Analysis of ship-side wave profiles, with special reference to hull’s sheltering effect

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ARCHIEF

-&e otQ

iblioheek van bouwkunde Onderad'Iin nische HogeschooI DOCUM EN T AT lE D AT UM g OKT. 1973 HIATIE

ANALYSIS OF SHIP-SIDE WAVE PROFILES,

WITH SPECIAL REFERENCE TO HULL'S

SHELTERING EFFECT

by

Dr. Kazuhiro Mori,

Prof. Takao Inui,

and

Prof. Hisashi Kajitani

Department of Naval Architecture

Faculty of Engineering

University of Tokyo

9th SYMPOSIUM ON NAVAL HYDRODYNAMICS

Paris

21-25 August, 1972

Lab. v.

Scheepsbouwktn

Technische

HogeschooL

(2)

Attempt is made to find out the effctive wave-making source of a ship from the measurement of the hull-side wave profiles.

The integral equation of the source distribution function is simplified and solved numerically under the

specific limitation, (a) rectangular, vertical central plane,

and (b) draughtwise uniform.

Two muid models M20(B/L=O.0746) and M21(B/L=O.1184)

whose hull-generating sources are optimized to give the

minimum wave resistance at Fn=O.2887 (K0L=12) , are tank-tested and wave-analyzed.

The obtained source distribution m() shows a clear

discrepancy from the hull-generating source m() in a

similar way to the so-called p-correction, or

a () = m()/() = i - p

(1-j) , (p=O.4)

Wave profiles, wave patterns and wave-making resistance

are calculated in two ways, (a) from hull-generating source

and (b) from wave-analyzed source m()

The gap between experiment and calculation (a) is

satisfactorily filled up by calculation (b)

From experimental results it is proved that the design

procedure where a(e) is taken into account is very significant.

(3)

to give the theoretical basis for the correction function ()

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Introduction

The wave analysis has two objectives, i.e. (a) to

determine wave-pattern resistance directly, and (b) to find

out the actual wave-making mechanism of a ship-like floating body.

This paper deals with the problem (b) by means of the

measurement of ship-side wave profiles rather than by free

wave patterns in the rear of ships.

Method of Analysis

The co-ordinate system as shown in Fig. 1 is adopted

throughout the papers.

All quantities in the following equations are

dimen-sionless, where L(=L/2) , half length of the ship, and U, the

velocity of the uniform flow are taken as the units of

length and speed, respectively.

Let us assume that the hydrodynamic singularity(source) is distributed on the surface

(1)

Then the perturbation velocity potential at an arbi-trary control point P(X,',2) is given by

-.çt

(x ; ?Z

) '(.S

(2)

(5)

where

r=

(x-

t (tt-t

-f

r=

()Z

(-)

(t;)Z

H = - AA-SO

[c)

(c.,&t

&)

j

k -

KoiSec

ec&

By making use of the well-known free surface condition

K09. ax.

(4)

(5)

zo

on the distribution surface S. The Green function

is

Q-(x.,; ,,)=-- --k-

+ H (3)

the integral equation(2) can be converted to

Jfvfl%sx

(6)

where (z,) denotes the surface elevation in general.

For simplicity, let us confine ourselves to the specific limitations,

the distribution surface is the rectangular,

vertical central plane (-ll, -t.çO)

the distribution function is draughtwise uniform.

Further, the ship-side wave profiles are

selected as the given information of the wave elevation

Thus we have the fundamental integral equation

o

(6)

where and

;.

L

(;+

) 2 (9)

Using above expression for m(), we divide the

distri-bution plane into M number small meshes, and assume that within the meshes the source strength is constant, then the

wave profiles are given by

J

(c)=

(x,) t

()

with (10)

c(.)=

COQ P1t (x.. '. I pl

(z)=

si

nit;'

(. *

i:

t0)

o f

;

₍₁₂₎ & J, a. Izo

For numerical solution of Eq. (7) , the modified Fourier

expansions are introduced as follows

1

m()=ZOrLc.os rtìt'

i- rnt' (8)

P=1 ru

where

By preliminary studies of Eqs. (8) and (10), five term

truncation N=5 are found suitable. Then

5x2=10

numerical

coefficients {b} (n=l, 2, , 5) are determined by

the least square method.

Table 1 shows an example of such preliminary studies.

Starting with the wave profiles which are calculated from

the hull-generating source ffi) of the model M2l at the

speed of Fn=0.2887(K0L=12), the wave analyzed sources m()

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bution function G (r,;,t0) as expressed in Eq. (12).

3. Models and Wave Profile Measurement

Among a variety of hull form characteristics, the

beam-length ratio(B/L) is supposed as the leading parameter for the sheltering effect.

Therefore a set of two muid models M20(B/L=0.0746) and

M21(B/L=0.1l84) are prepared as shown in Table 2 and Figs.

56.

The hull generating sources ifi() of M20 and M21 are

optimized to give the minimum wave resistance at the speed

of K0L=12(Fn=O.2887) under the following restraints

o.oi

=

zo

d 1'

U)

=

0.036 _(MzI)

to

and (13) ìYLt)

( =i)

Three kinds of tank experiments, i.e. (a) towing test,

(b) wave-profile measurement, and (c) wave pattern

measure-ment, are carried out with M20 and M21 for the speed of K0L

=7-'-'20 (Fn=0 .3780 0.2236)

As the typical examples, the results of the wave-profile measurement at the speed of K0L=8, 12, and 16 are

reproduced here in Figs. 7-9, where the two kinds of

calcu-lation, (a) from the hull-generating source () , and (b)

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dotted lines and by plots, respectively.

4. Analysis of Measured Wave Profiles

The proposed method of analysis is applied to the

measured wave profiles of the two tested models M20 and M21

for the twelve speeds K0L=7 through K0L=20.

In this procedure, the measured wave profiles at twenty

positions x=-0.95, -0.85, , 0.95 are adopted as the

principal input data.

In addition, some selected readings of the wave recorder

on the longitudinal cut line y=O.25(x=l.O-2.0) are adopted

as the supplemental input data, which are useful for the definite determination of the source around the stern.

Figs. 10-12 show the wave-analyzed sources m() of M20

and M21 for the three selected speeds K0L=8, 12 and 16,

where the hull generating source ff() is also given for

comparison.

Because of a very low level of wave elevation, the

accuracy of wave analysis is rather poor with the thin model

M20, particularly at the lower Froude number Fn<0.2887(K0L> 12)

In Figs. 13 and 14, the similar results at K0L=lO, 11 and 12 are summarized with M20 and M2l, respectively.

From Figs. 10 through 14, a clear discrepancy, which is roughly proportional to the beam-length ratio of the models,

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is observed between the wave-analyzed source m(E) and the hull-generating source ii()

For the convenience of further studies including the

effect of Froude number, the ratio of the two kinds of

sources, or the correction function

(E)=m()/i() is

calculated with the wide model M2l.

The results are reproduced in Figs. 15-17.

Here it must be remembered that the relative accuracy

of cx() is poor around midship, because of T() being null

for =O.

With respect to the sheltering effect, the present

authors [l [2] [3] suggested a simple, empirical correction,

like

c)= t

,Ah_(I

-) (14)

with

,v.O4

+or

8/Lo.I2o.l5

In Figs. 15-l7, Eq. (14) , which we call u-correction, is

also given for comparison with the wave-analyzed result of

M21(B/L=O. 1184)

It is noticeable that the general tendencies of the

wave-analyzed correction function are of quite similar

tendency to the simple, empirical relation (14), except the higher Froude number Fn>O.30(K0L<1l).

At the higher speed range, the inclination of c() is

getting steeper with increasing Froude number.

Therefore Eq. (14) is modified here to a more general

expression, like

(10)

The results of the generalized straight line

approxi-mation, which is applied to the wave analyzed correction

function of M2l,are given in Fig. 18 together with the original proposal (14)

5. comparison with Tank Experiment

Before entering the discussions on the theoretical

basis for the obtained correction function a(), its

justi-fication and usefulness are examined by comparison with tank

experiments on four items, i.e. (i) wave profiles, (ii) wave

patterns, (iii) wave-making resistance, and (iv) amplitude

functions.

(i) Wave Profiles

By direct integration, the model's side-wave profile is

calculated from the wave-analyzed source distribution m()

which is reproduced by the plots in Figs. 7--9.

Its satisfactory agreement with the measured wave-profile (full-lines) makes a striking contrast with the

rather poor result of the existing theory(dotted lines)

A slight phase shift, however, is observed with the

first wave crest of the wide model M21, which appears to be

attributable to the non-linear flow effects in close vicinity

of the stem.

To find out the accuracy of approximation, similar

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.i-correction (0.4) , whose result is also presented in Figs. 7-9.

(ii) Wave Patterns

The measured wave contours of the tested two muid models M20 and M21 are obtained at a single speed Fn=O.2887

(K0L=12) by cross fairing of the longitudinal cut wave recordings of every 5cm intervals from y=O.25 (close to the model's side) through y=l.75(tank side wall).

The final results are reproduced in Figs. 19(M20) and

20 (M21) respectively.

The corresponding calculations are carried out in three

different ways.

by existing theory, or from the hull-generating source

,

from the wave-analyzed source m()

by u-correction (i=0.4)

These calculated wave patterns are presented in Figs. 2l-24.

The existing theory (a) (Fig. 21) shows the poorest

agreement with experiment. Particularly, the transverse

waves are tremendously exaggerated in this calculation in accordance to the author's previous suggestions1'

The clear disagreement between existing theory (a) and experiment is again improved successfully by the present

approach (b)

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details of the measured wave patterns of the tested models

M20(Fig. 19) and M21(Fig. 20) are beautifully reproduced in

Fig. 22(M20) and Fig. 23(M21), respectively.

With respect to the effectiveness of the simple

correc-tion Cc) , Fig. 24 shows its relative merits and demerits in

comparison with (a) and (b).

(iii) Wave-Making Resistance

Ordinary towing tests are carried out with M20 and M2l by fitting the plate-stud stimulator at 0.05L behind the stem.

The wave-making resistance coefficients

= Rw/--

UZ Lz (16)

which are obtained by adopting Schoenherr-line with form

factor K=0.07(M20) and K=0.15(M21), are presented in Figs. 25 and 26.

In these figures, the three kinds of calculations are also given for comparison with the experiment,

Cw calculated by existing theory, or from the

hull-generating source t()

Cw calculated from the wave-analyzed source m()

Cw calculated by u-correction (i=0.4)

Calculation (b) again shows the best and the most

satisfactory agreement with experiment in contrast to

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(iv) Amplitude Function

Comparison is also made on the amplitude function of

the total free waves for the speed of Fn=0.2887(K0L=12). Figs. 27(M20) and 28(M2l) show four kinds of amplitude

function, i.e.

amplitude function calculated by existing theory, amplitude function calculated from wave-analyzed source distribution,

amplitude function obtained from the measured free

waves by longitudinal cut method,

amplitude function obtained from the measured free waves by transverse cut method.

As easily observed, the amplitude function (b) shows

the highest average level of free wave amplitude,

particu-larly in the transverse wave range. It appears that the

difference between (b) and (c) or (d) may be partially

explained by the wave-breaking resistance.

In Figs. 25 and 26, Cw at the specific Froude number

Fn0.2887, which is obtained from the longitudinal cut

method (c) , is also presented.

6. Search for Theoretical Basis of Correction Function c(E)

The practical usefulness as well as the experimental

justificaion of the obtained correction function

_{() are}

clearly demonstrated in the preceding sections.

From a theoretical point of view, however, its hydro-dynamical mechanism still remains open for further investi-gations.

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T

-

[ sec& e (K0pAec&) 9 (19) '9

3=_4ç

ci9.Ç

J_9

)

p= (x-i)

c& t

(-'L)/(-fl (18) e - ec48

From the standpoint of the boundary condition which is

used to obtain velocity potential in Eq. (2) , two possible

causes for the discrepancies between theoretical and

wave-analyzed results can be mentioned. Namely, (a) finite Froude

number effect for the hull-surface condition, (b) non-linear

effect for the free-surface condition.

For the time being, preliminary calculations of these

two higher order terms are carried out with respect to the

amplitude functions.

6.1 Hull Surface Condition

Throughout the present paper, the hull-generating

source () is derived from the so-called double-model

approximationm which is correct only for limiting case Fn-*O or K0L-*co.

To find out the effect of finite Froude number, the

Green function G(Z,,Z;,n,) expressed in Eq. (3) is

rewritten as

%1Zt3

(17)

(15)

In Eqs. (17) through (21) , the second term G2 denotes

the free wave component which propagates oscillatorily to

the rear of a ship, while (G1 + G3) represents the local disturbance.

Differentiating Eq. (17) as to x, y, z and integrating all over the distribution plane, the velocity components

u,

y,

w are also written as

u = u1 + u2

+ U3

y = y1 + y2 + y3

(22)

w = w

+ W2 + W3

In finite Froude number problem, the free wave terms

u2,

y2,

and w2 and a part of local disturbance

U3, y3,

and

w3 play important role which is shown in Fig. 29 in the case

of M21 and K0L=12.

Based upon the "exact" hull-surface condition,

V Lt- (23)

where y=f(z,z) denotes the half-breadth of the hull, the

"exact" hull-generating sources for finite Fraude number K0L=12 are obtained with M21.

This result is presented in Fig. 30 together with the double-model approximation r() and the wave-analyzed

source m()

The corresponding amplitude function is also given in

Fig. 31.

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effect of finite Froude number for the hull-surface condi-tion can explain the cause of the correccondi-tion funccondi-tion c()

partially, but not completely.

6.2 Free-Surface Condition

Before discussing the problem (b) , we assume that the second order term for the free-surface condition is indepen-dent of the higher contribution of the hull-surface

condi-tion (6.1) . The velocity potential in Eq. (2) is obtained from the linearized free-surface condition,

UZ'

₊

ax.1 ¿=0

Assuming that the second order term for the free-surface

condition is independent of the hull-surface condition, the approximate calculations are carried out as follows.

Using small parameter E (we can choose

E=B/L) ,

we can

get the well-known expansions

£-E'c

-r3&4t

ç =

1- -t

Then the kinematical condition of free surface

(U-s-)

-

-a4' o is written as ¿ U

'J

¿2 2---x. -

r -_ (27) + o

(24)

(17)

is expressed by

2

(29)

L

+

Paying attention to order of E, we have following two

to and E2

equations as E respectively.

where

Similarly, the pressure condition

(J t U _4'

z J

Eq. (31) , which is the second order term of the

free-surface condition, can be written as

+t

' _az Z

=

('e, ») U r z i ..a - +

H

ax _j

-

f ¿ ( -a4 \2 (34)

The second order term of the velocity potential 2 is

given in accordance with well-known Fourier double integral

UZ

-(30) a L2 - u j

(_?..

(L\Z

a ax

--

ax) k a _j - \ j (31) (28) Iz

(18)

J';(X'')

o)

ZQ'

₍₃₅₎

where G(z,,;z',',O) is given in Eqs. (3) and (4)

Once is known, ' (z,) can be calculated by Eq. (33)

all over the z=O plane, then 2 is obtained by Eq. (35)

Eq. (35) means that the velocity potential of second order

term is just that of source distribution y (z,), which is

distributed over the free surface.

The second order amplitude function corresponding to

2 is given as follows after some approximations.

C2($)/L

_{= 4-}

5ec& ') c.o ( o1secL&vt&)- Q (&')

n (sesi)

_J

o

(2ss*') -Q(&') CRO (.2cc',te')J

(36)

where

=

-f

K.tcec& %Si'

(.hec&x') d.z'

(37)

(c')= -.--

KoiSc _cs

(4sec&L') ¿tX.'

Fig. 32 shows the first order amplitude functions, C1 (o)/L, s1 (e)/L of the model M21 at the speed of K0L=l2 which are calculated in three different ways.

by existing theory, or from ir(E)

by wave-analyzed source m() by p-correction (pO.4)

The difference between the result (a) and (b) is

presented in Fig. 33 together with the second order

(19)

Eq. (36)

Fig. 33 suggests that the second order correction for

the freesurface condition is important only for the

diver-ging wave range, where the wave-slope is predominant.

Consequently, it appears that the remarkable

discre-pancy which is observed in the transverse wave range cannot

be explained by this kind of non-linear effect.

By summarizing the preceding discussions (6.1) and (6.2) it may be safely concluded that the real mechanism of the sheltering effect cxU) should be investigated not only by

the second order considerations but also from some different

kind of approaches such as suggested by

Brard41

or Pien and

Chang151.

Application to Hull Form Design

Although the theoretical basis of the sheltering effect still remains unsolved, the presently obtained results will be of practical use for the hull form design with least wave resistance.

As an example, an asymmetry muid model M21-Modified is

designed under the same geometrical restraints as M2l except that

(a) The approximate correction function c.() which is given

in Fig. 18 is applied in the process of minimization of wave resistance.

For example, at the designed Froude number Fn=O.2887 (K0L=l2), we have

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10.3 + 0.8 J , tore body

(38)

0.7 + 0.3 , aft body

The correction function a(E) being asymmetry fore and

aft, the optimized hull form is also asymmetry under the

following restraint,

Total source = = = 0 (39)

The modified Fourier expanJions are used for iñ()

It is expected that the modified asymmetry model

M2ì-will be superior to the original model M2l as far as wave

r.sistance at the designed Froude number is concerned.

Figs. 34 through 36 show the calculated results. i.e.

(a) hull-generating source, (b) load water line, (c)

ampli-tude function, respectively.

The towing tests of M2l-M were run on May 9th through

12th, 1972 and Fig. 37 shows its result. As far as wave

resistance is concerned, the agreement between experirtient

and the present theory is noticeable. Moreover, M2l-M is

much less, as expected, than M21 in wave-making resistance

at the designed Froude number.

These results may suggest that the principal leading

factors of the sheltering effect are B/L and the shade of

water plane as discussed in the preceding sections. In this

connection, it must also be remembered that the simple

Ti-correction of Eq. (14) was originally proposed not with

muid models but with Pienoid models.

Therefore, the correction function a(e) which is obtained

with M2l(B/L=0.1l84) will be applicable not only to

muids

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8. Conclusions

Analytical method for obtaining the effective

wave-making source of a ship is developed, where the measurement

of the hull-side wave profiles is adopted as input data,

instead of the free wave patterns in the rear of a ship.

Two muid models with different beam-length ratio (B/L

0.0746, 0.1184) are wave-analyzed.

Clear discrepancy is observed between the two kinds of

sources, i.e. (a) hull-generating source () based upon

double-model assumption, and (b) effective wave-making

source m() analyzed from measured wave profiles by means of

the proposed method.

The correction function cï() = m()/()

which is obtained

from the wide rnodel(B/L=0.1184) is almost identical with

the authors' proposal for the sheltering effect(l968),

= 1 -

ii(l-IJ),

(p=0.4)

The second order calculations for hull-surface condition

as well as for free-surface condition are found insufficient

to give the theoretical bases

of

the obtained correction

function c.(E), which suggests the importance of the hull1s sheltering effect.

Considering cz.() as a correction function, M21-M is

designed under the same restraint condition as that of M2l..

From towing test results, it is proved that M21-M is much iess

than M21 in wave-making resistance and such a design

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T. Inui, H. Kajitani, N. Fukutani and M. Yamaguchi

"On Wave Making Mechanism of Ship Hull Forms, Generated from Undulatory Source Distributions", Journ. Soc. Nay. Arch. Japan, Vol. 124 (Dec. 1968)

Do. : Selected Papers from Journ. Soc. Nay. Arch. Japans.

Vol. 4 (1970)

T. Inui and H. Kajitani : "Sheltering Effect of

Compli-cated Hull Forms", Proc. 12th mt. Towing Tank Conference

(Rome, 1969)

R. Brard : "The Neumann-Kelvin Problem for Surface Ships",

Report 11CST (1971)

[51 P.C. Pien and M.S. Chang : "Potential Flow about a

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Remarks m() = Wave-analyzed Source

CE) = Hull-generating Source

m()

_N=4 _N=5 _N=6 N=8 -1.0 0.3207 0.3172 0.3159 0.2546 -0.0791 -0.9 0.2927 0.3015 0.2936 0.3112 0.2772 -0.8 0.3740 0.3796 0.3735 0.3712 0.3891 -0.7 0.4911 0.4980 0.4907 0.4979 0.4865 -0.6 0.5920 0.6042 0.5922 0.6036 0.5872 -0.5 0.6437 0.6595 0.6439 0.6442 0.6649 -0.4 0.6297 0.6444 0.6294 0.6291 0.5992 -0.3 0.5468 0.5577 0.5465 0.5611 0.5327 -0.2 0.4027 0.4108 0.4028 0.4219 0.4328 -0.1 0.2134 0.2218 0.2137 0.2123 0.1909 0.0 0.0000 0.0110 0.0001 -0.0231 -0.0434 0.1 -0.2134 -0.2007 -0.2136 -0.2281 -0.2153 0.2 -0.4027 -0.3911 -0.4029 -0.3863 -0.4881 0.3 -0.5468 -0.5376 -0.5466 -0.5198 -0.6691 0.4 -0.6297 -0.6197 -0.6294 -0.6324 -0.5650 0.5 -0.6437 -0.6254 -0.6438 -0.6717 -0.7467 0.6 -0.5920 -0.5578 -0.5922 -0.5853 -1.1521 0.7 -0.4911 -0.4410 -0.4910 -0.4251 -0.2983 0.8 -0.3740 -0.3198 -0.3740 -0.3400 -0.5465 0.9 -0.2927 -0.2514 -0.2936 -0.3322 -4.4894 1.0 -0.3207 -0.2909 -0.3138 0.1284 36.416

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Item M20 M21 L(m) 2.001 2.001 B(m) 0.1492 0.2368 D(m) 0.2055 0.2424 d(m) 0.1405 0.1724 (m3) 0.0178 0.0347 S(m2) 0.5390 0.6686 B/L 0.0746 0.1184 d/L 0.0702 0.0862 Designed Speed, K0L=12 Condition of Restraint

v=O.O18

V=0.036

Distribution Plane T/L=0.04 (muid)

a1 1.08722 2.17444

Source

Distribution a3 -2.05643 -4.11286

(25)

(EP)

I

o

Fig. i Co-ordinate System

1.0

(26)

0.0

t I I

M2O

M21

_1.0

(AP)

KL=12

1.0 _(A.P)

0.01

0.0 ° FieLd Point of M 20

----DQ

ofM2l

1.0 (A.P)

Fig. 2

(27)

Fig.

3

The Dffective

7ave Height of Finite Sheet (2)

YL12

M20

M21

_1.0

(A?)

&- Fietd Point of M 20

1.0

(28)

-1.0

(EF?)

-1.0

(EF?)

- Field Point of

M 20

of M 21

KoL=12

Fig. 4

The Effective Wave Height of Finite Sheet (3)

(29)

0.12

0.14

0.16 zu

9

(30)

3.02

3.04

0.06

0.08 Ojo

0.12

0.14

016

0.18 z/[

9

/

81/2

/

(31)

3/L xlOO

2.0r

_2.0-

_0.0

-2.0

-1.0

-(Fi?)

M21

0.0 Fig. 7

Wave Profiles of M20 & M21 (K0L=8)

Measured

CaIcuated

- 02 with /-Correction(M=0.4)

o

DQ from Anatyzed S.D.

2.0 -I'0L8

/

0.0 _-2.0

(32)

3/Lx 100

1.0-

0.0 _-1.0--

1.0

0.0 -1.0--1.0

- (FI?)

-lo

-(F.P)

M20

N

Measured

CalcuLated

- DQwith U-CorrectionCU=0.4)

°

DQ from Analyzed S.D.

1.0 Fig. 8

Wave Profiles of ?I2O

& M21 (K0L-12)

F0L12

(33)

5/LxlOO

1.0 r-1.0

_0.0

_-1.0

Fig. 9

Wave Profiles of M20 & M21 (K0L=16)

Measured

Catcutated

-DQwith ,a-Correction(p.=0.4)

o

DQ from Anatyzed S. D.

1.0 K0L=16

0.0 _-1.0

(34)

m()

_O5r

_0.0

-1.0 (FI?)

Double Model Approx.

Analyzed M20x2

m()

DQ

'M21

05 K.L=B

Fig. 10

(35)

m()

0.5

0.0 -lo

(FP)

-05

K0L12

Double Model Approx.

Analyzed M 20x2

09 M21

Fig. 11

Wave-Analyzed Source Distribution

(K0L=12)

(36)

00 KoL=16

N

Fig. 12

Wave-Analyzed Source Distribution (K0L16)

Double Model Approx.

Analyzed

M20x2

m()

DQ

M21

0.5

1.0

(A.P)

-/

(37)

m()

0.5-0.0

M 20x2

Double Model Approx.

K0L=10

11 12

0.5

Fig. 13

Wave-Analyzed Source Distribution (M20)

1.0 (A.P)

m()

-1.0

(ER)

(38)

0.0 -1.0

(EF?)

M21

Fiq. 14

Wave-Analyzed Source Distribution (M21)

Doubte Mode' Approx.

K0L=1O

11'

12

t

(39)

0.5 0.0-

1O-o

00'-

o

O5

0.5

I i

-1.0

(F.P)

0.0-KL=S

o o o e o o o

o

M21

o o I L ¡ i ¡ I i i t ¡ i I _i

i

i j

-0.5

0.0

0.5

1.0 (A.F?)

Fig. 15

The Ratio of m()/()

(K0L71O)

Wave Analysis

1h-Correction

(0.4)

O o o o

0.0-

1.0-o o

(40)

10-0.5

00 tO

0.5

0.0

1.0

0.5

0.0

L=11 K0L=12,

°

K0L13

o _o o

Wave Ana[ysis

Correct ion

(iL

=0.4)

M21-o

t

i I I i i I i j i J i I

00

05

1.0 (14.P)

(41)

0.0

1.0

0.5

0.0

10

0.5

00

10

05

00

o L

j1

K.L20

I I I

-0.5

o o o

M21

i i J I t I I

00

0.5 o

Wa

Analysis

Jh-Correc t ion

(z=0.4)

o

Fig. 17

The Ratio of m()/()

(K0L=152O)

I J J I

lo

(42)

LO

0.OL

-1.0

(EP)

I i

K0L=12

(=0.4)

Fore Part

K0L9 lQ

KoL=7.8

Aft Part

K0L=7 8

K0L=1415,1618

K0L=14 20

IL=1516 18

J

)

j L I i i t J I

0.0

0.5

1.0 ()

(AP)

K0L=1O 11

Fig. 18 The Correction Function for M21 Type Ships

K0L=11 12

1.0

(43)

-1.0

(F.P.)

YA

-v

ii

)

4

"II

---'s iii

I I

J //

'j ,

-?

-2

,

/ir -0

00

0.0 ()

Fig. 19 Measured Wdve

r,

_{// ., / ,} - )

//

,;',/

-1.0

(A.P.)

c7

Pattern of M20 (K0L12)

2.0 ---C)

-2

'

---/

'I

/

O

( L'

2

-2

'2

-

--_

-2,-lç

M20

KoL12

Measured

(

in mm )

1.0

0.5

(44)

M21

KoL12

Y/i

Measured

C

¡n mm )

,'

/

I, '

1 6 -2

,

/

\

ji

,,

,-4

/

I

,,,

, I,

2 /

/

i

I

JI,

0

1/-2

/

/ 8

/

)

J

4

-6 /_

//

4

(_

- ,-(

--/ --/

1/

ç

__

-

,-I

-

L__-, /

,

.._...-6 f

/

/-' ,/1-;'

J (

f

---.--,...-.., 4/' )

'

,->/

,

-'

/

-I

-)

/7

f-2

1)7/',,

,_-.-ç

,)

,j

6

1:'

'(

()

,__ --- _, % , I

/

f f

,

c-_--4.

'2

6

7 -, i

j 2

'i

__-> - r

,/

/

i

-_;;.:::;:

II ///,

6

,,,,,,-16"/-2'

-- .----2 6

r) f,O\\

( __

/2

-s

/////

(__

-o

Fig. 20

Measured Wave Pattern of M21 (K0L=12)

o o

2.0 X,1

3.0 -1.0

0.0 -1.0

(F.P.)

cm)

(A.P.)

1.5

1.0 _0.5

(45)

1.5 -1.0

(F.P.)

1.0

0.5

Y,'1

I

O'

-,--..'

/

₁

\

C s.'

\

/

#1/

/

_/

-f

t a

'

f

-D--\ i

I

/

I

/

/,'iO

I/f

III,

-12 I

/ / / /

/ 1/1(1 ///-16l /

/,

I

/(¼... \

"ø (

/

.'2/4(

-0.0

()

M21 & M20x

KoL12

Calculated

(in mm )

---10

(A.P.)

-__-4 /

/

-Fig. 21

Calculated Wave Pattern of M20 and M21 (K0L=12)

2.0 /

_{,__I}

/

-T

/

-2

/

_/

/

_/

/

.--_____-4 )

X/1

(46)

y/L 1.5- 1.0

M20 K0L=12

,;:v C2'

X'

_\ , ,;.'

,'

,-) j I

I,, ,,

/

I,

F

,

I I I/I / r 'L. ,' , (t _ ---I --i-- _.:_.

-III'

- /

tu, I

I

/ /, ,

I I .

/////,,,'

J F

-- 7//(

-,.&/"('.( (__'-_' t'_5 (

;;7u/'/,'

-/

¼'

-p

,

--::--'

-i ('

(in mm)

(AP)

I,

-I

,1 li I

₎

/, -3

/,

_I,

--i / /

¿-ç,

Fig. 22

Calculated Wave Pattern from

Obtained Wave-MakiflcT Source

Distribution m() of M20

(K0L=12)

QO

(47)

1,0

1.5

1.0 F.P.)

y

L

0.0 ()

-2

M21

KoL12

(in mm)

/

_r

-2

t

4

/

,

/

I

,

.-/

/ r'

/

r

I)

I ,' ,-2..io

__ r I'

I /í

% I//I

'('-I'll

/

i..

.

-?2 -' -- _)t (

'_7 .5

(-1.0

_(AP.)

2.0 X/L

F10. 23

Calculated Wave Pattern from Obtained Wave-Making Source

Distribution m() of M21 (K0L12)

4

(48)

-Lo

(EP)

lo $

0.0 ()

M20x2

& M21

_K0L=12

Catcutated with /(-Correction

(in mm)

»)

¼_-,, -4

-,_ -,___ /

/ I

/

Il

Ç /

--, /

---4

\

II / '---,'-

_f

e

---)

)

cc

_ç#'

ç_S_

I)

/'

..

-,-.

-6

\

i

,,_1

-,,,

V//

/

r-/ __,

)-8)

__

I,

/'

4

I (

I.. (__ ¡

/

2

-\

)

(1 '

'

),, L

,_-_--'

-i,

c-c-'- fl

\ II

t_____)

2

,'-;z

( 7

-_-- 2

,c8 ¡ i?

((\'

2 C

/d_fl) ¿

'?

'

-0 8 /

i

6

I

___________________________________

,1--7

-21

'-

---t

c-.

1.0

(AP)

Fig. 24

Calculated Wave Pattern of M20 and

M21 with p-correction

(p0.4,

0Ll2)

2.0 x/t

y/L

1.5-

1.0

(49)

1.0

i-o

M20

Calculated

Calcu(ated( 1U0.L)

Towing Test

Present M

Longi. Cut M

I i i J

0.35

0.30

CaLcuLated

Fig. 25 Wave-Making Resistance of M20

IT.

Fn.

.5

00

(50)

1.0-05

00 M21

I I I i J

0.25 CauLated

Calculated ( AL0.4)

Towing Tést

Present M

Longt. Cut M

I I I

0.30

Fig. 26 Wave-Making Resistance of £121

I I

(51)

A(0)/L xlCO

0.5

0.4

0.3

0.2

0.1

-Prosnt "1ethod

LonçJLC!Jt Mthod

-Trans.Cut Method

C ulculated

M20

.ol.i2

i"

₇₀

k

80 decree

Fig. 27

(52)

xlOO

05

10

20 ¿:0

60 C.'Icu!td

Present Method

Longt.Cut Mthod

___

-Trans.Cut Method

j-

80 degree

Fig. 28

(53)

u/U xlO

2.0 r

M21 KoL=12

0.0 _{_20L}

j

v/Uxio

2.0

0.0 i

-1.0

-(EF?)

-2.0 -

w/U x1O

2.0 r-

00 -2.0

-1.0

(

-0.5

I 4 t I t 4 4

-0.5

0.0

0.0 u-Component

ExcL. Free Wave Terms

IncL Free Wave Terms

v-Co mpone n t

4 t 4 4 f t I f f

0.5 w-Component

Fig. 29

The Velocity Component of Hull-Surface (M21)

0.5

(54)

M21 K0L=12

Double Model Approx.

WaveAnatysis

0.5 Exact Hull Surface

Condit i on

F±cj.

30

T1e Source Distribution C'btaine

from Dxact hull-Surface Coniition (M21)

(55)

A(e)/L xlOO

_0:5

-0.4

0.3

0.2

0.1

O

Exact Huit Surface

f., J

10

20

30

40

50

60

70 80 degree

Fig. 31

The Amplitude Function of the Source Distribution Obtained from Exact HullSurface Condition (M21)