On the hydrodynamic forces and moments acting on the two dimensional bodies oscillating in shallow water

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keports of search Institute for Applied Mechanics

VoL XXIV. No. 7S, 1977

ON THE IP DRODYNAMIC FORCES AND MOMENTS

ACTING ON TITE TWO D!MENSTONAL BODIES

OSCILLATING IN SHALLOW WATER

By Mikio TAKAK1*

This paper presents the solutions of radiation problems and the diffr-action problems in water of finite depth. The hydrodynamic farces and

moments acting ou the cylinders of the arbitrary cross sections are calculated

precisely by adopting the close fit method.

In the next place, the forced oscillating tests for heave, sway and roll are carried out by using three cylindcrs of Lewis form cross section. It becomes clear from the comParisou with the expermental results that the twoimensional hydrodynarnic forces and moments acting on the arbitrary cross sections without bigek-eel can he precisely obtained by the close fit method.

Finally numerical examples for submerged circuiar cylinder are shown.

And the effects of water depth and the viscous effects for roll are discussed.

1. Introduction

Because of increasing dimensions of ship in recent years, several sea de-pths of her trade routs and her working area have been becoming relatively

shallow. The reduction of sea depth may lead to hit or grand the bottom

of sea. It therefore has been becoming a great important problem to predict

the ship motions and the hvdrodynamic forces acting on the ships in water of finite depth, in which the ships can operate safely.

When the strip method is available for predicting the motions of ships in

water of finite depth, the two dimensional hydrodyramic forces and moments

acting on the body of ship are required according to the assumption of the

strip theory. As for the calculating methods of the h3-drodvnamic forces.

and moments acting on the two dimensional oscillating body in water of

finite depth. Porter discussed about tile hvdrodynarnic forces acting on tile heaving cylinder '. Yu and Ursell calculated the added mass and the wave

) This paper is translated from two papers which have been published at the Society .

J of Naval Architects of West-Japan on May 1975 and 1976.

* _{Research Associate, Research Institute for Applied Mechanics, Kynshu University "}

amplitude ratio of the heaving semi-circular cylinder and confirmed the availability of their calculated values by the experiment of he wave

ampli-tude ratio for heave . Sayer and Ursell obtained the added mass coeff

ici-ents of heaving semi-circular cylinder at w-Oby numerical calculations C. H. Kim and H. Keil independently calculated the hydrodvnamic forces

and monients acting on the Lewis form cylinder for heave, sway and roll by

adopting the method of multipole expansions which was an extension of the method of Grim and Tarnura cf the water of infinite depth to that of

the water of finite depth. Black & others S) and luma & others solved the radiation problem of the rectangular cylinder by using the method of

dividing the fluid domain. These methods above mentioned contain a

weak-point, by which is limited the shapes of sections of the ships or the

struc-tures. There exist on the contrary the finite clement method and the close

fit method which do not limit the shapes of cross sections to calculate the

hydrodynamic forces and moments acting on those. As for the finite element method, K. J. Bai Seto & Yamamoto and Takashina & others solved the radiation problems. As for the close fit method, Maruyama

Kan 1517) and Ikebuchi' obtained the two dimensional hvdrodynamic forces

and moments in water of finite depth.

As mentioned above, espacially there have been few experimental resuics. Consequently the correlation works between the theoretical and the

experi-mental investigations have been scarcely performed.

In this paper. firstly we shall extend the relations among the hydrody-namic forces in the case of water of infinite depth to those in the case of water of finite depth using the potential 2) of water of finite depth, and solve the radiation problems and the diffraction problems on arbitrary two dimensional cross section using the close fit method in which singularities are distributed on the surface of the clinder. The hydrodynamic íorces

and moments acting on an arbitrary cross section are calculated precisely

by using the above method.

In the next place, the forced oscillating tests are carried out by using three cylinders of Lewis form cross section, and the experimental results

concerning added mass coefficients, damping force coefficients and wave amplitude ratios are compared with the theoretically calculated results obtained by the close fit method. The availability of the theoretically calculating method is confirmed experimentally.

Finally we shall discuss the effects of water depths as for the hydrodyna-mic forces and moments.

1. 1 Notatica

L Length of cylinder

B Breadth of cross section C

C' FJh,ffk E,, e, Pi MH, M5. it Ñ, Ñ5, LVR

HYTJRO DYNAMIC FORCES IN SHALLOW WATER

Y, Oscillation

çj(x,y; f, y'

b,(x,y; x', y

(1j(X, y)

amplitude in the j-th mode

) Velocity potential normalized by velocity amplitude in the

j-th mode motion (j=1, 2. 3)

Velocity potential normalized by amplitude of incident wave (j=O, 4)

') Stream function correspoñding to velocity potential Density of source singularities distributed on sectional

con-tour

Integrai constant for the j-th mode equation

Hydrodynamic force or moment of the k-th mode by the

j-th mode of motion

Wave exciting force or moment of the j-th mode in

beam-sea condition

Pressure of the j-th mode

Added mass coefficient for heave, sway and added mass

moment of inertia coefficient for roll Damping coefficient for heave, sway and roll A _{Volume displacement of cross section}

Water plane area

Cß Block coefficient

GM Metacentric height

Radius of transverse gyration for roll

Breadth of bilge-keel

C _{Submerged part of cross sectional contour in rest position} ds Line segment along cross sectional contour

n Unit normal to cross section C

iJ _{Beam draft ratio (=B/2T)}

e _{Sectional area coefficient (=4/BT)} g Gravitational acceleration

KT _{Non-dimensional frequency (=} (02

T)

p Density of fluid

h _{Depth of fluid}

o, _{Circular frequency of wave}

il _{Wave iìurnber in water of infinite depth (=}

m0 _{Wave number in water of finite depth (a real root of}

equation: °' =m0 tanh m0Jz)

n,, (j=1, 2, 3..)

Imaginary roots of equation:

g =m0 tanh m2b Ç _{Amplitude of incident wave}

j(x,y; fl,

(x,y)

hi

AH, Â5, AR Wave amplitude ratio for heave, sway and roll

(Amplitude of radiation wave normalized by amplitude of oscillation)

1SR Lever of added moment cf inertia

Lever of damping moment

L

Mass moment of inertia for roll

Phase lag ori the j-th mode

Subsciipts

c Real part

s Imaginary part

A Symmetric motion

B Antisymmetric motion

i

Motion along x-axis (swaying mode) 2 Motion along y-axis (heaving mode)

3 Rotation around the origin O (rolling mode)

4 Diffraction wave

O Incident wave

2. Formulation of fundamental »rc,biems 2. 1 Coordinate systems

Consider a cylinder, whose cross rection C is symmetrical about a center.

///;///////////// //////////

y

HYDRODYNAMIC FORCES IN SHALLOW WATER 5

line, winch is forced into a simple harmonic oscillation with a small ampli-tude in water of finite depth as shown in Fig. 2. 1. Let a horizontal axis, which coincides with an undisturbed free surface, be x-ccordinate axis and

a *ticai axis be v-coordinate axis, which is measured downwards, and a

normal direction n on hull surface be directed into fluid. Considering the

motions of fluid with a constant depth h to be linear and two dimensional

phenomena, it is assumed that the fluid is invisid, irncompressible and

irrata-tionaL and the effect of surface tension is negligible.

2. 2 Fundamental equations

Let the velocity potential , the pressure P and the displacement of free

surface be represented as follows:

t) =Rc( (x,y)e},

P(x,y;t)=Re{p(x,y)e},

(2.1)

,(x,y;1) =Re

(x,y)et}.

and we have the relation between wave frequency w and wave number m0

in fluid of finite depth:

(02

K=--j-=m0 tanh m0h. (2. ¿)

Furthermore, let the unitØ velocity potential (x,y) with respect to radi-ation potential (x,y) oscillating with the amplitude Y1 be put as follows:

o1(x,y) = iwj Y1, (1=1,2,3), (2.3) where ç satisfies:

EL] the Laplace equation, in the fluid domain:

ôx2

y2'

ES] the dynamic free suface condition:

,.

ziØj+--- =u, dy 1o

[B] the water bottom condition:

-.

[R] the radiation condition:

+

[N] the ship hull condition:

(2. 4)

(2. 6)

6 M. TAKAKI

r'

Oj

_{on C,(j=1,2,3) (2.8)}

where vj are normal components of velocity on ship hull and can be expressed as follows:

v1=_Ox v2=_p. v.=yP-_x--.

_{On On On on}

Rewriting the equation (2. 9) by using the stream function conjugated with the velocity potential j,

we have

(2. 9)

where c(j=1,2,3) is unknown constant.

The velocity potential of incident wave and reflective wave can be put as

follows:

(x, y) = -a. ç

(, y),

(2. 11)

where a is wave amplitude, and then we have

on the hull surface C. The pressures with respect to the radiaticn potential and the diffraction potential can he expressed as follows:

p=p2Yj1,

(j=1,2,3), (2.13)

(2.14)

and the displacement of free surface with respect to the radiation potential

can be expressed as follows:

2. 3 Velocity potential and stream function

When a periodic source with a unite strength is put at an arbitrary point (x',y') in water of finite depth, the velocity potential, which satisfies

condi-tions (2. 4)--(2. 7), at the point P(x,y) have been given by Thorne19) as

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

r1 . , 1°( cosh k(h--v) cosh k(Jzy'

± 2.1-. 'L jo 1(K_{cosh khk sinh kh) cosh k}

i.

t

4'1=y±c1,

=x+c2, s!,3__(x2±y2)±c3, (2. 10)

=0, (2. 12)

ed ed as :ial :ial 'int idi-where

¡IYDRODYNAMIC FORCES IN SHALLOW WATER 7

esinh kysinh ky'1

_{k(x--x')dk (2.16)}

k cosh kh

4r cosh rn3(hy) cosh rn (hy')

c

ri0x ')

2,n0h+sinh 2n0h

r1=/ (x_x')2+(y_y')

r2=/(x_x')2+(y±y)2.

And the stream function, which is conjugated with equation (2. 16), can be

expressed as

S(x,y; x,y')=S(x,y; c',y')+iS,(x,y; x',y')

O O +2.P y

iI

sinh k(hy) cosh k(hy')

- 1

2 Jo l(K cosh khk smb k/i) cosh uk

e" cosh ky sinh ky' . ,

s1nk(xx)aR

k cosh k,z

dr smb nz,(hy) cosh nz0(hy')

_{m ---'}

2rn0h+sinh 2m0h Sifl

(2. 17)

3. Two dimensional boundary-value problems 3. 1 Integral equatioas and solutìns

Distributing line density of source oj( j= 1.2, 3) on the surface of cylinder, the velocity potential and the stream function at an arbitrary point P(x, y)

in fluid can be ecpressed as follows:

j(X,y) =Jci(x'.y').G(x,y; x',y')ds(x,y'), (j=1,2,3), (3.1)

j(x,y) 01(x',y').S(x,y; x',y')ds(x', y'), (j=1,2,3). (3.2)

Solving these integral eqations and evaluating the line density of source

j, the velocity potential function can be obtained. It is, however,

com-plicated to perform the numerical calculations of equation (3. 1) involving

differentiations on these velocity potential. So we should evaluate the line

density of source cj (. y) by using the stream function (3. 2). Rewrite

equation (3. 2) by dividing into a real part and an imaginary part, the fol-lowing equations can be obtained:

ç(P)

₌ j {sj(Q) .S(P, Q) cj(Q).S(P, Q)}ds(Q),

1.(P)= j {c1(Q).S(P, Q)+e(Q).S1(P, Q»ds(Q), (j=1, 2,3).

One can evaluate oj. and , by solving these two sets of integral equations,

but in the case of symmetrical cross section of cylinder, one can svmplify (3. 3)

the numerical calculations as follows21)

The case of symmetrical motion (j=2, heave): There are following

rel-ations given by

2(x',y')=r(x',y'), S(x,v; x',y')=S(x,y; z',y'),

and the imaginary part oi stream function can be obtaìned as

p 4 sinh m0(hy)

S1(_ ,

Q) =_h_mSlfl ?n0x.cosh m0(ny) cos in0x.

(3.4)

The case of anti-symmetrical motions (j=1, sway; j=3, roll): There are following relations given by

cJ(x',y')=.31(x',y'),(j=1,3),

S(x,y; x',v')=S(x,v; x',y'),

and the imaginary part of stream function can be expressed as follows:

4.'e sinh ;z)(hy)

S1(P,Q)=

_{2rn0h+sinh 2mi°5}

rnx cosh in0hy) sin rnx.

(3.5)

We consider the following functions given by

and P2'1_f (c2(x',y')

PJJ

Lc1(x', y')

f (r,(Y,Y)

Pj)J L,rj(x', y')

.cos sin .cos sin

rfß(x, Y')-oj(X', y'),

m0x" cosh _{rnO(h_ûdS(XY.)} rn0x'J cosh m0h

rn0x'j cosh m0(hy') _{ds(x',y),(j=1,3),}

m0x' cosh m0h

jA(X,Y)

=rj(x', y') ±-j1(X', y'),

(j=1, 2,3),

and so equation (3. 3) can be iransformed into the following

:J= J ': jS (x. y; x', y')ds (x', y'),

x',y')ds(x',y')

(j=L3).

forms: (3. 6)

(3.7)

4

sinh m(hy' cosh m1h ((sin m0x)

( O

cos m0x )LiL'. -

(3. 8)

The above equations (3. 9) can he rewritten by the values of and 0J,

(j=1,2,3) as follows:

P2A'_f (a2(x',y') cos PJA)J CLaJA(,.,Y) sin

P2,

_{D 11 I}

( (a2(x',y') cos

_f.'

'\

AJß2 jc'-fJi',1)

in (j=1, 2,3). 'Jk I 11k _pFY1 y'), rn0x) cosn mQh

n0x'i cosh m(hy')

_ds(x',y'),

n10x') cosh rn0h (j=1,3).

(3.9)

J

(3. 10) Substituting these relations into equations (3. 9), we have:

D PIA

fc

"h1c'Pia,

(»-=1,2,3)

Furthermore the distributions of sources densities o(x', y') oscillating with the j-th mode can be obtained by the relations of equations (3. 7) and (3. 11) as follows:

cj.(r', y') =a,fA(x , y')

P1.a1x'. y')

(3. 12) (j=1, 2,3).

Substituting these values (3. 12) into equation (3. 1), the radiation potential

functions can he obtained.

3. 2 Integration of pressures

Since the hydrodynamical force of the k-tb mode by the j-th mode of

motion may be defined as FJke, the force normalized by the velocity

am-plitude can be represented as follows:

(3.11)

(3. 13)

HYDRODYNAMIC FORCES IN SHALLO W WA TER g

Solving the above equation (3. 8). we can evaluate the values of a4 and aiR

(j=1,2,3). In the next place. we take the following functions given by:

PJA=PJC+

fi

(j,k=1,2,3).

defined as follows: In the case: j=k

the coefficient fj expresses the added mass or the added mass moment

of inertia,

the coefficient fjj. expresses the damping force or ti-te damping moment.

In the case: ik

the coefficient f.and f, express the coupling

force or the coupling

moment.

Table 3-1 Hydrodynarnic force & moment and the non-dimensional coefficients Inertial force force Heave (j=2) MJL=pf2:c M3=pf11 Sway (j=l) Roh (j=3) = Pf31 IN,r=PC-f--,

N=Paf,1

I N,,=pwr

3. 3 Wave amplitude ratio

A group velocity of wave in water of finite depth can be expressed from

the relation (2. 2) as follows:

2m0h _{_'}

(_-_tanii in0h).(1-- _{sinh 2rn0h).}

Since the average total energy per unit surface area are expressed to be

ogf1, the average total energy of propagating wave per unit time E0 can be expressed as

(3. 14)

Inertial moment Mp.=pf

Damping moment N, = - pwf1 NR = - pwf Added mass coef f. j = M/pJ I =Ms/pA

Added mas in. t. of

inertia coetf. MRMR,P4T

Damping foi-ce coef f. Ñ11=Ii/B/2g

Ñ=--t/72

Damping moment

coeff. - R p4B2 g

Wave amplitude ratio Aff =/Y

Added moment arm 'SR _{IVI I} M

r

rom

I

HYDRODYNAMIC FORCES IN SHALLOW WATER 11

' ¡ 2mh

E0= pg.2( g tanh m0h.)

sinh2nzh), (J=1, 2,3). (3. 15) m0

Since this energy is equal to the average amount W0 of work performed by the wave damping force acting on the cylinder, which is given by

w0= po)3 Yj /jj

the following equation can be obtained

Â2_{= =K2.F(rn0Jz) j fI,}

I fI},

(1=1,2,3),

where

2 cosh2 ,n0h F(m0ñ)_{= 2rn0h4-sinh 2;n0h}

With aid of equations (3. 13) and (3. 17) the relation between the wave

am-plitude ratio A5 and the damping force coefficient N5 can be obtained as

A'

Çko = F(rn0h) ,

(j=i,2,3).

(3. 19)

This is the so-called "Haskind-Newrnan Relation" in water of finite depth.. 3. 4 Wave exciting forces and rnomerLts by atera1 waves

The wave exciting forces and moments of the j-th mode are defined as

E5et, then according to equations (2. S) and (2. 14), the forces and the

moments normalized by the amplitude of the incident wave are defined as

follows:

'ga =

J(04-4)

ds, (1=1,2),

(3. 20)

e3= _{pga1T =}

J,(co±c4)__ds.

The velocity potential of incident wave with a unit amplitude is given by

ccsh n(hy)

_e*OOo.

(3.21) cosh m0h

be

can Since the function q and (j=1,2,3) satisfy Laplace's equation, according to Green's theorem, there is a following relation

(3. 16)

(3. 17)

And we ha e

-

from equation (2. 12). Furthermore. substitu-ting these relations into equation (3. 20), the following expression can be

obtained

where

ds=Jc',

ds, (j-=1,2,3). (3.22)

e1= H1 (m0h), (j=1, 2.3), (3. 23)

Hj(nz0h) f cosh m0(hy) eox ¿s,

)

Jc\ On (ynJ cosh mh

Hj*=Th+jH#:,

(j1,2,3).

(3. 24)

On the other hand, the velocity potential at infinity can he represented by

using of equation (2. 16) as follows:

jiF(m0h) .H (m0h) .ç40'.

H1 (rn0h) = Hç (m0h) I =Hj(m0h), (j=1, 2,3).

The wave amplitude ratio can be expressed as follows:

Âj=K.F(rn0h) .H1(,n0h), (j= 1,2. 3)

and by equation (3. 25) the phase lag j can be written as

h is clear from this equation (3. 25) that the function H14 is proportional to the amplitude of the wave propagating to the positive direction of x-axis, and that the function H5- is similarly proportional to the amplitude of the wave propagating to the negative direction of x-axis. Suppose that the

cross section of cylinder is symmetrical, we have

EJ=

tan(

(j=1, 2,3).

(3.25)

(3. 26)

(3. 27)

Substituting the above relations into equation (3.23), the wave exciting forces and moments can be represented by the wave amplitude ratio and the

phase lag as follows:

e1 =

K'h)

e'i, (j= 1, 2, 3) (3. 28)

HYDRODYNAMIC FORCES IN SHALLOW WATER 13

of the discontinuity of its first order derivative the relatica between the

function H, and the source densities distributed on the surface of cylinder

K2.F(nz0h) I f.iJ =42 .1C2 .F2 (m0h) . (P +F),

I fihl /{F(m0h). (PJ +P)} 4rn?, (j-1,2, 3). (3.31)

This formula is a convenient equtidnto examine the accuracy of numerical

calculations for hvdrodynamic forces and moments acting on the cylinder. 3.5 Coefficients of reflective waves and transmissive waves

We extend the relations of waler of infinite depth according to [19 to the case of water of finite depth. With aid of Green's theorem by using (2. 16)

the radiation potential in finite depth of water is represented as

J= ç&a&n)G(x,y; x',y')ds,

(j=1,?,3).

1

1a2

a

2(x,y) _2rJc

Oiz ¿s

F(m0h)H,(m0Jz)

cosh ,n(hy)

COS m0x,

cash m0h

Ø21(x y) f a ¿s

±F(m0h).Ht(rn0h) cash m0(hy) rnx.

cosh m0/z

Taking into account the symmetrical cross section of cylinder, the radiation potential can be expressed by dividing into a real part and an imaginary

part as follows.

In the case of ç?2(x,y)=p2(x,y)

(3. 32)

(3. 33)

tu-be

can be given as follows:

Hj(m0h)

2.Jci

coshrn0(Jzy) (j=1, 2,3). following relations: H3 from equations (3. be obtained: (3. (3. 17), 29) 0) (3. 21) =

And from equation (3.6) we have the

H2c(rnoh)J9C P?C j_Hi(m0a)

_Pic

H2(m0h)_7

(P

Hj(n20h))

LPJ,

.

Furthermore by eliminating the function and (3.30), the following equation can

by anal axis, the the iting I the 1 and

In the case of ç(x,y)=ç'j(--x,y),

(j=i,3)

ds

+iF(m0h) .H (nz0h) coshnl0(JiY) sin "lox,

F:

iF(m0h),II(m0h)

coshrn0(hy)

Furthermore, from the condition =0, (j=1,2,3), and equations (3.33) and (3. 34) by taking into account the singularity of the function G. the condition for an arbitrary point P(x,y) on the surface of cylinder is repre-sented as follows: '2(x,y)=

F(m3h)

ds cosh m0(hy) cosh m3h cos m0x, 0(xv)

=_JçarG. ds

ir

cosh ,n0(hy) cosh m0h Cos fl20X,

1(x,

ds

cosh m0(hy) _{sin m0x,}

cosh m0n

In the next place, since a diffraction potential ç is given by 013(x,

y)=;J

cosh m0(hy)

-cosh nz0h Slfl m0x.

454 *44 ± 4B

where the function and are complex valued functions. ations (2. 12) and (3. 21) it follows

(3. 34)

(3. 35)

(3. 36)

equ-3)

the

pre-

equ-i

HYDRODYNAMIC FORCES IN SHALLOW WATER 15

& I cosh m0(hy)

I

-i cosh rn0h

cos m0x

a cosh m0(hy) _{sin rn0x}

an -

an i cosh ,n0h

(3. 37)

The diffraction potential can be written with aid of Green's theorem as

= 21J(

)G.ds=

-

(3.38)

and similarly the incident wave potential can be expressed as

Since it is possible to recognize that the function and are independent each other, the following equations can be obtained:

i

f Ç cosh in0(hv)

) a

c cosh iiJz

- COS

- 2

J

S coshrn,(hy)

Qx+4B}-G.ds.

And the fllnction H4 can be expressed as

H4t (m0h) r

a.

a cosh rn0(hy)

J an

an)

cosh ds

a

cosh rn(hy)

_{etmox ds}

-

_an

_coshmh

= H + 1H B.

Furthermore, since the function H4t has a symmetric relation and an antisym-metric relation, we have the following expressions:

HA(mOh) =HA(m0h)₌

_{-J (i+ç'4)}

_öna cosh m0(h y)_{cosh m0h}- cos m0x, i

J

(3.41)

(3. 42)

Ha(moh) = -HB(rn0h) = iJ(oB+c4P) & cosh m0(hv)

&n cosh ,n0h sin m0x.

(3. 43)

--)G.ds=o.

(3. 39)

Substituting (3. 39) into (3. 38), we have

Now we divide the function and into a real part and an imaginary

part as follows:

Ç444c±

Ç4ßÇ4BC+ tÇ4ßS.

From the above equations and (3. 37) we have the following

±4AC) =

H (»0h)

Ht43 (m0h)

(n0h) - 1I4 (mh)

±

21J(A±4Ac)Gc

cosh m (hy)

cosh m0h COS n2x{i

/)4AS---GCds±F(mA,z) .Ht4

J

expressions:

According to the above equations and (3. 41), we have the following integral equations satisfying the boundary conditions (2. 12):

F(rn012) ii4(rn0h)}, , cosh m0(hv)

(me,,) _{cosh m0Jz} COS 7fl0X,

/

(3. 45)

J&oB±Ç4Bc) ds

cosh m0(hy)

+ _{cosh m0h} sin m0x{1±iF(rn0h).Ht.3(rn3h)},

(3. 46)

(3. 47)

&=-

JcSus G,ds_iF(moh).Htc(nzoh)C0Sh in0(h-y)_{cosh m0/j} Sifl P?X.

On

Since these equations (3. 33), (3. 36), (3. 45) and (3. 46) are the second kind

of the integral equations cf Fredholin Type, it is able to recognize that these equations have solutions respectively and the dependent relations among these solutions can be obtained as follows:

/ 4AC= -=0,

---6 0 cosh m0(hy) (3.44) O O On an0A= O O;z

&coshrn0(hy).

cosh m0h CUS m0x, sin mox, cosh r,20h cn

!ary gral max, / mr,x. kind that rn ong )

HYDRODYNAMIC FORCES IN SHALLOW WATER 17

Ht(m0h) HtB(mCh)(.1

_{_) C 48)}

H(m0h)

HIßC(mGh).

J

J.

similarly

(*oA+*4Ac) .F(m0h) .HAc=çS4A{1F(nzOh)

(oE±4Bc) .F(rn0h)

=iç{l+

iF(m0h) .HB3}.

From the above equations and (3. 43) the following couations can be obtained

2S

=F(rn0h) { 2 + (HAs) 2).

Ill

=F(rn0) {(H)2± (H3)2}.

J

By putting on

114-is

=tan a,

---=tan a2,

2

1c 2C

from (3. 47) and (3. 48) the function H can be represented as

i

F(mh) (e2 sill a2-re1' (22).

Concidering that the functions FIL. and HT are imaginary and the functions

and H3 are real, we have

2;

a2=12 ₂

The function H can be rewritten according to the above relations as íoilows:

H4*

= F(rn0h)

(iez cos 2e' sin

(3. 53)

The outgoing wave amplitude at infinity, according to the diffractin potential

ç, are given as

-..jJJ4.F(0J)

cosh rn0(/zy)_{co-sb m9h}

e'o

(3. 54)

When the incident wave with an unit amplitude incomes to a rigid body from a possitive direction of x-coordinate axis, the wave amplitude at ini

i-nity, x*±, can be expressed as follows:

(3.52)

(3.49)

(3. O)

F(rn0h) cosh m0(hy)

e'

cosh nz0h

{1+iH4.F(nz0h)}

cosh m(hy)

e*irnoX

cosh mh

V/lien there is a rigid body in water of finite depth, the coefficients of

reflective wave and transmissive wave can be consequently expressed as

follows:

C,?= jiH;(m0h).F(rnh) , _(3.56)

C. = i + ill; (m0k) .F(m0h) . (3. 57)

4. _{Cernparion experimental resils with numerical results for two} dimen-sional hydrodynamic forces and moments

4. 1 Numerical method to find Green's function and stream function Since it is very difficult to soiv analytically the integr?.l equations in the

previous sections, we suppose that the source densities on th cross section,

where the surface of cylinder are divided into N-parts, are constant

respec-tivcly, and tolve two system of linear equations with (N--1)-unknown variables which are unknown source densities oj and c and integral constants C.. and G. (j=1,2,3). Thus the distribution of source densities can be

evaluated by the above method.

ir

/

inj(j. / X I X

cl

0* rn0

-irj(jl-S'

Fig. 4. 1 Line of integration in the complex plane -- rn

HYDRODYNAMIC FORCES IN SHALLOW WATEF 19

In this case the numerical method to find Green's function and stream function are as follows.

The term of logarithms in (2. 16) is transformed to the expression of

infi-nite integral 22) and the principal value integral taking the line of integration in complex plane shown in Fig. 4. 1 is carried out and is transformed to the

infinite series given by

GC

4r cosh rn(hy) cosh ;n(hy')

2rn0h+sinh 2m0h

rn0Ixx'

; cos n1(hy) cos n-(hy') e'i''

-

npz + sin 2nh

In the next place, as for the stream function corresponding to the above

Green's function we consider similarly the following two domains, we have

SC= sgn(xx')

sinh rn9(hy) cosh m0(hy')

_{20h+sinh 2rn0h}

cos nz(xx')

4

Jj

., sin

fl(hy) cos n(hy')

,

...flJPL-YSIfl flj except for the above one.

It is necessary to calculate the infinite series G, and S. Now putting the errors of the sum with N-terms in infinite series GER, SER respectively, the functions GER and SER are represented23; _as

-cosj(a)cosj(ß)

eT

GER=

_{cos n5(hy) cos n(hy')}

i i i 2nh ± sin 2njh (4. 1) (4.3) (4. 4) n the tiofl, spec-flown ;taflts an be and

4t. sinh m0(hy) cosh m0(hv)

-S-=r

sgn(xx)

2m,h±sinh 2nz0h sin nj(h--y) cos n1(hy')

cos in0(xx

.'

- 2-(4. 2)

2n1h+sin 2n1z e

for xx'<O, y-y'<O.

Is of

as

Il-SER=

1f

---(tan

_{1_e_r cos a}

ersifla

+tan

.1

sInJ

±-

S

2r j

.!,

sin nj(h-y) cos nj(h-y) e"''

2njh+sin 2nh where

a=---(y+y),

=---(y_y'),

The errors GER and SER for instance x-x'I/--=O.2, KT=O.5 are shown

in Table 4-1. According to this table it is obvious that the errorsGER and

SER become large as the depths of water become deep. In order to

calcu-late the values of the Green's function and the stream function within 0. 1% with maximum error, we calculate those taking the number of terms N=20

in infinite series.

Table 4-1 Errors of infinite series (Ix-x' ,'---=O. 2, KT=0. 5)

eTsin

1_e_T cos

\

fa-fl\

)COSJ 2 1 _{e_ir} -J

r=

Ix-xi.

(4.5) N

JT

ERR

-..

I 1.1 1.5 2.0 4.0 I 6.0 GER 0.88x10' 0.24x102 0.54x10 0.76x10 063x10-1

O _{SER 0.76x103 O.28xl0} 0.59x101 0.95x1OE' 0.30x10-1

GER O.29x1O- 0.10x10' 0.19x103 0.62x103 O.45x102

SER 0. 14 x 10 0.28 x 10 0.85 x 10 0. 27 x 10 O. 97x102 GER 0.35x10- 0.27xi05 0.33x!05 0.88x10-3 0.12x10 SER 0.21 x 10 0.44 x 10 0. 56x 10 0. lOx 10 0. 21x10-2 GER 0.35x10 0.30x10 0.95x105 0.33x104 0.21x10-3 SER 0.21x10 0. 56x105 0. 15x10 0. 70x10-5 i 0. 72x10' 100 GER_SER 0.35x10-0.21 x 10 0.30x1050. 36 x 10 095x10-'0. 15 X 10' 0.87x1060.20 x 10 0.68x10-'0. 17 x 10

In the next place, the error of GER and SER become large because of a

slow convergence if the number of terms N are not taken to sum up much more

in the case And so we transform the equations (2. 16) and (2. 17) to the following forms in order to reduce the computing time. The velocity potential is written by

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

x',y')

f pk(YY')

=Iog_!1L±2.P.v.J

'.

cos k(x-x')dk

o od

leu-1°/o

= 20

HYDRODYNAMIC FORCES IN SHALLOW WATER 21

+2.P y

í

e'(Ksinh ky'k cosh ky') (kcosh kyK sinhky)

k(K/e)(K cosh Rhk sinh kh)

xcos k(xx)dk

dr coh nz0(hy) cosh in0(hy')

_{cos m0(xx')}

± I

2m0h+sinh 2m0h

The stream function is represented by

S(x,y; x'.y')=S(x,y; x,y')+iS(x,v; x,y')

-t C 1f + V')

sin k(xx')dk

+2. P \T

1 e(K smb ky'k cosh ky (K cosh kyk smb ky)

J

k(Kk) (K cosh khk smb kh)

I

xsin k(xx')dk

1(Y' 4it siiih

oshn(hy')

= (O - Û2) ±J0. +1k + iSa. (4. 8)

10- i um 1h-30,

w-3

-

I (4.9)

uni S'2r e

(V*1f') sin K(xx'). )

1O

The third terms I in (4. 6) and 1k in (4. 8) become to zero as the depths of water become infinitely deep, and the function G (x,y; x',y') and S (x,y; x', y') coincide with the Green's function and the stream function in the case

of infinitely deep water respectively. Namely by adding the third terms

'k and Jh to the equations of infinitely deep water, these equations satisfy

the conditions of water of finite depth. Making a computer program of equations (4. 6) and (4. 8), the computer program to calculate the hydrody-narnic forces in water of finite depth is completed and at the same time the computer program in the case of water of infinite depth is completed. On

the other hand, if we have a computer program to calculate the hydrody-namic forces of infinitely deep water by the so-called "close fit method", we can make easily a computer program to calculate the hydrodynamic forces

in water of finite depth by adding the third terms 'h and 1k to the eqations in water of infinite depth. Therefore these (4. 6) and (4.8) are very

con-(4. 6) =log ±L.+Ik±iG. 2 _{(4. 7)} Ii

_{hm G-92e}

;i»y' t

K(xx').

10-i 1O-of a more 2. 16) -ne.

venient equations.

The calculating methods of the first term and the second term of (4. 6)

and (4.8) in water of infinite depth have been studied by many researchers2'

21), we omit the calculating methods of those and deal with the method of calculation of the third terms I,. and J, in each equation. We use the fol-lowing notations in Fig. 2. 1.

V=Kh, u=kh, ß1=tan X

Yh1 Y/ h,

y,

y,=y'/h,

xh,=x/h,

x2=x'/h,

rhl=(x2 ±y2) liz/h,

r..

= (z'2±y'°) l2/h.

2=tan°

j

(4. 10)

The third terms I, and h are transformed to the following double series

which involve the singular integrals F21(V), F223(V) respectively.

-

_F

₊₍

_y

y2 rkI2321.r+22't ccs(2s-2t-1-1)?1.cos(2t±1)fl2

_{(2s-2t+1)! (2t+1)!}

+

22'2 cos (2s-2t4-1)

_{(2s-2t+1)! (2t)!}

. cos 2t32

J =F2,1 (V){V2±

rh122.rh22211. cos 2t9 cos (2s-2t+1)$ +

_{(2s-2t±1)! (2t)!}

-

r,Ll2t_2.r2ot.cos(2s_2t)31.cos 2t92 } (2s-2t) ! (2t) F22+3

(V)J V2

rhl-2.rh-.sin(2s-2t-1-2) t9.sin(2t+2)2

_{(2s-2t+2)! (2t±2)!}

+ y

'

_{(2s-2t±2)! (2t+1)!}

(2t r 222.siii(2t+1) A1.sin(2s-2t+2)2 L-O

(2t±2)! (2t-+-1)!

.,

rft12

°'.r22t.sin(2s-2t±l)91. sin(2t±

1)32

(2s-2t±1)! (2t+1)!

(4. 11)

r+12I.r,2.sin(2s2t+ 1)9. cos(2t+1)A,

(2s-2t±1)! (2t+1)!

V2 ,

r1 r2

cos 2tfl) sin(2s-2t±2)492

-

(2s-2t+2)! (2t)i

y

r,.sin (2s-2t±1)

cos 2tfl2

-

L

(2s-2t+1)! (2t)!

1. 6) of fol-- ___________ t £ F2+1(V)=P.V. J

HYDRODYNAMIC FORCES IN SHALLOW WATER 23

r2 r2'. cos 21ß,

sin(2s-2t±1)ß2

(2s-2t+1)! (2t)!

F2j+3(V) IV r,l2-'.rh2212-2.cos(2t+1)Il.sin(2s-2t±2)fl2 t

(2s-2t+2)! (2t±1)!

-

._4-J ,,

r221.sin(2s±2t+2)j1.cos(2t+1)Z

(2s-2t±2)! (2t+1)!

rhl222 rh22. sin (2s-2t+2) cos 2t92 +

_{(2s-2t+2)! (2t)!}

,. 2t+1 2

2'.cos(2±1)1. sin(2s-2t±1)2 1

h1 P2

(2s-2t+1)! (2t+1)!

Y m

Fig. 4.2 Line of integration in the compiex plane

where the singular integrals F,1(V),F23(V) involving the non-dimensional frequency parameter V only can be represented by taking the line of

inte-gration in the complec plaiie shown in Fig. 4.2 as follows :

e'

du

(Vu) (V cosh uu sinh u)

-

J

(4.13)

b23'

-

-N21=

_{- 2}

V2{e sin(-!__b)}

+bV{cos( S b) sin(--.--b)}

b2 sr 'sir

±----(e cos

2

_b)}J

b±co.s b) -bV(sinh b-sin b)

+--(cosh b-cos

b)1. F₂₊₃₍

_{) -}

- e".u2'3 du 1 J0 (V-u)(V cosh u -u smb u) J0 D23 ° N21

_{:.:Cv2{e-s cos4-±cos(--_b)}}

-.bV{cos( _b)+sin(-_b) Sir

.(Sí

2 sin D2+3= D23+1. (4. 18)

In the numerical calculating equations (4. 13) and (4. 16) with aid of Simp-son's rule, it is necessary to determine the piesh size of Simpson's rule, because the behavior of F231 (V) for tile mesh size b=0. 02, 0. 05, 0. 1 and

0. 5 are shown in Fig. 4. 3 as the functions of frequency V and it is seen that the effect of mesh size of Simpson's rule depends on the accuracy of the singular junction F231 (V) in low frequency range and it leads to the

wrong calculated results consequently.

Table 4-2 Mesh size b br Simpson's rule

s=O s=i S=2 S=3 O V0.015 0. 015< V0. 04 0. 04 <V0. lo 0.10 <V 0.20 0.20

<Vl.O

1.0 <V<o O

Vi.O

1.0 <V<c

O <V0.2

0.2 <V<c

o V<co

0.001 b=0.01 0.02 b=0.05 b=0.10 0. 50 Û=0.50 (4. 14) (4. 17) (4.15) (4. 16)

Frequency (V= Mesh sizc (b)

g

b=0.10

b=0. 50 b=0. 10 b=0. 50

imp-rule, and seen y Of the

HYDRODYNAMIC FORCES IN SHALLOW ¡VATER 25

2.0 L b005.0.l.0.5

r

/

¡.0-

b0 f005 r o t J \ 0.1 02 0.3 _.2 04

L\

----.--.

\

\

-2.0-/

.5

Fig. 4.3 Behavior of the singular function F2.1(V) for varying b at low frequency

C. H. Kirn also has recently corrected his numerical results by taking into

account this effect 26i _{The mesh size b used in our calculations are shown}

in Table 4-2. _{And the limits of integrations by using Simpson's rule are} determined within 0. 1% with maximum error. _{The number of suffix S} in double series I and L are taken from S=O to 7 because the numerical

results obtained by taking S=7 coincides with the ones obtained by S=8 up

to five significant figures.

4. 2 Accuracy of the numerical calculations

26 M. TAK.AKI

dividing the surface of cylinder become much. The accuracies of the numeri-cal numeri-calculations of the hydrodvnamic forces for heaving semi-circular cylinder with the valuable dividing number of surface cf one side N4, 8, 12, 16,

20, 30 and 40 are shown in Table 4-3. It is obvious that the accuracies of numerical results become good as the dividing numbers become much and

the depth of water does not depend on the accuracies of numerical results too much ar1d the hydrodynamic forces of very shalljw water h/T=1. 1 also can

be obtained with a good accuracy due to this table. Since the maximum error for calculating with N=20 is no more

than 0. 2% at the most, we

think the accuracies of results obtained with N=20 to be enough and have calculated with the dividing number N=20 hereafter.

Sayer and Ursell have recently calculated the added mass of heaving semi-circular cylinder in water of finite depth at the low Írquency KT=0.

1.0 0.9

0.8-0.7 0.6 0.5

-i

Semi- circulcr

Scyer & UrscII(KT0)

° Tckcki

(KT0.005)/

/

/

0.4-Ti2

¡'0

N k' '

KY 1.2 1.5 j 2.0 4.0 6.0

FIYDRODYNAMIC FORCES IN SHALLOW WATER 27

Table 4-3 Accuracies of hydrodynamic forces for heaving semi-circular cylinder

10.0

Their results and our results (our results are hot KT=0 hut KT=0. 005) are shown in Fig. 4. 4. It is seen from this figure that we obtained good numerical results at such a very low frequency range by this caicuIatd

method.

4. 3 Analysis of forced oscillation test

The forced oscillation test was performed in the small experimental tank

(LxBxD=60 mxl. 5 mxl. 5 nl) of the Tsuyazaki Sea Safety Research

Labo-ratory belonging to the Research Institute for Applied Mechanics of Kyushu

University. Photo 4-1 shows the coidition of forced heaving test and Photo 4-2 shows the condition of laterally forced oscillation test. Three model chips

used in this experiment have the two dimensional Lewis form _{cross section} shown in Fig. 4. 5, and the lengths of models are ali 1. 45 rn in order to

keep the two-dimensional test condition. _{The forced heaving test and the} laterally forced oscillation test (swaying and rolling) were carried _{out in the} different test conditions respectively. _{Namely, the forced heaving test was}

performed in four kinds of water depth (h=130.Ocm, 59.2cm, 31.4cm and

21. 5 cm) by using only the model with a rectangular cross section. On the

other hand, the laterally forced oscillation tests were carried out in various depths of water (h=124.Scrn, 66.5cm, 32cm, and 22.5cm) by using three

ship models shown in Fig. 4. 5. _{In these ecperiments the test of the} rect-angular model was carried out with a bikeel of 10 mm breadth and the

tesi í:circular model were ried out ith and without a bilge-keel.

We measured the displacements of the cyLnders, the hydrodynamic forces and moments acting on the ship models and the amplitude of outgoingwaves.

0.50 1. 00 1.02081.0152 1.0036 0.99770 j 0.99488 0.98890 0.99758 0.99755 099720 1.0026 0.98907 j 0.9828 0.98760 0.99361 8 0.50 1.00 0.999660.99755 0.999520.99754 0.999750.99829 0.999870.99874 0.998911.0001 0.999920.99798 1.00721.0036 0.50 1.0001 1.0000 1.0001 1.0002 1.0002 0.99910 0.99883 12 1.00 0.99980 0.99958 0.99989 1.0002 1.0002 0.99878 1.0017 0.50 1.0002 1.0001 1.0002 1.0002 1.0003 1.0003 1.0004 16 1. 00 1.0006 1.0002 1.0003 1. 0005 1. 0008 1.0011 1.0011 0.50 1. C003 1.0002 1.0002 1. 0002 1.0002 1.0011 1. 000.4 1.00 1.000S 1.0004 1.0004 1.0006 1.0006 1.0028 0.99806 0.50 1.0001 1.0001 1 0002 1.0001 1. 0600 1.0003 0.99748 0 _{1.00 1.0007} 1.0095 1.0004 1.0005 1.0002 j 1. 0013 0. 99597 0.50 1. 0003 1.0002 1.0001 1. 0601 1.0002 1.0008 1.0012 40 _1.00 1.0011 1.0005 1.0004 1. 0605 1.0004 1.0008 1.0015 eri-der 16, of md too can urn we I,

'? %,:..t&

t:1

;J

_.;;;

't

A'

'T'

4

«U, t

rr

-, -- -i

1ic.

'-':

f ._,.z_tfl_&.

f

,... i I 5 ' - --.- .1 - ..

-'---.

Photo 4-2 Condition of laterally forced oscillation test

U

The swaying force, the roiling moment and the coupling force acting on the ship model were composed with three forces which 'rc-ere measured by

the three strain-gauge dynamometers2. The amplitude of the outgoing waves was measured by the wave height meter which was set at the point

from the ship model by 5m. Thus the forces and the wave heightmeasured

PIYDRODYNAM!C FORCES IN SF/ALLOW WATER 29 Rectangular cylinder 8/2

Jf=M,2= P4sR,

T Elliptical cylinder

----.2\T=.2Vd=- !V,.l,,, T

i,

Semi-circular cylinder 0' -1.0 0' 0.7654 0' 0.7054 H-I.25 H.-l.25

H.-l.0

B-040m B-C.40 B_040m T -016m T-016m T -C.."Cr be.K. - ¡Omm

Fig. 4.5 Cross sectionai forms of models

were analyzed with aid of Fourier expansion, and we compared the numerical results with the experimental results of the first order with the circular

frequency co.

Now, the heaving equation and the coupling equations between swaying and rolling about O (shown in Fig. 2. 1) are represented as follows:

(p4+Mff)+Nff +pgA

=ZA sin(cúi+ez), (4.19)

((P4+Ms)±Ns±(M*P4

)ç±Nç= YA sir(cot±E).

OG)±N,=L4 sin(wt-),

(4. 20)

where the direction of moment is c.ockwise,

-f

Z4: Heaving exciting force, : Phase lag between heaving displacement and heaving hydrodynamic force, YA: Swaying exciting force, : Phase lag between swaying displacement

and swaying hydrodynamic force, L4: Rolling exciting moment, EL Phase lag between roiling displacement

and rolling hydrodynarn ic moment.

g on d by oing point 3ured

i

(i )

The forced heaving test

The origin of two-dimensional model ship O is forced to a harmonic

oscil-lation =ZA sin wt. Then the coefficients of hydrodynamic forces with

the first order of circular frequency w can be obtained from equation (4. 19)

as follows:

ZAcossZ w Z4 N Z4 sin_{) ZA}

p4+

Addedmass of heave Damping force of heave,

J

Z4=1. 5cm: Heaving amplitude. (ii) The forced swaying test

The origin of two-dim'flional cylinder C) is forced to a harmonic

oscil-lation =y4 sin wt. Then it means that ==0 and the relations between

the coefficients of hydrodynamic forces and moments with the first order of circular frequency w can be obtained from equation (4. 20) as follows.

Addedmass of sway,

i

L4 cm pg4.G-M . Virtual mass moment

J#-rM_

+ _{w2 of inertia of roll,}

I

(4.21)

22)

WYA

₎

y4=2cin: Swaying amplitude.

(iii) Forced rolling test

The two-dimensional ship model is forced to a harmonic oscillation

sin cút about the point O origin of ship model.

Then it means that t==O

and the coefficients of equation (4. 20) have the following relations.

N Y4 Siti ty Damping force of sway,

LA cos EL _{+ pl Coupling moment of sway}

(4. 23)

s

w YA

L sin CL _{into roll.} WY4

(4. 24) N L4 sin CL : Damping moment coeff. of roll,

j

YA COS Ey

-

W2A Coupling force coeff. of

N,

YA Sfl Cy roll into sway,

(4. 25)

Y4cosE _p4:

il-th 9) cil-een of ç=5°, 10°, 15° Rolling amplitude.

4. 4 _{Comparison the experimental results with the calculated results} The calculated results and the eeperimental results obtained by the above

mentioned method are shown in Fig. 4. 6-s-Fig. 4. 35.

(i )

Added mass coefficient for heave: M

M11 increases in the full frequency range as the depth of water becomes

shallow. This tendency is agreement with the Tasai and Kim's result

which shows that the heaving natural period of ship becomes large as the

depths of water become shallow. In the range of low frequency Ivi11 of infinite depth increases in a logarithmic way, but 1Ç1 _{of finite depths for}

KT--O vic-Id finite values respectively according to our calculations and

e-periments. Sayer and Ursell have calculated MJf of semi-circular at KT=0 as for this characteristic . The calculated results are larger than the ecperimerital results at very shallow water, but the tendencies of both are in good agreement. (Fig. 4. 6)

Q-=l.Q

Ho= 1. 2.t5 N

f

0.5 (.0 ¡.5

Fig. 4.6 Ví11: Added mass coeff. of heave for rectangular cylinder

(ii) _{Damping coefficient for heave: Ñff and wave amplitude ratio: Â}

NH increases in the full frequency range as the depths of water become

shallow. The wa,'e amplitude ratios A» also have similar tendencies. Since

the damping coefficient obtained by the potential theory is ail compose. of

the wave damping force, the damping coefficient Ñ11 at RT=0 has _{to yield}

to zero because of Afl=O. But since the gradient of Aif at KT=O becomes

HYDRODYNAM1C FORCES IN SHALLOW WATER 31

I O - rh rh -.1 A ' - -- _' Q o

h/r.l.344

h/T-:.962 hIT-370

h/r-8.t25

Cal. Exp. o

---

-f

Q N

large as the depths of water become shallow shown in Fig. 4. 8, ÑH does

not yield the zero value near at KT=0 shown in Fig. 4.7 that is different

from one of other oscillating mode. This characteristic is comfirmed by calculating very near at zero frequency KT=0. The calculated results of the damping coefficients and the wave amplitude ratios are in good agre-ement with the eperimentai results.

d'= ¡.0 Ho L 25 k Ca! hIT-1.344

h/TI.9e2

-hIT-370

hIT -6.125 X 9'. Ó

;c

---o 0 0.5 ¡.0 ¡.5 ¡<T

Fig. 4.7 Ñ11: Damping coeff. of heave for rectangular cylinder

(iii) Added mass coefficient for sway: M

decreases generally as the depths of water become shallow, but increases coversely in the range of low frequency as the depths of water become

shallow. The above tendency is independent on the cross sectional form of

cylinder and appears uniformly. As shown in Fig. 4. 9, 4. 10 and 4. 11, the

numerical results agree well with the eperimentai results.

(iv) Damping coefficient for sway:

Ñ, increases in the range of low frequency as the depths of water become

shallow but has a tendency decreasing conversely in the range of high frequency as the depths of water become shallow. The rectangular model

especially of which beam draft ratio 119 and the sectional area coefficient o

are larger than those of semicircular model shows intensely the above tendency. Because the wave amplitude ratios become large in the range of high frequency

as the depths of water become shallow (Fig. 4. 17), when H and o of the cross section become large. The theoretical calculations coincide with the

ioes rent :d by ts of agre-eases come rm of , the :ome high model ient e .dency. uency of the :h the i z <q UI 0. 5

HYDRODYNAMIC FORCES ZN SFL4LLOW WATER 33

0= i.o -1-Io - I. X

h/re

¡344 h/T- ¡.962 h/Trn 7Q /

h/re.r23

L ¡ 0 0.5 1.0 1.5 - ¡<T

Fig. 4.8 Ä,: Wave amplitude ratio of lìeav for re;taguIar cylinder

0.5

'o

=0.7854 ¡.0 Cc?. Exp. h/T.I.,25

hIT-i6O

---h/r.3.325 -

---h/T6.225 - O o X 0.5 1.0 !.5 ¡<T

34 M. TAKAKI

.,

OE

¡.0

0.5 C-)

Fig. 4. 10 i: Added mass coeff. of away for elliptical cylioder

¡.0

Ci. S o

-I

& Q. - 0.7854

:25

Cal. Fxp. h/T-l.40G X o

- --

A I i I ¡.0 ¡<T

- ¡.0

Ho = ¡.25 Cal. Exp. h/T-t.4o6 X h/T.2.00

hIT4.156

-hIT-7.781 C) i I 0.5 (.0 (.5 K1

Q, <T

f

.: X X o - -

-.---r

/

¿I J 'o h

"

J,

/ /0

C

Ob

1.0

j.0

fl. 4.12 Ñ: Damping coef î. of sway for semi-circular cylirider

RYDRODYNAMIC FORCES IN SHALLOW WATER 35

(1 =0.7854

Ho = ¡ .0

=0.7854

Ho = 1.25 Ccl. Exp. h/T-I.125 X h/T1.6O o h/T=3.325

h/Ts.225

o Cal. Exp. hIT-1.406 X

h/r-2.cO

o

hIT4.156

-0.5

1.5

KT

Fig. 4. 13 Ñ: Damping coeff. of sway for eIlipicai cylinder

<2:

I.0

0.50 0.25 o t.-,

¡.25-<q 0.75 -o 0. 5

k'

k.

o.2

.. .. = 0.7854 Ho 1.00

Fig. 4.14 Ñ1: Damping coeff. of sway for rectangular cylinder

o hir-1.125 h/1 -1.60 h/T- 3. 325 h/I -6.225 Cal. Exp.

-

o 0.5 ¡.0 ¡.5 k. o - KT

Fig. 4. 15 Â1: Wave amplitude ratio of sway for semi-circular cylinder

1.0 =1.0 Ho = 1.25 h/T.I.06 hiT-2.00 hiT-4.156 hiT-7.781 Cal. Exp. X k. O

-

--k. 4 L o. LO 1.5 KT

t

1.0-0.5

X

F:g. 4.16

HYDRODYNAMIC FORCES IN SHALLOW WATER 37

experimental results shown in Fig. 4. 12, 4. 13 and 4. 14.

(y) Wave amplitude ratio for sway:

The tendency of A, with respect to the depths of water is completely

similar to that of the damping coefficient composed of the wave damping

force which increases in the range of low frequency and decreases conversely

in the range of high frequency as the depths of water become shallow. The experimental values show a little scattering, but tile calculated results

are in fair agreement with the experimental values shown in Fig 4. 15, 4. 16

and 4. 17.

(vi) Added nass moment of inertia coefficient for roll:

-Since the virtual mass moment of inertia (J±i11) composed of the mass

moment of inertia J and the added mass moment of inertia M has been measured by the forced rolling test, it is better to indicate the virtual mass

moment of inertia (J-i-M1) than the added mass moment of inertia MR because of involving the analytical error. So the figures show the virtual

mass moment of inertia coefficient (T+M).

Fig. 4. 13 shows the virtual mass momement of inertia coefficients of semi-cIrcular cylinder. In the case of the scmi-circular cylinder without a

o HS

¡.25

FA

A <q Â: Wave amplitude A # Cc!. Exp.

h/r-t.406

X 0/1-2.00 0

h/T4.t56

-hIT-7.761 O

0.5

¡.0

1.5

- KT

sway for elliptical cylinder ratio of

A

o

0

.,I.o0

H 1.25

4.17 A: Wave amplitude ratio of sway for

Ccl. Exp. hIT-1.406 X

h/T-2.00 - -

G

h/r4.1.6

-hiT-7.781 o 0.5 1.0 KT rectangular cylinder

bilge-keel rolling about the center of the circular O, the added mass moment of inertia obtained by the potential theory is equal to zero and then the

virtual mass moment of inertia for roll coincides with the mass moment of inertia of the cylinder. The experiments of the conditions with a bilge-keel (bBN.=lO mm) and without a bilge-keel have been carried out. The virtual mass moment of inertia of the semi-circular cylinder without a bilgekeel obtained by the experiments agrees almost with that obtained by the

poten-tial theory in the water of deep depth (k'T=6. 225). The virtual mass moment of inertia of the semi-circular cylinder with a bilge-keel, however,

increases rapidly in the range of low frequency as the depths of water

become shallow. In the forced rolling tests by Vugts , it has been seen

that when the rolling amplitude is changed, the added mass moment of

inertia MR decreases in the case of the small rolling amplitude, on the

other hand, in the case of the large rolling amplitude MR increases in the

range of low frequency. He has explained these phenomena to be caused by his experimental error. But it seems that these phenomena caused by the viscous effect of the fluid, because the radius of transverse gyration

and the center of gravity of this cylinder are constant at the variable depths of water in oar experiment and the above tendency appears more strongly in the case that the bilge-keel is set and the depths of water become

sha-mt he of e1 tal el SS r, er of ¡y ¡n [y 0.10

mass marnent of inertia

o c'J

0.12-s. = 0.7854 H0 = 1.25 Cal.

a 30

.

A e withak. without s.. h/I- L 125 X tI/I - ¡CO o h/ -3.325

I

h/7 -6.225 o o X 0.20 L o X

-

X

o.

o

0.10 mass moment ot inertia

HYDRODYNAMIC FORCES mr SHALLOW WATER 39

0.4O cf = 0.7854 H 1.0

h/I 1406

o

hIT-2.00 - -

--X

hIT-4.Ise

--o.':

t./r7.78:

-X o o O

-Fig. 4.18 j+Ñ: Virtual mass moment of inertia coeff. for

semi-circular cylinder with and without bilge-keel

o

0.5

¡.0 1.5

Fig. 4.19 J+ZIR: Virtual mass moment of inertia coeff. for

elliptical cylinder without bilge-keel

o 0.5 I . O i.

Ex p.

X

T

o 0.5 1.0

- KT

Fig. 4.20 jÖ+j1R: Virtual mass moijient of inertia coeff. for

rectangular cylinder with bilgekeei

how much more. it however seems to be problem which we ought to study hereafter.

The tendency that the added mass moment of inertia M increases in the range of low frequency as the depths of water become shallow appears in

the ecperirnent of the elliptical cylinder and the rectangular cylinder in

shown Fig. 4. 19 and 4. 20. But the theoretical calculations are in agreement

with the experimental results in the range of high frequency. It is seen that MR increases as the depths of water become shallow and the cross sectional form becomes full according to these figures. The added mass

moment of inertia of the rectangular cylinder especially increases in overall range of oscillating frequency as the depths of water become shallow. _The

above phenomena coincide with those of twai's experiments _{in which the}

natural period of rolling becomes large as the depths of water become

shal-low and the block coefficient of ship body becomes large.

(vii) Damping coefficient for roll: ÑR

As the example of the condition without a bilge-keel the wave amplitude

ratios of the rolling elliptical cylinder are shown in Fig. 4. 21. _{The tendency}

as for the effect of depths of water is the same as that of the swaying: the

t

*

cri _a

i.o

Ho ¡.25 _h/T=I.406 Col. Exp._X h/r-2.00 0.22 A _o 4-e- _h/T4.I5G o

20 -o

y.

h/T7.7eI

O 0.18 o X

40X

Ö \ 0.16

-'-r----

o o o - - 4 0.14

moss moment of inertia

0.12

0.10

i

t

f

I

¡

î

HYDRODYNAMIC FORCES IN SaALLÛW WATER 41

'

025L

15 r 0.7854 I -

HonI.25

<q t

020L

4

0.I .5 OJO

005L

/ /

0.5

¡.0

Fig. 4.21 AR: Wave amplitude ratio of roll for elliptical cylinder

without bilge-keel 0.00 5 0.004 0.003 0.002 0.001 O, I / X A o X / T

05

ç= 0.754

Ho

¡.25

1.0 CcL hIT- ¡ .4O6(!G0)

h/T2.OoC 5)

h/r4.l6(Pt5)

-h/T7.CI(PA. 5)

I

Cal. Exp. h/T.I.406«PA-10) X hIT-2.00 (L-I5) G hIT4 .ç5G(A,5r) A h/T-7.78s(?4-l5) 0

'.5

Fig. 4.22 ÑR: Damping coeff. of roil for elliptical cylinder

without bilge-keel Exp. G A o ., . . w,

I-wave amplitude ratio AR becomes large in the range of low frequency and becomes small conversely in the range of high frequency as the depths of

water become shallow. The theoretically calculated values are in good agreement with the experimental values as shown in Fig. 4. 22. Since

the damping coefficient of the rolling elliptical cylinder without bilge-keel NR is almost composed of the component of wave damping force and does not involve almost the component of viscous effect of fluid in the range of

low frequency, it can be estimated by the potentiai theory. The viscous damping component, however, becomes much as the csci!lating frequency

becomes to the range of high frequency, and the rolling damping components

become to he composed of the components of the wave damping iorce and the viscous damping force in the range of high frequency. And the rolling

damping values obtained by the experiments are much more than the theo-retically calculated values obtained by the potential theory and the effect of water depth does not appear clearly in the experimental values at the same

t i me.

In the next piace, in order to study tli effect of the hilgekeel the forced rolling tests have been carried out by using the semi-circular cylinder with

the bilge-keel (l'ß.J:.=lO mm). At the first time the wave amplitude ratios of the rolling semi-circular cylinder without a bilge-keel are shown in Fig.

4. 23. According to the potential theory, the wave amplitude ratio of the rolling semi-circular cylinder is equal to zero, but there are outgoing waves actually with aid of the viscous fluid. Krap and Kotick j _{have obtained} the following experimental formula which expresses the amplitude of the outgoing wave, when the semi-circular cylinder is rolling about the center of the cylinder O in the water of infinite depth written by

where r: radius of semi-circular cylinder. : fluid kinematic viscosity.

The values of the above formula (4. 26) are represented in Fig. 4. 23.

According to this figure it is seen that the wave amplitude ratios AR become

larger in the range of low frequency and conversely smaller in the range of

high frequency than that of the above formula as the depths of water become

shallow in wzter of finite depth.

As shown in Fig. 4. 24, the wave amplitude ratios of the rolling

semi-circular cylinder with the keei become larger than that without a

bilge-keel and the order of the wave arnp!itude ratios take a figure up one place. And so it is seen that we can't neglect the outgoing wave amplitude by the effect of the bilge-keel and the effects of water depths as for AR become generally large as the depths of water become deep.

The rdling damping coefficients Y, of these two conditions with and

without the bilge-keel are shown in Fig. 4. 25. It is seen from this figure

C=4(Kr)2. V/cù d (4. 26)

n

0.010

O. 005

HYDRODYNAMIC FORCES IN SHALLOW WATER 43

<' 0.C6I- 0.05-C.04 0. 03-o Sy Krcp Kotic 0 0.5 1.0 ¡.5

- Kl

Fig. 4.23 Â: Wav3 amplitude ratio o roil for semi-circui&r Cylinder without bilge-keel

= 0.7854 Ho 1.0 r -020m -0.205m baK. ¡omm o A A o x X 00?- G

LXt

0 0.5 ¡.0 ¡.5 KT Fig. 4.24 Â1: Wave amplitude ratio of roil for semi-circular

cylinder with bilge-keel

o hIT-1.125 X o X X hiT-160 C. = ¡5e hIT-3.325 A - h/T-622f o X o h/T.II25 )ç C.

(jo hIT-160

o 2 hIT-3.325 A 0.7854 o= 1.0 0.015 o X X X s cl'

Q' r ₀₇₈₅₄ Ho° 1.0

Ix h/T;J25

0.01CL-

_h/r-i.so

T4 _{= ¡5°}

H hIT-3.325

o Q.008L _hIT-6.225 o WITh 8K. 4 (h8.K.-tOmm) X 0.O06 C

0004-x

0.002

x

___4--

4

e without &K. 0 _{0.5 ¡.0 ¡.5} -KT

Fig. 4.25 _{ÑE: Damping coeff. of roll for semi-circular cliader}

with and without bilge-keel

that the ÑE without the bilge-keel are not affected very much the effect of water bottom. We suppose that we negiect the _{wave damping force withcut}

the bilge-keel above menticned as the small component and the rolling dam-ping forces are completely composed of the frictional damdam-ping component.

Namely we consider the oscillation of the semi-circular cylinder about the center of the cylinder O and the flat plate in the uniform flow as the same

condition and obtain the frictional resistance _{acting on the flat plate using} Hugh's frictional resistance coefficient 32 given as

G1 = 1. 328R-°.5 + O. 014R;°."4. _{(4. 27)}

where R,=-1: Reynolds number. L: Length of plate,

v=l.Ol 106m2/sec (2OC), V: Speed of fluid.

We transform the term of the _{square of the velocity to the equivalent linear}

equation as follows:

R1=

1 ₈

'2

HYDRODYNAMIC FORCES IN SHALLOW WATER 43

where S=rL: Surface area of cylinder.

The frictional resistance nioment N2. obtained by the above equation are

exprcssed as

N2.c,=LJr.dRí=

pC.r Cú2ÇA2T4 L cos wt,

p1wçr4L,

(4.29)

Ñ-v'B/2g =

(1.328 R°.5±0. 014 R°.'14)/KT, (4. 30) where T=r: Radius of semi-circular,

V=wr,

p4=15: Roll amplitude.

It has become obvious that the values obtained by the formula (4. 30) are in

good agreement with the experimental values. The rolling damping

mo-ments of the semi-circular cylinder with the bilge-keel are not significantly affected the effect of water depth shown in Fig. 4. 25. IL is necessary to

take into account the direct pressqres acting on the bilge-keel because the N2 with the bilge-keel are larger by taking a figure up one place than that

without the bilge-keel.

Fig. 4. 26 shows the wave amplitude ratios A2 of the rectangular cylinder with the bilge-keel (bE.K.=lO mm). The wave amplitude ratios A2 of the rectangular cylinder without the bilge-keel, which are shown with a mark

(0) in Fig. 4.26,

are sufficiently estimated by the potential theory, but

the wave amplitude ratios with the bilge-keel become larger than the theore-tically calculated values because of the outgoing wave occured by the

bilge-keel. The tendencies of A2 as for the various depths of water become

small constantly as the depths of water become shallow. The rciling

damp-ing moment coefficients N2 of the rectangular cylinder are shown in Fig. 4. 27. The order of the rolling damping moment without bilge-keel obtained by the experiment is almost equal to that obtained by the potential theory. On the other hand the order of the rolling damping moment with the bilge-keel takes a figure up one place. Furthermore it is an interesting problem that the rolling damping moments are not affected very much the effect of

water bottom until the ratio of_water depth and draft hIT becomes to be

equal to about twice, but the N2 increase rapidly and the values with the roiling amplitude Ç.i='0° are equal to those with the rolling amplitude &= 15° as the depth of water becomes very shallow h/T=1. 406.

The effects of water depth as to the rolling damping coefficients obtained

from these experimental results may be summarized as follows.

(a) The component of the wave damping force is affected significantly the effects of water depths, and it is not able to neglect the wave damping

46

o. i

0.08 0.07 0.06 0.05 0.04 0.03 0.02-0.01 M. TAKAKI Cal. Exp. 1/TI.4O6 X (with BK.)

hIT-2.00 - - -

C )

hIT-4.156 ---

C ) hIT-7.781 o ( IA (without 8K.) - 1.0 1.25 Exp. o A o p.

0.3

A 0.5

¡.0

¡.5

- KT

Fig. 4.26 Â: Wave amplitude ratio of roll for rectangular

cylinder with bilge-keel

h/T.I.406 X (.l01 hIT-2.00 hIT-4.1G A hIT-7.781 C X. A O O O

Xo8

A A h hIT. 4J55, T 2.00 hIT-1406 s ?o=I0°(withou?8.K.)

Fig. 4.27 ÑR: Damping coef f. of roll for rectangular cylinder

with bilge-keel 'ti x A

Li

(with SK.) A

oI=10°

(beK.-tomm) A

Ai

* A A s s

-.

0 0.5 ¡.0 ¡.5 KT

r

HYDRODNAMIC FORCES IN SHALLOW WATER 47 force occured by the bilge-keel.

The firictional damping component and the direct pressures acting on the bilge-keel within the viscous components are not affect the effect of water depth at all.

The pressure distributions on the surface of the cylinder occured by

the bilge-keel do not change until the ratio of the water depth and the draft is equal to -te (h/T=2. O) and then change very much and produce

a large pressure damping force as the depth of water becomes very shallow (h/T=1. 4).

(viii)

Cross coupling terms: M=.44, Ñ=Ñ.

Firstly the symmetric relations of cross coupling term 1f=M and N=

of the elliptical cylinder without a hilgekeel are satisfied and the nume-rically calculated results almost agree with the experimental results shown

in Fig. 4. 28, 4. 29, 4. 30 and 4. 31. The tendencies of cross coupling terms as to the various depths of water are similar very much to those of the

added mass and the damping coefficients for the swaying eilitical cylinder

respectively.

In the next place, the symmetric relation of cross coupling term vI=M

.5 Ø= 0.7854 1.25 h/i. 1.406 i/T -2.00 h/i 4.156 h/T 7.781 0 0.5 ¡.0 ¡.5 Kr

Fis. 4.28 %t: Coupling force coeff. of roll iOto sway for elliptical cylinder without hilg'-keel

.5

O.l5\

L

¶

0.05 IO .

006-(

0.05-O.03L 0.02 0.01

Fig. 4.29 Ñ: Coupling moment coeff. of sway into roll for

elliptical cylinder without bilge-keel

o C.' o 'A A X.'

0.5

0-

0.7854

Ho

¡.25

Cal. Exp.

h/y!.4o6

X hIT-2.00 o h/T-4.156 A hIT-7.781 -

---

o o

'ç

C =0.7854

Ho = ¡.25

---AA

TT A A _A Cal. Exp.

h/T-1.406(-I0°)

X

h/T-2.00 (9A5)- - -

G

-- -

A h/T-7.781('A-5°)

¡.0

'.5

KT Fig. 4.30 _{Ñ,7: Coupling force coef f. of roll into sway for elliptical}

cylinder without bilge-keel

t

t

i

0.5 1.0

i

t i

ç

0.06

<;

0.041-I/f,

oc

OE 1 I i 0.5

l.a

1.5 ¡<T

Fig. 4.31 Ñ: Coupling moment coeff. of sway into roll for

elliptical cylinder without bilge-keel

0.05

0.10

HYDRODYNAMIC FORCES IN SHALLOW WATER 49

A o X I,,

c /

<) o - 0.7354 Ho = 1.25

X

.., rnc.. h/I.I.40t3 ¡;/T-2.00 h/I .4.1 6

n/r

-7.78! h/T= ¡.406 h! r = 2.00 h/T=4.156 h/r =7.78! C(2!. Exp. X o A o X Cc!. Exp. X o

Fig. 4.32 J1: Coupling force coef f. of roll into sway for rectangular

cylinder with bilge-keel

Cc ¡.0.

H0 = 1.25 0.5 1.0 KY 1.5 X

50 M. TAKAKI

0.10

Fig. 4.33 Ñ: Coupling moment

cylinder with bilge-keel

-0.05--0.10

-0.15

-0.20

Fig. 4.34 Ñ,,: Coupling force coeff. of roll into sway for rectangular

cylinder with bilge-keel

HoI.25

o

X

L

o

coeff. of sway into roli for rectangular

1.0 1.5 = (.0 Ho= 1.25 Q 0.5 X o X hiT- 1.406 hiT - 2.00 hi T - 4.' 50 h/T-7.781 -s - -- ----*____ ._

_4

X X o X o O o o A A X o (without 8K)

t

_o 0.5 0 1.5

XX

_XX

i I x Col. Exp. hIT.! 406 (with 6K) o ) hIT-2.00 A ( hIT-4.156 hIT-778, o ( )

QQ

o A A 0.05 A o

i

- C. 04

HYDRODYNAMIC FORCES IN SHALLO TV WATER 51

cr= 1.0 Hoe' 125 CcL Exp. h/T!4oS X (wit)B.K.) h/r-2.00 Q C ' J hIT-4,56 A _{( J}

hIT -7,78 t (without o.K.)

1.0 1.5

KT

Fie. 4.25 Ñ: Coupling noment coeff. of sway into roll for rectangular

cylinder with bilge-kect

of the rectangular cylinder with the bilge-keel (bB.K.r=lO mm) is satisfied in comparatively shallow water. But the theoretically calculated values do not coincide with the experimental values as the depths of water become

very shallow shown in Fig. 4. 32 and 4. 33.

Tite experimental results without the bilge-keel satisfy the symmetric

relation N=N and are in good agreement with the theoretically calcul-ated results. But the absolute values of the experimental results with the

bilge-keel are larger than the numerically calculated absolute values.

Especially the coupling terms N of the rolling into the swaying become

large. The bilge-keel affects very much not only the rolling damping

mo-ment but also the coupling terms which are proportional to the velocity shown in Fig. 4. 34 and 4. 35.

4. 5 Examples of numerical calculations

( i ) Comparison with Kim's results

C. H. Kim has calculated the hydrodynamic forces and moments acting on the rectangular cylinder which has a Lewis form cross section with the sectional area coefficient =1. O and the bearndraft ratio I-I=1. O, by using the method of multipole expansions. The results comparing our values with

Q

\\_

A Q l:- ___' --0.06 o QQ A Q _Q Q -0.06 A A A A A A -0.I O

r