Nt
MITSUBISHI TECHNICAL BULLETIÑ
No.90
1*_i
Prediction of the Wave Resistance of Ships
by Statistical Analysis
June 1974
MITSUBISHI HEAVY INDUSTRIES, LTD.
Bib$iotheek van de
Order dehna derScheepbouwktmde
Technisch
H o e.cho,
OOCUMENL\Ì
Prediction of the Wave Resistance of Ships
by Statistical Analysis
1. Introduction
In designing ships, it
is important to find a hull form
with excellent propulsive performance under givencon-straints and to make an accurate prediction of the perform-ance. The performance depends on wave resistance, viscous resistance and self-propulsion factors. These factors vary in a very complicated manner with hull form and ship's speed.
In full ship forms it can be assumed for practical
pur-poses that wave resistance is influenced mainly by the shape of the entrance alone, and, therefore, the wave resistance is
estimated from the comparison of the entrance forms of
type ships1» In fine ships, e.g. cargo liners and container ships, however, the wave resistance depends on whole hull forms and therefore the estimation of wave resistance is far
more complicated than that of full ships, though the wave resistance plays the most important role in the propulsive
performance for this type of ships.
Under such circumstances, attempt has been made to obtain an estimation formula by combining the theoretical
formula of wave resistance and test results of ship models
with statistical analysis. Examples of estimation of wave
resistance of various ship forms, e.g. a cargo liner, a general cargo ship and a passenger cargo ship, are given. The for-mula is found very useful for preliminary design of ships.
This formula is also effective for improving ship hull
forms. As examples, the prismatic curves of Series 60 hull
forms (Ci = 0.65 & 0.75) are modified by referring to the influence lines obtained from this estimation formula. The model tets show that their wave resistance is reduced by
one-half at the service speed.
In the following are given the basic idea of this method
and examples of the prediction of wave resistance and ship form improvement.
Resistance and Propulsion Research Laboratory, Nagasaki Technical Institute, Technical Headquarters
TECHNISCHE UNIVER$flEIT
Laboratorium voor
Scheeoshydmhj
Archief
Mekelweg 2, 2628 CD
Deift TeL 015-786873. Fax:015. 781838Hiroshi Tagano*
A method to predict the wave resistance of ships at the preliminary design stage is presented. The effect of principal dimensions and prismatic curves on wave resistance is investigated theoretically. From the investigation an estimation formula is obtained statis
rica//y by combining the theoretical formula of wave resistance and test results of ship mode/s.
Examples of estimation of wave resistance of various ship forms, e.g. a cargo liner, a general cargo ship and a passenger cargo ship are given. The formula is found very useful for preliminary design of ships.
This formula is also effective for improving ship hull forms. As examples. the prismatic curves of Todd Series 60 hull forms (Ci = 0.65 and 0.75) are modified by referring to the influence lines obtained from this estimation formula. The model tests show
that their wave resistance is reduced by one-half at service speed.
2. The Methods of the Prediction of Ship Wave Resistance
lt has been one of the most important problems of ship
hydrodynamics to find the laws of variation of wave
resist-ance according to hull form and ship's speed. The history of the model experiment started with W. Froude's test in 1872 and of the wave resistance theory began with J.H.
Michell's study reported in 1898. Since these great works, many researchers have investigated the laws theoretically and experimentally.
There are several methods at present to predict wave resistance or residual tesistance for given ship forms and
ship speeds, which may be classified as follows.
The method of estimation on the basis of comparison
of the resistance characteristics of type ships.
The method of estimation by charts derived from sys-tematic series test results.
The method of estimation by regression formulae based on tank test results of various ship forms.
The method of estimation based on theoretical calcula-tion of wa'ie resistance.
The first method is most popular and is useful for the
design of ships when appropriate type ship data are avail-able. It is not easy, however, to find a way to improve a hull form better than type ships without enough information on the relation between the resistance and hull forms.
For the second method are avalable the charts of
Taylor2) vamagataÇ
Todd4 SR455 (The Shipbuilding
Research Association of Japan) and so on, of which Taylor's chart is the most popular. In this chart, residual resistance
(r) is represented graphically versus speed-length ratio with
C, C, and B/H as parameters. These parameters being varied in wide range, the chart is applicable to majority of merchant ships. The shape of prismatic curve, however,
MTB 90 June 1974
being defined uniquely according to prismatic coefficient C in the chart, the effect of shape of the prismatic curves under given G cannot be investigated, though the variation of residual resistance according to the position of center of buoyancy is graphed in some charts.
The third method is applied for analysis of the results of model basin experiment of fishing boats by D.J. Doust and
T.P. OBrien6) D.J. Doust JG. Hayes and LO. Engvall(8)
and T. Tsuchiya A.S. Sabit(1o)(i 1 reported the regres-sion formulae obtained by the analysis of the Series 60 and
B.S.R.A. (British Ship Research Association) Series. These regression formulae are generalized as follows.
C,=f(xI,xP,x3,
'2
-
ax,x*
i-O
where
C total resistance coefficient
a : regression coefficient
a, /3 : O or positive integer
m. : the number of the terms of regression equation independent variable
The introduced independent variables are grouped into three classes.
Ratios of principal dimensions
Llr, LIB, Bld, Cb, G, C
Variables representing shapes of prismatic curves and water-lines
Cp,
IC!b, +a,, ±a,
trimwhere
j-aV: The angle which the water-line makes with the
center line of the model at the stem.
-i-a,: The maximum angle of run up to and including
the designed floating water-line. This angle is measured at a section 5 percent of the
water-line length forward of the aft end of L.
UBS: Maximum buttock slope of the 1/4 beam buttocK measured relative to the floating water-lines.
(c( Others
Blockage correction term
Section area of a bar or wooden keel
In this prediction method, many model test data a re used effectively, and the equation can be applied to im-proving ship hull forms. These merits are not found in the previous methods [(1) & (2H .
lt was reported that the
method was used in designing trawlers and they showedgood performance relative to vessels designed by the pre-vious methods (1) and (2(.
As, however, the regression equations used in the above method are determined with purely statistic analysis, each term of the equations has not always a physical meaning. Therefore, this method leaves some room for improvement by means of theoretical consideration.
A method which fell under the category of the forth
method was first reported by T.H. Havelock.(12)(13) He
re-placed the wave-making characteristics of the ship form by a traveling two-dimensional pressure disturbance and calcu-lated the wave resistance of such a pressure disturbance. After some simplification, he found
Raet+ß(1- 7cos-)e
whereR : wave resistance of a ship
c : speed length ratio
rn, n universal constants
a, /3, y : constants depend upon the form of the ship
He showed that notwithstanding the limitations of
theo-ry and the difficulty of interpretation of experimental data,
a good quantitative agreement was found in several cases
with the published results of tank experiments on models when suitable numerical values were given to the coeffi-cients in the formula.
Recently, Oortmerssen,(14) confining himself to small
ship like tugs and trawlers, proposed a pressure distribution considering the stream-lines around a ship, and presented the wave resistance of a two-dimensional pressure distribu-tion. After some statistical analysis of the resistance data of tugs and trawlers, the final form of the resistance equation was presented as follows,
= L
-F'
2 C4e cosF, whereCf(1c,b, G, LIB, CWL, Bld, G)
m = bG
- h.GWL : a parameter for the angle of the entrance of the load water-line
i hese works indicate that wave resistance theory is very
effective for predicting ship's wave resistance. However, these formulae should be improved further for application to hull form design, especially in that the relation between
prismatic curves and wave resistance is not presented satis-factorily.
Recent development of electronic computer has enabled us to calculate wave resistance of ships theoretically using
the technique of finite element method5X16) With this
method, the ship wave resistance can be represented as a function of ship's hull form and ship speeds. But there is a
large discrepancy between theoretical value and
experimen-tal results, and some researchers are trying to modify the
theory to reduce the discrepancy. This method has not been
put into practice to predict the wave resistance and hull form design, but it will be possible to connect ship's hull
form to her wave resistance by wave resistance theory.
The method of prediction proposed in this paper has
been established by combination of the advantages of the above mentioned existing methods.
3. The Proposed Method for Prediction of Ship's Wave
Resistance
3.1 The Outline and the Characteristics of the Proposed Method
There are many variables which affect ship wave
resist-ance. The variables are too many to be introduced in
estimation formulae of ship resistance. In type ship andchart method, some variables are selected and the relation between ship resistance and the variables is examined, and the other variables of type ship or mother ship are adopted unchanged. In this method, principal dimensions and
ordinates of prismatic curves are selected as variables, and
the characteristics of this method is that ordinates of
prismatic curves are the main variables.
First the equation of theoretical wave resistance of ships is so symplified as suitable for practical design, considering
the characteristics of actual merchant ship hull forms and
their wave resistance. An estimation formula is obtained by
combining the above mentioned theoretical formula of
wave resistance and test results of models with statisticalanalysis. In the selection of models above mentioned tech-nique of type ship and chart method is considered.
The characteristics of this equation are as follows. (1) The wave resistance coefficient is represented in a type
of product of a function of principal dimensions and a
function of prismatic curve.
(2> The functions are similarto those derived from theoret-ical consideration.
The prismatic curve is represented by a set of discrete points instead of a formula, thus being applied to various shape of prismatic curves.
The influence function of frame line shape is omitted. In ordinary merchant ships, the variety of the shape is
not so wide except the shape of bow, and the remarkable correlation is found in the shape of bow and the
sec-tional area ratio at forward perpendicular. The influence
of frame line on wave resistance
is included in theprediction formula indirectly by introducing the
statis-tical method described in the section of 3.4.
The form factor of viscous resistance is introduced to
obtain the wave resistance from the total resistance
measured in
model experiments. The form factor
isexpressed in terms of principal dimensions. The total
resistance, therefore, can be expressed by principal di-mensions and shape of prismatic curves.
3.2 Theoretical Formula of Wave Resistance of Ships
As shown in FIg. 1, the coordinate system O-xyz fixed to the ship is employed such that the xy-plane coincides
with the still water surface and the x-axis indicates the
upward direction perpendicular to the still water surface.
The wave resistance of a ship can be expressed as
follows.
Rn 2+ Q2) sec3ûdO (1)
Fig. i Coordinate system
where
= eK,Zsec'S ( Kox sec O) dx dz
Q
j
Oxg : acceleration of gravity
In this expression ships are assumed to be wall-sided by the above mentioned reason.
There are several methods of expression of sectional area
curve. Here the curve is expressed approximately with a
connection of third order polynomials, considering the
process of drawing lines.
If the i-th ordinate of the prismatic curve is denoted by
f, the distribution of ship breadth is expressed as follows.
Th Cnj
= aoI(Ç)fo+aIA(C)fl+az)f2+
+a+2,(E)fi*z
(2) at Çi E E+i where=_y_
R/2aj(E) :
third polynomials of E
midship section coefficient
Pand Q can be expressed from equations of (1) and (2). K,dsec5 n
P) B11e
'ç-'Q1 2 Kolsec2O S
C I C
s, =
(Kol C secO) dEThen, the wave resistance coefficient C can be expressed by Rn + pv 2 V 2Cm2(1 5'1* (l_e_Ko ec5)2 c r
\Cò L dl
i+
(s,)2}cosO dO
where : ,::ì:
bc'_
(ai'!i(Ei)
_d1(E3)) sin(,q)
+
e) -
( e1)) cos (E1 q) _(a!i( ji)
MTB 90 June 1974 at
jl
andda
a =
q Kolsec8 Theref ore,C=
A00f02 +Aof,f1 +A02f0f2++An fi2 +Al2f1f2+
where
X cos(1q)
-
i(çi-i))sin( ,i q)
2C(i
(1_e_Kdec'O)2(2bcbci
A,
\CbL dl
+ 2bsbs,) cosß d9
and2C2(i
.-'-' A7rCoLdl
xcosû dO (4)Thus the wave resistance can be obtained by the
expres-sions of (3) and (4) using the principal particulars and
prismatic curves. As examples, numerical calculation of the wave resistance is carried out for two ship forms, selected from Series 60. The results are shown in Fig. 2, where the
test results of those ship models conducted in our towing
tank are also shown for comparisonY Fig. 2 shows that the wave resistance coefficients obtained theoretically are four or five times as large as the test results. Therefore, equations
(3) and (4) should be modified for practical application,
though they reflect the nature of the wave resistance curve
of these ship forms relatively in a wide range of Froude's n um be r.
As another example, the wave resistance of two ship
forms, of which the prismatic curves are shown in Fig. 3, is estimated theoretically with equations (3) and (4), and the results are presented in Fig. 4. This figure indicates that the theoretical formulae are more sensitive for the variation of prismatic curve than the experimental results.
Therefore, it is necessary for effective estimation formu-lae of wave resistance that these defects should be removed.
AP 0.010 0,005 O
//
C, = J_p2v2.'3 C, = J_p2v2.'3\\
R, R, 0.19 0,23 021 0.22 0.23 0.24 0.25 026 0.19 0,23 021 0.22 0.23 0.24 0.25 026Fig. 4 Wave resistance Fig. 4 Wave resistance
/
/
/
/
/
/
EXPERIMENT EXPERIMENTFig. 2 Wave resistance of Series 60 Fig. 2 Wave resistance of Series 60
6
s.s.
6
s.s.
Fig. 3 Prismatic curves Fig. 3 Prismatic curves
/
\
FR FR (5) (5)From the expressions (31, (41 and (5), c is expressed as follows.
2
(1_e_K00c')2(Foofo2+Foifofi +
B
ir
From the expressions (31, (41 and (5), c is expressed as follows. 2 (1_e_K00c')2(Foofo2+Foifofi + B ir
+Ff,) cosO dO
(6)following non-dimensional expression of vsave resist-ance s introduced R +pv2B2
+Ff,) cosO dO
(6) ósi Xsin(,-iq)
(ii1
1 (4)a,.,(Ç,-i))cos(,-iq)
(ia»i( E)
jaj( ¿i)) cos (j q)
0.010 0.005 R, -.pvçj213 THEORY (X(
,
-.pvçj213 (X(,
THEORY+A0fof,
+A1ff,
+Af,2
(3) 1.0 0.8 0.6 0.4 0.2where
F (2bcbc+2bs1bs1)
F = bc+ bs
In the above formula (6), the term of
K,dec'Oexpresses the influence of the draft of ships, F is a function
of Froude's number and ff ¡s
a function of prismatic
curves. F,, and fL are independent of principal dimensions of ship.
In order to investigate the relation of C, and draft by
means of experimental results, the wave resistance of the
following ship forms is introduced from Taylor's chart.
LIB = 7.0
C,, = 0.60
Bld = 2.25, 3.0, 3.75 and 4.5
The Cu,, is plotted against Bld, and compared with the
curves obtained by the following expression.
Csk(1e
Kd)2 (7)The curves of Fig. 5 show that the expression (7), in which
the value of k ¡s determined so as to coincide with the
experimental value at Bld = 3.0. The formula, as a whole,
gives
a fairly good approximation of the experimental
results. This may be interpreted such that the relation be-tween the height of elementary wave and the draft of ship in the range of O = 00 to O = 90e' can be approximately substituted for the relation at 6 = 0°, (0 denotes the angle of propagation of the elementary wave). c may therefore
be written as
This equation indicates that wave resistance coefficient of ships is composed of two kinds of function; a function of
principal particulars and the other of prismatic curves.
..pv2B2
TAYLOR CI-IART
2C71 B B
__)
(1_eid)2(Boofa2+Boifùfi+
(8)
There are several empirical charts for prediction of the
resistance of ship forms. In predicting wave resistance with those charts, it is
often found that the resistance
isestimated accurately when the prismatic curve of type ship
is adopted without modification. This can be understood
easily from the expression (8).
B,, in expression (8) is a function of Froude's number only, and is determined with statistical analysis of the
model test results as described in the next section. 3.4 The Method of Statistical Analysis
Step 1. Calculation of simple correlation coefficients Let n be the number of ship models prepared for the analysis and x, (j= 1---p) be the elements
presenting ship hull form and wave resistance
coefficients of ship models, then,
Sums: Sums of squares: (C) Means: L n MTB 90 June 1974 3.0 40 Bld Fig.5
CWBB/d
-
) 2 Standard deviations:S
n1
Cross-prodUct sums:S =
Cross-products of deviations: Dk1(Xkk) (x)
Simple correlation coefficients:
Dk,
- l Dkk I
Step 2. Calculation of regression formulae
Let us put the regression formula as follows.
y = ao+alxI+azx2+ + a,x,
then
Matrix of :
Inverse of
r"
(i) Regression coefficients:a (, 1)» ,
C
-(k) Sum of squares attributable to regression:
RS a D
Sum of squares of deviation from regression:
DS=DRS
Coefficient of determination: R2 =
Multiple correlation coefficient: R
2 DS
Variance:
S -
n_q-1
Standard error of estimate: SProportion of total variance added:DSR I - DSk
D.
(I) (m) n) 0.015 0.0 10 0.00 5 0 17 0 2,0"TB9O June 1974
3.5 Form Factors on Viscous Resistance of Fine Ships
Three dimensional extrapolation method is introduced
for separation of wave resistance from total resistance with
the Schoenherr's friction line as the basic line of frictional
resistance.
The form factor on viscous resistance is usually obtained in low speeds.
lt
is difficult, however, to determine thefactor by the test results since the measured values are of lowest accuracy, and, therefore, the accuracy of the
predic-tion formula of wave resistance is influenced by the error of form factors. n order to eliminate the above defect, a prediction formula of the factor is obtained statistically on
the basis of theoretical study.
The expression (9)
is proposed for the form factor of
fine ships.
(91
where
k :constant
maximum sectional area
S : wetted surface area In the expression (9), k'
LIB
J Co where k' constantis introduced and the following expression (10) is derived.
A = 1.5b
-L / B
s/ d
This formula is used for separation of wave resistance co m Po n en t.
3.6 Prediction Formula of Wave Resistance of Ship The expression (8) in the preceding section, shows that wave resistance coefficients can be represented by principal dimensions and a number of ordinates of prismatic curves.
Here let us investigate the process of determining B statistically. The ordinates used for the prediction have to satisfy the following conditions.
Representing the features of prismatic curves. Being well correlated with wave resistance. Being independent of each other.
Correlation coefficients between the ordinates of
pris-matic curves and wave resistance coefficients were evaluated
by (g) and the proportion of total variance added by (ii) was examined. As a
result, the following ordinates of
prismatic curves were selected as independent variables.
C1, 0.65 : S.S. FP, 9 1/2,9,8,6 1/2,3 1/2, 1
0.65 <Cò 0.75: S.S. FP, 9 3/4, 9, 7, 3, 1
The distribution of these ordinates are presented in Fig.6 and Fig. 7. Fig. 6 shows an example of the prismatic curve of Cb = 0.60, and the figure also explains that the ordinates
near bow are effective for wave resistance. On the other
hand, Fig. 7 shows an example of Cò = 0.70. In this case,
1101 1 .0 0.8 0.6 0.4 0.2 1,0 0.8 0.6 0.4 0.2 AP
o used point for estimatiolt of resistance
6 7 8 9 p
S.S.
Fig. 7 An example of prismatic curve, Cb 0.70
almost all ship hull forms having a parallel part, the used ordinates for the prediction are distributed fore and,. aft
parts of the prismatic curves.
Anayzing the test data of about 100 models whose
block coefficients are under 0.65, the regression coefficients
are obtained. The Froude's numbers introduced are from
0.13 to 0.27 with a step of 0.01.
On the other hand, when block coefficients are from
0.65 to 0.75, about 70 models were analyzed. As the
designed speed of these models is lower than the formergroup, the range of Froude's number is from 0.1 1 to 0.23.
The standard errors of the wave resistance coefficient calculated by the prediction formulae range mostly from 0.0002 to 0.0003 and therefore
it can be said that the
formulae are accurate enough for estimation of wave resist-ance of ships in hull form design.
Selecting four models at random, the predictions by the
above formulae are compared with the experimental data
in Fig. 8. These four models have various ship form and designed speed, therefore the curves of wave resistance
coefficients are different from each other. Namely. B and C models are designed at high Froude's numbers, D is designed
at a low speed, and the designed speed of A is situated
between C and D.
Next, let us investigate which element of ship form is
effective
for the accuracy of prediction formulae and,
moreover, the variation of the effect according to Froude's
number. An example of the results is shown in Fig. 9, which presents the proportions of total varianco added of
ship hull form parameters to residual resistance coefficients based on Froude's number. The figure shows that B/L and B/LO (L, is the length of entrance) are effective for
regres-sion of residual resistance coefficients at low Froude's number region, and block coefficient (Cò) is important at
high Froude's number zone. In the medium Froude's
number zone, the effect of B/L is reduced and other factors become effective. It may be reason why the factor B/L is
effective
in low Froude's number region that the form
factor is
affected by B/L. In the high Froude's number
used point for estimationof resistance
AP 2 3 4 6 7 8 9 FP s s.
zone, most of the wave resistance curves used in the analysis
show a hump, and the increment of the resistance coeffi-cient is related to Cl. This fact corresponds to the above
mentioned effect of C1. The blank zone of Fig. 9 is the area corresponding to the variance of residual resistance coeffi-cient.
The factor of B/L. is one of parameters representing the
characteristics of prismatic curves, and, therefore,
it
is recognized that the sum of the contribution of B/L. and ordinates of prismatic curve shows the effect of prismaticcurves. The proportion is considerably large in all the range except the zone of Froude's number of 0.25 and above, and
this shows that prismatic curves play an important role in
predicting residual resistance coefficients.
o o E o o 0.5 ç) 1.0 0 0.005 0.10 0.00 5 0.005 o 0.12 4V27,,23 A B BIL 0.14 PREDICTION TANK TEST 0.19 0.20 0.21 0.22 0.23 0.24 0.20 0.26 0.27
Fig. 8 Comparison between predicted wave resistance and measured
ORDINATES OF PRISMATIC CURVE)
_-'
B/L, ,,' I -c1i
Bld 0.16 0.18 OLWL 0.20 0.22 0.24 0.26 0.28Fig. 9 The effect on residual resistance regression
3.7 The Applicable Range of These Prediction Formulae The prediction formulae presented in the expression (8)
of the previous section 3.6, are derived by the statistical
analysis of model test results based on the theoretical
investigation. Namely, in
the formulae, the theoretical
relation between the wave resistance and principalparticu-lars of ship
isintroduced and the prediction formulae
express the resistance curves of various merchant ships
accurately. Therefore, the relation may be applied in the
range of common merchant ships.
A
'B
6
Fig. 10
Cw'fj
In the expression of the effect of prismatic curves on
wave resistance, the coefficients of B are the function of
Froude's number and the selection of square stations
cor-responding to f,. On the other hand, B obtained by the
statistical analysis of test data, are the function of not only Froude's number and J, but also all the correction factors, e.g. the effect of viscosity of water on wave resistance and
other higher order correction and so on. Though these
correction factors are included, the wave resistance
coeffi-cient (CJ is estimated accurately by the expression of (8).
This means that C, is related with f by a second order
homogeneous expression, despite the large discrepancy between theory and experiment. Namely, a theoretical
rela-tion between f. and C, is expressed n A of Fig. 10. On the contrary, the relation in the estimation formula is varied to
B of the figure, owing to the modification of B -
Theapplicable range of regression formulae may be commonly as follows.
where
cn
: the standard deviation of f.
In this case, considering that the regression formulae are obtained based on the theoretical expressions and the formulae are very accurate, it may be permitted that the formulae are used over the usually applicable range to a
certain extent. The line C in Fig. 10 expresses an example
X
'
MTB 90 June 1974 C C 0.005 0 fi mean fiaMTB9O June 1974
that the optimum value is not in the applicablerange. Even
in
such a case, the applicable range of the prediction
formulae can be increased further by conducting the towing
tests on a model designed in an outer range of J and by
returning the test results into the formulae.
The mean and standard deviation of the ratios of princi-pal dimensions are presented in Table 1.
Table i Mean and standard deviations
A :. C 0.65 B : 0.65 < Co 0.75
4. Improvement of Ship Hull Form
lt
is very valuable to know in the design of ship hullform whether a given hull form has some room for improve-ment in propulsive characteristics or not.
Methods of improvement of ship hull forms may be
divided into two kinds. The first is to vary ratios ofprinci-pal dimensions and the second is to improve the shape of
prismatic curves. The former is conducted based on the results of series model test, or by the chart for resistance estimation derived from series model tests. In the latter, It
is important to evaluate the variation of resistance owing to
the modification of prismatic curve in hull form design.
However, there has been no definite method to reply this requirement, and it is the present state that the relationbetween resistance and modification of prismatic curves is
predicted by comparing experimental data. The present estimation formulae can also be applied for this purpose
as they relate the wave resistance with the ratio of principal dimensions and the prismatic curves.
In this section, an improvement method with influence
lines obtained from the prediction formulae is presented.
On the basis of the expression (8), the effect on wave
resistance of partial
modification of prismatic curve
ispresented. In equation (8), putting J, to J±J and
elimi-nating the second order terms, the wave resistance coeffi-cient corresponding to the modified hull form is expressed as follows.
C
Co+(Bofo+Bi,Ji+Bzifz+
+2Bf+
+B,,,f,,) tJ,
= c0+
whereC0: the wave resistance coefficient of the original
Mean Standard Deviation 6.829 0.382 2.352 0.253 0.7050 0.0441 hull form In the expression (11),
aCOBf+BJ+Bf+
+2BJ+"-+BJ,
(12)corresponds to the influence line introduced by Hogner$18)
By the expression (12), the influence line is obtained and
on the basis of the line the effective parts of prismatic curve for reducing wave resistance are presented.
The resistance of the hull is estimated and the effect of the modification can be evaluated. In the repetition of the
above process, the excellent ship hull form can be obtained.
The characteristic of this method is that the variation of
wave resistance can be estimated with high accuracy. The influence line is expressed in the form of a series of
the sections used in the estimation formula as shown in
Figs. 6 and 7. By those points, the followingsare expressed. The size of bulbous bow
The hollowness about bow The shape at fore shoulder The shape at aft shoulder The fullness of run
By these information, the characteristics of prismatic curve are presented.
5. Design System of Ship Hull Forms and Some Examples
5.1 Design System of Ship Hull Forms
Prediction formulae of self-propulsion factors and wet-ted surface area can be obtained in a way smiIar to the
wave resistance. A ship hull form design system are
there-fore composed as presented in Fig. 11. The main parts of this system are the data files of electronic computer. The
data files are composed of two parts (A) and (B). Results of
model experiments are stored in the file of (A), and (B) contains the coefficients which are derived from statistical analysis of the data in the file (A). Wave resistance
coeffi-cient C, form factor K, self-propulsion factors t, w
, eand wetted surface area S are estimated with this data file. In addition to these propulsive characteristic elements,
influence lines are calculated.
When ship hull forms are designed satisfying the design
conditions, the model tests are conducted and he results are stored again in the data file (A). The coefficients of estimation formulae are renewed by statistical analysis
including the new data.
This system is introduced in the initial design system19)
The formulae which compose this system are named
Statistically Processed Power Estimation Formulae, abbre-viated as "SPEF".
5.2 Some Examples of Ship Form Design
The three ship forms which are designed with the above
system are presented, comparing the estimation and the
Group A B Items
L/B
B/d CbLIB
Bld Cb 0.353 0.283 0.04 10 6.885 2.53 5 0.6070M0DIFICATI0N
ND
DESIGN CONDITION
PRINCIPAL PARTICULARS
.'_ [HULL. FORM)
-& PRISMATIC CURVE
DATA FILE (B)
PREDICTION
FORMULAE
C.K. I.
INFLUENCE LINE PROPELLER DESIGN & POWER CALCULATION
PROPELLER DESIGN & POWER CALCULATION
ANALYSIS OF TEST DATA
DATA FILE (A)
TANK TEST RESULTS
Fig. 11 Block diagram of "SPEF" system
results of experiments.
5.2.1 A liner, C = 0.60
An example of typical hull form of liner
s given. Thishull form is the first one which is designed by this system.
In
Fig. 12, the resistance curve as obtained by the
prediction of this ship
is compared with the measuredvalues in the form of residual resistance coefficient based on Prandtl-Schlichting's friction line. The figure shows that the prediction agrees very well with the results of model test.
0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 YLWL
Fig. 12 Comparison between predicted residual resistance and measured
5.2.2 A general cargo ship, Ch = 0.60
This hull form is designed on the examination of the
effects of the following parameters on the wave resistance under certain design constraints.
Entrance angle
The shape of fore shoulder The length of entrance and run
MTB 90 June 1974
In Fig.
13, the resistance curve by the prediction
formulae is compared with the model test results. The test results are slightly larger than the predicted curve.
0.0 10
0.005
GENERAL CARGO SHIP
Ch =0.64
- PREDICTION o TANK TEST
Fig. 13 Comparison between predicted residual resistance and measured
5.2.3 A passenger cargo ship, C) = 0.64
This ship was designed for special purposes. The charac-teristics of this ship form are as follows.
The designed Froude's number is relatively high.
The center of-buoyancy is situated relatively
back-wards.
In this ship form design, prismatic curves are represented
with polynomials and the relation between the wave
re-sistance and the coefficients of polynomials
isinvesti-gated by the prediction formulae. On the basis of the above mentioned study, this hull form is selected.
Fig. 14 shows that the prediction agrees with experiment except low Froude's number zone.
The above examples show that this prediction method
is very useful for excellent hull form design and for predic-tion of the wave resistance.
F,-IOLWL
Fig. 14 Comparison between predicted residual resistance and measured
6. Improvement of Series 60 Hull Forms by the Present
Method
As examples of the application of the present method,
0.0 10 v22R,, CARGO LINER (=0.60 0,005 PREDICTION "J Q-o, uJ z
-o TANK TEST C:, CI) O "J cLPASSENGER CARGO SHIP
Ch 0.58 R», J. PV2 V21 PREDICTION o TANK TEST Q, IJJ -z 12. Q, u 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.010 o
MTB9O June 1974
the modification of ship hull forms of Series 60 ( C = 0.65
& 0.75> is presented.
6.1 The Case of Cu = 0.65
The mother ship form of series models is selected as the
object of this improvement. The original model number in
DTMB is M.4218. The principal particulars of this hull form are as follows.
L,XBxd :
145.00x20.00x8.00m
Displacement: 15495 tL/B, B/d
: 7.25, 2.50 C1, C9, C, : 0.6507, 0.6626, 0.9820 Service speed, F: 1 7.37 kt, 0.235The service speed is determined by the Troost's formula. The towing test of the original hull form was also
conduct- 1.0-0.8 0.6 0.4 0.2 o 0.03 0.02 001 AP i
Fig. 16 Modification of prismatic curve
SERIES 60 = 0.65 ORIGINAL MODIFIED 2 3 4 S.S. --ORIGINAL MODIFIED AP
ed in our Experimental Tank.
The influence line of the hull form is calculated by the
previous formulae and shown in Fig. 15. The figure shows
that the
hullform has some room for improvement.
According to this information, improvement of the hull
form was attempted to decrease the wave resistance by one half at service speed.
The modification of the prismatic cunies is shown in
Fig. 16 and the original and the modified prismatic curvesare presented in Fig. 17. The difference is maximum at F.P., adopting a bulbous bow of 6% of maximum sectional area. The other part is modified within 2% of maximum sectional
area. The body plan and the profile of stern and stern are
given in Fig. 18 comparing with the original hull form.
The model experiments were conducted n our
Experi-mental Tank with a model 7 meter in length. The results of the resistance test are presented in Fig. 19 c;omparing with
the experiments of the original model. The figure shows
that the prediction formulae are very effective and the wave resistance is decreased by one half at designed speèd. The
wave resistance is obtained with Schoenherr's friction line
and the form factor is 0.122 as presented in Fig. 19. The shaft horse power versus ship's speed is estimated
with these experimental results and shown in Fig. 20. The figure shows that the service speed of the improved hull
form increases by 0.32 kt.
SERIES 60
C1 0.65
-TEST RESULTS OF ORIGINAL FORM DF OF MODIFIED FIRM
PREDICTION OF MODIFIED FORM
K=0.122
Fig. 18 Comparison of body plans, Series 60 (C1, = 0.65)
WI o-'4 Ó W z e, I) Ui FP J
/
018 019 0.20 0.21 0.22 0.23 024 0.25 0.26 0.27 = ,1 L/
-0.01 s s -0.02 F. = 0 .23 - 003Fig. 15 Influence line of Series 60 (Cb = 0.65)
SERIES 60
= 0.65
2- 4 f= ÍMOD'4EDIOR6INIL
Fig. 17 Comparison of prismatic curves Fig. 19 Comparison of wave resistance
7 PP Z -1 -2 0.0 10 0.005 R, lp,223 Full Load
7000 a) o-u, 6000 5000
Fig. 20 Comparison of SHP curves
---ORIGINAL
MODIFIED
SERIES 60 C5 = 0.75
TEST RESULTS OF ORIGINAL FORM
D OF MODIFIED FORM
---PREDICTION OF MODIFIED FORM R,, 18 9,
/
3/ ---'ìj----0.14 0.15 0.16 0.17 0.18 D.19 0.20 0.21 0.22 0.2 F,ILL
Fig. 24 Comparison of wave resistance
Fig. 23 Comparison of body plans, Series 60 (Ch = 0.75)
MTB9O June 1974 6.2 The Case of C4 0.75
The model M.4213 in DTMB s selected as the object of
improvement. This model is the mother ship form in the series of the position of longitudinal center of buoyancy.
The principal particulars of this ship are as follows. LPPXBXd : 145.00x 21.48x 8.59 m
Displacement: 20618 t
L/B, Bld
:6.75, 2.50Cb, Cp, Cm : 0.7514, 0.7590, 0.9900
Service speed, F,: 14.0 kt, 0.189
The influence line and prismatic curves of the original hull form and the modified form according to the influence line are presented in Figs. 21 through 23. The aim of this
modification
is to decrease the wave resistance of the
original hull form by one half at the designed speed. Themodel test results are shown in Fig. 24 and it is presented that the aim is achieved.
SERIES 60 C, 0.75 1.0- 0.8- 0.6-ORIGINAL 0.4-MODIFIED 0.2-0AP I 2 3 4 6 7 8 9 FP o-= 5000 4000 3000 SERIES 60 C, = 0.75 ORIGINAL MODIFIED Full Load 04Okt
i'
\. ,,/
w o /Fig. 25 Comparison of SHP curves
0.03 0.02 0.01 0 -0.01 -0.02 -0.03 F. = 0 .19 Ap 1 2 3
9\Ç
Fig. 21 Influence line of Series 60 (Cb = 0.75) Fig. 22 Comparison of prismatic curves
13 14 15 V7 (t) 0.010 J, 0.005 SERIES 60 C4 =0.65 ORIGINAL MODIFIED 8000 Full Load K= 0.166
MTB 90 June 1974
The shaft horse power versus ship's speed is estimated and shown in Fig. 25 and the figure shows that the increase of the service speed is 0.40 kt.
7. Conclusion
The results of this study are summarized as follows. The theoretical expression of wave resistance is derived as follows.
2C2 (1
le
Kd)z(Bf2+Bff+
C=__Ld
+B,f2)
whereB is a function of Froude's number. f. is an ordinate of prismatic curve.
Prediction formulae of wave resistance are obtained
statistically by combining the above theoretical formula
Taniguchi K., Watariabe K. and Tamura K., On a New Method of designing Hull Form of Large Full Ship, based
on the Separability Principle of Ship Form, J. of the
Society of Naval Architects of Japan, Vol. 120 (1966)p.36
Gertler M., A Reanalysis of the Original Test Data for
the Taylor Standard Series, DTMB Report 806 (1954) Yamagata M., Senkeigaku, (1953)
Todd F.H., Series 60The Effect upon Resistance and
Power of Variation in Ship Proportions, SNAME Vol. 65 (1957) p.445
The 45th Research Committee, The Shipbuilding Re-search Association of Japan, Design Charts for the pro-pulsive performances of High Speed Cargo Liners, (1964) Doust D.J. and O'Brien TP., Resistance and Propulsion of Trawlers, NECI Vol. 75 (1959) p.355
Doust D.J., Optimized Trawler Forms, NECI Vol. 79
(1962) p.95
Hayes J.G. and Engvall LO., Computer-aided Studies of Fishing Boat Hull Resistance, FAO (1969)
Tsuchiya T., New Statistical Regression Analysis for
Fishing Boat Hull Resistance, J. of the Society of Naval Architects of Japan Vol. 132 (1972) p.63
Sabit AS., Regression Analysis of the Resistance Re-sults of the B.S.R.A. Series, ISP Vol. 18 (1971) p.3
References
of wave resistance and test results of ship models. As examples of prediction by these formulae, the wavE
resistance of three ship forms whose block coefficient
are 0.58, 0.60 and 0.64, are estimated by these formulaE and the experiments show that the formulae are usefu for preliminary design of ships.
The process of improvement of ship hull form by thesE
formulae is given and as examples, two hull forms ol Series 60 (C = 0.65 & 0.75) are improved by referrinc
to influence lines.
8. Acknowledgement
The author wishes to express his appreciation to th
members of the ship designing department of Kobe ShinYards and Engine Works and of the Experimental Tank ol
Mitsubishi Heavy Industries, for their cooperation in carry
ing out this investigation.
Sabit AS., An Analysis of the Series 60 Results,
Part 1. Analysis of Forms and Resistance Results, ISPVol. 19 (1972) p.81
Havelock T.H., The Wave-making Resistance of Ships.
A Theoretical and Practical Analysis, P.R.S.-A VoI. 82 (1909) p276
Havelock T.H., The Wave-making Resistance of Ships.
A Study of Certain
Series of Model Experiments,P.R.S.-A Vol. 84 (1910) p.l97
Oortmerssen G., A Power Prediction Method and Its Application to Small Ships, ISP Vol. 18(1971) p.397
Breslin J.P. and King Eng., Calculation of the Wave
Resistance of a Ship represented by Sources distributec
over the Hull Surface, Davidson Laboratory, R.No.972
(1963)
Gadd G.E., A Method for Calculating the Flow over Ship Hulls, RINA. Vol. 112 (1970) p.335
Ohira K., Re-Tests on the Series 60 Models, J. of Seibu Zosen Kai, No.25 (1963) p.5
Hogner E., Influence Lines for the Wave Resistance of Ships-I, P.R.S.-A Vol. 155 (1938) p.292
Aihara K. and Sugawara N., An Interactive Real Tim Design Procedure n Ship Initial Design, COAS 9
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