REPORT No. 776 THE THEORY OF PROPELLERS. II-METHOD FOR CALCULATING THE AXIAL INTERFERENCE VELOCITY By THEODOEI+JTHEODOESEN SUMMARY A technical methodis given for cukulutina the axi.a.?inter- jertwce vekwity of a propiiler. - The m&d kvolveg the we of certdn weightfu.nctiarwP, Q, and F. Numemk+?wuhLes for i%aweight junctwna are given jor two-blude, i%ree-bid, and six-bladepropelkm. INTRODUCTION lt has formerly been the practice to use the G1auert-Lock simplified assumption that the interference velocity is pro- portional to the loading at the point considered. This assumption is obviously inadequate since the interference flow depends on the slope and curvature of the loading func- tion as well as on the local magnitude. A method is developed herein for calculating the axial interference flow for any loading. The method is accurate to the fit order and therefore gives the interference flow in ratio to the loading for small loadings. It can be shown that this accuracy is adequate for all technicid applications. The present paper is the second in a series on the theory of propellers. Part I deals with a method for obtaining the circulation function for dual-rotating propellers (See reference 1.) U1 w v P n w r K z Z1 o h R P, (z) g(z) Q r SYMBOLS axial interference velocity at ZI [v.(zl)] rearward displacbrnent veloci~ of helical vortex surface (at infinity) advance velocity of propeller number of blades order number of blade (Os ns p— 1) angular velocity of propeller circulation at radius x (-+) circulation coefficient to first order ~Trww nondimensional radius in terms of tip radius reference point at which interference velocity is calculated angular distance of vortex element from propeller advance ratio (V/coR) tip radius of propeller function defined in equation (1) function defined in equation (3) used for PI(x) in tables and figures; ref era to other blades (n#O) used for Q1(z) in tablea and figures; refers to blade it9elf (n=O) () ~n= phase angle of nth blade — P P1 ‘ehmgleat’@%) WEIGHT FUNCTION P,(z) It can be shown that the axial interference flow is given by where the summation is over the number of blades O to p– 1. The important function PI(z) is deiined as J m P,(x) = o d zkP+’’’-’+31+(al where n=Oj 1, 2, . . . p— 1, the number of the particular blade. The problem is thus essentially solved by giving the function P,(z) for each point along the radhw It is convenient to make P,(z) finite by subtracting a quantity that is independent of z. The function PI(z) may therefore be redefined as J . P,(z) = o [ 1 ( 31+(9 CF+X,’-2IZ, Cos e+ (1) It is noticed that, in the integral P, (z), the integrand changes from + o to — o at Z=Z, for 0= O. This difEculty, which occurs only for n’= O (that is, for the blade itself), is overcome in the following manner: The expression (2) which is integrable and equal to may be subtracted horn P1(z) to yield a finite and smooth 53
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REPORT No. 776
THE THEORY OF PROPELLERS. II-METHOD FOR CALCULATING THE AXIAL INTERFERENCEVELOCITY
By THEODOEI+JTHEODOESEN
SUMMARY
A technical method is given for cukulutina the axi.a.?inter-jertwce vekwity of a propiiler. - The m&d kvolveg the we ofcertdn weightfu.nctiarwP, Q, and F. Numemk+?wuhLesfori%aweight junctwna are given jor two-blude, i%ree-bid, andsix-bladepropelkm.
INTRODUCTION
lt has formerly been the practice to use the G1auert-Locksimplified assumption that the interference velocity is pro-portional to the loading at the point considered. Thisassumption is obviously inadequate since the interferenceflow depends on the slope and curvature of the loading func-tion as well as on the local magnitude. A method isdeveloped herein for calculating the axial interference flowfor any loading. The method is accurate to the fit orderand therefore gives the interference flow in ratio to theloading for small loadings. It can be shown that thisaccuracy is adequate for all technicid applications.
The present paper is the second in a series on the theoryof propellers. Part I deals with a method for obtaining thecirculation function for dual-rotating propellers (Seereference 1.)
U1
w
vPnwr
K
zZ1
ohRP, (z)g(z)
Q
r
SYMBOLS
axial interference velocity at ZI [v.(zl)]rearward displacbrnent veloci~ of helical vortex
surface (at infinity)advance velocity of propellernumber of bladesorder number of blade (Os ns p— 1)angular velocity of propellercirculation at radius x
(-+)circulation coefficient to first order ~Trww
nondimensional radius in terms of tip radiusreference point at which interference velocity is
calculatedangular distance of vortex element from propelleradvance ratio (V/coR)tip radius of propellerfunction defined in equation (1)function defined in equation (3)used for PI(x) in tables and figures; ref era to other
blades (n#O)used for Q1(z) in tablea and figures; refers to blade
it9elf (n=O)
()
~n=phase angle of nth blade —
P
P1 ‘ehmgleat’@%)WEIGHT FUNCTION P,(z)
It can be shown that the axial interference flow is given by
where the summation is over the number of blades O top– 1. The important function PI(z) is deiined as
Jm
P,(x) =o d zkP+’’’-’+31+(al
where n=Oj 1, 2, . . . p— 1, the number of the particularblade. The problem is thus essentially solved by giving thefunction P,(z) for each point along the radhw
It is convenient to make P,(z) finite by subtracting aquantity that is independent of z. The function PI(z) maytherefore be redefined as
J.
P,(z) =o [ 1( 31+(9CF+X,’-2IZ, Cos e+
(1)
It is noticed that, in the integral P, (z), the integrand changesfrom + o to — o at Z=Z, for 0= O. This difEculty, whichoccurs only for n’= O (that is, for the blade itself), is overcomein the following manner: The expression
(2)
which is integrable and equal to
may be subtracted horn P1(z) to yield a finite and smooth53
., —-. — .I
54. REPORT NO. 77 6—NATIONAL ADVISORY
integrand. Thusj by subtraction, a quantity
‘72%=E7F (3)
is obtained. Finally, for the blade itself (n =0),
P,(z) =Q,(z) +Fwhere
The integral Ql(z) is convenient for graphical integration antiis, in fact, small in comparison with the function F.
No discontinuities arise in the P functions for the oth~bladea (n# O). The P functions are therefore used directlyin the cdcnlation for the other blades. It should be notddthat the functions P, Q, and F are all symmetrical in z
ddxl. The use of the subscript, which has been used to in -cate reference to the point xl, is therefore discontinued. %the following discussion, the functions Q and .F refer to theblade itself and P refers to the other blades.
Since the weight function is needed in the form z%, it ~
written asdP dQ dF
‘h– ‘% ‘Zd;
It is to be noted that by far the largeat contribution com~from the logarithmic function F since it really represents t~eentire field in the neighborhood of the point considered. Indeveloped form,
dF%Z’–
d ‘kA+%$T310g’z-’1’ ‘4)NUMERICAL EVALUATION J& ~GHT FUNCTIONS Q, +’,
IThe weight functions Q, F, and P are shown in a series bf
tables and figures. The first step of integrating against @e
angle Ois omitted for simplicity.The ~nctions ~Q and dF’
dx &have been obtained by graphical diHerentiation of the Qand P functions with actual calculation at the end pointsz= O and 1 for accuracy. It should be noted that thekefunctions and their derivatives are continuous and smooth.The results are given in the following order:(1) Table I and figure 1: Q against z (O= z= 1.00; O.IMM~
zl~ 1.00; k=~, 1, and 2), obtained from equation (~)
(2) Table II and figure 2: ~ against z (O~z~ 1.00;
dQ .0.1564sz,S 1.00; X=$ 1, and 2), where ~ B ob-
COMMJ?M?EE FOR AERONAUTICS’
tained by graphical differentiation of Q except for
.=0 and 1, for which $$ is obtained analytically
(3) Table III and figure 3: –~ against z (0s zs 1.00;
1, and 2), obtained from equation (4)Table V: P againstz for T=60° (OSZS 1.00; 0.1564S
Zls 1.00; A=~~ 1, and 2), obtained from equation (1)
Figure 4: P(z)–P(l) against z for 7=60° (0S ZS 1.00:
0.1564~z,S 1.00; k=~) 1, and 2)
Table VI: same as table V for T= 120°Figure 5: same as figure 4 for T= 120°Table VII: same as table V for 7= 180°Figure 6: same as figure 4 for r= 180°Table VIII: same as table V for r=240°Fi
Ye 7: same as figure 4 for r=240°
Ta le IX: same as table V for 7=300°Figure 8: same as figure 4 for 7=300°
Table X: ~ against . for A=; (T=600, 120°, 180°,
240°, and 300°; z=O and 1.00; 0.1564SWS 1.00),obtained analytically
Table ICC: same as “table X for ~= 1Table XII: same as table X for A=2
Table X1l.1 and figure 9: –x ~ against z for A=;
(~=60°, 120°,180°,240°, and 300°; 0.1564s Z= 1.00;0.1564S z,= 1.00), obtained by multiplying vahms intable X by –z
Table XIV and figure 10: same as table XIII andfigure 9 for X=1
Table XV and figure 11: same as table XIII and figure9 for ~=2
Table XVI and figure 12: ~–z ~ against x for three-
blade and six-blade propellers (~= 120° and 240° forthree-blade propeller; 7= 60°, 120°, 180°, 240°, and300° for six-blade propeller; 0.1564 S z= 1.00;
OSZ,S 1.00; ~=~~ 1, and 2); it may be noted thmt
these values for two-blade propellers me given bydP— for T= 180° in tables XIII to XV and in
‘x dziigures 9 to 11
APPLICATION OF METHOD
Steps to obtain the induced velocity expressed M ~ are
a9 follows: 2
(1) Plot the quantity z $$ against the circulation coo5-
cient K and perform graphically the integration
THEORY OF ~ROpDLLERS. II-METHOD FOR CALCULATING THE AXIAL INTERFERE NCE VELOCITY 55
(2rL) Plot similarly the functions &~ against K and per-
1form the integration
——
szd;dK
imm zdF~ becomes irdinite at z=zI, it is necessaxy to exclude
kgap from xI–~Ax to zl+~~ and to consider this gap
separately by use of a Taylor expansion.1 (2b) The contribution from the gap AX becomes
‘=-’[z’K”+(’-iclO@l&l&where
AZ=21Z-XII
b=ib=ti“xl’
c=A2+z?=c0s2~1~nd K’ and K“ are the derivntivea of K with respect to z., (3) Finally, there is rLcontribution from the other bladea.
This contribution is obtained by plotting x~ against K for
the other blades. Since the value
directly from the tables, this workwith a single graphical integration
JZ&ix
~–za~ can be taken
contains only one step
By addition of the results of steps (1) to (3), the totalinterference veloci~ VI in the axial direction is obtained.The relationship between the axial interference velocity fi atthe radius z, to the axial displacement velocity w of the~ortex sheet may be seen from the sketch in figure 13. Therelation is
ol=& cd$bl
cr, conversely, the displacement velocity w of the vortexehect may be obtained from the calculated axial interference~elocity al by the relation
1 VI-w=—2 Cosldl
WhiCII giveE the axial displacement velocity at the propellerdisk, I?or the case of the ideal loading this axial displace-ment velocity must come out as a constant, thus permittinga check on the weight functions. Cases of nonideal loadingam evidently of more practical concern. It is the purposeof this paper to give a method for calculation of the tialinterferimce and displacement velocity for any (light)loading,
LANGLEY MEMORIAL AERONAUTICAL LABORATORY.
NATIONAL ADVISORY Cohmrrmm FOR AERONAUTICS,
LANGLEY l’IELD, VA., September 19, 1944.
REFERENCE
Theodcmen,Theodore: The Theory of Propellem Indetermina-tion of the Circulation Funotion and the Mass Coeffloient for
Dual-Rotating Propellers. NACA Rep. No. 775 1944.
.10
0
Q
-.10
-.20
.08 I I I I I
.04
0
1 \ 1 1 I 1
\
-.04 R\hl \
Q \ Q \ I
–.08 Y ‘ .45@7\
\1
‘ .5878–.12
L Q ,.7071
\ \ \-.16 .8090
-20
-.
Q
-.
+./2 I I I
o .2 .4 6 .8 Lox
(a) + (b) x-l. (0) A-z
FIQUM L—FMotton Q a@mt r.
;. ----- ____
56 REPORT NO. 77 6—NATIONAL ADVISORY COMMT13?EE FOR AERONAUTICS
I..3090
(a) +.
o
— .xl = o./5&?
>-. / ~ — — _
— , .30 %7
dQ / - ‘z=
L_ ,,.4540
/ “> .5878
:2’ ~ — — — — p . –---- --.7071
—>.89/0
–.3
2 .4 .6 x
(b) A-l.
ha- Z—Fundbn dQ/&agahut z.
THEORY OF PROPELLERS. II-METHOD FOR CALCULATING THE AXCAL INTERFERENCE VELOCITY 57
-a
x
(a) A-:.
FIQUEE 3.—Fmlction -rdEQagdnst%
-x
.30
xl = /.O(X27
..25 - ‘.. I] .<951 I
/
/
/.8090
.20/
dQ. / / ‘z
d ~/ “, /.5378
./5 -A ~. ‘ / /
.4540
Ay / /’
{ /
/
/ / ““ / ‘ .3 030
./0 4
d / ‘. //
/ / / ‘r / / . /&w/
-m -/ ‘ / ‘
0)o .2 -4 .6 .8 f.o
x(b) X-L
Fmmm a—omltilmed.
.
——. . . . ._.—....— ----- . . .. . ...—-—-— .
58 REPORT NO. 77 6—NATIONAL ADVISORY COMMTITEE FOR AERONAUTICS
.15 -I I
x= = /.0000
.951fA ‘
/ / .8090
./0-/
/.5878
-+$ A/ / I
.4540/ / ~
.05- .3CA0I~
.1564— I
(c)
o .2 .4 .6 .8 LO
Y
- .80:0
.4‘- “y ‘
\
~ ~ ~ 1
\
I1.0000-
o ..2 .4 6 .8 /.0
(a) A+.Fmmm 4—P(z) —P(l) -r for 7-OF.
.20 \
\ \xl = 0./564
.3090\
.15-
\\
.5878~ \
\ \
~Q
/.00> > \
N \
\
.\
h.05- K \ \ \
\ \
o(4
-8 .9 10
x
(b) x-l. (o) A-a.
Pmwrm 4.—Chtlnmd.
I \ I 1 I 1 I 1 i , 1\
20a+ ‘= O.;564
\
\\3090 \
.15- \
-
y ,0
~%
s \
/.00% \\\ \
I
.05p111t
1 1 I
(c)
.8 .9 Loz
(8) x+-condnded.FMUEE4-onntInRMI.
x
(o) A-!tOamfnded.
Fmwm 4.-c!QnohkL
THEORY OF PROPELLERS. II-METHOD FOR C&LCIJLA T13TG KKE AXIAL RWW@ERENCE VELOCITY
.4
I (4o .2 .4 .6 .8 /.0
x
(a) ?.-*.FIQuEE6.—P(z) -P(1) againatz for 7B1ZP.
iv\
z, =0./564
J5 -\\
..?090>
~ Jo \ \
N . \\ \
\.05-
\\
o(a)
.8 .9 . /.0x
(a) A-@onclndwL
Fmum 6.J2adirrned.
x(b) A-1–Ccmolndeii
~GUEE 6.~thnHL
.95/F.4 Is
(CJ
Q 2 .4 .6 .8 /.0x
(c) X=2~GUEE 5.-cOUthn&i.
60
._—. .. .-k-. . . . . . . . —— . . .
REPORT NO. 77 6—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
20
\
UI = 0./5641.6
/
J70so2L2 y \
y
~\
\\
1= \090
4 -% ,1 \\ ‘ \
— -1.000s -\- ~
~
(a)
o .2 .4 .6 .8 {.0r
(4A+.FIcturuzIL-P(.Z)-P(l) againstx m T-lEm.
27
\-xl= Q 1564
.15-’ \
.3090’~ \
. \~Jo
\
v. :8090L \
<\
\ \\ .
.05-
\
(a)o.8 9 Lo
x
(a) A-@andndeiL
Fmmm 6.-ChtinrM.
20
1.6 \
~ \
T<.8
0 .2 .4 .6 .8 /.0x
(b) A=LFIGVBE 6.-ConIhm&i.
20
1.6 \
ZI.z
y\
z.30
~ .8i.
\
4.95/l-’
0 .2 .4 .6 .8 /.0x
(0) ?.-2
FmuEE 6.—I3onthmwl.
.20
--xl = 0.1564
.15-\\
.309P\\
y,,.\ \
\ .:5878\
T . 4?0> \k- X
\I
\ \.05- I Y
/.0000.’ ~\ \
\\ \
\
(c)
‘8 .9 10x
(a)Ad-Conoludeil.
FrmrRrr 6.-Concladed.
20
1.6
\
ZL 2
~_s
.58,7.LI,
# Ia909~
.4
1
(a)
o .2 .4 .6 .8 /.0.-
(a) +.
Fmum 7.—P(z) -P(l) against r for T-MT.
THEORY OF PROPELLERS. J1-MJ?PHOD FOR CALCULAT12W THE bxIAL INTERFERE NCE VELOCITY 61
a-
(a) A.@amlndd.
lhmm 7.4onthmi.
I95( &
1.Oqoo’ wo .2 .-4 .6 .8 /-0
x
(b) .X-L
Fmurm 7.-Contfnnod.
20
\
-xI = 0./5 64
I.q \
\
,
i?.X.6
. .:5878b ,
.B040\
.4 - L
.95) P’●
(c)
o 2 .4 .6 .8 /.0r
(o) ,X-2FIGUEE 7.-C!mdada.i.
843110-o~
.?4
20 o
-----x= = o./ 5s#
\ .1.6
/.2 / \
49090\
~ .8\
T\
,x — — \ \
4
/45878
/ I.8090-. / ‘
— —
o/ - 1
951 A / ; ~
>/ ‘
-4 6 -(al
0 ..2 .4 .6 .8 LOz
(a) +.
FIamt f+P(z)–P(l) againstr for l-m.
.30 \
..-\
-.2?1= Q9090
\
\
.25 - +78 78\
>./5W. -
Y \\ \.20 \ \ \
\
.80 9> \
7.15 -\
\ \\\
x \ \
K\ \
JO35 1/ T ~
/ -Y
.C5 -\
I.000aI 1 (4
O& 9 /.0x
(a)x-ij-kMML~13UmS.-C0ntinmi
62
...~,. ..: -— .. -..1 . . ——— . . . . . . —
REPORT NO. 77 6—NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS