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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 ARBITRARYAIFOIL 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 eF'Ui.ITICO 'FU T'U THCDORSEII r PITR ARY AIRFOIL POTENTIAL THTEORY 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 arbitraryairfoil potential theory. Er hsr'nonric 3srnti.heis the corresponding conjugate trigo nonmtric seies is used as an approximation to the Ef'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 jhere 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.inaticn 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). Bcause 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 '.ecnrmnined 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 rference 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 seris given by :'I') = A + A cos c + .+ An cos (nl)cp+ An cos np + 3i sin + + Bn_1 sin (n 1)C nl = 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 2n1 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 iltermr!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 :Luution (4) uses only onehalf 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 qpr.cints ridway between the given hpoints. 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 40point method of reference 5 and requires only onehalf as much computaticnal work. Ho.' to handle small irregularities or bumps in the ~rcurve 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 Evalaes are given by the sum of E and Ae. This method cannot be ju3tfied 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. Iiiman, Irven: Numerical Evaluation of the cIntegral Occurring in the Theodorsen Arbitrary Airfoil Pot.rtial Theor>. NAC.' AHR o10. L'27a, 1944. o NACA ARR No. L5H18 TELE 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 NATI' I AL ADVISORY COh"TiTTEE FOR AERCN OUTICS  UNIVERSITY OF FLORIDA 3 1262 08103 27419 UNrIVERSITY OF FLORIDA DeLiCUMENTS DEPARTMENT 1 'J M.'.,ARSTON SCIENCE LIBRARY P.O. BOX 117011 '.AINESVILLE, FL 326117011 USA Ci 
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