INVESTIGATION ON THE EFFECT
OF THE PROPELLER RACE
UPON
THE PERFORMANCE OF RUDDER
( Hydrodynamical Research of Ship's Rudder )
The Second Report
By Shojiro OKADA, Ba. Eng. (Kogakushi)
The Technical Research Laboratory
Hitachi Shipbuilding & Engineering Co., Ltd.
January,
1959Lab.
...,CiseetrAc i
Techv
g°4JSCiirlt,Or
INVESTIGATION aN. THE EFFECT OF THE PROPELLER RACE
UPON THE PERFORMANCE OF RUDDER
(Hydrodynamical Research of Ships Rudder .
= The Second Report
By Shojiro OKADA, Ba.Eng.
The Technical Research Laboratory
Hitachi Shipbuilding & Engineering Ca. ,Ltd.
Table of Contents
Abstract
1. Introduction
2.. Theoretical Consideration 4
2.1 The force acting upon the rudder placed in the propeller race
2-2 Calculation of propeller race 13
2.3 Results of the numerical calculation of force acting upon the rudder
a
Experiments on models3.1 Experimental apparatus and method
3.2 Results of the measuring of propeller race
and the consideration for it. . 44
3,3, The Result of measuring of the force acting
upon the rudder in the propeller race and
its consideration -H.- -7- 6-0 Conclusion --- 95
K
1 3 5 33 40 40ABSTRACT
In the previous report (the First Report), the
per-formance of open rudder, particularly the effect of the
velocity of steering, was dealt with, clarifying its
various specific features. In this Second Report,
description will be made on the effect of the propeller
race bearing influences upon the rudder performance,
which is also the principal object of the present
inves-tigation, from both angles of theoretical calculation
and model experiments.
The points made clear as the result of theoretical
calculations are as follows:
The most important factor governing the force
acting upon the rudder in the propeller race is the
velocity increasing coefficient of the propeller race,
succeeding to the advance speed of the rudder and helm
angle, in the scope of the present series of theoretical
calculations. On the rudder, there is acting about 70
80% of the total energy of water which has been
accelerated.
The effect of flow angle of the propeller race
against rudder is insignificant in an instance such as
the present calculation where the rudder has a
symmetrical section.
Even when the longitudinal position of the rudder
may shift backwards from the position of the propeller
disc by 0.5-1.50R (R:radius of the propeller), no
significant effect will be produced on the force acting
upon the rudder.
In consequence of the experiments on models conducted
in the circulating channel, the following points have
been made clear:
The coefficient of increase of the normal force
of the rudder caused by the propeller race multiplies
in proportion to the 1.5 power of the slip-ratio of
of the propeller.
By the propeller race, the position of the
center of pressure travels forewards, and the amount
of travel will be the larger, as the slip-ratio is
the greater.
Even though the longitudinal position of rudder
may shift backwards from the propeller by 1.00-2.00R
(R:radius of the propeller), it may well be presumed
that the performance of the rudder will not be changed
to any significant degree.
When the position of the rudder is transversely
(port or starboard) biased against the center of the
propeller, or when there exists a marked difference
between the diameter of the propeller and the height
of the rudder, it is enabled to know about the
(2)
(c4
.01
"I - (a) (b)-quantitative tendency by making examination separating
the area upon which the propeller race directly affects, from the area other than the above.
(e) The effect of angular velocity of steering is
noted to be acting even in a rudder placed in the
propeller race.
1. INTRODUCTION
In the First Report of the present investigation,
the author described about the performance of open
rudder, and various factors affecting such performance.
Particularly in connection with the effect of angular
velocity of steering, he carried out theoretical
calculation along with the experiments on models in the
circulating channel, and by bringing both data into
comparison, the relevant characteristics have been
made clear.
In the present report, the problems pertaining to
the performance of the rudder located behind the
pro-peller, namely, the main subject of the present
investi-gation will be treated from both angles of theoretical
2. THEORETICAL CONSIDERATION
In discussing the performance of rudder behind the
propeller, it is ideal to consider on the basis of the
effects on the rudder produced by the resultant velocity
potential of the propeller and the rudder. However, a
problem of such mutual interference cannot be treated as
a stationary theory, hence in the present instance, the
effect which the propeller brings upon the rudder alone
will be considered on the assumption that the propeller
is not to be affected by the rudder in whatsoever way.
In case the rudder is located behind the propeller,
the velocity and flow angle acting upon respective parts
of the rudder may diversify in accordance with the
pro-peller race respectively. For the reason, this cannot
be handled as a problem of two-dimension in which the
rudder is assumed to have an infinite depth. On the
contrary, it should be treated as a three-dimeaional
problem that needs due consideration for the depth of
the rudder, and naturally, the trailing vortices should
also be given consideration as they are generated from
respective parts of the rudder.
Prandtl developed his "lifting line theory" in which
he presumed that trailing vortices flow infinitely
backward from both edges of bound vortex which is fixed
to the aerofoil, thereby to satisfy Helmholz Law that a
vortex does not terminate in the fluid, and that
series of vortices are consistent in their circulation
at any point of their section.
It is also a known fact that Prandtl theory holds
true no longer, when the aspect ratio of the aerofoil
gets smaller (generally below 3). To replace this
theory, the "lifting surface theory" made advancement,
and more recently, Mr.JonesMand Mr.Lawrence(2/developed
the theory applicable to the instances where the
aspect-ratio is small. When the aspect-ratio is so very small
as an instance of a rudder, it is suggested that these
improved theories should be applied. But these theories
accompany difficulties to strictly satisfy various
conditions attached to them, if ever complex propeller
race is to be treated. Hence "lifting line theory"
comes to be considered sufficiently simple and effective
to know the tendency of the force acting upon the rudder.
This is why the theory was adopted in starting with the
calculation in the present case.
2.1 The force acting upon the rudder placed in the
propeller race
In case of an aerofoil put in a uniform flow, the
"lifting line theory" assumes that the trailing vortex
disregarding the mutual effects of the trailing vortices
themselves. As shown in Fig.1, in the reverse direction
to the uniform flow u0, the axis x is asslimed, and let
the ads
be drawn in the direction of the height ofthe rudder, and the axis y,perpendicular to the x-z
plane. When the rudder is replaced with a vortex line
conforming to the axis z, the trailing vortices will
extend infinitely backward in the negative direction of
the axis x.
When FR represents the circulation of the rudder,
the intensity of the trailing vortices flowing out of
dF
the point z' is
dz,ozc
R and the velocity of the directionof y induced at the point z by the trailing vortices of
semi-infinite length is given by Biot-Savart theory as
follows:
1 dER
4K dz' z - z
Hence, by integrating this to the range of the whole
height of the rudder, the induced velocity at the point
z is found. dz' 1 b CITA Vi= ( 1 )
47rJrb
dz' z -z'
2where, b is the height of the rudder.
In general, when the velocity uo flows in against
the rudder, the velocity at z will be u, by the
afore-(6)
-mentioned induced velocity vi, whereas the angle of
incidence a decreases by al(Refer to Fig.2) In
respec-tive positions of the rudder, the force acting upon the
rudder will be PuirR, when the theorem of Kuttd-Joukowski
in 2-dimentional aerofoil theory is applied on the
assumption that the stream is 2-dimentional, and its'
direction is perpendicular to ul. On the 'other hand
When the tangent of the lift coefficient CL against the
angle of incidence a' is assumed to be -cAL
k, the
circulation round the rudder
rx
will be expressed by thefollowing equation:
rR__,i-uock( a
=a1)
where* is the chord length of the rudder.
Hence,
i)
= iu ck ( ty - (2)
vo
Now, in
order to find the rudder performance of3-Fig.1
Fig .2
c
dimentions, it is necessary to find the FR for
respec-tive points of the rudder in the direction of the height,
or the distribution of circulation. For the purpose,
the integral equation consisting of (1) and (2) above
must be solved.
As this equation is rather hard to solve analytically,
the method to approximately solve the equation by
developing it into Fourier series is employed, although
there are in practice such methods as Mr.Glauert's(3),
Dr.Lotz's4), Prof.Tani's(5), etc. These methods are almost
similar to one another, and are considered that either
one will meet the requirement, but for the present, Mr.
Glauert's method is utilized:
Firstly, z and z' are set as follows:
Z = -
COS 0 ( o 0 )(3)
z'=---cose' ( <01<n.
)
2
z represents the coordinates as measured from the
center of the rudder upwards. They vary from 0 to it
the whole height of the rudder from the bottom
edge to the top, and so the circulation around the
rudder can be expressed in the Fourier series as
shown in the following equation:
FR = 2bUIAnsin nO (4)
The values of coefficient An are determined by the
(8)
III
two fundamental equations connecting rR and vl, and the
series should be chosen for the circulation TRso that
it satisfies the condition that the circulation falls to zero at the bottom and top edges of the rudder.
U in the equation (4) is the uniform flow assumed so as
to make it possible to calculate the induced velocity,
and then the induced velocity at the z point of the
rudder, that is, the point 01, is given by the following
equation:
U InAn cos n O' sin nO,
v1( 01) d 0 U InAn
7r4iu cos 6e-cos 0, sin 0
cos nO'd0' sin
no
dr7 =
cosOLcosp
sin 0
Accordingly, the equation below is established for
the Aeneral point z on the rudder:
visin 0 -U InAnsin n (6)
In this case, the direction of vi gives rise to
question, but it is considered most adequate to assume
that the direction of the vortex flowing out from the
rudder corresponds to that of the stream line.
However, on this assumption, the direction of the
vortex flowing out is caused to vary at respective
points along the direction of the height of the rudder,
whereby it is made entirely difficult to solve the
(5)
-above integral equation. In the present case, as it
seemed not to be deviating much from being adequate to
presume that the vortex flowing out from the rudder
would coincide with the mean of the directions of
incidence into the whole height of the rudder after
the vortex has flown pasta certain distance rearwards,
that is, the direction of the longitudinal center line
of the vessel, such a condition was decided to be
assumed for the present calculation. Then the direction
of vi is perpendicular to the longitudinal center line
of the vessel and also along the horizontal direction.
In this case, the components of the flow of the z
direction are of the values mutually offsetting between
the upper and lower halves of the rudder, and are
considered not to be affecting the normal force and
moment of the rudder, hence they are disregarded here.
In consequence, Fig.3 below shall replace the
afore-mentioned Fig.2:
Fig.3 (10)
u = mU + UsinwInAn
sine
(8)In this case, rR is FRiuick(cy-al), therefore, the
equations (7) and (8) are substituted, and so,
where, co =the angle that is constituted by the
longi-tudinal center line of the vessel and the
proprller race.
ai =the amount of diminution of the angle of
incidence by vi.
aR =the angle formed by the zero lift line to the longitudinal center line of the vessel, that is the helm angle.
Now, assuming that for the angles aR and co, the
anti-clockwise direction as measured from the longitudinal
center line of the vessel is positive, and also vi is to
show positive or negative depending upon the value of
(aR-co), the amount of diminution of the angle of
inci-dence by vi is expressed by the following equation:
v
cos w (7)
u1
If uo=mU,(m is to represent the velocity increasing
coefficient by the propeller race), the following will
be given:
u =u,
vi sin sosin no
-ck
sin n 0
ckmU FRUcoscpI RA.
2sin 0
2ck
sin n 0
±-UincoXnA
2sin 0
On the other hand, by the equation
( 4 ) ,FR=2bU ZA,, sin nO
and writing that the right side of both equations are
equal to each other, the following equation is obtained:
MAnsin nO (nit( cos yo - asinw) + sin
0 ) =m iice sin
(9)
ck
where,
=4b
By substituting the known. values for respective
points on the rudder, for example, p-number of points
on the rudder in the equation
( 9 ) ,as obtained above,
p sets of simultaneous equations are constructed, and
by solving them, the coefficient An can be determined
to
n =1p .
When Anis decided by this procedure, the lift aL and
the drag OD can be expressed by the following
equation:
o L =p b2 U2 { insin 11-41 sin nO
singo( Ink sin no ) (
Asin no )
(10)
OD= pb2U2 cos( lik,sin no )(
nAn sin nO ) dOThen, ON and OT will be worked out by the
equation
below:
(12)
-aN= aLcosce + eDsin a 1
(11) aT = aLsina - oDcoscy
When the distribution of the propeller race, or the
velocity increasing coefficient and flow angle
obtained by the above procedure is available, the force
acting upon the rudder in the propeller race can be
found.
2.2 Calculation of propeller race
There have been many publication as to theories
applicable to the calculation of the propeller.
Mr.Pollock(6), Dr.Horn(7), Dr.Kawada0, Dr.Kondoh(9), and many
other authorities, proposed their methods in which they
used the "simple blade element theory", or the Yvortex
theory", and as to whether or not the results of these
theoretical calculation represent the actual propeller
race of the vessel accurately, Dr.Yamagata(10, Mrs. Taniguchi and Watanabe", Mr.Leathard00, and others
carried out the measurement of the propeller race
partially, but there is still no data available which
are based on the results of systematic measurement in a
broad range, therefore it is rather difficult to make
presumption.
To ascertain the effect of the propeller race on the
number of blades, the mean value of the induced velocity
generated in the course of one revolution is worked out
by making use of the equation for the induced velocity
for a propeller with an infinite number of blades. It
is generally known that when the induced velocity at
the propeller disc is 1, the induced velocity at a
point infinitely backward will be 2 times as large.
The rudder is assumed to be located midway between the
propeller disc and the infinite rear, hence in order to
know what value of the induced velocity will be shown
at the rudder position in relation to the above value,
the following simple calculation was performed to
ascertain the velocity field in its rear as generated
by the bound vortex and trailing vortices of the
propeller as follows:
(1) Induced velocity due to the bound vortex
The induced velocity dy generated by the element of
line vortex dS which has the intensity of Fp at the
unit angle, will be as under-mentioned in accordance
with Biot-Savart theorem:
pp
J.S.r)
dv-47L- R'3
whereR'---\/(x-x' )2 + (y-y')2
+ ( z-z )2
r
is the vector from the position of bound vortexz) to the optional position(x,y,z,).
(14)
Accordingly, as expressed in terms of respective
When the number of blades of the screw propeller is
set as infinite, the bound vortices are distributed
over the disc centering the axis of the revolution of
the propeller, where the vortex has a radial axis . The
induced velocity by this vortex element is divided into
components of cylindrical coordinates as shown in Fig.4.
= -dv3 sin 0 +d-4 cos 0 dv!. = dq- cos 0 +d-v- sin 0
dx = 0 dyi = dr' cos 0' dz' = dr/ sin0' (14)
= cos'O zi = sin 0/
y= rcos0
z= rsin 0 hence, Fpdr/ d-cr; - rsin(0- ) 47r1V3 - F dr/ dV4 COS(0-- 0/ ) 471"R'3 (15) component,dv=
d..q dv; = it will PP be: dy' (z-e )
dz' ( ) dx1 3r-Yf ) - ( y-37' ) - dx' ( z-z/ ) - dyi ( x-x/ ) } (13) 477R'3 P 47uRf' 4-7L-R 3 dziFig.4
P
dr'x-x') sin( 0-0') 4nRi3
When this is integrated in terms of the whole surface
of the vortex, the induced velocity due to the bound
vortex is obtained as follows:
1
R-
cos 0'ITO - -f
rp(X-X1 )f2 dO' dr'47E-
0R' 3
In this case, ['is independent to 0, and is the
function of //only, the field of flow is symmetrical to
the axis of the revolution, and since
sin O'
r27r dO' = 0
Jo R/3
v, and vr are zero. In the equation (16), at x-x'
and (x-x' )_ -oc , namely, at the position of the
pro-peller disc and at the infinite rear,
v;
will alsotend to zero. In short, the induced velocity of bound
vortex is generated in the direction of circumference
(16) Fig.5 (16) ( IT A A \ x r )
alone, and this also disappears at the position of the
propeller disc and in the infinite rear.
When the equation (16) is rewritten in
non-dimen-tional form so as to facilitate the calculation, the
following will be given:
J1
2x cos 0, de'
rirp(
f
d5'
8 7r ° (
(C -
)2+ e2+ e2-2
ee'cos01(
17 )
where, V-
(n:revolutions, R:radius of propeller, nRU:advancing velocity )
pp
TP=
h U
(I-11=g, Q:velocity of revolution )x'
C =
ci=
(2) Induced velocity due to the trailing vortices
Since the trailing vortices are the vortex layer
distributed in the locus where the blades of propeller
have passed, they form a group of vortices with spiral
surface. Henceforth, in case of an infinite number of
propeller blades, a group of cylindrical surface of the
vortex may be assumed. Now, let dS stand for the
element of spiral line vortex, and ds its absolute
value(Refer to Fig.5).
dS =ids sine - ids cos e sin Y+kds cosecos 0' (18)
where,
1, j, k
are the unit vectors of the coordinates, r
-(x,y,z) respectively, and
e = tan-1-Dr'
Assuming that the circulation of trailing vortex at
the unit angle is di-, the components of the induced
velocity due to the element of line vortex will be given
by substituting the equation (18) in the equation (12).
dr; {-cosesinb(z-2 )-cosecos Hds -4nR13
dy =
dr cosecoso(x-X )-sine(z- )}ds 47c1R ' 3 dirz -dr {sinE(y-y ) ±cosE.,sino(x-5, );ds 47:R' 3where (2,9',2) and represent the orthogonal
coordinates of the trailing vortices and the cylindrical
coordinates, respectively. When this is rewritten in
the components of the cylindrical coordinate system, the
following equations will be given:
dv
-dr, cos e
r-rcos ( 64-) }ds
47rR'3
dip
dv =
x-i) cos ecos( 04,)-i'sinEsin( 04')
}ds42-cR'
drp
= e-(x-i)cosesin(
04)
+sine{r-i4cos(0-o)})ds42-cli'3
When this is integrated all over the trailing
(18)
( 19)
(20) 3
-vortices, the induced velocity due to the trailing
vortices will be obtained.
Namely, for ?, it suffices,
dr;
to integrate from 0 to R as drp
dr, for e from 0 to
fd
dr
27T as =
and for
from ,00 tO XI
case
However, in integrating
the diStribution density of
line vortex shOuld be taken into consideration.
The propeller advances for 27ch1 .during the time it
makes one revolution.,
The circulation of
trailingvortex
generated in the meantime is 271----e- dr, which is
dr
left behind in a straight line in the directiofi of the
axis x. Hence, the distribution density of line vortex
drp
in the direction of the axis x is
dr per unit
length.
Therefore, the induced velocity due to the
whole trailing vortex will be;
.1 dfp -t"COS
VX
4rhlo
0dr -00
R1.31 drp
,A(x-idicosecose+issin'esin0
rRr27-c
r
rdxdOd
47rh1 cosejo
di4 *I-00 R'3dr
k-X) sine case ±(r-i.cos id}sine)
=
-P
47rh, cose o
dr
x' IdddI
(21)
By use of the following formula,
=ds ,
rdxdOdr
-sin()
,dO = 0
R 3tx
dx
( )(2+ a2 )3xdx
JO Vi( x2±a2 )3 2 12+ a2a
Th/x2±a2the equation (21) is simplified to be,
1,
i''.,(-rcos)
-x-Y
R,
]
d'ddi"/R
- n/.
Vx1-
47rhisiodi'
.13r2+ r
-2ri'cos'o
n2 1
ad rp
1>'coslif-
, 47ch3710 dr o R 1 a dri; ( r - r c o s 0 ) X-x"47rJ0 dr ,10
r2+ i- 2`ri'c os 'ON 1 R/Further, making, use of
the formula,i
1
r
reosO
0,r<r
2ir 0
r2+-2r?'cos
r
(20),
d
(22),
and the consideration
is given td the .fact
that rp
disappears at the tip of
aerofoil, and so:
F Q
1 d Fpi.(r -
rcosro) 'x-x'
P ,de al- ( 25 ),i.,
r_._ _ ____2 ^2
^ 2U47ch1,0 dr 0
r2+ r -2rrcos0
R' 1 drpr - TCOS
'e X-XP,
,. rpvo.
-
CO' di-(26)
2r
& 47c 0dr 0
r2+^r2 -2ri-cos 0
Rf 1dr') ?C0S3
,, r 47rhjlo Rdi' jo
R'( 27)
101 It ( 23)(24)
-1
-This is rewritten in a non-dimentional form, as
follows:
= =
r2Ir
rp I drip
Fa
_ecos)
47'10 di -fo e2± 'e COS
c
- c'
x , dd'e (28)
(C-C' )2+ e2± e2
-2e e
cosrp drp - cos g
U 87r2-*/o dF Jr77
e2+
e "E cosio = c
- c'
d'eod ( 29 )Vcccf-f- e 2-42
-
2 e cos '61 1 ti drp r27, 'e cos gj
j.
dde
= U 47r 0de
v (C -C'e2 +
-2 e cos (3o) At the position of the propeller disc,rp
qx
-(31)J'rp
q0
47ff(3) Boundary condition and the characteristics of
the propeller
Regarding the boundary condition, similar to the
assumption as made at the time of the rudder, the
flow is approximated as two-dimentional for every
aerofoil element, and the lift acting upon it, is
assumed to have the lift coefficient CL which is
2-2
47re 2 2
'e
corresponding to the instance of the attack angle a
where e=arctg
S2/4 and the induced angle
au.
of the aerofoil are substracted from the pitch angle ap, namely, as given by the formula:
a=ap
e
---v, aip= u24_02 r2 e X =XWhen the equation (31) is utilized to compute alp , it
is given by:
Ji7p
crip- ( 32)
4;7e
When the solidity of blade element
T
( Number \x(chord
kof blade) llength)) is used, the lift at the 2nr
position of the radius r is:
P 2 9 9
L = --(U
) CL 2r.r dr2
On the other hand, from the Kutta-Joukowski theorem,
the following is obtained:
L=p(U2+Q2r2) Fp.2ndr
Therefore, assuming that both are equal, the following
equation will be given:
1VU2+ D2r2 r.T.0 =pp 2 L dCL . If is constant, da (22) (33) '1
As plotted in non-dimentional form, it is expressed as follows: 2 ''' 2
r =
P 2u
r dCL ( ap-e - ail) )da
1 )2-
1 TP= -2-o- ap -ar c s in
\/1-1- ( L-7re-)2 Vrp dCL 47re / da/1 42
7te It, /al) arcsin\ J'
r
1± (27r,e J ) 1±-1 4 (27re )2 dC, dBy the above procedure, the distribution of the
circulation of the propeller can be computed. (4) Numerical calculation
As already mentioned by the equation (17), the
induced velocity due to the bound vortex has been
found, and the induced velocity due to the trailing
vortex has also been obtained by the equations
(28)-(30), while as shown in the preceding paragraph, the calculating formula for the distribution of the
circulation of the propeller has further been
ascertained. By these data, the induced velocity at
(34)
(35) \
any optional point of 'the velocity field behind the
propeller can be calculated, but the integration by,-0'
aS given in the equation (17), (28)(300,is rather
difficult to perform in the form as they are given,
therefore in order to alter to a calculable form, the
elliptic function Of Legendre-Jacobi is used for
representation. If
1
_.I-ecos"6
C'e
M( e , =
°dO
4n-'0 e24\2-2e ecos 0 (c
e2±*2_
2.e-e cos-d4x 0 ( 36 ) 1 Nce cos d0 ( 371 \/(C-C' )4f2W-2eicos0 cos 0ce 1 jr27r
( 38)r
4n
° \AC
)2+e242-2aCOS
3then the equations to express the induced velocity can
be transformed as follows:
f-rp(c
o(
2n .0 -,1rp
d , -j °M( e , 2 0 de' Jen. drp E\Lmc F, e)cil
47re
27cilo d. d-rpe ?),
o de 0In the equation, (36)m writing as 'cos the
( 24 )
-= -
-
)= d
following is obtained by transformation: C- C' 1 du 27re .V/( e f v/(1.--u2 ) (1-k2 u2 ) ( 1 du 1.
(e±e )2 V( c_co2+ e+-E)2.1,
(1_,u2
where, k2 4eF cc
)2±(F±)2
±e)2 4e1(e+e)2
c
-NI( e,e) = k[(k)+ 2e V (Cc' )2+
(e4
)2 e_f,where, K:complete elliptic integral of the first
kind.
II: elliptic integral of the third kind.
k:modulus
c:parameter.
Of the equation (43), the elliptic integral of the
third kind is not calculable, therefore it is
trans-formed as follows:
7rdz
11 ( ,c,k)= f'
(i_cz2 )
i_z2 )(i_k2z2 )
In the equation (44), if z= snp , then
dp 11 =
f
2 K ° 1 - c snp and further, .7., ( -(43) (44)K c srfp dp
- K
Jr
2°
- C
sn -QHere, if c = k2sn'a ( a = const) , and is called
cn a dna go(K,a,k) ( H - K) sn a sn2p dp k2 sn a- cn a dn a 2 2 2 ° 1-k sn a sa p
then it will be known that there is the relative
equation:
hro(K,a,k) - H0 (a,K,k) K-E(a,k) - a-E(K,k)
where, E:elliptic integral of the End kind.
However, 2 sn p d-c II (a,K,k)--- sn K cn K-dn 2 2 2 - 0 0 1-k sn K snp and so sn a 11= en a dn a (K-E(a-k) - a-E)+K (45)
In this manner, the equation (44) is transformed as the
equation (45) , but a is an imaginary number, so it is
written as a K 1A(where A is a real number) . Thus,
-1>
sn a> 1.k=
1
sn a - sn(K+ iA) =
dn(A, )
where k' = complementary modulus .
By the above equation, A is rendered calculable,
( 2 6 ) K
and consequently a can be found.. 'From this, E( a ,k)
mill be worked out., but the coefficient of II of the
equation sn a cna .dn a. beforehand.. Writing then, sn a - = dn(Ak' cn a odn a Id2 -1) ( 1-d2 k2 ) sn iA - en IA E(a,k)= E(K+iA,k) =E(iA)+E- k2
:dn IA
This is transformed by Jacobi 's imaginary trans-formatiom;
ic-127-a
sn a
E( a )= A-E(A,kf
will have to be calculated
d N/l-d2k
} -FE
-\/(1-ek-( 46)
(4'7)
By substituting the ,equatiOnS, (46) .and (47) Th the.
equation K45) ji
will be
found as follows:a.
,
{K-EAW'
A(K-E))d2 -1) (1-d2k2 )
Substituting the equation (48) in the equation (43), 1 \2 where e
/
k C- )A--(e-4 )2 d ,\/d2 If= )-(48)
1 1N(e3)
where .1c2=
N(
(c - )2--F (e±? 7rV( C- C
In the same way
1 0( , e) = 7,r/(c_c, )2,4.:(e+e)2);:y 1
4ee
(28)-
c' e±,e)2:K(kri:{KE(A me,F
(49)where, in consideration of (C-
n
being negative, { iswith /
positive
to
complynegative
Next, in order to work out N(e, ) in the equation
(37), write. 2= cost similarly to the previous
instance and transform as follows:
(2z2-1)(az
(c,c1)2+(e-i=e)2j°
ril-Z21-k2Z2
2(K-V_ e-F)2 [ k2,-A(K-E)d
(50) where D.= 2D } 7C{(C )2 ± +.,e )2} kf2 K-E4ee
k 2 c')2 e-'-e' )12Accordingly, by substituting, respective equations (491
(.51) in the equations. (39)-1(42), induced velocity is
converted calculable.. ( 51) 111 II k2k/2 E-k'21C -2 1
of
1 1 , (e-
} 2K
The above calculation is applied to the propeller
of two kind. By way of example, an instance where
a propeller of Troost B°4-40 type with the pitch
ratio of 0.800 is working at the advance coefficient
_ U
of
=0.60, is given in Fig.6, illustrating thepropeller race at respective points at 0.7R in the
rear of the propeller disc. In Fig 6, c =
'
namely R
the non-dimentional value representing the distance
rearwards from the propeller disc in the direction
of x is plotted along the axis of abscissa, and the
ratio of the induced velocity at respective point to
the uniform flow (the velocity of flow as before the
incidence against the propeller), gx, go, gio and Cll.,
on the axis of ordinates.
According to the figure, the induced velocity qx
of the direction of x as caused by the trailing
vortex is the greatest, go of the direction of 0
comes next, and other two components are rather ^
small. Among them, gr gives a negative value,
indi-cating the presence of the flow of contraction.
Next, when the values as on the propeller disc
(ix and go, which are directly related if the
perfor-mance of the rudder is to be considered, should be
given as 1 respectively, the values of gx and go at
in Fig.?, in which these ratios, namely,
ix
x )at C ----0
( 61 )
at C
are plotted along the axis of ordinates. It is
generally known that the induced velocity at the
infinite rear is two times of that at the position of
the propeller disc, but the status of variations of the
induced velocity between the two points is as
illust-rated in Fig.?.
In the same figure, besides the results of
calcula-ion for the Troost-type propeller (increasing type),
those of the decreasing type propeller of the vessel
wY-Maru', who is actually in service, is also indicated
as obtained in the practical working condition (advance
coefficient:0.69). Even in the case of mutually
different distribution of the propeller pitch, no
significant difference is noticed in connection with
the status of variations in Eix and ie. Therefore, the
induced velocity of the propeller in an ordinary single
screw ship at the position of the rudder (ordinary,
c
--0.5--0.7) may well be considered as io to be about
1.7-1.85 times and ix to be about 1.60-1.75 times of
the induced velocity as at the disc of the propeller.
Now, with a view to finding the distribution of the
incidence velocity against the rudder, the propeller
10
/Or
Fig.6
The curves of induced velocity,
x
Fig.7
-15 .TROOST 54-40 TYPE P/D- 0.800 3-0.60 'Y"MARU J- 0 6q 0 0g%L.) - 0.800
J060J.
3 2 Lt. g00
TROOST B 4-40 TYPEDISTRIBUTION OF PROPELLER RACE
IC 0.5
00
0 05
RADIUS OF PROPELLER
OR
race in respective positions of longitudinal center
plane of vessel are worked out as shown in Fig.8 and 9.
Fig.8 represents the instance of the Troost type B-4-40
with an advance coefficient of J=0.60, and Fig.9, the
instance of the aforementioned "Y-Maru", where the
advance coefficient was J=0.69. In both figure, the
velocity increase coefficient in of the propeller race
at 4 sections behind the propeller, that is, sections
behind the propeller at the distances of 0.25R, 0.50R,
(32) < 3 1.5 2O0
/
. . Y MARU ---J- 069 m IT 10 5" 05 0 0.5 IOR RADIUS OF PROPELLER Fig.8 Fig.9 0.25 -- -- 50 --1501.00R, and 1.50R and the flow angle 49 to the center
line of the vessel are used to express the data.
According to them, the increase coefficient is the
greater as the position is further backward, whereas
the flow angle does not show any heavy variation.
2.3 Results of the numerical calculation of force
acting upon the rudder
In the preceding paragraph, the distribution of the
propeller race at respective points on the rudder as
in various rudder positions, have been made known. These values are to be substituted in the equation (9)
to set up the simultaneous equations. Now, in order
to know the tendency of the force acting upon the
rudder placed in the propeller race, calculations were
carried out with the following five instances. In
the rear of the Troost B-4-40 type propeller with 200
mm diameter and P/D= 0.800, operating at the advance
coefficient of J=0.60, the same rudder as used in the
experiments described in the First Report (N.A.C.A,
aerofoil
NO.0018
symmetrical rudder, with a height ofb=200 mm, and a chord length c==120 mm) was set with
its center of height conforming to the position of
the propeller shaft center, and when the gap between
at 15', 25°, and 35° respectively, and when the gap was
1.00R and 1050R, the rudder angle to be at
35'respec-tively.
In this calculation, the rudder was replaced with
the lifting line located at the point behind i of the
chord length from its leading edge, hence this position
of lifting line was to represent the position of the
rudder. As to the points to be selected on the rudder
in establishing the simultaneous equations, 9 points
were presumed to be sufficient including the top and
bottom edges. In respect of the tangent value k of
lift against the attack angle, it was decided to use
k=5.24 (See foot-note), which was obtained from the
results of the open rudder tests, about which report
was made in the First Report. By solving the
simulta-neous equations thus constructed, the coefficients A,
A2, 'A7 were found as shown in Table 1.
When An is obtained by the above procedure, oL,
op,
oN, and OT can be calculated by the use of the equations
(10) and (11), and then by applying these to the whole
height of the rudder, L, D, N, and T for respective
instances are found. As tabulated in the form of
Note: This value is slightly smaller than the value
found by K.E.Schoenherr (k= 5.46).
Table 2. C , a Ar'LN\
c=-0.5
a= 15'a= 25
c=-05
a= 35
c=-0.5
c=-1.0 a= 35° C=-1.5a= 35
Ai 0.1654 0.2756 0.3898 0.3927 0.3996 A2 0.0673 0.0627 0.0394 0.0549 0.0539 A, 0.0209 0.0348 0.0654 0.0518 0.0534 A4 -0.0148 -0.0142 -0.0279 -0.0147 -0.0129 A5 -0.0023 -0.0076 -0.0059 -0.0060 -0.0063 A6 0.0018 0.0020 0.0053 0.0026 0.0023 A7 0.0013 0.0023 0.0067 0.0027 0.0036 \C\\(\y C \, CL CD 1 CN CT -0.5 15 1.139 0.199 1.135 0.174 -0.5 25 1.895 0.441 1.890 0.507 -0.5 35 2.675 0.826 2.665 0.891 -1.0 35 2.760 0.869 2.753 0.934 -1.5 35 2.875 0.907 2.870 0.962respective coefficients (
C=
, where A: thep A
-area of rudder, U:uniform velocity, which flows into
the propeller, and so on), they are as indicated in
Table 2.
Table 1. Aii
By plotting ON and CD for respective rudder angles in
the
case of c=-0.5
along the axis of ordinate, the curveshown by the solid line in Fig.10 will be obtained. In
the same figure the experimental results in the case of
open rudder which was previously reported in the First
Report(rudder stock position ratio: h=0.25, and
V-0.99 m/s) are also given. By bringing them into
comparison, the condition of the increase of Cx and CD
by the effect of propeller race will clearly be seen.
In the experimental results, as it began to stall
around 20% CN decreased at any greater angle than this.
Meanwhile,
a stall is not
taken into consideration in thecalculation of ON in the propeller race, so ON is shown
as though continuing to increase even at greater angles
than above, making it no longer adequate to be the
object of comparison. Accordingly, ON was similarly
calculated for instances of open rudder without giving
consideration for a stall (The empirical value of 5.24
was used as tangent value k of lift against attack
angle), which result is given in the same figure in the
dotted line.
The tangent of CN against a in the case of open
dC
rudder --14(aspect ratio A =1.67) obtained by the above
da
calculation, almost coincides with the experimental
result, being 2.49. On the other hand, the tangent of
CN against a of the rudder as located at the point
C= -0.5 in the propeller race g.c, (aspect ratio: A==
1.67) is 4.34. Therefore, it will be learnt that the
normal force increases for about 74% by the effect of
the propeller race. This is, however, smaller than
the increasing coefficient for the force
It=
2.01which was calculated from the mean value of velocity
increase coefficients E(1.42) of the propeller race
at the position of the rudder as found in Fig.8. In
other words, according to this experimental result,
of the energy imparted to water by the propeller, the
energy at the position of the rudder is not
effec-tively utilized as a whole, but about 75% of the
whole energy is working effectively to the rudder.
On the other hand, comparison is made by use of
Table 2 in the changes of CN and CD between the
instances where the longitudinal position of the
rudder is changed as -0.5, -1.0 and -1.5. The further
the position of the rudder departs from the propeller,
the more both CN and CD would increase though slightly,
which fact agrees with the tendency of increase in the
velocity increasing coefficient m of the propeller
race.
RUDDER ANGLE
Fig.10
CN
C* iPA(TILT
f
the values for the cases where c are -0.5, -1.0 and
-1.5 respectively are plotted in Fig.11, which gives
almost constant values, showing that 0, is proportional
to the square of the increasing coefficient Fn. of the
propeller race.
Presuming that 0N in the propeller race increases in
almost linear proportion to the increase of the rudder
angle as shown in Fig.10, the effects of the angle of
(38) 14 13 1.2 1.1 10 -05 1.5 30
BEHIND PROPELLER (CAL.)
OPEN ( E X P ) (CAL.) 20 10 Cr
z
OPEN 20 30incidence c9 appear at respective points of the rudder
to ascertain degree when calculating for a rudder
having such a symmetrical section. But in case the
rudder is considered as a whole, these effects are
reverse mutually between respective points of upper
and lower portions, whereby only slight effects alone
are thought to manifest.
Generalizing the above results of calculation, the
following can be mentioned.
The most important factors governing the force
acting upon the rudder placed in the propeller race
are the advancing velocity, and rudder angle, then
follows the velocity increasing coefficient m in
the propeller race within the scope of this
calcu-lation. On the rudder, about 70-80% of the
acce-lerated energy of water is working effectively.
In case of a rudder having a symmetrical
section as used in this calculation, the effect of the angle of incidence is slight.
The condition of axial changes of the induced
velocity of the propeller taking place in its rear,
does not vary significantly even when the pitch of
propeller may gradually decrease or increase. Fig.7
may serve to represent this axial changes for the
propeller of an ordinary vessel.
(4) Even when the longitudinal position of the
rudder may shift for 0.50-1.5R(R:radius of propeller)
behind the propeller, the normal force and drag will
not be much affected.
3. EXPERIMENTS ON MODELS
3.1 Experimental apparatus and method
;rust as in the case of First Report, the present
series of experiments were conducted at the circulating
channel at the technical research laboratory of the
Hitachi Shipbuilding and Engineering Co.,Ltd.
Regarding the dynamometer for the measurement of the
forces and moments acting upon the rudder, detailed
description was made in the First Report, therefore it
will not be repeated here, and instead explanations
will be given here principally about the driving
apparatus for propeller and the 5-furcated Pitotis tube
for measuring the propeller race.
For the purpose of driving the propeller, a
column-shaped displacing body having a stream-lined section
as given in Fig.12 was fixed at the center of the
measuring section in the circulating channel. The
propeller was fitted at the rear end of the propeller
shaft projecting horizontally rearward from this
displacing body for about 0.8m. (40)
The fore-end of this propeller shaft was engaged with
a bevel gear through a thrust block. On the other
hand, inside the column along the vertical direction,
another shaft was provided, the lower end of which was
engaged with the propeller shaft by the medium of a
set of bevel gear, while its upper end was connected
with a ill) driving motor by the medium of pully and
belting. The revolutions of the propeller were
counted by the revolutions of the another shaft which
were reduced to 1/20 of the revolutions of the
pro-peller shaft by means of a worm and a worm gear,
utilizing the system of making and breaking electric
circuit which was related to each revolution of the
another shaft, so as to give the revolutions of the
propeller.
In order to examine the race of a propeller working
at definite revolutions (definite advance coefficients)
by a driving apparatus such as above-mentioned, the
5-furcated Pitotis tube as illustrated in Fig.1 was
used. This had an analogous form to that which was
used by Mrs.Taniguchi and Watanabe(11) to measure the
race of the propeller fitted at the stern of a model
ship. It is provided with a static pressure tube in
the center and 4 pieces of total pressure tubes were
right and left sides of the central tube, each of which
was inclining towards the center at an angle of 45%
Namely, this 5-furcated Pitot's tube consisted of 5
pressure measuring tubes combined in one.
The calibration of the 5-furcated tube was carried
out at the towing tank of the Naval Architecture
Depart-ment of the Faculty of Engineering, Osaka University.
The 5-furcated tube was mounted at an inclination of
given angle against the advancing direction of the
towing carriage, and the indication of respective
pressure tubes were read as the carriage advanced at a
given speed. In this case, the angle of inclination
was set at every 5' up and downwards and towards the right and left for the range from C--30 'including the
angle at which they were combined to the both
direc-tions. If the differences in pressure between the
upper, lower, right and left total pressure tubes, and
the static pressure tube are written as
4, y_, z
z_, and the sum total of all the four as M, then m
is considered to be greatly affected by angular
deviation in vertical direction, and y+-Y- by angular
deviation in horizontal direction. Hence, the
cali-bration curves obtained were as shown in Fig.14, as
these data were plotted along the axes of ordinates and
abscissa, by which it was possible to learn the angular
(42)
deviation of both directions. Regarding the velocity
of incidence, since the value of M(M0) at the time
when the two angular deviations are both 0% is
pro-portional to the square of the velocity, so the ratio
of M at respective deviations to Mo were put together
into the calibration curves as shown in Fig.15, from
which the velocity of incidence was to be calculated.
In the actual survey of the propeller race, the
5-furcated Pitot's tube was mounted on a base which
could be moved up and down, and also to the right and
left at a low pitch so as to facilitate the 5-furcated
tube to shift to respective measuring position as
might be required, and every consideration was given
so that the velocity of flow within any required section which was at a right angle to the mean flow
could be measured speedily and handily. The status
of actual measuring operation of the propeller race was as shown in the photograph of Fig.16.
Prior to the measuring of the propeller race, with
a view to checking whether the effect of wake
generated by the pillar-shaped displacing body mounted
in front of the propeller so as to drive it, was
extending to the propeller position, the distribution
of the flow velocity was determined with the section
at a distance of 100mm behind the propeller, by
-conducting the measurement with the propeller removed.
From the results of these measurements, it was affirmed
that there would be no objection to regard the flow
distribution as uniform, except that at localized points
the flow was observed to be inclined to the center line
of the channel at an angle of about If- 2°. Therefore
necessary correction was made for such inclination with
the results of measurement of the propeller race
enforced thereafter.
3.2 Result of the measuring of propeller race and
the consideration for it
In order to facilitate the comparison with the
results of the numerical calculation as carried out in
the previous Chapter, the measuring of the propeller
race was also carried out by use of a propeller of
Troost B.4-40 type, having a diameter of D= 200=4).
Including the instance of the pitch ratio of P/D=0.800
and the advance coefficient of J=0.60, of which
calculation was made previously, with the propeller
having a pitch ratio of P/D= 0.800, the 3 advance
coefficients of ,T= 0.40, 0.60 and 0.80 were chosen,
and with the propeller having a pitch ratio of 1.000,
0.40, 0.60, 0.80 and 1.00 were selected, and the
measuring was performed with 4 sections as at 0.25R,
0.50R, 1.00R and 1.5R (R= radius of the propeller)
behind the propeller. However, inasmuch as the Troost
type propeller used in this case had a blade rake
angle, the propeller surface adopted was the section
that was at a right angle to the propeller shaft, and
which also included the intersecting point of the
maxi-mum thickness line of the aerofoil section at 0.7R
with the pitch surface.
The measured points included in a section were 97
in number within a square of 250mmx 250mm as shown in
Fig.17. Fig.17-19 indicate examples of the results
of measuring. These figures were prepared in the form
of resultant vector of vertical and transverse
compo-nents, vz and vy respectively of the velocity at
respective measuring points at the time when the
propeller was viewed from its rear side. Fig.20 is
intended to show the status of distribution of the
ratio of the resultant velocity composed not only of
the vy and vz 9 but also of the axial component vx to
the general velocity of incidence. (The same method
of representation was employed as in the case of m
utilized in the numerical calculation of the propeller
race in the preceding paragraph.) By the figure, it
is clearly known that the velocity around 0.5--0.6R
is most accelerated.
-Fig.17--19 are indicative of the measuring results
when a propeller with pitch ratio of 0.800 was working
at the advance coefficient of J=0.60, over the
sections at 0.25R, 0.50R and 1.00R behind the propeller
respectively. According to these figures, the
circum-ferenceially accelerated condition of water can well be
observed. The direction of flow at every point outside
the projected area of the propeller's actuator disc, is
pointing more towards the center of the propeller than
the direction of the tangent of the cocentric circle
which passes the point. This fact gives exact idea
that the propeller race is a flow of contraction. At
the section at 0.25R directly behind the propeller, the
centripedal velocity is at the largest, and it decreases
in intensity as it gets further backward. This may be
deemed to show that the intensity of the flow of
contraction is at its maximum at the point most nearly
behind the propeller.
With a view to ascertaining what changes the induced
velocity generated by the propeller will undergo, as it
departs farther from the propeller disc, the induced
velocity at 0.7R of respective measured sections were
computed from the results of these experimental surveys,
-and represented in the form of
4.,
go and q, accordingto the similar method of representation as used in the
numerical calculation in the preceding Chapter, that
, in the form of the ratio of respective induced
velocity to general uniform flow, as shown in Fig.21.
In the figure, the results of numerical calculations
as mentioned in the preceding Chapter, are given as
well. The comparison of these data reveals that the
empirical values of qx, ir are presented larger
than calculated values near the propeller. This may
be due to the lifting line theory us ei in the
calcu-lation. As it moves farther backward, the values of
,
q0+% and qr shows a tendency of approaching the
Fig.13 The apparatus to measure the race
of Propeller.
^
values of numerical calculation, but qx alone is large
in its value when it is close to the propeller, and
diminishes as it goes farther back behind the
pro-peller, which is a trend quite different from the
calculated value. However, within the extent of the
present experiments, respective induced velocity
settles down to a constant value at positions farther
-70
11411
MMEOMOMMIN mommMiumm HOOMMUNIO IIMINOMMumen IIMOMMOOOON UUOMMOOm/Ailiktik
2' 2 0 g-toFig.14 Calibration diagram for the
5-furcated Pitot's tube.
backward than 1.00R.
On the other hand, in order to ascertain what
distribution the propeller race makes in its radial
direction , or in other words, what changes it shows
for the direction of the height of the rudder, the
radius of propeller was plotted by the axis of
abscissa, and the velocity increasing coefficient m
and the deviated angle of the flow (the deviated
Fig.15 Calibration diagram for the
5-furcated Pitot's tube.
Fig. 16
.7-514°-.
The Model Propeller and its race
Measuring apparatus.,..
(:50
OA 111ALSM
I
II
rdi
1111
oolb
r,Pli
Mieladd
Sill
TiOOST B.4-40 TYPE
D-200'1 moo
GO-2514, 7a60
NOTE:FIGLIRE SHOWS THE RESULTANT VELOCITY VECTOR°
OF AND 1 DIRECTION, VIEWED FORE WARDS
FROM THE AFT. AND THE SCALE OF VECTORS AT THE MEASURED POINTS ARE SHOWN IN PER-CENTAGE OF THE RESULTANT VELOCITY OF THREE
DIRECTIONS, OF WHICH SCALE SHOWN IN FiGua
SCALE OF THE RESULTANT VECTOR
Fig.19 The result of propeller race measurement(1)1
F/D=0.800. GAP- 50mm, 3=0.60
( 51 )
Yig.18 The result o propeller race measurement(2)
rim
itirgistimbs
wins_ainnom
.
i
Ei_
mrAwaringiiiiim
NEVINAMMEIBIN
inemigioninn
11111MLIMENVA
18111.*S11,
Mi
,1
Fig.19 The result of propeller race measurement(3). GAP i002'1, 3=0.60
nummoimmi
eiraiteraim
111110111M161111111
arMill1111110111
minimmanri
sammonams
IRILIEMIP/11=1111
111111111111101111111111
II= NM=
,
gdrA
ZILI\
1112H1111111M11111
NNW 1,111ri
rjFr
-iirtramorow
Fig.20 The distribution diagram of in
(P/D= 0.800, J=0.60, GAP= 25rnm)
1-<craz,
<DT
<cr-4
Fig.21 Experimental results of induced velocity
compared with theoretical ones..
FA,-=0.800. J-CALCULATION 'EXPER PENT CO- 0.25R 05OR' 1.00R 5 0 R 05 'RADIUS OF ,PROPELLER
Fig.22: The measured results of propeller race.
00
EXPERIMENTED CALCULATED 05 DR RADIUS OF PROPELLER (54) 10 10 5 0.500
D- 0 800, 3- 0.65 RADIUS OF PROPELLER GAP' -0 .3t/P 86R ---- I.36R 5, e 15 LS %- 0.800. J- 0 60, GAP - 0 50 R M /..---i/i
/ `.... ..."...N 1 10 1.0i\
\\
5 0.5Fig.23 Measured results of Fig.24 The diagram of the
propeller race com- propeller race
pared with calcu- according to ,T.F.
lated race. Leathard's experiments.
and respective measuring points) by the axis of
ordi-nates, and so Fig.22 was obtained. As shown in Fig.20,
it is clear that the velocity around 0.6R is most
accelerated. With a view to making contrast with the
result of calculation, the data in the case of J=0.60,
at gap= 0.50R were written in the same figure which
the velocity increasing coefficient in and the
variation in the angle of incidence co are fairly
con-forming to each other, while in their absolute values,
in is smaller in the empirical value, but co is larger
in the same value.
With regard to the result of measuring of propeller
race of this Troost B4-40 type propeller with a pitch
ratio of 0.800, Mr.J.F.Leathard published his data as
he obtained by use of hot wire with a propeller of 8"
(208mm) dia in a state of J=0.65. As rewritten in
the similar form to Fig.22, it is as given in Fig.24.
However, the author's experiment was conducted at
J-0.60, while Mr.Leathard's at T 0.65. Moreover, there
is a difference in the position of the section used
for measuring, thereby making it impracticable to make
comparison in a strict sense. Even then the author
dared to bring the two data into comparison inasmuch
as there had been no available data of this kind so
far as he knew.
Since the author's data show a smaller advance
coefficient, that is, its slip ratio is larger, it is
quite natural that the velocity increase coefficient in
is larger. With respect to the position where the
maximum value of in is shown, it is also considered
were measured at the nearly point to the optimum
efficiency, shows its presence in the proximity of 0.7R,
whereas the author's data gives it to be located at a
position nearer to the propeller boss than the former.
Furthermore, both data are found well conforming to
each other in that the velocity increase coefficient
directly behind the propeller is larger than those
which are farther in the rear of the propeller.
In the author's experiment, J was made to change in
the same propeller, therefore the mean value of
velocity increase coefficient Ft of the section at
1.00R in the rear of propeller was plotted along the V
axis of ordinates and the slip ratio s ( s=
1-nP
V:advancing velocity of the propeller, n:revolution,
P:pitch) along the axis of abscissa, whereby Fig.25
was obtained. In this case, in consideration of its
effect on the rudder, the mean velocity increasing
coefficient of the radial direction is used as M.
When the aforementioned Mr.Leathard's data as of the
section at 0.86R behind the propeller are marked in the
same figure with
"x"
mark, they give values a littlesmaller than the results of the present experiment, yet
it will be noted that they can be said to be in good
conformity.
In Fig.25, the experimental results in the case of
(56)
1.0
.3
A ( SUP RATIO OF THE PROPELLER)
Fig.25
s-7, diagram.
15 I.0 .7 .5 .20 RESULTS OF THIS EXP.
x RESUCT OF IF LEATHARD'S ExP. -2 .3 .4 -5 .CEXPERIMENTED THEORETICAL 02 3 .5 .7 1-0
the pitch ratio of 0.800 and that of 1.000 are
con-jointly given. This reveals that in an instance of the
Pitch ratio difference at this degree, the mean of
velocity increase coefficient may be regarded to be
almost on a common level when it is viewed in terms of
slip ratio s.
Besides, when the increased portion of the mean of
velocity increase coefficient M, that is
m= -1
isconsidered and plotted against the slip-ratio s on
logarithmic cross-section paper and so Fig.26 will be
obtained. On the other hand, when Am is calculated
in accordance with the "simple momentum theory",
dis-regarding the radial velocity and its changes,
- (52)
1 - s
When this is rewritten in Fig.26, it comes forth as the
theoretical curve in the figure with 'which the
empiri-cal values compare a little smaller, but they may be
said to be in a very good agreement with each other in
their general trend. Within the scope of s= 0.2-0.6
which is also within the range of the practical
condition in ordinary vessels, the curve in Fig.26
indicates an almost straight line, hence the following
can be established as empirical formula:
(58)
(52)1
I
Lm= 2.4,6 s1-5
Summarizing the above results of the comparison
between the results of measurement of propeller race
and the simple approximate calculation in accordance
with the "lifting line theory" , the following
con-clusion will be reached.
Among various induced velocities generated by
the propeller, the general trend of the results of
measurement of the radial component CI', and the
cir-cumferential components ol-4-q coincide well with the
results of calculation, but the axial component
ix
does not correspond, 'where the velocity directly
behind the Propeller is greater than that at position
farther backward.
The status of the radial distribution the
velocity increasing coefficients in and the angle of
incidence 92 show a tendency of conformity between
the experimental data and the results of calculation,
but in in the calculated values are found a little
larger than experimental value, while in So, the latter values are a little larger.
Regarding the propeller race, at the point
backward for about the propeller radius from the
propeller disc, it reaches an approximately constant
value, and for the extent farther backward by another
propeller radius, the value may well be assumed to
remain constant.
(4) The velocity increasing coefficient will be the
higher, as the slip ratio is the greater. The increased
portion of the velocity well corresponds to the result
of the calculation by the "simple momentum theory".
3.3 The result of measuring of the force acting upon
the rudder in the propeller race and its
consideration
The measuring conducted for the purpose of knowing
the effects of propeller race on the rudder performance
was carried out on the undermentioned conditions, so
that the calculation of the propeller race and the
results of actual measuring can be used for reference
The data of the rudder and propeller used are as
mentioned in Table 3 below.
Table 3
(60) Speed of flow at the time of
experiment about 0.8 m/sec
Propeller used Troost B.4-40 type
Rudder used
Form of section NACA-0018 type
aspect ratio 1.67
Position of axis
of rudder
0.30 the chord length from the
leading edge
-I
-Note: As to the size of the propeller and
rudder, a propeller with a diameter of
200mm and a rudder with a height of 200mm
were used in usual experiments with
exception of special cases.
Experimental Conditions
Firstly, in order to know about the effect of
variations in the advance coefficient J, for the
propeller with 200=0 and P/D= 0.800, three conditions
were chosen, namely, J=0.40, 0.60 and 0.80, and for
the propeller with the same diameter but P/D 1.000,
4 conditions, namely, J= 0.40, 0.60, 0.80 and 1.00.
So as to ascertain the effects of changes of
the rudder position along the longitudinal direction,
( the rudder position is represented by the point
behind i the chord length from its leading edge.),
or the effects of changes of the gap between the
propeller and the rudder, the gap/R (R: the propeller
radius) was changed in three ways, that is, 0.60,
1.00 and 1.50 with the each of the 7 instances
above-mentioned, respectively, that is 21 conditions in
total, to which one more condition was added, namely,
an instances where the propeller had only the boss
With each of these 22 conditions, the rudder angle was
set at every 5' for the range from CPto 400 onboth port
and starboard sides, and the measuring was carried out
as usual.
With a view to affirm the effects of bias
conditioryof the rudder towards transversal direction,
that is the instance where the rudder is not right in
the rear of the Propeller, the experiments were
con-ducted with the instances in which the rudder position
a
varied for (R:the radius of
3 ' 3
Propeller) in both Port and starboard directions
respectively.
In the abo'Ve experiments of (i), (ii) and (iii),
a model propeller with 200mm diameter, and a model
rudder of the light(h) 200mm xthe chord length(c) 120mm
x the thickness(t) 21.6mm vere used. Lastly, in order
to know about the effects of the propeller on rudders
with different sizes, three instances of different
rudder heights, that is, 200, 150 and 100mm, were
experimented with the 200mmo propeller, and another 3
instances of different rudder heights, that is, 200,
150 and 100mm, with the 100mmcb propeller, 6 different conditions in all.
For the purpose of investigating whether the
effect of the angular velocity of steering, about
(62)
(iii)
R and R
(i70
which full description was given in the First Report,
could also be witnessed with a rudder in the propeller
race, a several experiment was conducted with the
instance of P/D =0.800.
Concerning the non-dimentional form of
represen-tation of these experimental result, there was employed
where,
V : Uniform velocity which flows into the
propeller (m/s)
p : Density of water, A
: Area of rudder(M),
: Chord length of rudder (m).
the similar procedure to the instances appearing
the First Report, that is, as used in the
Resultant force coefficient
Cp-Non_al force coefficient
CN-Tangential force coefficient CT
Lift coefficient
Drag coefficient CD
Moment coefficient CIE
-equation in (53).