ACE No. LilJlO NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIMK REPORT ORIGINALLY ISSUED NoTember 1944 as Advance Confidential Report L4J10 THE THEORY OF PROPELLERS III - THE SLIPSTREAM COHTRACTIOW WITH NIJMERICAL VALUES FOR TWO -BLADE AOT) FOUR -BLADE PROPELLERS By Theodore Theodorsen Langley Memorial Aeronauticetl Laboratory Langley Field, Va. r- NACA 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 - 155 DOCUMENTS DEPARTMENT Digitized by the Internet Arcliive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/theoryofpropelleOOIang ]?ACA ACR No. Li|--T10 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE CONPIDENTIAL REPORT THE THEORY OF PROPELLERS IIT - THE 3LIPSTRE:\IV! CONTRACTION WITH NUMERICAL VALUES FOR TWO -BLADE AND POUR -BLADE PROPELLERS By Theoaore Theodorsen As the conditions of the ultimate wake are of concern Loth theoretically an^'i practically, the magnitude of the slipstream contraction has been calculated. It will be noted that the contraction in a representative case ic of the order of only 1 percent of the propeller diaiiieter. In consequence, all calculations need involve only first- order effects. Curves and tables are given- i'or the contraction coefficient of two-blade and four-blade propellers for various values of the advance ratio; the contraction coefficient is defined as the contraction in the diarr-eter of the wake helix In terms of the wake diameter at infinity. Tbe contoui" lines of the wake helix ere also shown at four values of the advance ratio in compalrison wltli the contour lines for an infinite number of blades. INTRODUCTION Since reference is often made to the wake infinitely far behind the propeller, it Is desirable to establish certain relationships between the dimensions of the propeller and those of the walce helix at infinity. The present paper considers the relationship of the propeller diameter and the vvake di am.eber, or the problem of the slipstream, contraction. Tne discussion is restricted to a consideration of first-order effects, that is, to the determination of the contraction per unit of loading for infinitely sm.all loadings only. It will be seen that the contractions are indeed very small, of th:=- order of a few percent of CONFIDENTIAL NACA ACR No. L^JIO the prooeller diameter, and that the high-order terms are therefore not of concern. The interference velocity accordingly is neglected as s^all compared with the stream velocity. The wake helix lies on a oerfect cylinder cmd the pitch angle is everywhere the saine . It is noted that the assujnption of zero loading corresponds to that used by Goldstein I'or a different purpose. SMBOLS R tip radius of oropeller r radius of elerrsnt of vortex sheet Ar contraction Ar^ total contraction or contraction at ~ = o T angle between starting point of soiral line and point P H pitch of SToiral G angular coordinate on vortex sheet advance ratio X ratio of radius of element to tip radius of vortex sheet (r/R) vpj radial velocity V advance velocity of propeller w rearward displacement velocity of helical vortex surface — w w = - p number of blades K mass coefficient CONFIDENTIAL NACA ACR No. lJ-JIO CONFIDENTIAL 2kw r K(x) circulation at radius x TfcXVR K(x), CO circulation function Tor single rotation angular velocity of propeller, radians per second ZttVw radial velocity at Doint P due to a doublet element at 9,x except for a constant factor / / _\b cos (e+T^ - s^.n (9+tV Ti- 2x^+\2e2+ ^ cos (S + t)] Yl - \ 1 + x2 + \292 _ 2x cos (9+t) 5/2 72 = Yi = ^2 _ K(:0 where n = 0, 1, 2,..op-l n /'I 'JO Jo dx angle of contraction, except for a constant Y, factor Y2 = / Y-|_ d9 contour line of contraction, except for a constant factor Ix h- fy^ = ' Yp d9j K LL ' ' \ ' '70 Ar o R / c - > 5 ^ total contr'action in terms of radius ( — ^ -^ Y ' K k ■5 X3 — contraction coefficient -r- Yz CONFIDENTIAL li- CONFTDENTIAL MAC A ACR No. L^JIO ^ T — w. \^ I - k) F(k) - I E(k) ix=l J THEORY Tbe radial veloc.1ty is obtained by using the Biot- Savart lav and integrating ovor tno entire surface oi' di3ccntlnui ty. If Z^rg pro>:)ler;i is to determine che ratio is tae total CDntraction, tlie o P. for various nurii"be r s of b 1 ad e s several advance ratios. Simple expressloiiS referring to zero loading are used throughout. The radial is calculated, from an element on a spiral q:).^ dicular to the The sngle betwe and the noint belov; the plane is designated frorri the starti distance h be inv/ard (See fi of circ radiur axi s and en the s P is de to infi H, the ng point low tlie dVK' velocity g s . 1 and 2 . ) at the point P This velocity results ulatiOii f dsj which is located r th'it starts in a plane perpen- containing tlie reference ooint P. tarting point of the spiral line signated t. The spiral extends nity. If the pitch of this spiral element at a projected angle of the spiral is then at a reference plane where h T (1) c-j introducing the nondimensional quantities \ = - X = -^: R V (2) in the Biot-Savart law, the follovving expression is obtained for the rac'ial inward velocity dvg' due to an element on the vi/ake helix o± strength f r COInTPIDENTIAL KACA ACR No. iJ+JlO CONFIDENTIAL 5 1 f d9. P cos (e + t) - sin (6 + t) ,^> GVj^' = Ajc — . :^ — J- (5) ■'I'T R [l + x2 + ^20 2 _ 2^, CQS (e + t)J5A By di.f f erentiating equation (5) virith respect to x, the field of a douolet eleinent on the helical vortex sheet is obtained, the doublet eleinent consisting of two neighboring singlet elements each of strength f . Setting f dx equal to p and dividing tlrirough by the stream velocity V gives dlB.= 1 r d6 -^ [e PCS (9+T) - si n fG +T J [l - gx^ + \^9^ +x cos (B-t-T).! ,,. ^' " hrr RV ■ r^ ^ ^2 , ^202 . ^x cos (e+T)]5/2 where vj^ is the radial velocity at the point P. Equation ([;.) may be written in the form 'Vr 1 r d9 where _ [p cos (e+T)-sin (0 +t)1 [l - 2x'^ + X.^9^ +x cos (B+t)] ,., fl - -^ ^^ -^-^ 2 ^ i ° ) [i + x^ + \^q'^ - 2x cos (9+t)]5/^ The function j-i is plotted against 9 + t for four values of X. and various values of t and x in figures 5 to 6. With rn ^ 2rrVw ^^,^^^ pcO ^ 2rr£w ^(^) pw rrVRX c where 2k:w - K(x) (7) = 2kw (8) CONFIDENTIAL 6 COKFIDEITTIAL NACA ACR No. li^JlO substitution in equation (S) gives alf< = . V' ^ m y, de (9) 9 If the T>oint P is at a distance h = H — below the propeller, integrating equation (9) over the wake yields IR. =.^5i I il(.xlYV, dz d9 (10) J-9 Jo It is noted that, v/ith equally spaced blades, the function X ■/! (11) is an odd fxinction of 9 and fte y / Vt d9 = (12) ■:i' '.-'-9 Equation (10) can therefore be rewritten as n-^ nl Y = -l~- j ^l.Ji^-^^ (15) CONFIDENTIAL NACA ACR No. L)+J10 CONPIDENTIAL 7 Let - n -^ Yt = 1^0 Y: = t y-T ajc Yi de i^e (11+) Values of Y-i and Y;., mrltiwlied by a constant factor for convenience in TDlotting, are given in tables I to IV for two-blade and four- blade prot'ellers for which \ and G take on various values. Theae functions are plotted against B in figures 7 and 6. Equation (15) becomes \2 c. Y- 15) Nov; V _K V dr dh ^'h d' h 1 '1^ \ d9 (16) Therefore V d8 R \^ Os h - Yp (1?) v/hence Ar _ R2 - Rl . R R - 1 R pR^ dr = IX dx = - \^ c. cfR- J. Ll k Yo d9 (13) p A Tjm T rMPT-.Tm t /i t 8 CONPIDEIITIAL NACA ACR No. iJ^JlO If R2 is the radius at the propeller and R]_ is the ultimate radius of the wake (32 = 0, 9x = "= 1, Jo where Yz = / Yo de I/O Values of Yz are given in tables V and VI and are plotted in figure 9 f<^r t\*o- blade and four-blade propellers for v/hich X ar.d 9 take on various valu.es. After all substitutions are made, the conTjlete multiple integral for the total contraction is obtained as ,'.'o nrc ,1 — = - -r- I I / -v(x) ->_y. (9,x) dx d9 d9 CO t;0 I/O '^ (See figs. 10 and 11. ) INFINITE Nm'BER OF BLADES For purooses of comnarison, it is useful to obtain the contraction for the case of an infinite number of blades. Ey resolving the circulation into components parallel to and osrpendicular to the axis of the v/ake, the helical vortices can be replaced by a system of vortices parallel to the axis and another of ring vortices having centers on the axis. Only the ring vortices contribute to the radial velocity. The field due to a vortex ring of strength f and radius r = Rx, located at a distance h belov/ the CONFIDENTIAL NACA ACR No. liiJlO CONFIDENTIAL reference point P, is given by Lamb (reference 1, p. 257)' In the notation of the present paper it is fR 2^/2 ,\ _,., . 2 ,.,, V (20) where ii(k) and ?(k) are the complete elliptic integrals and .2 M-X h^^ R + (1 + x)' As before, a doublet ring is obtained by differentiating equation (20) vjith respect to x. 3y setting f dx = E the following expression is obtained for the 'field of a doublet ring: r 1 ° Sir H I 1 - k'^ I ' 2 i E(k) + x^ 2 - k^ 2?(k) - ^ ^ E(k) 1 - k^ >(21) In order to obtain the effect of the entire vortex system, equation (21) is integrated, with respect to h/R and X as r.i n PR" •Jo J -h/R BttH k < X 1 ? k2 1 - k2 Efk) + xS 2F(k) - 1 - ^^^^L U|dx (22) The radial velocity 1 6v VR = — -^ R CONFIDENTIAL 10 CONFIDENTIAL NaCA ACR No. IJ+JIO Equation { 2Z) may be v;rltten in the form so that Now '-'0 -J-h/R ^0 ^r 1 / dr ,, — = - / ■— ah R t. I dh nh/R ,/ V fi /'I ;'■•' /, N 3y substituting yj (/h/R TTCoX-VR r 3: — ^- — K(.x) (26) where K(x) is the horizontal comnonent of the circulation coefficient, wiilch may be expressed as K(x) W--^;^— V" (27) CONFIDENTIAL MAC A ACR No. LiiJlO CONI-'IDENTIAL 11 the follo'wlng equation is finally obtained; P 16ttK v\2.x2y x/2 1^^ dx I k, 2-l_ E( k) . x2 1 - k"^ 2 l1 2F(k) -^^^E(k)Hd^ l-k2 JJR (28) For convenience in usin,^ the Legendre tables, the second integral is written in the form ■' z T d-- 1 R (^9) (ih/R where r / 1 1 ,\ A\ zi = sin ^ d^ (See table VII and fig. IJ . ) CONFIDENTIAL 12 CONFIDEFTIAL MAC A aCR No. lJ+JIO INFINITE rj:.!3ER 0? BLADES FOR DUAL ROTATION The contraction for a dual-rotating propeller v/ith an infinite number of blades is next obtained. In this case ,K(z) = 1, and the radial velocity is *^R 2ttH r ,2 U^ - k) F{k) - ^ E(k) > x=l x=0 Since the value at the lov;cr limit is zero axid K(x) = 1 for an infinite nu"iber of blades, it follows by substituting Ar R ^.v■ Jh/R w-i d— ^ R where \(rj - ^) P{^) - I E(k: I \v. x=l (See table VIII and fig. ll;..) CONCLUDING RSI ARKS The contr and four-tlade values of the tri-nle Integra and susceptibl methods are de hoped that the purDose. It i contraction, and advance ra terms of the r action coefficients are given for two-blade single-rotating propellers at four specific advance ratio. The calculations involve tions and are therefore somewhat laborious e to numerical errors. Until more convenient vised to perform this integration, it is values given In this paper will serve the s well to notice the small magnitude of the A four-blade orooeller with normal loading tio is shown to have a total contraction in adius of Ifcss than oa^ percent. The CONFIDEICTIAL NACA ACR No. lIlJIO CONFIDENTIAL IJ first-order treatmezit embodied in the paoer is therefore adequate for all technical purposes. Langley Memorial Aeronautical Laboratory National Advisory Coiraiiittee for Aeronautics Langley Field, Va . RLPERSNCE 1. Lamb, Horace; Flydrodynamics . Sixth ed., Cambridge Univ. Press, 1932. CONFIDENTIAL NACA ACR No. LiiJlO CONFIDENTIAL 111 TAPLE I FUNCTION \^ il- \\ FOR TVv'O- BLADE PROPELLER 9 (de-) \3 i p (deg) 'h K --1 \ = 1 -^i 50 -0.0002^^ ! 1 -0.000133 -0.000683 100 .000148 2 -.000213 -0.000282 -.001033 150 .001906 I -.000293 -.001367 150 .004815 -.000382 -.oocLSS -.001232 170 .000679 5 -.001235 200 .OO5OS2 -.000R70 -.000925 -.001316 220 .002008 : 7 -.000^82 -.001358 250 .000730 1 p -.00078'^ -.001353 300 .000,'.' so i 5 -.000898 -.001350 550 .0013U5 , 1 10 -.000984 -.001792 -.001358 i+OO .000622 20 -.001159 -.001571 -.001286 -.001567 500 .000270 ho -.000999 -.001030 550 .oookoo 60 -.000802 -.000S62- -.000759 600 .OOOli+S so -.000359 .000^60 -.000321+ -.OC0551; 650 .000051 100 1 .000312 .0u008'+ 700 .00 oil' 9 120 .001730 .000709 .000271^ 750 .000175 ll.:.0 .003151 .OUOS09 .00054.7 800 .00 00' 58 160 ISO 1 200 1 .003351 .00071+0 .000521 900 .000079 .002638 .000^92 .000152 .ooi5ii-'-^ .000257 .000100 220 .000625 .000093 .ooooili- .000073 2I..O .000160 -.000015 260 -.Q00CS6 .ooooli.9 .000005 .000008 280 -.000013 .000036 300 .000213 -.oooool .0000li:7 ^20 .000363 .OOOOOli .000036 3L0 .000390 .0003I1.3 .000008 .000050 360 .000051 .000001 380 .000252 .000003 -.000003 Uoo .000115 -.000005 .000001^ 1,20 .000027 .000020 .000007 ldi.0 -.000011 6.000011 , .. , . 1 k6o .000011^ NATIONAL ADVISORY COT.TVJITTSE FOR AERONAUTICS fAfTiPTDFWTiTA''", NAG A ACR No. l1|J10 GONPIDENTIAL 5 15 A- TABLE TI.- FUNCTION -y-Yj_ FOR FOUR-BLADE PROPELLER n 'i?'. 9 (dee) \5 (deg) '=i . = i \ = 1 X = il 2 10 50 60 0.000500 -.0206S0 .097700 1.251120 1.'?5S120 1 2 '5^ -0.001+211 -0.000001 -0.000275 -.000250 -.000088 .000083 .000251 70 80 65 90 2.879450 3. 81; 9270 ,'+.59L.6oo )+. 967600 i)-. 538300 6 9 10 -.011371 .000002 -.000013 .000501 .ooofoij. .000905 .001263 .0011+32 95 100 110 120 160 3. 5^! 9380 2.685010 1.^30560 .821510 1.017150 20 1+0 60 80 100 .006150 .I57OOI1. .563018 .3I-12699 .2185I1.8 .001786 .0061+66 .006i+2^ .ooiii7l+ .002161 .002981+ .00 5 5 28 .005108 .001577 .000552 200 250 ^00 350 1+00 .902770 .i4.3696o .523530 .217700 .15kl90 1 120 1 lixO 160 180 200 .112755 .091686 .075117 .053218 .O4O86I .001281 .000865 .0005^5 .0005LS .000221+ .000293 .000221+ .00 02 1I+ .000166 .000073 i+50 500 600 .15I4.OIO .119570 .038I150 -.029500 220 ^)'0 260 280 300 .050025 .0211839 .021139 .017093 .0lll.0l|8; .000176 .000130 .000100 .000062 .0000l,'.2 .000101 .000066 .oooos? .0000I+8 .000037 320 3I4-O 560 1+00 .0111714. .009069 .007762 .0061+39 .001+650 .000052 .000016 .000052 -.000018 -.00001:1 .000025 .000002 .OOOOOlj. 1+20 J+ko [(60 1 .0037i|8 .00588^. .oorjJ+28 -.00000^ -.000001+ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS CONFIDENTIAL NACA ACR No. L/+J10 CONFIDENTIAL 16 TABLE III.- .PTINCTION \ h'' FOR TWO -BLADE PROPELLER e %^ ( e 1 ' (deg) >-.i (deg) ^ = 1 \ = 1 X= li ^ ! 2 2 8.138900 1 i 0.377250 -0.00)w'i9l4 -0.003156 20 8.159900 8.219I1.00 i ^ .377650 .373580 -.ooM+55 1+0 ^^ -.0014.280 60 8.29llj.00 1 6 .^8011^0 -.003990 80 8.351)4.00 i 8 1 .382320 -,005Il90 100 8.^58900 \ 10 .385170 .C02ii2C -.002790 -.002[;.66 120 8.223900 20 .000595 -.001651 iLo loO 7'7k5300 p -.000876 -.000139 6.570900 1.0 .U37620 .006355 18 4.917900 i 50 1 .OOOI1O7 200 3.iLi2900 1 60 .U67020 .010675 .000907 220 2.512900 30 .001302 2l|0 2o0 2.232900 .1.86770 .013075- .001577 2.202900 90 ..001717 280 2.21.5900 100 .L.85670 .013075 ,001724 300 2.215900 120 .L.53070 .011070 .001504 320 2.026900 1.80 .573150 .008050 .OGII29 5JaO 1.7^^900 loO .267250 ,001+870 ,000718 5oO I.k0li900 i.ioSIiOO i3o .169!: 00 .002li70 .000[>li)| 380 i 200 1 .102200 .001070 .000313 I|.00 .890900 .78014.00 1 i 220 .067520 .oooii70 .000207 1x20 2l;0 .055?io .000280 .000178 ao . 7ii5L.oo 260 .055i.).6o .000270 .000 18 9 1+60 .7Lol^'oo 280 .05281Q .01,8820 .000280 .000162 500 .671.1+00 i 500 1 .000290 .000112 5Uo .50114.00 ! 5f^o .0/4.0500 .000290 .000065 530 .32^1^00 5lp 560 .02863c .000260 .000025 620 .2L.61i00 .016630 .000150 .000007 660 .2^lliOO ^30 j .007610 .000090 .000007 700 .18914,00 .0021^00 .000080 .000006 7'fO .112000 lj.20 .OOOI42O .000070 780 .030000 kko .OOOlOu £20 -.006000 660 -.OOliOOO _ CONFIDENTIAL NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS NACA ACR No. Li|J10 CONFIDENTIAL 17 TABLE IV.- FUNCTION —Yp FOR POUR -BLADE PROPELLER T 1 n 1 9 (deg) V'. 1 e (deg) \5 1 1 1 1 ! X = 1 X = 1- 2 OJJdlklk 0.017253 0.007918 0.005175 10 .0-dllhlk 2 .017259 .007913 .005107 20 .02i+li.lb ij .017261 .007913 .0014.728 50 .021^.380 .017266 .007918 .00k08li ko .021^.256 b .017276. .007913 .003^1!^ 5^ .02^969 ' 10 .017287 .007918 .0023 1+3 6o .025353 1 ^^ .017^17 .007908 .001699 70 .021953 ko .010676 .006875 .001235 30 .019555 1 60 .013899 .001^520 .000902 90 .oi6oIj.9 i 80 .009902 .002668 .000675 100 .0iJ5ii0 1 100 .00681+5 .001591 .OOOS14.1 110 .012166 1^0 .oo';;099 .ooLi-Oio .OOlOlli. .oooI'04 130 .010889 , liiO . 000 bl,lj. .00031!^ 150 .009821 1 160 .00^112 .0001,07 .0002L.0 170 .U08lfi4Jx iGo j .002383 ,000255 .000173 190 .0069SI 1 200 220 .001859 .000157 .000135 210 .00S705 .00^773 .001^86 .000091 .00U102 230 21,0 260 .001186 .000087 .000075 250 .ooLo92 .0009119 .0007i.4 .OOU01I.9 .000052 270 .00352B 2SO .000028 .000035 290 .00303s .002594 1 ^;00 .ooo;3q6 .000010 .000020 510 5f-0 .000i,72 -.000002 .OOOC09 530 .002210 1 3i-o .000363 -.000010 .000005 550 .001337 1 360 .000^71 -.000019 570 .001612 1 530 .000192 -.000021 390 lio .001576 koo .00U132 -.000015 .001166 I;20 .000087 -.OOOl^'OI {^50 .000959 .oooo1l6 L.50 .00075I; 1 I170 .000550 i 1 • U90 .000567 1 510 .000205 1 550 1 . .000075 1 i _-i 1-7 AT 1 NAL AD VI SOR Y COMMITTEE FOR AERONAUTICS CONFIDENTIAL NAG A ACR No. L'iJIO CONFIDENTIAL 18 X 3 TABLE v.- l^TKCTION ~^' FOR TWO -BLADE PROPELLER ^ ^ /\",. 1 1 -z 9 (deg) '4- -^ ! e i (deg) k - i_ X =^ \ = 1 v = il k 1 1 2 ^ .o[,c6i5 i o.oi99J4i+ 0.002li^.l O.OOG602 20 .0U739 2 .019800 .00215],^ Uo .cko85Q k .015656 .002163 6o .0^6926 .01S511 .002180 So ,03 29';^ 5 8 .015566 .002191 100 ,029007 10 .01^,221 .002201 .000714.3 120 , 0250^)6 20 .oi!^)'7li. .002217 .000855 lM.0 .0212I18 ?^ .000921 i6o .017300 ho .01291$ .002111 .0009SO .00094J.t l3o .0] 50^.7 50 200 .013062 60 .0x1197 .001852 .000910 220 .011670 80 .000854 21^0 • .0105U9 .0093S2 .OOII4.9O ..000780 260 .009301 90 .000695 2'^'0 .,oo8al:£ 100 .0075^5 .00109:^ .000607 300 .0073^2 120 .00^752 .00l,.l'^-8 .000728 .0001^39 $20 .00 6$ 81 140 .0001+37 .000303 3iL0 .C05l;&3 .C0ii7~^If 1 160 .002927 .0002^3 ,000208 560 180 .002099 .000136 ,oooil+8 5 So .ooi;i39 1 200 .00159k .000087 .OoOlll l+on .00366L 220 .001277 .00006s .OOC086 Ii20 .003208 2I4O .OOIOL9 .000055 .000066 llliO .002003 .002.1^9 260 .000837 .0000117 .O00Ql|8 UbO 280 .000632 .OOG039 .000030 300 .001.8^2 500 .OOOli-il .000031 .000017 5I1O .001^18 ^20 .000276 .000022 .000007 5.:0 .000 9 22 1 -^ko .00Gl.'x6 .0000 06 .00c 018 .000002 620 .cooS'-^i 360 ,000003 . CO 00 i 660 .000^.25 ^80 Hoo .000021 .OOC005 .000001 700 ,000232 .000005 .000002 7:u0 1 .00008s Uo .000001 730 -.oooocS 320 -.C000U5 NATIONAL ADVISORY C0:.1MITTSE FOR AERONAUTICS NAG A ACR No. liiJlO CONFIDENTIAL 3 19 TABLE VI.- FUNCTION '•■— Yz FOR F01TR-3LADE PROPELLER '^^ i p (deg) .- .1 , 1 ' (deg) L^5 1 \ = ± 1 ^ 1 \ - - 2 X = 1 X = 1- 0.0677&6 0,0^^5'^2 0.010515 0.005286 10 .06332s .059264 .05S006 2 .0"';2969 .010256 20 4 .052567 .009900 ?o .05176]! .009684 .050757 ,q -> .051162 .009407 30 • 0li:sl9 i 10 .0"'^0553 .009151 .ooi+539 ?0 .0i|2l;20 1 20 .027537 .007750 .005552 V^ .0331,50 i .U02760 30 .03)^2^2 1 .02i531i ,005153 .002151 .031732 50 .001650 100 .029167 60 .016211 .005155 .001297 110 ,026892 70 .OOlofiO 150 .022S85 80 ,012055 .001916 .000856 150 .019265 1 90 • .000722 170 .016059 100 .009157 ,001185 .000618 190 .013358 1 .oiiili.8 1 120 .007100 .000758 .ooo.'i65 210 I'lO .005512 .0001^5^ .000510 230 ,009505 1 160 .UCL,26S .0C351[l .000267 ,0002li.5 230 .0077^-5 ! i3o .C00151 .000175 270 .0061155 200 .00257/-:- .000086 .000122 290 .003288 i 220 .002005 ,000052 .000080 310 .004^00 1 2iL0 .0015L5 .000022 .000014.9 330 .C03I.61 260 .001165 -.000000 .U00029 530 .0027l;6 230 .000865 .00065!!. -,000012 .U00015 570 ,00215s 500 -.000019 .000006 390 ,001621 ^20 .oooi.52 -.000021 .000002 .ooiiai 1 lh-0 .000507 -.000019 -.COOOlf; [00 ,000325 1 560 .000196 ^50 ,0005^2 p3o LOO .OC0116 -,000003 kJO .000306 .000061 -,000002 k90 .000155 L^O .000023 510 .000052 i N.J^lTIONAL ADVISORY C0I.TITT2E FOR AERONAUTICS CONFIDENTIAL MCA ACR Ho. rJ+JlO CONFIDENTIAL \-^ K| Oh O PI Ph C5 ta M EH .0^ O O -O '■~J CO -d-ovo-j- \ 1-M _-J-MA'Aj CM rH r-^ ,H , erico -Al_-j'-vi) i>--0 '^M rj KA r-OjO^ rH " rH lTnCO CT.OvJ ^ rrv^-rH rH \r\ ro>tX) '^ O O rH Al _\KO rJ On 11 o o o o o O rH M OJ K-v -t-^ ^« '3 o o o o o o o o o O O ^ B • o iM r h/R O OMD r\)a) -J-OOTVO-J- -A) rH -ri K^AJ rg rH r i rH , C M3 O OACO vT/ t^ONt^rvJ tr-HTo MJ " rHi'AJ -o oj 1 rv---- n:\ VOVO a-wt rH a\o.) "; O O '^J tOiU^iCO (AJ fj\-:i(\| rr\ cJ^^r^ II o o o o o r-l r-\ rvj KV^- uAr-- ^. - o o o o o o o o o o o o ^ O O O B • o e » • e « • o ^^ o o~o f\ico —J^ O CO ^O _z)- OJ rH \. ^ --rJ-KNOJ Al iH rH rH 1 1 Lr\K-N[^a:)vo ra MTN ■-•lO rH|-d- _:j vO _d-_Lj' a\ a. K^ o r\i r»-r-i o- o-^ o- 1.1 z-t a <':! r-i, , \ ai_:J- ^K f< Cjr:> o H CJ K Hi o LxJ oO «rij M ;>• crt O o — -- -- "" ' ^ T" ^ >- ^ >*_ f r . 'V -. '^ ^^ V- 'n' -" ; — — h- r- - -- ' — *^ ^ . "-^ ■*■ - . L- , . ^ • •» •*- -, !*'• ■-- ■"• -J •"t. ^ - ^ i'^ ■ . " ■» S. 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O , - U ' 4, — ^ n ■^ 1 \\ . - ** / \\ <-• 1 ii c — _ " - = 1 r < \]- - - \ , , % .-^ : 1 - \> ' \^ A \ — — '- 'o \ a I ■ / 1 J ^ \ ,^ " 1 'a> ■^ 1 1 \ \ 3 ~ ■ ■- - — — -__ — _. >2i.B. .^ _. _ ' 1 Cd O a ■H NACA ACR No. L4J10 Fig. 7a H & ? •^ all 1 1 \ \ ) hl< / J ~- _ ( H iM K^ / y ) ^ / _^ — -^ l\ — <^ / k - ^ /■ ■^ • A r ' h— i / i •^ ^ ^ 1 ', "S. — — — — — ^ — --' V \ 1 l__ ■< 1 L ^ -^ ■J < E-1 Z Q I— I O O o 3 O "> 00 o o CM o o 8 o '^<|<^ NACA ACR No. L4J10 Fig. 7b •^ 1 — — — — :._. — -^ — , , -- — ~ -" — — - 3 ^i " 1 «] S s s S — -- 1 1 ' 1 J <; Eh - s o 1— I 2 o o / / I / / \ __ ~ 1 - 1 1 - 1 — - 1 ■ / 1 I \, / • \ 7 / — , ... _, — \ / — - _ — ;^ ^ A /- / , — »* t^ -^ - ^ ^_ i^ / - C " r KJ. ^ 1 / =- -t-r L-j ^ ^ / L _._ - "' 1 — — 4. ^ y 1 '— / 5 \' ' - 1 1 1 -3 - - ! ■" 1 i ; i — . . _.. i ;^ :) < Q O 5 c .') o NACA ACR No. L4J10 Fig. 8a ' — — p ■ 1 ' «c O la t s e 1 ' 1 1 < z 1 i 1 =1 — ( - 2 , / 1 / / / /- / / -" r tMf y J II ^ / ^-.- ■< ^ y / i - ^ — HP // ^ > ^ 1/ / ^ / / / — — — MICV \ \' V \, w \ s w 1 ti \ \ < I— I •z, Ed Q ►-4 •z. o u o 1-1 b 1-1 o >-• U f P CO d >?■ en 3. O H O 00 o o «3 O O o £ II o ■ O Ho' o & .» o a ^^= r~i c 1 o o ■i:> » O H C o 3 o « •«-l 8 H O Ik* 01 u p o NACA ACR r^o. L4J10 Fig. 10 ^ 1 i^ H ^ ^ ^ ^ ^ gS m i* SPS jia -Tf]|i!{^:|l*- i '^-r- .s: ■- .: 1 ■ -- — ^ - -'£ TTT' 'H^ rnl" ttn — X'-~- :.d ^: „ ^ -" -- "S— mm ¥> fej -"-■ -; ^' TT" --- ;-' nS U 3r ^it ¥^ m iH^ -■^ -s* i '? pr?Pi^ —. 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NACA ACR No. L4J10 CONFIDENTIAL Fig. 13a AT 1 TTcT - ..- •12 _^:-- _; _ - "" I um 361 o f ' - _ b Luu iuu .10 — i ' V r . u . . .08 - - - \ '\ 1 O ■ .06 1 - V \ . i l\\ - ^^^ .04 \ V \ 1 - 1 k \ ^ - .02 V N WTI )NAI A[ wm \ N y ■>, 1 ' ;UMI biij pth JKfl LixONAUT us ^ V •^ :^~ ~-J^ n 1 ^ -^^_ K - ^ — -1 rH ^ t-J — — ' 1 — — 2 h/tl (a) A= J. CONFIDENTIAL Figure IS.- Contour lines of wake helixes. (See table VII. ) NACA ACR No. L4J10 CONFIDENTIAL Fig. 13b AT JL_ R c. .10 .08 .06 ,04 .02 tE 14-^: ±^''r' -1 ' '- -i — i-r ^: .-OT i444- T^-n- is t+Jl: Ti+r ;ffif -jxt;' :??;-:■ -:-.-.- jT^" :^:- r'jj!^."' -:,_ ; ;"■■'. ;-:-a :'Tt?ji ^^^ — — 1 : . T ,■: " ;^"_." :-: K imT tSi ol . e: 1 1 , , - ^;.:'. : :: ::-;: In- W ^ek m "TTq --" 4' — r- 1 1 ' '- -'-"- -■-u, -:'^ — — * 1 -f, ■■ ■, ;;I I-" ' -rr- ---- -,- : -- ■ ■ -'■, ^b Is -;■- ■ rr \ \ -- -- -— - \\ rr— ,"7, " " " H -- " rr— V L ■ -- t— - — r— - b -- - - T--- ^7T _~"' -- F — \' \ ■ ■" v-S h— — ^^ L ■ ■ k rr— Nsr SNA . 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"^ ^ ^^^^ ::::^ _^^ ■r K' 2 h/R (b) A= I . 3 4 CONFIDENTIAL Figure 13. - Continued, NACA ACR No. L4J10 eSEEEMMlAt Fig. 13c ' H , J ^ E-H 2 1 Q as € 00 2 g 5 U :-: t: ■ I- ;: ■ < to -0 •H Z -' _ O u V • t — 1 H M - r a . 2; 1 U OJ ' b ■a< • — CO — I " 7 f / /j / // ^-- // /, '/ / / tH • ^ / ,'• ; / ,' ^ ^, ,' '' 1 ,^- 1 c ( § c S -^1 ^1 n as 3 5 ? • c D NACA ACR No. L4J10 Fig. 13d ► ^ s ■■*. c: ' a 1 1 f! <; i| 1— 1 E-" o a ed U ■5 « N * i "if Q i SI ! 1 1 (X. ^ t z II c J i! /p /;' /; ■— ; / / ; ; / ; — ; / 1 / r 1 f 7- // 1 / y 1 1 r > ^ \ / • V .^ *' • 1 ,^' •^ \ <: Cd C3 I— I Ix. o HJM 3 I Ho" NACA ACR No. L4J10 Fig. 14 — — -"■ -^ ~^- --- — --■" — ^ — ""^ '-'■-'■ 'i"~ "^ •~* i . s — -— g ■« O i B ,- S a — " < ' ^ : ■P' I" \- Q r *"* ^ 25 - o ' * — o — ^ — • 1 — / ~i / t 1 / / --- / / / / ^ / X „-' ■^ -^ 1 — — ■ Q O O 9 O SP' UNIVERSITY OF FLORIDA 3 1262 08105 003 lljiv}i\/§%iTV C)F FLORIDA DGGUMENTS DEPARTMENT 120 MARSTGN SCIENCE UBRARY P.O. BOX 117011 GAINESVILLE, FL 32611-7011 USA