ARR No. LlH»7a 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 WARTIME REPORT 
 
 ORIGINALLY ISSUED 
 
 April 1°M as 
 Advance Restricted Report lAD27a 
 
 NUMERICAL EVALUATION OF THE € -INTEGRAL OCCURRING IN 
 
 THE THEODORSEN ARBITRARY AIRFOIL POTENTIAL THEORJf 
 
 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 
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 nically edited. All have been reproduced without change in order to expedite general distribution. 
 
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nil 2 
 
 
 
 \l> 
 
 NACA ARR No. L4D27 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 ADVANCE RESTRICTED REPORT 
 
 [erical evaluation of the e- integral occurring i 7 t 
 the theolorsen arbitrary airfoil potential theory 
 
 Irven Naiman 
 
 SUMMA3 " 
 
 A more precise ra I of evaluating the e-integral 
 occurring in the arbitrary airfoil theory :f Theodorsen 
 ( t\CA heps. No a . 411 and 452) is developed by retaining 
 / her order terraa in the Taylor expanaion and by use 
 of Simpson's rule. Formulas are riven for routine calcu- 
 lation of the e -integral and for the necessary compu- 
 tational coeff icients. The computational coefficients 
 are tabulated for a 40-point division of the range of 
 integration from t"> 2tt. With no increase in compu- 
 tational work the systematic error in the numerical 
 value of e is reduced from the order of 1 percent to 
 approximately 0.1 percent. 
 
 I] TRODUCTI ' 
 
 The solution of the general problem by means of 
 conformal transformation for the flow about an arbitrary 
 airfoil (references 1 and 2), a syi etrica] lattice 
 (reference 3), and a biplane ( cence 4) involves t] t 
 
 determination of the imaginary part of a cor-] lex trans- 
 formation function, given the real part. As ~' ! wn in 
 references 1 and kJ the real part may be e ded in a 
 Fourier series and the imaginary part is the conjugate 
 Fourier series. It is else shown in these references 
 that the imaginary part e may be obtained from t] e 
 real part \!/ by the following functional equation: 
 
 r2TT 
 e((pi) = _^- / *(<p) cot * " CP ' do (1) 
 
NACA ARR No. I.4D27 
 
 This integrs 
 transformatio: i 
 
 fuucti.ons oil the 
 evaluation of thi 
 This meth- 
 
 ane! a. 
 fives an error of 
 division of the r 
 in the accuracy :' 
 if the labor invo 
 given herein is f 
 that previously r 
 about C.l percent 
 cise method, have 
 table I. 
 
 1 occu 
 obi ens 
 circle 
 
 s i ite 
 od, wh 
 ab c ut 
 ange o 
 s ther 
 lved i 
 ound t 
 equire 
 , Con 
 been c 
 
 ra f re 
 
 invol 
 
 A p 
 
 gral i 
 
 " - . i 3 
 
 quently i 
 
 the 
 roce&ure 
 
 s riven 3 
 currentl 
 percent fo 
 
 ii. te 
 efore 
 s not 
 o invo 
 d and 
 stants 
 : )ute 
 
 gration. 
 
 very desi 
 increased 
 Ive a lit 
 to give a 
 
 for use 
 d and are 
 
 n conforma] - 
 
 evaluation of the 
 for the numerical 
 n reference? 1 
 y in use at LMAL, 
 r a 40-point 
 
 An improvement 
 r ab le , p a r t i c u 1 a r 1 y 
 A revised method 
 tie less work than 
 n error of only 
 in this no re pre- 
 
 pr e son ted in 
 
 EVALUATION 0? TIC e- INTEGRAL 
 
 The evaluation of the ^-integral is complicated by 
 the discontinuity at cr = o' . This difficulty nay be 
 surmounted by a separate solution across the discon- 
 tinuity. When s = cp - cp' is substituted in equation (1) 
 
 £(<?') = -57? 
 
 i r 2rf -w 3 
 
 / h{^? + s) cot ; 
 
 •cp' 
 
 or, because of the periodicity of this function, 
 
 e(cp') = 
 
 ..2- / 
 
 2tt- 
 
 '| r (cp' + s 1 ) cot 2 ds 
 
 (2) 
 
 The discontinuity now occur? at s = 0. Tor purposes of 
 numerical evaluation this integral ma^ be broken up as 
 
 f cllov ? r 
 
 e(cpt) = -i- J \J/(cp' + s) 
 
 -J -s 
 
 cct ~- ds + i \|/(p' + s) cot -1 d: 
 
 J. 
 
 e i + £ ? 
 
 (3) 
 
NACA ARR lo. L4D27 
 
 v.'her? 
 
 rz 
 
 ■1 - 
 
 1 I 
 2 
 
 i \ 
 
 i 
 
 -'-a 
 
 «>(cp ? + s) cot % ds 
 
 (4) 
 
 md 
 
 r2TT-a 
 
 
 " 1 
 
 l|ffi>« + s) cot £ ds 
 
 (5) 
 
 Evaluation of c 'i • ~ The first integral c, includes 
 
 the discontinuity and the limit s may be taken as some 
 convenient small value. By a Taylor °eries expansion the 
 integral is easily evaluated as follows: 
 
 i!'!v' + a) = l'(v') + s\|/' (q>«) + 4-i|r»(<pi) + |^|fiii(<pi) + ... 
 
 When this expansion is substituted in equation (4) the 
 internals containing the even-ordered derivatives are 
 found to be identically zero. Equation (4) then becomes 
 
 _1_ 
 2rr 
 
 .'3 
 
 C Xj 7t \jl s 
 
 \|/"» i 
 
 7. I 
 
 CO 1 
 
 ds + 
 
 (6) 
 
 where the derivatives are evaluated at cp' . The Taylor 
 
 g 
 
 expansion for cot -^ is 
 
 cot -^ 
 
 3 
 
 6 
 
 
II AC A ARR No. L4D27 
 
 ind equation (6) is thus obtained as 
 
 fi = 
 
 ;rr 
 
 ■ R '-" \- 36 "3600 ~ •••/ 
 
 Vs 
 
 
 / 
 
 s°\l/"» [ 1 
 
 £0 1680 
 
 o5 V / Co 2 
 
 150 T V 84 
 
 'J 8820* V^ 108 ' " -y ' 
 
 7 ; 
 
 Evaluation of £o.- The second integral Co (equa- 
 
 tion (5)) nay be rearranged for convenience in numerical 
 calculation as follows: 
 
 "2tt 
 
 ~TT 
 
 / \|/ (cp ' + s) cot 75 ds + I 
 
 -'s ♦% 
 
 ,-2rr-s 
 
 ^ r (cp' + s) cot § ds 
 
 _1_ 
 
 "2tt 
 
 i r 
 
 'S 
 
 •vMcp' + s) cot ^ ds+ / \|/(c?' - s) cot - ds 
 
 m 
 
 "1 
 
 v|/(co' + s) - \J/(p» - s) j cot | ds I 
 
 J 
 
 ( ■■■■) 
 
 where -s has been substituted for s in the second 
 integral and the Units have been rearranged accordingly. 
 
 numerical ksth< 
 
 [ of re feren ce 1„~ In reference l the interval 
 
 to 2tt is divided into n equal parts of magnitude 
 
 2rr/n (n is an even number) . The values of \|/ are 
 d< signated \j/ «. \!r „ ,..<,, \J/ 
 
 Vn, \ 
 
 
 J- *1> ' ' ' > 
 
 w] ere ^ it the value of \j/ at cp = cp< and V: n = U.- n 
 
 "2 2 
 
o. L4D27 
 
 the value at cp = c?» ± rr. The integrations are per- 
 ked over intervals of width Srr/n with the ty values 
 the midpoint of the interval. The range of integra- 
 tion for e-i i3 frc;n s = -Tr/n to s = rr/n and for 
 e from njn to 2ir - — . 
 
 The first integral c^ i- s evaluated by retaining 
 only the first-order terns in s, 
 
 e l —^V = -|V ( ;) 
 
 where bhe slope vv ' is determined graphically a1 
 that is, at cp = <p ■ (s = 0). 
 
 The second integral £o is composed of the si 
 
 integrals across each interval 
 
 2k+l , 
 
 n-1 r~--- 
 
 £o - - 
 
 c " f~: i'2k-l 
 
 k=l •• TT 
 
 n 
 
 ^ '<tt 
 
 1 ^ n 
 7— / I *(<P f + s) cot £ ds 
 
 The function v!/ does not change much across the 
 interval and is therefore approximated by use of the 
 
 cp ' + — — ). Then , 
 n / 
 
 n-1 c — ——it 
 
 n 
 
 n 1 • 2k+l 
 
 , n-1 sin — ; — rr 
 
 - - -J ^> M/ v log ^_ 
 
 7~i sm — '- — TT 
 
 K=l 2n 
 
NACA ARR No. L4D27 
 
 or, by equation (8), with ^ n _^ 
 
 \b i 
 
 n 
 
 /2 
 
 NT 
 
 k 
 
 k=l 
 
 (10) 
 
 where 
 
 a k = log 
 
 n 
 
 = 
 
 .. 2k + l 
 
 cm — - Tr 
 
 _2ri 
 
 2k •• 1 
 
 s m — — — — r r 
 2n 
 
 (II) 
 
 ?he complete integral is given by € ~ c i + € 2> or 
 
 £ - -<§V + f/L a k(^ " *-*)f (12) 
 
 L ~ k=1 J 
 
 Values of the constants 
 
 for n 
 values 
 n = 40 
 
 and in reference 2 
 ese constants, tog 
 
 , are fiver,, in table II, 
 
 were given in reference 1 
 = '20. Revised 
 
 n 
 for these constants, together with those for 
 
 n 
 
 th 
 
 ev 
 
 Imp roved me the 
 evaluation of the £ 
 improved by the fol 
 Is divided into 
 designated as in 
 integral e 2 i 3 
 to * n-1 (M/ n-1 - *_ 
 
 is therefore twice 
 tion, that is, s ~ 
 in which only the f 
 used is insufficien 
 used. These deriva 
 by numerical differ 
 
 d 
 
 .- The nu 
 integral 
 
 merica 
 
 - 
 
 will b 
 
 1 
 
 owing met 
 
 hod; 
 
 
 equal par 
 
 ts and 
 
 e 
 
 previous 
 
 s e c t i 
 
 a 
 
 luated by 
 
 Simps 
 
 1 
 
 V The range 
 
 a 
 
 a large a 
 
 s tha t 
 
 
 -2tr/n to 
 
 2tr/n 
 
 
 i - 1- order- 
 
 t e rm 
 
 t 
 
 arid the 
 
 higher 
 
 t 
 
 ives are 
 
 1 c 
 
 e 
 
 itiation. 
 
 
 1 accuracy ox" 
 
 e shown to be 
 
 the 
 
 2TT 
 
 the ty values are 
 on. The second 
 oil's rule from \l'-j_ 
 
 of integration for < 
 
 in the previous sec- 
 , The approximation 
 of equation (7) is 
 
 de r i vat I Tr c s mu s t b e 
 onveniently obtained 
 
NACA AH7 io. L4DI 
 
 rhe Newton-Stirling formulas for derivatives (refei 
 ence 5, p. 75) are 
 
 
 1 *, 1 E 
 
 gS^fll 
 
 C. -T 
 
 - 5 3 '" - 
 
 ** * 
 
 ■' - ' = 6% - 
 
 J 
 
 (13) 
 
 ore s = tabular In 4 (2-Tr/n) . The mean central dif- 
 ferences 6'lc can be expj of the tabular 
 
 values as 
 
 26^ = \|/ n - ty 
 
 -1 
 
 
 o, I , , o» I 
 
 ■-1 " -: 
 
 (14) 
 
 
 = \I/-z - 4tyo + 
 
 2 + sxi;^ - 5V_i + 4V_2 - V_3 
 
 J 
 
 gives 
 
 substitution of relations (13) in eq I * sn (7) 
 
 £i = - 
 
 4, 
 
 L 
 
 g^«6'H2 
 
 s 
 
 13 
 
 . 
 
 - 25 
 
 '^ ~ 
 
 O.J * "3 
 
 ^ / 13 
 
 1512 
 
 o • / " • 
 
 (15) 
 
8 
 
 NACA ARR No. L4D27 
 
 The further substitution of relations (14) in equa- 
 tion (15) gives 
 
 e l = 
 
 J- Ai, 
 "2i\\ c l 
 
 '-l) 
 
 (, 
 
 19 
 
 2 + ■£ + ■—■ + 
 
 90 • • -y 
 
 4/x + iL + A 
 
 18\ 15 G4 
 
 \^c V -2;ilg + oo 5 + ...1 - S ^ &70 + 37 q + • . .^ - 
 
 + (^3-^3) 
 
 19 
 
 450 + *• 7 " S "l 
 
 f 1 
 
 x^j. . 
 
 + 
 
 'J 
 
 i 
 J 
 
 or 
 
 e l ~ b lO-l ~ *l) + b 2(V-2 ~ Uf 2) + t 3 (-i/_ 3 + Mr 5 ) (16) 
 
 where 
 
 b l = ffl 
 
 / 4 19\ p/l i 5 \ 
 
 y 9 90/ " s ^18 135 1512/ 
 
 1 /239 167 2 
 
 ' 2ttV90 2520 3 
 
 - ... 
 / 
 
 ^E-^r(Mi)- 
 
 2/11 
 
 S U?0 378 
 
 1 /83 2 2 
 
 Tn\225 ~ 315 s 
 
 \ 
 
 h 1 /l9 s 2 
 
 / 
 
 / 
 
 ° c = 2ir\450 1512 
 
 1 
 
 > (17) 
 
 > 
 
NACA ARR No. L4DI 7 9 
 
 e 3ecor.d integral r - is evaluated by Simpson's 
 
 rule from \i/-, to ^ n ._i 
 
 c 2 - -:•-'— ~-!'-'i cot ~ + 4\!/ c cot 
 
 2ir 
 
 .t or, I 1 
 
 11 
 
 + :'v: , cot — - 
 n 
 
 
 or, by cquatS •/.•,,•: 
 
 n/ 
 
 /o 
 
 
 
 wttere 
 
 CO 
 
 TT 
 
 c . kir 
 
 ■5— cot - - 
 Oil 
 
 c n E 
 
 ^ 
 
 (19) 
 
 where (except in the first term) c = 2 for k odd and 
 a - 4 for k even. The complete integral is gj 
 
 c = Cl + £ 2 , or 
 
 2Z? 
 
 j.. - X 
 
 (20) 
 
10 
 
 NACA ARR No. I.4D27 
 
 where 
 
 At, - 1>- + Ci 
 
 !1) 
 
 Values of Av for n = 40 are Riven in table I. 
 
 ACCURACY 0? EVALUATION 
 
 The accuracy of the two methods of evaluation 
 described may be determined by integrating various 
 harmonics. The results are presented as ratios of the 
 integrated value to the correct value so that a valve 
 of unity is a correct evaluation. Values of this ratio 
 for the harmonics are: 
 
 Harmonic j 40-point method 
 
 of reference 1 
 
 4 0-point pre sent 
 method 
 
 1 
 
 2 
 3 
 4 
 5 
 10 
 
 1.01434 
 1.02662 
 1 .03700 
 
 1.04 547 
 ] .05280 
 
 1.03112 
 
 0.99977 
 
 I -CO ')44 
 r : 04.0 
 
 1.00060 
 .9 J 8 iO 
 
 .95~::0 
 
 Inasmuch as the higher harmo: ics • ~ ~. 
 in a much smaller proportion than the 
 3uch contours as are encountered 
 error 01 i ; ' ■ ".• 40-point method of 
 order of 1, rcent, whereas tr. 
 presence' - ?in is approximately 0.1 D.eroe: 
 
 ' : j .nter 
 
 for 
 
 ■ s - the 
 
 is y. 
 
 the 
 
 met] 
 
 LanrJ r " i orial Aeronautical Laboratory, 
 . v 3 - Dry ' ". cee for 
 Lar -:i c y Fi el d , Vs , 
 
"A ARR No. L4D27 11 
 
 REFERENCES 
 
 1. rheodorsen, Theodore: Theory of V. : ir~ Sections of 
 
 Arbitrary pe. NAG A Rep. No. 411, 1931. 
 
 2. Theodorsen, Theodore, and Garric 1 :, I. '."..: General 
 
 Potential Theory of Arbitri :■" ctions. 
 
 . fc.Oii lie ... • i.O. --•- C , X1'«-" , «J . 
 
 3. Garriclc, T. ~. : Ct, the Plane Potent: lc past a 
 
 Symmetrical Lattice o. Ltrary Airfoils. flACA 
 
 ARR ? T o. 4A07, " 
 
 4. Garric 1 :, I. E. : tenbia] pbitrary 
 
 Eiplar.e Wli : ' :tions. EJACj . No. £42, 1936. 
 
 5. Davis, Harold T.s Tables c n the ! • Mathei atical 
 
 Functions. Vol. I. The Principia Press, Inc. 
 (Bloomington, Inc.), 1933, p. 75. 
 
NACA ARR No. L4D27 12 
 
 TABLE I.- VALUES OF A k FOR USE WITH EQUATION (20) 
 
 k 
 
 A 
 
 1 
 : ^40 j 
 
 J 
 
 H: 
 
 n = 4 ) 
 
 1 
 
 1 
 0.62827 
 
 ! n 
 
 0,01423 
 
 o 
 
 .14824 
 
 12 
 
 .02422 
 
 5 
 
 . v ,'614 
 
 1? 
 
 .01021 
 
 4 
 
 . 10259 
 
 14 
 
 .01698 
 
 5 
 
 .04024 
 
 ! 15 
 
 .00690 
 
 6 
 
 .06542 
 
 1 16 
 
 .■'1083 
 
 i 
 
 .02720 
 
 17 
 
 .00400 
 
 8 
 
 .04E i 
 
 18 
 
 • OOP 2 8 
 
 9 
 
 .0.1951 
 
 1 ] 
 
 .00066 
 
 10 
 
 . 03333 
 
 : 20 
 
 ' . 
 
 
 
 i 
 
 TABLE II.- VAU~r DP a k P C : USE WITH EQUATION (12); 
 
 ; ::: r ": or i ^ ss l and 2 
 
 
 
 ak 
 
 ' ' 1 
 
 k 
 
 
 
 
 n = 10 
 
 n = 20 
 
 n = 40 
 
 1 
 
 1.06544 
 
 1.090::'" 
 
 1.09056 
 
 2 
 
 .44211 
 
 .49426 
 
 .50671 
 
 3 
 
 .23117 
 
 .31141 
 
 .33028 
 
 4 
 
 .10302 
 
 .21750 
 
 .24303 
 
 5 
 
 
 
 .1577? 
 
 .19028 
 
 6 
 
 
 =11449 
 
 .1545? 
 
 7 
 
 
 .0! 
 
 .12 041 
 
 8 
 
 
 .05115 
 
 .10326 
 
 9 
 
 
 .02493 
 
 .03207 
 
 10 
 
 
 
 
 .07362 
 
 11 
 
 
 
 .06714 
 
 12 
 
 
 
 .05711 
 
 13 
 
 
 
 .048ie 
 
 14 
 
 
 
 .04004 
 
 15 
 
 
 
 . ^, \J i-J \.> I 1 
 
 16 
 
 
 
 . Kj <0 O *_/ 1_/ 
 
 17 
 
 
 
 .01837 
 
 18 
 
 
 
 . 01245 
 
 19 
 
 
 
 .00613 
 
 20 
 
 
 
 
 
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