2,1i, so that the integration
path can be shifted until s proceeds from i W2 f!
to i c2 + wo (fig. 2), which corresponds to C = -
to + -, when IY(s) -e, = sin E that is, ds =

d_ (AE = 21 K sin3 ) Moreover, let

x = -= n (58)
C'l Ea C

so that, because of
Ve2- e
X(wi + iuj) = ((i+ i+a3)= e2) (59)

and, according to (33):

Enm(u i+ U )= i BEnm()_ (60)

the downwash function on the elliptic disk becomes

w m
E nm )()-i- Pn Qcos + nsine)Mnm()AE d-- (61)
TT 2". ..

The coefficien'tsobtained in the _'olynomial of (
and n are complete elliptic integrals of the first and
second categories. The calculation of the downwash function
in the case of n = 1 is expressed as:

IHACA Technical Memorandum No. 971

E .A) = J/1 As ,

Pi(Y) = Y, (

From (61) follows

w11( = 1 K2 Q' (n) 1
V 2

that is,

Wi' (r ,'7 0( -

M11(C) = 1

x) = In + 1


/ (( cosc + n sin ) -- d C
C cos C


T:-e lifting surface.- The shape of the
given by
z = z(x,y)

sraced i


surface is

The slope of the surface in x direction must agree with
the direction of the flow at the same point, that is,

az(x,y) = w(x,y)
cx V


from which follows, for z = z(x,y)

z(x,y) = w(xy)d


the lower integration limit being arbitrary; we equate it
to zero and add to the value of the integral an arbitrary
function in y:

z(x,y) = w(x,y)dx + g(y)


.-I 1n
L %

The lifting line, the induced drag, and the suction
force.- *ie merely refer to the corresponding chapters of

TNACA Technical :Memorandum No. '971

Kinner's report (reference 2), where these problems are
treated in detail. The results are readily applicable
to the elliptic disk. The lifting line, by which the
lifting surface- is assumed replaced, is obtained by cor--
relating the lift elements through integration parallel
to the x axis:
+xR +xR

anm() = (pu- Pob) dx = 2Sp nm=1 dx

-xR -xR

It affords, for instance,
o 4 c (-ri )
agl(n) = 4 2

a,(1n) 41T P_ V 2 K ) (K2 p) (64

a4(Tl) .,0

The potential function of the second type.- The fore-
going potential functions lend themselves in any way to
linear combination and yield the corresponding linear com-
binations for lift, lift moments, and downwash w. The po-
tential functions dealt with so far afford the aerodynamic
quantities of a correspondingly curved wing by shock-free
entry of flow, that is, at a certain angle of attack where
no flow around the leading edge occurs. The'arbitrary
angle of attack is obtained by superposing a flat elliptic
disk with its flow, where, as is known, the leading edge
is suction edge, that is, the lift density approaches in-
finity. All the potential functions of the first type
approach zero, however, on the disk edge with the root
from the edge distance; hence the task of finding poten-
tial functions that have these qualities. They are
achieved by applying on the potential functions of the
first type at constant x, y, z, and K2 the following
boundary transition (reference 2):

S. d= d 1 m(xy 0,sc)] (65)
n cn- i dc n

:IACA. Technical Memo-randum No. 971

These potential functions on possess the quality
of becoming infinite on the whole border of the disk.
Because of the condition of smooth efflux on the trailing
edge, iz later is necessary to combine the functions

The downwash function of the second type.- In con-
formity with equation (65), the dow-'nwasn functions are

obtained by applying the operator 1 d -cn to equation
cn- 1 dc
(61), while observing the interrelationship:

1 d [n d Pn ) d Pn-i(r)
cn- 1 [ Pn(n)]= n Pn() -i
Sd d d at

The same applies to Pn(Y) an&. (n). Equation
(61) then gives +a
wn mi (r:d qn-i 1 d Pn-(Y) M M( d
V n d 2 ./ *d Y n cos E
a (66)
In dealing w-.ith the second fundamental problem, that
is, in the calculation of a prescribed ving, potential
functions are used, the downwash functions of which are
independent of x on the disk. According to (46), this

implies, since = 0, that

E" 0 (67)

With coefficient bn still to be defined, we put

_n(x,y,z) = L b .nm( ,y,z) 0-n(X,yz)= bn-mn n (x,y,z)
m -m

Correspondingly it is:

Wn = Zb Wn mi w n br wn ml (68a)

Wn(x,y) is designated as downwash function of the poten-

TIACA Technical-Memio-r&'nd No.; 9.71

tial function of the second iy "p .Adccording to (68a)
and. (66), it is: -

w(x,y) bnmEnm n-)
V m dn


2 d Pn1(Y d E
+ d Y 2m n4n(C) C os C



The coefficients bnm are now so defined that wn
is a function of. n only. The first term in (69) already
depends on n only; hence the second term itself, which
usually depends on E also, must be a function of T only.
dn-i is a polynomial in Y with terms of the form
d Y

yn-zp = [ cos c + n sin C] n-2p

hence is a sum of terms of the form

(CosE)n-2p-1a -(sin O) tn-ap-a' Il


For the following arguments it is assumed that Mnm (c)
is even in c ; so that all terms with odd powers of sin

disappear in .the integration from 2 to +Z. But if a
a2 2
is even, (sin c)a = 1 + sum of cos terms. When this is
written in (70), Yn-2p .consists of the following.sum-

(cos e) n-2p-a 1a


and the condition that

n 2p a. n 2 q <. n 2 p


c = and = n + 1, respectively (" is an
2 2 4 '" % integer)

and stipulating that

f (cos)n-q bnmMnm(E) A C d ;
m = 0;




I-ACA Technical Memorandum No. 971

causes all terms containing the powers of cos C, i.e.,
po-'ers of to disappear. The sole nondisappearing sum-
mand cf yr"-p is obtained when n 2 = 0, that is,
when first n is even and, according to (71) n 2p a = 0;
this then reads, according to (70a), 7nn-2p depends
no longer on c, so that it can be put before the integral.
Combining the summands before the integral, which nov has
the same value for every p, there is obtained conformably
to d F1') in (69) for the integral
d Y

d Pn-1(n) 1 i ,bnm E[m a-
/ Cbn m n P A"
d n 2 m Cnmos C

(The integral disappears for odd n.)

Mn-" (C) contains the factor sin c; hence it is odd
in r. Considerations corresponding to the foregoing then
give the condition
Cos Cn-2q-1 m _m/ d C
(cos )- sin e bbnm n- 1m) AEd = 0

= 1,2,..., (T 1); T= n and = n 1i (74)


'he 7-alue of the integral in (69) is other than zero
only if n is odd. Summed up, it affords, by attention
to 2,(K) = Eg+1() = 0 (equation (16)):

1 t -r( ) 1 ( = i* Par- 1i( )
- War+i(TI) = k2r+i r r(T) = 2r
V d V d n

1 d Par(n) d (,4) r d sr- ('r,)
1- w.(ar+ 1)(T) = J2r+i ; War() = lar
- dn d n


NACA Technical Memorandum No. 971

k2r+1 = b2r+1 mEr+i ()

1ar = 1 /^ mr IMr (E) Le d c

a (75)

J2-r+-r =L sinE Z b;m+1 M2r+1(E)AE E
2 T -m r r COS E


bnm and bn-m satisfy equations (73) and (74), re-
spectively. They are a 1 homogeneous equations for the
r, unknown bn and T 1 equations for the T unknown
bn-m, respectively, which can be determined therefrom up
to a common constant factor. The latter is so chosen that

k2r+1 = 'ar = /I K2; isr = Jnr+1 = 1 (75a)


1 (d \'%r(Tl) 1 \r(fl Par-i(n)
-w(r+1f) (;2r ar()

- .n a d T

Lift. lift moments, and the lifting lines of the po-
tential function of the second type.- These quantities are

obtained by the application of the operator 1 cn
cn-1 dc
to the corresponding quantities of the potential factor of
the first type (42, 43, 44, 64). It is pointed out that,
during the differentiation, the real content of the
ellipse Fell emb'-dies the factor c2. Then, bear in
mind that (equations (68), (75), (75a))

gives the lift A = 8 V Tr
2 ell


Z"ACA Technical Memorandum No.. 971

O2 the pitching moment k= 8- aV Fll (8)

*-a the rolling moment L = c -- V F (79)
3 2 i

2 1>
E ;I- 2K2 sin2E d C

and for the lifting lines

az(n) = 4 i Vc 1 K Po(n)
2 0
as(n) = 0
a3(n) = 4 E V2 c l K2 P2(n)

a4(ni) = 0

The second fundamental problem of airfoil theory.-
Thiz involves the calculation of the aerodynamic quanti-
ties of any given elliptic wing by means of the foregoing
potential functions of the first and second types. The
boundary condition to be met is given by
(x.y) = w(^xy)= dx (62a)
8 x V / d

which, differentiated, leads to (46):

w = V (46)
cx z
Posting as a linear combination of potential
functions of the first and second types, leaves

7 = i anm n + n 'n +n + Dn -n (81)
n=- m n n

leaves on the disk (p = 1)
"3,' m am nm na I 6
8- an m since -. n 0 (81a)
op= Z p=. az p=

NACA Technical Memorandum No.:971

where then, however,

a- nm J En m() Enm() d d Fnm(p) (82)
z c p v La( 4 ) P=

Now, if is developed according to the Lame

---n m n n

n. ( m Em(p)= I2 Mnm()

then, because of equations (46), (81a), and (82), based
upon the orthogonality"of Lame's product,

gnm = dFm()1 anm (83)
Ld( p_ 1p=3

This defines the coefficients anm. of the potential func-
tion of the first type. On expanding z.= z( ,n) in a
series of ( and n, the anm 's are determined by a
comparison of coefficients in all terms affected with pow-
ers of | in th-e form of a screen method (reference 2).

From equation (46)

-w S an dx= w(n) (84)
V n,m o z V

is now only a function of i. This residuary condition
is complied with through a suitable linear combination of
the downward function of the second type, which depends
solelyon in. The condition reads

n n ( )_ + Dn w-_n() .= w(T) (85)
n n
This equation is integrated from n = 0 to n, and
then multiplied by P2,-i(,) and P2y(n), respectively -
integrated again from n = -1 to +1, and so yields
through the intermediary of the inter-relations

NACa Technical Memorandum No. 971

+1 d 2 2

a Pr-i ) P2,-n) d 1 = 2(2ac-1)+1 4c-l
- r o

/ 2r(T,) P2 -i(T) d T =

if r = a

if r a

a(2a- 1) r(2r+ i)

two infinite systems of equations for the coefficients
Cn and Dn
----- +1 -/
E C+r+ C 2 C / w(n')dn' P2a-1(n)dn
r=o a(2o-l)-r(2r+l) 4-1 /
-t o a = 1,2,...
,'/1 E i 2
Davy, 1 + 7 Dar = ~ ; / w (r,') dn'P2Y (n ) dn
47Y+1 r= 1 Y(2Y+l)-r(2r-1) V / / r( 1)d 'pav(
-.o 0 = 1,2,...

The outflow condition.- Apart from -the compliance of
the equation systems (86), the coefficients Cn and Dn
must also satisfy the outflow condition, that is, they
must be so chosen that the lift density disappears on the
trailing edge. The potential function of the first type
satisfies this condition without that, since they disap-
pear on the whole edge. The behavior of -'*nm on the
boundary is known with the application of the differentia-
tion process (65) to (41).
[Qnm]-, c Mm() + ((./l-^))

1= f1 X2
a=. /c2 y2-

(The term ((/ p2)) disappears with the root from the
edge distance.) We post, respectively,

Zbnm m() = ()

and bnm qnm(w) = Mn(P) (87)

whence the potential function of the second type becomes

=[n]p= bnm ) c Mn() + ((.1/ )) (88)
P=1 C-y2 -
1- K

Now the functions M n(c) have orthogonality proper-
ties similar to the trigonometric functions, as may be

NACA Technical Memoran.dum. No. 971

proved by (7?3) and (74) if it is remembered that Mn(qP)
can be written in the form

An(c) = (sin p) C3[ao(cos )n-3-1 + ag(cos )n-C-3 + ...]

It is
-- ii1i(qp) Mn(cp) Acp dc = 0, if n- = 2, 4,... (69)

The outflow condition reads, according to (88):

7 Car Ilr(CP) + E0 C.r+i A2r+1(Co) 0 I<2cp< 3_L (90a)

SDr M.-Cr(C) + Z Dar+1 ?-(ar+i)() -0 Di<%< L (90b)
r=i r= i 2

which, after multiplication by Msa-1(c) A 1) and I4-s2y(cp)6Ac

respectively, and integration from .= n T to 3T-, give
2 2
Cg_11ia-_1,_aM + Z- Cr iara-i = 0 a = 1,2,... (92a)
D2V I-a.-Vay + Z D^r+l I-(ar+i) ,-sY = 0 7 = 1,2,... (92b)
whereby 3zT

V/ liy(cp) 1i(CD) Acp dcp (93)

The infinite equation systems (86) together with (92)
then enable the determination of the coefficients of the
potential functions Cn and Dn of the second type.
These coefficients in conjunction with the previously com-
puted coefficients anm of the functions of the first
type make the pr.ed-iction of the aerodynamic quantities of
the given wing possible, as will be illustrated on a model

Note.- The calculation is made on the assumption that
the flow strikes the elliptic disk parallel to the minor
principal axis. The length of the principal axis in y
direction was c, in x direction cl- i3. The calcula-
tion is readily applicable to the case of elliptic wing
in flow parallel to the major principal axis when K2 Is
assumed negative, that is, supposing the aerodynamic cen-
ter on the y axis. The conversion formulas for the el-
liptic integrals with negative K2 are:

ilHCA Technical Memorandum No. 971 25

n TI
dc 1 ____ d- _

J. -I+(- )sin2 c 2- J _C
*o 1--?- S in
> (94)

/ J1+ (- )sin2 dE =E /1-K /l--- sin C dC
'./., 1-K
o *-0

The integrals are as far as the factor before the
integral, the same as for the reciprocal axes ratio.

The flat ellintic disk in straight flow.- Let the
angle of attack be 0o, that is, the elliptic area is
given by

z = -x tan ao = -x Co

Then we have, according to (62),

1- ( x, y) dw -
= o; = 0
V dx

on the disk. According to (83) therefore, the coefficients
anm of the potential function of the first type are all
zero; those of the second type must be computed conform-
ably to (73, 74, 75a), as exemplified here for n = 2 and
n = -3.

For an axis ratio of
'(c)2 4
K2 KK = 2 = 0.96, that is, A = -6.37
5 Fell TT- 2

the complete elliptic integral is:
F = / d = 3.01611;
Jo l 0.96 sin2 T

E = / J1 0.96 sin2 dE = 1.05050
In the case of n = 3 and E1 = 1; Cg = cu = 0

Es() = /1/TD (V2 Ps2) s = 1;2

M3 (E) = K2 sin2 p62

MS ( ) = 2 Ps

NACA' Technical Memo'randum No. 971

and according to (20). it is:

1 1 3 =
PS 2- Ps Ps -1

that is, pi = 0.19792; p32= 0.97008

Condition (73) and (75a) then read:
b33' / (0.96 sin2E- 0.19792)LEdE

+ b3 IT (0.96 sin2 e 0.97008) A eC.d = 0

- r= 1) l K2(0.96-0.19792)-b, 3 1-K (0.96




= 1.317,

= 0.3095

i. e., M3(cp) = 1.627 K2 sin2 c 0.561

For n = 3 and C = C2 = C3 = 1, we find

E3-1() = 1 2 -.. M 1_) 2 -j;

Mv3i () = cos e sin e

Equation (75a) reads:
--1 d3 -1
,1 = -. sin E b3 cos E sin CAC e = b -/ sin2 ACd
Scos e /
"' o
that is,

I = 2.654
b 37 = 2.654


M._3 (p) = -2.E54 cos p sin Cp,

/K 2- = -K cos P (')

The determination of the other necessary functions
proceeds in similar manner. The integrals (93) can be com-
puted by means of the functions ILn(;), being either ellip-
tic or reducible to elementary. The coefficients ly 8 of

IJACA Technical Memorandum No. 971

(92) are herewith known, Calculation of the right side
of (86) yields
+1 Ji -1
/ ,nao0 for a= 1
-ao I d n' P2a-(n)dT=-Co T1 for aa-()=
/ -0 for a >1
-1I -c J-
For Y = 1,2,... the right side of (86) is always
zero, whence no asymmetrical potential functions occur in
y. In the effected calculation, the series (31) of the
potential function was stopped with n = 4; hence eoua-
tions (92) and (86) must be taken for a = 1; t. It gives

2.101 Ci + 1.5217 C0 + 0.527 C4 = 0 (a = 1),
0.2410 C2 + 0.5132 C- + 0.9?24 04 = 0 (a = 2)J

1 2 1 2
C5 C + C2 c + --c- = -3 o (a = 1) I
-i 9 1 (3s)
-1 C3 + 04 = 0 (a = 2)

The direct solution gives

C = 0.53 a0o, C_= -0.7785 ao, C3 = -0.342 cao, C4 =-0.0135a0

and the lift, according to (77), at:

S= 0.568 aox V Fell = 4.55 a.o Fell

that is,
d co

The moment about the y axis is,according to (78):

i, = -G.7785 a x 0.9524 X c A/I K2 R 7 ?2 -
0 3 2 ell

i = -1.98 ao c l Ka 2 ll

the center of pressure is at x = -0.435, that is,
cl :2
at 28.3 percent of the maximum wing chord.

Incidental to the calculation of the induced drag, it
is emphasized that the lift distribution of the lifting

ITACA Technical 1'emorandnm No. 971

line does not disappear when:allowing only for an infinite
number of series terms in (81) at the wing tips.(fig. 3),
and hence must be included as a substitute lift distribu-
tion. In this case the elliptical is most suitable, giv-
ing for the induced drag the wellrknown formula

a Fell
Cwi = Ca ..

For the axes ratio

SK = = 0.75, A= 2.55

the procedure is the same, the quantities bnnm being as-
certained from (73, 74, 75a). This affords the functions
Min(Cp) for the integrals ly 8, wherewith the coefficients
of (92) are known. The solution gives

01 = 0.3741 ao, C2 = -0.6347 ,o, 03 = -0.2347 ?o,

C4 = -0.0138 ao

the lift being

A = 2.99 ao P. V2 Fell and d a- = 2.99

the pitching moment

M = -1.397 ao 1 Fell

and the center of pressure at
= -0 467 or 26.7 percent of maximum wing chord
oYl K2

For I -K 2 = 2' = 3 A =- 0.637

it gives

Ci = 0.124 ao, C2 =-0.5245 mo, C3 = -0.066 mo

C4 = -0.011 CLo

that is,
that is, d a = 0.99

d a.

NACA Technical Memorandum No. 971 29

The center of pressure is situated at

-- --- = -0.584 = 20.8 percent of the chord

For comparison the values for the flat circular disk
are repeated (reference 2):
Sd c = l.E2
S- = 0, A = 1.372
d m.o
center of pressure: z = -0.515, i.e., at 24.3 percent of

T'e calculation method used here permits even a bound-
ary tr-,.nsition to the lifting line (K' = -.1 = 0).
It affords, when two series terms are taken into account,

d ca
Sca = 2 n c.p. at 28.8 percent of chord.
d ao

The latter result corresponds to an elliptic spanwise
load distribution with a center of pressure at 1/4 chord
in each airfoil section (c.g. of a homogeneous semiellipse).
Development of all quantities appearing in the equation
systems vith respect to a affords for small K':

d Ca 2 T
d ao 1 63 i \ 4 7 1 ,2
1 + 3 K + ln 2-
d c4
Then d ca is calculated according to linear wing theory
d ao
w'Lere, as is kno wrn,

1 d Ca 2 n A
Aeff = 0o + Ca --'
f.'. d do A + 2

there appears a marked discrepancy at small A with re-
spect to tnh ve.lues computed in accord with the theory of
th5 lifting surface (fig. 4).

On the other :-and, the agreement with Jeinig's re-
sults is good. (See reference 6.)

Figure 5 shows the center of pressure position plot-
ted against aspect ratio.

The elliptic winz in yaw.- This problem can be treated

NACA Technical Memorandum No. 971

under the same assumptions as the wing in straight flow.
The so-called angle of yaw B is defined in figure 6.

row the streamlines are straights in
z = 0, defined by

y = -(x xR)tan P + 7R

= -x tan 0 + const

From equation (i)

(V cos 0 + u) _- + (-v sin P + v) 8- +
8x ay

the plane


)w 1 P
w a
8z p 82

under simplifying assumptions, there is found

V cos V sin W 8 -=V2
ax 6y d5

But because of (95) it becomes

dw 8w 8w
I--= tan ,
dx ox dy

that is dw-cos P=w-cos 0- -sin w
dx ax dy

Hence eouation (96) can be written in the form

cos V = V
dx 8z

(y = x tan P + const)

The downwash follows from integration along a stream-
line again assumed as a straight line parallel to the di-
rection of flow

w = 1
7 cos

S- dx


The potential function .*. is again assumed as a
linear combination of the potential function nanm, n, _-n'
in the form (81), whence the same forumlas are obtained
for lift and moments.

The downwash function is computed on the basis of an
oblique-angle system of coordinates in the xy plane,
givenaby the ellipse.diame.ter parallel .to the stream direc-
tion ( axis and the related.conjugate diameter -
n axis. Posting



NACA Technical Memorandum No. 971

tan c: = ,/l 2 tan 5,

tl/l 2K sin B
that is, sin !3 =

cos =
003 *.'p = -----

we have the quoted diameters given by

tan c n
y= -

tan. O' + -
and y = ,I respectively (99)
'l K

On tiie disk, i.e., p = 1, it is according to equa-
tions (6) and (17):

= cos y;
cV/l K ./l- nK

Similarly, we post

/ 2 2
= 7 cos (, 0 );

that is,

cos 6 /l- K sin S
- = z- -nTl

S= = = L sin CP

r, = I sin (P q )

/- sin (cos
S^= -0 + (loo)

In this instance the potential function nm con-
formable to (32) is utilized. During the respective dif-
ferentiations and integration along a streamline, the
fact that y = x tan 8 + const should be borne in mind.

Sm1 v'Y(t) e x
m2- -I e iea e3 c

1A (t) e,3 Y
+ L-;e
-e e 3 'e

/Y(t) -Te z
-+ nm(t) d t
Vei eea e/ i e3 -



,'ve find that

a on(X) dCn(Xt)1 1e
82 d X c v/e ea /e e


32 IACA Technical i4embrandum 0o. 971

and _
daQ(x) adQn(x) 1 E f (t) ea l'Y(t) e tan
d x dX c L er/e2- e e3- eg e3- -eg

8 n(x) cn(x)
n3 thus can be expressed again by -- ; posting
8 z dx
the result in (97) and integrating over x gives

wm 1 1 Y(t) -eg
wn 1 1 -
V 27Ti cos -/ e
co2i c J sex eg/ex e3

x 1.. .. nm..(t) dt(101)
JY(t) e Vy(t) e3 tan
veg e lIe e3 /e3 e iez e

With equations (51) and (52) Qn(X) is now replaced
again by

1 F pn(Y(s)) dY
Qn(x) M = I iT ds
.2 X Y(s) as

The integrand has poles at t = s, where X(t) Y(s)= 0
and at t = to, where

I e "'2 t) e e (t) 3e tan = 0
sea eL/ea e3 e3 ei es ee


1Y(tg) e3 = i Ve7 e3 sin qc ; 7(tg)- e3= e3- e3cos q

The denominator at this point is as

t t cos 9O i sin CP tan s +
Li ie e2 e1 e J

The residuum at t = t is therefore

11i.CA Technical Memorandum No. 971

2niQn (- (t ))

nm (t6)

i cos cU + i tan sin yp

= 2n i B- En (- Kcos ng) n (')


with a view to the fact that, according to (100)

a(tp) = sin C. + cos Y = 1
Cl K

and according to equation (33)
Enm('S) = 1 Enm )
Sj eo=- KCOS

For the residuum at t = s, the same holds true as
in the case of straight flow. It affords:

,r 1 iA c
2n i i P( cosc + n sin e) CS- 2s 'iMn
/n cos s yl-"2 tan 0 sin E

m( ) dE


By replacing 1l K2 tan "by tan Vc and posting the
coordinates E and n by means of equation (100), the
values (102) and (103) from (101) lead to

: m )
- / U CO[ S cos(c-4p )+1n.sin(ra V)] Lc de (104)
A- 2 ./ cos(Z+, ,)

For the terms of Pn' containing powers of g, the
denominator cos(C + C%) in the integral disappears, af-
fording complete elliptic integrals of the first and sec-
ond types. To compute the others, numerator and denomina-
tor are expanded -,ith cos(c y)A1 P, which, with

cos(c + og) cos(C ) A2, = cos2 A sin3 C



"NACA Technical Memorandum No. 971

Swnm ('.). = Enm(-K.cOS cpP.)n(n()

1 + 2 nc s A
-4A / Pnl pcos(^ )+ni(c+q)) (-])Mnma)
J_ cos2-1apsina (104a)
The existent integrals are reducible to complete ellip-
tic integrals of the first and second types and one integral
of form

7 sin2 C dC
r cos2 2 A -p sinae Ac

which, as a complete elliptic integral of. the third type,
is reducible to incomplete first and second types (refer-
ence 7.).
(l-K2) sin A sin' C de_
cos / cos32-A20 sin2c A
0 0

= E(-)P() E s F(0) A tan F( T (105)

The calculation of the downwash function for 'the po-
tential function 1 is given as a model example.

E() = IA a M (C) = 1

Pi(Y). = Qj(X) = In .- 1
2 X 1

E' (-K cos cp) =1 K cos =1

Then,-accordin-g .to (1.04a).:

.i.eCA Technical Memorandum No. 971

1 1 1 / 2

+n +H
i 1 :, [i cos('+ n sin( cos + n cos d
S / cos2c-L'-sin2C

Al( = 2. 2 )
S3 / T cos sin2

.he evaluation of the integral Lives

w1 l L )2 p TT

.'he downwash functions of the potential functions of
the second type,- According to (68)

Sn = n n

-,nm 1 d [cn (m ,y,z,c)]
cn-' dc

according to (65). The downwash function wn B',l) is
computed from wn (I(,p) that is,

- bnm Enm(- Kcosy )

+ 1( + Ed (106)
2 LP dY cos(c + (P)

in the same manner.

36 ITACA Technical Memorandum Xo; 971

On the disk we have a- p =0, that is,,on a

streamline (i = const) the downwash function is constant;
hence wn is a function of np only, and we accordingly

put = 0 in -d- P (Y). To find the summand
dY n-1

n n-p Lsin (e + cp )]n-p from Yn-2p

n n- 1
we put T = 7 and. respectively, whence
[sin (C + y )]n-2p =[sin(E + p)]n-ETCsin( +p)]T2-Tp

=[sin(C + cpo)]n'2T(l + sum of cos terms)

Factor cos(e + Cq) can be extracted from the sum,.
thus becoming shorter with respect to the denominator. The
new sum, however, multiplied by [sin(C + q)]1n-2 yields
only terms which either satisfy odd in C or else condi-
tions (73) and (74), respectively, and accordingly disappear.
A proportion other than zero is afforded only by
[sin(E + cp)]n-2r x 1, which is, however, no longer depend-
ent on. exponent, p. In consequence, all n,-,P before
the integral can be combined conformably to

d Pn-() into d Pn- (io ), since the integral for
dY d-o
every p is the same; i.e.,

Urn( T1) 1 a 1 dqn-iflp)
--- = E KbO1 CnP) ---o.--
7 60 m d 7p

+ I s [sine ccosc + cos e sin -

cos C cos r? + sin c sin cp n
xMn(c) ede (107)
cos2 As sin2 C

Hence the downwash functions for the wing in yaw have
the following form

IL~CA Technical Memorandum. No. 971

V wn() d k(B) +dQ-( in) n_() (107a)
d np nTI
1 W'n() = (-'n' d^ V^ ip( (10d_)

w_ = () d .In.d + j n(P) d T(107b)

The second fundamental problem for the case of the
wine in sideslip.- The procedure is the same as in the
case of the wing in horizontal flow, by putting

dw(x,y) V m ()E
7- 1- ^u-co s dw gn m .) m (n) (108)
dx c n m
1 w(.x,y) = cos z(x,y) (108a)
V dx
and y = -x tan ) + const; \ to be given again by
S= a m + Cn + Dn -n (109)
n m a n n n n (
Equation (96a) tien gives

g 1 n I an (110)
S.. .., a

d(,p2 1) =
and with it the coefficients of the potential functions of
the first type. For the determination of the coefficienrts
of the functions of the second type we take (97) in the
following form:
x x
; 'n dx + Zf dx
+ C + Q Dn [ ; cos
Sn / 8z cos n n n cos

i ,/no'm dx 1
w(x,y) a = w() (111)
V n m 5z cos V

The right-hand side is a function of n only because of
the determination of the anm from equation (96a). The
rest of the formula then reads:


NACA Technical .KMeiorandum No. 971

n "n d Ti dnn

-(F) dn-() ( + -) w(
+ n D In + nJn() W(J) (112)

This equation is integrated from n = 0 to n and
then multiplied by Pa- (-n) and 2y(.n) again
from T = -1 to +1. This affords on the tasts -of the
orthogonality characteristics of the spherical functions

7 (Ckr+x aer+i( )+D2r+ 1 2r+i (P( )) 2a._l) 1r (2r +)

+ (C2ai2Ca() + Doaj2(P)) 2---
4a 1
+1 TjP
S / / w(n' )dn'pPa-(np)dn; a = 1,2,... (!13a)

-1 0

rx (Csr k2r () + D3 lar 12r 1) 1

+ (C2+i1 is'+1 (P) + D2sg+, jsy+ij()) 1
I 4+ + I
1. f, (.1i.
S/ w(Ti ') d-n Pay(np) dna; Y = 1,2,.. (113)

-1J /
-i "

The outflow condition.- The formulation of the outflow
condition is prefaced by the following note. The downwash
functions (107) on-the wing in sideslip have the quality of
becoming infinite by n = 1 like Qn- 3) and
d Qn-1 ()
respectively. But in = -1 are marginal
d n ,
points of the elliptic disk in which parallels to the flow
direction touch the ellipse (fig. 6). Accordingly, it is
to be assumed that the so-called "vortex tails" in these
points leave the disk parallel to the direction of the
stream. Hence it is postulated that no flow around the
trailing edge occurs between the points Ti = 1. In
other words:

lACA Technical Memorandum No. 971

SCnlin() + D3n -n'() = 0 2 + < < + 3

This equation, multiplied by

aX (p) A Lp and 2y(cp) LA, respectively

and integrated from 1- + 0 to +-3 + gives the fol-
2 0 2
lowing systems of equations:

Ca-~I ac-1,S1l2-L+ S C2rI r, a-i+ Z Dar.-2r,2 20,-1= 0;
r=1 r= 1
a = 1,2,...

+I WC (W0) ( 0 (114)
D2rI_ ) 7, C2r-12r_1,_2Y+ Z D2r+1U=
r=y-r1,-Y -(rr+1),-Y
V = 1,2,...

= My (c) (iC ) Ac. dQ (114a)

These two eatations ((113) and (114)) make it possible
to determine the coefficients Cn and Dn. Together with
the previously defined ann 's the aerodynamic quantities
of an elliptic wing in yaw can be computed. As to the
equation systems themselves, they form a coupling of the
systems (82) and (92) for the case of straight flow, as is
readily apparent from the similarity of the corresponding
coefficients and which thus affords a first simple mathe-
matical check.

'The flat elliptic wing in yaw.- The calculations are
carried out for the axes ratio

and the angles of yaw

= 150 and P = 30

Again only the potential functions of the second type
conformable to (108) and (110) are required. To simplify
the voluminous paperwork only the functions up to the de-
gree n = 3 are taken, i.e.,

ITACA Technical Memorandum ITo. 971

i, a2 -as, -3 (-i .does not exist)

The five unknown coefficients of these potential func-
tions require five equations. Two -are taken from the con-

dition 1 w(n) = cos P tan so on the disk, equation
(113a) for a = 1, and (ll3b) for Y = 1. For the other
three the outflow condition (equation (114)) with a = 1.2
and Y = 1) is used. The Ln,(qc) are those previously de-
fined in the case of straight flow. The integrals
1) obtain now only the limits 1. + p and + 3 pT
while TIY remain unchanged on account of the periodicity
of the integrand. The values of the incomplete elliptic
integrals of the first and second types necessary for the
determination of the coefficients kn (), in(p), etc.,
were taken from Legendre's tables (reference 8)

For p = 150 it affords

C1 = 0.5204 a0

On = -0.7194 a0 De = 0.0144 ao

C0 = -0.3343 co D1 = -0.0065 ao

i.e. ,

A = 4.16 ao V. F
0 2 ell-1
M = -1.83 a0 c 1 K2 2 V2
o a Tell
2 ell

For P = 30 it is:

0C = 0.407 ao

C2 = -0.562 .o D = 0.0277 a.0

C3 = -0.265 a0Q D3 = -0.0124 a

A = 3.26 ao V2 Fell
M=-1.43 a V2 e

L = 0.074 a. cE a
2 ell

NACA Technical Memorandum No. 971

The new additive moment L is positive according to
the above calculations, which is synonymous with the fact
that the leading wing half receives greater lift. The
coordinates of the centers of pressure are:

= 150: X = -0.440;
c,/ K

S= 30. x = -0.439;
cl K2

- = 0.00925

-= 0.0227

(:p = 6002')

(p = 13051,)

For comparison we repeat the values obtained at P = 00
when the expansion is stopped uith n = 3. The bracketed
terms contain the change in percent with respect to the
quantities computed with the four expansion terms, and from
which inferences can be made regarding the convergence.

Ci = 0.563 ao (-0.9 percent)

C2 = -0.776 o.o (-0.2 percent)

C3 = -0.365 ao (+7.0 percent)

A = 4.50 ao P V 2
o 2 ell

1 = -1.98 ,o cll K2 E V2 Fell

D2 = 0

D3 = 0

Center of pressure: = 0.4Z.8
c./JI K2

(+0.7 percent)

The results are correlated in figures 7 and 8.

For great A an approximate formula for the rolling
moment in relation to angle of yaw and axes ratio
K' =,/1 K2 is again expedient:

dcL 15
d =- ~ '; sin 2
do, 128

4 3 i tan + -2
1 2 sin
n 243
1 + ---- -- K'
cos 0 128


The formula indicates that the moment on transition
to the lifting line (K' = 0) disappears. But if the

i.e. ,

NACA Technical Memorandum No. 971

moment with the half wing chord-instead of-half the span
is made nondimensional, thus voiding the factor K' in'
(115) the moment coefficient becomes logarithmically in-
finite on limiting transition. In the extreme case the
lift decreases with cos2 0, for the reason that the
flow velocity in x direction is V cooe .

Horner's results on wings of different plan forms in
sideslip are in very close agreement with the values given
here. The assumption that the rolling moment is, aside
from the angle of yaw, largely dependent upon the aspect
ratio rather than the chord distribution appears therefore

Translation by J. Vanier,
National Advisory Committee
for Aeronautics.


NACA Technical Memorandum No. 971


1. Prandtl, L.: Beitrag zur Theorie der tragenden Flche.
Z.f.a.M.M., vol.16, no. 6, Dec. 1936, pp. 360-61.

2. Kinner, U.: Die kreisfirmire Tragflchie auf potential-
theoretischer Grundlage. Ing.-Arch., vol. 8, 1937,
pp. 47-30.

3. Keine, E.: Handbuch der Kugelfunktionen. vol. I,
Berlin, 18738, p. 3,17 ff; vol. II, 1881.

4. 4'hittaker-Watson: Modern Analysis. Cambridge, 1920,
p. 549 ff.

5. Hobson, E. W.: Spherical and Ellipsoidal Harmonics.
Caibridge, 1931, p. 476.

6. Weinig, F.: Leitreg zur Tjeorie des Tragfligels endlicher,
insbesondere kleiner Spannweite. Luftfahrtforschang,
vol. 12, no. 12, Dec. 20, 1936, pp. 405-09.

7. Schlomilch, D.: Comp. d. h6h. Analysis. vol. II,
Braunschweig, 1865, pp. 336-39.

8. Legenrdre, A. M.: Tafeln der ellipt. Normslintegrale.
Stuttgart, 1931.

9. hoerner, S.: Kr5fte und riomente schragangestromter
TrarflGel. Luftfahrtforschung, vol. 16, no. 4,
April 20, 1939, pp. 178-83.

IACA Technical M.iemioreanium No. c'7


/* /C(, /-K
.-- -*

/. ,

Figs. 1, ,3

- i.- '

- r.

~-i= -w-I -

V -__

I -,


4. '~y. rrl
L -. ./

I- -- -

S-- ,-. 0J
r .11 j

C,.9 .4

a J. ,'.r
Svi- K-i/.s

Conrutel lift jistriouti.n of the
lifting. line.
Elii.:,tic lift iistrib'uti.n wi. -
e..-al total lift.
Fig. .3

W 4i.u.-,
I 12

"jj-J 1i
1 32

L ~i1




C- 4-


'dIo i

qi.1 1ase e 1 i^E '

NACA Technical iMemoranium n Io.71

~711 I

4. C


1 .5

U.4 0. 1 1.- 1
L i rcu lar i i 'H:

Liftin!l-.- sun.-.& *-- -
Li f.i:, line -

Tizo'irt 0.


U. .r 1.2 1..
Circul:-,r Jii"

Figure 5.

Fils. 4,5

NACA Technical Mennraniurn ;o. 4I



Figs.A, ", S

V 91

/ V cosB
I/- i
-V',s i:,3

-- "

Figure 6.

0. 'i7-"

Fig-ure 7 L

daL 2


U I-- .' :' u-

pl)tte! against angle of yaw p

as.ect ratit /\ =- ..?7

LC.- 24 "0o

plotted agist angle of ,'av 3

aspect ratioA\ = 6.37.

Firure S.


1111 11 2 01 1 III
I 1262 08106 312 4