Numerical evaluation by harmonic analysis of the epsilon-function of the Theodorsen arbitrary-airfoil potential theory

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Title:
Numerical evaluation by harmonic analysis of the epsilon-function of the Theodorsen arbitrary-airfoil potential theory
Alternate Title:
NACA wartime reports
Physical Description:
8 p. : ; 28 cm.
Language:
English
Creator:
Naiman, Irven
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Conformal mapping   ( lcsh )
Harmonic analysis   ( lcsh )
Aeronautics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: A finite trigonometric series is fitted by harmonic analysis as an approximation function to the psi-function of the Theodorsen arbitrary-airfoil potential theory. By harmonic synthesis the corresponding conjugate trigonometric series is used as an approximation to the epison-function. A set of coefficients of particularly simple form is obtained algebraically for direct calculation of the epsilon-values from the corresponding set of psi-values.
Bibliography:
Includes bibliographic references (p. 7).
Statement of Responsibility:
Irven Naiman.
General Note:
"Report no. L-153."
General Note:
"Originally issued September 1945 as Advance Restricted Report L5H18."
General Note:
"Report date September 1945."
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 - 003595904
oclc - 71010257
System ID:
AA00009377:00001


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Full Text

ARR No. L5H18


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WARTIME REPORT
ORIGINALLY ISStED
September 1945 as
Advance Restricted Report L5H18

NIMERICAL EVALUATION BY HARMONIC ANALYSIS
OF THE -FUNCTION OF THE THEODORSEN
ARBITRARY-AIFOIL POTENTIAL THEORY
By Irven Naiman

Langley Memorial Aeronautical Laboratory
Langley Field, Va.


I
I.


WASHINGTON
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 pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.


L 153


DOCUMENTS DEPARTMENT







































Digitized by Ihe iiernel Archive
in 2011 With Iunding Irom
University ol Florida, George A. Smathers Libiailes wilh support from LYRASIS and the Sloan Foundalion


hllp: www.archive.org details numericalevalualang






"^ '-

TACA ARR No. L5H18

Hi.-TI.CJAL ADVISORY COMMiITTEE FCR PEr OCN.UTICS


ADVANCE RESTRICTED REPORT


I]U!";ERICAL EV'rLU..TION BY H.~;iCI.IC AITALYSIS

OF TFH e-F'Ui.ITICO 'FU T'U T-HCDORSEII

r PITR ARY- AIRFOIL POTENTIAL THT-EORY

By Trven Naimar


S U !I..1 P Y


r ftl:ite trigonometric series is fitted by harmonic
sa,-i~ sis as an _pFprcxi.rmaton function to the *-function
,:f the Theodorsen arbitrary-airfoil potential theory.
Er hsr'nonric 3s-rnt-i.heis the corresponding conjugate trigo-
non-mtric seies is used as an approximation to the
E-f'ur.ctio' r!. C set of coefficients of particularly si...?le
fr-). LI is Lttained algebraically for direct calculation of
th-... -.-cilis from the corresponding set of *-val:.es. P'e
[ ,rr..ula for C is


E() cot ( -
/ n 2n -k k
k=l

j-here the. suruiation is for odd values of k only nd,

f kT
\ + L_ + )



I:iTR DDUCTION


In the determination of the flow about an arbitrary"
_.ir'fili referencess 1 anc 2) the problem arises of trans-
f-.~lrin, cirve, nearly circular, into a circle. This
rtranftoratior., a basic problerr in conformnal nmapinr is
fui. t-.e- red'ai ed ti the d.-,term.inatic-n of the folloV'inr: a
twvo c2onjuL:ate Fourier series:







2 NACA ARR No. L5HlS



a a, + (am cos mcp + bm sin me)
m=l
(1)


=1
E = / ^p sin my bC c)s "r "



(See references 1 and 2 for significance of notation.)
Tive following integral relations are equivalent to series
relations (1):



2r 2 2




s V P in relation+ .Bcause of t e cyclical






be i.ritten -1T to Tr. \then the integral is broken Into
o ao
> (2)


(q) = --- @(p') cot -' 9 d ,
2n 2


It is convenient to introduce a new variable
s = cp' ( in relations (2). B-cause of the cyclical
nt ure of these frictions, the lirits of integration may
be written n -IT to n. henry; the integral is broken into
two parts, -n to 0' ard 0 to n, and -s is substituted
:or s in the first part, the following relations are
obt ined:
Tr^

7(cp) = 2,. L p ('0 + s) E(c s cot -ds



e(c) = _- ( s) q(o + s)1 cot a ds
-Io

Thus, by use of relations (1), (2), or (3), c may
be determined if 4 is known or w may be determined
if E is known.








rACA ARE rto. LSH18


In the, airfoil. problem is specified as a
functionI :-f cp by means of a curve End e is to be
d.eter1,ined. In theory the Fourier coefficients may be
'.ec-nrmnined in relations (1) but in practice, because of
t!. ui..'.no'.:n analytic nature of the curve, it is neces-
sary to iesorut to socme type of numerical approximation.

T.I refer.-rcer 1 and 2 an appr-:xinate method of han-
dling the integrals .f relations (2) is presented. In
r-ference 5 a refinr.emnt of this method is given for the
..::e i:ntcr.i10. An. alternative procedure is to approxi-
mate relations (1) by a finite trigonometric series and
tLhn to determine the coefficients by harmonic analysis.
. dcevlc.-prnt of this method is now given.


iiARMONIC ANALYSIS


The *-function is to be approximated by a finite
trigon'-rimetrii seri-s given by

:'I') = A + A- cos c + .+ An cos (n-l)cp+ An cos np

+ 3i sin + + Bn_1 sin (n 1)C
n-l
= A + cos mQ + Bm sin mp) + An cos nP
m=1

If "r is specified at 2n equally spaced intervals in
n 2iT
the ranf.e 0 < cp 2wr that is, 0, Y 2 ,
then
n

2n-1

S 2n r
r=0

In practice, 4 is given as a function of
9 = iand therefore 4 is taken as a function of cp
as a firsi: appro,,mation. An it r:.tion process is neces-
saoy t,,o ilte-rmr!ine both t and c correctly as func-
tions ,.;f C.




























- n.


os r.-
'-', S- --


z Mo-


I\ ( lr


- COS + .-, zi r.c


-7-
M t, Cos n
ir* \p


I+ :I


- cos -. -:', sin


n.o
nC ,' (-I)'


si r


Sr-


-Ain (-i)v l"
/ r
r


..-- :- r -.-' su.-.-_-: .. i3 interchar.ged,


=t'


r-


:NACA NAR :o. L5H1G


1"8




















ST '" i =' ~- -- 1 .
__-___-- _


/I '---Xi


Z.
-L-- - -





-I


- .,r -.f i = ., ..*


cl.1*,rI


r--


. .: =---c

_.. -- cecai...e _-,f t- --e ----------_. -r .:: : -


S --
i- :n


-- --












C, _

\. -. -' *.-
-~ -- L
a -


cL Z


r







ITACA ARR No. L5H18



n
<(,) = 1 ) cot 2n
kl1


Finall:-, then,


CE(P) =
k=I


Ck(-k -k &


"- k+k


and for odd values of k

S1 kT
C = cot -k
k n Oa

an- for even values of k

Ck 0

Equation (4) thus gives the same result as is
obt.'ined by harmonic analysis aEd synthesis. Co:m.parison
with equi.tions (3) indicates that equation (L.) may also
be interpreted- as the evluation of thick integral by ihe
oidinury rectangular summation formula usin,- intervals
of width 2Tn/n and usi*-g the value cf the integrand at
the iri point of each interval, thia is, at s -
where k is odd. n


PR'CTICP L 0BSE1 V.AT IONS


:Lu-ution (4) uses only one-half the available
information. It is evident that all the points may be
used because all the given n points may he considered as
alternate (odd) points of a system; of 2n points. The


where


(1)






NACA ARR No. L5H18 7


: le.s Cf so computed is, of course, to be plotted at
q-pr.cints r-idway between the given h-points. The n vsl.i.e
:f i. therefore give values of E corresponding t-:'
*.~o'.,'rcY.i.ibtion function consisting of a trigonometric
3?r.ies of n 1 harmonics. Values of the coefficient 1
n : 10, 20, 4,0, and 80 are given in table I. For
S.-in curv?s the present method for n = 20 is more
accurate than the 40-point method of reference 5 and
requir-es only one-half as much computaticnal work.

Ho.' to handle small irregularities or bumps in the
~r-curve is of interest. One procedure is to fair through
the burni. and to designate the faired cuive j4. The devi-
ation frcm Vir is & AL'-curve. The conjugate 7 miay be
datcrmined in t he usual manner and a conjugate ,IE may
be determined by use of a vry- small interval, say,
n = 200. Ta' desired E-valaes are giv-en by the sum of
E and Ae. This method cannot be ju3t-fied on strict
mathematical grounds but is probably :n.,re than adequate
for engineering purposes.


Lngrley Memorial Aeronautical Laboratory
National Advisory Commit ee for A.-ronautics
Langley Field, Va.








REFER ":TCES


1. The;odorsen, Theodore: Theory of Wing Sections of
Arbitrary Shape. NACA Rep. ITo. 411, 1391.

2. Theodorsen, T., and 3arrici:, I. E.: General Potential
Theory of Arbitrary 'ing Sections. NACA Rep.
No. -52, 1955.

5. I-iiman, Irven: Numerical Evaluation of the c-Integral
Occ-urring in the Theodorsen Arbitrary Airfoil
Pot.-rtial Theor>. NAC.' AHR o10. L-'27a, 1944.






o NACA ARR No. L5H18

T-ELE I.- V.'L'JES OF Ck FOR USE WITH EQUATION (4)


k Ck


1
5
* 51
7
i-
15

17
-II


21
25




I 35
-7



i 9
59
55

wi



51;
553
55

U1
:, i




1 i

75

79


1 = 10

0 5i1 5


.3I09 L


n = 20


- p


j.65551
.20827
.12071
.0o159

.'4270
.05062
.0.2071
.012,.0
.005.


n = 41


0.65629
.21122
.12 F6
. u'8b4
.0.6777
.0b542

.05742
.'5171
.0270C.
.02311
.01 71
.01670

.0141 5
.00922
S.00705
.00'97

. 00')98


n = SJ


0 .6654
.21196
.12(91
.09037
.06999
.05697
.04790
.04121
,D36 Qr
.03194
.02853
.J2577
.o02339
.02551
.10155
.01951
01794
.016 51
01525
o.0iLc7
.01500oioo
.01202
.01111
.01026

.oo8oo71

S00755
.00o68
,0 06



o00 79
.05126
OC0274
.00223
.00173
.00125
000C74
.00025


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COh"TiTTEE FOR AERCN OUTICS


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