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Re-evaluation of Resistance Prediction for High-Speed Round Bilge Hull Forms
Conference Paper · September 2011
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11th International Conference on Fast Sea Transportation
FAST 2011, Honolulu, Hawaii, USA, September 2011
Re-evaluation of Resistance Prediction for High-Speed Round Bilge Hull Forms
Prasanta K Sahoo1, Heather Peng2, Jae Won1, and Dileepan Sangarasigamany3
1
Dept. of Marine and Environmental Systems, Florida Institute of Technology, Melbourne, USA
2
Faculty of Engineering and Applied Science, Memorial University, St. Johns, Canada
3
National Center for Maritime Engineering and Hydrodyanamics, Australian Maritime College, Australia
ABSTRACT
Predicting the resistance of a high-speed monohull has been
of interest to Naval Architects for several decades. Even
though considerable amount of research has been carried out
in this area, there remains a degree of uncertainty in the
accurate resistance prediction in the early design stage.
This research paper attempts to investigate a method for
enhancing the accuracy of resistance prediction methods for
high-speed round-bilge monohull form vessels for a wide
range of volumetric Froude numbers (Fn). While a number
of systematic series are in existence, their data are either not
readily available or scattered in various internal reports and
publications which makes it difficult for practicing naval
architects to exploit the knowledge base. In this paper the
following high-speed systematic series hull forms have been
considered for regression analysis, namely:
Numerous studies have been carried out on predicting the
resistance of high-speed round bilge vessels. Various
methods, factors and assumptions were employed by
various researchers at various times which have influenced
their analysis. Owing to different techniques, these methods
would have significant difference among their predicted
results for a particular vessel. Therefore, people who may
consider using one of the resistance prediction methods to
predict a vessel’s resistance must carefully choose the most
appropriate method that suits the vessel’s geometrical
characteristics.

NPL (1969)

S-NPL (1994)

SKLAD (1972-1980) and
The aim of this paper was to investigate the NPL, S-NPL,
SKLAD, and AMECRC series which have undergone
exhaustive tank testing and combine them in an attempt to
find a superior solution in predicting the resistance for a
broad range of round bilge high-speed vessels parameters.
While this was not feasible while investigations were under
way, separate regression equations have been developed for
each of the series pertaining to their own unique geometrical
characteristics.

AMERC (1984-2000)
2.0
Earlier objective of this paper was to obtain a common
regression equation for a wide parameter space which would
be encompassing all the above systematic series. As this
was not feasible due to lack of data in areas that were
considered crucial, hence separate regression analysis has
been carried out for each series. The new regression
equations have been proposed for a broad range of
geometrical parameters so that a designer has an instant tool
to make a decision regarding powering prediction in the
design stage.
ROUND BILGE MONOHULL SYSTEMATIC
SERIES
Table1 presents some of the series’ parameters and list of
original references. The performance of round bilge hulls is
strongly dependent upon the slenderness ratio, L/1/3,
Savitsky et al. (1973). Figure 1 illustrates the L/1/3 range
covered by each of the series while the Figure 2 presents the
body plans of the round bilge hull systematic series
described in Table 1.
KEY WORDS
Monohull, Resistance, High-Speed, Regression
1.0
INTRODUCTION
Fuel economy and environmental concerns are two
dominant factors in this century that demand that resistance
be accurately predicted in the early design stage, so that
there is no undue penalty due to high fuel costs throughout
the life of the vessel. This has its own implication in
choosing the most appropriate propulsion system to suit the
vessel’s resistance characteristics.
© 2011 American Society of Naval Engineers
Fig.1: L/1/3 Ranges for Round Bilge Systematic Series
[Bojovic (1998)]
311
3.0
NPL SERIES
Resistance data for high-speed round bilge form obtained at
NPL were originally presented in 1969. The work was
extended to examine the effect of the hull parameters on
calm water resistance, Bailey (1976). Experimental
investigations involved testing of 22 models where the bare
hull models were bereft of any keel or appendages. The
water line length LWL and the block coefficient CB of the
models were set at 2.54 m and 0.397 respectively, where the
B, T and the displacement of the vessel were varied. The
model was also designed to have the LCB at 6.40% of LWL
aft of amidships. These vessels were divided in to 7 groups
according to their slenderness ratio, L/1/3. Figure 3
represent the parent hull form of NPL series and Figure 4
describes the distribution of geometrical parameters of NPL
and S-NPL series.
4.0
SOUTHAMPTON EXTENDED NPL SERIES
Ten slender round bilge models were derived from the NPL
series and this extended series was deemed to broadly
represent sleder hull forms suitable for catamaran
applications. The calm water resistance testing of the S-NPL
has been described by Molland et al. (1994). The models
were tested as monohulls and in catamarans configurations
with different hull spacing. The body plan of the hull forms
are shown in Figure 5.
Table 1: Round Bilge Hull Systematic Series
[Bojovic(1998)]
Series
(No. of Models)
Nordstrom
(12*)
De Groot (31*)
MarwoodSilverleaf (30*)
Series 63 (5)
Series 64 (27)
L/1/3
5.657.72
5.237.75
S-NPL
extended (10)
YP (3)
D-Series (13)
4.5-6.4
8.0412.4
LCB
6.0-8.0
4.478.30
6.3-9.5
5.575.72
7.0-15.1
3.975.17
4.5-8.5
6.627.93
6.36.93
0.9-2.0
4.0-8.0
5.687.05
5.416.25
6.586
4.313.1
4.0-12.0
4.3-8.7
4.0-8.0
0.9-2.2
3.04.0
3.04.0
1.52.5
1.52.5
3.065.05
3.05.0
3.05.0
3.03.75
4.396.90
2.55.5
2.54.0
0.350.55
0.1-1.5
0.4
0.397
0.397
1.0-2.0
6.4% L
aft
6.4%L
aft
Fn=0.3-1.2
Fn=0.1-1.05
Fn=0.1-0.6
0.350.55
0.350.55
0.48.52
0.450.60
0.350.55
0.400.50
Fig. 4: Range of Parameters Covered in NPL and S-NPL
Series [Bojovic (1998)]
Fig. 2: Round Bilge Systematic Series- Body Plans [Bojovic
(1998)]
Fig. 3: NPL Series Parent Hull Body Plan [Bailey(1976)]
312
Fn
.45-1.12
2.5-5.75
8.4518.26
4.628.20
3.337.50
VTT (4)
MARIN
HSDHF (40)
AMECRC
HSDHF (14)
CB
0.3730.41
0.8-2.7
SKLAD (27)
NRC (24)
B/T
3.163.57
5.2-8.2
SSPA (9)
NPL (22)
L/B
4.836.94
© 2011 American Society of Naval Engineers
1.0-3.0
Fn=0.2-1.0
Fn=0.150.80
0.6-3.8
Fn=0.1-1.2
Fn=0.1-1.0
Fig. 6: SKLAD Series Parameter Space [Radojcic et al
(1999)]
Fig. 5: Southampton Extended NPL Series [Molland et al
(1994)]
5.0
SKLAD SERIES
The research on SKLAD series of models were carried out
at the Brodarski institute, in the former Yugoslavia, over the
period from 1972 to 1980. Twenty seven high speed roundbilge, transom-stern, semi-displacement hulls were
developed and used for the research. The displacement
volume was kept constant at 0.230 m3, so that the length of
the models varied from 2.7 to 6m. The models were divided
in to three groups each, according to their block coefficients
CB, L/B ratio and B/T ratio. The ranges of varied parameters
are outlined in Table 2 and the series’ parameter space is
illustrated in Figure 6. The parent hull form of SKLAD
series has been shown in Figure 7.
Table 2: SKLAD Series Parameters and Range
[Radojcic et al. (1999)]
Parameters
L/B
B/T
CB
L/1/3
LCB %LWL aft of
midship
Range
4.0 – 8.0
3.0 – 5.0
0.35 – 0.55
4.5 – 8.5
8.8, 9.3 and 9.2
for each CB
© 2011 American Society of Naval Engineers
Fig. 7: SKLAD Series Parent Hull Plan [Radojcic et al.
(1999)]
Constant values were taken for the position of the
longitudinal centre of buoyancy so that LCB = 8.8%, 9.3%
and 9.2% of the LWL aft of amidships for CB=0.35, 0.45 and
0.55 respectively. The round-bilge models had a sternknuckle (chine) for approximately 20% LDWL and a built-in
spray rail for approximately 40% LDWL. The after body
bottom was both flat (12 degree deadrise angle) and hooked
(wedge is incorporated) and provided enough space for
propellers, a low shaft angle, and low dynamic trim.
Forward sections were of a deep V-form with a small angle
of entrance that ensured good seakeeping qualities and
reduced resistance. All models were without appendages
and were ballasted to even trim at rest and towed
horizontally at the centre of buoyancy.
Each group had a constant prismatic coefficient CP of 0.715
and a maximum section-area coefficient of 0.621. The series
models were tested over the volumetric Froude number
range Fn form 1.0 to 3.0. A comprehensive regression
analysis have been performed by the authors, Radojcic et al
313
(1999), to determine the residuary or total resistance
coefficients.
6.0
AMECRC SERIES
The AMECRC systematic series is based on the High-Speed
Displacement Hull Form (HSDHF) systematic series
developed at the Maritime Research Institute Netherlands
(MARIN). The HSDHF project was major research project
on combatant vessel design, jointly sponsored by the Royal
Netherlands Navy, the United States Navy, the Royal
Australian Navy and MARIN. It was initiated by the
growing belief that a significant improvement in the
performance of transom stern, round bilge monohulls could
be obtained, especially with regard to their Seakeeping
characteristics.
Fig. 9: AMECRC Systematic Series Parameter Space
[Sahoo and Doctors (1999)]
The parent hull form of AMECRC series was based on the
parent hull form of HSDHF series and subsequently 13
more models were developed by systematic variation of L/B,
B/T and CB, Sahoo and Doctors (1999). Table 3 presents the
geometrical parameters range of HSDHF and AMECRC
series and Figure 8 depicts the parent hull form.. The
‘parameter space’ or series ‘cube’ of the AMECRC series is
given in Figure 9. All the 14 models have the same length of
1.6 m and the influence of change of the series parameters
on the hull shape are illustrated in Figure 10 and parameter
range in Table 4.
The models were tested in the Ship Hydrodynamics Centre
at the Australian Maritime College. All models were
constructed with a water line length of 1.6 m. Calm water
tests were conducted at speeds from 0.4 to 4 m/s,
corresponding to Froude Number (Fn) from 0.1 to 1.0.
During testing, the models were free to sink and trim, and
resistance, trim and rise of centre of gravity were recorded.
Table 3: Parameters for HSDHF and AMECRC
Systematic Series [Sahoo and Doctors (1999)]
L/B
HSDHF
4 – 12
AMECRC
4 -8
B/T
CB
2.5 – 5.5
0.35 – 0.55
2.5 – 4.0
0.396 – 0.50
Fig. 10: Change in Hull Shape of AMECRC Series [Sahoo
and Doctors (1999)]
7.0
REGRESSION ANALYSIS
The general purpose of multiple regression is to analyse the
relationship between several independent variables or
predictor variables, called x, and a dependent or criterion
variable, y. The experiment data model doesn’t have not
only several independent variables but also have several
valid cases.
It is easy to deal with data that have some linear or
nonlinear relationship between them. A straight or a curve
line can be fit through those points. From the equation of
that line the same parameter can be predicted within the
valid range. Simple regressions like this can be done with
the help of Microsoft Excel program.
Hull form parameters


Fig. 8: AMECRC Parent Hull Body Plan [Sahoo and
Doctors (1999)]
314
Analysed hull form parameters should be in nondimensional form.
Data should give a uniform coverage of the space
defined by variations of the analysed hull form
parameters. The density of the uniform coverage is
not vital factor, but it is desirable to obtain a full
© 2011 American Society of Naval Engineers


combination of at least three values of each varied
parameter.
All the parameters that may have a significant
effect on the dependent variable should be included
in the analysis, and any parameters that are not
included should either be constant or should have
an insignificant influence on the dependent
variable.
The extreme values of all varied parameters should
be carefully defined.


Table 4: Systematic Series Parameter Range [Sahoo
and Doctors (1999)]
Model
L/B
B/T
CB
Model
Disp.(kg)
L/1/3
1
8
4
0.396
6.321
8.653
2
6.512
3.51
0.395
11.455
7.098
3
8
2.5
0.447
11.454
7.098
4
8
4
0.447
7.158
8.302
5
4
4
0.395
25.344
5.447
6
8
2.5
0.395
10.123
7.396
7
4
2.5
0.396
40.523
4.658
8
4
2.5
0.5
51.197
4.308
9
8
2.5
0.5
12.804
6.839
10
8
4
0.5
8.002
7.998
11
4
4
0.5
32.006
5.039
12
8
3.25
0.497
9.846
7.464
13
6
3.25
0.45
15.784
6.379
14
6
4
0.5
14.204
6.606
Assumptions Regarding Regression Analysis Application
 The principal parameters of the hull whose
performance is being predicted must fall within the
range of parameters’ values covered by the data.

All parameters that are constant in the analysed
data set, must have that same constant value in the
proposed design (comment: if a certain parameter
is constant, it does not reduce the prediction
accuracy, only prevents the investigation of the
effect of that parameter).
Selection of Independent Variables
Independent variables are generated as functions of varied
hull parameters and/or speed.
 Independent variables should be in nondimensional form. When there is theoretical
evidence as to the form that the independent
variables should take, an attempt should be made to
utilise that form.
 When a regression equation has two highly
correlated variables as useful independent
© 2011 American Society of Naval Engineers
variables, it is wrong to further include their
product them as an independent variable because it
will lead to some instability in the equation. And is
unnecessary as it will not add significantly to the
accuracy of the equation.
It is possible to have two highly correlated
independent variables which if one is included in
the regression equation without the other is not
effective, but if both are included then the equation
is more accurate.
It also possible to have two highly correlated
independent variables in a regression equation
which
both
have
significantly
non-zero
coefficients, but which predominantly explain the
variance of each other rather than the variance of
the dependent variable. Each of these could
become insignificant, if the other is removed from
the equation.
Production of Good Regression Analysis
 Each independent variable used in the regression
equation should have a high significance level,
generally not lower than 95%.
 It should not be possible to improve the accuracy
of the equation by introducing extra independent
variables.
 It should not be possible to exclude an independent
variable from the equation without significantly
reducing the accuracy.
 The regression equation should not contain more
than ten independent variables, Fairlie-Clark
(1975). Fung (1991) concluded that residual error
starts to stagnate after inclusion of 1 to 17 terms.
More terms in a regression equation may contribute
to a better fit to the data, yet give a poorer
interpolation result, Savitsky et al. (1976).
8.0
REGRESSION ANALYSIS TECHNIQUE
Several other techniques have been tried to predict RR/ in
terms of L/B, B/T, L/1/3 and CB. The method used by
Radojcic (1997) has been adopted here. In his method
principal hull form and loading parameters were
transformed in to another set of variables with a range from
-1 to 1.
The method has two sets of similar equations, one for
AMECRC and SKLAD series hull forms where the
parameter space has varying L/B, B/T, L/1/3 and CB; and
the other for S-NPL series hull form which has all but CB is
fixed. The equation developed for S-NPL is similar to the
equation developed by Radojcic (1997) for NPL hull forms
which have 27 terms, and the equation developed for
AMECRC and SKLAD series have 48 terms. They are as
follows:
315
9.0
S-NPL hull series:
 L ( L / B) min  ( L / B) max 
 

B
2

x1  
 ( L / B) max  ( L / B) min 


2


(1)
 B ( B / T )min  ( B / T ) max 
 

2
T

x2  
 ( B / T ) max  ( B / T ) min 


2


(2)
 L
( L / 1 / 3 ) min  ( L / 1/ 3 ) max
 1 / 3 

2
x3  
 ( L / 1 / 3 ) max  ( L / 1/ 3 ) min 


2





(3)
FORWARD STEPWISE REGRESSION
PROCEDURE
This is the method used in the paper for regression analysis.
The method starts with a single independent variable in the
regression model. At each succeeding step, additional
independent variables are introduced based on significance
testing using the t-test. Those variables that possess the
greatest statistical significance relative to the dependent
variable are added first. This procedure is repeated until no
significant independent variables can be found outside the
regression model. Also, if a previously added independent
variable becomes insignificant due to the addition of new
variables, the previously added variable is removed from the
regression model. The acceptance and rejection of each
independent variable is purely based on the F-test (criteria
F-to-remove and F-to-enter in the software).
RR   a0  a1x1  a2 x2  a3 x3  a4 x1  a5 x2  a6 x3 
2
2
2
a7 x1x2  a8 x1x3  a9 x2 x3  a10x1 x2  a11x1 x3  a12x2 x1
2
2
2
 a13x2 x3  a14x3 x1  a15x3 x2  a16 x1  a17 x2  a18x3 
2
2
2
3
3
3
a19 x1 x2  a20x1 x3  a21x2 x1  a22 x2 x3  a23x3 x1  a24x3 x2 
3
3
3
3
a25x1 x2  a26x1 x3  a27 x2 x3
2
2
2
2
2
3
3
2
(4)
AMECRC & SKLAD hull series:
 L ( L / B) min  ( L / B) max 
 

2
B

x1  
 ( L / B) max  ( L / B) min 


2


This model has been used for all of the analyses described in
this paper. F-values, mentioned above, where selected so
that the final regression model contains no variables with a
statistical significance (p-level) greater than 0.05 (5%).
Specifically, p-level represents the probability of error that
is involved in accepting observed results as valid.
When conducting regression analysis results could become
unstable if highly correlated independent variables are
included in the regression model. Control over this matter
was achieved by setting the tolerance level from 0 – 0.001
(0% - 0.1%). That means that variables whose tolerance was
under this level were considered redundant with the
contribution of other independent variables already in the
equation.
(5)
10. FINAL REGRESSION MODEL AND RESULTS
 B ( B / T )min  ( B / T ) max 
 

2
T

x2  
 ( B / T ) max  ( B / T ) min 


2


(6)
 L
( L / 1 / 3 ) min  ( L / 1 / 3 ) max


 1 / 3
2
x3  
 ( L / 1 / 3 ) max  ( L / 1 / 3 ) min 




2






(7)
Linear and non-linear interpolation analyses have been
performed to arrange the results for the specific Froude
numbers where needed. Therefore an uncertainty is
expected in organizing the final experimental results for
regression analysis. Thus assuming the uncertainty is very
small, it has been ignored in this regression analysis.
(8)
Regression equation has been obtained for a wide range of
Fn. The ranges of Fn are for AMECRC systematic series
1 to 2, S-NPL systematic series 1 to 2.5 and SKLAD
systematic series 1 to 3. All three series have increments of
0.25 for Fn. The regression equation obtained for all range
of Fn have a higher degree of accuracy with R2 = 0.9999 or
higher while obtaining the coefficients of the equations. The
predicted values of RR/ for all the existing models are quite
close to the actual experiment values. Regression equation
obtained by Bojovic (1998) for calculating the wetted
surface area coefficient (CS) has been reproduced.
(C )
 (CB ) max 

 CB  B min

2

x4  
 (CB ) max  (CB ) min 


2


RR   a0  a1 x1  a2 x2  a3 x3  a4 x4  a5 x1  a6 x2  a7 x3
2
2
2
 a8 x42  a9 x1 x2  a10 x1 x3  a11 x1 x4  a12 x2 x3  a13 x2 x4 
a14 x3 x4  a15 x1 x2  a16 x1 x3  a17 x1 x4  a18 x2 x1  a19 x2 x3 
2
2
2
2
2
a20 x2 x4  a21 x3 x1  a22 x3 x2  a23 x3 x4  a24 x x  a25 x42 x2 
2
2
2
2
2
4 1
a26 x42 x3  a27 x1  a28 x2  a29 x3  a30 x4  a31 x1 x2  a32 x1 x3 
3
3
3
3
3
3
a33 x1 x4  a34 x2 x1  a35 x2 x3  a36 x2 x4  a37 x3 x1  a38 x3 x2
3
3
3
3
3
3
 a39 x3 x4  a40 x4 x1  a41 x4 x2  a42 x4 x3  a43 x1 x2  a44 x1 x3 
3
3
3
3
2
a45 x1 x4  a46 x2 x3  a47 x2 x4  a48 x3 x4
2
316
2
2
2
2
2
2
2
2
2
2
For the purpose of developing the regression equations,
initially RR/ has been derived from the experimental data.
AMECRC and S-NPL systematic series had their own
published experimental data, where SKLAD systematic
series had a regression equation developed by Radojcic et
al. (1999) based on CR.
(9)
© 2011 American Society of Naval Engineers
Table 5: AMECRC Series Regression Coefficients
It is to be noted that regression coefficients which do not
play a significant role in the regression equation have been
ignored in the equations as shown. Table 5 depicts the
regression coefficients for the AMECRC Series which needs
to be read in conjunction with final equation (10). Table 6
represents the regression coefficients for the SKLAD series
as per the equation (11).
Equation for AMECRC Series:
RR

 a0  a3 x3  a4 x4  a6 x2  a7 x3  a8 x42 
2
2
a10 x1 x3  a13 x2 x4  a14 x3 x4  a15 x1 x2  a16 x1 x3  a17 x1 x4
2
2
2
 a18 x2 x1  a19 x2 x3  a20 x2 x4  a22 x3 x2  a23 x3 x4 
2
2
2
2
2
a25 x42 x2  a27 x1  a32 x1 x3  a33 x1 x4  a35 x2 x3
3
3
3
3
(10)
 a36 x2 x4  a37 x3 x1  a39 x3 x4  a40 x4 x1  a41 x4 x2 
3
3
3
3
a43 x1 x2  a44 x1 x3  a48 x3 x4
2
2
2
2
2
3
2
Equation for SKLAD Series:
RR

 a0  a1 x1  a2 x2  a3 x3  a4 x4  a5 x1 
2
a6 x2  a7 x3  a8 x42  a9 x1 x2  a10 x1 x3  a11 x1 x4
2
2
 a12 x2 x3  a13 x2 x4  a14 x3 x4  a15 x1 x2  a16 x1 x3 
2
2
a17 x1 x4  a18 x2 x1  a19 x2 x3  a20 x2 x4
2
2
2
2
 a21 x3 x1  a22 x3 x2  a23 x3 x4  a24 x42 x1 
2
2
2
(11)
a25 x42 x2  a26 x42 x3
Equation for S-NPL Series:
RR

Fn
ai
a0
a3
a4
a6
a7
a8
a10
a13
a14
a15
a16
a17
a18
a19
a20
a22
a23
a25
a27
a32
a33
a35
a36
a37
a39
a40
a41
a43
a44
a48
1.00
0.021256
0.010086
0.000000
0.000000
-0.090351
0.000000
0.000000
-0.000362
0.000000
0.000000
-0.033111
0.000000
0.000000
-0.001802
0.000000
-0.001931
0.000000
-0.000580
-0.001214
0.000000
0.000000
0.000000
0.000000
-0.002368
0.002357
0.000000
0.000000
0.000000
0.108373
0.000000
1.25
0.041582
-0.052430
-0.004230
0.000000
0.000000
-0.000432
0.021886
0.000000
0.000000
0.003119
0.000000
0.010262
0.000000
0.000000
0.002726
0.000000
-0.025257
-0.000830
0.000000
0.000000
0.000000
0.000000
0.006008
0.000000
0.002361
0.000000
0.000000
0.000000
0.000000
-0.007058
1.50
0.057983
-0.018881
0.000000
0.000000
0.000000
0.000000
0.000000
0.573707
0.000000
0.011745
-0.027641
0.007585
0.000000
0.000000
0.000000
0.000000
-0.024854
-0.004635
0.000000
0.000000
0.000000
-0.000525
-0.019674
0.000000
-0.008204
0.000000
-0.549618
0.003440
0.000000
0.000000
1.75
0.062536
-0.047772
0.000000
0.002433
0.000000
0.000000
0.074728
0.297682
0.000000
0.007869
0.000000
0.007979
0.000392
0.000000
0.000000
0.000000
-0.024806
-0.002786
0.000000
-0.067811
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
-0.293514
0.000000
0.000000
-0.003685
2.00
0.064701
-0.052937
0.000000
-0.002000
0.000000
0.000000
0.021788
0.000000
-0.012685
-0.006719
0.000000
0.008686
0.000000
0.000000
0.007845
0.007021
-0.025552
0.000000
0.000000
0.000000
0.001858
0.000000
0.000000
0.000000
0.038912
-0.002973
0.000000
0.000000
0.000000
0.000000
Table 6: SKLAD Series Regression Coefficients
 a0  a1 x1  a2 x2  a3 x3  a4 x1  a5 x2 
2
2
Fn
a6 x3  a7 x1 x2  a8 x1 x3  a9 x2 x3  a10 x1 x2
2
2
 a11 x1 x3  a12 x2 x1  a13 x2 x3  a14 x3 x1 
2
2
2
2
a15 x3 x2  a16 x1  a17 x2  a18 x3
2
3
3
3
(12)
Equation for NPL Series, Radojcic (1997):
RR
2
 a 0  a1 x1  a 2 x 2  a3 x3  a5 x 2 

2
2
3
3
a12 x 2 x1  a13 x 2 x3  a16 x1  a19 x1 x 2 
a 20 x1 x3  a 22 x 2 x3  a 23 x3 x1  a 26 x1 x3  a 27 x 2 x3
3
3
3
2
2
2
© 2011 American Society of Naval Engineers
2
(13)
ai
a0
a1
a2
a3
a4
a5
a6
a7
a8
a9
a10
a11
a12
a13
a14
a15
a16
a17
a18
a19
a20
a21
1.00
0.513250
-3.306930
-1.235130
5.562080
1.124810
0.108200
0.104930
-0.390920
-0.125760
-0.332970
0.397760
0.172350
0.113140
0.058230
0.124810
-0.011950
0.042470
0.001910
0.000000
0.037150
0.000090
0.078230
a22
a23
a24
a25
a26
-0.000490
0.043660
-0.031620
-0.009960
0.018660
1.25
0.032441
0.000000
0.000000
-0.060165
0.000000
0.000000
0.002861
-0.187583
0.004264
-0.000285
0.123824
0.000000
0.048196
-0.001019
-0.037893
0.000000
0.000000
0.000000
-0.003115
0.000000
0.000000
0.000000
0.006592
-0.009150
0.000000
-0.001581
-0.003189
1.50
0.290520
0.459060
-0.408120
2.491560
0.681920
-6.292570
-0.354660
-3.807000
0.296720
-3.444920
12.973620
1.101930
2.683190
0.002800
1.168150
-0.162990
-1.883950
0.000000
0.168550
-0.221780
0.004360
0.556630
1.75
-0.945600
10.679900
3.011100
-11.914900
-2.018900
-12.501300
-1.193600
-7.143100
0.981800
-6.180200
24.520100
1.534300
5.922500
-0.039800
1.892700
-0.285900
-3.649800
0.000000
0.325700
-0.624300
0.003400
0.879400
2.00
0.154870
0.000000
-0.269920
1.349990
0.277620
-2.358710
-0.007070
1.011810
0.051650
-0.985530
3.479220
0.615690
-0.045110
-0.031620
0.643570
-0.177010
-0.529810
-0.032470
0.027300
0.007380
-0.004810
0.180780
2.25
0.440900
-2.522410
-0.826620
4.726010
0.919760
-0.641400
-0.236280
2.478460
-0.031980
-1.335920
0.000000
0.241520
0.922390
0.264600
1.017340
-0.155620
-0.080960
-0.081400
-0.049170
-0.016400
-0.014400
0.416740
2.50
-0.848500
6.844500
2.113600
-10.093300
-2.041200
-3.248100
0.013600
2.75
-1.026700
8.281800
2.557500
-12.212900
-2.469800
-3.930200
0.016400
3.00
-0.070970
-0.151850
0.000000
0.926100
0.179000
0.000000
1.120600
0.216600
0.000000
4.356800
0.914000
-1.043000
-0.182300
-0.476500
-0.261000
-0.530500
-0.035400
0.010700
0.041700
-0.008300
-0.083700
5.271700
1.105900
-1.262000
-0.220600
-0.576600
-0.315800
-0.642000
-0.042800
0.013000
0.050400
-0.010100
-0.101300
3.330640
-0.293710
1.963290
-5.405990
0.394460
-2.727690
-0.007310
-1.168100
-0.178380
1.088780
-0.056290
-0.160160
0.296080
-0.012530
-0.302420
0.172920
0.598090
-0.195200
-0.012480
0.274470
0.377300
1.080300
-0.271500
-0.013000
0.506000
0.104600
0.506520
-0.076720
0.006180
0.078040
0.025110
0.777610
-0.005090
-0.025520
0.068100
0.218600
0.425400
-0.094400
0.031000
0.087800
0.264500
0.514700
-0.114200
0.037600
0.106200
0.121380
0.147860
0.000320
0.015740
-0.068550
317
-1.006460
-0.358190
2.069300
0.450970
For NPL and S-NPL Series, the coefficients ci are shown in
Table 12:
Table 7: S-NPL Series Regression Coefficients
ai
a0
a1
a2
a3
a4
a5
a6
a7
a8
a9
a10
a11
a12
a13
a14
a15
a16
a17
a18
1.00
0.145663
0.000000
0.000000
0.121710
0.000000
0.000000
-0.466263
0.071608
0.588435
0.142337
0.092976
0.000000
0.000000
0.057891
0.000000
0.000000
-0.038131
0.000000
0.000000
1.25
0.248405
0.000000
0.001822
-0.112798
0.000000
0.020294
-0.001685
0.000000
0.000000
0.000000
0.000000
0.000000
0.024790
-0.271027
0.000000
-0.106479
0.000000
0.000000
0.167585
1.50
0.334126
0.000000
-0.010633
-0.136986
0.000000
0.023584
0.044714
0.000000
0.000000
0.000000
-0.006864
-0.040654
0.000000
0.000000
-0.006026
-0.013507
0.000000
0.000000
0.000000
Fn
1.75
0.391436
0.000000
0.000000
-0.146568
-0.100553
0.029722
0.000000
-0.009405
0.150517
0.000000
0.000000
0.000000
0.000000
0.000000
0.019188
0.019186
0.000000
0.000000
-0.028958
2.00
0.266540
0.000000
-0.155760
0.257630
1.657930
0.071770
0.000000
0.606780
-1.104850
0.000000
0.264390
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
-0.082020
2.25
0.464022
0.000000
0.000000
2.50
0.525205
0.000000
0.000000
-0.179423
-0.188960
-0.183378
0.035417
0.000000
0.000000
-0.004029
0.037020
0.000000
0.000000
0.349322
0.000000
0.236978
0.000000
0.245393
0.000000
0.000000
0.023702
-0.239572
0.000000
0.435946
-0.794819
0.046974
0.000000
0.000000
0.000000
0.367524
0.000000
0.000000
1.008678
0.000000
0.000000
L
C S  c 0  c1  
B
2
2/3
2
1
2
B
L B
B
 L B
   c 2      c3    c 4    
T 
 B T 
T 
B T 
L B
L
 c 6      c 7  
B T 
B
2/3
B
 
T 
1.0
0.031092
0.000000
-0.041840
0.007983
0.002904
-0.003601
0.000000
0.000062
0.005626
0.002059
0.000000
0.002059
0.005375
0.010811
1.2
0.057789
0.008445
-0.042841
0.015331
0.010337
0.000000
-0.000655
0.000000
0.004361
0.010917
-0.003984
0.006095
0.016808
0.008090
1.4
0.070863
0.004134
-0.044967
0.013614
0.043687
0.005111
-0.013832
0.002578
0.004098
0.001183
0.011616
0.001874
0.006989
-0.023857
1.6
0.080384
0.000962
-0.041259
0.014880
0.009803
0.012468
-0.038512
0.004118
0.004899
0.000667
0.030670
-0.004834
0.001094
-0.057101
1.8
0.092592
0.005733
-0.043200
0.022856
0.008203
0.012205
-0.043373
0.005143
0.004763
0.002649
0.033934
-0.006093
-0.001804
-0.064610
2.0
0.105658
0.010769
-0.046379
0.031878
0.007646
0.010455
-0.050962
0.006964
0.010023
0.002493
0.044572
-0.009345
-0.010127
-0.081529
2.2
0.113350
0.007299
-0.042913
0.031266
0.013292
0.000922
-0.050318
0.008068
0.012955
-0.003659
0.047574
-0.010929
-0.017531
-0.080688
2.4
0.118892
-0.001703
-0.039950
0.023441
0.026694
-0.018860
-0.036356
0.009177
0.012276
-0.018867
0.025806
-0.015162
-0.033868
-0.063470
2.6
0.123105
-0.014977
-0.034003
0.008990
0.043975
-0.043354
0.000000
0.009824
0.006514
-0.033644
-0.015288
-0.020789
-0.050023
-0.014102
2.8
0.120589
-0.041899
-0.017179
-0.022888
0.057718
-0.046133
0.051148
0.009804
-0.008769
-0.057179
-0.047272
-0.019478
-0.060285
0.040157
3.0
0.115058
-0.077104
0.005255
-0.067378
0.071086
-0.048405
0.057769
0.008532
-0.034102
-0.101116
-0.062657
-0.051664
-0.118198
0.073258
(L/B) CB
0.744941
-2/3
(B/T) CB
0.352265
-1
2/3
(L/B) (B/T)CB
-1
R2
St. error
0.02013
(B/T)1/3CB-1/3
Table 10: SKLAD Series – CS Regression Parameters
and Coefficients
ci
2.456288
(L/B
Since the wetted surface area determination has already
been carried out by Bojovic (1998), these are also being
reproduced in Tables 9 to 11. The regression model for the
wetted surface area coefficient would have the following
form as shown in the following equations. The wetted
surface area coefficient is given by:
CS 
S
(14)
2 / 3
0.046307
0.037945
1.367162
0.999543
)-2/3
11. WETTED SURFACE AREA COEFFICIENTS
(BOJOVIC (1998))
(17)
3.328344
-2/3
2/3
Fn
0.8
0.012677
-0.008102
0.000061
-0.002725
-0.001185
0.004254
-0.025702
0.001395
0.004913
-0.005901
0.015476
-0.008887
-0.008427
-0.033521
2/3
Table 9: AMECRC Series – CS Regression Parameters
and Coefficients
ci
(L/B) CB
ai
a0
a1
a2
a3
a5
a12
a13
a16
a19
a20
a22
a23
a26
a27
B
 c5  
T 
2/3
4/3
Table 8: NPL Series Regression Coefficients (Radojcic
1997)
2/3
2/3
2/3
(B/T) CB
)-4/3
(L/B
-1
(B/T) CB
0.434391
(L/B)-3
0.013612
0.000188
R2
0.99862
St. error
0.04145
Table 11: NPL & S-NPL Series – CS Regression
Parameters and Coefficients
For AMECRC Series, the coefficients ci are shown in Table
10:
ci
4.445787
L
C S  c 0  c1  
B
L
 c 4  
B
4/3
2/3
C 
1
B
C 
2 / 3
B
B
 c5  
T 
B
L
2 / 3
 c 2  C B   c 3  
T 
B
1/ 3
2/3
)2/3
(L/B
B
1
 C B 
T 
C 
1 / 3
(L/B (B/T)
L
 c3  
B
3
2 / 3
B
 
T 
2/3
(L/B)-2(B/T)-2/3
(B/T)2/3
23.016071
(B/T)-1
C 
2/3
B
L
 c2  
B
4 / 3
1
B
  C B 
T 
(16)
0.252716
186841481
14.463151
131.22276
)2
(L/B (B/T)
)2/3
(L/B
(B/T)
2
318
-1
(15)
B
For SKLAD Series, the coefficients ci shown in Table 11:
L
C S  c0  c1  
B
(B/T)
)-2
2/3
-0.008127
0.528452
R
0.99892
St. error
0.03395
© 2011 American Society of Naval Engineers
12. CONCLUDING REMARKS
The purpose of this study is to provide a set of regression
models for various round bilge high-speed hull forms at a
glance, which would be of immense benefit for a practicing
naval architect. As the results are based on pre-existing
model test results of the various series, it has been shown
through this study that the developed regression equations
are able to give relatively accurate predictions of resistance,
with little input data about the hull and minimal time for the
calculation process. It is therefore assumed that the
regression equations would provide viable first estimates of
the resistance characteristics of hull-form in early design
stages. The regression models are to be used with due care
with regard to type of hull form used in monohull
configuration.
ACKNOWLEDGEMENT
The authors would like to express their sincere gratitude to
The Australian Maritime College, Australia (specialist
institute of University of Tasmania) and Florida Institute of
Technology, Melbourne, USA for their support,
encouragement throughout the course of this research work.
REFERENCES
Bailey, D., The NPL High-speed Round Bilge Displacement
Hull Series, RINA, Maritime Technology Monograph No. 4,
1976.
Bojovic, P., Resistance Prediction of High-speed Round
Bilge Hull Forms, Australian Maritime College, 1998, n.p.
© 2011 American Society of Naval Engineers
View publication stats
Jin, P., Su, B., Tan, Z., A Parametric study on High-Speed
Round Bilge Displacement Hulls, High-Speed Surface Craft,
September, 1980.
Lahtiharju, E., Karppinen, T., Helllevaara, M., Aitta, T.,
Resistance and Seakeeping Characteristics of Fast Transom
Stern Hulls with Systematically Varied Form, Transactions
SNAME, Vol.99, 1991, pp.85 - 118.
Mercier, J.A., Savitsky, D., Resistance of Transom-Stern
Craft in the Pre-planing Regime, Davidson Laboratory
Report 1667, Stevens Institute of Technology, June, 1973.
Molland, A.F., Wellicome, J.F., Couser, P.R., Resistance
Experiments on a Systematic Series of High-speed
Displacement Catamaran Forms: Variation of Length –
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