The conformal transformation of an airfoil into a straight line and its application to the inverse problem of airfoil theory


Material Information

The conformal transformation of an airfoil into a straight line and its application to the inverse problem of airfoil theory
Alternate Title:
NACA wartime reports
Physical Description:
62, 14 p. : ill. ; 28 cm.
Mutterperl, William
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:


Subjects / Keywords:
Aerofoils   ( lcsh )
Transformations (Mathematics)   ( lcsh )
Aerodynamics -- Research   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


Summary: A method of conformal transformation is developed that maps an airfoil into a straight line, the line being chosen as the extended chord line of the airfoil. The mapping is accomplished by operating directly with the airfoil ordinates. The absence of any preliminary transformation is found to shorten the work substantially over that of previous methods. Use is made of the superposition of solutions to obtain a rigorous counterpart of the approximate methods of thin-airfoil theory. The method is applied to the solution of the direct and inverse problems for arbitrary airfoils and pressure distributions. Numerical examples are given. Applications to more general types of regions, in particular to biplanes and to cascades of airfoils, are indicated.
Includes bibliographic references (p. 48-49).
Statement of Responsibility:
by William Mutterperl.
General Note:
"Report no. L-113."
General Note:
"Originally issued December 1944 as Advance Restricted Report L4K22a."
General Note:
"Report date December 1944."
General Note:
"NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003614298
oclc - 71252746
sobekcm - AA00006295_00001
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Full Text

'PccA 1


December 1
i:,: Advance Restricted




944 as
i Report L4K22a




By William Mutterperl

Langley Memorial Aeronautical Laboratory
Langley Field, Va.


L. WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
ie research results to an authorized group requiring them for the war effort. They were pre-
,. heldi under a security status but are now unclassified. Some of these reports were not tech-
yfifed. :Al hav eep. produced without change in order, to expedite general distribution.

ili iff:ll fe" all .

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ARE No. L4K'C2a

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ITACA rL.FR lie. LL4:.2 a F:R.7 !, T 'iL .






By W'illiam liutterperl

S Ji' 1? 1. ,y

e. t!.nhd c coni om:.l"m trans'orm-, tion is developed
that maps an riixf'oil i.nto a strSi-tt line, the line
being chrse:i cs th-e e-.-:tenrdeCd chore' ii-. r.'o the airfoil.
The flapping ic acconls?..*. by opcratini d'irec ti- ul thi
the ord'inates. The .Abs.:-nce of an-' pre-ll.inary
transformation is found to slh: irt.ii1 the .c.r:: subs taint l ally
over th' t :'f pfrev c.Ius :;.ethods. Use is ima-ie c'f tile
superpos tion o of obtain I rti-~.:irous counter-
part of the appro:xirmate rmeth!os of thin-:!irfoil theory.
The method is applied to the solution nf the direct and
inverse problems for ar-bitrary airfroils and pressu-re
distributions. Ilumeiricsl ex.-,mples are ti' -n. appli-
cations to mrcre genere-al types f regions, in i.articular
to biplanes ,rid to cascades of airfoils, are indicated.


In an attempt to set up an efficient nurierical method
for finding the potential flow, through an arbitrary cas-
cade of airfoils (reference 1.) a I;et.-jod of conriformal
transf.oriimation "was developed that w:.s found t. apply to
advantage in the cass ,.f isni-,t:~e.d airfoils.

The method 7.onsists in t ans.r-lrmin t-he isolated
airfoil directly to a straight line, nmrely.. the ec:tended
chord line of' the airfoil. The absence of the hitherto
usual preliminary transfoi.j-,ation of the airfoil into a
near circle rakes for a decided simplification of concept
and procedure.



The exposition of the method, followed by its appli-
cation to the direct problem of the conformal mapping of
given airfoils, is given in part I of this paper. In
part II the method is applied to the inverse problem of
airfoil theory; namely, the derivation of an airfoil sec-
tion to satisfy a prescribed velocity distribution. A
comparison with previous inverse methods is made. Addi-
tional material that will be of use in the application of
the method is given in the appendixes. In appendix A cer-
tain numerical details of the calculations are discussed.
In appendix B extensions of the method to the conformal
mapping of other types of regions are indicated. The
relation of the methods used for the mapping of airfoils
to the Cauchy integral formula is discussed in appendix C.

Acknowledgment is made to "T rs. Iois Evans Doran of
the computing staff of the Langley full-scale tunnel for
her assistance in making the calculations.

S', ; OLS

z = x + iy plane of airfoil

S= ( + ir plane of straight lines

p plane of unit circle

P central angle of circle

Ax component of Cartesian mapping function (CMF)
parallel to chord

Ay component of Cartesian mapping function perpen-
dicular to chord

Ax0, Ayo particular CiT's, tables I and II

T displacement constant for locating airfoil

r = 2R diameter of circle, semilength of straight line

c = an + ibn coefficients of series for CPT'

PN negative of central angle of circle, corresponding
to leading edge of airfoil

,TA, A ARR Io. LLI22a 5

p. central an:le :,f c'rci? niinu.s t.0D corresponding
to trailing ed;e cf c Lirfo'i

c air-cil chord

c z section lift coeffrciernt

vz vslocity at surface cf air.c-il, fra.-tion of 'ree-
streaf!,, velocity

vp velrocity =-t surface of circle, frazt3on of fr'ee-
strear, velocity

V fre.e-str'-:am vlccit7

ds element of on airf: jl

F c ir-culstion

ut thiA.kness fact. tor

U1 cal' t.';- r' .1':a i;ctr
Stici:.Lnress rati,

X nror'n lizling c~nr tarnt

Sdenomin;: tor :-f equation (17)

C ccarib r, rentcet

x, 6y incler.ental C:i?'s

U positive areas. under a;pro::imate vp(C) curve

L negative are-i unler s;prroxImat a p( ) curJve

a angle of attack

aI ideal angle of attack

Y = +

Ft true potential

0a approximate potential

8 central anole of near circle

c = e

4 NACA ARR No. Li4K22a


N leading edge (nose)

T trailing edge

c camber

t thickness

o, 1, 2 successive approximation in direct or inverse
CIIF methods



The Derivation of the Cartesian Tapping Function

Consider the transformation of an airfoil, z-plane,
into a straight line, t-plane (fig. 1). The vector
distance between conformally corresponding points such
as Pz and Pt on the two contours is composed of a
horizontal displacement Ax and a vertical displace-
ment Ay. The quantity Ax + i Ay is only another way
of writing the analytic function z t; that is,

z = (x + iy) (S + irn)

= (x ) + i(y r)

SAx + i Ay (1)

By Riemann's basic existence theorem on conformal
mapping, the function z t connecting conformally
corresponding points in the z- and c-planes is a regular
function of either z or t everywhere outside the
airfoil or straight line. This function will be referred
to as a Cartesian mapping function, or CHF. In order to
map an airfoil onto a straight line, the airfoil ordi-
nates Ay are regarded as the inam in'iry part of an
analytic function on the straight line and the problem
reduces to the calculation of the real part Ax.

IACA ARR Ho. L4?K22a 5

The calcul:-t.ion of the reil part of an analyrtic
function on a closed contour fror- the kno.jrn x\"lues cf
the imaginary: part is well kI1,own. It is convenient for calculation to consider thc- straight line as con-
forimall. iy reited to i circle, p-pl ne, by: tihe f'aiiili'ar
trains or-mrat ion

S- = + 2a)

where the constant disp:.laceraent T has been inserted for
future conver.ience in locating the ai:-fcil. 7or corr.e-
sponding points on tnh str:irht line ant- the circle,
equation (2a) reduces to

S= TT + r cos 0 :

Considered as a function of p, therifor:- the C.i -
is re- ular e E',.'r.'her.o ,uats Li .m i;'le ?,nd is therefore
expressible b:: thle inverse iov-' sEriEcs

z : "

T he analogy, of equation (3) with the T'h:eo:idr.sen-Glar'ic1'
trarnsfornation (refer-encer. 2)

log = --
P 1i pa

which relates *conforr.mall, a near circle, pt-plane, to a
circle, p-plane, -ay be noted. Oini thie circle proper,
where p = Re' and deF'ning c = + ibn, equa-
tion (5) reduces to two conjugate Fourier series for the
C-F; namely,

Ax = a + a- cos nc sin nQ (I-)
a n n
1 0 1 R


CO bn ao
y = bo + cos nO -n sin nc (5)
1 Rn 1 R

These series evidently determine Ax from Ay or vice

An alternative method of performing this calculation
is possible. It is known that if the real and imaginary
parts of a function are given conjugate Fourier series,
as in equations (4) and (5), with the constant terms
zero, two integral relations are satisfied. (See, for
example, references 2 and 5; also, appendix C.) These
relations are

1 2Tr (
Ax(cp) =- Ay(c) cot --- de' (6)
27r 2

Ay(p) = L f Ax(p') cot ~' dp' (7)
S2n0 2

Before the detailed application of the CT' z -
to the solution of the direct and inverse problems of
airfoil theory is made, some necessary basic properties
of this function will be discussed.

Airfoil Position for Given ClIF

It is noted first that the regions at infinity in
the three planes are the same except for a trivial and
arbitrary translation; that is, by equations (1), (2a),
and (5),

lim z = AxM + i Ay = co o ao + ib,
Z, -- co
lima = p + T

Secondly, if an airfoil is to be mapped into a
straight line, it becomes necessary to know the point on
the straight line corresponding to the trailing edge of
the airfoil. For a given CMF, ax(C), Ay(P), and
straight line of length 2r located as in figure 1,

ITACA ARPR To. 4 tV22a

the sa loil coorinates ::, y eir obtained frcrn e':Iuia-
tion, 1) and (2L) as

x = T + r c0 c,' + ~ ::L(L ) i" )


The leadin-' and tr-allinr ed.lfes of' the airfoil will be
taken as the points '...rles.:ontin. to tl.e e-:treriities of
the airfoil abscissas. Thie ccrr-s[oonr'irng locations on
the circle -::.e therefore deter.miirned by tma'riniziiig x
with respect to C in equav Li (). Thius

dx ,. d :x
= j -r sin + d

d', d c"

sin = (11)
2 d,:"

The condition ( I'1) yelds, usuallyy, by gr hical deter-
rin action; tne an:-.le.1 corrOs.- po;nL,...-ng to the leading and
trailing edp:es (iig. 1

S} (1 )
q T _- n + TS

It will be found, convenient to so alter th-e position
and scle of a derived airfoil that, for ex:aple, its
chordwise extremities ar-s .ocgt.c-d. at : 1 and tle-
trailing edge has the ocrdiinate : :r 0 L to be peflrred
to as the normal for:ni). The cinrd c of a dei'ived air-
foil is by definite on the difference in airfoil abszissa
extrer:ities, -r b:- equ.tionrs (12) an:d 1'),

c = -r ,os -,. ?) + x L(w) (;1)

The incre-ase in scale fr'om a t': sone des red co is
obtained simply by ,milt iplyin r, x, d i by the
facto-- c..,/c. The translation neicessar- to bring t the
trailing edge of the airfot'c, to its desired location is
then accornrplished by adjusting the t-rinslation constants T
and ho.


Velocity Distribution on Airfoil

Once the C.77 Ax(p), Ay(c) and the diameter of'
circle r of an airfoil have been determined, the
velocity at a point on its surface is obtained in a well-
known manner as the product of the known velocity at the
corresponding point of the circle and the stretching
factor from the circle to the airfoil that is,

vz() = r p() (14)

where vp(P) is half the velocity on the circle (since
r = 2R) and ds is the element of length on the airfoil.

The velocity on the circle v,(C), which makes the
point c = n + PT corresponding to the trailing edge
of the airfoil a stagnation point (Eutta condition), is

vp(c) = sin ( + a) + sin a + T) (15)

where a is the angle of attack. TI-e velocities vp
and vz are expressed nondimeAsionally as fractions of
free-stream velocity. The stretching factor ds/dC is
obtained from equations (9) and (10) as

d2 d_ r sin 2 (16)
ds (dx) + Ody=)

The velocity v,(cp), equation (14), therefore becomes

z sin (cp + a) + sin a + ) (17)
Vz (() (17)
dL- sin c + d -
Sd (r dcp/
This equation is the general expression, in terms of the
CF?, for the velocity at the surface, equations (9) and
(10), of an arbitrary airroil. The denominator depends
only on the airfoil r sometry, while the numerator depends
also on the angle of attack. Equation (17) is similar
to the corresponding expression in the Theodorsen-Garrick
method except for the absence of the factor representing
a preliminary transformation from the airfoil to a near

FTACA ARR I!o. LiK22a 9

The exrressions for the lift coefficient and ideal
angle ci attack : nay b-. not. ci. The c-ircul;tion P around
the airfoil is (V is free-stream velocity)

r = ir RV rin (a + 3,) (1)

The lift coefficient c, is defined by

Scc V =


= !,nr sin r (a + '3 (19)

where the airLfoil cir-d .c is gven by eqi ation (1).

The ideAl angle j att.) (,'e;er.:nce is defined
as that i.n tle of aftut ':k fnpr '!:..a; a sa r'.r t onr poin.:t
exists at te leaeiifn edge; that :., vz = 0 fnr = -
in equation (i1 ). -ence,

CT = (2,)

Superp.oition c' Solutions

The sun of tn;, analytic functions is an analytic
function; therefore, for a given f.-plane ci:-cle, the sun
of two Ci--'s is itself a CI S' a -,. also evident fro-c
equations (4) to (). Thus, s i'x- + i A-- and
Lx2 + i Ay2 of tw" crompionent airfoils rC, f'or the sae r,
be added tog- th-r: to .'ive a CiT (xjl + A:' + i (y + Y
and thence, by equation (17), an exact velocity distribu-
tion for a resultant airfoil. The resultant profile and
its velocity distribution is a superposition in this
sense of the component profiles and velocity distributions.
Thus, without sacrifice of exactness anC: w.itl no g-rerat
increase of libor, airfoils nm-a oe analyzed and synthe-
sized in termr? of cnorionent syrn_-:etrical thicl;rkess distri-
butions and mean camber lines. This result provides a
rigorous counterpart of the v,ell-knoin approxiinate super-
position methods c-f thin-airfoil vrrtex and source-s inI
potential theory.


As a particular case of superposition, a known CM?
Ax + i Ay may be multiplied by a constant S and the
resulting CMF S Ax + iS Ay determines a new profile by
the new displacements S Ax, S Ay from points on the
original straight line. It is evident that, except for
the corrections (S 1) Ax to the airfoil abscissas,
this new profile is increased in thickness and camber
over the original profile by the factor S. The effect
on the velocity distribution is that of multiplying the
derivatives in equation (17) by S. By virtue of a reduc-
tion in scale by the factor 1/S this profile may also
be regarded as obtained from the original one by using
the same Ax, Ay but a length of line 1/S times the
length of the original one.

The use of superposition as well as the application
of the Ci':' to some particular airfoils will be illustrated

Application of the C:.F to Some Particular Airfoils

Symmetrical thickness distributions.- The Cartesian
mapping function was calculated for a symmetrical 30-
percent thickness ratio Joukowski profile from the known
conformal correspondence between a Joukowski profile and
a straight line. I1;- CMF is given in normal form in
table I. The associated constants To and ro are
given in table II and the profile itself, as determined
either from the standard formulas or from equations (9)
and (10), is shown in figure 2(a). The symmetry of mie
profile required only the calculation of Ax('), Ay(P)
for 0 cp 1800. i"-i-h corresponding velocity distri-
bution (fig. 2(b)) was obtained from equation (17) by
use of the computed values of the derivatives. At the
cusped trailing edge the velocity as given by equation (17)
is indeterminate; however, the limiting form of equa-
tion (17), determined by differentiation of numerator
and denominator, is

limr v = Icos + a) (21)
cp-T 2 )2 / \2
P` )T d' x d\
c 0 r dp2/ \r dcP2

FTACA APR Po. LT.r22.a i

It is s-en f'r-comi this cx::r-es.: ) tih"t tle v-'locit- ,t
c.i.S -id e.-.d depends on the co0 ,.1' der' -.:.ti'- .af tn-.
napping function, that is, On tii'- ci-i.r'aturl e -t tlie i .::-- .
The ,c:,:1.uT ed second d, riv-'.tives : :,-:.. 'i d' -L,-,t/',IY
of the CiF : .of table I are plo tt.d in i 'u.- f'r '. '
of valuL-:3 of. ear 1.'
f "l..' .
T.=. ".s '.oi' .Tru L" etr2 ,;O i ",r- 11il, :' do:if'-r: nt

JTu.. D Ii -.,-,-f'.i1. s. in-;i tzJ cr. -: i. tIe j tin
us uperI siti..n i Soltion::" The L_ f ih

L) p, -- '7 L ,' 1- I
.to sultil to obtain .: r 1i .: tf in
rotio _. ,'btrin,: fLon

Ut Ayo


it- .--.- 1)-- ---
whe re s,- i t r.,a.. i:jl i: rlinat- t e :an
CIlF (t:. 1 an.. 1 the c.l n eintd o tl. r. s Lt the seil'icnrd
o; the der.tve'l ,., i... T"i t2. ri ltt. ic.:n s ,w o r o 1 n-

iL .1 sn tde t -A D

+ T -

.,alues ,.-._ uat ','e e calcul..ti e,-t r'"oo;., th,_,z f' rru.,i _a- ;-or
th ic :ne- -i ,.ic f' ,if pe ;-. e ,_-t ,n,-ni 12 p-. re -^ 1- .t /-,'l E.,rIe
riven in t hble II. 1i'9 re-- ,Itin.7 C.'. '_- .ere.- i:!.,,rn n,
mali z-d 's indici t, d i. i- l-: e s e t, .i.n ".-kr.c' i P,:."ti,.,n
for Gi'-n C -'- So that th- actual f;tfrs :." -i.t -:h t1o
..iultipi:- the 'o n i-a n:l L. -.,,--, w '- ThE: "aSes
are 1'i '- -, t- b-e II, t-. eth: ith i-th.:. : s oci9tcd.
cons tants T sail r- T!h r profi i- S thus l d. et ,'-r Li'-d 1 i.e
slhown in -i,.:.-e -( .-) a nd tin e c,'l-,y.Spri.InL' t V eloc'.it-' dIS-
tribut LonWrs in Ii -'re 2() .

Th.- ds .'ved : rci '-.--: r-e n t TJ.u';:,.- s;'i pi, .-.fi1e .
Th pr'oint o x" M x-.v':,, thI' :nl'ss is s i ftt.-.d "--,il.: 'long the
chord so r'. e' t as the Lhi.C':1.EZ rv tiao .-i.ct-aes. CZo-
v'erseli,,, the point of irxi.u. thi l i-,ess v.OLid C e shifted
for -ar ,d by -!oin.i fr'-omi a thin Jo-ut:. vsW i l' to a

J:CA ARR No. L4K22a

thicker one. (This result was the reason for starting
from a thick section.) The C;',F for the 12-percent thick
derived profile is illustrated in figure 4. It is to be
noted that the horizontal displacement function AXot(CD)
is symmetrical about CP = whereas the vertical dis-
placement function AYot(p) is antisynretrical about
CD = iT.

Mean camber lines.- The CIMF was next calculated for
a circZu-Tarc profile of 6-percent camber from the known
conformal correspondence between a circular arc and a
straight line. The normalized C-7 and its derivatives
are given in table III. The C:7T is illustrated in fig-
ure The symmetry in this case is with respect to
CD = 900 and C" = 2700, the Ax c () being antisymmetrical
and Ayoc(c~) symmetrical. The circular-arc mean camber
line is shown in figure 5(a) and the corresponding
velocity distribution in figure 5(b)'

Derived mean camber lines were obtained front the CI'F
for the circular arc in a manner similar to that for the
symmetrical profiles. The expression determining the
factor uc for a desired percent camber C is
Uc 7r
uc Omax

2jro cos CPN + uc Ax9)

with the solution for uc

2Cr. COs CN
Uc = cs (25)
Lgy 2C Ax( c)
max I

The angle cP in equation (25) (as in equation (22))
corresponds to the extremity of the derived mean line.
Because the factor uc is to multiply the derivative
dAxo(c )/dc, the angle CN as determined by the maxi-
mum condition (11) depends on u,. One or two trials
are sufficient to determine uc simultaneously with C
from equations (23) and (11) for a given desired camber C.
Values of uc and T.r (also CT by symmetry) are
given in table IV for derived cambers of 5 and 9 percent.
The actual multiplying factor to obtain the derived
CIiF's in normal form is given in table IV as Xuc.


The derived camber lines. are shown in figure 5(s)
It is seen that the derived canmb'er lines have been
separated into distinct upper and loie-e surfaces. Fur-, for the q-oercent camber line the "lower"
surface, that is, the surface correspondin3 to the lower
part of the strai:iht line or circle, lies above the
"'upper" surface. Lltnough such a scanber line is physi-
cally meaningless b: itself, nevertheless its Ci.T can be
compounded with that for a thickness distribution t. giEve
a physically real result (if the resultant profile is a
real one). The velocity distribution of the ,-percent
camber line is given in figure 5(b). The velocity : dis-
tribution" of the 9-percent camber ilne is included in
figure 5(b) for arithlm!etical com-;arison although it is
physically meaningless for the reason just mentioned.

The velocities at the cusped rnxtremities of the
camber lines are given by equation (21). The second
derivatives of the CiF of table III were computed. They
are plotted in figure 5 as d2Lx ./dc'r2, d2A' /d2 for
a range of c near 1300'. These- second derivatives, in
combination w ith those for the s -LnMe trial profile, can
be used to !.ive a more accurate determination of the
velocity at and near a cusped trailing edge than is
obtained by using equation (17) near the trailing ed.e.

Combination of s3y.L.i trical nrc-file and. mean .camber
line.- The CliT's derved for the symcLnetrical .pro'fil-es and
for the mean camber lines can nowv be combined in ,'-ryving
proportions to produce airfoils having both thickness
and camber. These airfoils may be useful in themselves
or, as in the folloviing sections, ray be used as initial
approximations in toth the crect and inverse processes.

As an illustration of such combinations, the CDiF
of the 12-percent thic!: s:y.inetrical profilee ,;f fi.giure 2(a)
and the CMF of the 6-percent a -iber circular arc of
figure 5(a) were added together. The airfoil profile
thus determined is shn-.;n in figure 6 (a). For comparison,
the airfoil obtained in the mannerr of thin-airfoil theory
(see,for example. reference 4) by superpDsition of the
same symmetrical profile and a 6.5-percent camber cir-
cular arc (in order to duplicate the cg,,mber of the exact
airfoil more closely) is indica ted in the figure. The
velocity distribution of the dotted airfoil should,
according to thin-airfoil theory, be. the sum of the
syrr-metrical-profile velocity and the increment above the


free-stream value of the camber-line velocity. This
velocity distribution, determined from the two component
exact distributions at zero angle of attack, is shown
dotted in figure 6(b). The exact velocity distribution
of the "exact" airfoil of figure 6(a) was determined
for the same lift coefficient (cL = 0.88, a = 10131)
from the known CIJ. This distribution is shown in
figure 6(b). The two velocity distributions differ ap-
preciably, although in the directions to be expected
from the differences in shape of the corresponding air-

It appears that the CMF's of a relatively small
number of useful thickness distributions and camber lines
would suffice to yield a large number of useful combi-
nations of which the (perfect fluid) characteristics could
be determined exactly and easily in the manner indicated.

The superposition of solutions can also be used with
the airfoil mapping methods based on the conformal trans-
formation of a near circle to a circle. There is a
decided advantage, however, in working with the airfoil
ordinates directly, both in the facility of the calcula-
tions and in the insight that is maintained of the rela-
tionship between an airfoil and its velocity distribution.


The direct problem for airfoils is that of finding
the potential flow past a given arbitrary airfoil section
situated in a uniform free stream. This problem can be
solved by a CMF method of successive approximation some-
what similar to that in reference 2.

Method of Solution

Suppose an airfoil to be given as in figure 6(a).
The chord is taken as any straight line such that perpen-
diculars drawn from its extremities are tangent to the
airfoil. For example, the "'longest-line" chord, that is,
the longest line that can be drawn within the airfoil,
satisfies this definition. The x-axis is taken along
this chord and the origin is taken at its midpoint.
Suppose, in addition, an initial CMF Axo and Ayo,


straight line ro, and choi.ovwise translation constant To
to be oven such that the corresponding airfoil has the
sa5e chord and is similar in shape to the given airfoil.
(At the worst the initial airfoil could be the given
chord line itself.)

At the chordwise locations ;:r(cfO) of the initial
airfoil, corresponding to an evenly spaced set of -
values by equationl (9), the differences 6yl(i() between
the ordinates Ay-tl) of the given airfoil and Lyo ( '')
of the initial airfoil are measured. The ordinate dif-
ferences 6y1 (Z,) determine a conjugate set of abscissa
corrections uzx1 ( r1 in accordalii c- either withl equa-
tions (4) and (5) or equation (6). The details of this
calculation are given in appendix A.

The initial sernilength of straight line r corre-
sponding to the initial airfoil is then corrected to r
and the translation constant To adjusted to T SO
that the use of ri with the first appro':imate CorF
Ax1 = Axo + 6x1, L:.1 = Lo + by1 yields a first approxi-
mate airfoil of v'w.ic.h tlhe chordwise e.:tremnities coinc-ie
with those of the given airfoil. This correction is
described in detail presently. If the first approximate
airfoil is not satisfactorily close to the give r airfoil,
the procedure is repeated for a second appro.r;:inmate air-
foil, and so on. The successive :irfoils thus deter-
mined provide a very useful criterion of con.vergence to
the final solution; nai.iely, the given airfoil. Evidently,
the fundamental relation between an airfoil -nd its
mapping circle

c1 c2
Z p = + + *+

can be used in the manner indicated to effect directly
the transformation of an airfoil into a circle. It
appears preferable, however, to subtract RB/p from the
second term on the right and thence to introduce the
straight-line variable = p + F.

Ths exact velocity distribution of any of the
"approximate" airfoils (hence the approximate velocity

16 NACA ARR No. L41K22a

distribution of the given airfoil) may be obtained from
equation (17) using the derivatives of the corresponding
CMF. The zero-lift angle PT to be used in equation (17)
is determined for each approximate airfoil along with the
corresponding correction for r.

The correction for r is necessary because if the
chordwise locations of the first approximate airfoil were-
computed by equation (9) with the original values of r
and T, Axl () being used instead of Axo(c), the re-
sulting chordwise extremities would in general not be at
x = 1. It is therefore necessary to adjust ro and To
such that with the derived Axl, Ayl,

xl(i N = 1
Xl( T1 = -1

where a' 1 and CPT1 are the angles on the circle corre-
sponding to the extremities of the desired airfoil. This
operation was mentioned in the section "Superposition of
Solutions." It may be termed a horizontal stretching of
the given airfoil. The condition given by equations. (2;,.)
applied to equation (9) yields

1 = Ti + r1 cos N1 + AXl(o1)

-1 = T1 + r1 cos PT1 + Ax(QcT) J

Subtraction of the second of these equations from the
first gives for r!

i +
rl = (26)
cos PN1 cos T1
11 1l

!TACA ARR ITo. L4-!22a

Addition of equations (25) gives for T

cos Fp,1 + cos0 A1xQ 1) + L.: i TT)
T = r1 ----- + (7)
12 2

The angles Ce and mTD in eq.uations (24) and (.27)
correspond to the extremities of the desired airfoil.
They are civen b.- G.rphic'4l solution .,f equation (11)

sin = -- ( )
r1 d *

Equation (11) mn'st be ol'e,1 sirlultaneoinl-- ,;ith equn-
t on (2) fonr r~ 1 and In prr.'tice only a
few siccessive trials re ijecess-"-r. Thence T is
obtained by equation (27). The argle ':T determined in
this process 12 equivalent to the zero-lift anwic of the
airf:,il, equa.tion (12).

Illustrative Example of Direct Meti:od

AS a num-erical illustration cf tihe direct method the
velocity distribution of the NA"i. 6'512 airfoil was cal-
culatea. In order to obtain an initial airioil, the C1!F
of the 6-percent camber circular arc (tabls III and IV)
was added to the CI.F of the 12-ocrcent thii-: sr-i.Ltrical
profile, derived from that of table I as inrdicalted ii a
previous section. Before this addition was r;wade, the
C0.hF for the circi.-lar arc was increased in scale (multi-
plied) by 1.092S/1.0072 to correspond to the same length
of straight line r as the sy'.m nrtrical pr.-file GrFT'. The
normalized result:-nt CLIF andl the associated constants are
given in tables V(a) and VI, respectively. The initial
airfoil is shown in figure 7(a).

The given airfoil, il'CA 6512, was so rotated through
an angle of -0.88'" (nose .-do n) as to be ta:;1ent to the
initial airfoil at the leading edge. The cornvergence
near the leading edge was thereby accelerated. The given
airfoil is shown in this position in figure 7(a). Two
approximations wore then carried cut in accordance with


the procedure given in the preceding section. The numeri-
cal results are given in tables V and VI. The first
approximate airfoil is indicated by the circles in fig-
ure 7(a); the second approximate airfoil was indistin-
guishable to the scale used (chord = 20 in.) from the
given airfoil. The velocity distributions of the initial,
first, and second approximate airfoils are given in fig-
ure 7(b), together with those corresponding to one
approximation by the Theodorsen-Garrick method (refer-
ence 5). The second approximation velocity distribution
differs appreciably from that of the Theodorsen-Garrick
method on the upper surface but agrees fairly well on the
lower surface. The discrepancy for the rearmost 5 percent
of chord on the lower surface appears to be due to lack
of detail in this region in the Tliodorsen-Garrick cal-

The convergence of the CTF method is seen to be
rapid, considering the approximate nature of the initial
airfoil, although two approximations are required for a
satisfactory result. The second approximation could
probably have been made unnecessary by suitably adjusting
the first increment 6yi(p) near the leading and trailing
edges on the upper surface before calculating 5x!(9).
The direction in which to adjust the increment is obtained
by comparing the thickness of the initial airfoil with
that of the given airfoil in these regions. Because a
thicker section has a greater concentration of chordwise
locations toward the extremities, for a given set of
T points, than does a thinner section, the chordwise
stations would be expected to be shifted outward as the
thickness of the section is increased. The ordinates
Ayl(P) should therefore have been chosen at chordwise
stations slightly more toward the extremities than those
given by equation (9).

The accuracy of the velocities is estimated to be
within 1 percent. It was expected, and verified by pre-
liminary calculations, that the results would tend to
be more inaccurate toward the extremities of the airfoil
than near the center. This result is evident from equa-
tion (17). A given inaccuracy in the slopes dAx/dr and
dAy/d can produce a i&lee error in the velocity near
the extremities, where sin p approaches zero. This
disadvantage does not appear in the Theodorsen-Garrick
method, in which sin C is replaced by one. Exces3lve
error in these regions can be avoided in various V;ys.

N.ACA ARR ,To. LiLT22a

If the initial airfoil, for which the slopes dxo,/d?'
and dAyo/dc have presumably been computed accurately,
is a Zood approxiriatioln in these regions, as evidenced
by the smallness of o5l, 6y1 compared to Ax0, 7y,
the effect of inaccuracy of the slopes d5xl/dC", dbyi/dc,
will be reduced, since the:T are added to. the initial
slopes dAxo/dr', dAyo/dC. It was to reduce the magnitude
of the incremental CIF near the leading edge that the
TACA 6512 airfoil was drawn tanoent to the initial air-
foil in this region.

The error in the derivatives can also be avoided by
computing tlher, front the differentiated Fourier series
for 6x1, 6:.i. (See appendix A.) This calculation was
made in the illustrative example, after it was found that
an error of about 5 percent in the velocity on the upper
surface leading edge could be caused by unavoidable
inaccuracy in measuring the incremental slopes.

The fact that the computed derivatives do not repre-
sent the derivatives of the CIRF but rather the deriva-
tive of its Fourier expansion to a finite number of
terms may introduce inaccuracy. (The derivative Fourier
series converges more slowly than the original series.)
A comparison of the computed derivatives which the measured
slopes will indicate the limits of error, however, as well
as the true derivative curve.

The importance of knowing the CMF derivatives ac-
curately may make it desirable to solve the direct
problem from the airfoil slopes, rather than from the
airfoil itself, as given data. This variation of
technique enables the CMT' derivatives rather than the
CLIP itself to be approximated initially. Further
details are given in reference 1.


The inverse potential problem of airfoil theory may
be stated as follows: Given the velocity distribution
as a function of percent chord or surface arc of an unknown
airfoil to derive the airfoil. Before the questions of
existence and uniqueness of a solution to the problem as
thus stated are discussed, several CP'T methods of solu-
tion will be outlined and illustrated by numerical


examples. Various previous methods of solution will then
be described briefly and their inherent limitations and
restrictions on the prescribed velocity distribution will
be compared with those of the CMF methods.

The prescribed velocity distribution is assumed to
be either a double-valued continuous function of the
percent chord or a single-valued continuous function of
percent arc. (Isolated discontinuities in velocity are,
however, at least in the percent-chord case, admissible.)

CMF Method of Potentials

This inverse method is based on the fact that, if
the airfoil and its corresponding flat plate and circle
are immersed in the same free-stream flows and have the
same circulation, conformally corresponding points in
the three planes have the same potential.

Consider first the case where a velocity distribu-
tion corresponding to a symmetrical airfoil at zero lift
is specified as a function of percent chord. If an
initial airfoil is assumed, the prescribed velocity can
be integrated along its surface to yield an approximate
potential distribution as a function of percent chord.
This potential increases from zero at the leading edge
to a maximum value at the trailing edge. Of fundamental
importance to the success of the method is the fact that
this potential curve depends mainly on the prescribed
velocity distribution and only to a much lesser extent
on the form of the initially assumed airfoil. The chord
line of the initial airfoil taken as the x-axis is next
sufficiently extended that, in the same free-stream flow
as for the airfoil, the potential, which in this case
is simply V1, increases linearly from zero at its
leading edge to the same maximum value at the trailing
edge as exists for the approximate potential curve derived
initially. Horizontal displacements Lx between these
curves are then measured as a function of the straight-
line abscissas and, hence, as a function of the central
angle c of-the circle corresponding to the straight
line. These horizontal displacements Ax(f), together
with the conjugate function Ay(p) computed therefrom
and the length of straight line previously determined,
constitute a CMF fnr an airfoil that is a first approxi-
mation to the unknown airfoil. The approximation is
based on the use of a more or less arbitrary initial


airfoil to set up the first approximate potential. The
exact distribution of tie derived first approxi-
nate airfoil can now be computed and compared with th,e
prescribed velocity. If the agreement is not sa-tisfac-
torily close, the procedure is repeated, with the airfoil
just derived taking the place of thie one initially assumed.

The complication introduced in the general case in
which the prescribed velocity distribution corresponds
to an ur.synmentrical airfoil w\lth circulation can be
resolved as follows: It is convenient in this case to
discuss the potentials in the circle plane. The pre-
scribed velocity distribution is trEansferred to thi, circle
plane by means of the stretchingr facctor, pre.Eured l:nowrn,
of the initially aszuLmead airfoil: that is, equ.tion (lI)
is solved for v,.) (. The first appr?.:irtate potential
distribution as a function of the c-n.-trAl 9.n, lo is
obtained b integrating Vo(c) throut1-h a (-rr..n ne of 2n
radians (around the ni.rfoil), starting from. tile v'.ilue
of C) near zero for which vp(CD) is zero I the front
stagnation point). This apnroxirmate potential curve has
a r.inim.un ialue of zero at the front stagnation point,
rises to a rmaximr,-aui for the v;ilue o'f 0 near T: corre-
sponding to the reer sta-Rn- tion point, then f:lls to a
minimum for t-he final vrlue of 0 (the front stn 'net ion
point), which is ar. anrle 2n radians from the starting
c-point. The difference between the final and the initial
potential minirurms is a firrt approximation to tle circu-
lation r.

A circle of such diameter is now derived which, with
this circulation and the same free-strearl flow as for the
airfoil, yields a potential distribution (henceforth called
true potential distribution) th1,t h-As the sane mrnaimlul
and minimuiri values as the approxiimats potential curve
just derived. If the maximur- approximate potential is
denoted by roU and the decrease of pote ntial (considered
positive) from the maximum to the final value by r,-L,
where ro is the diameter of tne circle corre spending
to the initial airfoil, the para.:eter y is first com-
puted from

= J- (28)
2(Y + cot y) U + L


by means of figure 8. The desired diameter r is then
given by
ro(U + L)
r = (29)
4(cos y + y sin y)

The parameter y is actually the sum of the angle of
attack and zero-lift angle of the unknown airfoil, to a
first approximation; that is,

y = a + PT (50)

It is related to the circulation P by equation (18).

This procedure for the calculation of the diameter
(see, for example, reference 6) follows easily from the
expression for the potential distribution on a circle,
obtained by integration of equation (15) as

t(r) = rof v(CP)dCP

= r [cos y+ y sin y- cos ~+1 a) + (C+ a)sin y| (51)

If the diameter r of the derived circle is much
greater than the diameter ro of the circle correspon-ding
to the initial airfoil, it is desirable to increase the
CMF Axo, Ayo of the initial airfoil by a factor suffi-
cient to modify the initial airfoil such that it corre-
sponds to a circle of diameter r. A new approximate arid
true potential distribution is then obtained as described
but by using the modified initial airfoil.

The first approximate horizontal displacement func-
tion is now determined as the sum of the horizontal
displacement Axo(cP) corresponding to the (modified)
initial airfoil and an increment 6x1(0) produced by
the noncoincidence of the approximate potential distri-
bution 1a and the true potential distribution Dt.
This horizontal increment may be measured between the
two potential curves, both considered plotted against
chordwise position in the physical plane. With sufficient
accuracy this incre~,-n nt may be computed as the vertical
distance between the potential curves divided by the

!..C. JalR Ho. iL4:22a

slope of the g ppro.ximnt potential cu? nmely,, the
Drescribe.d velocity '. If, therefore, all quantities
are considered as functions of '

Ax1 = Lxo + 6x1

= Ax, + (32)

The ordinste function on yl(C') conjug-ite to LAx1 )
can nc'., be computed and, together with Axp ) a nd the
diameter r obtained previously, determines the first
approxir'mae airtoil by: ecuations (Q) and (10). CaiLu-
lation or measurenent of the CfIF derivatives dx1//dcQ,
dAyl/drm and the use oi equations i11) and (17) tL.en
determine the zerc lift unjlecle 3 arnd the ex:-ict velocity
distribution of the first approxim'-te airfoil. The angle
of attack, to a first &approximation, is given by equa-
tion (50), tihe value of y derived fror., equ.,tion (23)
being used. This exact velocity distribution is compared
with that prescribed and, if the agreement is not close
enough, the procedure can be repeated with the first
approximate airfoil as the initial 9aircil.

In the case where the prescribed .velocit:y is speci-
fied as a function of percent t;-c, then b:" line integra-
tion of the prescribed velocity along the percentt are,
the true potential distribution of ti-o: urInknnon airfoil
is known as a function of qr-c (e;:ceet for a trivial scale
factor). The maximum and rniniir'auja values of this potential
distribution then the .uniqc..e determination, by the
calcui.tion previously described, of the circle corre-
sponding confornr.All: to the un.:nc..n airfoil. Correlation
of the 'potential distribution of t:-is circle with the
potential distribution as a function of arc initially
calculated therefore iel-ds exactly ti, potential distri-
bution of the un:n'own airfoil a.- a fu-rn tion of Lhe central
angle 0 of the circle. This fsct has been noted by
Gebelein (reference 6). The calculation of the diameter r
as outlined above for the percent-chord case is thus
unnecessary. The remainder of the proced'lre is the,
the successive approximate airfoils no." being adjusted
to correspond conformally to tnis circle before corre-
lating their percent-arc lengths vitn the prescribed
velocity distribution in preparation for the next
approri nation.

:T.AC ARR No. LlK22a

The successive contours determined by the method of
potentials are, of necessity, closed contours, whether or
not the sequence of contours converges to a solution
satisfying (mathematically) the prescribed velocity dis-
tribution. The closure of the contours is a consequence
of the method of setting up the horizontal displace-
ments, Ax(C), and solving for Ay(cp), by which the
contour coordinates are obtained as single-valued func-
tions of 9. The necessity for closed contours does not,
however, exclude the possibility of deriving physically
unreal shapes; namely, contours of figure-eight type.
This point will be discussed at greater length later but
it may be remarked here that it is the extra degree of
freedom introduced by the class of figure-eight type
contours that admits the possibility of a unique solu-
tion to the inverse problem treated here.

It will have been noticed that, whereas in the direct
method a Ay is determined from the given data that is,
the airfoil and a Ax is computed therefrom, conversely,
in the inverse method of potentials a Ax is determined
from the given data that is, the velocity distribution -
and a Ay is computed therefrom. Similarly, just as
the direct problem can also be solved by deriving dAy/dp
from the given airfoil slopes and thence computing
dAx/dc, so, conversely, can the inverse problem be solved
by deriving dAx/dc from the prescribed velocity dis-
tribution and thence computing dAy/dc. This inverse
method of derivatives will be discussed after some
numerical examples are presented, illustrating the nethod
of potentials.

Examples of CMF Method of Potentials

Symmetrical section.- The method of potentials was
applied first to the derivation of the c.,r.etrical profile
corresponding to the prescribed velocity distribution
shown in figure 9(a). As an initial airfoil the 12-
percent thick profile derived from the 30-percent thick
Joukowski profile in part I was used. The initial OMF
and associated constants are given in table VII. The
initial airfoil and its velocity distribution are siLown
in figure 9. The first increment CMF and the resultant
first approximate airfoil and its exact velocity distri-
bution were calculFted by the procedure of the preceding
section. The incremental slopes d6x1/dp, d6yl/dp
were computed and found to approximate the measured slopes

;HACA ARR Io. L4K22a

very closely. The results ;re presented in table VII
and figure q. It is seen that the change in velocity
and profile accomplished by, one step of the inverse
process is large; that Is, the coe.vErgnce is rapid.
The high velocity cf the first point on the upper sLrlface
(c = 153') is due to lackl of, detail in the calculation.
(Tvw.elve points on the upper surface wE-re calculated.)
'or niprcticnl pu-poses the nose could be easil- modified
to reduce this velocity if desired without rcing through
a conmlete second a.ppro::imati .on.

ean car.ber line for ,aniforri ';elocitnr increment.-
As a second examule of tihe inverse CM F rethod, the profile
producing uniform equal and opposite v:elocit: increments
on upper and lower surfaces .was derived. 3By the methods
of thin-airfoil theory this -elocity distribution yields
the so-called logeriitihnic carlberi line. The prescribed
velocity distribution is indicated in figure 10(a). The
velocity peaks at the extre:iities of the F:rescribed
velocity curve were assumed in o.d'er to coLm:ensate for
an expected rocuding .-f. of the velocity, in this region
in worl:ing up front the initial velocity' distribution.
The convergence to the prescribed i.'nifcrrm v.,locity dis-
tribution would there" be accelerated. The initial
airfoil was taken as tle S-percent ca-mber circular arc,
discussed in part I. The initial C;.TF and its associated
constants are given in tables III and IV. The circular
arc and its velocity distribution are shown in figure 10.

A first approximation was calculated d as outlined in
the previous section. A n.u.ierical difficulty appeared
in the process of solving equation (11) for the zero-
lift angle of the first al:.rcrxinmate airfoil. It appeared
that a 2L-point calculation (12 points by sy-mnetry) did
not give sufficient detail in the range TT < C < 1 T
to yield a reliable solution of equation (11) for the
zero-lift angle. This res.ilt was a consequence of the
prescribed velocity discontinuit Pt the extremities with
the consequent lrge but local changes in CIF and profile
shape required in these regions. The solution obtained
for the zero-lift angle wan ;T = 6.1', which by equa-
tion (19) with r = 1.3045 and al = 0 yielded
cz = 0.67. The desired cl, however, is .80,' which
would correspond to = 7.27. It w/s considered
that a relatively minute change in the shape of the
extremities of the derived caliber line v.ould alter the

,TACA ARR No. L4K22a

slope dAxl/dcp in the desired range sufficiently to
yield a zero-lift angle of T = 7.270. On the other
hand the effect of such a local change on the CMF as a
whole would be small. The velocity distributions of the
derived profile were therefore computed for both zero-
lift angles quoted previously.

The results are given in table VIII and in figure 10.
Included for comparison in figure 10(b) (vertical scale
magnified) is the logarithmic mean line of thin-airfoil
theory, computed for ct = 0.80. The velocity distri-
bution of the derived shape as calculated for the desired
lift coefficient of c = 0.80 is seen to be a satis-
factory approximation o the desired rectangular velocity
distribution. The profile itself is seen to be one of
finite thickness:as.compared with the single line of
thin-airfoil theory. Airfoils obtained by superposition
of this type of camber line with thickness profiles would
therefore be increased in thickness over that of the
basic thickness form.

The changes in velocity distribution and in shape
of profile are again seen to be large; that is, the con-
vergence was rapid. As is to be expected, the rapidity
of convergence of both the direct and inverse methods in
comparable cases is about the same.

CMF -iethod of Derivatives

Instead of approximating by the method of potentials
to a CiiF that, when differentiated, yields the prescribed
velocity, the CMF derivatives may be obtained directly.
The controlling equations are equations (17), (9), and
a modification of equation (7).

Vsin (co + a) + sin (a + T) (17)
jz(C ) = (17)

dVy 1 dAx ,p' d
A-- sin (P ) A ( --

= cot dp' (7a)
dcp 27T dcp' 2


x Q + Ax' Ao) (9)
= CS CO + A ) (9)
r r

These equations, tog-ether with the au-:iliary equations (11)
and ('1), constitute a set cf s-i:rultaneoius equati.ons from
which, the CIT derivative d.x/d'3 may be determined from
a prescribed volcity distribution v-. The corresponding
airfoil is determined by integration cf dLAx'/dr and its
conjugate dy//de.

Consider first tie case wn-re the velocity is speci-
fied as a function of percent -arc. As explained in the
previous section, the constants r and y of the final
circle corresponding to the Lunl.aiowrn airfoil can in this
case be deter.nined initially.. Points of equal potential
along the arc and circle are then found, which yield vz
as a function of :. The angle of attack a in equa-
tion (17) is taken is some re'ascnable value and dA:'/r d,
determined by successive ayproxlmration. In the first
appr-oximation dLy/ir dg may, for ex-rample, correspond to
some l-nowLn Cio ,L
some known CiO?. Equation ( 17) is then solved for
dAx/r dr, for which the conjiuate dAy/ji dC is calcu-
lated next and used as a basis for a better determination
of dAx/r d(. The airfoil coare6spondin; to any- approxi-
mation is obtained by intePraLion of d'x/d.d and its
conjugate day/, 'd. ('The nethod: of derivatives may be
regarded as based on the use of the function
p z This function is regular ev.'erywhere outside
the circle p = Re approaches zero at infinity, and
reduces tc + i on r;te circle itself. )
QQ dfo
In general the d':/d' as det.ermincd in any approxi-
mation will huve an aver geuevalue other than zero. The
Ax(c) obtained, sa-, by integration of .ts Fourier
series -.;ould therefore contain a tern pr.-oportiornal to cO
in addition to a Fourier series. Thus, iQ() would
not be a single-valued function of rc anr. the resulting
contour w;iould not clcse. Si.iply subtracting the average
value of dAx/dcr (the constant term in its Fourier series),
however, vill close the derived contour. If the method
converges, this average value approaches zero in the suc-
cessive approxisrat ons.

A preliminary r over-all adjustment of an initially
chosen CIiF ma-; be desirable. Thus, if dx1./aQ is


calculated in terrs of the dAyo/dcP of a previous approxi-
mation and is found to be larger than dAxo/dp by some
factor, dAyo/dc can be multiplied by this factor and
the calculation of dAxl/dc repeated.

Although the angle of attack may be arbitrarily set
initially in this calculation it should be so chosen that
the final airfoil will coincide approximately in position
with the initial airfoil. After each calculation of
dx/dcp, the zero-lift angle PT can be calculated,
equation (11), which thereupon fixes a, since y= a+ T
is known.

If the prescribed velocity distribution is speci-
fied as a function of percent chord, vz(c) must be
determined in the successive approximations by use of
equation (9). The quantity y = a + PT may be deter-
mined in each approximation as in the method of potentials
or, in physically real cases, by equation (19). The
diameter r is so determined that the successive airfoils
are of a standard chord length.

It is evident from the structure of equation (17)
that near the airfoil extremities where sin T--)0, and
in particular at the nose of the airfoil where dAy/dp
is comparable to dAx/dcp in magnitude, the convergence
by this method (and by the method of potentials) will be
comparatively slow. If modifications to the airfoil only
in the immediate neighborhood of the nose are required,
it may be more expedient to apply a preliminary Joukowski
transformation, that is, to use these methods with the
Theodorsen-Garrick transformation.

An example of the use of the CMF method of deriva-
tives to solve an inverse problem is given in reference 1
for the case of a cascade of airfoils.

Method of Betz

In the inverse method of Betz .(reference 7) an air-
foil and its velocity distribution are assumed known
(fig. 11) and a desired-velocity is specified as a func-
tion of percent arc. The new velocity and length of arc
are specified in such a way that the extremities of
potential are the same as on the known airfoil. Both
known and iLujknown airfoils then transform into the sare


circle end, in particular, the velocities 't points of
equal potential oi. the two rrof?-les can be found.

In orDier to determine the prcfile co:-responrdirn to
the new v: -icity, the corjplex dris;lIcerien t z2 z
between points of equ.i'l potent al oi the tw'o p00i1' les is
expressed d !s a fur.t -on o:" th- c :rr.sp:cni : l- omp;;lex
velocities (denoted b:' vz) thus,

d z2 d/dz V71
._2 z rI-- =
dzI ') z7 _-c/ v

z V_
z z = ( 1 dz1 (55)
-T v

where the ".ntegr- tion is cerrie,; ,jout -10o.: tn? knl.wn pro-
file fro.n the tr'silin. ?dge, w-.ich ia t::>ein as coincident
for the t'c Sirfoilq to the point The complex
function v, / is Cdet- r.rinecd pi:.rjo:' from the
1 ,
known rctio corresrond.In to the :points of equal
potential by the er-.ig',ent tb-t, inssi..uch as the t,.o pro-
files have nearly tie s,,n :ilo6e. s corr E:n ondirng points,
z t i S
the real Dret of 1 is -iven by 1. (This
Vz2 v2
assumption, like the rppr'oxiinations ins tip Ci'.iF methods,
is least valid at the nose- of the airf'oil. The function
z2 z1 is in fact a Cartesisn i:npping f -nction.) The
imaginpsr part is then computed as uhe c:nnjug~te function,
equation (7).

In addition to the resztr-ict ions on the velocity dis-
tribution mentioned initially, futrtier con.'iit ions must
be nmet in this ;::ethod, if clossI co:it-urs are to b? ob-
teined. Thus, the condition for closure nf contour,

d(z z1 = f( 1) dzl 0
(V, 1(.5

NACA AE No. I41<22a

and the required coincidence of v 2 and vz1 at
infinity, lead to the following three restrictions on
the real part R(p) of the integrand in equation (54)
considered as a function of p in the circle plane,
p2Tr =21T p21n
R(C")dp= R(c) cos C dcp = R(cp) sin cp d( =O (55)

Method of '.loinig and Gebelein
The method of Weinig and Gebelein (reference 6) may
be described essentially as follows: The given data are
the same as in the Betz method. Consider the function

log log i 6)
log v = zl (z2 )(6)

where Pz2 and "Z are the slopes at corresponding
points of the two airfoils (fig. 11). Since !vz21 and

I vzl are known functions of p with the data as given,
and since log 2 is regular outside the circle,

2 P1 can be calculated as the function conjugate to

log -V2 The angle Z1 being known, z2 is thereby

determined and hence, by simple integration, the unknown
airfoil coordinates are obtained.

As in the Betz method, the condition for closure of
the desired contour

dz w/ dp = dp = 0 (7)
JC Cj dw/dz V
JC Vc /c z

leads to the additional restrictions on the prescribed
velocity distribution,

ITA'A A'"7 ?To. LLTj22a

-1 log vz(o) dcp = 0

J2 log vz( )I sin ? dp = -sin 2y (58)


n- log IVz()1 cos Q dr;: = -'(1 cos 2')

where y is given by equation (3B).

Discussion of the Various Inverse Hethods

The methods of B3etz and aof V'einig-Gcbelein may be
somewhat narrower in scooe than the CLF :ncthods. The use
of mapping functions such as in equations (.55) and (56)
is based on the ability to srecify- dz2/dz1 unambiguously
in the corresponding regions. This requirement appears
to restrict the contours obtainable by these methods to
those bounding simply connected regions, l'urther investi-
gation of this point is necessary, however. By the CIIF
methods, figure-eight contours Lave arisen in the course
of solution of both the direct and the inverse problems.
(See the 9-percent chamber derived mean line (fig. 5(a))
and the illustrative examples in reference 1.) Such con-
tours were first encountered as preliriinary results
(unpublished) in using the method of potentials with the
Theodorsen-Garrick transformation. The CI.F apparently
makes no fundamental mathematical distinction between simply
connected and figure-eight contours, for although z -
must be a single-valued function of z, t, or p, the
coordinate z itself is of the same character as t and
the latter has two Riemann sheets at its.disposal in
consequence of the Joukowski transformation front the I-
to the p-plane.

The methods of Betz and of Vcinig-Gebelein require
the numerically difficult closure conditions equations (55)
and (38)) to be satisfied in advance. If the methods are
worked through for prescribed velocity distributions
which do not satisfy these conditions, it appears that

32 TACA ARR No. LT4K22a

open contours result. In the CNF methods, however, there
is either no closure condition (method of potentials) or
a numerically simple one (method of derivatives):
)2T, dAx d 2m = 2T dAy

J0 d d 0 d c
[This simple closure condition in the method of deriva-
tives is fundamentally a consequence of the fact that
the required absence of the constant term in the inverse
power series for the C1F derivative mapping function
ipd(z t)-, mentioned previously) automatically ex-
cludes the inverse first power (the residue term) from
the power series for d(z t)/dp. Thus, physically
impossible velocity distributions lead to open contours
in the Betz-Weinig-Gebeiein methods and to figure-eight
contours in the CMF methods (if the latter converge).
From the practical point of view in these cases, it may
be easier to obtain the airfoil corresponding to the
"best possible" physically attainable velocity distri-
bution by the CI. methods than by the others. If the
succession of airfoils determined by an inverse CTU method
is seen to tend toward the development of a figure-eight,
the successive approximations can be stopped at the "best
possible" physically real airfoil.

As to the existence and uniqueness of a solution to
the inverse problem as stated, a rigorous discussion of
the solutions, for a prescribed velocity distribution,
of the controlling equations (17), (7a), and (9) is
lacking. For physically possible velocity distributions,
however, specified as a function of percent are, the
' !inig-Gebelein method shows that there is one and only
one airfoil as a solution. If, however, the velocity is
specified as a function of percent chord, some further
condition is necessary. This requirement is evident from
the fact that one velocity distribution for an airfoil
can, for differently chosen chords, be expressed as a
different function of percent chord in each case. One
chord with a given velocity as a function of percent
chord can therefore have more than one corresponding
airfoil. There is reason to suppose that the further
condition for uniqueness of solution in this case is,
the chord being defined as in the section "The Direct
Potential Problem for Airfoils," that the ordinates to
the airfoil at the chordwise extremities be specified.

"TACA ARR o. L4122a

From the experience '.vith tle C".TF nmeth-ds g hiredd to
date, it i tblleved thst tc a velocity Jictrio.ution
specified as at the buFginnrirg j part II, ard w'.th thie
further condition nentione- i-n tLe opecent-chordl case,
there corresponnds onie end jnly ,one c-lsed ccnt'-iur satis-
fying the rlF s:-ste::a of elqu-,tioni s. t is "urth- .r ,o- e
believed .hat ihe Ci' r nethc-r. s are fle-::ble enough to
converge t.: thi.s ~C: it on ir. at lea:t tnt-sc cas.-s of
aerodyn n.1ic int-rest.


1. Ti: onfior-rmal trarsform-tion of -in airfoil to
a strain t lin7- bTr the Carteslia;-, m i: .' ir. lfunc t io (CIlT)
method results ir. single.' nu-."rical -olitions of tnr
direct and inv-rse potential problems for .irfriWls than
have been hither-to. availa'.le.

2. Tlhe us- of super.-osition w tb th-, CrIF t.iethod
for airfoils vrovi].ds a ri orous coLunter;--'t of the
approx.mae mi-cl- cds o' thin-airfoil th-. or7.

Langley :vleriorial Aerorautical Laboratory
national A.dvisory, Coru.:ittl.:. for Aeronautics
La--riley Field, IF.





The basic calculation for the type of mapping func-
tion treated in this paper and in reference 2 consists
of the computation of the real part of an analytic func-
tion on a circle, given the imaginary part, or vice
versa. To this end the conjugate Fourier series, equa-
tions (4) and (5), or the conjugate integral relations,
equations (6) and (7), are available. This type of cal-
culation appears to be fundamental in many kinds of
two-dimensional potential problems. For example, the
solution of the integral equation relating normal induced
velocity to circulation in lifting-line theory can be
solved easily by a method of successive approximation
if the transformation from the lifting line" to the
circi! is Inown. Quicker methods of calculating a func-
tion front its conjugate than those given in this appendix
or in reference 2 would therefore be highly useful.

The use of the Fourier series rather than the
integral relations in the calculations of this paper was
based on the following consideration. Because the func-
tion 1/z is regular outside the unit circle, the real
and imaginary parts of 1/z on the unit circle, namely,
cos Q and -sin 9, satisfy the integral relations (6),
(7). The substitution of -sin cP for Ay in equa-
tion (6) and subsequent numerical evaluation by the 20-
point method of reference 2 gave results that were higher
than cos P by a constant error of 2.8 percent. Evalua-
tion by a 40-point method reduced thecerror by half, or
to 1.4 percent. By the Fourier series, on the other hand,
the first harmonic (a one-point method) suffices to give
exact results in this case. It appears, therefore, that
when the given real function is expressible in terms of
a small number of harmonics, as is the case in airfoil
applications, the Fourier series method is preferable to
the use of the integral relations.

The Runge,schedule offers a convenient means of
carrying out the basic calculation of mapping functions,
namely, the analysis of a periodic function into its

-ACA A 1R Io. L~4T22a

Fourier series and the sy-cI.esis of a ou'rier series
into a function. Ta3 theor a'.l. use of the schedule is
described, for example, in reference C, -herein are also
given schedules for 12-, 2'-, 56-, and 72-point har-ionic

The n=cess.ry analyses and s,-nthecses in the direct
and inversL Ci.F methods are carr:.ed oit in accordance
with equaticns (1I) and (5) and their derivatives.
Table IX contains the scher!e of substitution into the
Runge schedule, table X, for the various C2T^1 methods.
In the dir-ct method, for e:xamr: e, the set of vr'lues
6y/12 corresponding to the evenly sp..ced O-values is
substituted into the yn spaces at the beginning of the
sum-table. The suns nnd differences of these qu'nitities
are then obtained as directed at the left of the indi-
vidual tables substituted into the succeeding tables.
In tnis wa- tlie -nti-e sLun-table is filled out. Before
the product-table is used, the s.ulm-tah:-le should ce checked.

The quantities sirroun6ed b:, thie heavy lines in the
sun-table are ne::t rmultiplied by t!.h proi"-r factors at
the left of the prodJuct-table al.ei, t.- results entered
in the spaces as indicat-dc by the letters
at the left cf the individual prodLct-spaces. L, heavy
horizontal line at the lower le't edge of ? product-
space indlcictes that thle correspronding pro.-luct has
already been ob!': ined in a previous -p.ce in the same
row. A heavy vertical line Q&ong the left edge of a
product-space is used to .o.iphas.ze that the negative
value of the product of the sum-table quantity andr the
product-table factor is to be entered. The I'ums of the
product-table columns are then entered in the I, II, III,
and IV spaces. ;. che.c on the v'irk of the product-table
up to this point is provided by the columns at the right.
The sums and differences of the I, II. III, and IV quan-
tities complete- the product-table and give the Fourier
coefficients an, bn corresponding to 'y.

In order to perform a s-nthesis calculation front a
set of Fourier coefficients an, tn to the values of the
corresponding function aL the even r-points, the coef-
ficients an, bn are entered in the d and D spaces,
respectively, in the sun--table, and the ret.ainder of the
sun-table and the pro .duct-table worked thro,.igh as before.
The final values in the a: b spa-es of the product-
table are then entered in thel d and L spaces at the
beginning or the sun-table r:nd the suns and differences

36 VACA ARR No. LQK22a

obtained as indicated by the synthesis column at the
left. (Note that do and d12 are to be multiplied
by 2.) The resulting Yn quantities are the desired
values of the function.

The numerical values in tables X(a) and (b) illus-
trate the process of obtaining 6x1(p) from 6yl(P) in
the first approximation by the direct CMF method for the
NACA 6512 airfoil.




Simply Connected Fie.Zons

If the Cl,'IF nethod is app4I-d to the napping of a
simply coni;ected boun.dary v.ith a vertical discontinuity,
such as a recta;ngtle or an infinite line with a vertical
step, the nr'bigu itv "f tilh ondiJ.ite Sy at the discon-
tinuity will prevent an automat a an rapid convergence
of the m-thi-,od. ithour.h the dil'ficultv could be lessened
in particular cases such as for reci-tn.igles. byt taking the
diagonal as x-axis, thus rem' tlie vertical discon-
tinuity, or by using syiu'netry,, as with squares, it is
evident that in general a reference shape particulilc-ly
suited to the contour under investigation is needed.
The circle has been shown in reference 2 to be a good
reference shape, for the square. It could be expected d
therefore that an ellipse wruld be a ood refer-ence shape
for the re ctangile' ' just as the apping
function based on the circle ,vs formed of an angular
displacement an..d j rPdial di.slacennt, theF appin
function based on the ellipse should be forried of dis-
placements elorn and ortho'c.ntal to the -ellipse, that is,
should be specified by elliptic coordinites. The speci-
fication of a figure byr elliptic coordinates k(P, 0) in
the physical plane z is equCiralent, however, to the
transformation of the figure to a t'-plane by the two

1 +iB
z = p + where p' = e

t' = log p' 'where t' = W + iG

and specifying the transferred figure by the Cartesian
coordinates of the t'-plane h(, 8). Th_ rectangle under
considerat ion vrill be a near-- circular shape in the- p'-
plane and a near-strLaig t line shape in the t'-plane.
The mapping of the rectangle by. means of an elliptic
mapping function in the physiclc:i plane 12 therefore seen
to be accomplished b- the- Th -odorsea-Garck method in
th- near-circle p'-plane and b : the CMNF method in the


near-straight line t'-plane. Fro- this point of view,
therefore, the Theodorsen-Garrick method consists of
specifying an airfoil in the physical plane by elliptic
coordinates, forming the corresponding elliptic mapping
function (' 4o) it, which conformally relates the
airfoil to an ellipse or Joukowski airfoil as a basic
shape, and expressing' the elliptic mapping function as a
regular function outside the circle. On the other hand,
in the t' = log p'-plane the Theodorsen-Garrick method
consists of the transformation of the near-straight
line *(8) to the straight line C4 = Constant by means
of what is now the CMF (f xo) ic. Thus, the
Theodorsen-Garrick method may be regarded as a form of
the CMF method, in v'hich log p' takes the place of z
and log p, the place of t.

The mapping of simply connected regions by dif-
ference mapping functions based on the curvilinear co-
ordinates appropriate to the particular reference -hape
considered is therefore equivalent to using the CTU dif-
ference function z t in the of the near-straight
line into which the reference shape is initially trans-

T.,pping of the Entire Field

The Fourier series representation of mapping func-
tions, equations (i) and (5), enables the calculation of
corresponding points in the two regions to be made, once
the correspondence of the boundaries has been calculated.
By the latter calculation the coefficients an, bn and
the radius R of the circle of correspondence have been
determined. If now a larger radius R' > R be substi-
tuted for R in equations (4) and (5), the resulting
synthesis of the Fourier series will yield the mapping
function for the circle of radius R'; that is, will
determine points in the given plane corresponding to the
points in the circle plane at the distance R' from the
origin. It is necessary, of course, to use the mapping
function in conjunction with the shape in the physical
plane corresponding to the larger circle. In this way the
entire corresponding fields can be mapped out. It may
be noted that substitution of R' < R for R in equa-
tions (4) and (5) enables the mapping of those corre-
sponding points inside the original contours for which
the resulting Fouri er series converge.

IPA'JA ARR To. IJ.1Ti22a 39

It appears 1: o ror ,r ':ifif'lcult to finr-i the point.
In tlhe circle rlrre.p.nd.Ln. to-. .poi.t of the L.'iv.en
plane than ,ice rs:a. Ti.s Cl.clatL on ai:, hoeveCr,
be acro..Tplished by r:ethod o:' uccessive apFpro':i:a' tins.
For cxamrle, if the i- ven pln: is that 0o' s near circle
the pool r c,:.ordi. iates of tlie i.ver.;- r.,int in the neal'r-
cir-PZ; pine a~s,: azsstU,'ed to be a fir-st approxir;ia tion to
the coordir. te- R' and ,' of tih- '"V -siired point in ti-he
circle rlRne. aubstitutiop f" t2 -' values into eq'.u-
tions (i 4 and .T I i=lds a i:" t a' prox;im j te mapping;
function wh'icn can be used to *:orrl..ct the coordinates it'
and rn, etc.

ip lanes

In the case of the biplane arrr.nge,,lsnt the CiFP may
be set up directly in the physical pl._ne in the same way
as for the sinw'le airfoil. In place .f the simple trans-
formation froi straight line to circle, !n'-wever, the
transfor-cimation f'r:-o the two e-tend-:' chor-d linEs )of the
airfoils to twc c conentrc circles i.' c-d. 'ihis trans-
format ion is deri'e-'id in r.ef *,er,-ce "'. Tie MF'' me thod for
biplanes bears the se rs c relating to t.:. netho.: of ref-
erence 9 that the ClF rviethodl f.r- onoplae ifeils bears
to the Thlieodc-rsen-G'-.-rie ethod (r-efer r--'.e 21.

For biples (fi.. 12) the C.' a C, being regular
in the reY4ion out i. tde the two str'ai.lht, is regular
in the anrnui-r region of the p-plain:- and consequenti"- is
expressible as a Laurent series in p

where (Il')

,n = a,- + ib,

If, for the inn-er circcl,, the relationship; is written

z = L:1 i

p = Re


and for the outer circle

z = Ax2 + i Ay2

p 2e R P


there is obtained, upon substitution into equation (40)
and reduction

an + a-n b n b-n
Axl() = ao + cos np + b n sin ncp (43a)
R1 n RIn
1 1

Ax2(cp) = ao +


an+ a_n cos nQ+

Sbn + b-cn -an a-n
Ayl() =bo + cos n- ------sin ncp
R1n Rln
1 1

E bn + bn \ an, an
Ay2(cp)=bo+ b+ n cos np -\ sin nCP
R2 R2
1 i 1



These equations are the generalization to the biplane of
equations (4) and (5). The corresponding integral rela-
tions may be derived as in reference 9.

The solution of equations (43) in either the direct
or the inverse problem may be accomplished as before by
successive approximations. For example, in the direct
method the two airfoils are given. If no initial approxi-
mation biplane were available, the two chord lines would
be taken as the initial straight lines. By the trans-
formation of reference 9 this fixes the chordwise loca-
tions on the straight lines corresponding *to a set of
evenly spaced Q points on the concentric circles. The

- b-n sin np (43b)

FACA ARR io. L':22a

or6din.-'.tes u f ;'- c ;n tnerc fore be mIasur:r hr ich
deteriL:ijines Ly2() by analysis r'd yrtne.esis of equ'-
tion.r (5al) rand ( .d), resp- ctiv-lyr. (The radius ratio
2/1 is fixed by the initial Ltansformation f',ron the
stra.' I t lin-i.e tn the concentric ire'l s. ) Thes; .ly; (,')
values then de teri-,ine set of Lx2-('?( values u'" t"he
givr3n slhaprj of she second .,irfoil sand the known chlorldwise
locations of its first aprt';.roximtion str.-iht line.
Analysis or L::( ") and su.bse.quent zynthesl- s of Lx: ')
by eiuationns (+4b) arid (5 ) respect el'-, det, 'urmine s a
correction to I1 b- a J. hor.izoItal s tixetch!-ng proiccess
(constant Lx, A;: adjustment of r ) to rtaintrin the
given Esi;'f,:il i ic .rd. The r'DC.- 'LL re i, n1 ow re:,Peated 'I.J- th
AY2(Pl aS the initial set of ietas..ur:-d ordirtnate th:t.
determines y S c), Lx1( ;, and A,1"`Q) as before. The
radius R2 can now cb siri.ilarl- corrected. Ths step
completes the ~. rt appro '- ir'at ion !. Fior the second l-l. d roxi-
mation a new corresnondence bet'-er-ier ti e c:'rrected t-'traigtht
lines and the once riL-i. c '.rcl is ."-.1cul;. ted and tlhe
proce-'.,ure repeated.

The iP verrst problem could -ls_ L.e soSied b'r methods
similar. to those iv ;7-n for the isolated arii oil. Sup-
pose, for exa'r':'le, a v;ini' section wivN e gi'"An and it were
desired to der' '.e a slat of c .-k 2 r'd an-i giv.'n a'pro.:xi-
mate loctioti san,' hab-ing a pr.scribe.- velocity distri-
bution. Thei method of surface p>ote"t ials, for example,
enailes the ca .i ilc L' _u o,'I a 'sproirte I1 i A '1
(sub'-cript 1 r I e-rs tro sitt). The initial. correspondence
of points betwreer the st:- i,1't lin n:-s ad concentric
circles, a.nd. tli- before also R2/, b inL det errined by
the initially a.esi.nume straiI-it i nes, thie function .. ( 4)
is thlereup-on o'ta inod by. analysis ajn' synthesis of e.r.a-
tions (i5a) and (I b), res].,e active ly. 'Te horizontal dis-
placemrtent L.x2 ) thence determine L :2j) by the
known shape of the rmain wing section. The dieterm..ination
of L y-(D) br analysis anty. s ynthes of cu. tions (J4d)
and (1i 3c) :.ominpites thie caicul ition of the first ap)proxi-
rmate slat section, for vhhLch the ex'-ct vo.locit;t oistri-
bution can now also be calculated. If tihe main win:l
section were al o unknown then the a~i n' section abo'e is
regarded as an initial ap:nroxiiri.tion, the role of the two
airfoils is revei-:rsd, and the procI-edure repea.t-d to com-
plete thn first eaproxirn nation.


The CIF method can be generalized in the same manner
for multiply connected regions. The transformation from
the n reference shapes (such as straight lines) to n
circles being presumed known, the CMF can be set up as a
series convergent in the region between the n circles,
and the mapping function for each boundary explicitly
expressed by allowing the coordinate vector to assume its
value on each boundary in turn. A method of successive
approximation for the solution of the resulting equations
depending on the particular problem under consideration
would then be established.

Cascade of Airfoils

A simplified but practically important n-body problem,
namely, the cascade of airfoils, may be mentioned finally.

The reference shape into which the cascade of air-
foils, figure 13, is to be transformed is chosen as the
cascade of straight lines coinciding with the extended
chord lines of the airfoils of the cascade. The trans-
formation from the cascade of straight lines to a single
circle is well-known, reference 10. The CMF chosen as
indicated in figure 15 is therefore expressible as an
inverse power series in the circle plane and the resulting
procedure in either the direct or the inverse problem is
seen to be essentially the same as for isolated airfoils.
The detailed application of the CMF to cascades of air-
foils is given in reference 1.





The foregoing nrthrods of confer, ;l t--.rrsforr iation
have been presented from the point of' viewl of represen-
tation of the various mapping functions as infinite
series. In pa-rtcular, cho expression of the C'iartesian
mapping function as an inverse p,;wvvr series vJi:.d every-
where outside aind on a cir-cle led to the Fourier- series
representation for the C'?IF on the circle itsFelf. The
integral formula represe.itn.taion vas ;:hin obtained from
the Pourier series by the tietihod 0of refer ence 5. It is
of interest to see hov, the integral r-lations (6) ar.-I (7)
can be derived dir.e-ctly from the Cauchy integral formula
for a function regular outside a circle. (These integral
relations have also been derived -1oy 7- etZ, reference 7,
by a hydie-,o:Jyna!ic Al1 argiunent.) Sin:,ce the ar plicrbility
of the Cauchy interpIr iforrula i.s iot rstrictertd t)
circular boundaries, however, the results vill be capable
of generalization, in principle at least, to arbitrary
simply and multiply ccnnec;ted regions.

The Caucny integral formula gives the values of an
analytic function f p) within a s-i.t:ply connected do-
main D in terns of its 'ralueZ f(t) on the boundary
of the domain as

f(p) = rit (Ut)
2i1 t p

where the path cf integration is cointerclockwise around
the boundary. Consider the domain D outside the simple
closed boundary C in the p-pine (fig. L). This domain
can be made simply connected b" an outer boundary B and
the cuts betwenc- th-j two boundaries, as indicated by the
dotted lines. The Cauchy integral formula for the func-
tion f(p) at an interior point n of the domain D,
in terms of its values on tihe boiudary is

:fp 1 = f(t) 1 + f(t)r
f(p)- f(t) dt + --- (t dt (45)
2i i t p 2ni t p


where the equal and opposite integrals along the cuts
have been omitted. The paths of integration are indicated
by the arrows in figure 14. The function f(p) is as-
sumed to be regular everywhere outside the boundary C
and in particular to approach the limiting value f, as
p---. If the boundary B is enlarged indefinitely,
the integrand of the second integral of equation (45)
approaches f,/t and thus

lin f(t) dt = f (46)
02lTiB t p
t-- BV B

The variable p will now be made to approach a point t,
on the boundary C, and equation (45) will consequently
reduce to an integral equation for the boundary values
of a function regular everywhere outside and on the boun-
dary. In order to evaluate properly the contribution of
the remaining (first) integral of equation (45) in the
neighborhood of t', the boundary C is modified as
indicated in figure 14. The point p is made the center
of a semicircle whose ends are faired into the original
boundary. As p-- t', the modified boundary approaches
coincidence with the original boundary. The integral
over the modified boundary is now evaluated as the sum
of the integral over the semicircle, which in the limit
is half the residue of the integrand or -f(t'), and the
integral over the rest of the path, which in the limit
is analogous to the Cauchy principal value of a real
definite integral of which the integrand becomes infinite
at some point in the interval of integration. Equa-
tion (45) therefore becomes, in the limit,

.-f(t') i f(t) dt + f (47)
2 2 i C t ti

In addition, there is the auxiliary condition that

1 P f(t)
2 j t dt = f (4)
2ni L1C C

which follows from the fact that f(p) is regular every-
where outside the boundary C. Equation (47) is well
known in the theory of functions of a complex variable.
(See reference 11.)

I:.'2 .1 -; 'o. L7K22e

If, now, the function fp) is taken as the Cartesian
nappin funi-ctio.-i z 0or -, o:- the Li., und:-ary,

f(t) = Lx i y, (hc9)

and if, furT.Lher, thLe bounda-:, C is tAe, n as a circle
with origin :t the center,

t = C

t' = e -

substitution of equations .,4) anl (') into equation 1 7
and using equation (48) (with f = ') lea-ds to th-e inte-
gral relations (A) andr (7). if uhe poiar alL..pi: fiiunc-
tion log = i (. i i(p ) rcferencz 2)
is substtitted for ft), the Theod ) e n-i:,l icl integral
relations are oht.ined.

The ite '. 1 has alrsead:'r bee-n applied
(reference 12) t -ro",Lc-rn: .:f c :,nfo-rnial. n- ,pring ".rn the
m nner just lrdi iated. E, rrnan rs included in refer-
ence 12 (charter XI) contributions of' two Russian authors,
Gersicori- and Kr-lo'.-. In ref'e;-.nce 12 the mapinnL funrc-
tion from a cir-cle to s near cir.-le was ta:en in the iorm
log p. The resu;.itj n, nteT;-.l equ'.tion does not appear
to be a.s convenient a ths of te of e C'-' mT.ethods. The use
of forms ,ch s log -- or z I are not only accurate
nur:ericraliy sinr:e tr.-i. e:rxpress -lapnges in the coordinates
of the boundaryi s, but siso th-,?- to pairs of integral
equations wjhi-ch conta-in tl: solu'-..l; ions of bothl the direct
and the inverss :)otential problems.

Prom the an-ilysis given it appears possible to trans-
form conforrnilly i'from', on-e bound -Jlr to another wi-hout
requiring the. trai'nsforrl::a tion from e either bounrdary to. a
circle, sine:- the boundur.l C in equation (47) can be
rather arbitrary and '(t) can be t:ai::en as a ma,pping
function frorim this boundary. to Anoth..r arbitrary one.
The resulting integral equation for the napping function
is, however, not as easy,- to solve numerically as v hen the
bound ry C is circle.


Once the conformal correspondence between two boun-
daries is known, corresponding points outside the boun-
daries can be determined by the Cauchy integral formula
(l.). It is noted that the Cauchy integral gives the
correspondence of individual pairs of points rather than
the correspondence of entire boundaries at once, which
is given by the Fourier series representation. Further-
more, the possibility exists of determining pairs of
corresponding points inside the given boundaries by the
Cauchy integral, that is, of analytically continuing the
conformal transformation beyond the original domains.
For if the transformation-from a boundary C in a
p-plane to a boundary C' in a p'-plane were known, the
outside regions corresponding, then the correspondence
between a boundary A internal to C and a boundary A'
internal to C', if it existed, could be determined by
an application of the Cauchy integral formula to the
region bounded by A and C.

For example, if the boundaries A and C are taken
as concentric circles and the mapping function as

f(p) = log t-
= ie = (I X) i(( ) (51)

in the notation of figure 15, the Cauchy integral formula
applied to the annular region in the p-plane (assumed
free of singularities of the mapping function) yields,
in the limit as the variable point p approaches the
inner circle A,
1 -2T 1 C~1
1(cl1' ) =- 2 (cP) cot 2- dcl

1 ,23 co(Po) sin (
-L n{21 ESCPo) sin (0 R) -j(cp) sinh (X-X (52a)
2JO cosh (X -) os (o0 1)

1 P2 l I '
l2 01r') ) cot C 2 dcp1

+ I2n o(po) sin (c 0 ) + o(Co) sinh( X1) (52b
2J00 cosh (X X1) -cos ( cq') 0

HA.CA ART 'To. L4-!.22a 47

In addition, the condition of regularity of the function
f(p) in the annular region yields the auxiliary condi-

1 2TT 1 2Tr
1 1o --1 = yode
2 T
0o 0

2r (53)
1 P2* d /2,
74d P = kj eo ^ o

In the problem under consideration, the mapping function

P (co) io(co)

for the cuter boun::aries is cnown. The radii e", e'
of the concentric circles are given.. The second integral
of equations (52) are th',.s k!nowvn fi.unicticns of nl'' Equa-
tions (52a) and (52b) thfiefore cos.-titute a pair of
integral equations, similar to those of Theodorsen-Garrick,
for the function () i ), pertaining
to the inner boundaries.
It is noted that if the variable point D of the
Cauchy integral forrr.ula for the annular region is made
to a,-,oroach the outer boundary C, then two additional
integral equations similar to equations (52a) and (52b)
are obtained. These equations, together with equa-
tions (55), are a generalization to the case of ring
regions of the corresponding Theodorsen-Garrick
equations for simply connected regions and can be used
for the conformal mapping of near circular ring regions.



1. Mutterperl, Ailllrmi: A Solution of the Direct and
Inverse Potential Problems for Arbitrary Cascades
of Airfoils. NACA ARR No. LiK 22b,, 19JL.

2. Theodorsen, T., and Garrick, I. E.: General Poten-
tial Theory of Arbitrary Wing Sections. NACA
Rep. No. 152, 1953.

3. T1ilii.:-n, Clark B.: An Extended Theory of Thin Air-
foils and Its Application to the Biplane Problem.
NACA Rep. No. 362, 1950.

Allen, H. Julian: General Theory of Airfoil Sections
Having Arbitrary Shape or Pressure Distribution.
IT.CA ACR Ho. 3G29, 1943.

5. Garrick, I. E.: Determination of the Theoretical
Pressure Distribution for *':'-nty Airfoils. NACA
Rep. No. 465, 19533

6. Gebelein, H.: Theory of Two-Dimensional Potential
Flow about Arbitrary Wing Sections. NACA TM
No. 886, 1939.

7. Betz, A.: Modification of Wing-Section Shape to
Assure a Predetermined Change in Pressure Dis-
tribution. NACA TM No. 767, 1935.

8. Hussmann, Albrecht: Rechnerische Verfahren zur
harmonischen Analyse und Synthese. Julius
Springer (Berlin),1958.

9. Garrick, I. E.: Potential Flow about Arbitrary
Biplane Wing Sections. NACA Rep. No. 542, 1956.

10. von Karman, Th., and Burgers, J. M.: General Aero-
dynamic Theory Perfect Fluids. Application of
the Theory of Conformal Transformation to the
Investigation of the Flow around Airfoil Profiles.
Vol. II of Aerodynamic Theory, div. E, ch. II,
pt. B. W. F. Durand, ed., Julius Springer (Berlin),
1935, p. 91.

TNAA ARR No. TLtTK22a 49

11. Hur-witz, Adol f, ard Coil.rant, ?.. AlP.l'erei ne
Funl:tionenthcorie und elliptische Fur.!:tilonen,
and Ceometrizche l.iirdutionerntheorie. Bd. TTi of
VtchermaLischen ."!is enschaften. Jillnus ringer
(Berlin), 1?29, p. 5.

12. 2erymran, Stefan: Partial Differential Equations,
Advanced Tooics. Advanced Inrtructicn and
Research in mechanicscs, Bron UTniv., Surmaier 1941.

HrACA ARE No. L4K22a


C0rTESTh' r..'AP..II3 "U:DTI,,?: F?" SY'T..IETRICr,-.L

O-PEr'-F'I THI. '.iFS. JOTTj:O -TI3 Pr.F'ILL






Profile T T u r T u
_____________t (g_____) (.deg)
Joukowski 0.30 1.100. 0.087 1.250 0 10G 1.000

Derived .24 .3 o .0716 1.155 0 180 .355

Derived .12 .1022 .0357 1,09231 0 1s80 .55






(radBans) Ao dAx 0/dc dA o/dpc

6 x 0 0.120 0.108 0
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11i .oko .0160 -.0o344 -.0781

12 .01L2 .00169 -.115 -.0279

15 -.0170 .0021o6 -.117 .0346

14 -.0o49 .019o4 -.0352 .0926

15 -.0587 .0490 -.0239 .123
16 -.0552 .0328 .0506 .125

17 -.0555 .110 .113 .0756
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C (P aI ]
Profile (per- uc Xur T r (de) (deg) (deg) c ideal
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arc 6 1.000 1.000 0 1.0072 -6. 18 186.84 .75
Derived 9 1.502 1.1-99 0 1.0050 -10.26 190.26 ------

NACA ARR -No. L4K22a

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S r-49-I r-4-4 -4 -i4

'NACA ARR No. L4K22a




r -

',o r--CDo'a o





Process Entry in schedule Result

Direct method

Analysis Enter a for Yn a, b
12 an' b

Enter in Enter in
dn spaces Dn spaces

-bn an 6x

Synthesis nan nbn d6x/dp

nbn -nan d6y/dp

Inverse method of potentials

Analysis Enter for yn an, bn

Enter in Enter in
dn spaces Dn spaces

bn -an 6y

Synthesis nbn -nan d6x/dp

-nan -nbn d6y/dp

Inverse method of derivatives

1 dAx
Analysis Enter 12 d for Yn an, bn

Enter in Enter in
d spaces Dn spaces

bn -an dAy/dP

Synthesis -bn/n an/n Ax

an/n bn/n Ay


59 NACA ARR No. L4K22a

x i

x .j 1 S;

i c__ T -

T : 1
CD a

o ',

w -a
-J o' to 5- .

I- _In

0u +" 1
h : 1- -1

hi -? _ii -, -+ S


j-. ?? "^H -- ^

NACA ARR No. L4K22a 60

:;, ~^^~^7 IS

4 4 4' .

-!' x
.4 4).

o'~ ? ^ ^
1 4 4 44.4 -. 4
44 4 ,- y ) ^
.o 4 i I I I I; : **' -' '. 4.
-"vl 8'4i~ a ai 3

*. 4.. I I I ? '*' *
4 4 ->_. _

4. C
i, 4

I4 I* < I *
4) ;* Cii -t i' *, a> ,
*, 4 4f .. 4' *
*i. ~~ ~ 4, 7-Q i *. ** .

4 -- -; I i i '

'* "'

F i tilt
,-, I

I) 4 ) 4 4
4. [ : '^ s <

4 ? 4 -. i '

4. a 4- *.4 4) c

4)' : -

4. 4 t ?

i -- I I II I : 1 1' 1 1 *s ?

-d l --

61 NAQA ARR No. L4K22a


Z q

a Z



W 4.
S|- ,
4 I-

Li fl 2! I

LaJ ^ ,- _,,* i 1

m H

NACA ARR No. L4K22a 62

-- ~- *

% Io
S- is a

I a
t .

r CO

C'- ** 9t t s

\ .- I I I ,- I ." -^ ,

I. IP'. I 1 .:
-' --i <

~~YY\-~~ --^-^^




Figure /

- flustra/ion of the Carleskan
mopping function.

Fig. 1



S/ .2 .4. .- .6 .7 .8 .- /0
() V'ie/oct/y dsfrbuftons.
f/j're 2. dbAoJi/ f/7//cAnewS forn.< derived //7/c/e75 fore'5,
and ye/oe4y d/S/d/os/b/ y ettecd/ c
Ctatejo'n /y/yoppinlg fs'wchor'.

Fig. 2

/oj Piar!es.


/gure. Second d~t-i//i^,4s of/ /~F~o.

Fig. 3


- /2-percenf thickness symmetrica/ profi/e
--- /2-prcen/t 1/tnss'6s modifiAd by inverse method
-- 6-p.erent comber circular-arc orof/le
-- 6-perce'/t comrber modified by /nyerse method

Figure 4. Carlesian napp/ng functions.

Fig. 4


- Upper surface
---Lower surface

(a) /'ean comber hnes. COMMiT"EE F IAMNCS

0 ./ .2 .3 .4 .5 6 .7 .8 .9 /0
(b) yVeocdy d/str/ibuton5s o<-0.
79ure 5 .-C/rcu/ar-arc 7mean /ne, derived mean //'ea,
and re/oc/ty d/s/'rib/to/ns by mefAod of
Cot tewJ n i,,n/n4 y .f7nct/on.

Fig. 5


- Eract superpos/tion,
--- Approx/maoe superpos/tion



~"1 Upper surface
.- ..

f Lower surface-^ -

EXct sup erposif/o
Approimrate superpos iion
-_= 0o8-8B

-- --

o J .2 .3 4 .6 .7 .8 .9 10

(b) lVe/ocity a/slr/bluton

w~io/ ^/ z^y

Fig. 6


- 4,ACA 65/ o/rfo//
'- /n,' '/ airfoi/
' First *approxif'i<3f/o0
If /l i l 9ii


(o) Profi/es.

- Theodorsen- Garr/ck retod
/-frst approx/latilon
---CMF met od, zero approximat/on
o CwF method first aJproximat/on
EI CMF method second approx/Imt

.2 .5 4 .3J f.
(bj Velo fiy do'trlou t/ons.

, yr. 7.- a-ec/ C/f ,-F //mweh/ or /A4C4 6S'/2 o#rAo//

Rig. 7


Figure 8. Deferm/inat/on of y = o( +6r

Fig. 8


f .- ...--- .-- ..--- ---,.-- Z -- O -- -- -- -- -- -- -- -- ---- -- -- -

.a -- -

- Prescribed veiocity
Initial ve!oc/ty
__ 0 Firs t ~ADroxlfGtO/-/O

S .3 o .6 .7 .9

(?) Ve/octgy d/ s5ir/bt/ons5.


- -Intal orof ile
-Der/red profile (/iapproz)

(o) Prof-es..

'/~y/ry 9f /r,,,.Y <'/7-~ 7<7or- dys/ /// 4 ess

-- .6 ..5 d .3 .2 i _
-H -2- 2 4 .4 .6 .8

Fig. 9


-- ---
J #"1 /
C,- -- ---! .; -: _.--

6 I
S--Prescre o-- c-

^ _Lower surface i

-- Prescrioecy ve,'oc, '.

'.itij=. ____

A .5

-/rf//la/ Vel'oc/"y (c.'cv u'r rc,c~.,6 L
F/rsr approxmrr.o.f or < = .67
F/rst approXlmrnoion Imodif 'e~',.&S

.6 .7 .5 .9 .10


- 'P, .t.Z/ C'rOf//e(C,', -cu/ ac~, G. 7-
-- Fi.~ ~,c;~r,oinr;o..mal
-- Th..n~r-~~.z.rfo..'/ t,?eory ,C7 =.8O

fad ProfileJ.
//iure /'. /-wI .e 6r7F/77 ,7//1od f r --/ ?Arr ,//,, .6-O.

Fig. 10 -

'111ilr7rr 1P

I~ v~/oc./5c d~c'riD~ton~.


Figs. 11,12








Figure /13- Carlesin maop/ng
funcf/on for cascades,

Fig. 13







(5 1j



Figs. 14,15

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