V^/\-Cp^L'^f(9 ACBNO.UH03 "^ NATIONAL ADVISORy COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED August V->\h as Advance Confidential Report IAHO3 THE THEORY OF PROPELLERS I - DETERMINATICK OF THE CIRCULATION FUNCTICK AND THE MASS COEFFICIINT FOR DUAL-ROTATING PROPELLERS By Theodore Theodorsen Langley Memorial Aeronautical Leboratory Langley Field, Va. 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- mcally edited. All have been reproduced without change in order to expedite general distribution. L - U90 t)OCUMENTS DEPARTMENT \ NATIONAL ADVISORY COMMITTEE FOR .^i'PONA'jTICr; ■7/V ^Of ^^ ADVANCE CONFIDENTIAL RFPODT T^IE THEOEY OF PROPELLER!? ■ DETERMINATION OF THE GIRCTILiiTION FFNCTION AND T^IE; F.ASZ COEFFICIENT FOR DNAI,- ROTATING FPOPELLERS SUMMARY Valiaes of the circulation function have been obtaMied for dual-rotstin,;^^ propellers. N'upierioal values are given for four-, eieht-, and twelve-blade dual-rotatlm'!: pro- pellers and for advance ratio?- from 2 to about 6. In addjtion, the circulation function has been deterrdned for f3in{rl3-rotr2tina- propellers for the higher values of the advance rfd:lo. The masfi coeff icl^^nt , another Quantity of significance in prop>eller theory, has bzeri introduced. This nass coefficient, vhlch is actually the ir.san value of the ci"^culstion coefficient, expresses the effective area of t!ic cGluTr.n of the medium acted upon by th<5 pro- peller in ter-Ts of the propeller-disk area. Values of the mass coefficient, rhich have been dete^r.'ined directly by special ■measurements and also by integration of the circulation function, are given for the four-, eVpht-; and tv.jelve-blpde dual-rotating propellers. The mass coefficient has also been deter:nined for several cases of single-rotating propellers, partly lor the purpose of comparing such experimental values with theoretical results in the knov^n range of low advance ratios and partly to extend the results to Include a range of high advance ratios. 'Fne effect of stationary countervanes on the m.a3s coefficient has also been determined for several cuses of •cractlcal interest. INTRODTTCTION i r c u,l .'.: 1 1 o n Fun ■■•. t j '^ n K Cx ) In 1929 Goldstein (reference 1) succeeded in solving the problem of the ideal lift distribution of single- 2 COI'TFIDirTIAL NACA AC^. Fo. li'T-T05 rot^.ting propellers. Goldstein's vjork ir; r-ss+rioted. to the case ci a lij'^it loading and. also, in effect, to & small advance ratio. Numerical values given oj Go].dsteJ. n for the oftirurr circulation distribution are reproduced in tabls J and figure 1. So'-^.e additlona] values calcu- lated by r'rav.'er (reference 2) for hi.gher advance ratios are given in table I and have been super! m.pos':-:d on the Croldstein results in figure 1. Numerical results by L-ock and Yeatman (reference J"* for the four-blade pro- peller are reproduced in table IT and figure 2. The parameter K used in tables I and TI if: the tangent of th.- tin vortex an,.rlc in the ultimate waV'.'' X 1 "<' ! + V TT n^ where v is the rearward dipplace-'-.ent velocity of the helico.l vorte:-" surface at infinity. (A list of the symbols used throughout the paper is given In appendix A. ) These data have been used, for comperlson v.'ita results con- ta^'.ned in the present paper. It ehould be emphasized that a distinction has been made oetv.'een dimensions and conditions o.l the slipstream at +he p^-op roller and those in the ulti^^'.ite wfiPe, a die- tin'.; t ion that does not occur in the treatment of lightly loaded, propellers. The present oaper is concerned exclu- sively with conditions in the ultimate v.'ake ; in fact, it can be shown that thrust, torque, anti efficiency are all uniquely given as functions of thp ultir:ate wake only, no knovjled.ge ci the propeller bein^: necessary except for purposes of actual desij^.n. It should be pointed out that both the diameter b.nd the advance en^.le of the ultimate helix are different fromj tne values at the propeller, the diameter being smaller and the advance ratio larger. It can be shcim that the distribution function depends on the advance angle only. The ideal disti-ibution function is therefore identical for light and heavy loadings pro- vided both' refer to identical helix angles in the ultimate wa'''e . Tn i'igures I and 2 the quantity XCx) is a char.".ct:i r- istic function rels.ted to the circvilation V(y.) along the bl?.'d.c 3 3 follov's: COlJFIDEi'TIA.L NAGA AOF No, lLi.HOJ CONFIDE^ITTAL vix) pr^-o 2tt(V + vOw where F is the potential dirference across the helix so.r'faoe Pt a radius x, p is the number of blader. , and w is the angular velocity of the propeller. The quantity V + w w 00 Pan is the potential drop for a velocity w throurh a length H _ V + VI p ~ GO wliich is the a:; al."o a photograph, shows the equipment used in the manufacture of the helix surfaces. The vertical insulated cylinder is an electrically heated oil tank. To the top center of this tank is attached a simple die or guiding device with a spiral slit through which the heated plastic sheet material is pulled at a uniform rate. A fan Is used to supply cooling air at a uniform rate. '."1th certain precautions an alm.ost perfect helix is produced. Two models of single helix surfaces thus obtained are shomnn at the left and center in figure 6, A preliminary type of built-up model of laminated construction, which was abandoned as too inaccurate and expensive to build, is shown on the right in figure 6., In figure 'Jfa) are shovm examples of dual helix surfaces used for the main investigation. A four-blade dual-wake model is shown on the left and a six-blade dual-wake model is shown in the center. On the right is a four-blade single-rotation helix surface with four- blade "guide vane?:," In figure ']{'b) are other examples of high-order multiple dual-rotation wake m.odels. Some additional examples of single-rot? tion v/ake m.odels vdth guide vanes are shown in figure C The m.ethod of building the dual helix models is indicated in figure 9. TTnit surfaces were cut from light- and left-handed helix surfaces and glued together to form a miultiple dual helix. Tortunately these comrlex built-up dual models were needed only for determining the mass coefficient k and did not have to be too accurate in detail. CCNFIDENTIAL FACA ACF. Fo. iJiFOJ CONFIDEFTIAL 7 For the r"ual-ro tat 1 ng-propeller field a significant property is to be noted: The field repeats itself not only along the axis but also circamf erentially. A "unit cell" consisting of the helix surface between two succes- sive lines of intersection is representative of the entire helix. It nay be seen that the boundary condition is tal-en care of by inserting two insulating planes contalnin^^ the axis and the two intersecting lines, respectively, and by using conducting end planes perpendicular to the axis which contain the sarne two intersecting lines. The vessel rray therefore be given the form of an open I'-shape tray -/;i th the electrodes at each end. The representative helix nay be obtained sinply by stretching a rubber nern- brane from, one corner of the tray to the opposite corner at the other end. The rrenbrane is equipped with stiff radial spokes and is securely damped in place. It aato- natically assumes a spiral shape, the effects of gravity being of secondary order. The entire tray is arranged on a machine lathe with the helix axis along the center line and the exploring needle is attached to the carriage. This arrangement affords convenient reading of the voltage at any point on the spiral surface. In order to increase accuracy, the trays v/ere made of considerable size, 6 to 10 feet long. 3y changing the length and the angle of the tray, all values of \ and the effect of the number of blades could b, investigated. In figures IC and 11 are shown experimental setu^.s for measuring the potential distribution KCx). The con- nections leading to the exploring needle m.ay be seen i.n figure 11. Figures 12, 13, and ik show the general arrangement for determination of the potential distribution on dual wake m.odels. Figiire 12 shov/s a unit cell for very low advance ratio. Note the V-shape test tank and the adjust- able end plate to change the advance ratio. Note, also, the rubber m.embrane stretched between opposite corner^. Figure IJ shows the arrangepient for supporting the exploring needle. In figure iL^ is finally shown the complete experi- mental setup for dual helix surfaces of very high pitch. Wall, Hnd, and Thickness Corrections The similarity betireen the electrical test method and tVie conventional wind-tunnel method may be extended also to include certain corrections. Obviously there is CONFIDEfTTIAL 8 CONFTDENTTAL NACA AC' No. ll|.TT05 a correction that corresponds to the custorrary v^all cor- rection. This correction is readily ascertained by use of vessels of different diaineters, a procedure that can- not be easil^r utilized in wind-tunnel practice. It Khould be noted, further that the wall corrections are obtained with great accuracy since each reading by the el<':ctrical method is rnore precise than its aerodynamic counterpart. 3y using tube diameters about three ti^nes the diaTieter of the test spiral the error in the results was reduced to less than — percent. A correction not appearing in aerodynamic practice is the end correction. This correction occurs only with single-rotating propellers and is therefore of :ii.inor Importance in the present investigation. Dual-rotating propellers possess planes of constant potential perpen- dicular to their axes, and the ends therefore cause no difficulties. By cutting the dual helix at a plane of constant potential and by inserting a conducting end plate in the cylinder the boundary condition is satisfied. For sJngls heliy surfaces, tests on two lengths of the same helix must be used and the difference observed. This procedure was used to measure the mass coefficient k. To m^easure the potential distribution I'(x) a long helix is required, the m:ea3urements to be m.ade near the middle. Another source of error exists for which the cor- rection has been referred to as the thickness correction. This error results from the fact that the m^aterial of the helix sheet must have a finite thickness. This error may be determined by using sheets of two or more thicknesses. It Is readily seen from theoretical considerations that ar.'proximately one-half the thickness of the sheet m.ust be added to the diameter in order to obtain an equivalent infinitely thin sheet. It should be mentioned finally that there is an error resulting from inaccuracies in the ^'i-.odel vortex sheets. The error in K(x) can be minim.ized by using mean values from a large number of readings over a con- siderable portion of the helix. Fortunately, there is no effect on the mass coefficient hc since this coef- ficient is a mean-value param.eter. COFFIDEIvTIAL V'.nt. ACP Fo, LhW^ OONFTDFFTIAL 9 Proof ITriat the Mass Coefficient k Ts the Blocking Effect of the (Infinitely Long) Helix Surface The irasj!' coefficient k is obtained e.xperimontally by treasuring the change in resistance caused by the helix surface v'hen inserted in the cylindrical container. On inserting the (infinitely long) heljx in the container the current between the end plates T^^ is decreased by a definite amount ^T , and it can be proved that kF _ ^ "■ " lo or il S where F is the projected cross section of the helix itnd S is the cross section of the cylindrical container. oj Green's theorerr / ("v^r - ':;vu) a- ^ la = /([JV^w - 'Arv^u) dT Tf ^ ,2, V'^U = v^^' = it follows that (•[JV'y _ iryj) do = Let W bo the distance z alonp the helix axis measured frorr' a reference plane perpendicular to the COITFTDE'^'TIAL 10 CONFTDT::MTI,\L NAG A ACR TTo. rJj.HuJ \ axis; W is therefore a unit vector in the dlrectioa z, L^t 'J be the potential resulting from the applied, volt- ag-e and the local gradient VTt may be written where IJ is the constant voltage difference betveen the ■b-id plates, rhicn are placed at an axial distance L apartj and i is the local current and i^ the current at Infinity. If the surface integrals for the entire "nclosecl hell? surface A and the end surfaces S, respectively, are taken, the following relation is ootained I V dA„ = a ^ /(^-'^^^ or fc /" "a "s ^" where the integrals art. to be taU'cn over both sides of the p helij^ sarff'ces End over both end plates. Hov'/ever , k /•■ n- W ^ ^^z A may be \irritten in the fcr-T #/f'^. - ^'^) dF vihere the integral is ta>en over one turn of one helix for one ?^ido only, as U-, - 1^2 is the dif^'eronce C07JFIDENTIAL NACh AC^ Fo. riiHOJ CONFIDFi'^TIAL 1.1 potential between the tvo sides of the sheet. The Volt- aire drop per sheet is TJ _o_ H L p B^' definition Ffx) = -i ^ TT _2 II L n 'xC'jMi. TJ, A TT dA. = / F(x) dF ^ 2F / F(x) > d>: = kF Also, •T z i dS = / - ^Us = ^ 8 nd tlr e r e f o i^e kF _ AX 's "■ I,, \';here J.^ Is the totsl current betx-^een the end plates v'ith a unlfor^i gradient — ^ in the field. L EXFFKIMENTAL DATA Mass Coefficient K I'TuTierical values for the mass coefficient k, obtained en dual-rotstion v;aVe models, are shovn in COFFIDENTIAL I- 12 COnpIDENTIAL NACA AGP. No. tJ.lHOJ '.'1 t'ure 15. This chart is probably adequate for all rtistical purposes, as a large range of advance ratio has been covered. The designation used for the propellers comprises three digits; The first digit refers to the number of I'irht-handed blades, the middle di£it to the nuniber of guide vanes, and the last digit to the nuiTiber of left-handed blade:^ - for instance, 5~0~3 repre- sents a dual-rotating propeller with three right-handed and three left-handed blades. The highest number of blades tested was for a 6-0-6 or twelve-blade propeller. As a matter of interest, it is very fortunate that the ifethod. and equipment could be tried out in all its ramifications on the known case of the Goldstein curve for a two-blade propeller. The Goldstein curve is the curve in figure I5 marked "Theoretical." The test points, vdiich have been corrected for wall, thickness, and end effects, are shown. Except in a very few cases, the test points lie on the theoretical curve for the two- blade propellers. The somewhat lesser consistency in the cases of d\:al-rotating propellers is not due to in- herent test inaccuracy but rather to a necessary limita- tion on tim.e and equipment for making the m.odels, per- form.lng the tests, and obtaining the corrections. The thickness correction for the hijh advance ratios is con- siderable. Fote that three thicknesses have been used for many of the test points. A glance at one of the composite models shown in figure 7(h) will suffice to indicate the labor involved in producing the models, F^ach test point in figure I5 represents a different com- plete m.odel: some fifty models thus are represented by the results shov.Ti. This number was necessary in order to include all propellers and all advance ratios of interest at present and in the future. Results for single-rotation va^ce models with guide vanes are shown in figure I6 for two-, three-, and four- blade propellers, respectively. Such guide vanes are supposed to represent stationary vanes arranged immediately i-n front of or behind the propeller to straighten the flow. It should be noted that the cases shown correspond to those of an ideal thrust distribution both on the propeller and on the guide vanes. The uppermost curve in each part of figure 16 is reproduced for purposes of com.parlson with the correspondi;"-g dual case. COI'IFIDENTIAL r;AC ..'... /.CR Fo. lIlHOJ confidential 13 Distribution Function K{x,d) The measured potential distributions on dual v;a^ 1 \ 1 ^ 5 '-i \ = 1.0 \= 2.5 0.1 .252 .l6U 0.0919 0.0283 o.ooljoL ,p .U18 .505 .412 .1758 .0552 . 0097U i • pW .2U6 .0795 .01U15 .S29 .U86 .297 .0999 .01806 .Ii5 .655 .510 .1082 .01976 .671 .528 ■ 331 .3U5 .1155 .0212U . 6 .679 ■ 5U0 • 1259 .12U3 .02JU2 . 021x25 .7 .65U .317 .1193 .338 :P .625 • 325 .1213 .02596 .580 .U57 .505 .1156 .02310 .85 .^28 .U3 .276 .1061 .02lli7 .9 .551 .235 .0919 .0187 .0166 •925 .395 • 511 .0817 •95 .329 .260 "173 .0687 .Ollj.1 .975 - .190 .0U97 .0105 CONFIDENTIAL NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS \ 7;CA ./iCR i:o. lIi,H03 C0NF]DEMTIAL 13 TABLE II OFTT?mM 3TrCUI.ATT0N DTSTRITiUTIOW FOR THE FOUR- BLADE P ROPELLER 1 1 Calc\ dated by Lock and Yeatman (referenca 7, 1 1 i X -ir X 3 z x=l X ^• = fe '".JOO c . 1 50 0.209 ^.0667 ') . 66 0.100 -.06k 0. ill 20 0.0 61 .2C00 .^0 6 . 5;>; ^1553 , 1 70 .200 .175 .2857 .1,1286 ,1^,0 •49 , J200 .LlOO? .706 .i;-o . 6h0 . 2000 •297 .300 .203 .81i.3 -;•?- .2667 .k05 .I;00 .577 .5711; .310 . S''CO .Ii.5'^ .7ul '5535 .ii97 C ""f '^ .kU9 .7i1l3 .529 ,s6oo . 871 .50-^ .77^' .021 ./i.coo . ^-72 . 600 .k92 •X^5( ' 0571 .9206 .317 . k'O'.; .£81 .625 .!).667 . 630 .700 ./|6o .. 2c2 , "^^ "'"'■'"■ .£3'7 . 700 .821 .^555 - 672 . too .215 . 7 600 .1:72 •7^0 .675 ,806 , 6000 ,698 . ^-00 .370 , wCOO .P50 .6S0 ., 6667 .706 i C ■'^■'^■' ■. .7i^ .050 .1+88 .0335 , 627 1 .'^e^^^^ r ,-■■ -• ,0335- .^L.y " NATI 01: AL ADVT SORY COmflTTEE FOR AERONAUTICS COrFIDET'TI/.L \ NACA ACR No. L4H03 Fig. 1 S(x) 1.0 nm rTn cmn i^'-Li-miij r=njW7: l-'IIUIL! r~nq UJJI ■^"' '- r !'^ TT" "' ' ■ rL "' 7- uu u;i 'rW^\ - CONFIDENTIAL IF ^*? - - : "^ ^ '■^ h" " == -; ■>, „ ■^ ^ 7 \ .9 ^ i - .— ",' s s. " ,". , ■ 1/ r-' ^ ^ \, s N " "■Tr: ■;: "_^ \? / ^ 1^ ; s ■ ■N '' -- . ■ ri' ':- ~ _ ' -'t '- ,' / y 1 -" \ s. \, -^ : .« ^--1 ;■■ / r\ . 5 - N s V -> / / f — 1 ^ k. ^ 1 V - - \^ / ■ - ^ ' T: *! ■s. s N > " -T—_-^ -"-- / t'^ X 1 _ N • \ •V -~^ ■') / " / ' ' ■■_~^ 5_ u_, "" \ \ : J / / ^ r" ■^ ^^ : \, y ■ , J ■/ ',-' i> - ^S ;^ >! N, \ V I^ ' _'-}.i - f ;.. / / - f' y - \ \ \ .6 - " y 1 r" '^^ j^ ■* : ' 7 ) \ '\ A / / / / ' 5, \ > N> " ~ "f ' fr:: J / v/ ;_ -^ ^ _- Si "' \ V I ■ ■ -x-d - r r ^ -^^ "" "^ -.. \ \ \ \ ■■5 •i — ^ - ~~4 1 / r— -/ — -2_ _ ^ /- ; — l^:^ — — .-_ ^_ Si .-:— V A J T - ^'"' -■- — — f t — 7 — — y 7 ^ — — - "^ ^ ^;:-^ ^- — - ^ ^-^^ '^ \ — -N s \p ^- — — +i [ J 1 / ,< / ^ ^, \ ^ \ - .1+ 's I \- / / f'- -7 ;- 1 : ■ '=£ N s \ - \\ :v^ I ./ / - 2 -P" 'OT K, 'U •r N \ \ \ . '^T t - 1 -^r " - -: ■ r s« •^ =^ u =iH (f ili 1*« w< i f \ ^ .... -_ ^ i / ts== s=^ N • } , '■ r / " ~' r^ =<»* ' ~- ifc; ^: \, \\\ "7r j 1 / r ft ^ - .: -v; fr )(<\ r<) A r N ^ \ S- 1 f f y L'". ; k^ 'v \ : i j j 1 - r ^ ^ - " : : L- _• nN 11 L / / / ' - \ \ 7 (1 -. . J y - - \ s h l^ / " f l/. ': " ' 1 ~, ■ y i 1 f '/ ■/ : 1 ^ — — ~^ -^ - \\ :. In// / y ^^ ^ — , -•*. •^ \^ Si / --- ^- i I m «* m w ^>1 . fpy ^\. '■ (Dt H Ul « ™ N n f't-^ — ca -,^ — - V — - — . -- — • — 33 ;«.- ~ ""■ ~ ^d^ ete sa Lu . 1 I — 1 1 — 1 ■~ \— .1 .2 .5 A .5 .6 .7 X Figure 1.- The function K{x) for sovaral values of X for two-blade propellers .8 .9 1.0 CONFIDENTi AL \ NACA ACR No. L4H03 Fig. 2 ttt hr^. ' TT-rrr "^ ^~^- "■■' i ~^' --XT -' ^■r T~~ '■ ' tf , i, ' ' iSi i $\ ''.- , . ■ ■ . r=; g 3 ^ = ^ ^ ^H 1 y "1 ,— ■— ^ — ^ -^ ^ r- . ■-!^" ' -: 1 -^ ^ -^ ^ — ,^ ^ ^ ■ i.v • rig- I ■ ' "1 ^ ^ -*^ r ^ ■^ ^ -^ / ^ ^" j^ ^ ,r^ >' Y" / % tr' : . r^ ^ y ^ / / 1 fe. f'-'' ." /^ / . / / \- / / 1 ^^, / / t y / y W y^ / / '- 1 = 1 \\ r--' ■ -■ / 1 / j a P ^' is tr: \/i •'j 1 \ :-' "^ ; ^,- j 1 ■ -- — r- 11.: ■-'- r. .1 M. 1- .-L ; < . 1 — 1 z . M Q . 1 — 1 . o \ ' J , >^- \ ■ \ [ 5 ► — 1 r \f^ Y I \ \ 1 — t ^ { \ \ \ \ fi t d ' \ \ •" \ \. - 1 ! ( =3 1 \ \ \ s. \ ' ( X4 \ ^ \ \ N 4 ; \ \ V \ \ K V f \ \^ ^ \ ■ N \ \ . v^ \ \ \ ' . ;• \ S" \ ' -- s. ' '"• ^^i" \ ■ ^ \ V i-l< \ s HI \> if> ^\ N ■^ \ ■'-" ^- :- '■ It \ \ \ S \ r^ \, — 1. , /<: N ^ s s \: \ s ^ :-- ■ k "■ k \ N. \ *:■ 's s ■\l ■> X N \ N ■" T; ■. ;-. I ,-. ■' k N" '^ s . k S ^ y fr -i '^ ■-^ ■^ k. N \ ^-1 it, ■ ■ ^ '--. ^ V, N \ \ it'. i:^ :' '■' "~ --, t^ -^ V X \ \ \, bn- r " ' : - - ■ '-^ -^ *-- •\ S \ s \ 1 ^-' - ' "--, -~-, ^ \ s, \. " tv ■-■ - 1 "" '-, <- s V s\ 5m s \ ^^: t ' 1 ■^ s\ to k^ .: ''■■-! _! . Uu '■^■ii ■i ; -'; '' ^ i4 ut" i^ "■■f ' bi p* ic. ---: L_ 1 '^^ \ NACA ACR No. L4H03 Fig. 3 o tr\ • -d- n o o O -=t a • 1 C ITN a » 3 tf\ o • tt n 0) Ci > rH O o r-l S> O O K> d * u. + o «« c a > 44 « • •P P o (\i Q u 8> • „ » •H V) o • + > •p c o •H O Vi 4-1 o Vl l/N o • o r^ o a a O 1 • • »CV r-l • 1^ xp X (X) 2 = H \ NACA ACR No. L4H03 Fig. o •H 0) o a CO CO (C e a) TD >J tio G G q; •H +j > o O Q.^ to QD G .^ ■ r^ CO ;-i r-t 3 CD w -a (6 o OJ e e (U u J*! o n3 <»-l > a. C 3 o •.-> ■H (U ■►o w (d ij 1 o . u 1— 1 1 r-H 0) r-H 0) tlD u c 3 ■H .0 CO \ NACA ACR No. L4H03 Kig. 12 Figure 12.- Unit, cell for very low advance ratio, \ NACA ACR No. L4H03 Fig. 13 Figure 13 - Unit cell for very low advance ratio, showing ^arrangement of exploring device. \ NACA. ACR No. L4H03 Fig. 14 -G O +J ■■H (X Xi bo U o Cm (1) o 3 I ao \ NACA ACR No. L4H03 Fig. 15 '.'■■' ^ iiii: iSi i;i^ t4l ■Si ^ Ht t^l !'H p:i ■;'! :i\ i|h rt1 Pi- ;:!! ff^fFi !tif:f¥ ffitltff .1=tiilt|#.p!|tf.:j|itt: m wft t{ ■ - ' . ; 4 '■ ; 1' i i HI 1u P' V> 1 a ng IBi "»«, i ;; 11 ' ' ':1 ' li' ]'■ r . 1 ' i i .,.1 Id 'M ff< i ' ^ y ' 1 p k' •iO I't ■'t 'i W' ^ ^ M- i. + .r M M '■! I [ ■ T ■1 ' ( * !l.| i| r; I i ;-.j. rl ; !" ti 'i-'i '/[ ,.; r .: .-I .96 u'^ ^ ^^'-l •-■ "jf 'ii ::':! '- ■'" ^J!i ■:' ^m^ -7.'" ; ,\: ■ !i ' . V t \ i- ' ,;■ [ ■ 'i -.1 ■ '' it- M iv 1 ': '~fil iili. \l\ ' -. \ '■ 1 I-] ,■ 4 1 •>2 V i Thickness of material .- of helix surface i; (In.) -^ + 0.060■^ ii; X .030 - EStperlmental ' --. .020 J ■ 4^ D .000 Extrapolated '^f 1 .... tJ^ 4 ^ ^ ■> . ' ' !-;-4 :i \ \ \ \ ' > J \ ^ \. \ :i _' :' i i .8U \ \ N ^ !;. {| i"'' \: k ^' M V - :1 :l'\ .| i ? \ \ S' k .- r; .n\ !^. , I ' ■■ \ ■ s^ s ■s s <:' ^t f m ^' i^ii *lil i|li^ ^ *!5 illt ^3 ■H't ■ ■ . t T- j \ \ , s S 4 u i.. \ \ ^", '^ t' s N. : ■'• ■ ! ; t 1 "-T -i •72 ; \ \ \ \ \ . i \ ^ V s* 4 M :r. ".'r ■^i" 1 i :'r A .00 \ \ \ \ J \ 1 ' r :....- a .6k i-; ':■ ■ \ %' \ I-. .1 " \i j| f^S i \ \, N \ + i . ^\U C':B 1 .601 \ A ■-/ " ] s *■ \ \ S :♦ T- -! fe^ ".T-. 7". 1 I, ■ ■ '•i. \' N. V » 11 S .56 '- \ 1 \ - \ ^ \ m '■ N, .. '1 i*" 1 \ v; : \ \ is V S-i 1^ ? -: r 4 -H \, ^ r'.' r ^ N 1 •52 f \ \ \ S s s> 1 .1^8 i _^ " 1 \ f-l !;H > ■ \; (( +^; ■K +-- 4 -I jJ ''1 j \ ; ; ■; ■ 1, ■■ ;■ 1 \ ■C' ^.J t s N, fF .1*4 K' :' : \ . ' ■ ( "i i- ' :-| \ < 10. N 1 ■■ '■ il +1 v . \ ' \ \ -•^v >; .ko n\ ■ ■ ■■■ V \''' *^f \''A 'za K' --1 '^ s. ;"^- Si li. t-fl \ 4 \ 'tt i.;-- 1,-! \ s >, f s. <,■ ^ *| .36 ;- ( ; 'i''' " i: : ^;\i '^r 1 \ 'S "^li V ' -J-.' .1 ' 1 k ^ N, "fi. . s*^ s + , ' :~ ■ ' \ ^ "'■■ ■' , . ' - •32 1 ^ V \ 5^ )t! X h*" \ \ .■ ■ -^ - • '; ^ \ X ■^ -n i \ - Ss ■v. ^ ^It — \ j' ■■1 s < > '-<, -'* _ -t \ \ ' s V '--v T \, ..,, X N 2- i'i ^ s^ t k \ X •~ ( ) I .16 N 1 "*-. ~s - ■^ I > \ i> 1 t^ ^ ^ ■; .j 'A Wrt 1st 1C( 1- " H ■nf .08 1,' *■; ■ ^ ^ 1 w nitfl A IVW m .] !:,-^ - -~^ 4i COM Mil £EF OR. EKQ m ttt .Ok ;t ^r M ■iJ -^ .£t il ""^ - 4 U lliu 13£ IS*. 112 m \^ L- u L_ , • _J 4 3 1.6 2.U 5.2 k-0 li.8 5.6 6.1; V * w nD Figure 15.- Measured values of mass coefficient k for dual-rotating propellers with various numbers of blades. 7.2 \ NACA ACR No. L4H03 Fig. 16a .72 p (a) SlngXe-rotatlng two-blade propellara. Figure 16.- Measured vaiuea or mass coefficient k against ^ "*".* nD showing the effect of guide vanes. \ NACA ACR No. L4H03 Fig-. 16b (b) Single-rotating throe-blade propellers. Flgiu*e 16.- Continued. \ NACA ACR No. L4H03 Fig. 16c 2 5 U 5 y •>• w nD (e) Single-rotating four-blad* propellers. Figure l6.- Concluded. \ NACA ACR No. L4H03 Fig. 17a ' w T—T r^i^ST T- l" -- " r m t f ■M 1 P Pi'tj 1 K-i g:1 «-. o SSbS • •HO-^: j:tj:: -' ;•) ■i • 5 T^ ;i»^ --,. 'O. > > ■> i^ ' 31 J ■■.A' '^ N > - > s ■i^ N- ■# 1 ?^ r' > s '^ ^ 1^ S x" 4-X,- ,4 ill / >< — 1 ,> / ^^ ^ W- : "iXi;T7 *K 3 i ' ! ^ / S, », ■^^ ;;;. ^ r- '\ 5, s V-.1 y ( ' N sL> \ V . / - A 'x — ir.¥^: -- _j - i _| f --, / f '\, > h N 2._ ^. ''■ - ' \ . ■ '" /, l\ y. :,1. ■ ; — 1 -, : - i ' 1 '-4- ■ 1 ■ - / ^ \ , A ■ :^ 'T"^ - 1 I /-^ i / 1 ► ■ y V I* i ^"1 •- . - - i / / / / " y . . . . -3? vj - ' / / y r — -^ — \ 4 ; / / _^ - ~l^ N\ ~\ ;r~^ ■ 1 1 y V > , ■ ] J i / . A U '" -t^ -:-.. •■_ % 1: i I ■ '; ; \- -V I T ,- 1 "■: r.i? ■ :" _i f ' '•■'A i- -":; , .- \ -^~ ill j 1 1 -_: X"^ r "" y - ■>: / J f -;:t -^i ^^ i / / : / 1 - '; , ; i,' ' . 1;-' _4 -L J_ 1 \ ' i I . h^ — ■ ■ - i \ ■ __ '' \ rM " 1 ! -; — -- +^4 ® * ^^ [ j\t r " ir ' \ — -) i i ■"■:-, L " ■ T '1 . r^ : \:\ . t - '- ■' i ',' 1 ' ;-i ■ . , ' ' ^ ^ i > ■ .. : : ; " -. _■ . ' -' 1 J . " IJ- - i ' i ■ \ Ih - -- - — ,-^ -■.- • --: , V _|_ v: " ; .. 1 ii* -'■' ,■- ^:-: iV-14^ vi-4.ttf '•■''A I'.r. IL. '■'J -■" ^ iiii - ■'-'a 1-.: E ■ li ','M\ j i ■ XI93 ^TUn SuOfB XBI5"®^0d XBUO^^OBOJ \ NACA ACR No. L4H03 Fig. 17b i±mm -i ; i--4r.;t ^ Xl»o ^fun 8aoxB T'T5«»^od xv^oT439*U \ NACA ACR No. L4H03 Fig. 17c XI^o 'iltin 9uoxB x'T^uer^od x^ao^^oBOj \ NACA ACR No. L4H03 Fig. 17d 11 i irlyt.. 1 ! f" '': ' II 1 Et t:. -:'-['- L : ' ..: 1 : -r !? i 1 ^__.._ In O ^ ;r J O 04^r^ *< O M : -H a c « «--IO-HO(\l ftICO OrH-HB-d'CyN OSrH a *a • t^ ^r^ M^ t- fa^O B • • • . aFs^ ° ° ' ■ o + □ X < —' ; ^":H h • :- : ■A 4 "L-^ -; i '" ■ — ■* - — — I^t 1 ' h 1 ;;[■ r - ^ -- Ii •'}} M' i: ! '1-- J :„ -- •~ -j^ 1 ¥i ■■ - ' h' — 4-: ^- t U '- ' ^ ' t'jH "~il 1 1 1^ ^- --' -^ 1 1; ^ M ^ o C , 1 ■^' 1.- .-M^, M,,-iH r:4irsK^-Mt};;'' l. 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' 1 '.;fi ~ 'rt _ ; ■-' ;■ [ ' ■ '..I. 1^ r- — -^ — -T^ r- K ^ -: :'■ ^: 1'^ \~ -' — ~~i — ~- — — — — - - :t^ i - 'I Sili^l. -J. ^ UiL . __ -^ -A.L J » :^ ^ ^ Li-^' -J- fTN nj \ NACA ACR No. L4H03 Fig. 17e Tiao ^fwn Suo-pB t«T3"s30<1 T'aoT^^'-M \ NACA ACR No. L4H03 Fig. 17f %M$ a : ^m pi m P fif Iff m 1 S ±iilHiii±ili±ttytHmf4j4+t+W mM trttU tiJiiJ atHi-HWH) tj.' i 1 1 1 ill (f: '^r ; -"'T 159 1 ^'i'A 1 P7- '" ■ -' SrH SrH t< • ^ OCrH "" 4J ® a 1 ^•^©■^-d-OMD \£> CO ^ -^ ^ ..ts" • • •• .• .• .• n : -J .-■■-;. — n -. — rm^.|t-.-rt-'-'t-hH-f+' rt.t",-! I 0: 1 ' 1 '■ '• -- Jl Wl,": ' 'i^ \'''\.: r.' f ^ ;■' H'! ,.;! ■''t V I'i ,|t, 1 3 ~ t .'j if ' p ' '■:^ ■j;^ ^T'- --"■ ig' :": ^!i? • : ' Jt . ■-»:! ■ liM nHt ;.]-f t?- ! ■ 4T P ^_:: -— t. . ■- + 1 i,u - 1 -1 ' •IT~r i-l Hi 1 -■ r so- rt — '^: — 1 i y ■4 'S s k - s 1 ^- 1 \\ ■] ': ; \' -. : 1 ■ / , 1 1 Vft ^ i fear iJi- : ,« L L 1« *s f ^""■ 1 - s' ^ '• !^ ^ Z' ^^ t^ -■ / / s - ^''v / V \ / y \ / %' U'': / s l" '. ^■■■^ / / y \ , ) \ •- - ^/ / f &: " M I L hrf r-' '-- - r - : \ \ / / V / T t ■-': j ■ 'iX t^ ( i '; r V _^ r- / '^ 'J : \ 'S t\r. / s \ / ^ f \ 1 n i*^-' r :."■_ \ '." \ ■ if \ — ~ - :\ - ^ I Y ■'1 V :■ 1- ■ ■' ^^ }■ ^-■■r ■ ,\ ■ ■i 1 I '} " ' V V "1 1 - .i 1 1 *it - \ ~J A »4 T 1 ] ■ , ; '" ' / \ /l ■ ■■ i i^ f I-: ■' ' _..t-_ K 4 z — / r »■'' 1 — -- --- T — — — — \ V / // f -— ^ 1 -.J H / ^ "T _: I ■ '"• il '' i . k / ■ t- ii 1 " -- \ / i 3 / \ / r ! ■• i „. / > V - - — i| ,ji. r 1 1'. -''1 1 , , . ■ . '1 ' : ' / / ■ ' r •^ ' \ H 1 "- .' - 1 / I 1 \ jj % "-. -r / _ ^^ y n I ''■" ii \,y. / 1 ! \ -; \ 1' 1 i " 1 h ' ' ? rf' ~1- - v:: — V - • — ? ' . , -- \ I -1 1 ' . , 1 ■k r^ [- r -— ' — ;!■, ^ — *^i ' - i -"j ' ■ --- r4 ' i M ' |r ( ' i — ■■1 *'! {' (, ' 1 k L-] — -- ■i ■1 :' '4 J r-:; ; ii 1' -t ; "■ . ' " I ' "■ '1 , " ! — +T ^: — ■: ' _ : ; ^ lit M Ul H\ mi illt .■rf\;i m h'i-. .,4 1^ iill iji iiiX 'lit ; .: Ai i iiil kl _-Li ^ S'rt h !'^ i1t, ,' iiiti iiai tT©o ^T"^ SuoxB tBT^ug^od xwuo-j^oaad \ NACA ACR No. L4H03 Fig. 17g TTrr r. ■ ■ -- irjt ^ § i PT ■*;! ":; ~ ^ _.. EF "'■ THT wr- ^\ jrrr IT-I ^ — "^ »r- ni ! , g w\ ' : ••-> a fl « OrH*4 ■ NO ffs o, 4> 8 NO ^o KO • b h « O O I «4> h)H 1'' ,»*l ^ tL.' f ~T' ■ ". •'^i r ■ ■ '■' 1 1- 1 ; : 1 -; ■ 1 : - ' \i :,-: I I 1 ■ p+tB^-r fflt H p :-i- ff- titi: 1 i -- I I r •--: '•".- ^-i H'l '--- :,-.[ ^r- ;s — -■• — ~ r- — " I ■- i: g 1 1^" ;. i f Fl M : J__ :. — r -- - U -■ . -■- i f [^ . 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""-- " .: -r -7- 1 ¥^" fHT -: ! ■' < ■ '■ ^- "j__i_- " — — - — - rK, -., "" '■'%'' '■ - : ■' - i ' : — - -j — — - i ; ■ :- -- .„ -~ -^ — '-' "■ 1 t ~ --- - {■- - ^:J ' i ■-' -- ■ ;■ ' r ■- ;- ; 1 1 - 'i. ■ ■ ' i-- ]•- ^-f i-4-^^ ^ T ^ --( . ■„: -_ . ; I'p }'-]' -4 ! i ■ ; 1 '• '■ ' ! ' !' ' -"- : ■ ■ H ■r '% ' '^ i .:: ; :■^ t ■:'^-:"r ;-;i irij ;: ■ , £j t i 1 K i. i 1 i2 ' \'i^ m, :,tf ^ m 1^ . L- — c li^ ] ■-■ ii:J _i ■ ■ _[ ^ uU u-i ^ lii. 11... ^ -y 1^ i: i.1. "I i4i -H ■^>k 5, K WOO IT"" SuoxB X'T5ti»5od x»uoT^3«*M \ NACA ACR No. L4H03 Fig. 17h m\ If ■1 ffi T+H I'ifiii iiiiFiiHiiiiir fn li" W i P li i i m m fmfflt W"\^ ^s I "i ' 1 [.■, r-' ]-r -^ — 12 ' 1 «-» o a (^ *? O O 4-.^ •H O bfi 1:^8 ^1 ^ 1 ? KO • • ■ ... aiSn ° o . . . 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Si N, ^ '^i* -■/ y i 1 /^ 7 y \^ "H ^ ■r^ k s s h' ■-- r)n N ii ■ \ \, V fef ^"'/^ / ^ Pt y / ^ / /■ / ^ : :i 1 . -:_ — Sl; s ^t" s :- ^ fc 1 ■ """ I / / '/ ;' ■■!: -r- "N s N ^ — ti^f ■;; 1 j i /- / |« fS" ■ ^: --■ \ ^ } ^ — — — i.: - 1: ' > "s, •s \ : f I -'4--- / i^ — ■ N S'- V ^ \ : :■ r i // r -- ■• ; \| ^ V — ^ 1 f ': :l U- ',■' -- V* v i-VI \ . ; ' . ! : V ■,; \ ' 4 ' / ^ - \ : ^1 ' r i r -- H 5 ; ; -— — — — — - H tl ~ 1 — \ ^ ^ +-— ^- m ] 1 "" — / III i \ '" I !-' ■ - ■:' ■ ; ■ 1 ■ 5lL 1 ..T ^ ' ! ■1 1 ft 1 1 i 1 J T -T -■ — - — ^ — — ^ -r — -^ ■ '!'! I '■'i " : .'■: t ,' j . 1 i' ' i r ■ 1" '— — — ::_ 1 1; ^ r,- — ;! ■ t t"t -^it: m .1 - .<• bt ^ ■ ' ' '■ , 1 ' ■ " 1 ' . ' L ^ \ . ' ■ '! i'" T- — — ^ f^ ■^ r*'- 1 Lii ■ ' ■ ill; k. ■ nit jS • ■±2 iii "--I- :.- -1-1 n L. ntr fc. m ^ Lli M lUI -^ lfN^{ ^ jii ^: iiii 1 i; t M: L , 1. nM M f\J sD O XX95 QTun SuoxB xeT^ue^od t^uoitioBJj \ NACA ACR No. L4H03 Fig. 17i \X90 ^jvm 3uoxB x^T^aQiod X'°°T5^'«M \ NACA ACR No. L4H03 Fig. 18a y: ;r;:r '"'; " — j ■^ pr^ m ;-.: '-''] -'i* ,■■-,,1 :l.i -:;; ^■' ,:.! f"' ■■\r\ ■,' 'mi >ii ^-T- p>. t,t; Pi *5 SiivjtSf KJtt ?A& wpi ...-: ^ a ._j-^ ■- --- J V ^- -aP •■ -: ,- ,.^ ^^ ^ -^ ^ 4 t ■ ^ -■ ! -^ ^ y I. :!'] ■',-^ ■** ' -4 — ; X / / !■ • ■ I. .^ (•' J r r —i ' ■' ; t: 'r ^ ■ : -'- — ■ ■- 1 / ,-' / - / / : i ^ / / ii-^ / - "^ ; / / ■ f- - ■y r : \ ' t i ■ / f - :; / — . !- vj ■^■ a r- ^ 1 I ^:'- e / '"■■ ^ p.- ■{ J r '* ' " i ^ ','.' ■ / •n / T in- — '" ^ -.- — I "^i ;:i' — r:t / ''[W ' 1 f t ' X -^ — -T R rl t I / ^ 1 -: . lu ; 1 ! f" !: :._- " 4- ' 1 - i 1 ' ! . .' L i :S L .- 1 t-, 1,1 - : .' 1 r;- 1 ■ if — 7 j ^ I'-. : / il J -1 ■ i-*; i:^ — — -- J — 1 1 — — — ~ — - ':! .-: - ■ -f ■ / ■ * .,^ 1 / ^ ,-;; ■ ■ ■■". r^' ■Jj . _- :- ■'•ft --i S : '■A f '; — -; — ; ^- _.;. , '" ■ E o F O k > -. .' ' !.' 1. , ■,} 1 .;; ' J- II '-■-:. ^v" tiffl 1 * %. § f I ;-; : -U .1 „ 1^' S'i }* U r' 1- :-' " -- - ' i '- kl f--: -1 - rH 1 a a — " ■~ T^^ :,;; !■ > :i ■'■'; ul ['"■ fi " r— ,. -" r-- ... ~ ■ - .[ .- i — =^ ;:t: i|,. iil '■ ' __ ~ ~ —J ;-- ) • -r — — — 1 ; ' , - f-;t y l; , , • i"| i.':i &'- I - — : -"r<\ )-:-■ -■ — ^^ -: — -i "T " * ■r r- - — t '■■' Ej4 ^' ■"tT W; t ; ■'^ -' ill ''i' "-1 • ■ M '\!^ 41 fi;' iiii ' . ^ ■ji . P,| ill! ' t'' : - !■■ ;.,{ ,,?t .-! / ^ i: IT . 1 ■ : ~n ■-i i;K - - 1 ■- :" — ' 1 ; 'i ; i *j :'- t JT J- a fcj u yh iisi L- •'-. 1 _ ,; ll.. ^ 3 J) r^ 3 C -H 5- C « « *^ v. c • >* o I -o o \ NACA ACR No. L4H03 Fig. 18b NJt#BiJi « tftr TT- ::T:i }4r +- — ~ -4-^ \ NACA ACR No. L4H03 Fig. 18c R m '1 r| m 33 ':;■ i.:14 ;".: P| It^ m P> r- \ 1 • 1 ■ " r " t — ■ '^- -': - 1 = !"■■:- • rt'" r - r Pvr: i— - '' '^ '- - ; *^ ^ ■St=» ** ^ "^ s fe» "1 -M i ■"1 '■■ li^ H- •-^ --J ••— ' - r"^ ' -t f'^ 1 ■: ! 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