The theory of propellers

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Material Information

Title:
The theory of propellers
Alternate Title:
NACA wartime reports
Physical Description:
21, 41 p. : ill. ; 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: As the conditions of the ultimate wake are of concern both theoretically and practically, the magnitude of the slipstream contraction has been calculated. It will be noted that the contraction in a representative case is of the order of only 1 percent of the propeller diameter. In consequence, all calculations need involve only first-order effects. Curves and tables are given for the contraction coefficient of two-blade and four-blade propellers for various values of the advance ratio; the contraction coefficient is defined as the contraction in the diameter of the wake helix in terms of the wake diameter at infinity. The contour lines of the wake helix are also shown at four values of the advance ratio in comparison with the contour lines for an infinite number of blades.
Bibliography:
Includes bibliographic references (p. 13).
Statement of Responsibility:
by Theodore Theodorsen.
General Note:
"Report no. L-155."
General Note:
"Originally issued November 1944 as Advance Confidential Report L4J10."
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 - 003638511
oclc - 71833731
System ID:
AA00009428:00001


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Full Text


ACR No. LAJ1O


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WARTIME I REPORT
ORIGINALLY ISSUED
November 1944 as
Advance Confidential Report L4J10

THE THEORY OF PROPELLERS
III THE SLIPSTREAM CONRACTION WITH NUMERICAL
VALUES FOR TWO-BLADE AND FOUR-BLADE PROPELLERS
By Theodore Theodorsen

Langley Memorial Aeronautical Laboratory
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-
nically edited. All have been reproduced without change in order to expedite general distribution.
L 155
U .uM, NTS DCF K/ ', ,


~ci 1

hAccr L


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Digilized by Ihe Inle nel Archive
in 2011 willi luinding Irom
Univers ly of Florida, George A. Smathers Libraries wilh support Irom L'RASIS and the Sloan Founda[ion


lillp: www.archive.org details lleoryolpropeIlle00ang







NACA ACR No. L T10O

NATIONAL ADVISORY COMMITTEE lF;R Al-:RO:IAUTCTCS


.AVAIICE COT'IDENTTAL REPORT


THE THLnl:.Y -'F PRP..ILLERS

IT' THE SLIPSTREA'.' CTI'FEACTTI1i WITH UifrLRITCAL

VALT.S POR T,'O-ELADE AID FOUR-PLADE FRPELLEiS

By Thec~ore Theodorsen


S M. AR'


As the conditions of' the ultimate wake are of concern
both theoretically and practical-ly, the magnitude of the
sliptr-ea!i contraction r.as ben calculated. It will be
noted that tLe contraction it a rcpr:-sentative case is
of tne order of only 1 percent of the propeller dia.neter.
in conscq,.ience, all calculations nEed involve only fi"ist-
jor, r f ff'e : t ri.rve srnd table are aivein. -or the
c,-ntraction coe"ici ent of tLv'o-blaCe und founr-I.lade
pro-Jtr .er3 for various valui ? of tLe advance ratio the
c'ntracti 'n coeff ciiint is defined as the c-ontraiction in
the di Ar et, -t o-f Y.; a '.(.a- h l i n. in t,'Lr'll of the wake
di t'e:-r at nflr 'i ty. 'ift 'cont.T f' liint. if tle wake
hr-li-x iE also. 3Cln at f:mr 1Ailc- s tof tne aIvancc ritio
.,in c )Ulur'isi 3n witi the c'"nt>iur lints for an infinite
nurmbe.r of blades.


T :TR L TDUC .,TI


Since reference is ofteu ir'-de to the anilke infinitely
far be;iin the ID'nrellFlr, t is -lsirable t) Lstaolish
certain relationships betwe;.in tb; 6iin-,m.sii'ns of the
prcpell r and those f the "*.Jal' he.-lix at. infinity. The
present paceri cnjnlders the rclat.,-nship of the propeller
-armret.-rr and the ;er. ;! .]ameber, *;r t.h probl~lm of the
Slir trr'amT contraction.

Tne ciiscuFsion is restrictrcd to a ronsiitdration of
first-order effects, that i3, to tie detc.rminatio'n Af the
cintraction pr-r unit of loadnrig fo infinitely small
loadings only. It will be setn c.ht the contractions
are indeed very small, of' tn..- ard;r of a iC'e percent -)f









FACA ACR No. L4J10


the prooell]r dianmeter-, and that the high--order terms
are ti.er.--:'re not 'f concern. '7_ e Interference velocity
acc)ringly is ni-elected uS 3s'a.ll compared with the
streak vl!ocit-. ft-he wa':e heli. lies on a r erfect
c.yli.nder .nd t:ie pitch angle is ev-rywnwhert the sane. It
is noted that Lhe asruminLion of" zero, loading corresponds
to that used by GOlrJrtein lor c different purpose.


S'i" ?BOLS


R tip rarius :'f ornpeller


Ar,


h = -
cTT


- w
jr


ras ius of ;-1..rent .f vortex:-: seet

contract :,n
h
total contraiction or contraition at 0
11
angle betw...n st'.-tlin, point of sniral line and
p-lnt P

pitch of Ir-Iral

an gul.-r co.ordinAt' on vorte:.: seet




ac'ance rjtir o )-:

ratio of rjdlu.s of ele;!ent to tin radius of vortex
she t (r/R )

radial velocity

advance velocity of' propeller

rEarwvard disp.rlaccient velocity 3of helical vortex
s-.1rfa c



number of blades


mass coefficient


C0,1TFIDEiJTIAL


CONFIDENTIAL








NACA ACR No. LLJ10 CONFIDENTIAL 5

Cg = 2KW
( ncskVVR
F circulation at radius x c -- K(x
K

K(x) circulation function for single rotation l
2 rrVw/
w angular velocity of propeller,radians per
second

yl radial velocity at ooint P due to a doublet
element at O,x except for a constant factor


y = !-9 cos (6+T sin (9 +T)r F- 2x% +*.62+ x cos (9+T)
1-
B1 Y2 + kq2 2x cos ("+TI +5'2


Y2 where n = 0, 1, 2....p-l
n
I'1
Y1 = y2 dz


Y2 angle of contraction, except for a constant

factor 2 = d1


Y contour line of contraction, except for a

constant factor = / Y2 d)



-- total contraction in terns of radius Y
R ( c 4

ro s /5 c
K contraction coefficient 3Y
R Cs \


CONFIDENTIAL









CONFIDENTIAL NACA ACR No. LJT1O




2 2 =
zi = sin tan' x2- x /E(',-) + 'x [ (k) E(k)



1 [ )\ rk) 1
w = T E(k)J


T EO Y


The radial veljocty is. Detained .ty ju-ling the Biot-
Savart laL and ir'te;rati n.- ov:r' trc. centir'e surface of
di .;:cr, ti i i if tr, ti.e total Crtra'ctl on, the

problem tl, t.:, determine che ratio for various
nuir,:ors of bl:ces. ut sevcra&] *.iv:nce ratios. Simple
e.xLr.essiiIns referric! to zern lot.ring aoe used throughout.

ThL ra~'Ll in;ard 'e-locty dvh' at the point P
is calculated. (See fi.'. 1 and 2.; Th.LS velocity results
from an e-lement of clrc'11atiJn f ds, wnuich i. located
on a si'"al *" radiu." r th -i starts in a plane perpen-
d'ci:.lar to to- a xis andl c:.ntiirn jin t.ie r'rfere.nce ooint P.
Thie ngl, b-t, .~c': t'i, s't rtinL pflint .sf Lhe so.ral line
and the rint } 1I2 dcsir.snateu T. The spiral extends
oelnw the plan<: tc 'nfini'ty. if the pitch of this spiral
is designated H, the clemenL at *i projr.cted angle 0
fror the startIng oi.nt of the sc.iral is ti-en at a
distancp 1t llew t;l.e reference pl..ne where


h = :--- (1)
_rr

Ey intrnr'ucin the nc.ndimensi-.r.al ..iiantities

:,R
(2)



in the 3iot-S.-ar't i.aw, the follo- .,'/ing expression is
o'Otairne.d for chj racial inwari velocity dvR' due to an
elet.ment on, the wake helix of strength fr


CONFIDENITI AL









IACA ACR Ho. LlJO10


1 f d9e @ cos (9 + T) sin (e + T)
dvR' = A -
r R + + 2o2 2x cos 3 (6 + T7

By differentiating equation ( w) with respect to
the field of a douolet element on thl helical vortex;
sheet is *obtained, the double elei-met consisting of t\
neighborinr;- singlet elements each of strength f.
Setting f:dx equal to an:d dividing through by the
stream velocity V gives


53)


wV


d 1 r c e ('3 +T) -sin -1 +T'] -+x2 T2 + x c.os (9 T7 1 .
v I-V T 1 + x2- \2 ix cos (P+T


where vR is the radial velocity at the point F.

Equation IL) may be written in the form


R r 1 P d ,
SLrr V


(5)


where


y 9 c ns (r+T) -
Il +
L


sin (F +T)] [1 D2 +
-"- + 2" ,x cos


\22 +x cos (9+T)]

(A+T]5/-


The function i1 is -olotted agiln-t
of X and various values cf T and
Wi th


++T for four
x in figures


'IT I.i



ST'.'2f

TiVRX, cs
- 0 '


where


2KW
es -~V


(8)


= 2CON
CONFIDEINTTAL


(6)


values
3 to 6.


CONFIDENTIAL









NACA ACR No. L4J10


sub-titution in equation (5) gives


V( ?.2 cs Y('1
d--= L l de
v p yi


A
If the Point P is at a d5strance h = H-- below
the propeller, integrating equation (9) over the wake
yields


It is nntcd that, -ith
functi ,.n


rr
I
-e 0


.--L dx d9
p / 1
n


(10)


equally soaced blades, the


(11)


is an Aj:d function of 9 and


i .
21 -- j


7y, d9 = 0


(12)


Eluationi (10) can therefore be rewr tten as


V1 = .
~ r
'-I


1J :/


-. ,>i y d dx d
1a


CONFIDENTIAL


(13)


COIJ PDENTIAL









NACA ACR No. L4.J10


Let


2 i:(x)N
n Yl
]-,


=r
CR9


y dx



I1 d@


Values )f -,and rti ed a constant factor
for conv-enienrce in nlottingr, aru iven in tables I c.-. IV
for two-ilade and four-b-luce nr'teoI1er s 'fr which ;.
and 6 take on various vulue. The:- functions are
plotted against 9 in fisure2 7 and -

Equation (13) becorn es


\- K\ s





h jir d} 1 dx
'V
1- Jh .


(15)





(16)


Therefore


d: : ?_ vR = C. -
d r 9 'V ~ 2
d~j ~ 14


(17)


r 2 -. 1R
R R F'


I
JRI


dr =


x ^ u c s


9'
-, d9 (13)
ji


Pr, v.r IrT nTri rmT '. T


(14)


whence


CONFIDENTIAL


'.i- ';









NACA ACR No. L4J10


If R2 is the radius at the propeller and R1 is the
ultimate radius of the wake (2 = 0, el = ),


Ar c, \35 p5
-2= Yd= s 4K 3
(iO


(19)


where


Y = Y2 d
u'


Values of
plotted in
for which


Y3 are given in tables V and VI and are
figure 9 for ti.o-blade and four-blade propellers
K a.d take on various values.


After all substitutions are made, the complete
mulltir.le inteiesl f:.r tht total contraction Is obtained
as


R
-7-


K f (x) y (Y,x) dx d9 de
-4'O 00 n


(See fiSz. 10 snd 11.)


TN INVITE NUIrTBZ3R OF PLADES


FPr purposes -.f co-narison, it is useful to obtain
the contraction for the case -f an infinite number of
blades. Ey rcsolvirng tac circulation into components
parallel to and Derpendicular to the axis of the wake,
the helical vortices can be replaced by a system of
vortices parallel to tne axis and another of ring vortices
having centers on the axis. Only the ring vortices
contribute to the radial velocity.

The field due to a vortex ring of strength f and
radius r = Rz, located at a distance h below the


CONFIDENTIAL


CONFIDENTIAL







NACA ACR No. LlJ10 CONFIDENTIAL 9


reference point P. is given by Lamb (reference 1,
o. 257). In the notation of the present paper it is
1
fR x2 k) Ek)] (20)


where E(k) and F(k) are the complete elliptic
integrals and


k2 4.x
+ (1 + x)


As before, a doublet ring is obtained by differentiating
equation (20) with respect to x. By setting

f dx = -
H

the following expression is obtained for the'field of a
doubl'et ring:

2 1 2 1 2
R k -- 2 k2
r k x 2 k' (k) + xF 27(k) -E(k) (21)
0 d. H 1 1 k2


In order to obtain the effect of the entire vortex
system, equation (21) is int grati-d with respect to h/R
and x as



h I k2 2 k2 Ih
a- k x k2 Elk) + x [2w(k) k2 V k) d dx
o0 _-h/R
(22)
The radial velocity

1 i6
vR R2 6h
R


CONFIDENTIAL







NACA ACR No. LLJ10


Lq.'at n 12 i) .n;iy be -.:rLtten in the .o)rm


r. r
"'.) J-h/5i


1




1


1


I
i:3

91
I ,

liJ


h \
dx = d:


r .(-, x )


-I \
MQ x)!
.1 t..


Ph



;-.
= I
., V :'
'Icy,


1


I'

ib.


/h d x
V' ^-^


(25)


3y substituting

ry .\VR
= .. (26)


where ;'(x) is the horiz-nt&l cnr.nonert of the circulation
coefficient, w-:cb ;ais be e-:Dresetd as

/ 2_ 5(/2
7 ; =z ( (27)
+ x


CONFIDENTIAL


s. t;at


(23)


Ta 'I


(24)


C 3N.IL ETTI.AL








YACA ACR No. LJTJ10O


the following equation is finally obtained:


Ar CS
P 16wT


2+ x2
2+ x


/2
dx


x 2k2
1-k2


E(k) +x2 2F() 2- k2
1 -k2


Ek) d
J R
(28)


For convenience in using the Lt-gendre tables, the
second integral is written in the form


where


. hh/


(29)


zi = sin


r
S tan2


/ 1


Ci 1
- 2 (k) + 2y. F(l:) E(k)


and


\(


i- n x
s-in4


f 1in-1f.


(See fig. 12 for plots of Z] against l/R.)
The final ex-ression is


Ar C
R loTrK


Ii0
JO


/2 /2
2 + x2


dx /
vJh/R


sin -itan( d x 2
~L


- x) E(k)


1
+ 2x2F(k) E(k)1 dh
R

(See table VII and fig. 15.)
CONFIDENTIAL


CO'IFIDEUITIAL









12 CONFIDENTIAL NACA ACR No. LcJ10O


TN'TNITE RT~3BFR OF LADESES FPOR DUAL ROTATION


The contraction for a dual-rotating proneller with
an infinite number )of blades is next obtained. In this
case K(x) = 1, and the radial velocity is


k. 2
1 r/ x=1
p = =-- 2 () E(k


Since the ''value at the lower lir.it is zero and 1(x) = 1
for an infinite nu-ier 1i bla its, it follows by substituting
the value of that



Ar C s h



'N hc r e
I 1X=1
where


) (') E(1


(S3e table VITT and f-i I;..)


C?^CLULrI O PE.' ,'""S


The contraction coeffici-ets are given for two-blade
and four-tlade singlc-rotating rr:')ullers at four specific
values .f tl'.z advance ratio. T-.e calculations involve
tr l.le- int, ,ratins and ,re ti;--erefor sj~. what laborious
an, susc-:rtit le to numerlcr.al trr-orzs. 'intil mrire convenient
meti,.cto3 are dcvise2 to perxor1n this interation, it is
honLd- chat th-- values given in this paper v ill scrve the
puro :se. It is well to notice the !-:sll magnitude of the
conit-acti.-:n. A four-tladre Drocllcr with ncrimal loading
and advance ratio i smnajn to have. a ttaal contraction in
terrs of the rad.ius3 of less tchan one percent. The


CON TIDE!,T IAL









NACA ACR No. LjJ10 CONFIDENTIAL 15


first-order treatment embodied in the paner is therefore
adequate for all technical Durposes.


Langley Memorial Aeronautical Laboratory
National advisoryy Com.ittee for Aeronautics
Lan,;ley Field, Va.















EL.FER iCE


1. Lamb, Horace: Hydrodynamics. Sixth ed., Cambridge
Univ. Press, 1i52.


CO:FIDE3":TIAL





NACA ACR No. LJ1O10


TAPLE I.- AUUTCTION --Y1 FOR TWO-BLADE PROPELLER
1


9
d( e)


50
10oo




200

25U
301-1
350'


hoo
500
55.0
600
60o

700
75u
330
' 0, ,"


%I
Y

S (deg)
I


*o; o i"' i
r a o4'' :

r.00 1
.000750



. 1.O i,-.,'
.0006x22


.COOO7130
.*00':50



* 007:1d73

* 30;30''j


.j Ij,
.000175
I 8I~* c
0007


2
I


5







20
LO
tic





IcC
On^'-


2Lo.
460


Lo


;"'o
U.O


760

.1,o


1 I
2


-.JJO15
-.o ,':'2

-.0ju' 3

- .3*U0c.113
- 7 ..i



- .'9,2150
-... U :


-. .r r *i
-.00 3'59
-.. 2 ?


-.0 0., 9




.3':"Dl,










* 300;"2

.,o00115


-1
t'1


\ =


-.000923



-.001702 -

-.O 171.
- .0, 1226

-.000 2




.0 -07'4
. CO', OL 92
.0002,,o

. '.J 0 1
.;00014
.'u.2005
-.3"0015
-.ujUCO0



-. JA 'QO l
.0000G03

-.000005


.00,0027 .000020
-.0'00011 6.000011
. '00014


1
2

-0.Ji'06.i"
-. 00! 1 '.
-.001u:
-.oox16 2
-.0u1: 5

- ,01 U 1'4
-.0. i15

-.:.0175-

-.00u, 137








-7I
-. 00 I5
-.Ol5oov
-. 001567
-. 00103




.OiL'7321

-.0"U075
.00C, 10
.000136


.OUCI047
.00'c75


.0 .c'L -" .:'


.0,cC 35.3

.. 00 ":00
-.0'0000

.001007


'TATTOIAL ADVISORY
COT.y.ITTEE FOR AERONAUTICS
Or n P'T OPM T AT.


-,----


CONFIDENTIAL





NACA ACR No. L4JlO


TABLE II.- FUNCTION F-Y1 FOR FOUR-BLADE PROPELLER
7-y


0.000500
-.0206830
.097700
1.251120
1.:253120


5.51i959o
2.685010
1.350560
.8215 10
1.017150

.902770
.456960
.525530
.217700
.154190


.119570
.035450
.029500


-0.002211


-.011574

.006150
.15ool.
.565018
13 70.Ol,.

.2185', "

.112735
.0916 56~ 6
.075117
.059218
.0 0_2 61
.0!J02
.02L159
.0o17 i


.D I,,01. 3
.01117L4
009069
.007762
.006459
004650

.003748
.00-r 1-
..) +4 2 8


-0.000001


.000002


-.000015

.00176
.00o 6166
.006425
.00L174
.002161

.001281
.0o0865

.000535
.000224

.000176
.00':1350
.000100
.0 C"06 .2

.000052
0 000016
.000052
.000015
-.0u0001
-.o00000
-.000004


-0.000273
-.000250
-.ooo000
.0000251
.000251


.000501
.0007o 0
.000905
.001265
.001-52

.002984
.005523
.005108
.001577

.ooo002
.000552


.000 214
.000214
.00o166
.000073

.000101
.oooo66
.000057
.00004 3
.ouooo4
.000057

.000025
.000002
.00000o


NATIONAL ADVISORY
CO,,?ITTEE FDR AERONAUTICS


CONFIDENTIAL


CONFIDENTIAL


13





NACA ACR No. LUJ10 CC

TALE TIT.- I TTCTTON


)NFIDENTIAL 16
0x
'-2 FOR TWO-BLADE PROPELLER
4


5.1L129900
2. 12900
2. F2900
,,
2.202.q00,
2.,59500

2.215900
2. 2 6100
1.75 6900





2,5 013
.10. 00



.7L oo00

.o7l, l0

- IO ,,30
.2L.6.L OO


.112000
.0;00030
-.0.I6000
-.001 000


0.577250
.577650



2 2720
7 0i


.1.670 20

.li6770

.?.I 670

.L5070
. 7 c r15
.267 0t
.102200




- r-
.o 6752



.027' 10


.002400

. o'00 20
.00010,j


-o. ooLL94
-.ooti5

-.oo1300
-.03 ,,'90

-.0027o90
.000.95


.003555


.010 S75

.015075

.oi3075

.011070)

.00700
.o0 70
.001170



.0300270
.000 :60
.000200

. ooo2o9


. O0u ,iQO
.00 S00 0

.000070


-0.00515O


-.002466
-.'00u13.
-.300jo76
.ooulj o


.000907
.O i2. 02
.o 1577
.00, 17
.3j1724

.j01129
.00071-3

.000'u15

.000207
.00 0178
.00 18



.0001625

) 300007
.000005
.000000


CONFIDENTIAL


iL .


NATIONAL ADVISORY
COMMITTEE FOR AERONAUTICS





NACA ACR No. LiJlO


TABLE IV.- JUNCTION -Y2 FOR FOUR-BLADE PROPELLER
4


*. oillb-1
.02146
.0245 3
.021j256

.0255969
.o0253






0 C) 5
. 0u19582






.0702166
.00 o .77

.00L:21u
.0017E5S
.oo00o36
.00 552

.oojoi



. 001 57


.00035
.0010'7f
,ooo,754

.000567
o.30205
.000075


0.017259

.0176: 1
.01766
.o'17 76.

.0172E7
.017317

.o13 C9

.O00u85


.000x6
.030 .72
.000 71


.OO ul2
.00)0k7
.00o06.
.aooooL 6


0.00713i
.007 1
.007 91

.037919

.007*'03

.007U60
* 0,7c7 13







.000157

.oeoj5.
.00OOu01
.o 0000








-. O0L0
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CONFIDENTIAL






NACA ACR No. L4J10


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NATIONAL ADVISORY
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