Numerical evaluation of the epsilon-integral occurring in the Theodorsen arbitrary-airfoil potential theory

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Material Information

Title:
Numerical evaluation of the epsilon-integral occurring in the Theodorsen arbitrary-airfoil potential theory
Alternate Title:
NACA wartime reports
Physical Description:
12 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 more precise method of evaluating the epsilon-integral occurring in the arbitrary airfoil theory of Theodorsen (NACA Reps. nos. 411 and 452) is developed by retaining higher order terms in the Taylor expansion and by use of Simpson's rule. Formulas are given for routing calculation of the epsilon-integral and for the necessary computational coefficients. The computational coefficients are tabulated for a 40-point division of the range of integration from 0 to 2pi. With no increase in computational work the systematic error in the numerical value of epsilon is reduced from the order of 1 percent to approximately 0.1 percent.
Bibliography:
Includes bibliographic references (p. 11).
Statement of Responsibility:
Irven Naiman.
General Note:
"Report no. L-136."
General Note:
"Originally issued April 1944 as Advance Restricted Report L4D27a."
General Note:
"Report date April 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."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003605363
oclc - 71124488
System ID:
AA00009417:00001

Full Text


ARR No. L4D27a


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WAlRRTIME REPORT
ORIGINALLY ISSUED
April 1944 as
Advance Restricted Report L4D27a

NUMERICAL EVALUATION OF THE c -INTEGRAL OCCURRING IN
THE THEODORSEN ARBITRARY AIRFOIL POTENTIAL THEORY
By Irven Naiman

Langley Memorial Aeronautical Laboratory
Langley Field., Va.


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 136


DOCUMENTS DEPARTMENT








































Digilized b, the Inlernel Archive
in 2011 *ilh landing Irom
University ol Florida, George A. Smalhers Libraries wilh support from L',"RASIS and the Sloan Foundation


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1I;.'T'.IC. L rVALT-Tl''- C1.F IF!E -7T .' nL C'CIrRT TT: T'T

LT. i Ti.-j", C"t.tf ART l i 1, V _L ,T:.. L-TIAL T',CRY

B Irven : a'r, .n


C'IT .. "'TL .


A r.c rr ,r cis- ,rs t:Ir-:. 0,c hat.:-l'at' .r tic e :.te -"ral
ncc 'rrinr- in th"e anrb" tr r-'" ]r-,]- ] t.n-" r"y ,:1 0 -l .'..,i -,r; n
( .CA lei 3. :o 11 d CL 4C")I i e E'elope ,' r ct -:.:lli.
c c ".. s c e c. e r EI.. L', i/
I'5 hr :rdtr te: i.- th're rC-- y- ex ;i n d "1-' r u.'
of c3i. pson's r'l. ''r'ula aire riven f.r r- tinc ca.cu-
lation of the F-interrat l a -1. 1o the reces .c.sr-. c 1-.ru-
ttti '-l cne ffi ien r.t- '2'1-0 c m' t t t.sl r- eff: c E t-.t
are ta'cvlatc fPrm a 4'i-roi' di' :ion of the ran,' :1f
intee-rati .n-,. frc.i t- C'T, "ith :- inef-lr"-p in c.:,pu-
tat ional v''-,r: t-ir- ".1: c '-a tic err r in the r:..r ".nl
value r f C is reduce'.\ frr:. tie c'rd r of 1 r rcr-r.t to
appr;:.'iatcely 0.1 percent.





r' e _oluti n :f t .e gei.er al r 1ebler; by n_,rnr cif
ccnftr:.al trar.. forr.ron fp t-.r f'lo-" rbcjt anr. ay-i tri ry
airfoil (ref 'rc i. c s 1 and -l ) a -:. *.uti cal I 't ra 'c
(reference 7'), !'U a bipla.c (r'cfYennr 4) IvlVr r t'h
determinatnI n of the irc. in..r. ". -rt o' a c;,r. if;: tra:.-
.Vc rrrmation funct Inn, f- VIEn t'e r".'il pi rt. %,' '.,'r in
ref e rences. 1 and ; the real f-art .a-r ,' Eb eLC..d in a
Fourier series and ther ina..ir.ary parL io tr,. con.7u-ate
Four:ir series. It is :12 sh-..,rn in ti'-scS refec rincc-
that the i"arinnr: pi art E may bp i:,btair:e:-1 fro'n r I e
real nart by the following. fuec mtiU nal EcL-:qtiort:



rs.. (1)
2,n







ITACA ArR No. T4D2'7


This ir.te -rI] oc- evir frequ;iint1- ir n c fnfor-La] -
trsi .n c'for.atilo.- 'cb-. i'rbl .s in b'l fi n j i trio evaluati in of the
fi'c :' n o tie cii' ".: : p-r-t c;: ,,'F, for the ijiu er' local
cv'al-.:t in n of th s Ei r, ) 1 F -: "ii refereic.e. 1
arC .? Thi' -s-eh, cd, C.l".ic s currl-.-t ly ir, ;ve at L'TIAL,
iv'e'.- e.n r-r.3r *.f abc t I.F prcrrt cOr a 40 -.rI- nt
aivisi n f, : tie ranfe of in.tc r, t i n. An ir.prc..rnie'e't
in the acurac-' is t'.E.ref.re very Je irab-lc, n rticul Erly
'f th ab- r in"l': Ive-i not inc reaneri r revised me'thod
'iven here-n i- f'ouni to in,'ol',e a little less -.'ork: than
'.hat rre'v .l- 'sl- reqi..red -.n t Vi' r: .IrT--r -'f cr.1y
abLCou 0.1 ..pre'-nt. C' t.,-ts 'r us3 in t "'s r]:.re p e-
cise : thod have- be en cc :.'i tcd .d arec pre'!s:ntod in



S'.C. 7. ..L .' -.' _O .


TL'e e'rlu.ti'crn .' t'I r- n'-it .i 'ro cor..licr t ed y
t. Ge d isccIti uLt.- at 'r" C-1? .U1'f-icr'lty ru- 7 Le
Lur.:rcot-ur.ted b" a .ec a-rat e -oluti i: arc.ss 1h-e adi-con-
tinu'ty. ',en = c' i sU.::st tuted in equatior- (1)


c /t -'1
e(,'i) = -, -(.:,r + .) co:,t ds
.'.. J


or, because of th-e iric.tc it-i t tols f:..-t.tl.n,



J-c


ri(') 4 1. + '. t)
'e d.-contf'.rui'"v 'I cc'ir 't r Fcr u'. rr.c-F-. r f
nuier l', 1? evc I'v'it on .1 LYis intel rn1l. .:m.' be brk':ern i, aCs
follo'.:,f:



(C''l = -" / *;'(" + s) c t d2 + -' + cot do


= C1 + c2 (3)









!'ACA AF.FR 1. L4T'Z7


,3
:1 2T- OI + s) cot aL
-


and


(7' + s) cc. K '


Tval'.at i n of


c .- ,'he f-irst i'tec rrl


includes


the dl scont inf lit: ian'il th litP s r ay be- ta':en as _cf.e
convenient small value. :vy a "a'-!.o' -eri' e ex':.r.nson the
integral is easily evpluatcd a.- follovn



r( C + + s( ') + ri'( ') + ""+"(t ') ( c( ') + ...




V.'ben thi cExpansion is substi uted in equation (4) t',e
interrals containin; Lthe even-)rdereQ derivatives a~?
founLdl to be idertically zero. Equation (4) t':ei- '.coines


-s I
1 iY 7
S- cft is + s' cot ds + .. (C)
J-" '" --3


v:herr. the derivatives are evaluated at cp'. The Taylor
expansion for >ot 9 is



s 2 s s3
cot ...
,-7 3 6 360


T r .1',

ry I .
or" "F ~ r








4 1,ACA ARR :.1. r4D2'


;:nd c u.-t!7 ion ('3) is rchu o,'tait r-,ld as


1 ,-4 1.
c- t .. 5 I4'
~ ~r I+


ic:


/

(~1..


r .t
4^"


r


II /
I' :1..


/ -2
(1-K


- 4
1~G^D


.7)


. .) + .


.. alu:-t ;n of (.-..- j- : c '- .:i i tc. c l
t.'.on ( )., a e. rEarr. -l cc -. n.ve,.icn'e e in
calcula.tion as s .'. ll.'.r .


C.-, ( equ&-
uni;er :ic-. al


e = -- / ( + s) cot c + ( + s)
-S TT


cot i d.E]


= -- / L (J ,' + s) cot ri.+ '.'(D' s) cot + ds _


--ns
ST
= j Li('rL' + 31 1 9) I C;t d (6)

-J


where -s P as bcer, s ubitituted r'' s in th'e -eccvnd
inter-ral anC the limits have ',.':F. -' -r.an.ed acc rdin-ly.





i' '.i:' "' ,.,'-"r renr e 1 -- -r. t M fiu r :i e Lh2. ".i.Lerval
' to An i i.- 'cd int, o n e.iu l I:-rts :f r. itudec-
/n n s n ev/n (n is an vn n'iiber he v'uliues of P r re
designated n '.- : *

wh r i e th alu. of '2 att Q O and ni = ln
"^ n


1 L
c = .. |4 ,
-- -- ^" L









'ACA A.. !!o. I,4D2"


is the value at CD = ': Tr. The Inte-rs-tions are per-
'. r. 1e. rve: nterval].s .f ,'ldth On -n 2 th e v'al.es
t. ti".e ridcmint Uf the intr'. -51. The C ~ ne! cf it -ra-
t i n for E i frc.- = -1i', t = rn n'.d
r. fr- 1 ir n t ir -

_te fir-t i:t-rr.. l C1 is r.'. a trd h;' rK.. r."..
on y,- the f'r t- :r.- r t rrr.s -.n s,


_- 'I -'- ("-'



wh.re oI.E .l1re v :IL -':ti r ni. c.+ r.'"i. ll': L .),
that i at. = ( s = n .

?he Sez ,.-rl in, re" l c, .s r :,r t : th- .: .
inter-.1l ac;-.s: .e.ch :.r: erv-r,


i.-I ----r
S 1 x /
C =- -- / "
= 1 -.
~1


l.(-?t + oHl e,:t -. cs


The function 6 does not ca.r.sE rn.ijh na-cs-, the
interval an: is ta re f-re a .prc xi-ate-:! r uot of the
value at the nidm-ir.t Vk cc' ?n:r,


IT


n-1
L

!: 1


,-+1]
l Ti


/2 -_
J -


C
C-.t ,L^.
A.


n-i sin ---1

" / :.. 2k-!
S Is in ----g
-2 2n







6 !:ACA AER .0o. L4D27


or, by eaq'.aiin (C), with 4_- = k









nn
ChC c r e

+ 4 + f sI













I 7 .eth- '1 T c a I 1 (1 )
Sf (11)


an: Ci


The corirlete i."nt-._rral "s .tven h- c -- Ce + '2' 'r



L -:


Valucsz of the *contaits a zre gi.'cn in refreenre 1
for n = Id anld n rfc er ence 2 for n = Cle Revised
value- fIr these Cr:LantOt tvcG their 'with those f'or
n = 4'', are ric"r. in tble II.

Im pr-v'rd ',.eth:,d.- T'ie nu..neric.tl accuracy of the
ev- u.-tatior, of the C-inte ra.l *',ill be sho.:-i to br
irmpr-v'.'e br' tha fo cllm in.- h'etL'-i,: T:"ie in:c-rval 0 to 2n
is d'.'iided into n equal parts End :the valluer. arc
desirnated a.s in the re'.'ious ct ion. Tcio second
inte-r-l is v.ali..tl.ed by 3 ti,.p s*n' r.le fr.ori r
to C n-il n-1 1 r l L. fL, ra ae -f irLterratjcrl for E1
is thz-reft:re t'."iee A. Ia'"n as that in tnh. previous sec-
ticf that iS., C -1-r'n tc 21:',n. The c!pr.:-.ir.:atior
in 'h]:-h only the f- -t-'-,rder t,- rm eq'iati.nr: (7) is
use. is rinsuffl.iper.n ar:d the hi'; l-.r deri'.-ativc: must-'c be
uses.. The:'e de-rivat'.ve? are r.:ost conv'niently obtained
b" nurierical diffcre'Ltiation. *










::ACA ARE Fo. L4D2' 7


The :.e't-n- r, rl'.:r f r:-.ul' 2:... J,. 't.'.'.-1.-'e r ,' -
eP.r e 5, :.. "5) a-e


1 1 -
-' 5 = -' + --._ .. .



T'7 1

*? a o I
-- ^ --- + ...



L .- CL


.''ere = 1 "ul r-.t r -- 1. ( n '..) -, -.7 ca f-

valfe~' as


( 1 ')


,'



_- ** -- -2



= u 4"' + ["'m 1 -i
/ t -.^


+ 4',- -


2'-,/ s-.b't u c ':n .1 r l.Iat.c.-: o (1..') i. '.-ti n ( 7)
pglvc'


1 S
S1 -




+ 2 5 ----
,_ ,. i


A I- *
2.5'trQ~ ii~ .2
150(9 .'s.) I
/ I.


N
I
.
.. u


(15)


r *-'
-U


C-
C. rL.~
* I-,


C-







/.,CA ARn Eo. L4D27


The- flurter substitut'ionr of relations (14) in equa-
tion ( 151) ives




S= < It-1 + 4 + + .0


+ + ) + & ... .
l15 d 4


+ + )


+ +' I *.) 1
I-


* ~ ~ n
/.. I


1 bl 'l W) 1 -C-2 "'2) "'- + )


, he re


E -- 2+ i- )- s(- + T+r 5132) -""5


167 g \
2520 s'- .../


21 3 1
- Tr\9





S1 (83
- ~^\225


+ ) -
+ 2--*./ -

C 2
~315 s
13 SE


+ -F
T, C.


-'I


N2L~


1 / 19 se2
S2r\450, 1512 -


(1C)


> (17)


-s 1-


2- -L) + ...+









rI;.CA A'7-. No. 14D:'7 9


The secor.i integ-al t is evaluste- by Sir-,.-'on's


L r D .1


c' -c I cct +- + T cat



-.v --.
+ -" + ;"- .L


r-,' I
L...


I-


.(,, -_


1


'- s-l c ."I
IT



C'. 1 7r
U --- -o -


where (-xcep.t in *hEI
a = 4 for" er: ..
c = en + C or


f'rvt ter:-) c = 9 fr l.'id and
'T e c r.'. l ..t in1:--r-1A -v .' ,r


r '-
:. _L
r
i;: j


(20)


*_ K' '.,)


cr, b: CQ.J'-:i t i


- L- --


wr.ere


(I -3)


( 1 ;


rul. e r-"r i-',


- '4, .",
= p







IV IIACA ARR No. T4D27


where

A,_ = b1. + ci- (21)


q.-u r f A,. f>- n = 40 are ."'ven in table I.


Crr"C Vk CLT
nr-P' [ ...CY .-'- IVA-U .TT, :


The sc irsc f : .r t: r a ret ov.c of evaluation
.dc -cri. r -d af" c I r'.te rr.. -' I '-: rat r.' arr'ilouo
harrrcnic, Tl.e r c- lt -..-e pre "rited a- r-Lt.Y s -'f the
irn:tuer -.tod vJ, ue t th : c :'r.ect value that a v .ve
of L.nit: is a cortr--c't e, .iuati.:n. Val.iae o'f thi, ratbi
far' the 1.9r.'nicc? .r- e:


Rarm nI'.c 40-r : int me Lti-. 4C-roint .re rent
o' re4e er.ce 1 'etn.-




4 .1

10 1. 3112 .. ...


ITna -uch s the higher hsr -.- ..t *.n er
in a mu:i "r t-r] prt n tl. .. .- r
such c:'' rt. c c ur:te '" .. L:r the
errc r -r rt .:Ehv of-' c, e
ordFr c .r. nt, w1 .r' l t .'' '. :. ...-:
pr'e e r. Is a.p r'o.iT.t y U. 1 ; .. .



Lanrle' -r.- A'?rcnr..it cal T bir t r

-- i. .









:ACA Th.r 1o. 14D27 11





1. T?.-:- rseen, '.'-.dore. '... ory of "-ir.- section of
Arbitrary "o',e. ":.CA "-o.. c. 411, l 'l.

2. Tw7jr-dorz.en, Theodore, and G>rric'k, I. '.: General
Potential Theory of Arbitrary Wing Zcctions.
F- ,4.? 7*,
,.. .. r -. ", -. 4 j, I .1 ,

3. Garrickc, I. 7. "-- t:w Plane Potential Flow past a
cvi.e-trical Lat-.Ice of Arbitrary Airfoils. M'A"
4.F "3. 4.'.'", I .' .

4. Garrick, I. 7.: -- tenital w about '.-bitra:-:
E'r-lare "'." r: '.*-.t ons. ."..A -p. '3. 542, 1936.

5. Lavis, arold 'i.: Tales' of the T gher -sthemratical
?u'-nctions. Vol. I. T'e Principia Preps, Inc.
(Eloormingtor, _-..), p. 75.







*I.ACA ARR 1o. L41r27


TABLE I.- VALUES OF Ak


FOR USE 'WITH "hl'T.ION (20)


'I -_________ -- ______ -


_ A_
! :, : 40 ..r


Ak
n = 40


0.52827
.1-1'.C4
. 614
.10259
.04024
. '542


.01U1
`.'. 33


11 0.01423
12 .02,-2
13 .01 1
14 .01698
1I .C('C. .10
16 .01083
17 .004: D

19 .0 *066
20 0


TA3L-7 II.- VAL"W:' D' a,


PC" "t WITH E.U:.ATI:T (12);


,MT'CD OF ."_.1 CS 1 AliD 2


k


1
3
4
5
6
7
8
9
10







UNIVERSITY OF FLORIDA
II I I I I III III I II ,
3 1262 08103 311 9




UNIVERSITY OF FLORIDA
DOCUMENTS DEPARTMENT
120 MARSTON SCIENCE LIBRARY
PO. BOX 117011
GAINESVILLE, FL 32611-7011 USA