The added mass coefficient of a cylinder oscillating in shallow water in the limit Κ 0 and Κ

ARCHIEF

PAPERS

OF

SHIP RESEARCH INSTITUTE

The Added Mass Coefficient of a Cylinder Oscillating in

Shallow Water in the Limit K O and K co

By

¿

Makoto KAN

1i4edc( "7ÇÇ

C-L-7

(/c-t'-May 1977

Ship Research Institute --Tokyo, Japan

Lab.

v

Scheepsbouwkune

Technische Hogeschoo

De!ft

THE ADDED MA$S COEFFICIENT OF

A CYLINDER OSCILLATING IN SHALLOW WATER

TN THE LIMIT KO AND K+ *

By

Makoto KAN**

SUMMARY

An integral equation method is described for calculating the added mass

coefficient of an oscillating cylinder in the shallow water in the both limits as the frequendy parameter K tends to O añd tends to infinity.

it is shown that the heaving problem of a floating cylinder in the limit K - O must :be trêated oñly by thé limit process of the wavé making prob-lem and that the probprob-lems of the other kind of oscillation than heaving of a floating cylinder and of any oscillätion including heaving of a fully sub-merged cylinder in the both limits K -. O and K - co can be treated by set ting directly K.r=O or Ke without taking any accoúnt of the wâve making

problem.

Some nuthericâl results are presented for Lewis form linder, including

heaving circular cylinder for which the complete agreement with Sayer and Urséll's results in the limit K -. O, and also with Rhodes-Robinson's results in the limit K -. co is obtained.

A sirnlè forrñula fò± the damping coefficient of a heaving ylinder in the limit K - O is also presented.

1. INTRODUCTION

The calculation of the hydrodynamical forces acting on an oscillat-ing cylinder in shallow water with the flat bottom, was first carried out by Yu and tïrselP and then by Kim2. Porter also seems to have done the calculation of them, as the results are cited in the Rhodes-Robinson's paper Unfortunately some of both calculations by Yu and Ursell and by Kim, especially the added mass coefficient in the low frequency range, are inaccurate. This fact was clarified by the several recent papers by Keil4, Takaki5 6), Kan" and Ikebuchi8>, which showed that the heaving added mass coefficient does not tend to

in-finity in the limit as the frequency parameter K tends to 0, in con-trast with the tendency as shown in the Yu and Ursell's paper or in the Kim's paper, but tends to some finite value in case of the finite depth However, there are some dlscrepancles between these finite

values as K-O by the above-cited several paper&8. The main

pur-* _{Received on March 2, 1977.} ** _{Ship Dynamics Division}

pose o this paper is to find thé correct finite value of the added mass coefficient of an oscillating cylinder in the shallow water in the limit K--sO and

K*oo.

Recently the inaccuracy of the calculations by Yu and Urell and by Kim was recognized9 or revised partly by themselves.

The added mass coefficient of a heaving circular cylinder in the limit K s O was calculated by Sayer and Ursell9 and in the limit K-+

co by Rhodes-Robinson3 The problem of a swaying cylinder in the

limit K O was solved by Kan", Flag and Newman12, Taylor'3, Keii4 and Kim'°, in connection with the problem of the lateral translation of a ship in the shallow water.

The method presented here for calculating the added mass

coeffi-cient in the limit K - O or K --'

oc, utilizes rather a simple Green

function in order to reducé the problem to an integral equation for

the unknown velocity potential on the cylinder surface. The method can be applied to the case of cylinder with an arbitrary cross section, and also can be extended to the partly non-flat bottom problems

Numerical results obtained by the present method in the limit K

- O will be usefül to check the accuracy of the calculation of the

wave making problem in the kw frequency range. The added mass coefficient in the limit K-+co will be useful to deal with the problem of the ship vibration with high frequencies.

2. MATHEMATICAL FORMULATION

2.1 Formùlatión of the Problem

It is assumed that the two dimensional potential flow is caused by a cylinder oscillating harmonically with angular frequency on the free surface in an ideal fluid with the constant finite depth h The

problem in the limit Kp O and also K* co is considered here, where ÎÇ= ?/g añd g is the 'gravitational acceleration.

If the velocity potential of this flow field is expressed by the real

part of

'1e must satisfy the following equation and linearized

boundary conditioíis.

in the fluid domain (2.1)

=O on the bottom (2.2)

on the free surface, when K 0

(2.3)

0=0

oñ the free surface; when K co (2.3)'

0=

on the body surface , (2.4)

where V, is the normal velocity component of an oscillating cylinder

'P=Qí2h at t-=

±00 when K+O (2.5)

where

V4s. (2.6)

The conditions (2.3) and (2.3)' express that the free surface i regarded as the rigid wall when K- O and as the equi-potential surface when K=-* co respectively The last condition (2 5) will be derived later in the section 23.

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

X =O(Kc0) _{'L B} =O(K-O) j

_F(V$urfa)

- n

s

(Body Sifa)

=o.

B (BottOm) y

Fig. i. Boundary Còndtions

2.2 Green Function

In order to find the Green function of this problem, we consider the flow field caused by an infinite series of sources and sinks of unit strength as shown in Fig 2, where sources are placed on (x',a."+ 4nh)

and (x', --y'+2h±4nh), and sinks on (x', y'±4nh) and (x', y'-2h-l-4nh). If we denote the velocity potential due to an infinite series of sources only by G+(x, y, x', y') and siñks only b G_(x, y, a,', y'), then G(x, y, x', y') is expressed as follows

G±(x, y; x',y)= ± log {(x')2±(yRy'--4nh)2}"2

fl=

-± j log {(x.x')+(y±y'R:2h_4nh)2}"2. (2.7)

7l=O0

(x'.-6h.y') o (x',L.h+y') (x'- 2h.y') o (x-y') (x'. y') y=h,,,, _, Cx', 2h+y) o

y=px::

(x's _4h.y') y=6h ' 6-y') (x', 6h+y') o _Sink o (x', 8h-y') - y

Fig. 2. Image Séries Free Surf7

,-BQt torn

Ii+

x2 _çoshxcosa

fl--oo( (a2hr)2) _Icosa

expression (2.7) is reduced to the following simple form'

G±(x, y; x', y')

±1 log

_{2()2[cosh *(x_x')_cQs}

±4

log 2()2{cosh *(x.x')+cos *(y±y)}

and

G0(x, y; x', y')=I

log 2()2{osh f(x_x')_cos

(Y')}

±410 2(-)2(cosh --(x---x')=cos --(y+y')}

where the constant terms are not omitted because the exprêssions in the lImit h -- co can be obtained immediately, but they are not of any

significance too.

2.3 Integral Equation

We apply the Green's theorem to the veloëity potential and the Green function G (=G0 or G) in the region, as shown in Fig 1, bounded

by the free surface F, body surface S, bottom B and the imaginary

;-- log

(2.9)

where the cöfistatit ierth is dèteriined such that

um G(x, y; x', y')=±1 log {(x=x')2-I-(y±y')2}

2

but it is not of any significance.

If we denote the Green functión which satisfies the boundary con ditions on the bottom and on the free surface in the limit K=-. O and K* co, by G0(x, y; x',y) and G(x, y;x', y'), resTpectively, then the fol-lowing relatiors will be easily understood from Fig. 2

y; x', y')=G(x, y; x', y')+G_(x, y; x', y')

and (2.10)

G0(x, y; x', y')=G+(x, y; x', y')G.(x, y; x', y')

namely

G(x, y; x', y')=-- log2(

2h )2(cosh(x_x')_cos

_-(_')}

+f log

2()2 {cosh

*(x_x')±cos

(y+Y')}

_I

1og2(k)2{cosh

_{-(x--x')cos --(y+Y')}}

log 2(-)2{cosh --(xx')±cos j-(y_yí)}

boundary at x ± co L and R. It is obvious that the integrals over the free surface and the bottom vanish, and so the following expres-sion of the integral equation is left

- {G(P, P').ø(P')G(P, P').ø(P')}ds(P') (2.13)

7r SURUL

where P is the point on the boundaries as well as P'.

2.3.1 In Case of k>oo

In case of K co, it

is evident from eq (2 11) that G,. - O and

G,., = G - O as x' -* ± oc, and so the integrals over R and L vanish

too Therefore the integral equation (2 13) for 0 is reduced to that

on the body surface only, such as

{G,,.(Ï, P').0,(P')C,,.(P, P').O(P')}ds(P'). (2.14)

2.3.2

In Case of KO

We get the following limit from eq. (2.12)

G0 - 2log r h

G0=

-

-j

at x' ±oo

)

f ø,,ds*O, there remains some fihite ñormal Velocity component of the fluid at z' - ± oc, and evidently

jØ,_*

Q/2h

at -c'* ±oò (2.16)

consequently

O -* ±:Qx'/2h±C±

at z'

± oc. (2.17)

Sbstitiiting eqs. (2.15)(2.17) into the integral over É and L, we bve

(G0O,,G00)ds= -- log +2C (2.18)

where

2C= C + C.... a) Anti-symmetric Motion

In case of the anti-srmrnetric motion such as swaying or rolling,

L

the sum of the source strength over the body surface _{Q= 5} Pds must

be zero, and the constant C of eq (2 18) must be zero because of the anti-symmetry of the flow field Consequently in this case the integral over R and L vanish too, and the integral equation has the same ex-pression as eq. (2.14), namely

{G0(P, P').P(P')G0(P, P').(P)}ds(P'). (2.19)

As a matter of fact, in case of Q=0 including swaying and rolling of an asymmetric cylinder and any oscillation of .a fully submerged arbitrary cylinder; integral èquation (2.19) is valid because the cOn-stant C has no effect on the1 calculation of the added mass coefficient and pressure coefficient, as easily understood from the followings.

Added mass coefficient dc

5

PPds=5 (w+C)øds=5 çD.(1)ds.

Pressùre coefficient KP=K(çø+C) -p Ø

as K 0.

b) Symmetric Motion

In case of the symmetric motion such as heaving the sum of the source strength over the body surfàce does not become zero. If the constant term in. eq. (2.18) is not determined, the solution of this

problem is indeterminate It does not seem that we have any condi-tion to determine the constant C However, if we regard the problem

as the limit of the wave making problem of KO, we can determine

the constant as

C=Q.B/4h

(2.20)

where B is the breadth of a cyliñdet. The proof of eq. (2.20) will be shown in the next section 2.4.

Integral equation in this case can be written in the same expres-sion as (2.14) ot (2.19), namely (P) 2_ 5 (O0(P, P'). (P') Ò0(P, P'). ø(P')}ds(P') (2.2Ï) where

0=G0-2log1----

(2.22) r 2h

We have the following expression as h

-*

hm

-

I

log {(xx')2±(yy)2) +1log {(xx')2H-(y±y')2)

Gr+jGi -

-=1 .KT

2log---

_2h

The expressions (2.21)(2.23) are valid to the cylinder with an

asymmetric cros section.

2Ä Limit Process in the Wave Making Prtthlem

The simultaneous integral equations for solving the potential fioî field accompanied with the wave making phenomenon, namely 0< K< cc, are written as follows"

ør+-(ørGrn-eGin)ds='

ønGrdS (2.24)

i+__s

ø.Gds

(?;?5) [(2 i _12 -

-hK,±hK2K

+i-}-.

hK hK2 ± K cosh K0(yh) - cosh K0(y1 .h)eo'"

(2.26)

where and

KK0 tanh Kh and -K=K,. tan Kkh.

(2.27)

When K' 0, we have the following expressions from èq. (2.27)

K Kh and Kk

nir/h. (2.28)

Using the formula

.-coskx=--_log(1-2a cos x±a2),

_al1

2

±4 log 2{cosh f(x_x')_ cos

and

um G.

ir/kh.

(2.23)

(2.29)

we can get the follòviing simple expressions för G,. and G from èq.

(2.26)

UmGr4 log 2[cosh f(x_x')_cos --(-')}

The radiation conditiòn at x' -p ±00 in the originâl wave making

prob-lernis

um (P±iK0P)=0.

(2.31)

By using eqs. (2.17) and (2.31), we can get the following relation when

K-0

Q/2hK0. (2.32)

We can also get the following relations from eq. (2.26) when K=-* O

fKo(x-_x') and -+ (2.33)

By using the relations (2.32) and (2.33), the term of 5 in the left hand side of eq. (2.24) is written as

-5 &G= _-5 cP,ds.

(2.34)

s 2h 2h s

Consequently We have the föllowirig expressions from eqs. (2.24) and (2.25) as K-30

r±25

Gs=55

n(Grj)(1s

and (2.35)

5

Cornpatihg eq. (2.12) with eq. (2;30), w have

G0-2 log --=G(K* 0) (2.36)

and then

r+15

rGo4S_5 (Go2lo)ds.

(2.36)

This equation (2.36) is quite the same as eq. (221).

These are the reason why the constant C of eq. (2.18) is

deter-mined as expression (2 20) in the preceding section

We can also prove that the Green function G,,. represented as eq

(2 11) and the integral equation (2 14) Ifl case of K+00 are derived

from eqs (2 24) (2 26) by the limit process similar to the

K K0 and Kk± (2n-1)2r/2h

(2.37)

and we should use the following formula

-' cos(2k-1)x=--log 1+?ço±

_2k-1

JaI<1.

4

1-2acosx+a

(2.38)

3. METHOD OF SOLUTION AND NUMERICAL PROCEDURE In all eases the. integral equation is written in the same type, as shown in eqs. (2.14), (2.19) and (2.21). It is evident that the Green

functions G0 and G0. have the logarithmic singularity such as -- log {(x

x')2+(yy')2}.

I4owevet, the kernel functions G0 and G0.,. have no singularity7)wl4s). _{So we can Solve the integral equation numerically} without any difficulties, by transforming it into the simultaneous alge-braic equations The detailed numerical procedures are not presented

here since they are shown in author's other papers7""5. It should be noted that in the author's numerical procedures the distribution of the velocity potential on the cylinder surface can be solved continuous-ly without assuming the constant distribution on the small segment of the surface as used in the usual numerical procedures This is due to the fact that the kernel function has no singularity in two dimensional problem, and this enables us to use any type of quardrature formula

If we write the equation of motion of the two dimensional heav-ing cylinder as

(pA±M,.)+N,.±pgB=Z,. sin (t+e,.) (3.1)

the added mass coefficient C.. and the damping coefficient D,. are de-fined as

G,.M,./pA and Dh.=(Ñ,jpA)VB/2g (3.2)

and C,. and D,. are represented as

5

ørønd8 and D=4:()2

\/KP_

5 ø.ø,.ds (3.3)

where A is the setional area of thè cylinder under the free surface and is the amplitude of the heaving motion.

In case of the swaying or rolling of the floating cylinder, or any oscillation of the fully submerged cylinder, when K tends to O D,. ap

proaches to zero since 5 ,.ds=O. However in case of the heaving of

the floating cylinder, Dh approaches to the finite value such as

(3.4)

This can be obtained by substituting eq. (2.32) into eq. (3.3). In case of the heaving of the floating cylinder, it can be proyed that

Q/=B

(3.5)

and ultimately we have the following expression for the damping

co-efficient Dh

liÌn Dh=--- B

k-.O _A 8h

4. NUMERICAL RESULTS AND CONCLUSIONS

The numerical results of the added mass coefficient Ch for heaving and C. for swaying in the both limits K* O and K- 00 are shown in

Fig 3Fig 6

The results are given for the circular cylinder and the Lewis form cylinder with i=H0=1 0, where =AfBd and H0=

B/2d. These results are obtained by using the Simpson's rule as the

quadrature formula and the number of the division between the water line and the keel on the cylinder surface is less than 30 In case of

h/d=1.05 where d is the draft, £h (K' 0) in Fig. 3

C',, (K* co) in

Fig. 4 and C, (K 0) in Fig. 5 for the Lewis form cylinder did not

converge even when the number of the division was increased up to

30. Other parts in these tabÏes are confirmed to be accurate almost

by 4 figures by varying thè number of the divisioñ. In case of the

heaving circular cylinder, the added mass coefficient in the limit K 0, Ch (K-4 0) in Fig. 3 agrees completely with Sayer and Ursell's result°'. Also in case of the heaving circular cylinder, the added mass

coeffi-cient in the limit as K 00

Ch (K* co) in Fig 4 agrees completely with Rhodes-Robinson's result2' at h/d2.0 and 4.0 which are the only two cases presented in his paper.

With regard to the discrepancies of the added mass coefficient in the low frequency range between several papers which has been men

tioned earlier in this paper, it was confirmed by C,, (K 0) in Fig. 3 that both results by Keil4 and Kan" are the most accurate among

them, though the detailed comparisons are omitted here. And also concerning the damping coefficient, the calculation by Kan" is accurate in the range of the low frequency. Some nurnetical examples to

coñ-firm it are presented in Fig. 7, where the 'alties at K=0 were calcu-lated by eq (3 6) and other values were calcucalcu-lated by the author's

/+. 0

3.0

2.0

Ch

(K-.0)

t Agieement with Sayer ard WseI('s resjlt, is compete.

Mot fut (y conerged

i., Lewis Förm G Ho= 1. O

C ir cutar Cylinder

Fig. 3; Added Mass Coefficient c1 (K - O), Heave

hid

Circular Cytindert Lewis For =H01. O 1.05 1.543 '(7.324) 3.942 h262 1.1 5 t0.8ß .284.7 1.20 0.965 2.270 1.30, 0.802 1.656 1.4O 0.6.98 1.326

i 60

0..5 8 L 09.8.9 1.8L0 0.52,4 '0..824.. 2MO 0.498 0.733 250

0497

0642 3.00

0629

3.50 0.582.... 0.6.45 4. oÖ 0.6 35 7 3 5MO 0.742 0.741 .1000 1.i 63.. 1. 05 2 20MO' 1.657 -. i4.36....

5Q0

2.361 1.987 10Ö 2.91 Z.41 8 -1000 4..765 3.875 i:.00OÖ 6.630 ' 5.340 100000 8 495 6 805 1.0

h/d

20 '3.0 4.0 Ch f

t

(KDo)

t Anñt

with Rhe7s-Robinson's tesjItisréte. * Not fully convecged

=H1.0

Fig. 4. Added Mass Coefficient C (K

-

oo), Heave

hid Circulär CyLinder Lewis Form 6=H0=1 Ç -

..IoS

2k53.

(82.6).

482.1....

.i.i.0_2i 63

1.15

1.99

3.722 1 20 _1 847 3 1 39_ 1.313 .1664 251.3 - 2.1.74 1.60 .1.386 1.81Ö L80 1 292 1 618 ..i2..0O ._.1 .2.30 t 1 .50.0 .2.50 t14.1 . .1.345 3.00 1.096 1.271 _3.50.

i06.9..12 3.O.

-- .400 .1 .053 t

1.204 5.00 1.033 1.17.

.i00.Ö

i008

1.138 20.00 1 .002 1 1 2.9 50.00.. 1 1.1 2 i0Ö00 1.000 1.126 3.0 4.0

6.0-so 4.03.0 2.0 -CS

CircuLar /

Cylinder - Cs

(k-0)

Not fully conveged

Lewis Form

f

1.0 .

1.0 2.0

- hid

3.0

Fig. 5. Added Mass Còefficient Gs (K- O), Sway

4.0 h/d CircuEar Cytinder Lewis Fôrm 6=H0=10 1.05 9.38 (22.4 29r Li 0.. 6.030 11. 203 1.1,5

4630

7.727 1.20

386

6.009 1 30 2954 4 292

3z2

1.60 1.953_ _2572 1.8O 1.682 2.145 200 1 518 1 893 2.50.. 1.303. 1 569 .oÓ 1.201 1.419 3.50 1.144 1335 i283. 5.00 1 .0 68. 1 .22 4 10.00 1.017 1150 20 00 1 004 1 132 50-Ö0 1001 1.127 10000_1 000 1 126 1 000.00 .1 .000 1 1 26

0.4

Cs

0.3

1.0 20 3.0 4.0

- hid

Fig. 6. Added Mass Coefficient C (K -# ), Sway Cs

(K-c°)

Circular CyLinder Lewis Form =H0=1.0 h , d Ci rcular Cylinder Lewis Fbrm 5=H0=10 1.05 0.6735 0.5131

1.10 0.6023

0.4819 1.15 0.5595 0.4610 1.20 0.5300 0.4456 1.30 0.4918 0.4241 1.40 0.4682 0.4098 1.60 0.4416 0.3925 1.80 0.4280 0.3827 2.00 0.4202 0.3767 2.50 0.4114 0.3694 3.00 0.4082 0.3665

3.50 04068

0.3651 4.00 0.4061 0.3645 5.00 0.4055 0.3639 10.00 0.4051 0.3635 20.00 0.4051 0.3635 50.00 0.4051 0.3635 100.00 0.4051 0.3635 0.7 0.6 0.5

Fig. 7. Damping Coefficient Dj,., Heave

prograril". These seem to confirm the author's opinion that the

dis-crepancy between the theoretical and experimental results in the low

frequency range in Vugts' paper'6 is not due tô the inaccuracy of the experiment, but due to the shallow water effect The figure from the previous paper7' is reproduced in Fig. 8.

Conclusions of this paper are as follows.

(i) An integral equation method has been developed for ca..culatg

the added mass coefficient of an oscillating cylinder in the shallow

water in the both limits K O and Kp 00

The Green functions, with rather simple expressions such as eqs (2 11), (2 12) and (2 22), to be used in this method have been found

It has been shown that in case of the heaving of a floating

cylinder in the limit K' O, the problem can be determinate if it is

regarded as a limit in the wave making problem, while in case of the swaying and rolling of a floating cylinder or any oscillation of a fully submerged cylinder in either limit K O or K - co, the problem is determinate originally The physical meaning of the former case, namely the case of Q*O, has not been made clear.

A simple formula for the damping coefficient of a heaving

cylin-Lewisform cylinder

6 = H0 = 1.0 CircuLar cylinder .5 2.0 0.5 1.0 B g2

2.0..5 -CH 0.5-h/d r h/d IO CH. Kim) Za0.0!m o _002m h4802.25m Ö.03m h/d:I2I _j,

////f//*/jf//// 7/

Circle, Heave I.0

Fig. & Comparison with Vugts' Experiment

1.5

der in the limit Kp O has been obtained as represented in eq. (3.6).

(5)

Numerical results obtained by the present method have confirmed the accuracy of Keil's results and Kan's results on the hydrodynamical force coefficients of the wave making problem in the low frequency range.

Numerical calculations were carried out by using. the electronic computer TOSBAC-5600 in the computer center of Ship Research In-stitute.

REFERENCES

Y. S. Yu and F. Ursell: Surface waves generated. by an oscillating circular cylinder on water of finite depth: theory and experiment, J. Fluid Mech., Vol. 11 (1961).

C H Kim Hydrodynamic forces and moments for heaving swaying and rolling

cylinders on water of finite depth, J. Ship Res., Vol. 13 (1969).

P F Rhodes Robinson On the short wave asymptotic motion due to a cylinder heav

ing on water of finite depth: I and II, Proc. Camb Phil. Soc., Vol. 67 (1970).

H. Keil: Die hydrodynamischen Kräfte bei der periodischen Bewegung

zweidimen-sionaler Körper an der Oberfläche flacher Gewässer, Institute für Schiffbau der

Uni-versität Hamburg, Bericht Nr. 305 (1974).

M. Tàkaki: On the ship motions in shallow water, part I (in Japanese), Trans. of

the West-Japan Society of Naval Architects, No. 50 (1975).

M. Tàkàki: On the two-dimensiOnal hydrodynamic forces acting on an oscillating cylinder in shallow water (in Japanese), Research Institute for Applied Mechanics,

Kyushu University, Rep. No. 45 (1976).

M. Kan: Calculation of the two dimensional fluid forces acting on a cylinder oscil-lating in shallow water (in Japanese) Paper presented at 26th general meeting of

Ship Research Institute (1975).

T. Ikebuchi: Wave induced forces and moments in shallow water (in Japanese), J.

of the Kansai Society of Naval Architects, Japan, No. 161 (1976).

P. Sayer and F. Ursell: On the virtual mass, at long wavelengths, of a

half-im-mersed circülär cylinder heaving on water of finite depth, 11th Symp. on Naval Hydro.

(1976).

-C. H. Kirn: Effect of mesh size on the accuracy of finite-water added mass, J.

Hy-dronautics, Vol. 9, No. 7 (1975).

M. Kan: Calculation of potential flow about two dimensional cylinder in channel (in Japanese), Paper presented at 16th general meeting of Ship Research Institute (1970).

C. N. Flag and J. N. Newman: Sway added mass coefficients for rectangular profiles

in shallow water, J. Ship Res., Vol. 15, Ño. 4 (1971).

P. J. Taylor: The blockage coefficient for flow about an arbitrary body immersed in a channel, J. Ship Res., Vol. 17, No. 2 (1973).

M. Kan: Calculation of nonlifting potential flow about ship hulls in shallow water

(in Japanese), J. of the Soc. of Naval Arch. of Japan, Vol. 129 (1971).

M. Kan: Calculation of pressure distribution on the surface of two dimensional

cyl-inder oscillating harmonically in calm water (in Japanese), Paper presented at 20th

general meeting of Ship Research Institute (1972).

J. H. Vúgts: The hydrodynamic coefficients for swaying, heaving and rolling cylin-ders in a free surface, I.S.P., Vol. 15 (1968).

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No. 3 Increase of Sliding Resistance of Gravity Wälls by Use Of Projecting Keys under

the Bases by Matsuhei Ichihara and Reisaku Inoue June 1964

No. 4 An Expressionfor the Neutron Blackness of a Fuel Rod after Long Irradiation, by Hisao Yamakoshi August 1964

No 5 On the Winds and Waves on the Nothern North Pacific Ocean and South Ad jacent Seas of Japan as the Environmental Condition for the Ship, by Yastifumi Yamanouchi, Sanae Unòki and Taro Kanda, March 1965.

No. 6 A code and Some Resúlts of a Numerical Integration Method of the Photôn Transport Equation is Slab Geometry, by Iwao Kãtaóka and Kiyoshi Takeuchi,

March 1965.

No 7 On the Fast Fission Factor for a Lattice System by Hisao Yamakoshi June 1965.

No. 8 The Nondestructive Teting of Brazed Joints, by Akira Kanno, November 1965.

No. 9 Brittle Fracture Strength of Thick Steel Plates for Reactor Pressuré Vessels, by Hiroshi Kihara and Kazuo Ikeda, January 1966.

No. 10 Studies and Considerations on 'the Effects of Heaving and Listing upon Thermo-Hydraulic Performance and Critical Heat Flux of Water' CoIed Marine Reactors, by Naotsugu Isshiki, March 1966.

No 11 An Experimental Investigation into the Unsteady Cavitation of Marine Propel

1ers, by Tatsuo Ito, March 1966.

No. 12 Cavitation Tests in Non-Uniform Flow on Screw Propellers of the

Atomic-Power-ed Oceanographic and Tender ShipCompanson Tests on Screw Propellers De signed by Theoretical and Conventional Methods=, by Tatsuo Ito, HajÏrne

Takahashi and Hiroyüki Kadoi, March 1966.

No 13 A Study on Tanker Life Boats by Takeshi Eto Fukutaro Yamazaki and Osamu Ngatà, March 1966.

No. 14 A Proposal on Evaluation of Brittle Crack Initiation and Arresting Temperatures

and Their Application to Design of Welded Structures by Hiroshi Kihara and

Kazuo Ikeda, April 1966.

No 15 Ultrasonic Absorption and Relaxation Times in Water Vapor and Heavy Water Vapor, by Yahei Fujii, June 1966.

No. 16 Further Model Tests on Four-Bladed Controllable-Pitch Propellers, by Atsuo

Yazaki and Nobüo S'ugi, August 1966. Supplement No. i

Design Charts for the Propulsive Performances of High Speed Cargo Liners with C =

0 575 by Koichi Yokoo Yoshio Ichihara Kiyoshi Tsuchida and Isamu Saito August

1966.

No. 17 Roughness of Hull Surface and Its Effect on Skin Friction, by Koichi Yokoo, Akihiro Ogawa Hideo Sasajima Teiichi Terao and Michio Nakato September

1966.

No. 18 Experiments on a Series 60, CB=0.70 Ship Modél in Oblique Regular Waves,

by Yasufumi 'Yamanouchi and Sadao Ando, October 1966.

No. 19 Measurement of Dead Load in Steel Structure by Magnetostriction Effect, by

Junji Iwayanagi Akio Yoshinaga and Tokuharu Yoshii May 1967

'No. 20 Acoustic Response of a Rectangular Receivèr to a Rectangular Source, by

No. 21 Linearized Theory of Cavity Flow Past a Hydrofoil of Arbitrary Shape, by

Tatsuro Hanaoka, June 1967.

No 22 Investigation into a Novel Gas Turbine Cycle with an Equi Pressure Air Heater by Kosa Miwa, September 1967.

No 23 Measuring Method for the Spray Characteristics of a Fuel Atomizer at Various Conditions of the Ambient Gas, by Kiyoshi Neya, September 1967.

No. 24 A Proposal on Criteria for Prevention of Welded Structures from Brittle

Frac-ture, by Kazúo Ikeda and Hiróshi Kihara, December 1967.

No. 25 The Deep Notch Test and Brittle Fracture Initiation,' by Kazuo Ikeda, Yoshio

Akita and Hiroshi Kihara, December 1967.

No. 26 Collected Papers Contributed to the 11th International Towing Tank Conference,

January 1968.

No. 27 Effect of Ambient Air Pressure on the Spray Characteristics of Swirl Atomizers, by Kiyoshi Neya and Seishiro SatO, February 1968.

No. 28 Open Water Test Series of Modified AU-Type Four- and Five-Bladed Propeller

Models of Large Area Ratio, by Atsuo Yäzaki, Hiroshi Sugano, Michio

Takahashi and Junzo Minakata, March 1968.

No 29 The MENE Neutron Transport Code by Kiyoshi Takeuchi November 1968

No. 30 Brittle Fracture Strength of Welded JOint, by Kazuo Ikeda añd Hiroshi Kihara,

March 1969.

No. 31 Some Aspects of the Correlations between the Wire Type Penetrameter

Sensi-tivity, by Akira Kanno, July 1969.

No. 32 Experimental Studies on and Considerations Of the, Supercharged Once-through

Marine Boiler by Naotsugu Isshiki and Hiroya Tamaki January 1970

Supplement No. 2

Statistical Diagrams on the Wind and Waves on the North Pacific Ocean, by Yasufumi Yamanouchi and Akihiro Ogawa, March 1970.

No. 33 Collected Papers Contributed to the 12th International Towing Tank Conférence,

March 1970.

No 34 Heät Transfer through a. Horizontal Water Layer, by Shinobu Tokuda, February 1971.

No. 35 A New Method of C.O.D Measurement Brittle Fracture Initiation Character-istics of Deep Notch Test by Means of Electrostatic Capacitance Method, by

Kazno Ikeda, Shigeru Kitamura and Hiroshi Maenaka, March 1971.

No. 36 Elasto-Plastic Stress Analysis of Discs (The ist Report: in Steady State of

Thermal and Centrifugal Loadings), by Shigeyasu Amada, July 1971.

No 37 Multigroup Neutron Transport with Anisotropic Scattering by Tornio Yoshimura

August 1971.

No 38 Primary Neutron Damage State in Ferritic Steels and Correlation of V Notch Transition Temperature Increase vith Frenkel Deféct Density with Neutron

Ir-radiation, by Michiyoshi Nomàguchi, March 1972.

No. 39 Further Studies of Cracking Behavior in Multipass Fillet Weld, by Takuya

Kobayashi, K'azumi Nishikawa and Hiroshi Tamura, March 1972.

No. 40 A Magnetic Method for the Petermination of Residual Stress, by Séiichi Abuku,

May 1972.

No. 41 An Investigation of Effect of Surface Roughness on Fòrced-Convection Surface Boiling Heat Transfer; by Masanobu Nomura and Herman. Merte, Jr., December 1972.

'No. 42 PALLAS-PL, SP A One Dimensional Transport Codé, by Kiyoshi Takeuchi, February 1973

No 43 Unsteady Heat Transfer from a Cylinder by Shinobu Tokuda March 1973

No. 44 On Propeller Vibratory Forces of the Container Ship Correlation between Ship and Model, and the Efféct of Flow Control Fin ori Vibratory Foces, by Hajime

No. 45 Life Distribution and Design Curve in Low Cycle Fatigue, by Kunihiro lida and liajime moue, July 1913.

No. 46 Elasto-Plastic Stress Analysis of Rotating Discs (2nd Report: Discs subjected to Transient Thermal and Constant Centrifugal Loading), by Shigeyasu Amada and Akimasa Machida, July 1973.

No. 47 PALLAS-2DCY, A Two-Dimensional Transport Code, by Kiyoshi Takeuchi,

November 1973.

No. 48 On the Irregular Frequencies in the Theory of Oscillating Bodies in a Free

Surface, by Shigeo Ohmatsu, Jànuary 1975.

No. 49 Fast Neutron Streaming through a Cylindrical Air Duct in Water, by Toshimasa

Miura Akio Yamaji Kiyoshi Takeuchi and Takayoshi Fuse September 1976 No 50 A Consideration on the Extraordinary Response of the Automatic Steering Sys.

tern for Ship Model in Quartering Seas by Takeshi Fuwa November 1976

No. 51 On the Effect of the Forward Velocity on the Roll Damping Môment, by Iwào

Watanabè, February 1977.

In addition to the above.mentioned reports, the Ship Research Institute has another

series of reports, entitled "Report of Ship Research Institute ". The "Report" is published in Japanese with English abstracts and issued six times a year