The theory of propellers

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Material Information

Title:
The theory of propellers
Series Title:
NACA WR
Alternate Title:
NACA wartime reports
Physical Description:
10 p., 85 leaves : ill. ; 28 cm.
Language:
English
Creator:
31262081065491Theodorsen, Theodore
Langley Aeronautical Laboratory
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: A technical method is given for calculating the axial interference velocity of a propeller. The method involves the use of certain weight functions P, Q, and F. Numerical values for the weight functions are given for two-blade, three-blade, and six-blade propellers.
Bibliography:
Includes bibliographic reference (p. 10).
Statement of Responsibility:
by Theodore Theodorsen.
General Note:
"Originally issued November 1944 as Advance Confidential Report L4I19."
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 - 003807489
oclc - 126892013
System ID:
AA00009393:00001


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(VAuA


ACR No. L4I19


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS





WARTIME REPORT
ORIGINALLY ISSUED
November 1944 as
Advance Confidential Report L4I19

THE THEORY OF PROPELLERS
II METHOD FOR CALCULATING THE
AXIAL INTERFERENCE VELOCITY
By Theodore Theodoreen


Langley Memorial Aeronautical Laboratory
Langley Field, Va.


I -


[i .' L F 'ou _YCf ':






* 1
NACA




WASHINGTON
NACA WARTIME REPORTS are reprints of paper 'Iginallv Issued to provide rapid distribution of
Advance research results to an authorized group ar a n for the war effort. They were pre-
viously held under a security status but are now unclas: ,, of these reports were not tech-
nically edited. All have been reproduced without change In r. *. expedite general distribution.
L 157


- ~7

















. ... .



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-- .. i. IC .


'I. Die -. C .r e


S -. : -. I


aI -" ': .'. Ic .:
T : 7r -.-.
Ic


"ien or calt the
- .-.ity o a p-02"._ eLler. ."'-.- ethod
:.;'a:n '. .'" .t functions P, ,
s fo'D the a'~i'.-Ib function' are
S.ec-b''- ., and s'.-bladCe


.. L '.";1TC


It i" n ths pra c' ce to ue the 3lauert-
I:.>: '. .. on t at tne intc."erence veloci-
S. T. : .: at -. -inFt corn idere ..
S. a :" i-. ;. lyk. inadequi.ate since the iter-
far. :.C. f-i ." :'-e..-".- :." th-e slope r...< curva'ule of the
1Ti.._- : _:..?:l r:. r..!w. er than or the lozal rra-.:tud". A
.- '..-'. i .6 .- .. iE in for calci 'at ..- the axial
i .: ... .' .. o i- The mn thod is
FS":-T ... r '-r and therefore iveC the
irnt-rfer.. --:.-- "i-. i.. r 'io to o the loading for rvall
loiD : ..I. it .. .. -wr'n '".t thir acc'i rcy i ad: uate
f_ : :' i ? 1 tons.

ir '-r --t r .3 the seco:- .. rI: a series on the
St'- : r r. r .rt I ealq "ith a method for
obt:Lr.i.- t:. 7 .;'', n "t tion for dual-rDtatir.
-r -. 1 c --. r .-. 1.)


:-- '" '. W. ; it7 at v; | vzfxi1

t( *-:" .:* r... r1..---l em>cnt veloct1 : of h :" .cal
'.' : -: .. -ce (at .._ ty)










2CC..TDE TRIAL ::-A ACR "o. L4119


V advance velocity of propeller

p number of blades

n o.'.:r nuit:her -: blade (0o n p 1)

0) .. .';ar velocity of proneller

P circulation at r '-us x

K circulation "ocfficient to firft or Kr (-T7,-)

x nondimensional radius in terms of tip radius

xi reference poiit at vhic!. -nterference velocity
is calc3,:af-ted.

Sac-.-:.' ar d"Asta3- e of vorte-x ele .ent from
propeller

, advance ratio (V/i ')

R tip radius of pi--T': .er

Pi(x) function defined in equation (1)

Q (x) function defined in equation (Z)

P used for Pl(x) in tables ar-i. figures; refers to
other blades (n 0 G)

Q used for Q1(x) in tables and figures; refers
to blade itself (n = 0)

T phase angle of nth blade \

pl helix anle at x1 tan"-L

........ ,. .... .. 1 (x)


It can be hbo:'n that th- :.;-I;'.l1 interf' :.e -.C'c -1 ::-
is given by the expressi-:.


CC. I .. -AL









C I. IJAL


VI i
-- -/, x -~ "- x




':.'', ."-...'".&r tioi is over the iui:iber of blodes 0 to
S- 1. .': portent '" ::tion P ( i) s defined, as


d -


-J


-I E xxI 3_ 1+
'//.T '.^ 1 1 \ii p /J ^


. ,- : -, 2., ... p 1,
t -:]:. p.e problc n i
' ..- -i.' "-,- t.-'- function T (x)
r". ". 'J .


twe nu'uber of the ':r-
thius essentially solved
for each point alo- the


It n venient t ake FP(x) finite `-
tr:ct:. I-" .1rntity that is ind ndent of x.
f'.Ct-_. F-I ) :nay therefore be redefined a7


I + x1 2xx1 cc ( + ) (7 -)7



1,(1






I:t I n.:t:c, that, in the int-:,.'al P(), the
nr:c'.-r'r.d ch-.: e f.:*-'. +o to -om at x xI for


J.icl


"' ''


.










iACA ACiR No. L4119


6 = 0. T 'is difficulty, which occurs only for n = 0
(that i", for the blade itse' -), is overcome in the
follc:.'i 1' manner: T- e expression


1 \2
-.-. x 1 2
4 ,-o-'. I


\j !X-1 .
\ -- + --
4W rJ


I, xx 2
+ L2


ITT


which is integrable ana equal to





2 : x I
X2 +


may be subtracted fro ;
smooth integrand. Thus,


r ,. ) t- e l v f :, t .....
'by c i.' i.l- .c tif *i -E :.t:t:,


S -" TL 1 TA T


(2)


C TDE Ti', TAL








C :.-- IA


I. r 2111


1


/ xx
-:


+ /2


*1) ')(a


+!


Finally, for the blade itself


(n = ),


Pl(x) = '1(x) + F


S2
? = 1,- 1--I- x
k + x:1


Ti'r.e int-)r. l
ti-n an.d !i ,
function F.


i-(x) i converlent for gray .'cal int -ra-
in fact, srall in c:. -. rleon with "-.


j7L(7-i


\V2


- X


~___


i c t&ir..i


* .. -. A.LT










6 C:-7IDEHTIAL ':ACA ACR :'". L-IT19


Sdisccrntinities arise in the P fu.irtions for
the other blades (n 0). ':- functions are there-
fore used direct' in the calculation for the other
bade 7. It should be noted that the function P, Q,
arnd are all s r- xrrical in ;: and x!. Inr use of
the subscript, which has been us.ed. to indicate reference
to the point xl, is therefore discontinued. In the
following die US on. the functions and F refer to
the blade it elf uad P refers to the other blades.

Since the we'. -:t function i: needed in the form
; r
x--, it i p wrItten as
dxd


x-- :.-- + 7.-



It is to te noted that `r far the la.--' st contribu-
tion comes :-:: the loaarit~:.io function F since it
really represents the entire field in the ne1'hborhood
of the point considered. In developed form,


xxi
dr 1 x 1 2
-- + log x x1 (4)
Gx :- ~x xI ,
Sxxi --




T '.. .ICAL :, ..... C- Z KiTj U i ICOJ k-. P-


The weight functions an, ', and P are shown in
a series of tables and fi-ures. Th first step of
integrating ---. i-.t the rncle 9 s omitted for sim-
plicity. The functions -m and U. have been o' tsied
dx c;
by Graphical diff- rentiation of the Q and P functions
with actual calculation at the end points x = 0 and 1
for accuracy. It should "e noted that these functions
and their derivative are continuous and smooth. The
results are given in the foll- r -. order:


C,:..r T":T AL








: I L ., 7




(3) Tal. b 2 ':' f u: : t :
() Table "' and fi"'7irc 2: a a,-inr9t,

( x .0 ; C.Ir: x I .C : X n ,


is oF obta:ind ni'".].: tic ally

(F) Tat le 4:* f" .( -x-- *(1r. st x
Sx 1.. ; 0.14 ; ,,







(ia) ..a 17 f oi T n b
(7) "--e :" --- ..-5nst tx = < I.;

( 1. ; 1, C 2, .btaine :r


(5a) -.:,le ': 2 a :.t :: oS T = :
Sx i; .1564 i .C; =, .
obtained "': quac.-'on (3)

(b) r:..urL 4: P(x) P(1) .: -ainst x r T =.:
(o ^x < 1..; 0.15t4 h z2 1.'-,; X = ,1 1a-d

(Ca) T'..e ": are as table V fr T =




(b) .-ure C: aC..ce 'a- f-ir 4 for T =: 13


(..) ure 7: e .::e a fg.re: 4 fr T = 2 "

( ) :..-le X: ae ac *.-,* e V or T .

(. .) .-ure 8: SSa:.e as f"l T.-. 4 for T = .." .o


' -' 7 '.T









SCO 7T LI 'AL NACA ."R *>. L4I19


(10) Trble X: -. t T for 1 = (T = 60,
LL 2
120, 1800, 2".0, and 300o; x = 0 and 3.C';
0.-,14 1.00), obtained analytically

(11) -.:Ie .": seme as table X for X = 1

(12) Table XII: samc es table X for \ = 2

(13) .'Ble X7II -nd fi:lure 9: again-t x for
1 o -C ,
S= (T = 120, 2 end 00;
0.15,4 i : 1.00; C.1-: < x < 1.00), obtained
by multiply] values in table X by -x

(34) -'.le XIV and figure IC: samrre as table XIII and
f:-'re 9 for X -1

(15) Table XV and figure 11: sa ,c as table XIII a-i
fi -.: e 9 for X = 2

(16) Table :-:VI and fic'.re 12: -:.- arairst x for
three-blde and six-blade propellers T = 120
and Z400 for three-blade propeller; T = GO0, 1200,
180, 240, and J30O for six-blade pro;e-ler;
0.1564 x 1.00; 0 x 1.CO; X = 1, and ;
it nay oe noted that these values for two-blade
propeller avre given by -7zx for T 1 in
Cx -x
tables XIII to XV and in figures 9 to 11


APPLICATOI'T O7 T':.TTOD


Steps to obtain the induced velocity expressed as
VI
-- are as follows*


(1) Plot the quantity -d a:-.-it the circulation
coefficient K and perform .r.'.:;ically the integration



CO FIDET2" AL









C "I I L


d a-la 7t K


(.~j1 t~-' irt~~ 11


c..-ce x --7 ^**-o'x. cs In fn t at x xl, it i! '.c Cc-.
vary :: e:T.Iude a fron Ax t3 x, + x
:. t con ri~ -r t'.is .P.Ye a^^Clo b. use of a lajlor-
to


il: brcc?e r


S= -' "" + .-


\.X2 + x 2
'S
=+


Xc = + =
+
c, I "T


S; *th res-ct


from the other
plnttl.. x-


;" for th. other "'1.*.es z'.. e the valuF

can 1. ta':en dciretl thr: the tables, this w'ork
only ore ~1 .. t a s:.. Ic Er .:' l lea] nt --:'.tion


C-.


&:-.- and :" are the li-va" of
to x.


(3)
bl.. .-s.



cont i-.
c ntat:


r,_
i i,,,


(2b) ..iI : contr b~'.r tlon ii'o


''
"L.r


I':.'.ly, :"ere is a cont rijUticn

-'.'" comt .' 1'on -".: o :tained :,









10 CC..7F TTI.TIAL NACA AC-r !7. L4I19


S- addition of the results of steps (1) to (3), the
total interference vcloci-.: v1 in the axial direction
is obtained. --: relationship be cen the axial inter-
ference velocity v at the radius xI to the axial
displacement velocity a of the vortex sheet ray be
seen from the sketch in figure 13. The relation is


v = : cosy 2
V1 COS TO


or, ccnverseely, the I -,place. t velocity w of the
vortex sheet ;:a- 1e obtained fr*-( o tihe calculated axial
interference velocity l i b'y e relation

S1 VI

cose 12 c


which zives the axial disnlace:ent velocity at the nro-
r-ller diIk. r tihe case of tle ideal loaciT..' this
axial dis lacemrent vlcocity riuit corne out as a constant,
thus per-itti.. a check on the wei. :t functions. Cases
of nonideal loading are evidently of i:ore practical con-
cern. It is the purpose of this ; -er to give a method
for calculation of the axial interference and displace-
nent velocity for any (light) load .f .


Langley !moerial Aeronautical Laboratory
national Advisory CoDTittee for1 Aeronautics
Langley Field, V .h






1. Th.eodoren, Theodore: '; Theor of Propellers.
I ;eter_ ration of the Circ-lantion :...tion
and the .- .~ Cefficient for Dual-Rotsting
Propellers. 7-T AC: ro. L )3, 194.




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.7 US.1i L ,r 1 : |1.6;- I .




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1,'1 i

_ ___ ___ __ i __---__----


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ii
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". ,* L F ,. -. r V .' t S x A ': : : x R T h E -
; F j .;',- .'v" :J 3 ":,'- 'i "v 2: -, *'- -7 ; C on cl ud ed

LFor thre-b'ade rr.:r-llrr, T = -'' and .'.o; for six-
Sde lronfl1 r, T = .' 0, o, nd


S. ". ) :'.' "- 1 i:'. I.


(d) Six-blade pro-eller; =



-. .75 2.261 3.'-. li.367 6 ."7'
S 71 _. ~ .- .
.. ...... .1712 1.2 2.1,
l." i .*'[: 1:_7. '73 1.77 2.1


(e) Six-blade Irorller; = 1


0 0 0 0 0 0
.1 I, 2.166 1.022 h. h.- ''7& 4.' .:-:
.909 2.171 L.L', .354 ." .% "
.-c7 .27 .677 2.7- ?.407 3. 3 .7
."" .166 .37 1.-: 2.152 2.67 -7 -.7?3
.122 .276 .12 1.11 2.00+ 2.2066
1.00 .3'c .25?3 .'t'l. 1.312 1.1L6 2.7 7i

(!) SiY-t.! e -,rc !e r; = 2

0 0 3 0 0 0 0
.l J 4. 1 L'. l.".4 6.77' ':- ..:.
.1 = J'^lo-, 2 .~ .-- 6 .... .
S2. 7 3.L 4.36~
-- 1 -. "l 7-7: 3.7 3 .3 .L7
.0 .2 .1 2 -.L-5 2.7 :: 2.71
S .. 1. 13 1. 6 2. :6i 2.35_ 7
1.'' ,162 2 .1 ,L 7 I. ;l 2.....
iii ttrm U i .. .. .. _


~r-"7; TT' 1Y.? :r~:?-~::!


I;: _, TT~-







NACA ACP. NO. 14119


.10


x
(a) x-
Figue 1,- Function Q against x.
CONFIDENTIAL


_. .f. .-*__ ....... __.......:
---------I- -. --,-. .. -. ....





L
.'K '4 .~ : ...






---- -- ---- -
S0 ..
-- f .-- I ,. ,V- -.-' -




--,--A-T,...., -,4 .
-~~---------- ... -' <- ',---i^--O- -- .fc.o ,




,- -- I. "


'ATIONAL ADVISORY-
WIT IL-- 5, r4u-- ,.




.2 .4 .6 .8 1.0


Q 0





-.10





-.20


Fig, la


C'-- '- Flrr',TlAL









.A ; A'75 No. L4119


.04


.04


I:i- .TIAL Fig. lb


- I I
- I


'I II >-X


----4


j ^ N s q^ 0, 564 r
k t 1 i -.


SI ,

,3 o- i ; a,



04-- .


I" V':j" p U .4 5 4 0


-.12


-------- ---- ---,-------i------'j\\* |'- --'V-'^ivv ;--

161 I ; 1 '
... I.. I t ..


- .2C


0 .2 .4 .6 *8 1.0


(b) X 1.


: O:.FIDENTTIAL


Figure 1.- Continued.


: ___L~-.-.ll--.


L"


w !


'l 1 \ \ *1 N I I'


. -.-_ -- -1'--_ --



N 'r


-.









Fig. Ic


!.AC.A ACP No. L4119












C4





0- '





-.04


-.08




-.12


.2 .4


x

(c) X- 2.

Figure 1.- Concluded.


CON FILC ENTIAL


.. r.lF i' NTIAL









NACA ACR I.*,. L4119


0 i 4 -r
: | C- -c*i-
M ^ ijear-, *** a*
6w ..


N



r.7
Zar
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^' A
/






/


0 00


/ 6

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S/




/


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\


I '?








NACA ACR No. L4119


-1







=1




i-.
-1Z



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C







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F-
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. .. .-. -

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31


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a.,

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ar
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NACA ACR No. L4119 Fig. ,







n i > ,


I I I
i J i I j 3






i .. Ii 1
-I -- *- I I |
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.- .... .. ...
.- .. .. ... .. .. \ .. jx


.. ... .*
i i \






.1 \ C z
---I



I i "i


I C.
I


-i






A-- _______ .____
I I








NACA ACR No. L4119


- -i---


S: .oo
. .


.30




.25




.20


.2 .4 .6 .8 1.0

x CONFIDENTIAL
2
(a) r-j.
Figure 3.- Punction -*dS against x.
dx


-- -; ...---i--. '6I
I | 1 .
--------- -i`----- I -i---- i--i-i- -- -j.--f ---^---n
----- ------- j--+C-Lr/- -^



T-*


K :I
----- ',-I-;---- -..--




-----I--------



--- -- -_- __ j.. ....-- .... ---..._. ... ._ ___ .
-- -t----~-- / A ---------i-











T A I '


-dx
dx


0


--


I


CONFIDEIiTIAL


Fig. -i









NACA ACR No. L4119


.2 .4 .6 .8


x

(b) X 1.


CONFIDENTIAL


Figure 3.- Continued.


I____ ~___


Fig. ?It


C.- tF I iF I F TIAL








NACA ACR No. L4119
















.20






.15


-x.
dx


C'. .PIDi .'.TIAL


.2 .4 .6 .8 1.0


x

(C) X 2.

Figure 3.- Concluded.


CONFIDENTIAL


Fig. '







NACA ACR No. L4I19







2.0
I 1


1.6


-4-!-.3
3 -
1.2




S. I
De 8 [


CC N FIDENTIAL


0 .2 .4 .6 .8


CONFIDENTIAL
(a) x- .1
2
Figure 4.- P(x) P(1) against x for T 60.


1.0


Fig. 4-s







NACA ACR No. L4119


L0


I-.


--4--I--I--- -1--4---



S-_9 -. I "
"- < .. .. ... ... ... ..... i -.


7C.I4


I. i, I I

------------_




IinI


T ...
.05 -c^- m e-

SI _ii" i ,- '-,


.8 .9 1.0


x
(a) K -, Concluded.
2
Figure 4.- Continued.


CONFIDENTIAL


.20


.15





.10


CONFIDENTIAL


Fig. 4a C,-,r, .


.25 -


- --4 -.. .


,----~


I !







NACA ACR No. L4119









2.0










1.2



I .8
. 8 ,--i


CONFIDENTIAL


.2 .4 .6 .8 1.0


CONFIDENTIAL


(b) K) 1.


Figure 4.- Continued.


Fig. 4b








NACA ACR I.c. L4119


2.0





1.6





1.2



,--

1 W .8





.4





0


.2 .4 .6 .8 1.0

x
X CCrPFIDE;TIAL
(c) X-*2.


Figure 4.- Continued.


CN 1FI DENTIAL


Fig. 4e








NACA ACR No. L4119


.25






.20


.10






.05


.8 .9


1.0


(a) X a 2 Concluded,

Figure 4.- Concluded.


CONF I DEN TI AL


,:'CON F I DENT I AL


Fig. 4c C: r,.'.







NACA ACP NG. L4119


2.0





1.6





1.2





.8




.4


.2 .4 .6 .8 1.0


(a) X = .


CCN:FIL'E'.TIAL


Figure 5,- P(x) P(l) against x for T 1200,


CONFI DE T I AL


Fig. 5..







NACA ACR NO. L4119


h .\


0 ___'L--i-- -- --


-- ~ 7-~--- !


I
---rH-M---^-r-1-
r' i : I I





WfiAE IIS -I--.'



--I------, o-..mil ,MH. im<--.I .. i I-"-\- ---


I ~*&' I


1.0


() 1 CONFIDENTIAL
(a) a Concluded. CFIENT
2
Figure 5.- Continued.


'....- ^ .. ....... .... .. ..--- .-,
i ; i
I-~\--r-- -1-- ---
i . .. ., 1 .
,----- ---- -- ------- -- --- --- -- --- ---- i
f ,i,


.20


.15


.10





.05


Fig. 5- :., .


!
, I


I


CONFIDENTIAL


I


~-.~.I .. I I








NACA ACR No. L4119


.2 .4 .6 .8 1.0


CONFIDENTIAL


(b) K m 1.


Figure 5,- Continued.


2.0





1.6





1.2





.8


0
0


Fig. 5b


CONFIDENTIAL








NACA ACR No. L4119











.20






.15
-4-








S9
--- -




.10 --






.05






0


CT''NFIEENTIAL


Fig. tL Cone.


.8 .9 1.0


CONFIDENTIAL


(b) X m I.- Concluded.

FIgure 5.- Continued.








NACA ACR No. L4119


2.0





1.6





1.2





.8


.2 .4 .6 .8 1.0


x

(c) X 2.


CONF IDENITIAL


Figure 5.- Continued.


CCNFIDENTIAL


Fig. 5 "







NACA ACR No. L4I19


CC ( NF I E r T I AL


Fig. 5c Cone.


.25




.20




.15


x
(a) X= 2 Conoluded.
Figure 5.- Concluded.


C,'F NFIDENTIAL


I. -











I> -
-1-- -- ---- -
a d (FX --A- --&
I -.
*.c \ .. ... .. -




iK


8 1.0 '- S -: -. -. :
::4- --^_-- -^ -= ^ S--'.--:, ;;"ib l-l. -"i> ":- *- -













K8 .. 1.0
---- ^- : i --T- ---
~ -- -- -- -- --- -- --.:" ^,,, --- --:- :- ---- .:: -

-i "GHM O "F Ity t~tt~~ ~ ~'--.,^< .^ "* : -

.8*I.


.10


.05




0


I;








NACA ACR No. L4119


.2 .4 .6 .8 1.0


x

(a) .
2


CC NF I 'E;NTIAL


Figure 6.- P(x) P(1) against x for T = 1800.


2.0





1.6





S1.2
14









.4





0


CO NFIDENTIAL


Fig. C L







NACA ACR No. L4119


F 'V1 V 7 7i


.15T
----------- --- ----- ----- I- -


-Ix I\ -. --
-^^--- ----:!1"
I






-I *--i----'-





i ., I .u p a ,."" 'i,
1 : ,1 AD $' -__ .


i I


4 !
-i _- ,_-_:


1 r:i;


x

(a) X I Concluded,
2
Figure 6,- Continued,


. FI.E AL


.20





.15


-vv-"


,4
14
I
'4
%


.10


.05





0
.8


1.0


Fig. rL Cone.


CONF IDENTIAL


* ;i


----!








NACA ACR No. L4119


.2 .4 .6 .8 1.0


(b) A = 1.


Figure 6.- Continued.


2.0





1.6





1.2





.8





.4





0


CONFIDENTIAL


Fig. 6b


-.FI[ F'I TIAL







NACA ACR No. L4119


.2 .4 .6 .8 1.0


(o) X 2.


7 '.FI- : TRIAL


Figure 6.- Continued.


2.0





1.6





1.2
a4




.8


0
0


C014 FIDE NTIAL


Fig. r-:









NACA ACR No. L4119


.20






.15





r4

.10

H




.05







0


.8 .9 1.0


x

(o) k = 2 Concluded. CONFIDr'.TIAL

Figure 6.- Concluded.


CCI. FIDENTIAL







NACA ACR No. L4119


2.0




1.6


1.2


.2 .4 .6 .8


1.0


x

(a) 1-.
2


.;.FI 'r TIAL


Figure 7.- P(x) P(1) against x for T 2400.


I_


Fig. 7a


CONFIDENTIAL








NACA ACR No. L4119


t-T-


[.1


,--,-I-- t-.-l-- .- --^---.- ------ -- --- i-- -,.1

S-1 -i- 1- 1 1


11
.20 090




..15 -- -----
--------- -_-

I F I
H- ---------- -v -

,.__ '._____


_ I




T-07


F L i L AD Is I


0O


CONFI: ; '.TIAL


.25 --


.05


n


.
.8


(a) K Concluded
2
Figure 7.- Continued.


Fig. 7a .' :..-.


,- -,--


I


CC,- FIDE, 'TIAL


I--i--i---








NACA ACR No. L4119


2.0





1.6





^ 1.2

I
%-.
x

L .8





.4





0
0


(b) X 1.

Figure 7.- Continued.


CONFIDENTIAL


.2 .4 .6 .8 1.0


CCr.NF ILE' NTIAL


Fig. "tI








NACA ACR No. L4119


2.0





1.6





1.2
r-q

I


' .8





.4





0


.2 .4 .6 .8


1.0


x

(c) X = 2.


Figure 7.- Concluded.

CONFIDENTIAL


CONFIDENTIAL


Fig. 7c









CONFIDENTIAL


.2 .4 .6 .8


1.0


CONFIDENTIAL X

(a) 2 2
Figure 8.- P(x) P(l) against x for T = 300.


2.0





1.6





1.2


N
'-
14


-.4


Fig. E-,


NACA ACR No.


L4119







NACA ACP No. L4I19


.35


.30





.25





.20


- I ----- --


c .. .. ..
--^A---.....

---. .. -- ... -- -..- -.
I 'I
I\\ J F : I


-- -- --- --- -- -------i










I < : -- |T ....
I ,,
.I F
-'--i- tS-- -- ,





S- ---
I f ------ --


-----r---N^----

-T -- -- -- -t -- ^ r -- ---


x
CONFIDENTIAL (a) Concluded.
2
Figure 8.- Continued.


CONFIDENTIAL


Fig. Pa Cone.







NACA ACR No. L4119


2.0




1.6




1.2




.8




.4




0




-.4


0 .2 .4 .6 .8


1.0


CONFIDENTIAL


(b) X 1.


Figure 8,- Continued.


CONFIDENTIAL


Fig. Et


r








NACA ACR No. L4119


2.0





1.6





1.2





.8


-.4


.2 .4 .6 .8


1.0


CONFIDENTIAL


(e) k 2.


Figure 8.- Concluded.


CONFIDENTIAL


Fig. -C








NACA ACR No. L4119


1.2




1.0




.8




.6


xdP
-
dx


.4




.2




n


0 .2 .4 .6 .8 1
x

(a) 600.
Figure 9.- Function -- against x for ~ .
Cd 2

CONFIDENTIAL


.0


-I ----- -- ----------..... --i-- ----..


---


o -r- :

-: I i -
_I I I, ,7
i I 78













NA ONRL VS
-., ,- -.-u .
";-I





-- --/-f- -- -- ""b-- ;' .iei
- -' --t- -, --...
____ _


CONFIDENTIAL


Fig. 9a


v








NACA ACR No. L4119


SdP
-x-
dx


0 .2 .4 .6 .8 1.0


(b) t 1200.

Figure 9.- Continued.


CONFIDENTIAL


CONPI DENTAL


Fig. 9b








NACA ACR No. L4119














1.0 -




.8




.6

dP
-x

.4




0.2





0 .2


CONFIDENTIAL


.4 .6 .8 1.0


(c) T 1800.

Figure 9.- Continued.


CONFIDENTIAL


Fig. 9c







NACA ACR No. L4119


0

-.2
OC


1.2




1.0




.8




.6


CONFIDENTIAL


.4 .6
x
(d) T : 2400.
Figu*a 9.- Continued.


1.0


L-----------------------

^F--(-------^

--- J-2 -


-- --^ 4 ----

_L----i- --4 ^ --l --^ ---- 4
-- 4 -- /- 4--^---------------






SI









'[IL LL Z it/ii

-'--- -'----=--'-- ',, ',- --- --
: i

1A


dP
-Zd
dx




!I


Fig. 9d


CONFIDENTIAL


I









NACA ACR No. L4I19


.2 .4 .6 .8 1.0


(e) T r 3000

Figure 9.- Concluded.

CONFIDENTIAL


1.6




1.2




.8


dx
dx


4




0




-.4
0


Fig. 9e


CONFIDENTIAL








NACA ACR No. L4I19


0 .2 .4 .6 .8 1.0

x

(a) 60.

Figure 10.- Function -x against x for X 1,
dx


:NF IDEE:TIAL


1.0




.8





.6





.4




.2




0


CONFIDENTIAL


Fig. 1Ca








NACA ACR No. L4119


.2 .4 .6 .8 1.0


(b) T 120U .

Figure 10.- Continued.


CONFIDENTIAL


1.0




.8




.6


dP
-x
dx


0
0


CONFIDENTIAL


Fig. 10b







NACA ACR No. L4119












1.0 j !


0 .2 .4 .6 .8 1.0


(e) T a 180.
Plgure 10.- Continued.


CONFIDENTIAL


CONFIDENTIAL


Fig. 10l








NACA ACR No. L4119












1.0


iko---4 !-,
I-I -- -

.8






P-

.4 i




.2 I 4


0
0


CONFIDENTIAL


Fig. lOd


.2 .4 .6 .8 1.0


(d) T = 2400.
Figure 10.- Continued.


CONFIDENTIAL








NACA ACR No. L4119


.2 .4 .6 .8 1.0


(e) T 300o

Figure 10.- Concluded.


: -;.FILF' T AL


1.2





.8


dF
-x
dx


.4





0





-.4


CONFIDENTIAL


Fig. 10e








NACA ACR No. L4119












1.0




.8





.6





.4


dP
-x
dx


0
0


CC'NFIL'ENTIAL


.2 .4 .6 .8


1 .
1.0


(a) Tr 600.

Figure 11.- Function -- against x for K 2.
dx


2-r.FIDE".TIAL


Fig. lla








NACA ACR No. L4119













1.0



I
r



.-6-------
.e34


dP
-x
dx


CONFIDENTIAL


Fig. ilb


Lr-- _


.2 .4 .6 .8 L.O


X

(b) T 1200.

Figure 11.- Continued.

CO I FIDE TIAL








NACA ACR No. L4119















.8




.6 -,


CONFIDENTIAL


.2 .4 .6 .8 1.0


(e) T 180.

Figure 11.- Continued.


CONFIDENTIAL


Fig. llc


dP
-x-
dx








NACA ACR No. L4I19











1.0





.8
-

i
.6 ---- -
.6 i


dP
dx
dx


CONFIDENTIAL


Fig. lid


.2 .4 .6 .8 1.0


(d) T 240.

Figure 11.- Continued.


CCrFIDE:TIAL


0








NACA ACR No. L4119


1.2




1.0


1+--- 'I
F-T --I
-i- -- ----- --


.Rh A-


xdP
d-x

.4




.2




G


-L N


iI-T V I _..
!/ ,' __ 1_____6 ,578


k !:... I
'I I


.-- -- ---..l- .... .Beoo




4-7--- 7 _---^ ----








SNA1 L N un.


0 .2


1.0


w r Ii I AI


x

(e) 1 300.
Figure 11.- Concluded.


~-3~A


CONFIDENTIAL


/ I i t I


i 1


1-


Fig. lle


V








NACA ACR No. L4119


2.0 I I







II


Hi ) -Ir -e o l = .









Figure 2.- Valu s of --- against x for




xd
hree-blade and pblae p elle rs.

Figure 12.-Valueo of T .dP against: x for
tee-blade and six--blade proallers.


COhFIDENTIAL


CONFIDENTIAL


Fig. 12d








NACA ACR No. L4119


2.0




1.6




1.2


z-x


0 .2 .4 .6 .8 1.0


(b) Three-blade propeller; 1.
Figure 12.- Continued.


COrFIDENrTIAL


Fig. 12t


CONFIDENTIAL








NACA ACR No. L4119












2.0




1.6 -__


.------ -
1--.---


0 -
0


CONFIDENTIAL


Fig. 12c


.2 .4 .6 .8 1.0


(a) Three-blade propeller; A 2,
Figure 12,- Continued.


C :,FIDENTIAL


S-x d








NACA ACR No. L4119


.2 .4 .6 .8 1.0


(d) Six-blade propeller; k .

Figure 12.- Continued.


COl. FII E'.TIAL


CONFIDENTIAL


Fig. 12d


dP
7-x-
** dx








NACA ACR No. L4119


.2 .4 .6 .8 1.0


(e) Six-blade propeller; X 1.
Figure 12.- Continued.


C:'i.FIHE; TIAL


CONFIDENTIAL


Fig. 12e








NACA ACR No. L4119


.2 .4 .6 .8 1.0


(f) Six-blade propeller; X 2.

Figure 12.- Concluded.


CC: FIDENTI AL


0
0


CONFIDENTIAL


Fig. 12f









NACA ACR No. L4119 Fig. 13




















,-( <
\-H
A










04a
0.






H W0
\0
_.4 0


r o
So
0 0
\ \0 0



o\ W \ -4

0 4
\C \, < 0-(
\I \.
0 \
\ \ 3 ^ d t
\ \ ~a)






-r \\ r





~~~~ v a






























































I


I
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UNIVERSITY OF FLORIDA

3 1262 08106 549 1


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l *LARY
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