Naca Report 775

8/7/2019 Naca Report 775

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REPORT N(). 775THE THEORY OF PROPEIJ.ERS

I-DETERMINATION OF THE CIRCULATION FUNCTION AND THE MASS COEFFICIENTFOR DUAL-ROTATING PROPEI,J,ERSBy THEODORE T H E O D ~ R B E N

o . I .2 .3 .4 .5 .6 .7 .8 .9 1.0v:FIGURE I.-The function KCz) for several values of for two-b1ade propellers.

o . I .2 .3 .4 .5 .6 .7 .8 .9 1.0X

FlGURE2.-The function KCz) for severalvalues ofx for four-blade propellers35

SUMMARYValues oj thecirculotionfumction. have beenobtainedjor dualrotating propellers. Numerical oaluee are given jor four-,eight., and twelve-blade dual-rotating propellers and jor adoomceratiosjrom 13 to about 6. In addition, the circulationjunctionhas been determined jor single-rotating propellers jor the highervalues oj the advance ratio. The mass coefficieni. another quamtity oj significance in propeller theory, has been introduced.This mass coefficient, which is actually themean oalueoj thecirculation coefficient, expresses the effective area oj the column ojthemedium actedupon bythepropellerin terms oj thepropellerdisk area. Values oj the mass coefficient, which have beendetermined direcay by special measurements and also by integration oj the circulation junction, are given for the four-,eight-, and twelve-blade dual-rotating propellers. The mass coefficient has also been determined jor several cases oj singlerotatingpropeUers, partly jor thepurpose oj comparingsuch experimenial oaluee with theoretical results in the Icnouni range ojlow advance ratios and partly to extend the results to include arange oj high advance ratios. The effect oj stationary countervanes on themass coefficient has alsobeen determined. for seoeral.cases of practicalinterest.

INTRODUCTIONCIRCULATION FUNCTION K(;;)

In 1929 Goldstein (reference 1) succeeded in solving theproblem of the ideal lift distribution of single-rotating propellers. Goldstein's work is restr icted to the case of a l ightloading and also, in effect, to a smalladvancerat io . Numerical values given by Goldstein for the optimum circulationdistribution are reproduced in table I and figure 1. Someadditional values calculated by Kramer (reference 2) forhigher advance ratios are given in table I and have beensuperimposed on the Goldstein results in figure 1. Numericalresults by Lock and Yeatman"(reference 3) for the fourblade propeller are reproduced in table II and figure 2. Theparameter Xused in tables I and II is the tangent of the tipvortex angle in the ultimatewake

X=1:. V+w11" nD

where w is the rearward displacement velocity of the helicalvortex surface at infinity. (A list of the symbols usedthroughout the paper is given in the appendix.) These datahave been used for comparison with results contained in thepresent paper.

/.0.9.8.7.6

K(v:).5.4.3.2. I

1.0.9.8.7.6

K(v:).5.4.3.2. I

,\ 1/10" -;:., / -- 1/7--


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36 REPORT NO. 775-NATIONAL ADVISORY COMMlTl'EE FOR AERONAUTICSTABLE I

OPTIMUM CIRCULATION DISTRIBUTION FOR THETWO-BLADE PROPELLERCalculated by Goldstein (reference 1, p, 450)

1 1 >....! >...!. 1 >....!% >"-10 % >"-"'j % % % >"-"3 %6 4.--

0.020 0.126 0.029 0.126 o.GW 0.124- 0 .000 0 .120 0.067 0.111 0.100 0.092.040 .m .0S7 .:145 .080 .24.0 .100 .232 .133 .213 .200 .175.060 .35'l .086 .351 .120 .3 .150 .200 .303 .300 .243.080 .4.4.6 .114. . 3 .160 .4.31 .200 .4.18 .2ffl .m .400 .W ;.100 .526 .14.3 .623 .200 .611 .250 .4.89 .333 .4.40 .500 .sss.IW .593 J7J .500 .240 .576 .300 .548 .400 .4.85 .600 .341.14.0 .6.50 .200 .M6 .280 .626 .350 .592 .4.67 .514 .7llO .331.160 .698 .Z 9 .69t .zn .669 .400 .ezs .533 .533 .800 .:"-:r 20.1 0.232 0. 164 0. 0019 O. 0'l83 0.00494.2 .418 .303 .1768 .0M'l .00974.3 .548 .412 .24.6 .0i95 .01416.4 .629 .486 .297 .0999 .01806.45 .665 ..510 ----------- .1082 .01976.5 .671 .518 .331 .1155 .02124.6 .679 .MO .34.5 .1239 . 0 2 3 4 ~.7 .8M . .517 .338 .l24.3 .02423.75 .623 .493 .326 .1213 .02396.8 .580 .457 .3O.'l .1160 .02310.85 .628 .4.J3 .276 .1001 .0214.7.9 .449 .351 .235 .0919 .0187.926 .395 .311 ----------- .0817 .0168.9:1 .260 .173 .0687 .014.1.976 ------- .100 ----------- .0497 .0103

TABLE IIOPTIMUM CIRCULATION DISTRIBUTION FOR THE

FOUR-BLADE PROPELLER

Calculated by Lock and Yeatman (reference3)1 1 >..-; 1 1% >'-6" % >"-'4 % % >"-"2 % >"I:4

0.1200 0.300 0.150 0.299 0.0667 0.006 0. HIO 0.064. 0.1429 0.061.2000 506 .250 .r05 .133::1 .179 .200 .173 .2857 .159.3200 .706 .3:iO .649 .2000 .297 .300 .283 .4288 .249.4000 .786 .4.00 .702 .2667 .405 .400 .377 .5714 .310.llOOO .848 .450 .744 .3333 .4.97 .506 . 9 .7143 .329MOO .871 .500 .776 .4.000 .512 .600 .492 .7857 .317.6000 .881 .625 .821 .4.667 .630 .700 .501 .ssn .282.7000 .887 .700 .821 .5333 .672 .800 .469 .9286 .213.7600 .872 .760 .806 .6000 .698 .000 .370 ------ ------.8000 .8.50 .875 .689 .6667 .706 - - - - - ----- ------ ------.9000 .714 .9EO .488 .8333 .627 ----- ----- ------ ------.9600 .503 ----- ----- .9333 . 5 ----- ----- ------ - - - - - -

I t should be emphasized that a distinction has been madebetween dimensions and conditions of the slipstream at thepropeller and those in the ultimate wake, a distinction thatdoes no t occur in the treatment of lightly loaded propellers.The present paper is concerned exclusively with conditionsin the ultimate wake; in fact, it can be shown that thrust,torque, and efficiency are all uniquely given as functions of

the ultimate wake only, no knowledge of the propeller bnecessary except for purposes of actual design. I t shbe pointed out that both the diameter and the advangle of the ult imate helix are different from the valuethe propeller, the diameter being smaller and the advratio larger. I t can be shown that the distribution ftion depends on the advance angle only. The ideal distrt ion function is therefore identical for light and heloadings provided both refer to identical helix angles inultimate wake.In figures 1 and 2 the quantity K(x) is a characterfunction related to the circulation rex) along the bladfollows:

prwK(x) >==211"(V+w)wwhere I ' is the potential difference across the helix surfaca radius z, p is the number of blades, and w is the angvelocity of the propeller. The quantity

V+ww-w%-is the potential drop for a velocity w through a length

which is the axial distance between two successive vosheets. Each sheet has 2"'11" turns corresponding to a tim1 second and there are p separate sheets correspondingblades. The quant ity K(x) is thus the nondimensioexpression for the potential drop across the surface ofcontinuity as a fraction of the available drop in the direcof the helix axis.

I t should be noted that the coefficientK(x) differs fromGoldstein coefficient

in which the velocity 'w has been disregarded in comparwith the advance velocity V. The coefficients are identif referred to the same helix anglo of the ultimate wake.

MASS COEFFICIENT IC

A significant coefficient, which will be termed the mcoefficient /C and which may be shown to be one of the bparameters in the propeller theory, is now introduced. Igiven here merely by definition/C E zl 1K(x)z d

where z is the radius and the integral is taken from xto x=1. By inspection it is noted that /C is really the mvalue of the coefficient K(x) over the disk area.K(x)=l , then 1C=1, which is the limiting value of /c.


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THEORY OF PROPELLERS. I-DETERMINATION OF CIRCULATION FUNCTION AND MASS COEFFICIENT

obtained or, after integration,

x2K(x) = >.i+x2

,,=1->.2 log ( 1 + ~ )

For the present problem a direct measurement of the aerdynamic field behind a propeller presents insurmountabdifficulties; in contradistinction, the electrical methodmeasurement is convenient and accurate and, in additiopermits the determination of local as well as integrateffects. The arrangement may, in fact, be consideredspecial calculating machine for solving the differential eqution for given boundary conditions rather than a means fobtaining experimental solutions.Since the ideal flow (far behind the propeller) is identicwith the flow around a rigid helix moving at a cons tavelocity in the direction of its axis, the corresponding electfield is obtained very simply by inserting an insulating hesurface in a conducting liquid and applying a uniform fieldthe direction of the helix axis. The vessel confining tliquid is a long cylindrical shell, also of insulating materiThe vessel is closed at both ends by copper end plates thare used as electrodes to apply the potential. The tespecimen helix is placed coaxially with the shell. The confing shell is considerably larger in diameter than the test helFigure 4 is a photograph of the tes t setup for the diredetermination of the mass coefficient K. The cylinderthe r ight const itutes a dummy compensating resistancThe electrolyte used in these experiments consisted of twater from the local water-supply system. The sourcecurrent was a 1000-cycle al ternating-current generaproducing a rather pure wave form at an available voltaof about 100 volts, which was applied to the electrodes. Aexploring device consisting of a fine glass-insulated platinuwire with an exposed tip was used to determine the voltaat any point on the helix surface. This pickup deviformed a part of a potentiometer circuit used in a Whestone bridge arrangement with a sensitive telephone aszero indicator. When voltage readings were taken,current passed through the telephone and the exploratiwire. This type of measurement is inherently accurate; terror in the electrical measurements is estimated as no t mothan one part in 10,000.Figure 5, also a photograph, shows the equipment usedthemanufacture of the helix surfaces. The vertical insulatcylinder is an electrically heated oil tank. To the top centof this tank is attached a simple die or guiding device withspiral sli t through which the heated plastic sheet materis pulled at a uniform rate. A fan is used to supply cooliair at a uniform rate. With certain precautions an almoperfect helix is produced. Two models of single helix sufaces thus obtained are shown at the left and centerfigure 6. A preliminary type of buil t-up model of laminatconstruction, which was abandoned as too inaccuraand expensive to build, is shown on the right in figure 6.In figure 7(a) are shown examples of dual helix surfacused for the main investigation. A four-blade dual-wamodel is shown on the left and a six-blade dual-wake modis shown in the center. On the right is a four-blade singrotation helix surface with four-blade "guide vanes."figure 7(b) are other examples of high-order multiple duarotation wake models. Some additional examples of singrotation wake models with guide vanes are shown in figure

/.0 /. 5 E.O E.5 3.0 3.5 4.0 4.5 5.0(V+7tV/nD.5

.: ~ ' - . :'-... r-, Pf\: ... "-..:: co, ". -1-6' < ,

:"\:' < ;:> '< ,


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38 REPORT NO. 775-NATIONAL ADVISORY COMMI'ITEE FOR AERONAUTICSThe method of building the dual helix models is indicated infigure 9. Unit surfaces were cu t f rom r ight - and lefthanded helix surfaces and glued together to form a multipledual helix, Fortunately these complex built-up dual modelswere needed only for determining the mass coefficient IC anddid not have to be too accurate in detai l.

For the dual-rotating-propeller field a significant propertyis to benoted: The field repeats itself not only along the axisbut also circumferentiaUy. .A "unit cell" consisting of thehelix surface between two successive lines of intersection isrepresentative of the entire helix. I t may be seen that theboundary condition is taken care of by inserting two insulating planes containing the axis and the two intersecting lines,respectively, and by using conducting end planes perpendicular to the axis which contain the same two intersecting lines.The vessel may therefore be given the form of an open Vshape tray with the electrodes at each end. The representative helix may be obtained simply by stretching a rubbermembrane from one corner of the tray to the opposite cornerat t he o the r end. The membrane is equipped with stiffradial spokes and is securely clamped in place. I t automatically assumes a spiral shape, the effects of gravity beingof secondary order. The entire tray is arranged on a machinelathe with the helix axis along the center line and the exploring needle is attached to the carriage. This arrangement

"I11

\ \r/

affords convenient reading of the voltage at any pointspiral surface. In order to increase accuracy, the traymade of considerable size, 6 to 10 feet long. By chthe length and the angle of the tray, all values of h aeffect of th e number of blades could be investigated.

In figures 10 and 11 are shown experimental setumeasuring the potential distribution K(x). The conneleading to the exploring needle may be seen in figureFigures 12, 13, and 14 show the general arrangemdetermination of the potential distribution on dualmodels. Figure 12 shows a unit cell for very low adratio. Note the V-shape test tank and the adjustabplate to change the advance rat io. No te, also, themembrane stretched between opposite corners. Figshows the arrangement for supporting the exploring nIn figure 14 is finally shown the complete experimentalfor dual helix surfaces of very high pitch.

WALL, END, AND THICKNESS CORRECTIONSThe similarity between the electrical test method aconventional wind-tunnel method may be extended ainclude certain corrections. Obviously there is a corthat corresponds to the customary wall correction.correction is readily ascertained by use of vessels of di

FIGURE ~ - T e s t setup for direct determ1natlon of th e mass coe1llclent K.


5/17

FIGURE 5.-Equlpment forconstructing oollulohl bollxSIIrfllOOll. FIQUIlIlG.-OelIulold,slng!o helix surfllOO!l. (On right, preI.lm1naryIJIInJnntedconstructlon.)

iZHoZoI:j


6/17

/

,t

.. '

I.I /._

,- .,"~ " ' , t ' / II ,

,J-",/1"

,., ' .l. .r-- ' ~ ' I

. I

j ,

(b) H!gho()rdermultiple blades, dual-rotatlon.FIGURE T.-Conoluded.

I .,I!'1\. f \, /\ I ;'\ />(/1\t

/

Iv- I,

\ /"

(0) Left and center-!or four and six blades, duat-rotatlon, Rlght-for four blades. ~ r o t a t l o n , with four-blsdeguide 'I"aIlOS.FIGURET.-Dual helixsurfaces,


7/17


FIGURE8.-Two-blade alngle-rotatton hellx surfaceswith guide vanes.

diameters, a procedure that cannot be easily utilized in windtunnel practice. I t should be noted fur ther that the wallcorrections are obtained with great accuracy since each reading by the electrical method is more precise than its aerodynamic counterpart. By using tube diameters about threetimes the diameter of the tes t spiral the error in the resultswas reduced to less than percent.A correction not appearing in aerodynamic practice is theend correction. This correction occurs only with singlerotating propellers and is therefore of minor importance illthe present investigation. Dual-rotating propellers possessplanes of constant potential perpendicular to their axes, andthe ends therefore cause no difficulties. By cutting the dualhelix at a plane of cons tant potential and by inserting a

conducting end plate in the cylinder the boundary condis satisfied. For single helix surfaces, tests on two lenof the same helix must be used and the difference obserThis procedure was used to measure the mass coeffiK. To measure the potential distribution K(x) a long his required, the measurements to be made near the middAnother source of.error exists for which the correctionbeen referred to as the thickness correction. This eresults from the fact that the material of the helix sheetmhave a finite thickness. This error may be determinedusing sheets of two or more thicknesses. I t is readilyfrom theoretical considerations that approximately onethe thickness of the sheet must be added to the diameteorder to obtain an equivalent infinitely thin sheet.


8/17

42 REPORT NO. 775-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSI t should be mentioned finally that there is an error result

ing from inaccuracies in the model vortex sheets. The errorin K(x) can be minimized by using mean values from alarge number of readings over a considerable portion of thehelix. Fortunately, there is no effect on the mass coefficientit since this coefficient is a mean-value parameter.PROOF'::'THAT TH E MASS COEFFICIENT K IS THE BLOCKING EFFECT OFTHE (INFINITELY LONG) HELIX SURFACE

where F is the projected cross section of the helix andthe cross section of the cylindrical container.

By Green's theorem

I f

The mass coefficient K is obtained experimentally bymeasuring the change in resistance caused by the helixsurface when inser ted in the cyl indrical container . Oninserting the (infinitely long) helix in the container, thecurrent between the end plates 10 is decreased by a definiteamount 6.1, and it can be proved that

orKF sr8=1 ;

6.1 SK=[O F

it follows thati (U 'VW-W'VU)du=O

Le t W be the distance z along the helix axis measfrom a reference plane perpendicular to the axis i 'Vtherefore a unit vector in the direction z. Le t U hepotential resulting from the applied voltage and thegradient VU may be written

'Vu=UoiL 1-0

Figure 9.-8teps In the construction of the more complicated belli: surfaces.


9/17


-------- - - - - ~FIGURE lO.-8etnp fur measuring th e potential distribution K(z) for alngle-rotatlon wake, models.

FIGURE B.-Setup formesmrllJgpotentialdlstrfbntlon K(z) forsfngllH"Otatlonwlllremodels. showing detall ofexploringdevice.

43


10/17

REPORT NO. 775-NATIONAL ADVISORY COMMITTEE FOR AEnONAUTICS"). .'---------------,

i ' j.I i r

I II :. ifJ}11II

!ii"...._... ...

LFIGURE12.-Unlt cell for very low advance l'IItlo.

\

.


11/17

THEORY OF PROPELLERS. I-D ETERM IN A TIO N OF CIRCULATION FUNCTION AND MASS COEFFICIENTUo is th e constant voltage difference between th ed plates, which are placed at an axial distance L apart,

i is the local current an d iJ th e current at infinity. I fe surface integrals for th e entire enclosed helix surface Ad the end surfaces S, respectively, are taken, th e followingation is obtained

fA U dA.= fsC U - z ; ~ ) d S&0 fAU dA.= fsCg - i ~ ) dShere the integrals are to be taken over both sides of thehelix surfaces an d over both end plates. However,

ma y be written in th e form

where the integral is taken over one turn of one helix for oside only, as U1- U2 is the difference potential between ttwo sides of th e sheet. Th e voltage drop per sheet isUoHLp

FIaURE 13.-Unit cell for very low advance ratio, showing arrangement of exploring device.


12/17

46 REPORT NO. 775-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

1',\1'\(

//- ~ ~ . L . - - -.-' ,..,..r

.; .


13/17

THEORY OF PROPELLERS. I-DETERMINATION OF CIRCULATION FUNCTION AND MASS COEFFICIENT 4765

.8

\\\ Thickness o f maferial\>1", o f helix scr-foce, (iL)+ Q06O} I - - I - -\ 1\ -, -, x .030 xperimen fa l\ \ \ 0 .020 I - - I - --, 8-0-1 ' \ ~ . 3 - 0 - 0 '\ 4-0:0- 1.---4-8-0 r-,, f'\ 1\ -, \ r-,,

\ I\. \ ' \ ..3-4-0 ....... ---r--...,.....--- --- t -- --- 1'0 I--- --- I-(a) I I >- (b) (c).16

.24

.32

.40

.48

.56

.6 4

.72

.08

01234560 1234560 1 ,2 3456(V+w)/nD (V+1O)/nD (V+w)/nD(a) Blngle-rotatlng' two-blade propellers, (b ) Blngle-rotatingthree-blade propellers. (c) SlngIe-rotat!ngfour-blade propellers.

FIGURE16.-Meamred values or mass cootllolent K against :1r showing the e1Iootofguide vanes.


14/17

48 REPORT NO . 775-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

o . I .2 .3 .4 .5 .6 .r .8 .9 /. 0Frocti0701 radius, or(a) 2-Q-2unit cell; ~ " t 1 D -L8lI.

FIGURE17.-Varlallon of polentlal along a radlns for various angular positions.

o . I .2 .3 .4 .5 .6 .7 .8 .9 /.Froctional radius, .r(oJ 2 - Q - ~ nnlt cell; :1:" 0-FIGURE17.-Contlnucd.

Anqubrgpsition from symmet.ry l ine or cell(fraction o f cell semionqle)4- -0 .920 +- 0.333 0 - - -0567'17- .753 0- .000 Dr- -.833D-- .542 x - - . 333 I>- -.945

./I--"" Vf-"" I---i r-.Ia-'

il--=- I - - ..- ,.-1- r-'- - V'\'1>0l--'"f '-l - i,..-.'-- --....... ;--. ....... --... f-,..

f '. .......b..-' Ie) ...- l/>o

.91.0

. I .2 .3 .4 .5 .6 .7 .8 .9 /. 0Froctional rooios, .:I:(b) ~ n n l t cell' V+ao_3.J-L nD

FIGURE 17.-eontlnned.

-c......

'I.' jr/-- I- -... / ' /I-k:: r-:, I(t- -1- L--:: /" V rb':::6::< I..- --1- l-I" " :--v- I - - "< ,_l- t---.. )'vI'-.

.. - .. c ,". .-j- ---(b

.9\J.8

(j

Jl2..4. ~ . 3

. f


15/17

THEORY OF PROPELLERS. I-DETERMINATION OF CIRCULATION PUNCTION AND MASS COEFFICIENT 4

./ .e .3 .4 .5 .6 .7 .8 .9 /.0Fractional radius, .t 'V+Wcg) ~ u n l t c e l l ; nD -1.M.

FiOURE17.-eontlnued.

Anquhr oosit/on from symmetry line o f cell(fraction o f cefl semianq/e)0-0 .660 0 -0 . 000 x--0 .692.......

t'x. "" . 1\-, \I\" \\

1\\

1\\IitII I

1/ /// v- V I(g) -- - . /o.1

.9/.0

. / .C .3 .4 .5 .6 .7 .8 ..9 /.0Fractional rodius,.t '(e) 4-lH. unit cell' V+W-3.1L, nD

FiOURE17.-oontlnued

Anquhr .oosifion from symmetry Hne o f cell(fraction o r cell semianqle)x- 0.850 +- 0.250 'V-- aoool:r-- .7[}() 0- -.4820- .486 t> - - -.844

-.- I'-- "t- - - :-- t - - ; - - I - . ""- \- r--, -... i" --. e-.'" < , / ,A '(::::: ;::::. .....- ,. /v- A, :::::::'- ..- ..- .- < --- l- V'"-I--' j,(e) l.-o'o

/.0

o o .I .2 .3 .4 .5 .6 .7 .8 .9 1.0Fractional radius, or(b) s-e-eunit cell' V+ID -3.12., nD

FiOURE17.-Coutluued

Anquhr /?9Sifion from symmetry l ine o f cell(fraction o f cell semianqle)0 - 0 . 7 7 8 X-QOOG +---0.5040- .454 v- -.314 0- -.776

"" -.. -'r-.... "\::::::::; ;;;;:: ...... < , r-,"""'- r-, \.r-, "" \'i

\Vi'

/ /I


16/17

50 REPORT NO. 775-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

.9

Anquhr ppSition fr om symmetry line o f c e l l(fraction o f cell semldngle)~ - - Q 4 5 2 o - - Q 4 5 2 +---Q576x - .000 c-- -.843

o .I .2 ...3 .4 .5 .6 .7 .8 .9 1.0Fractional radius, a:(I) lHHI nnlt eell ; V +u> - Ml., nDFIOllRE17.-Conclnded.

o

2

. I

2.3

.9

.7

.8

.4

1.0 I I I.9 I rv"ur/nOx 1.89.8 0 3./40 6.00.7 r::-. .......0 -...,

< ,.5 "" ".4 \-- --.3 - . . . . " \"-.e t\...\. I

...... -- '1/.......9 - l "- r-,8 - -,--I--7 '< , - , '\


17/17


advance ratio of wake helix

advance velocity of propellerrearward displacement velocity of helical vorsurface (a t infinity)rotational speed of propeller, revolut ions psecondangular velocity of propeller (211'1t)diameter of vortex sheet

1. Goldstein, Sidney: On the Vortex Theory of Screw PropellProo. Roy. Soo. (London), ser. A, vol. 123, no. 792, April1929, pp . 440-465.2. Kramer, K. N.: The Induced Efficienoy of Optimum PropelHaving a Finite Number of Blades . NACA TM No. 81939.3. Lock, C. N. H. , and Yeatman, D. : Tables for Use in an ImproMethod of Airscrew Str ip Theory Calculation. R. &; M. N

vAPPENDIX

REFERENCES

LANGLEY MEMORIAL AERONAUTICAL LABORATORY,NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ,LANGLEY F IELD , VA. , May 7,19#.

SYMBOLS

six-blade dual-rotating propeller shows a value of "0.486 at the same advance ratio. H a dual-rotating ppeller is no t used, the double guide vane is undoubteddesirable in some cases. Actually, the induced lossreduced to about half as compared with the 1088 in the cawithout vanes. The difference in the effect of two or fovanes is relatively small,

wn

rK(x)

w

Rr

H

K(x,8)P"

8FS106.1

DV+w1J)i\=1. V+w

1r nDpitch of wake helix ( V ~ w ) 'circulation at radius rcirculation function for singlerotation

( prw )2'll'(V+w)wcirculation function for dual rotationnumber of blades of propeller or separate hesurfacesmass coefficient ( 211K (x)x dx

1 fl r- )or;:Jo Jo K(x, 8)x dx dOratio of radius of element to tip radius of vorsheet (rIR)radius of element of vortex sheet

tip radius of vortex sheet ( ~ D )angular coordinate on vortex sheetprojected area of helix (7rW)area of end plates of cylindrical test tankcurrent through test tank with helix removedreduction in current due to presence of helix

(V+w)/n'n1.890 3.140 6.00I-.v .1/4 -,

1'/'

x

- ... I.I - 4 "1/4-, > - 1 - a - '-I- -

1/2' -a:- 311-' . --.-.< V -cr: ..-1 1 4 . ~ 1 , . , . t 1 2

-"-;;: ..-I . . - -0f- - 314"(If)1-07) =11

- 112"- 1314..1/4'"- 1120"-,- 13/4, .II L-" -- l-I r-

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Naca Report 775