The theory of propellers

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Material Information

Title:
The theory of propellers
Alternate Title:
NACA wartime reports
Physical Description:
18, 34 p. : ; 28 cm.
Language:
English
Creator:
Theodorsen, Theodore
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
Publisher:
Langley Memorial Aeronautical Laboratory
Place of Publication:
Langley Field, VA
Publication Date:

Subjects

Subjects / Keywords:
Propellers, Aerial   ( lcsh )
Aerodynamics -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
Summary: Values of the circulation function have been obtained for dual-rotating propellers. Numerical values are given for four-, eight-, and twelve-blade dual-rotating propellers and for advance ratios from 2 to about 6. In addition, the circulation function has been determined for single-rotating propellers for the higher values of the advance ratio. The mass coefficient, another quantity of significance in propeller theory, has been introduced. This mass coefficient, which is actually the mean value of the circulation coefficient, expresses the effective area of the column of the medium acted upon by the propeller in terms of the propeller-disk area. Values of the mass coefficient, which have been determined directly by special measurements and also by integration of the circulation function, are given for the four-, eight-, and twelve-blade dual-rotating propellers. The mass coefficient has also been determined for several cases of single-rotating propellers, partly for the purpose of comparing such experimental values with theoretical results in the known range of low advance ratios and partly to extend the results to include a range of high advance ratios. The effect of stationary countervanes on the mass coefficient has also been determined for several cases of practical interest.
Bibliography:
Includes bibliographic references (p. 16).
Statement of Responsibility:
by Theodore Theodorsen.
General Note:
"Report no. L-490."
General Note:
"Originally issued August 1944 as Advance Confidential Report L4H03."
General Note:
"Report date August 1944."
General Note:
"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 previously held under a security status but are now unclassified. Some of these reports were not technically edited. All have been reproduced without change in order to expedite general distribution."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003622107
oclc - 71420982
System ID:
AA00009429:00001

Full Text

0 ACR No. L1H03

.. :


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS






WARTIME REPORT
.. UU.I:UUCUUEU".C K.qs....c


S,ORIGINALLY ISSUED
August 1944 as
Advance Confidential Report LIH03

THE THEORY OF PROPELLERS
I I Dr ERMUVATI' OF THE CIRCULATION FOhCTICW AND THE
MASS COEFFICIHET FOR DUAL-ROTATIIG PROPELLERS
4...:: By Theodore Theodorsen

Langley Memorial Aeronautical Leboratory
Langley Field, Va.






.. .. .





D CMNS..A WASHINGTON

I; .:..1ACA WART1ME 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-
| :;,,.... y.osly held under a security status but are now unclassified. Some of these reports were not tech-
,.W" f... lly edited. All have been reproduced wllhout change in order to expedite general distribution.
"--.." "0 ---...--9


: .: :. :: .: .. : ..: OCUHENS DEPARTUFK.-
L "n *:i.' ..
( : :'" i l r" l %, 'i ,..., '" J.. :... ',:- ': .. :' : ':t :. ,:, :: "





4















.4




-7/c : ': >-

'ACA ACP I'. .U1.

ITATIONAL ADVISORY COMMITTEE FOR .-AiPOi A'.TI.!


ADVANCE COITIDE!'TT!AL FFPOnT


T'Tr TTOPFY OF FROPELLER2
T DTTER:I' ITATITON ') TTF CTRCULATION I ]'TTInI AKiD TI.

"ASC COFFFICIENT FOR DiAL,-ROTATTI'H F POF'T.rERS

3-- Theodore Theodorsenl


S ;'.'1ARY


"'lu s of t.,e ci rccul ati on functi.'n ha'.'- be n ,c'ta:irEd
for' d.6al-rrt3tin' rIrop'- lers. ,meri rial values ,re A iv en
fir Vc'u -, e'..ht-, and twelx.ve-bl.drie a-I .-r -t.tlti p1 r -
,.. llr. icr v n a c rati oc, frrr, 2 to C'j'.t' s. in
;--d,.'it ri the L.r.-l..it.cn functirn r as en rc te-r."ine
f',r .-.; -le-ynt.. ,'.nc -. r.ell .s for t. '- ;.-- er valuEs .:'
-he v a.Jv nc 0 l Tn rrs'; coe.l' i nt Tiotn:.r qc ti ty
,! si niL f'icanr.c.. '. ro.,ellkr the 'r, h' u r 1tr u .
This "- s ff r nt vhi.h is 'snt:al.. t'r r L .
:.f t '-' -'1tt1. i co-;ific L, expr' rs es the L ect ."
are& th ,: c l, .n :. t ,'-f'ilm c t.f ,' n b L' tt i' -r
,:6 [:. r. 'i n- t:2-3 ,f t .- pro/. 1]-r-,1 ', .ir-is. .a s 1
th ': s c i. f I ;. nt, I'i.. h, av -: n .*:.etc" n-ri :_. tl r
ab, -.. : r' -s i.r' .,itv L-r.-. : s 3 :, ir, ..rat i n .[ t
; re 1.. t :., i unr ticr, .r- t. vin or r J',..r--, t-.
tnr' t- "'.Ive--'i '-:- 5u'1- c.,t Alr rn. Tii-e r*. a
cet'.fi.cient ..a- elso been determined tor s.'.eral cases
of t- ,:-.:le-r tr, t : o.. .. ro:.e. o e rr, p r, tl, ,r .i ,e .i r e f

.'.n the -n.-;-n ranLe .f low :cdvancr ratioC o d 9n] arMty t.
*-xteri ti: re.:nlts t: inclue i range of hr1. r.'J-nze
r..t .ns. The effect of stationary vcoun-tervunes m., th:'.
' ':- coe fi ient has ls' oecn cdeter- ired r. 've: r-,i
0 L t* ti i, -ntir- rt .


IFT ~D' TT '?
-yrcu..ti n F.entnj n 'x)


In r 12'-i GoldsteIn (rtferenrce 1) sue :-rrdesd In sol'sing
the trouble of th? id Scol lift n .i trib'.iti -..n .-f .iln-le-








2 COrTFIDiTIAL NACA A,33 Po. iJ TIO3


ro -' r:o:-ellers. Toldstein's wovnrk ii r.tstrl -ted to
ihe case of a !iMht l.ja,:-ing and also, in effect, tj &
small 'r id\ncxrie mat.'. Puperizal vJl -s YE.vcn jy 3-ld.te?".
f.r the 0to. lia-'.rcul at ton distribution are reprodu,.ced.
in t.-ble I as. fi -,r. 1. I .: F I.d:d tiona] valucs -lcu-
lat, by !i'rs-or f r-ferencc 2.1 or' 11 glr fdvunce rati cs
are ven in table T- anr. have been u: r'mi 'd on tr-
-o'1dcsteain r-..-ult.3 In figure 1. !-Jn ri3c l r:sIlts by
Look ancd .'.ntr:en (rcf-'":crce 3' fo. the foir-bi..d- :- ro-
p e l l e r .r .- r ; ,r ,.5 c : d L n t sa : l1 T .-n, d: f t y :. r. .. : h u
'ranreter X used i.r tn bli .3 a7 i'l I ir th tngnrit. ofJ
th, tip v:r tex n, -1 n th ulti'-l't, .".'.-


L + ':"
IT n.

where w is th- r'. arv r ,'--1 c' c r.t v.- l.o '. ry of the
helic 1 vorter :.'.rf':'; .t 'n i.Ln t-., '. l Zi t -.-f the
s -'c ls -se- thr .: "no' t-.e- p- er s ;:vc n a oerndix A. )
T ..e i. i bee, se'd 1 -r comL-ris.' vri '. .' re iults Cd An-
ta- .i in t u-: :r.'.zert 3." er.

It should be eph..:-izerd tir-.t n d.Ltjnntioron has been
-**',: bet -- : dlm n- i'ns ea-nd r?cn- ticns of the sli.r-'tr-ea
at t'' .-r" c -i er :-ir t s-tcrse te the ult t.? ,w1e a cis-
t t-. : 1';, 1t r'?es r t fc u'.-. inr the tr..-e ~' t of r htly
loaded pi '.c'.l-r ..,- r- sr .r.t ,::f r i, conternek .c> lu-
sively th -,nd-itions In the ulti.-at ,,n.ke: ir. fact, .t
can be shown that thrust, torque, anrl. efficie:n'y are all
un:iq, ly :-,iv:-. as funat'rons .l thr- ultX .-_-.te v.'-"e only,
no n:,-'. ~,- of the cro-eller being, nacecsary except for
,'o-nfes of qotual de3Si,n. It should br- pointed -ut tnat
both t'-r i-'eter ..d the advnre rr. ls of the ultimate
helix are li.ffer-:it fr .i, trn- v-ltue et the pr.rope LI-'r, the
diameter be'.. --s r F- nr.d tu.e OBv',r. ratio larger. It
can ...- .howi that thE-- d tr"' .uti -ln fun-. 'ion depi.nds on
the id' .rc c .,' cri'. 'I h' ..Cc:- d irt i.bution function
is t -eef r- Ii-ant i cu for j. -'"ht --. he-~iv in:s [.ro-
5.ded bo':' rsfer to i -: t L.:. h-..:.x an -.71 i .: the U]tL aite


In f1 .--.:- 1 and :: th- iv :n ': v '. .' s a -.hatr::-ot:.r-
istic .- :t a r roelath -, or tyi,.?i ) alo
the bl'-*. as fol.o:-s:


C")!TTD~ITT 4L








NAOA AZF No, LTHO5 COTIFIDENTIAL 5


=x) E
'Tn('V + v:)w

',here F is the potential difference across the helix
-.' I.ae rt a radius x, F i. the number of blade",
and a' is the angular velocity of the procei]er. 'he
quanti ty

+ w
w i
p -


is the -otential drop for a velcity-v w through a
len t'-.

FT V + y.

cT.


v;[ ".ch t the a :'i ]. distc:nce between t,'r,,' s .uc ?eseiv, vort'y
shec ts Each shl':t has urn.: c-rreponding to a
time of I secMind 'and theze ar.c g sepc-rAte sheets cor-
r s. cr.- r.; to p blade P'< ; quantity '(x') is tlu.x
the tr, id.-ensinal exp;rcssi o-n for the potential dro':
a ross tle surface if .-iiconl i-nui:y as a fraction ,f th.
iv. ]larola dr,[: in the d-Lrecti :n of the helix ayiso

It should be noted that thv cocff ic'i nt (x,
r.' ". -r fro" +h 1 -oldst in co .f ffi ti nt




"n "hich the velolty VV has bee.?n di sr:garded in co,-
".r s'on with the advance velocity V. The .7oe ffTi nt.
ar. 1 i9.ent Icil if referred to th.? sar-i h,.lix angle cf tn-
;. t -. ,";ate '81.,, .


COTFI DE17TI AL








,AWACA ACR N-. 2r-.HO


Mia.ss Coefficient K

A signifIcant coefficient, which v.411 be termed the
mass coefficient K and which may be shown to be one of
t'1', basic carGmeters in the propeller theory, is now
intro--uced. It is given here merely by definition

1
K = 2 K(x) x dx



where x is the radius and the integral is taken from
x = 0 to x = 1. By inspectic.n it is noted that K is
really the mean value of the coefficient K(x) over the
disk area. Tf K(x) 1, then K = 1, which is the
limiting value of K.

A rhybvical interpretation of K is Interesting. It
is possible to show that K represents the effective
cross section of the column of the 'medium "pushed" by the
oro'-eller divided by the projected proyeller-wa1'e area.
Tn other ',ords, the propeller imparts the full interfer-
ence velo7!ty "" to a 0olur n of air of cross section
K per 'unit area of the ultimate procpller vsk-s. The
diameter of such a column is therefore VT for a pro-
peller wa1-e of unit d1iareter. Although mathematical
ref- n-ments will not bE considered in the present -aper,
this physical interoretat-',n should suffice to indicate
the natur- of thr cofficieent and the designation adopted.
It will be shown herein that the coefficient K is
readily obtained by direct measurements, to be described
later.

Figure 3 shows curves for various values of the mass
coefTicient K for the cases for which K(x) is known -
that is, for th[- single-rotating two- and four-blade
:r' elders from tablEs T and TI as well as for the
linritina case of an infinite nur'o.r of blades. This
latter case is readily obtain d by integration. With

2

x2 + x2

1 2
K = 2 2 2 x dx
f 27


CO~TFDE':TT AL


CO, 0FIDENTIAL







"ACA ACR To. L4CHO5 CONFIDENTIAL


is obtained or, after intesgrtion,


K 1 \2 log +


The curve for this equation is shncwn as the ur'per limit
line in figure 5. Values for three- and six-thade pro-
;i:i] l'"-3, which .,were calculated iron' datr- ,' Lock and
:r:eta.nn (frcf.renc: 5), are also sho.n in figure 5. Th:
curves in figure 5 are used later for comparison with
data obtsinrd In the present investi.gaticin.


FLECTRTCAL METHOD AND E 'TIPFFT TOR

YEASUPTM') Klx) AND K

Description

Tt is vell krown that the flow of electric currr.nt.
i.n i field of uniform- resistance is mathemnaticrtlly
cz..onti cal -ith the finv of rfec.t fluid.. The velocity
SctL-i --1 ay e per.'rctly reproducEd by an e-lectrical
.ctentisl, provided the boundary condi tions ar.1 identic'T.l,

Ror the pr-sent problem' a dire-t measurement of tht
n roc'.ntrmic i fi- c behind a pro,-' 11 r prese nts tinur'ount-
able -.ifficultier; in contradistinctionr, the electri cl1
retcho of measurement is convenient and accurate and, in
addition, perm.itr the determination of lozal as '.l:I1 as
irteg:-rted effects. The arrangement may, in fact, b
con.ider'ed a special calculating machine for solving the
cifferent-.al equation for given boundary conditions
rathLer than a means for obtaining experimental solutir-ns.

Sinze the ideal flow (far behind the rroneller) :s
identical ,-Lth the flov around a rigid h?.. 'novin7 nt
a. con-tann velori!ty in tne direction cf it' ax.i, thc
c.nrrcponir.g ele-tric field is obtained very simply by
insrt.inr an insulating helix surface-: in sI conductifnA
li "i.d and Ipnlyin- a uniform field in tnr direction .f
4he --liix axis. Th-e vessel confining the liquid is a
lc-.T-. nylirn.rical shell, aiso of insulating r-aterial. The
\essel is closed Pt both end-Is by copper end. plates that
7re used as electrodes to apply the potential. The test-
sze:c'.ren helix is pieced coaxially with t;hc shell. The
confining shell is considerably larger in daa.neter thrn
the fest helix.


CONFT DEN TRIAL







NACA ACR No. LTi-HO0


Ei.,i-.re L is a photograph of the test setup for the
d'r=,t determination of the mass coefficient 1. The
cyl i. nd=r on the right institutess a dummy compensating
rESistance. The electrolyte used in these experiments
consistedd of tap water from the local water-supply
sy' tem. The source of current was a lO00-cycle
1.t,-rn-ting-current generator producing a rather ure
wave forr at an available voltage of about 100 volts,
,:- ch vw-.s applied to the electrodes. An exploring device
c.on1ist-ng of a fine glass-insulated platinum wire with
*rL exposed tip was used to determine the voltage at any
M.. int on the helix surface. 'This pickup device formed
r.rt of a potentiomter circuit used in a .'heatstone
c idC.~e r.rran-7gement with a sensitive telephone as a zero
i: c ator. VThen voltage readings were taken, no current
-s-ed through the telephone end the exploration "wire.
Tiis t Tpe of measurement is inherently accurate; the
':-ror "n the electrical measure-,entL is estimated as
not mo'e than one [:art 1n 10,00CO.

1 rure 5, -"1 a rhcto:r-ih, sho'rs the equipment
us- d ini th= r'-nufacturc of the hlix. surfaces. The
vertical insulstzd cylInder is an electrically heated
il tan'-. To the topa cPnter of this tsr" is attached a
si'--le nie o.r ,u'l in device with a spiral s-lit through
which the heated plastic sheet material is pulled at a
u!'f'?'i r -te. 4 fPn is used to s' ,-pl, ccolng air at a
unifo-- rptc. "'I th certain r:cP.,autions an almost
;erf t helix. is rroduzed. Two models of single helix
s.rf-a: t.hu obt-ined are shown at the left and center
I t, fure 6., A preliminary tyce of butllt-up model of
la:n-n--ter. construction, which was abandoned as too
i:-cc'-irte and exp: nsive to build, is shown on the
r.t; ht in figure 6

Tn figure 7ta) are shown examples of dual helix.
srf.)=- used for the main investigation. A four-blade
ridul-jo.'re mndel is shown cn the left and a six-blade
r)! .l ake mo'el i. shown in the center. On the right is
a f ur-blsde single-rotation hell.x surface with four-
.]_-- "guide vanerl." In figure 7(b) are other examples
of' 'I -order multiple cual-rotaticn wal'e models. Some
'"it- nl examples of single-rntrtion wake models with
- ., ".'jnes o,re shown in figure The method of building
the -ual helix models is indicated in figure 9. Unit
strf c-,:- wer,-'e .ut from ri, ht- and left-handed hela x
surfaces and glued together to form a multiple dual helix.
.ortunatel., these c:m.:iex built-up dual Todels were needed
u'nly for determining the mass coefficient K and did not
hale tc be too accurate in ,rta' 1.


CONFIDENTIAL


CONF IDENTICAL







PTAC& ACP 17c. LLH053


For the duel-rotat ng-propeller field a significant
property is to be ncted: The field repeats itself not
only along the axis but also circumferentially. A "un` t
celI" consisting of the helix surface between two succe'1 -
sive lines of intersection is representative of the enti -
h-'lix. It nay be seen that the boundary condition is
t r n care of by inserting two insulating planes contoi,,.n
the ex!s and the two intersecting lines, resfect:.vely,
a;cd by usingi conducting end r lanes perpendiz1ular to the
a:-.:i which c -ont.in the same twvo intersecting li-:e L
vzcswel rsav therefore be given the form of an open '-.h:.p.
tr?1 th the electrodes at each end. The r-r -Ese-ntati ve
1b ]ix r-iy be obtained sim''ply by store thin, a r.uo,' r ,"a-
br.ne from nne cornr of the tray to the o'posit-, Corn.r
at the other end. Thc, mrm brane is equipped p ,ith stiff
racial sfokLs --*nd is sE-cure]y clamrp d In place. It aito-
matically as-sumrr'.s a spiral shaoe, the effects of ,r 'vDit'
being of secondary order. The entire trry is arra.n-rd.
on a machine lath-: 'rith the belix axis ~loni the center
line and the'- exploring needle is attach':d to the carriage.
Thi s arrangement sffords. convenient readtding of the voltfng-
Et any roint on the s3 ral surface. Tn order to incr.-:.-
ac curacy, the truys vere rude of consicp rable size, '. to
1C. feet lon.17. By chan.rin.- the length ar.nd th-: anrle of
the tr'sy, :ll values nf \ and th. -ffct orf the- n,..rb-r
of blades could b investi at d.

Tn figures 10 and 11 :-re shovn e;:perirmental setu.s
for measuring the potential distribution .'(Y). The con-
necti.ons leading to the exploring needle may be seen in
figure 11.

Figures 12, 15, and 11i show the gei-.eral arrangement
for determination of the potential distribution on J.ual
wake models. Figure 12 shows a unit cell for very low
advance oraci. "ote the V-shae test tank and the adjust-
able end plate to change the advance ratio. 'ote, ailro,
t-e rubber membrane stretched bet':.een opposite cornerr.
T-'i _)ce 15 sho's the arranr 7.-r'ent for support in;: the ex; .lorir.,
:,-dle. TIn f -igure 1i. is finally shovtwn the co.-m lete *:-> p'ri -
renta' setup for dual helix surfaces of very hirrh pitrn,


Tall, Fnd, and Thickness Correcticns

The similarity between the el*-.trical test method
and the conventio-nal ,innd-tunn;l method may be extended
also to Include certain corrections. Obviously other. is


CONFIDENTTIAL


Cr, FID--I'TIAL








S COITFTDF.T'TAL NACA AC" To. rLYhO


. correction that corresponds to the customary vall car- r
r-. ct'on. This correction Is readily asc-rtain.-d b-; us
c.i .s:.e]s atof i5 ff -rent diameters, a procedure that can-
i- e ,asti&y -itilized in wind-tunnel practice. It
,hrn jl: be noted -urther that the wall corr..ctions ar.1
obtainedd with great accuracy since eazh reading by the
;el ctrcal method is more precise than its aerodynamic
crunterrert. o.' uing tube diameters Tabout thre.. times
thF5.d-. seter of th- t-et spiral the error in the results
1
"'as r'u'iUced to 0 1--s t'-ian percent.

A correction not appearing in aerodynamic practice
is th end correct. on. This correction occurs only with
ir;l-,-rotatin- r.ir.o elle rs and i r therefore of nincr
i:.r'ortince in the Fresent investigation. Dual-rotating
proel]ers possess planes of constant potenti-jl prer.en-
ilc*.ur to their fixes, and the endr therefore cause no
dLfficultieS. B;y -uttinE the dual nelix at a lane of
c,-nstaLnt -otertinl1 and by inserting a conductingg end
pl te :ir, the cylr-.er the boundary condition is satisfied.
b'.:r s, n.le helMy surfaces, tests on two l1neths of the
sF-re helix rrust be used. and the difference observed. This
[.rocedurs "-as. u.-td to measure the mass coeffi >ient K.
'To c-messur, the prntentisl distribution (x) a long helix
is requir-..], the ,resqsiur'ements to be made near the middle.

.noth -r source of error exists for which the cor-
rcction has be-n referred to as the thickness correction.
Thi rr'r r-esults from the fact that the material of the
P: lY 1 x t rust havy a finite thick! b- 'te.rm.t.n -d by usin;7 sheets of tvo or more thickn ssees.
It ..- rn'adily s.E-en from, thecoretic1l considerations that
' r roimat-..ly one-half the thiclmnecss of th. shee t must
br ad&.:d to the di.ameter in ord.sr to obtain on equivalent
infinitely thin che :t.

It should be Tentioned finally thrt there is an
error resulting fr.', I.ns Jiracies in the r'co'el vortex
sh =?ts. Thr err, r :in '-(x) c-an be mi nimi::erc by usin'S
mean values from a large number of rc-arln-s over a con-
' iiKrabhle portion of tb helix. ?ortun,-. t"lt, y, t:iere is
n.., e.."fat on the 3ass ,oetficie .-nt K sinre this co-f-
ficient i_ a rean-value par-ameter.


C 0 FIDF ITI AL








1 ,0/. '. CP Non., L HO5 T


Proof That the M'4ass Coeff ici.ent K T3 the

Floc;ring Effect of the (rnfi nitely Lon-')

-eljx Surface

The n:as- ce*ffici-.nt K is rbta_-.nrld e.,-erim T t : ll.-
by .an'urin-: the change in resistanc'E caused by th. n i:,-
.urfn.:e when inserted in the cyl-i .-dri.i! container. O'
insert:'n_ the !finfn rItel longE h) pix in the ontai -.er the
-irrent betw'e.r the end plates Tn is decreased b-, a
def inte e.nnunt 6T, and it can he oprnved that

K LI
2 Io


S= T C
V-


here t.h rr-ected cross FE3to nr,. o the helix trn
;3 i9 the :r,.s 3rctiftn, of the cyli n.drI cal .container.

SGrOrren's theorec-


.'*:V) ,10 = (U2J4' .'2') dT


V = V = (.


it followit that


('V"' "T )-' do = C


Let be Ir:e distance z alone L-' h.elix azi.
measurEd froT a r--ferenc.! plane rerLOpucicIlLr Lri the


CO3'Tr'TDF"TTA.L


r'ITT D7r''TITAL








COr'F7TDENrTI. ',L


:,.CA ACR "o. icjHU3


I''.s; T"' is therefore a unit vector n the direction z.
Ftt T be? the potential resulting from the arplied volt-
.e :-r. the local gradient VTI may be written


L io

-. : ris the constant voltage: difference betw.-en the
0 1
S:at,---, 'whicn are raced at sn axial distance L
.rt, fnd i is the local current and Io the current
at 'r.finity. If the sArface integral: for the entire
hnclos-:' h:-.! surface A and the end surfa.ces S,
-resa actively, are tak'n, the follow'idnp reElation is cotained:1


UJ A = U- z 5 dL


-- U dAz =
U A


/ dS
--/


i'h-re the inter als art. to be ta:cn ov-rr
the p helix surfaces End over Joth .-nd
'I Clever,


both sides of
piates.


It U dAZ


ma' be written in the fori



0H ( 2 )

wh;rr, the intrral is t aen over one turn of one helix
for one %`id.u ,nly, as I1" -2 i.s thm difcr'r-ncre


CONFIDENTT AL








!:'/'. ."C'? vo. LLHEO F5


r,-tentmi-. between the t,'o sides of the sheet.
.te drop c:er street is


The volt-


Un I-I

I. r


B" d f ri. n t r.ion


TL
7.F'


.T .-s


Sf dA., =
A


f }(x) dF = 21 f ( x) : --
? 0


f/- d -1 Q
/ -o r_ d I I



an:' tl er-efore



S I,

where T. is the total *..rront bet.,een the e.id plates
vwit s unifor r ru lIrt in the fiel:..
L

EXY. F 'nENTAL. D .TA

'ass CoeffI.cient K


"uLerical valur.s for the mass s.c.effic ient K,
trained crn dual-ro tstion v.sae models, are sIovwr in.
'OF1I DE,.TTI AL


COnFIDF.TIAL











-' e 15. This chart is probably adequate for all "
bicl :'i.rrposes, as a large range of advance ratio
h.)s been covered. The designation used for the propellers
cnmorises three '!gits: The first digit refers to the
nur.ber of rirht-handed blades, the middle digit to the
nu-,b. r of guide vanes, and the last di'it to the
nuLmber of lePt-h'nded blades for Instance, 5-0-5 repre-
sents a dual-rotaiting propeller with three right-handed
and three left-handed blades. The highest number of
blades tested was for a '-0-6 or twelve-blade propeller.

As a matter c-f interest, it is very fortunate that
th.= me.thod 9s:d e~'ui.pmer.t could be tric-d out in all its
rn.rifications on the known case of the Goldstein curve
lor a two-blade prop,-eller. The Goldstein curvs, is the
curva in figjre 15 irarked "Theoretical." The test
-oints, vhich hnve. been corrected for wall, thickness,
an.d ,nd eff-rcts, are shown. Exc,.pt in a very few cases,
the t:st points lie on the theoretical curve for the two-
blade pror..llr1.-rs. The sorx what lesser consistency in
the c3-ses of cdual-rotatin trrorellers is not due to in-
here.nt tcst inaccuracy but rather to a necessary limita-
ti' :-i on time and equipment for rmutking the models, per-
forming th- tests, and obtaining the corrections. The
thikr:-.ccss corr'-tion for the hirh adva:'ce ratios is con-
.iderabl?. F't-. that threr thicknesses have been used
for m".ny of the test points. A glance at one of the
ccoit :- models shown in f gurze 7(b) w-ill suffice to
ri.1)ic'te the labor involved In producing the mod- s.
:ch' test point in figure 15. represents a different com-
l-t, model: some fifty Todels thus are rcprescnted by
th. "csults shov.n. This number was necessary in order
to include all propellers and all advance ratios of
intere-st at present and in the future,

Results for single-rotation vake models with guide
vanes are shown in fi-ture lb for two-, three-, and four-
blad.,e [ropellers, respectively. Such guide vanes are
supposed to represent stationary vanes arranged immediately
in fruit of or behind the propeller to straighten the flow.
It should b- noted that the cases shown correspond to those
of an ideal thrust distribution both on the propeller and
.n t.- gud..e vanes. The uppermost 3urve in each part of
fit.r-. 1b is reproduced for piur;oses of comr-arison with
the ccrrespondi:.-, dual case.


CONDIDETT AL


F;.CA ACP Mo. rJ,.'T0)


COPIDE.TI!AL







I;LC' ACR Io. LLHO C


Distribution Function K(x,6)

Thei measured potential distributions on dual v,':
models nre shown -n figure 1.7. These test vwer- mrad? on
lar-:-3ra1le unit cells of the type describ-d earlier,
"iir 17 ror.rtains results on 2-0-2, L-C- .;nd .-0-6 cu 1-
rntatlnr pvno.ellIP-s, in each case for thrr-e advance r-Aiti.r.
The p.otentiEl irop is given in niornimensional form and
is ilotted E-.ailn-st the rasdius. ETch curve Irer=-sents a
ria.al line cn the hlIrx. The ar&ular O-S'tion of the
ra'ial r line is L1iven as a frac ti cn of the ;ell een.iangle
.-.s.tr0 frTor, thi middle or symmetry inr. of the cell.

Firu -F"e 12 show w K(xy) as the potential d.. ffer.--.re
vt the zero eng,.l! or midway between tvo suicessz.1e inter-
scctin-fr lines. The Cesults are rrran.'ed in order shov'ing
t.he four-, : ieht-, an_-' twelve-blede dual-rtatitn_ pro-
l.. llers at three advance rstio:.

The fu.nctin K(8) is sho7n In fig-unr 19 olotted
a7air,st the an.'1c: m.a-sured from th sZor.e zr ro r1fer-_noe
1lr,., R.:sults afr shown for the s,, thr-ec p'roPellcrs
Dt the sarme thr', a.t'vorcJ trti.s. :Clirvc are shown for-
thre-, valu-s of the radial dIstance -,= -, Lnnd
L. 2


DISCUS eTO '


The concect of a mass coefficient K defined as


K = 2j 2 x) x dx


has h,:- n introduced, where K.xi is a noni rtnmensional
dis'ri'.biton funt-.itn anr! x is the nronrlmensjonral
rne Lus. 1"u1ror-ical 'al'e. .f K f')r the known -aes of
sin'- le-rttin, p.roellers are zhown in figure 5. +
is noted that the -ass -orffi. Ient dr'r',ps r.v;:i..tly 'rtth the
advance ratio. For 2, the value of K is less
nD
!-;n C .5 even for an infinite number of blades and is
le'"s than ". for the. two-blad.e sin,-;le-rotating ro.eller.


CON TDFNTI AL


COF 'IDENTI .L








MACA ACR No. L1HO03


Ftr dual-rotating propellers the mass coefficient
defined as

1T
K = K(x,9 )x dx dB


is considerably larger. A 2-0-2 propeller has, in fact,
V + w
a larger mass coefficient at 2 than the single-
nD
rotating propeller with an infinite number of blades.
The 6-0-6 prcneller at the same advance ratio has a mass
coefficient K = 0.79 or near unity. (See fig. 15.)

The effect of guide vanes is of considerable
practical interest. These vanes are stationary and are
arranged either immediately in front of or behind the
-ropeller. The question is whether such a stationary
system in sore cases may be more acceptable than a dual
arrangement of counterrotating propellers. As an example,
consider a three-blade single-rotating propeller at an
advance ratio of 5. (See fig. 16(b).) The mass coeffi-
cient K of the propeller alone is seen to be 0.142;
the 5-2-0 propeller with two guide vanes shows a value
of K of 0.253, and the 5-4-0 propeller with four guide
vanes shows a value of K of 0.2&6. For comparison,
the three-blade dual-rotating propeller shov.s a value
of K of 0.486 at the same advance ratio. If a dual-
rctating propeller is not used, the double guide vane is
undoubtedly desirable in some cases. Actually, the
induced loss is reduced to about half as compared with
the loss in the case without vanes. The difference in
the effect of two or four vanes is relatively small.

Langley Memorial Aeronautical Laboratory
National Advisory Committee for Aeronautics
Langley Field, Va.


CONFIDENTIAL


CONFIDENTIAL







::ACA ACR No. LTHO5


APPENDIX A

SYMBOLS

V advance velocity of propeller
w rearvard displacement velocity of helical vortex
surface (st infinity)
n rotational speed of propeller, revolutions per
second

w angular velocity of propeller (2Tni

D diameter cf propeller
V + w
- + advance ratio of wake helix
nD
1 V + v
V nE:

H pitch of wake helix (V -n W

r circulation at radius r

K(x) circulation function for single rotation G(-,K+'J)
(2n(V + .-J)v:)
T.V(x,e) circulation function for dual rotation

n number of blades of propeller cr separate hilix
surfaces


mass coefficient

1 -2T
or K
Li0 JO


2 f K(x)x dx
(x,9)x dx d 9)


ratio of radius of cle:ii.it to tip radius of
vortex sheet (r/E)

radius of element of voitex sheet

tip radius of vortex sheet (-D)
\2 /


CONFIDENTIAL


CCNFIDENTIr,,








NACA ACR No. LH.H05


e angular cocrdinrite on vortex sheet

F projected area of helix (TR2)

3 area of end plates of cylindrical test tank

Io current through test tank with helix removed

AI reduction in current due to presence of helix




EFFERENCES


1. Goldstein, Sydney: On the Vortex Theor7 of Screw
Propellers. Froc. Roy. Soc. (Londcn), ser. A,
vol. 125, no. 792, April 6, 1929, rp. 440-465.

2. Kramer, K. N.: The Induced Efficiency of Optimum
Propellers Having a Finite Nmriber of Blades.
T1.-CA TI No. ou4, 1959.

5. Locki, 0. F. H., and 'eatman, D.: Tables for Use
in an Improved Method of Airscrew Strip Theory
Calculation. R. & Pi. No. 167k, British A.R.C.,
135.


CON 'ID7NTIAL


CONFIDENTIAL






NACA ACR No. L4HO3


TABLE I


OPTIMUM CIRCULATION DTSTRIBUTTION FOR THE
TWO-BLADE PROPELLER

Calculated by Goldstein (reference 1, p. 450)


x

0.020
.OLO
,00
.o8o
.080
.100
.120

.180
.200
.250
.280
.500
.550
. 400

.500
.600
.650
.680O
.700
.90oo
.900


1=
10

0.126
.2145

.526
.650
.698
.758
.772
.856
.878
.908
.Q27
.91o0
-----
.950
.955


. 890
.7368


x

0.029
086
Mn
.111
.1543
.171
200
.229
257
286
.357
.129
500
.571
.605

.57
.929
.971


1
-T

0.126
.245

.523
,466
S6941
-7)2
.76
926




.882
.858
.717
.554
576


x


.o0o
120
.16o
.200

.280
60
.500
630
700
.Boo
.900
.960


0.124


.511
O.i71




.519




.31
.770
.775

.671
.519
.551


x

0.050
.100
.150
.200
.250
.500
.550
.,400
.450
, 500
.525
.750
.875
.950


1

0.120
.252
.331
.18
.1489
.592
.628
6514
:670
.676
621
.486
334


0.067
.15533
.200
.267
4.00

.6oo
S67
.355
.955


1
1 = 1.

0.111
.2153
.303



.555
.537
* 225
.u27
.505


Calculated by Kramer (reference 2, p. 25)


x= =x { _- i = 1.. \--2.5
0. __________________ _________________ ___________________ ____________________


I.25232
,18
.548
629
.655
671
.679
S65L
.6253
580
c28

.529
-----


o.1641
.12
.1486
.510
.528
.540
S.517
U953
.h57
.413
.351
.ill
.260
.190


0. 09 L q
,1758
.2L6
297



.525
.505
.276
.235
.173


0.02853
.0552
0795
.0999
.1082
.1155
.1259
. 125
12135
115 6
1061
0919
.817
.0687
.0197


0. OOhoL
.01Q14
.oil5
.01806
.'1976
.0212L
.025L2
.02L L2
.02596
.02310
.02147
.0187
o 168
:10o
.o10i


NATIONAL ADVISORY
COMMITTEE FOR AERONAUTTIS


x

0.100
200
.400


.00
.900


L 2

0.092
.175
.2h3
.295
5329
.551
.2q5
.220


0.1


.45

.7
:P
.85
,9
.925
.95
.975


I


CONFIDENTIAL


CONFIDENTIAL







Y-/PCA CP T'o. LT 1G5


TABLE If

CFT.'IT"UM T CULAIjTON DTSTRI TUTTJI !'O F T.E

FOUTF-BLADE PROFE.LLEfr


', C'lullat?d by Lock and Yeatman referencese a.,


NATIONAL !DVISO1RY
O1r.,TTEl FO ,. AEROPTAUTICS


C'FTT DE1 TT..L


COTPI DEI'T .'L





NACA ACR No. L4H03


CONFIDENTIAL "-


1-1


r



I T,












S2 .5 .6 7 .8 .9 1.0








I CONFI[EtT AL
FI-gure 1,- The function Efii for mevoral values of N far toW-ble.e propelLer!.
-T nr^ t p ^ i ~ '- r n ^ ""'" T ^





~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~I I ;--T -- f fi"'".14'.''^''^-"""r '- r:3-~
7 -.- ^ ^ ^ i r a y ^ ; i 1 S g -: ? 'f ^ F -
---^-fe^ ---?&^?7^^^^r- -T I ^^ --- -3r l
isls~l~g~sl^^^SSL^^^::^^^L A:

;^E~l6^^f^::i;g|^E^;^7 rr^r
0 .5 .6 .8~S^ -- -^ -- -^^^ ^
-|- -^^g ^--^ -^^ ---- --- ^--5 ^,x
-F1 9 P T ef u n c t i o n K- -C N-F 1-- -F-r -( ^ ^ T A L
f or e ve ml al ue of X fr t w o -bl-a--- 7 rtP-- ^ r g .


Fig. 1







NACA ACR No. L4H03


Fig. 2


. .." _.













1415



-- .
E---- -- --


4 1! 4 ^-.... ,


I
a
o

.4

0
0.
,.















6.
0
a

0^
I..
1I1
0



0

b.

-4
U
a.
0






a
i.

3
aD


Si











.... ... ... ... ... ... ...
+ 44 . ......l S I






p-'np *^I+T TT a^^.1 ^ T^il t! Hi1 4
T--T-^ ^ Rl~ ~g tt? 4444; :1^ ;;j p

;:2l||||l|||ll'llilp
B||l|- JUiLlll. lll








-r -r r -. j 7+1 ^:!T, ^L- j,^. .. {- .JL.
.. .. .......

14-2 ::T t ^ ^ ^ M^ ,^ --.4 ITT4 -

^^lll~l^-lg~l~:a
4-4 i*g^Sgi--^-
SS~~gglm 41^^?
71.Mr p tt t^ c ^ o 1? -'- ...... .
^- ^r;-4 ssa ^^ --,-- -*i:--
^l~itit-t^r 4 4^
fel~i ils~z~s'^^:-4-"

4-^^rE^^^Jf^T.^3--.^-, -L4 A-.--
"-^ ^ '^tg 'Tr-- ^". ... .... ..-^


rp x (X) ? = m


NACA ACR No. L4HO3


Fig. 3


v
ca
0


A
0'

0
0r4





+
fC

U
> :




oP
0
o




a
ole




CM




,4D
LT 0
a


--4

PC%
l-4


i
a

o




rz4


0







NACA ACR No. L4H03 Fig. 4








x







O




o

m









o
*4







a)
u
















,1 1
'44
(D
0




21


44
0




















4)


$4-


4.3
102
.. .)



)
L




NACA ACR No. L4H03


44


- .1


K


Figure 5.- Equipment for constructing celluloid helix surfaces.


Fig. 5






NACA ACR No. L4HO3


/


A6


I-.
/

A





LI..


f






1~'


Figure 6.-


Celluloid single nelix surfaces. (On right,
preliminary laminated construction.)


4VACA ` f A I to


Fig. 6


no,






NACA ACR No. L4H03


fa) Left and center for four and six blades, dual
rotation. Right for four Llades, single
rotation, with four-blade guide vanes.


Figure 7.- Dual helix surfaces.


Fig. 7a





NACA ACR No. L4H03



.../












.:
: <:


-4.

9_^j^^


Fig. 7b


oV
my

-"s
4,


f

V
4-
-N
4-, -


(b) High-order multiple blades, dual
rotation.

Figure 7.- Concluded.





NACA ACR No. L4H03


r
/.\
*,


N

K





K


t -


1~
/


Fig. 8


K


K<


I


Figure 8.- Two--clade single-rotation helix surfaces wiLn
guide vanes.


7 F i,
4
F







NACA ACR No. L4HO3 Fig. 9











Z.4


0;
rd




U)
.-4







C.,


Er





o










(D
." J" -'-.4

-.

--- ':2-
0.-
,... :
"" .,"
La











~2





('a)






- NACA ACR No. L4H03


IrS

&41.


L


Figure 10.- Setup for measuring tne potential distribution K(x)
for single-rotation wake models.


Fi6. 10







NACA ACR No. L4HO3 Fig. 11









o u


0 0
-4 0


.4'
M










C d-d
3O


0 0

L4


Cr3 0
0 "d
*"'D








: .
L, r,
ip o:






.C .
U) o








o,=I






NACA ACR No. L4HO3


Figure 12.- Unit cell for very low advance ratio.


Fig. 12






NACA ACR No. L4H03


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


Fig. 13







NAC4 ACR No. L4HO3 Fig. 14














'*S
r0


J~.e
m t 2









4 T,
.a.


.c
tao









xL?
\ '-I





On
rx)
,k tr. .-4
,c



.-4












Il F ickei. ,I,,t ia -



Jl 1. fi" l' : 'of helix Wurftce
In fig l in

i j '-i-, @tt3, itl'n,

l l-I IllI | 1 'o1 h u '
.... .. .. ,. "- i-,- ~ i + o .o6o -
S-.03)0 Experimental
.. 4o,.020




4-' '.r
7, i-47 v' ,
V W,, ., C .0 ti, .-p '"
'li l f4' .-. i :i
4'- \~ 0 T ilt -. ---.


... .. ...
'i E U-*:1
1" ,KI 1. 1 w 1 Ij .. .. z i


A. .. r II :
propellers with various numbers of blde















T *









propellers with vai'ous numsbers or" blades.


NACA ACR No. L4H03


Fig. 15






NACA ACR No. L4H03


g, I;


-, + .II !l.. i. I I- .I 4 i I. | k- .4 -l 4 6.4I 11 H I


f : 111" ||.. T '1 i
,:,- 0 :xperimental points
^|(corrected values)

.. .. -. -. .





inI:^? !iii'sl^l^l^'!^ gi:











^lii!^ ~ ~ ~ ~ 4 ...^l^^^ ^.^J ^iiii
at j T..........
L7 ,W


V w
nD
(a) Single-rotating two-blade propellers.
Figure 16.- Measured values or mass coefficient K against
showing the effect of guide vanes.


V + w
nD


-11


Fig. 16a


iiT5






NACA ACR No. L4H03


.352


.241


.16


.12 1


.u4


m Experimental points
(sorreoted values)


PMR0,3


~fj2~: 4~1


0 1 3 5 5 6
V+w
nD
(b) Single-rotating three-blade propellers.
Figure 16.- Continued.


1 114 9140, 11


Mimiji,~~~ ;. -. I-MJ M ,M .Mjj


.. .- ..-

"l ,ll l lS.


^gMIBii 10fi^MI- sgP;^


. . .. .. . i . .. .
WHWNWHH'i'"HIN !'I-,'


n .. rrr->.jl R, 1


t4f Wt-444 -TH! 14411 tfM11 --


44=44:v


wafffl


ttetamwa


!


tPQTq4###Ti


#411 ',' I I-


Fi g. 16b







NACA ACR No. L4H03


3 ExperLmental points
(corrected values)
h-t-Trht i 11 tH I I FF .E-qH]-H- ,=


.72

.68

.64


.60

.56


.52


.48


P. .









1-








2.
ot .. 5 6 |















V~w
nD
(a) Bngle-rotating flour-blade propellers.

Figior 16.- Concluded.
i~ii :^IEII| |:I!!I! ^1 ^ iigl ^~ i~lin liii








|! +1+ S A+++++~iS'^ ^- | |^
-:= g t +- 4 i 4-i+l i i 5 l .t : :: :|i 1 ^
^;^g~~~~~~~~~~4 Zj_ ;;^; ^tg ^^|| |:;^.^^








ii~~~i~~~i... ..-- ....-^ ^^~ ge
.6


Y **44
-'4
(e> ~ ~ ~ ~ ~ ~ ~~~P Sigerttn or-ld rples
Figur l6. Concuded


V M 1111 IT IM,1,1, 11!j P
. .. ...


.CAL.


Fig. 16c








NACA ACR No. L4H03


02.
al. *
005 S...4a ^

i~ ^ 'C (U *' W'*


IP
In c


119 cu 0e- )o .o uo
II10 Isun SuGIB TBI2Ueod TBuI^U.Ld


Fig. 17.a

















i

.I

















* -

-


.4 h








n L
.
i






S S
-0
4 .


o


ru 0








NACA ACR No. L4H03


Fig. 17b


*3 -t




* iil
o 0
IJ !








.43



.-


ITeo lnun SUCoe TeTiuelod TwzotojtaJ










NACA ACR No. L4HO3


Fig. 17c


7 + o3 a


- I; .4 1Ja l _I .1 .L. iI..- -.t I I v I L 4 i I L .1 Ea ,


I eo Tun SuoTW ljofuelod lua CgiJa


rN -


S
L CO






,i

- :" -







--


C-


o C








NACA ACR No. L4H03


TTOO 31nn SUo01 jwulu'Iod jeuo i)hjj


Fig. 17d











NACA ACR No. L4H03


Fig. l7e


ml
a







a

a
a
~ -~

*



(3.


ITU atrn SUOTa leliuaod Twnoiue.rd







NACA ACR No. L4H03


Fig. 17f


.44bH44"+,4t1 H4444.H.4. 4+H~1A+FI ,,4+rm -~F~,T7flF 1 F I


3..O
0'- a CDc
' 0 a
A 0 0 ..


S 0 XC0


.TII


t7..
fii


WH1l-lia-t


TTeM Irun SUOIa Ta1eZod TIaoiulaJa


1n 'D It- wB "- f




< Ip. + 0 .
= aamm r-






+I+l


A IA it


:fl-Wt .


.0
II

:1


H-HWm


a
-. -

* a
o 0;
* [U

- I-


i|- i 2 _ia ^[j^ ^JL_^~~i^lalHa I


--i^' P, IV I 1' .


411"M f 4.


an,


11-4 L A i


i65=t14lii 5Hi'.l 1; mjti.r__.4i~


"I-'- I,'I f I"? :Ii ",,*. -A-.' ..I- *I + :-H -++ 4-*


"VIC i :.~ai'^T JT L- -- .
ii~s~ri ^^d^^ ^i:;:311.


7'. I TAF


:'.,- ma--:i .


M' .t T.. -. tt,'lti:


K'


1


.1 i .a I


"s


U-C.PU, V-K


t J
FFT-OFI









NACA ACR No. L4H03


Fig. 17g


0

-
w


ne10 I un RUOTV 7l3asieod twuosujj.A







NACA ACR No. L4H03


Fig. 17h


irW T| i4* t -. -4 *H 4- W 4 -i'~. A~j L H .* {WL

i.

*40 g,


0
^01u

g 0 a 4- -
a5 *
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