166
A N A P P R O X I M A T E POWER P R E D I C T I O N M E T H O D
by
J. H o l t r o p * and G.G.J. Mennen*
1. I n t r o d u c t i o n
additional
R TR
pressure resistance
o f immersed
transom stern
I n a recent p u b l i c a t i o n [ 1 ] a statistical m e t h o d was
presented f o r the determination o f the required pro-
model-ship correlation resistance.
R,
pulsive power at the i n i t i a l design stage o f a ship. This
m e t h o d was developed t h r o u g h a regression
of
random
model
experiments
F o r the f o r m factor o f the h u l l the p r e d i c t i o n f o r -
analysis
and full-scale data,
mula:
available at the Netherlands Ship Model Basin. Because
'
1 + ^ j = C l 3 {0.93
+ c^^{B
ILj^f-''^^''^
the accuracy o f the m e t h o d was reported to be insufficient when unconventional
(0.95 - C ^ r O - ^ l ' t ^ S (1 _
combinations o f main
+ 0.0225 Icbf-^^^^
}
parameters were used, an attempt was made to extend
the m e t h o d by adjusting the original numerical predic-
can be used.
t i o n model t o test data obtained i n some specific cases.
I n this f o r m u l a Cp is the prismatic c o e f f i c i e n t based
This adaptation o f the m e t h o d has resulted i n t o a set
on the waterline length L and Icb is the l o n g i t u d i n a l
o f p r e d i c t i o n formulae w i t h a wider range o f applica-
position o f the centre o f buoyancy f o r w a r d o f Q.5L as
t i o n . Nevertheless, i t should be noticed that the given
a percentage o f L. I n the f o r m - f a c t o r f o r m u l a
modifications have a tentative character o n l y , because
parameter reflecting the length o f the r u n according
the adjustments are based on a small number o f ex-
to:
is a
periments. I n any case, the application is Umited to
Lj^lL=\-Cp
h u l l f o r m s resembhng the average ship described by
the m a i n dimensions and f o r m coefficients used i n the
+ 0.06 CplcbliA
The c o e f f i c i e n t
= (r/i)°-2228446
method.
The extension o f the m e t h o d was focussed o n i m -
defined as:
^ h e n TIL > 0.05
= 4 8 . 2 0 ( 7 / 1 - 0.02)^•''^^ + 0.479948
proving the power p r e d i c t i o n o f high-block ships w i t h
when 0.02 < r / Z < 0.05
l o w Z / Ö - r a t i o s and o f slender naval ships w i t h a complex appendage arrangement and immersed
In
Some parts o f this study were carried o u t i n the
scope o f the N S M B Co-operative Research programme.
The adaptation
o f the m e t h o d
to naval ships was
when T/L < 0.02
= 0.479948
transom
stems.
C p - l )
this f o r m u l a T is the average m o u l d e d
draught.
The c o e f f i c i e n t c^g accounts f o r the specific shape o f
the a f t e r b o d y and is related t o the c o e f f i c i e n t C^j^^j^ according t o :
carried o i i t i n a research study f o r the Royal Netherlands Navy. Permission
t o publish results
o f these
studies is g r a t e f u l l y acknowledged.
c,3 = l + 0 . 0 0 3 C^,,^
F o r the c o e f f i c i e n t C^j^^^^ the f o l l o w i n g
tentative
guidelines are given:
2. Resistance p r e d i c t i o n
The t o t a l resistance o f a ship has been subdivided
into:
Afterbody form
'^stern
K-shaped sections
-
N o r m a l section shape
[/-shaped sections w i t h
Hogner stern
where:
Rp
f r i c t i o n a l resistance according to the I T T C 1957 f r i c t i o n f o r m u l a
I+ATJ
10
0
+ 10
The wetted area o f the h u l l can be a p p r o x i m a t e d
well by:
f o r m f a c t o r describing the viscous resistance
o f the h u l l f o r m i n relation to Rp
R^PP
resistance o f appendages
R^^i
wave-making and wave-breaking resistance
Rg
additional pressure resistance o f bulbous b o w
near the water surface
*) Netherlands Ship Model Basin, (Maiin), Wageningen, The Netherlands.
S = Li2T
+ B) V C ^ ( 0 . 4 5 3 + 0.4425 C^ +
- 0.2862 C^
+ 2.38Agj./Cg
In
this f o r m u l a
- 0.003467 B/T
+ 0.3696 C^p)
+
.
is the midship section coef-
f i c i e n t , Cg is the block c o e f f i c i e n t o n the basis o f the
167
waterline length L, C^^p is the waterplane area coeff i c i e n t and A^j.
c^
= B/L
is the transverse sectional area o f the
= I-0.8
The appendage resistance can be determined f r o m :
=
1.89
= exp(-
C2
sects the stem.
APP
when 0.11< .Ö/Z <
= 0.5 - 0.0625 L/B
bulb at the p o s i t i o n where the still-water surface inter-
R
.
when B/L > 0.25
VC3)
ApKBTC^)
I n these expressions
O.SpV^S^pp{l^k,X^Cp
0.25
is a parameter w h i c h accounts
f o r the r e d u c t i o n o f the wave resistance due t o the acwhere p is the water density, V the speed o f the ship,
t i o n o f a bulbous bow. Similarly,
S^PP
fluence o f a transom stern on the wave resistance. I n
the wetted area o f the appendages, 1 + k.^ the
appendage resistance f a c t o r and C „ the c o e f f i c i e n t o f
the expression Aj,
r
I n this figure the transverse area o f wedges placed at
1957 f o r m u l a .
the
Table
below tentative
1 + k.^
values
are
given f o r streamlined flow-oriented appendages. These
values were obtained f r o m resistance tests w i t h bare
and appended
turbulence
represents the immersed part o f
the transverse area o f the transom at zero speed.
f r i c t i o n a l resistance o f the ship according t o the I T T C -
In
expresses the i n -
ship models. I n several o f these tests
stimulators were present at the
the transom chine should be included.
In
the f o r m u l a f o r the wave resistance,
other parameters can be determined f r o m :
leadmg
1.446 C„ -0.03
L/B
when
edges t o induce t u r b u l e n t f l o w over the appendages.
\
= 1.446
A p p r o x i m a t e 1 + /c, values
rudder behind stern
1.3 -- 1.5
twin-screw balance rudders
shaft brackets
2.8
3.0
skeg
1.5 -- 2.0
strut bossings
3.0
2.0
2.0 --4.0
2.8
2.7
1.4
h u l l bossings
shafts
stabilizer fins
dome
bilge keels
Cp
-0.36
L/B < 12
w h e n Z / 5 > 12
= 0.0140407 L / r - 1.75254 V ^ ' ' V Z +
- 4.79323
-Cjg
1.5 -- 2.0,
rudder behind skeg
F^^ is the
Froude number based on the waterline length L. The
= 8.07981 Cp - 13.8673
c^g = 1.73014- 0.7067
m^=
C | exp(-0.1
The coefficient
512, whereas
+ 6.984388 C |
when Cp <
0.80
w h e n Cp >
0.80
F-^)
is equal t o - 1.69385 f o r i ^ / V <
=O.OforZ,Vv>
1727.
F o r values o f 512 < Z ^ / V < 1727,
is determined
from:
= - 1.69385 + (L/sj 1 / 3 - 8.0)/2.36
The equivalent 1 + k.^ value f o r a c o m b i n a t i o n o f
d=-0.9
appendages is determined f r o m :
The h a l f angle o f entrance i^. is the angle o f the
2^eq
waterline at the b o w i n degrees w i t h reference t o the
yr.
centre plane b u t neglecting the local shape at the stem.
The appendage resistance can be increased b y the
resistance o f b o w thruster t u n n e l openings according
I f ip, is u n k n o w n , use can be made o f the f o l l o w i n g
formula:
to:
'
= 1 + 89 e x p { - ( i / 5 ) ' ' - 8 0 8 S 6 (1 _ c^^^^.30484
(1 - C ^ - 0.0225 / c ö ) 0 "67(L^/5)0-34574
where d is the tunnel diameter.
(100
The c o e f f i c i e n t C g ^ ^ ranges f r o m
0.003 t o 0.012.
y/L^f-^^^°^}
For
openings i n the cylindrical part o f a bulbous b o w the
This f o r m u l a , obtained by regression analysis o f over
lower figures should be used.
200 h u l l shapes, yields ip. values between 1° and 90°.
The wave resistance is determined f r o m :
The original equation i n [1]
sometimes resulted i n
negative ip, values f o r exceptional c o m b i n a t i o n s o f
h u l l - f o r m parameters.
with:
Cj
The c o e f f i c i e n t that determines the i n f l u e n c e o f the
=
2223105 C 7 " 8 6 1 3 ( r / £ ) 1
= 0.229577 {B/Lf-^^^^^
07961
^,-^)-1.37565
when B/L < 0.11
bulbous b o w on the wave resistance is d e f i n e d as;
C3 = 0.56 4 - | / { 5 r ( 0 . 3 1
s/AT^+Tp
168
where /7„ is the position o f the centre o f the transverse area A^j,
above the keel line and Tp is the for-
increase
= (0.105 kjl^
I n these formulae L and
- 0.005579)/i
are given i n metres.
ward draught o f the ship.
The additional resistance due to the presence o f a
3. Prediction o f propulsion factors
bulbous bow near the surface is determined f r o m :
The
Rj, = 0.11 e x p ( - 3 P - 2 ) F 3 . 4 - | p ^ / ( l
where the coefficient
statistical p r e d i c t i o n formulae f o r estimating
the effective wake f r a c t i o n , the thrust deduction frac-
is a measure f o r the emer-
gence o f the bow and F^^^ is the Froude number based
t i o n and the relative-rotative efficiency as presented i n
[ 1 ] could be i m p r o v e d on several points.
F o r single-screw
on the immersion:
shipS' w i t h a conventional stern
ar-
rangement the f o l l o w i n g adapted f o r m u l a f o r the wake
Pg =0.56VA^KTp
- 1.5
hp)
f r a c t i o n can be used:
and
= V/^g(Tp
w = cg C j , ^ 0.0661875 + 1.21756 c^^
T^\
U -
- hp - 0.25 s / Z ^ ) + 0.15
I n a similar way the additional pressure resistance
+ 0.24558
due to the immersed transom can be determined:
0.5
R
The
+
pV'ApC^
B
0.09726
L{\-Cp^)
,
— ^
J
0.11434
0.95-Cp
0.95-Cp
C,,,^„C^+0.002 c
stem
coefficient Cg has been related to the
Froude
The c o e f f i c i e n t Cg depends on a coefficient Cg defined
as;
number based on the transom i m m e r s i o n :
Cg = BS/{LDT^
when F^^j. < 5
cg = 0 . 2 ( 1 - 0 . 2 F „ ^ )
)
when B/T^
< 5
or
or
c^=S{lB/T^
- 25)KLD(B/T^
- 3))
when F^^p ^ 5
cg=0
when B/T^
F^^P has been defined as:
=
CQ
V/^2gApKB+BC^^p)
w h e n Co < 28
= Co
or
when Cg > 28
Cg = 3 2 - 16/(Cg - 2 4 )
I n this d e f i n i t i o n C^^,p is the waterplane area c o e f f i cient.
when
The model-ship correlation resistance R^
R^
> 5
with
T^/D<2
or
Cji = 0 . 0 8 3 3 3 3 3 ( 7 ^ / i 3 ) 3 + 1.33333
=V2pV^SC^
when
T./D>2
I n the f o r m u l a f o r the wake f r a c t i o n , Cy
is the vis-
is supposed to describe p r i m a r i l y the e f f e c t o f the h u l l
roughness and the still-air resistance. F r o m an analysis
o f results o f speed trials, w h i c h have been corrected to
ideal t r i a l conditions, the f o l l o w i n g f o r m u l a f o r the
correlation allowance c o e f f i c i e n t
was f o u n d :
cous resistance c o e f f i c i e n t w i t h Cj^ = (1 + A:)
+
.
Further:
Cp^ = 1.45 C^ - 0.315 - 0.0225 Icb .
C,
= 0.006(7: + 1 0 0 ) - ° - i ^ - 0.00205 +
I n a similar manner the f o l l o w i n g approximate f o r + 0.003^/LpJC^
€2(0.04-c^)
mula f o r the thrust deduction f o r single-screw ships
w i t h a conventional stern can be applied:
with
C4 =
when Tp/L
Tp/L
^
0.04
t=0.00\919
L/{B
-BCp^)+
- 0.00524 - 0.1418
w h e n T p / i > 0.04
C4 = 0.04
I n addition,
might be increased t o calculate e.g.
1.0585 c^^ +
D^liBT)
+ 0.0015 C^j^^„
The c o e f f i c i e n t CJQ is defined as:
-10
when L/B > 5.2
B/L
the e f f e c t o f a larger h u l l roughness than standard. T o
this end the ITTC-1978 f o r m u l a t i o n can be used f r o m
w h i c h the increase o f
can be derived f o r roughness
values higher than the standard figure o f
(mean apparent amplitude):
Cjo = 0.25 - 0 . 0 0 3 3 2 8 4 0 2 / ( 5 / Z -
0.134615385)
when Z / 5 <
= 150 jum
The
relative-rotative
efficiency can be
5.2
predicted
169
well by the original f o r m u l a :
Cg 75 is the c h o r d length at a radius o f 75 per cent and
= 0.9922 - 0.05908'Ap/A^
Z is the number o f blades.
+
+ 0.07424(C^ - 0.0225
Because the
Icb)
A C ^ = ( 2 + 4(^^)0
log(co.75//^p))-"}
formulae above apply t o ships w i t h a
conventional stern an attempt has been made t o i n -
{0.003605 - ( 1 . 8 9 + 1.62
I n this f o r m u l a t/c is the thickness-chordlength ratio
dicate a tentative f o r m u l a t i o n f o r the propulsion fac-
and
tors o f single-screw ships w i t h an open stern as applied
For
sometimes on slender, fast sailing ships:
used as a standard figure f o r new propehers.
w = 0.3Cp
+ \OCyCg
is the propeller blade surface roughness.
this roughness the value o f
= 0.00003 m is
T h e chord length and the thickness-chordlength ratio
-0.1
can be estimated using the f o l l o w i n g empirical f o r / = 0.10
and
T)^ = 0 . 9 8 .
mulae:
These values are based on o n l y a very l i m i t e d n u m -
c^j^
=
2.013(Ap/A^)D/Z
ber o f model data. The influence o f the fullness and
the viscous resistance c o e f f i c i e n t has been.expressed
i n a similar way as i n the original p r e d i c t i o n formulae
and
(?/c)o.75= (0.0185 - 0.00125 Z)Dlc^^^
.
f o r twin-screw ships. These original f o r m u l a e f o r t w i n The blade area ratio can be determined f r o m e.g.
screw ships are:
Keller's f o r m u l a :
w = 0.3095 Cp + lOCyCp
t = 0.325 Cp - 0.1885
- 0.23
Dls/W
AplA^
D/sfBT
=K+(1.3
+0.3
Z)T/(D\p^
+pgh-p^))
I n this f o r m u l a T is the propeller thrust,
Vj^ = 0.9737 + 0.11 l(Cp - 0.0225 Icb) +
+ pgh is
the static pressure at the shaft centre line, p^
is the
vapour pressure and TsT is a constant t o w h i c h the
- 0.06325 P/D
f o l l o w i n g figures a p p l y :
1
AT = 0 t o 0.1 f o r twin-screw ships
4. E s t i m a t i o n o f propeller efficiency
J5r= 0.2 f o r single-screw ships
F o r the p r e d i c t i o n o f the required propulsive power
the efficiency o f the propeller i n open-water c o n d i t i o n
has t o be determined. I t has appeared that the characteristics o f most propellers can be approximated w e l l
F o r sea water o f 15 degrees centigrade the value o f
p^ - p^ is 99047
N/m^.
The given p r e d i c t i o n equations are consistent w i t h a
shafting e f f i c i e n c y o f
b y using the results o f tests w i t h systematic propeller
series. I n [ 2 ] a p o l y n o m i a l representation is given o f
the
thrust
and
torque
coefficients o f the
B-series
propellers. These polynomials are valid, however, f o r a
Reynolds number o f 2.10^ and need t o be corrected
f o r the specific Reynolds number and the roughness
o f the actual propeller. The presented statistical pred i c t i o n equations
f o r the model-ship correlation al-
lowance
propulsion factors
and
the
are based o n
Reynolds and roughness corrections according t o the
and reflect ideal t r i a l conditions, i m p l y i n g ;
— no w i n d , waves and swell,
— deep water w i t h a density o f 1025 k g / m ^
and a
temperature o f 15 degrees centigrade and
— a clean h u l l and propeller w i t h a surface roughness
according t o m o d e r n standards.
The shaft power can n o w be determined f r o m :
I T T C - 1 9 7 8 m e t h o d , [ 3 ] . A c c o r d i n g t o this m e t h o d
the propeller thrust and torque coefficients are corrected according t o :
5. N u m e r i c a l example
^r-ship = ^r-5.senes +
^ ' ^ - ^ ^
The performance characteristics o f a h y p o t h e t i c a l
single-screw ship are calculated f o r a speed o f 25 k n o t s .
-^e-ship= -^e-B-series ^ ^^D
0.25-5:^
Here A C ^ is the difference i n drag c o e f f i c i e n t o f the
p r o f i l e section, P is the p i t c h o f the propeller and
The calculations are made f o r the various resistance
components and the p r o p u l s i o n factors, successively.
The m a i n ship particulars are listed i n the
o n the n e x t page:
Table
170
Main sliip cliaracteristics
The
lengtli on waterline
L
205.00 m
length between perpendiculars
hp
200.00 m
B
32.00 m
draught moulded on F.P.
Tp
10.00 m
Icb
TA
10.00 m
displacement volume moulded
V
37500 m^
longitudinal centre o f buoyancy
2.02%aftofy2L
20.0 m ^
transverse bulb area
centre o f bulb area above keel hne hp
midship section coefficient
CM
r
waterplane area coefficient
WP
w e t t e d area appendages
^APP
RTR
LR
= 81.385 m
^12
"^13
s
Rp
10.0
8.00 m
4
0.20 m
clearance propeller w i t h keel line
V
25.0 knots
C3
mj
CA
1. Holtrop, J. and Mennen, G.G.J., 'A statistical power prediction method', International Shipbuilding Progress, Vol. 25,
October 1978.
2. Oosterveld, M.W.C. and Oossanen, P. van, 'Further computer
analyzed data of the Wageningen B-screw series'. International Shipbuilding Progress, July 1975.
3. Proceedings 15th ITTC, The Hague, 1978.
= 0.00 k N
= 0.000352
= 221.98 k N
= 0.5102
RA
= 1.030
^total
= 1.156
PE
= 23063 k W
= 7381.45 m^
Cy
= 0.001963
= 869.63 k N
= 1.50
= 1793.26 k N
= 14.500
^11
= 1.250
= 0.5477
= 8.83 k N
Cpi
w
= 0.1561
^10
= 12.08 degrees
t
= 0.1747
= 1.398
T
= 2172.75 k N
= 0.2584
= 0.15610
= 0.02119
= 0.7393
= 0.7595
= 0.9931
= 0.9592
^0.75
= -2.1274
^/^0.75
^2
= -0.17087
X
= 0.6513
= 557.11 k N
= 3.065 m
= 0.03524
= 0.000956
= 1.69385
References
= 5.433
= 0.04
= 0.001390
R-APP
^stern
D
ship speed
= 0.5833
1+^2
stern shape parameter
Z
Cp
50.0 m ^
propeller diameter
number o f propeller blades
PUT
0.750
16.0 m ^
transom area
= 0.2868
4.0 m
0.980
w i t h the statistical m e t h o d re-
characteristics listed i n the next Table:
breadth moulded
draught m o u l d e d on A.P.
calculations
suited m t o the f o l l o w i n g coefficients and powering
F r o m the B-series
polynomials:
= 0.18802
= 0.6261
Fni
= 1.5084
n
Rp
= 0.049 k N
^Qo
Ps
= 1.6594 Hz
= 0.033275
= 0.6461
= 32621 kW
3
A S T A T I S T I C A L R E - A N A L Y S I S O F R E S I S T A N C E AND PROPULSION
DATA
by
J . Holtrop*
I . Introduction
In
a recent publication
[1]
a power prediction
1+^j
method was presented which was based on a regression
analysis of random model
0.93 + 0.487118 Cj,, ( B / L ) ' ( 7 y i ) 0 '»«'"s
(L/L^
)0.121563 (^3
•)0.364a6( 1 _ q , ) - 0.604247
and full-scale test data.
F o r several combinations of main dimensions and form
In this formula B and T are the moulded breadth and
coefficients
draught, respectively, L is the length on the wateriine
the method
had been adjusted to test
results obtained in some specific cases. In spite of these
and V is the moulded displacement volume. Cp is the
adaptations the accuracy of the method was found to
prismatic coefficient based on the waterline length,
be insufficient for some classes of ships. Especially
is defined as:
for high speed craft at Froude numbers above 0.5 the
power predictions were often wrong. With the objective to improve the method the data sample was
Lji=L(l
-Cp
+ 0.Q6Cp IcbKACp
- 1))
where Ich is the longitudinal position of the centre of
extended covering wider ranges of the parameters of
buoyancy forward of 0.5 i as a percentage of I .
interest. I n this extension
the
The coefficient c^^ accounts for the stem shape. It
have
depends on the stem shape coefficient Cj,^j^ for which
of the data sample
published results of the Series 64 hull forms [2]
been mcluded. The regression analyses were now based
the following tentative figures can be given:
on the results of tests on 334 models. Beside these
analyses
method
of
resistance
and
propulsion properties a
was devised by which the influence of the
propeller cavitation could be taken into account. I n
addition some formulae are given by which the effect
of a partial propeller submergence can tentatively be
estimated. These formulae have been derived in a study
carried out in a M A R I N Co-operative Research pro-
Afterbody form
c
Pram with gondola
V-shaped sections
Normal section sliape
U-shaped sections
with Hogner stem
-25
-10
0
1+0.011C„
10
gramme. Permission to publish these results is gratefully acknowledged.
As regards the appendage resistance no new analysis
was made. F o r prediction of the resistance of the appendages reference is made to [ 1 ] .
2. Re-analysis of resistance test results
A re-analysis was made of the wave resistance. A
The results were analysed usmg the same sub-division into components as used in [ I ] :
new general formula was derived from the data sample
of 334 models but calculations showed that this new
prediction formula was not better in the speed range
up to Froude numbers of about
5^ere:
Rp
= 0.5, The results
of these calculations indicated that probably a better
= frictional
resistance
according
to
the
l T T C - 1 9 5 7 fomiula
prediction formula for the wave resistance in the high
speed range could be devised when the low speed data
1 + fcj = form factor of the hull
were left aside from the regression analysis.
RjlPP
~ appendage resistance
By doing so, the foUowing wave resistance formula
= wave resistance
was derived for the speed range F^ > 0.55.
= additional pressure resistance of bulbous
bow near the water surface
Rj-j^
= additional
pressure
resistance
due
to
transom immersion
R^
= model-ship correlation resistance.
A regression analysis provided a new formula for
the form factor of the hull;
• ) Maritime Research Institute Netherlands, Wageningen, The Netherlands.
where:
c „ = 6919.3 C ^ i - " « ( V / Z , ^ ) 2 - ° 0 9 " ( / , / 5 _ 2 ) i - ' ' M 9 2
= - 7 . 2 0 3 5 ( f i / i ) 0 - 3 2 6 8 6 9 (7-/5)0.605375
The coefficients c^,c^,d
and X have the same definit-
ion as in [ 1 ] :
* (MARIN, Wageningen, The Netherlantds, Reprintetd from International
Shipbuilding Progress, Volume 31, Number 363)
=
expC-l.SVcj)
Cj
= (1
\
= l A46 Cp-0.03
X
=
"If
-0.8Aj:/(BTC^)
Here R "'-'4 n 4
,)/l.S
resistance prediction for
No attempts were made to derive new formulations
for the transom pressure resistance and the additional
-0.9
wave resistance due to a bulb near the free surface.
The
C3 = 0 . 5 6 / l ^ f / { 5 n 0 . 3 1 V
w,, =
'^^ '^we
Rl
0.55 according to the respective formulae.
1.446C^-0.36
w h e n / - / £ > 12
=
0.55
L/B
when i / f i < 12
cf
Rw-A,,^OQF„-A){R^_.
available material to develop such formulae is
rather scarce. As regards the height of the centre of
0.4 e x p ( - 0 . 0 3 4 F - 3 - 2 9 )
the transverse bulb area
it is recommended to obey
the upper limit of 0.6 Tp in the calculation of the ad-
C j j = - 1.69385
wheniVv < 512
ditional wave resistance due to the bulb.
= - 1.69385 + (Z,/V "3 _ 8 ) / 2 , 3 6
when
512 <
iVv
3. Re-analysis of propulsion data
< 1726.91
The model propulsion factors and the model-ship
wheniVv
correlation
> 1726.91
allowance
were
statistically re-analysed
and the trans-
using the extended data sample. This data sample in-
verse immersed transom area at rest Aj. and the trans-
cluded 168 data points of full-scale trials on new built
The midship section coefficient
verse area of the bulbous bow Agj.
have the same
ships. In the analysis the same structure of the wake
meaning as in [ 1 ] . The vertical position of the centre
prediction formulae in [1]
o f / l ^ ^ above the keel plane is/ig. The
regression
value
of
was maintained. By the
analyses new constants
were
determined
which give a slightly more accurate prediction.
should not exceed the upper limit of 0.6 T^.
Because attempts to derive prediction formulae for
A point which has been improved in the wake predict-
the wave resistance at low and moderate speeds were
ion formula is the effect of the midship section coef-
only partially successful it is suggested to use for the
ficient C^j for full hull forms with a single screw.
estimation of the wave resistance up to a Froude num-
The improved formula for single screw ships with a
ber of 0.4 a formula which closely resembles the orig-
conventional stem reads:
inal formula of [ 1 ] . The only modification consists
0.050776 + 0.93405 c
of an adaptation of the coefficient that causes the
humps
and hollows on
" (1
formula, which is slightly more accurate than the
+ 0.27915 c,
2°
original one reads;
^w-A
= ^1
('"1 ^ « + '"4 c o s ( X F ; 2 ) )
with;
C^ = 2223105 c / ' 8 ' ' 3 ( 7 - / £ ) 1 . 0 7 9 6 1 ( g o _ , . ^ ) - ) . 3 7 5 6 5
c,
The
=
coefficient
=
-c^i)
depends
' ^ " ^ 2 "
on the coefficient
BS/{LDT^)
when B/T^
< 5
or
Cg
B/L
= S{,1B/T^
-25)KLD{B/T^
when B/T^
when 0,11 < B/L < 0.25
Cj
Vi(i
defined as;
= 0.229577(fi/i)0""3
when B/L < 0.11
C7
'Cp^)i
the resistance curves. This
-3))
> 5
= 0.5 - 0 . 0 6 2 5 1 / 5
when B/L > 0.25
when c„ < 28
m. = 0.0140407 Z , / r - 1.75254 v'^V^ Co = 32 - I 6 / ( c „ - 24)
4.79323
"16
= 8.07981 C„
B/L-c^^
when Cg > 28
1 3 . 8 6 7 3 C 2 + 6.984388C/
TJD
when Cp < 0.8
=
1.73014 - 0.1061
when
T^/D<2
Cp
c „ = 0 . 0 8 3 3 3 3 3 ( 7 ; ^ / £ i ) 3 + 1.33333
when Cp > 0.8
when
in^ ; as in the R^^ formula for the high speed range.
= 0 . 1 2 9 9 7 / ( 0 . 9 5 - C g ) - 0.11056/(0.95 - C ^ )
For the speed range 0.40 < F,, < 0,55 it is suggested
to use the more or less arbitrary interpolation formula:
T^/D>2
when Co < 0.7
or
Cg
C j , = 0.18567/(1.3571 - q ^ ) - 0.71276 + 0.38648 C^,
when Cp > 0.7
c,„=
1 + 0 . 0 1 5 C^,„„
Cp^ = 1.45
- 0.315 - 0.0225 kb
The coefficient Cy
.
is the viscous resistance coef-
ficient with
Cy = (1 +A:)C^ + C ,
As regards the thrust deduction
of single screw
ships a new formula was devised of comparable accuracy :
; = 0.25014(5/I)0-28«6(^/^)0.2624 /
/ ( I -Cp
For
+0.0225
lcb)°°"^^
+ 0.0015 C^,^,„
the relative-rotative efficiency an altemative
prediction formula was derived but because its accuracy is not better than that of the original one it is
suggested to use the prediction formula of [ 1 ] :
t)g = 0.9922 - 0.05908/I^/>1^ +
+ 0.07424(C^ - 0.0225 Icb)
For multiple-screw ships and open-stem single-screw
ships with open shafts the formulae of [ 1] were maintained.
The
model-ship
correlation allowance was statis-
tically analysed. It appeared that for new ships under
ideal trial conditions a C j -value would be applicable
which is on the average 91 per cent of the
-value
according to the statistical formula of ( 1 ] . Apparently,
the incorporation of more recent trial data has
reduced the average level of C^ somewhat. It is suggested, however, that for practical purposes the original formula is used.