ABR No. LitG05 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 WARTIME REPORT 
 
 ORIGINALLY ISSUED 
 
 July ishh as 
 AdTance Eestrlcted Report LUGO5 
 
 AIRFOIL-COHTODR MODIFICATIONS BASED ON € -CURVE 
 METHOD OF CALCULATING PRESSURE DISTRIBUTION 
 By Theodore Theodorsen 
 
 Langley Memorial AeronauticcJL Laboratory 
 Langley Field, Va. 
 
 J: 
 
 NA(5X 
 
 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. 
 
 DOCUMENTS DEPARTMENT 
 
Digitized by tlie Internet Arcliive 
 
 in 2011 with funding from 
 
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 http://www.archive.org/details/airfoilcontourmoOOIang 
 
■7'f 2^2- i^V 
 
 •TAG A ARR Wo. rJAG05 
 
 ITATTONAL ADVISOHY COMIvETTEE FOR AERONAUTICS 
 
 ADVAI'^C^ uES TRIG TED REPORT 
 
 me 
 
 AIRFOIL-CONTOUR MODIFICATIONS BASED ON e-CURVE 
 '''TETHOD OF CALCULATING TRESSURE DISTRIBUTION 
 By Theodore Theodorsen 
 
 SmmARY 
 
 A r.ethod, based directly on the so-called e-curve 
 ..--thod published originally in 1931 in N -G /^ Report 
 No. iill, Is presented for use in making modif Icatiors 
 to the shape and pressure distribution of a given air- 
 foil. In D articular, it may be desirable to remove 
 excessive irregularities or local peaks in the dis- 
 tribution. In this process it m.ay be required that 
 certain parameters of the airfoil be kept unchanged; 
 for instance, thv3 angle of zero lift, the ideal lift 
 coefficient, or the moment coefficient. From an aca- 
 demic viewpoint, an altered distribution cannot bo 
 "prescribed" because com.pliance .vith the requirement of 
 maintaining a Laplpcian flow field is involved. A 
 prescribed distribution can therefore not be obtained 
 b;V iteratioa. The process, however adequate, is 
 necessarily one of qualitative modifications. Several 
 num.erical example 3 illustrating the use of the method 
 are given iii the appendix. 
 
 INTRODUCTION 
 
 Tn 193^ 'the author introduced the so-called e-curve 
 method for calculating the pressure distribution on air- 
 foils of arbitrary shape. (See reference 1.) The 
 merit of this m.ethod depends essentially on the fact 
 that the resr.lting integral relation can be solved by a 
 rapidly conv^irgent process. In the present paper the 
 problem of effecting changes in a given pressure dis- 
 tribution is considered. The method is based directly 
 on the important velocity formula fXTl) of reference 1 
 rewritten as formula (3""'') in reference 2, a later 
 report. Peference to these papers is m.ade re-^eatedly 
 herein v;ith subsequent omission of the details con- 
 cerning the use of the e-curve miethod. 
 
2 
 
 KACA ARR No. li^.G05 
 
 Oorslder a r'^vpn ^.Irfo:'.!. ^rom the e-ourve of 
 reference 1, ij; ard c a-r^e cbtair.ed; end frorr. forTiUla 
 (XIT) of rsference 1 tLe pressure, or the equivalent 
 
 velocity, !•=! found. The velocity iiiay he written in 
 tr-rrr.s ^f thr- quantities cp, c, -If, if , c + 
 9.S in for^Tinla (39') of reference £;" 
 
 ano 
 
 V 
 
 ijr 
 
 ^= e^^ 
 
 fir (a + cp) + sin (a + €i) 
 
 (1) 
 
 y |_^inh- M/ .- sin'^ (0 - c)J U - -~J ^ ^— j 
 
 ■".'.'here the reader i? re:^erred to references 1 and 2 for 
 the moanin??; of the various G^'.nnibol^. 
 
 IJovj consider a sli^htlj^ altered pres?ur>e distribu- 
 tion. This new pressure distribution is obviousl?/ 
 related to a new airfoil contour. It is pertinent to 
 remark here that froTri a purel^T- matheinat: cal viewpoint 
 the new distribution cF.nnot be "prescribed" unless the 
 new airfoil contour al-^o is prescribed. In &. potrntial 
 flow v/ithout singularities there exists a unique I'ela- 
 ti'orsh'rj betvi/een the contour and the pressure distri- 
 bvtion in the flov>- field. A prosrure distribution 
 cannot therefore be prescribed (^.athematically) for 
 the siriple reason that the associated contour must be 
 given in order to pre'='cribe it. Thus the problem of 
 specifyire: rigorously a Treasure distribution is 
 reduced "ad absurdui;." Froin. an academic standpoint the 
 so-called inverse rroblem therefore doc? not exist as such. 
 
 Certain alterations of a qualitative nature may be 
 performed in spite of the fact that a pressure change 
 cannot be pr'= scribed. It i'' the purpose of this paper 
 to indicate a method by which, qualitative alteration 
 miay be performed. It w:ll be noted that the present 
 method of contour modilicatlon '.vili serve the intended 
 purpose of the inverse method. 
 
 NATIVE OF ALTERATION^ 
 
 It is useful to observe that several types of 
 independent alteration are possible. By reference to 
 the velocity formula (1), for instance, a change in \!/n 
 will appear mainly in the multiplying factor e'^ and 
 will thus effect an increase or a decrease in thi veloc- 
 ities everyvjhere on the contour. Thl'' change results 
 
MAC A ARR No. L'4.G05 
 
 simply in a series of alrroils of different thickness as 
 the major effect. It is interesting to observe that 
 neither the angle of zero lift nor the ideal angle of 
 attack has been changed in this operation. The quanti- 
 ties f and '^ occurring in the velocity forinula are 
 considered available for the original airfoil contour by 
 the e-curvp method of reference 1. 
 
 The effect of a change in the angle of attack a 
 is well known and need not be discussed here. In fact, 
 the main interest lies in ir.proving the pressure dis- 
 tribution at and near the optimum, or ideal, angle of 
 attack. In the following discussion, therefore, the 
 proposed changes are performed at the ideal angle of 
 attack only. In other words, the pressure distribution 
 is examined at the ideal angle of attack, tentative 
 changes are proposed, and results are compared at the 
 Ideal angle of attack. The restriction that the angle 
 of zero lift remain unchanged may or m:ay not be irrposed. 
 For airfoil contours of zero m.oment coefficient, as used 
 in helicopter blades, the restriction may be imposed that 
 the m^oment coefficient remiain zero. In the following 
 section the nature of such changes is examined with 
 several types of restriction used to fulfill specified 
 requiremeiits . Such changes may be perfo.:-med in the 
 pressure distribution subject to any one restriction or 
 to a combination of several sim.ul taneous restrictions. 
 
 METHOD OF GHAIIGIKG THE e -CURVE 
 
 The € -curve can readily be obtained as e(c?) by the 
 m.ethod of reference 1. In most cases it is desirable 
 to keep the Ideal lii't coefficient constant in order to 
 obtain improvement at the exact value of the lift. In- 
 asmiuch as the expression for the ideal lift coefficient 
 contains the factor 
 
 2 V ^' I^ 
 
 ^ 
 
 this restriction is equivalent to maintaining a fixed 
 difference between e,. and e , the values ol' € at 
 
 the nose and at the tail, resoectively. The absolute 
 values ■■■ay or niav not be keot the sam.e. "'"f both e. 
 
 and £ rp are kept constan.t in the process of change. 
 
 'N 
 
h 
 
 NACA AKR No. iI+^tO^ 
 
 the Irl.eal lift Toef r-'.c:' ent , the :^*.deal angle of attack, 
 and the angle of :'er-o lift ore retained. This change 
 IB purely local arid oxtre.iielT restricted in natare; only 
 minor changes will saonijt to this strlrgenc type of 
 restraint. In order to maVe a larger chancre, the con- 
 dition of constant an.^le of ^ero lift raay be relaxed but 
 the requirement of a constant ideal lift coefficient 
 retained. 
 
 An Impoi'tant ca^e of alteration is the case in 
 Vi/hich the morr.ent coefficient is ;<:ept constant. It is 
 shown in reference 2 that the moment depends on the two 
 lowest harnonics in the t {(T") -curve . I'y prescribing an 
 alteration Lc(o) containing higher harmonics than the 
 second the nress'-ire distrib-atl o?i may oe altered without 
 affecting the mordent coefficient. H^re, al so, farther 
 restrictions ir.ay or may not be imposed. In general, 
 the more restrictions imposed, the more manlpulatioxis 
 ai'e required to adjust the € -carve. 
 
 TENTATTVE PRSr^S'JRS CHAITGES 
 
 ^Tow a tentative pressure change is translated into 
 a chanp'e in the f -curve will now be indicat'^d. The 
 ff9)- and c'cp) -curves are assumed to oe available from 
 the method of reference 1. 
 
 A nressiire variation Ap along the contour may be 
 tentatively prescribed. Since this e>:act change is not 
 expected snyvay, e?;act relatlonshios involving Ap need 
 not De used. It is seen from the velocity formula that 
 
 — v'- or its equivalent, the pressure pr. m.easared from 
 the stagnation pressure, is given very nearly as 
 
 ^s - I (}J 
 
 = A (1 + 2^4 
 
 dCp; 
 
 where A is a function cf position only. With similar 
 accuracy, therefore, 
 
 ^±9.- 2r- (i€) 
 
NACA ARP Fo. LL1GO5 
 ■rd, finally, 
 
 v;here the Integral Is to be taken over the range in 
 'shlch the tentative oressure change is given. Because 
 this pressure change is irr.properly chosen, the value of 
 
 Ae = 
 
 for the whole range in v.hich the change is given v;ill 
 not, in general, C'.:come zero nor will the area under the 
 Ac -curve 
 
 / 
 
 /)2TT 
 
 AC dCD 
 
 
 
 become zero as required by the conditions on e given 
 in reference 1. It is of paramount importance at this 
 point to repeat that the originally prescribed pressure 
 is necessarily unattainable, as is shown oy the fact that 
 the two foregoing integrals are, in general, different 
 froir. zero. It v;ill be noticed, however, in the following 
 discussion that the essential "shape" effect iray be re- 
 tained. The process is simply to make the A^-curve 
 conforn: to the given requirements by a suitable adjust- 
 ment involving the least possible change in the general 
 I'or^ of the Ac-curve. This adjustment is made by 
 changing the location of the maximum and miinim.u:n points 
 on the curve or bj extending the curve beyond the 
 original range. The area under the Ac-curve can also 
 be made zero by changing the reference or mean value. 
 Two basic conditions m^^st therefore be imposed on the 
 c-curve; namely, that the two foregoing integrals be 
 zero. Several examples are treated in the appendix. 
 
 Finally, a pressure change for constant m.oment coef- 
 ficient must be considered. It will be seen from ref- 
 erence 2 that the moment depends on the two lowest 
 harmonics in the e(c?) -curve and the value of Em. The 
 
NAG A ARR No. iLGO^ 
 
 'process is as follows: Pre?cribe a tentative pressure 
 change Ao, find the oorresponding ae, adjust to comply 
 v;ith the tv;o basic conditions previously nisntioned, and 
 aeterunne the following, four integrals 
 
 A. = 
 
 ?•-, = i- / Af cos <? dcp 
 
 ^.^^r 
 
 2TT 
 
 ^i sin 2cp dcp 
 •^0 
 
 3y removing 
 
 A-| sin c? + 3^ COS <? + A,, sin 2cp + B cos 2cp 
 
 from the initial ^ f-f unction, the resulting ie is made 
 free 3f the tvo lov^est harmonics . 
 
 In general, ^rn will bo changed slightly by the 
 removal of the two lowest harmonics. By adding a third 
 harmonic 
 
 A;; sin 5Cp + B2; cos 5cp 
 
 both the m.agnitude and the slope of e at the trailing 
 edge may be left unchanged. This end is attained by 
 choosing the prooer values of A, and 3-,. Thus, in the 
 
 function 
 
 A|_£ = -A-i sin 9-3-, cos cr - Ap sin 2Cp 
 - B„ cos 2CP + A, s'n Jcp + B, cos 5<P 
 
TAG A ARR To .. LL1.GO5 
 the constants A-, and 3, are determined by making 
 
 A^€ = 
 
 for 
 
 v;here 
 
 #- A,e = 
 dcp 1 
 
 Cp = TT + P 
 
 To the second order in 3, there results for A, and 
 
 
 = B^ - B^ - (Ub^ - |32)p2 +. 
 
 Thus, the six constants are known and the desired change 
 in the ^e-curve is given. 
 
 CONCLUDING REMARKS 
 
 It has been pointed out that the so-called inverse 
 problem does not exist in a strict sense of the term, 
 because a possible pressure distribution cannot be pre- 
 scribed unless the new airfoil contour is actually given. 
 An airfoil corresponding to a given pressure distribution, 
 therefore, cannot in general be arrived at by an itera- 
 tion process or by any other method. It is shown that 
 only certain qualitative modifications may be effected. 
 Such alterations fall logically into several independent 
 groups. Attention is given to localized variations, in 
 which not only the thickness factor but also the ideal 
 angle of attack and the angle of zero lift are kept con- 
 stant. Of interest, also, are the pressure changes 
 
8 li;\CA AFR No iJiGO 
 
 c 
 
 perforiried v/ith the reatris ulon that the monent ^oeffioient 
 and the an^le of z,er-j lift remain uncharged. This case 
 is of ^nportance for airfoils used in helicopter hl&des. 
 
 I,angley '^''eTnorial fleroaautical Laooratory 
 
 National Advisory CoTKrc'ttee for Aeronautics 
 Langl 3y Fl3ld, Va. 
 
MCA ARR l\fo. Il4.'305 
 
 APPENDIX 
 EXAMPLES 
 
 Flv3 cases are treated as examples of the airf-oii- 
 contour-modiflcation method of calculating pressure dis- 
 trl Juti on; 
 
 I. R.A.F'. 15 airfoil (see table I); local changes on 
 lower surface; angle of zero lift, ideal angle 
 of attack, and ideal lift coefficient kept con- 
 stant 
 
 II. R. A.F. 15 f-irfoil; local changes on lower surface; 
 moment coefficient and angle of zero lift kept 
 constant 
 
 IIT. Airfoil contour renerated from e - 0.1 sin (a - 14.5 ), 
 'ifr) - 0.1 (see table II): ideal lift coefficient 
 kept constant; angle of zero lift and ideal angle 
 of attack changed 
 
 IV. Alrfoj.l contour same as in cace III; angle of zero 
 lift kept constant 
 
 V. Airfoil contour sa-re as in case ITT; restrictions 
 same as in case I as an example of too severe 
 restrictions 
 
 Case I is based on the P. A.F. I5 airfoil, for which 
 figure 1 shows the shape and the pressure distribution. 
 The purpose of the intended alteration is to reznove the 
 wavy line on the bottom surface. The first step as 
 indicated is to draw a tentative pressure distribution. 
 In the curve at the top of figure 2 the corresponding 
 tentative change in pressure is plotted against the 
 angle cp . The ne2"t step is to draw the adjusted curve 
 for v;hich the area 
 
 Thus a closed A^-cur'/e is assured for case I; this curve 
 is called the adjusted curve and is shown in the center 
 of figure 2. It is also necessary to make the area 
 unc'er the /\e -curve 
 
10 NAG A ARR No. JiiG05 
 
 This can 03 done readP^y vithout altering A ?|,t or A€rn, 
 as shovn by the line for case I, for which 
 
 r 
 
 t.i. dcp - 
 
 The corresooncllng A^J;-curve is shovrn at the bottor. of 
 figure ?-. The modified airfoil shape and pressure 
 distribution for case "^ are shown in figure 1. Note 
 that the pressure distribution actually obtained differs 
 frorr the tentative oressure distribution and that no 
 change has occurred in the angle of zero lift or m the 
 ideal angle of attac'c. 
 
 Care TI is also based on the ^.A.F. I5 airfoil, 
 with the requirement imposed that the raoment coefficient 
 and the angle of zevo lift remain constant. The tenta- 
 tive pressure distribution is the ssaTie ac that used in 
 
 case I. The ^r^-curve adjusted for zero area and the 
 
 Af -curve adjusted for zei'O area are therefore identical 
 
 with those of case I. In this case it Is necessary to 
 
 comply with the requireiaent that the first and second 
 
 harmonics be removed and that some third harmonic be 
 
 added to retain the value of £„.. • As shown in the dls- 
 
 i 
 
 cuss ion, the function 
 
 i-|_f: = -A, sin CD - 3-, co.- cp - A sin 2cp - B^ cos 2cp 
 + k-z sin Jcp + B-, cos 3cp 
 
 is to be added, where 
 
NACA AR^ Fo, lJ.:G05 11 
 
 1 / 
 
 n^^ 
 
 B^ = — .' Ae cos cp dcp 
 
 'o 
 
 '' ^l 
 
 1 P'" 
 
 A^ = ~ I Z\e sin 2cp d<^ 
 
 '2 
 
 and. 
 
 *3 = H'^l - ^'■2) * H'^1 - 532)3 * 7i^h - 5*2) 
 
 P^.. 
 
 1 - B, - (-»i - i^a)?"^^- 
 
 Bj = B. - 
 
 By adding i^e to Ae of case I, the Ae-curve called 
 
 case IT end the corresoondlng A>|/-curve in figure 2 are 
 obtained. The resulting modified airfoil contour and 
 pressure distribution are shown in figure J- This case 
 is best suited for maintaining zero moment coefficient 
 in airfoil sections used in autogyros and helicopters. 
 
 The following three cases, cases m to V, are based on 
 the airfoil section generated from c = 0.1 sin (c? -k^'^;, 
 \|/ =0.1. The original and the tentative pressure 
 
 distributions are shown in figure Ji.(a). In figure i+fb), 
 Ap/pg is shown plotted against the angle '?. The 
 tentative pressure curve is adjusted for zer'o area as 
 before. The corresponding A£-curve is marked ''Case 
 ITT" in figiire 5 '-"^'^d. the corresoonding A'l^-curve Is 
 narked "Gase III" in figure 6. Thus f&y three choices 
 have been treated: 
 
 (1) The exact shape oi curve TIT is retained by 
 changing tlie zero line, or reference lire, for a e , 
 which change? ooth e^ and p,^. but retains the dif- 
 ference and therefore the ideal lift coeff ic:lent , The 
 resulting airfoil contour and oressure 6.1 sti^ibution for 
 case III are shown in figure 7* 
 
c, 
 
 12 IIACA ARR No. L^aO^ 
 
 ( d) In case TV th'.^ are^i under the A£ -curve cor- 
 resoonUin^r Lo the tent-'itive iiS-cni've ha^i been xpade zero 
 
 J.- c- 
 
 b^ extendini^ the range affe''-ted beyond the nose. In 
 this case, only the an,;^] e of: zero lift is Icept constant. 
 
 The result as compaiad with the original is shown in 
 figure C. 
 
 CJ) Tm case V the restriction is purpo^iely made 
 too severe by sneoifylng that no change shall occui' 
 either in the ideal lift, the angle of zero lift, or 
 the anrae of ideal lift. 
 
 C^se V in f i ;javre ^ becomes distorted in attempting 
 to fulfill the zero -are a requirem-int and the final results 
 sljown in fifi;ure 9 are corresijondinfly unsatisf actoi'y. 
 The conclusicn frori; tnis example is that, although cer- 
 tain requirenents are desirable or required, it is not 
 alvi/sys possible to obtain a good solution within the 
 limitations of such requirements. In such a case an- 
 other basic t;;, ne more suitable to the purpose must be 
 selected. 
 
 nF:FEREi^CES 
 
 1, Theodorsen^ Theodore; Theory of I'lng Sections of 
 
 Arhlt--ar:^' Shape. lU-OA Rep. No. i;.ll, 19?1« 
 
 2. Theodorsen, T. , and Garricl^, I. .H . ; General Potential 
 
 Theory of Arbitrary'- "'.-iug Sections, NACA Rep, 
 No. 14-5^ > 1^33- 
 
V-iCA ARP. V.D. ll'}05 
 
 1^. 
 
 TABLE T 
 
 ORDI HATES ^0" r.A.r. I5 AIRFOIL .MW 
 
 ^TaDIFI CATIONS, CASES T AND TT 
 
 ("stations and ord:.natef^ i;i percent 
 Ox'' Viing cbordj 
 
 i 
 
 TJpoar 
 
 ' surface 
 
 Lower surface 
 
 station 
 
 1 
 1 
 
 
 
 
 
 
 
 Original 
 
 Cr.se l' 
 
 C&se ] T 
 
 Original 
 
 Case T 
 
 Care IT 
 
 i 
 
 
 
 
 
 
 
 1^ 
 
 i 
 
 
 
 1.25 
 
 1.65 
 
 i.ps 
 
 1.69 
 
 -.70 
 
 1 -.80 
 
 -.63 
 
 ^0 
 
 2.-0 
 
 2.54 
 
 -.■^b 
 
 -.82 
 
 5.0 
 
 Ii.50 
 
 3 . 60 
 
 5-70 
 
 -1.14 
 
 : -1.05 
 
 -.9'-) 
 
 7.5 
 
 h.22 
 
 -1.25 
 
 1 -1-35 
 
 -].30 
 
 10 
 
 ii .Gl 
 
 h..lh 
 
 t.37 
 
 -1.25 
 
 -1.23 
 
 -1 . 02 
 
 15 
 
 5-^'5 
 
 S-29 
 
 
 -1.05 
 
 -1.27 
 
 -.91 
 
 iO 
 
 5.5^ 
 
 >li6 
 
 -.75 
 
 -1.25 
 
 -.£1 
 
 !.l 
 
 5.60 
 
 >55 
 
 5-l»5 
 
 -.29 
 
 1 -1.55 
 
 -.73 
 
 5.L5 
 
 5.39 
 
 5. 2d 
 
 -.28 
 
 1 -i.17 
 
 -.63 
 
 50 
 
 ^,00 
 
 ^..05 
 
 
 -■ii7 
 
 ! -i.oU 
 
 -.60 
 
 60 
 
 
 -.71 
 
 1 --'i 
 
 -.h^ 
 
 7^ 
 
 Go 
 
 3.'J^ 
 
 >90 
 
 -.9)4 
 
 -•75 
 
 -.69 
 
 5.05 
 
 5. 00 
 
 5.12 
 
 -,d^ 
 
 -.60 
 
 -.68 
 
 90 
 
 2.01 
 
 i.iS 
 
 2.0li 
 
 ■•^^i 
 
 -.4^ 
 
 - . L.'z- 
 
 ■-:U 
 
 ^^5 
 
 1 , 5 & 
 
 1.55 
 
 1.3M 
 
 i --5^ 
 
 100 
 
 
 
 
 
 
 
 
 
 ! 
 
 1 
 
 
 
 N ATI FA AD"^.' I " R Y 
 GOJa^ITTEE FOR AEROKAUTICS 
 
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NACA ARR No. L4GC5 
 
 Fig. 1 
 
 Original 
 
 Modified 
 
 NATIONAL ADVISORY 
 COMMITTEE FOR AERONAUTICS 
 
 P/'i 
 
 Ori^inaf 
 ren/cjf/t/e 
 
 o Modified 
 
 lOO 
 
 Percent chord 
 Figure /.-Shape and pressure distribution, case I . (f?.A.r 15 airfoil) 
 
NACA ARR No. L4G05 
 
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 ^ 
 
 
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 ii 
 
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 ^ 
 
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NACA ARR No. L4G05 
 
 Fig. 3 
 
 Original 
 
 Modified 
 
 NATIONAL ADVISORY 
 COMMITTEE FOR AERONAUTICS 
 
 P/9 
 
 1 
 
 -A 
 
 
 
 
 
 
 
 Original 
 Modified 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 — 
 
 
 
 -j 
 
 
 -.^ 
 
 
 r ' 
 
 
 
 - — — — 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 20 
 
 100 
 
 40 60 80 
 
 PercenI chord 
 Figure 3 . - Shape and pressure dislribufion , case II 
 
 (R.A.F. 15- airfoil.) 
 
NACA ARR No. L4G05 
 
 Fig. 4 
 
 -Oric^inal 
 ■ Tentative 
 
 pA 
 
 ' " > 
 
 O 20 40 do 80 /OO 
 
 Percent chord 
 (a) Original and tentative pressure distributions . 
 
 Wp^ 
 
 4 
 
 
 
 
 
 
 
 
 Tentative 
 
 Ad lasted for 1 - o 
 
 
 
 
 
 
 
 
 
 
 
 
 
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 n 
 
 / 
 
 '^^ 
 
 
 
 
 
 
 
 
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 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3.0 3.5 4.0 4.5 5.0 S.5 6.0 
 
 (b) Tentative and adjusted pressure ctianges . 
 
 NATIONAL ADVISORY 
 COMMITTEE FOR AERONAUTICS 
 
 Figure 4- Pressure distributions and pressure ctianges, case [II . 
 
NACA ARR No. L4G05 
 
 Fig. 
 
 
 
 
 
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 ) 
 
 
 ^ 
 
 /^ 
 
 /' 
 
 ' 
 
 / 
 
 1 
 
 t 
 
 
 
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NACA ARR No. L4G05 
 
 Fig 
 
 
 
 
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NACA ARR No. L4G05 
 
 Fig. 7 
 
 Onginol 
 
 Modified 
 
 NATIONAL ADVISfJRY 
 CHMMITTEt FOR AtRuNAUTlCS 
 
 Onginal 
 Modified 
 
 pA 
 
 I 
 
 / I _ 
 
 20 40 60 60 100 
 
 Percent chiord 
 F/ijure 7 . - Sfiape and pressure distribuiion , case III 
 
NACA ARR No. L4G05 
 
 Fig. 8 
 
 Original 
 
 Modified 
 
 NATIONAL ADVISORY 
 COMMITTEE FOR AERONAUTICS 
 
 pA 
 
 -I 
 
 20 
 
 -10 
 
 - Original 
 -Modified 
 
 /^ 
 
 ^,_ 
 
 _ ^ 
 
 ^.^^ 
 
 
 
 . 
 
 
 
 
 --'■ 
 
 
 t 
 
 ^ 
 
 
 
 
 
 
 --. 
 
 ■^-^ 
 
 ^^ '! 
 
 y 
 
 
 
 
 
 
 
 
 
 J 
 
 '— .^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 60 
 
 QO 
 
 100 
 
 PercenI chord 
 Figure 6 - Shape and pressure disfribufion , case IV. 
 
NACA ARR No. L4G05 
 
 Fig. 9 
 
 Oric^inal 
 
 Modified 
 
 NATIONAL ADVISORY 
 COMMITTEE FOR AERONAUTICS 
 
 PA 
 
 — Oriainal 
 ■ - - Modified 
 
 X 
 
 . 
 
 
 
 
 
 
 
 
 
 h 
 
 ^^^ 
 
 
 
 .__ 
 
 
 
 
 
 ^.^.^ 
 
 
 ' 
 
 
 
 
 
 1 
 
 - — _ 
 
 
 
 ^ 
 
 
 
 
 ■* — — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 20 40 (>0 80 /OO 
 
 Percenf chord 
 Figure 9 . - Shape and pressure distribution ^ cose V . 
 
UNIVERSITY OF FLORIDA 
 
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