## 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. 781838}

### Hiroshi 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, where### Cf(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/2### aj(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

and### da

### 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,

_{\Cb}

_{L dl}

### + 2bsbs,) cosß d9

and### 2C2(i

_{.-'-'}A

_{7rCoLdl}

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 X### sin(,-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: S
Proportion 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 -c1### i

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 principal### particu-lars of ship

is### introduced 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

is### presented. 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 followings_{are 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 Cb### LIB

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 _{v22}R,,
CARGO LINER
(=0.60
0,005
PREDICTION
"J
Q-o,
uJ
z

PASSENGER 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 t### L/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

hull### form 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 - 003

Fig. 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