Manoeuvrability of ships in narrow waterway

ARCE..F

Lab. y.

SheeP5b0

Technische

Hogeschoot

Deft

3. Manoeuvrability of Ships in Narrow Waterway

Katsuro KUIMA'. Member, Hironori YASUKAWA, Member

(From j S. N. A. Jan. Vol. 156. Dec. 1984)

L Repiiht from Naval Architecture and Ocean Engineering Vol. 2

3. Manoeuvrability of Ships in Narrow Waterway

Katsuro KUIMA, Member, Hironori YASUKAWA*S, Member

(From J. S. N. A. Japan. Vol. 156. Dec. 1984)

Summary

In restricted waters such as harbor, bay or canal, it is necessary to know the precise manoeuvring characteristics of ship including the effects of water depth, channel bank or the another ships from viewpoint of safety of navigation. In narrow waterways., specially, the effects of channel bank and hydrodynamic interactions between ships are fairly significant.

This paper examines hydrodynamic behavior of ships during meeting and passing in narrow water

channel by using slender body theory Furthermore ship motions with rudder control during passing in channel by using these hydrodynamic forces are discussed. This paper concludes as follows.

During passing, the interactión forces such as lateral force and yaw momeñt between two ships are

affected by the distance between ship and channel wall, and water depth:

By simulation study of ship motion, the possibility of the collision with another ships on the channel

wall to be caused by hydrodynamic interaction was indicated. Some guides ön two way traffic in water channel were obtained.

Introduction

In restricted waters such as harbor, bay

or canal, a ship is generally in the proxim-ity of other ships or some fixed structures.

The proximity of these other objects are potentially hazardous, and it is important to keep the course of each ship in these waters. Therefore it is necessary to know the precise manoeuvring characteristics of ships including the effects of water depth, channel bank or the another ships from viewpoint of safety of navigation. In nar-row waterway, specially, the effect of channel bank and hydrodynamic interac-tions between ships are fairy significant.

In the past, the calculation method of the

hydrôdynamic interactions of two ships moving in shallow water were shown by Newman and Yeung2. Furthermore, the unsteady hydrodynamic interactions

be-* Faculty of Engineering, Kyùshu University

Nagasaki Experimental Tank. Mitsubishi Heavy Industries. Ltd.

I

tween ship and arbitrary vertical-sided fixed structure were examined by Yeung and Tan3. However, the unsteady hydrodynamic interac-tion of ships in narrow water channel has not been investigated in spite of the important

problem.

In this paper, by applying Tan's method4, hydrodynamic behavior of two ships during meeting and passing in narrow water channel was examined.

Furthermore, ship motions with rudder con-trol during, passing in water channel including the effect of these hydrodynamic interactions were discussed from viewpoint Qf safety of

navigation.

2.

Basic Formulation

Let o-xyz be a coordinate system fixed in

space and o-x y z, be a coordinate system fixed

at the midship of the i'th ship as shown in

Fig.1. Each ship is assumed to move in a

straight line in the water channel. The ships are assumed to be moving in calm water of uniform depth h and in water channel with vertical-sided wall.

2 Katsuro KIJIMA. Hironori YASUKAWA Y w o Ya t 'U -, _Ship2 XN 5M Ship N Yi t UI Ship1 Fig. i Coordinate systems.

uniform depth h and in water channel with vertical-sided wall.

By the assumption that the free surface is rigid wall, we can treat it as .a "double body" problem. In this case, the velocity potential\#

(x, y, z; t) must be satisfied the following

boundary conditions.

rs6o

(1)'

a

/an118=U1(n)1 i=1,2

N (2) (3)

Ia/anIc=o

(4)

6.O

atVxi2±yl2+z2_00 (5)

where U1 denotes the speed of the i'th ship. B

represents the underwater hull-surface of the i'th ship and C the submerged surface of the water channel. fl and n are the unit normal

vectors in the normal direction to B and C

'respectively. n4 is the' component of the unit normal vector n in the x direction.

It is a difficult

task to

solve the 'full

three-dimensional problem as formulated above

directly. In order to make it more tractable,

some assumptions are made which will permit its decomposition into two subproblems so-called "inner problem" and "outer problem". Let e be the slenderness parameter, that is

L=O (1), B=O (e), d=O (E)

where L, B1 and d denote the length, breath and

draft of the i'th ship respectively.

We make the 'assumption that the water is shallow, and the minimum lateral distance

between the i'th ship and j'th:ship is denoted by

UN

S,, and between the i' th ship and the fixed object by S0, and these parameters are the

fóllowing ordeEs f magnitude,

h0 (e), S,=O (1), S0O (1)

i=1,2...N,

j=1,2...N, i4j

2.1 Inner Problem

The inner region for the i'th ship is defied as the region where have the following ordérs

of magñitude,

xO(l), y,z=O(e), i=1,2,.., N

Using cI to denote the velocity potential in the inner region of the i'th ship, eqs.(1), (2) aiid

(3) can be replaced by the followings,

a2c

82i_

8YI2 aZI2

[a t/aN]

i)=U(flr)1

(ai/aZ1l±O

where N is the two-dimensional unit vector normal to the section contour L(X) of the i'th

ship.

The potential cI

in inner region can be

decomposed into two components and it is

expressed as follows,

+ V"' ( x, t ) (yi, Zi) +f ( xi, t

z; x, t)=U1(t)I'1(y, Z1)

₍₉₎

h1 is the velocity potential due to the longitu-dinal motion at unit speed and 2 is related to a cross-flow of unit magnitude. V1" is the actual

cross-flow velocity at the section L X) of the i'th ship.f(x, t) is an unknown constant inde-pendent of the y and z coordinates.

corresponds to the problem of a two-dimensional dilating body confined between

two parallel walls. corresponds to the

lateral-flow problem of a rigid two-dimensional

body in a uniform stream of unit strength

These limits are given by Newman and Yeung and they take the following forms,

hm

S'(x)

3/i cb)

Manoeuvrability of Ships in Narrow Waterway

With the understanding that we consider only

the leading-order potential o, we shall

hence-forth omit tle subscript O.

The outer solution can be represented by a

distribution of sources G(x, y ;

1/) and

vortices G (x, y E h/) along the body axis.

G'' and G7' are defined as follows.

G

(x,y;E.»7)=1n(x E)2+y1)2Jh12

+H(x,y;E,1)

G(x,y

E,

)=tan

(_;

)

+H1'7'(x,y;E,)

(16)

where H and H'7' are functions harmonic in

the physical domain and so constructed that the

no-flux condition on C is satisfied, nimely

[

aG]_0

.

In the preceding, (E, ?) is the source or vortex

point, and (ì, y) is a field point. The construc-tion of G is discussed in Appendix. Finally,

outer' potential (x, y; t) can be written in the

following.

(x,y;t)

=

-_-[

f C1(S,.t)G,(x,y;E,)dS1

+1

Y;(Sit)G'(xy;E)dSi]

(18)

L,,,»

'where 0, and Y2 are the distribution source and vortex strengths of the j'th ship,

respec-tively. dS1 denotes .n infinitçsirial element qn

'the x-axis while L1 denotes the ship axis and the

wake of the j'th ship. E and 1 are parametric functions of S,.

The inner limit (y-Q) of the outer solution

as expressed by eq.(18) are obtained by means

of making a Taylor expansion of for small

values of y.

um,

2=y±C(x)

(n)

Iyi I>>'

The total flux from the body can be shown to be S'1(x1) where S1 (XI) is the sectional area of the body.. The blockage coefficient C1 (xi) is a

hydrodynamic coefficient associated with the virtual mass characteristics of the section in

cascade flow5'.

In this paper, C (xe) was estimated by using the approximate formulas given by Taylor6.

Finally, the outer limit of the inner solution is obtained the following.

hm

''(y,

z;x,

t) UÇt)S'(x)

I,iI»

+W (xi, t) [y±C (xi) J

+f1 (x, t) (1

2.2 Outer Problem

The region which is far away.,from any of the

ships is called the outer region of the problem.

In terms of orders of magnitude of the

coordin-ates, this region is defined as follows.

XI, ))=O (1), ZI=O _(E)

i=1,2,»', .N

In this region, it is possible to express the

velocity potential as .a Taylor expansion,

(x,y,z;t)=(x,y,O;t)+b(x,y,O;t)z

+__ 9L(x, y, O t)z2+

=

o(x,y;t)+

i(x,yt)z

+z(x,y t)z2+

(13)

where eq.(1) has been used to arrive at the

second quality, and o and are potentials

independent of z. By using eq.(3), i is easy to show that

ax2 ±

ay2 -

(i

Hence, the leading-order outer potential o

(x, y, t) satisfiés Laplace equation in x-y plane.

This problem

is

the same äs

the

liñi

1(x,jl;t)

1,1)>' f1(Si t) GJ'' (xo,yo C J#1

+1

L,,,,, +fci(Si,t)

(xo,yo;C,)dSi

+fY(SJt)

8(xo,yo;E ,)dS1

}y]

+-4-f

(S, t)

in

.Ix

Ci

+!Ii

xo,yo ;E,17)1dS.

±

_f

v (S1, t I

+H'7(xo,yo ; E )1dS1

7 ( E, t) d C1

+ [----f

(Si, t) (Xo, yo; C, 17) ds1.

+---f71(Sit)' Xi_f

+

a'

(xo,yo ; E, 17) }dS]Y

The point denoted by (xo,yo) in the

space-fixed oxy coordinate system corresponds to the

point (x,y=O) in the movingOxYi coordinate

System.

2.3 Integral Eqtiatioñ and Hydrodynamic

Force

In the next stage, compatibility of the inner and outer solùtions is required that they match in the intermediate region E «y(< i, i.e. they must satisfy the following condition.

Katsuró KtJIMA. Hironori YASUKAW

um LÌ'. ; X,

t)lim

1'. (x,y; t) Ó)

i, i))' I,iI «1

Using the expréssions we havé öbtàinedfor

the outer limit of the inñ&r sòlùtion eq.(1 2) and

fòr the inner lirnitof the outer solution eq.(19),

the matching condition is shown by the

follow-ing équátions obtained by the term of similar nature. O (xi,t) U (t)S' (x) - 1) LI2

Vi'xi,t)i)7f

71(E1,t)dE W t) IL (S,,t) (xo,yo;E,11)dS1

+ f 7 (S

t) (Xx, yo; E 17)dS1] a (S, t) (Xx, yo; C , 1)) ds

+_--f

(S1,

t)[xi

+

'(xoyo;E17)]dS 3)

The integral equation for 7 can bé obtained

+---C(x) I1

(19) easily using eqs.(22) and (23).

2C1 (x) ix1 1 fiii 2 7 (E1, t)dE1

2it

f Y (Si,

t){

X-+ a1H(xoyo;ev)}dSI

j=1

f7,(S, t)

(xo,yo;C,17)dSi

N

i

C

aH'

2irJL (S t), (xo,yo E, 7 ) dS1

i

_r 2 _iLl ., (S,, t) (xo,yo;E»7)dSj

i=1,2

N

Notice that for N ships there will be N coupled

integral equations.

The additional conditions which are imposed

on 7 are the wake condition, Kelvin's theorem and Kutta condition.

7 xç t) 7, (xi) for

xl< L1/2

fL/2_00 71(E1,t)dE1=O

I'. L _'\

i

dJ

7 \X =

)

_{- 7T}

_dt

where I'

is the circulation of the i'th ship.

Using Bernoulli's theorem for unsteady flow,

the following expression for the linearized pressure jump across the xaxis is obtained.

4p(x,t)=P (

)4(x1t)

(28)

The difference in potential across the xi-axis

S6 can be obtained easily from eq6(19).

S6 (xi,

t)_-f2

71 (E1,t) dE1

Substituting eq.(29) into eq.(28), we have the 'following expression.

p t)

- P [-rf

71 ( E , t) d E1

+U7i(xit)]

(30)

The lateral force acting on the i'th ship F1 and

the yaw moment M are obtained by integrating

the pressure difference over the length of the

Manoeuvrability of Ships in Narrow Waterway 5

ship as follows, L/2

F1 (t)= -f_ p(íd, t)dx

L/2

M (t)= _{f_LI!2 X} (xe, t)dx

The above expressions for the lateral force and

moment corresponds to the "equivalent" two -dimensional bodies which means they are the values per unit depth. In order to get the total lateral force and yaw moment acting on the ships, we simply multiply eq.(31) by the water depth h.

2.4 Method of Numerical Sóhition

It has been found that the time dependent solution of the integral equation (24) can be obtained by a rather simple numerical proce-dure, a discrete vortex approximation Let the each ship be divided into M segments of equal

length x. The total vorticity within the n'th

element of the i'tli ship at the k'th time step (t

kit) is assumed to be 70, n= 1,2, : M. Let

7(n= 1, 2,

k) be the vorticity in the wake

element of the i'th ship where nk

corres-ponds to the wake element nearest the trailing edge of the ship. We can now write the system

of simultaneous equations derived from eq.(24)

in a concise form. For the i'th ship, we have

{ , A ,,7 7,(k)

+

B.,.71 _{} =g,,(k}

m1,2 M

(3

i=1,2...N

where i

i

X..,

E.

+

:. for j=i, E1,<x,,

i

i

8H17

2*

_{Xi,._E1.+ ayl}

i

+

_{2 C1'(x1)} ,, } (23) (6) (27)

B.=

for j=i,E>x

i dG11>

.-.for j4v

i

I

i

aH7

2ir ( dyl

i

dG7 for

---

y. . for ..

i

U S1' ( E)

T 2

2h

i .,

J..LS1'() aG«

+2rL...iL.1l

_jn-1

2h

_ay

i kz

Here, x, is thè control poiñt located. át n'th

segment of the :j'th ship while E

and E, are

the vortex points on the n'th segment of the j'th

ship and on the n'th segment of its wake

respectively.. The soUrce points are denoted by

The additional equations can be obtaiñed

from eqs.(25) and (26), and we have the

following equations. 5',,+io== Y.c1 for

n=1,2,...-,k-1

(33) k= 2,3,

i=i,2...N

Mi k.

7i>+

7(k)=o _{: i=1,2...N} (3

n1 n=1

By eq.(33), we obtain the following equivalent

set of equations,

Mi

j=1 ,i=1

i=i

i#i

chosen to be 1/4 and 3/4 of the segment length

respectively, the Kùttä condition is satisfied. In

this paper, the segment number of each shi M1

was chosen to be 40.

3.

Results of Numerical Calculation

3.1 Hydrodynamic Interaction of Two

Ships iii Shallow water

The usefulness of the present method were examined by comparing with the experimantal results carried out by the one of the authOrs7 lof hydròdynamic interactions of two ships in shallow water. The model test carried out by Using the 2.5m model ship shown in Table 1, and lateral force and yaw moment dUring the transit in the meeting of two identical ships

were recorded.Then, speed of two ships U1 and

U2 were chosen to be 0.3279m/s(F.=0.0625) and water depth h/d1.3 or 1.5. Fig.2 shóws

the coordinate system in meeting condition and the sign convention for the force and moment of ships.

Îable i Main particulars of ship fòr numerical

calcúlatiôn and model experiment.

ShipZ i

- U2

- F3L SCA SOI? I - - SHIP

11GTH X.pp (ml 153.0 I .2.3 0RZATO 0 (ml ¡- 26.0 - 0.419_ DRAFr 4, (ml 0.7 - .1 0.140 5.967 rid 17.057 -

o/a---

-

-Block ComfO. Co..I - 0.690

-Acp.00 Mojo IC I 0.112

Trim ._r,a -. 0.0-

--6 Katsurd KUIMA. Hironôri YASUKAWA

A BfrM5

j1

,i=1 Fi I Shipi U, B°15' (35)

m1,2...M,

i1,2...N

Fig. 2 Coordinate systems in meeting.

0.3-CF

I -0.31 0.2

Manoeuvrability of Ships in Narrow Waterway 7

l'-0 I.3

CF

Fig.3 shows the non-dimensionalized lateral force and yaw moment acting on ship 1. The stagger, non-dimensionalized by the average length of the ships S'T(=S/-- (L1+L2)), is used as the abscissa for the plots which follows. S'r values-1, O and i correspond to the bow-bow,

midship-midship and stern-stern situations,

respectively. The force and moment coeffi-cients on the i'th ship are defined as follows,

CF-- F1 --PUi U2L1d1 CM1 M --PU1 U2 L12 d1

-

C.I. 8tlIl Ita-lo S»

"kJ

/,,-.c\

lO

'r

'

-0.03 -0.03 -004 (d) .1.3

Qualitatively, the calculation results agree well with the experimental results in various separation distance of two ships S. However, the phase lag of the peak value of the lateral force and yaw moment are seen in the figures.

3.2 Hydrodynamic Interaction between

Two Ships in Narrow Water Channel

The interaction force and moment acting on

two identical ships during meeting and passing in narrow water channel were examined by the

numerical calculations. The calculations were carried out on the ships shown in Table 1. Fig.4 shows the coordinate system in meeting and passing in water channel and sign conven-tion for the force and moment of ships. The

(a) (b)

Fig. 3 Lateral force and yaw moment acting on ship i

lateral distance between the i'th ship and

channel wall is denoted by S, and the force and moment coefficients on the i'th. ship are, defined as follows.

CF=1

F

-U2Bd1

C M

--PUi U2Bd.L

This expression of the force and moment

coefficients are different from the definition in section 3.1.

Fig.5 shows the effect of S, on lateral force

!111ICc:l::rtc,I

CF° 1::

a. a,:

a. a: CF,

Katsuro KIJIMA, Hironori YASUKAWA

a. Sr II A2 I. OP SIIP Ç

J

''''

-aa 0.1

Fig. 5 The effect ofSPIon lateral förce and yaw

moment acting on ship i in meeting.

i;! z O

Fig. 4 Co&dinate systems in

meet-ing and passmeet-ing in narrow water channel. ",o:C,atT:1 (b) P.::, -.

':'";2\

: aTTRACTCI .a_Ia,i.. -o.La a

r"

a -l.a -'.o .5

caco t r ta. PASSIOC COSICOTIPN

-o. S a.

yaw

mo-(a)

Fig. 6 The efféct of Sr on lateral force and

ment acting on ship i in passing.

c.cs:-I. o.CtS - .5.103 0.500 C Fz o. no US. S ATTOOCT SOS. .1.5 .3:5 (a)

Fig. 7 The effect of Spi on lateral forcè ànd 'aw moment acting on ship 2 in passing.

SIS

Manoeuvrability- of Ships in Narrow Waterway

Ll/L2 CS p/LI 0.T 3/1.1 5 3.5 .3 .5 25 s (a) (b)

Fig. 9 The effect of water depth on lateral force and

yaw moment acting on ship 2 in passing.

I55ISL cICIT!I e-Io .0.750... (b-) Pitt 1.0 5.5 .2 1.5 . ¡.3 Sr (a) (b)

Fig. 8 The effect of water depth on lateral force and yaw moment acting on ship i in passing.

IO Katsuro KtjIMA. Hironori YASUKAWA

and yaw moment acting on ship 1 in meeting condition. Then the separation distance be-tween ships S, was chosen to be kept constant

0.5L1.

The channel width W was chosen to be 2.0L1,

the relative speed Ul/U2=l.0 and the water

depth hid=1.3. As S,1

is increásed from

O.25L1.the force and moment peaks become

larger.

Fig.6 and Fig.7 show the effect of S,1 on lateral forcé and yaw moment acting on ship i

and ship 2

in passing condition. Here, we

assume that ship i is the faster ship and ship2

the lower ship. The stagger is again

non-dimensionalized by the average ship length so

that-i, O and + i correspond to the

bow-stern, midship-midship and, stern-bow situa-tions réspectively. The relative speed Ui/U2

was chosen to be 2.0. Fig.6 shows that as S,i is

decreased from 1.25L1, the forces acting on ship i are shifted by an amount approximately equal to the steady wall-induced suction force and the moment peaks become smaller. Fig.7 shows that as S, is decrease from i.25L1, the

peaks of the force and moment acting on ship 2

become larger. In other words, when ship i passes at the smaller distance from ship 2 to the nearest wall, the interaction forces acting

on ship i become larger. Conversely, when ship

i passes at the larger distance, the interaction forces acting on ship 2 become larger.

Fig.8 and 9 show the effect of water depth on

lateral force and yaw moment acting on ship i

and ship 2 in passing condition. S,1 was chosen to be kept constant 0.75L1. As the water depth

h/d is decreased from 2.0, the force and

moment on both ships become larger.

4. Discussions

In this section, the effect of these hydrodyna-mic forces acting on ships in water channel are

discussed from viewpoint of safety of

naviga-tion.

It should be noted that the duration of the passing conditins generally longer than it in meeting condition. The possibility of collision or ramming of ships are greater in the passing transit. Therefore, we consider mainly the

passing condition. in water channel.

Manoeuvring characteristics of ships

includ-ing the effécts of the hydrodynamic interaction

are also examined by simulation study. It is assumed that the slower ship, ship 2, is

the constant size (L2= 155.Om) that is shown in

Table i and the faster ship, ship 1, is similarto ship 2. The channel width was chosen to be

31 0.Om(2L2), and water depth i3.92m(h/d 2

i.6) or 17.40m (h/d2=2.0). Hydrodynamic

forces acting on the ships in water channel were used the values that calculated by the present. method in simulation study. The rUd-der of the ships was assumed to be controlled the parameters of the heading angle S and the angular velocity r' in the range of +35deg. as

follow,

= 80+k (çb - çbo).+ kz r'

where &o and çbo are the initial rudder angle to

keep the straight course and heading angle respectively, k1 and k2 are gain constants.

FigJO shows the ship trajectories during passing in water channeÏ as a parameter of difference of length between ships. The speeds

Ll,L2.O. S -I O 0.0 rO T/L2 I.0

1l.ro,

7. TI '. 11.5' II 'I 10. 14. 14. i1. I0. 0. O-0 0 0.0 0 7/12

Fig. 10 Ship trajectories as a parameter of difference

of length between ships.

UI .4.5 II. U2 s. o IN. NO Cc.STOOL II .112 0.0 6 6.0 IO .15.6.0 UI.7.5 .12. .0 6 0.0 I 0 7/U

Manoeuvrab'ility of Ships in Narrow Waterway 11

112.5.0

7/1.2

Fig. 11 Ship trajéctories as a parameter of difference

of speed between ships.

6.0I 0

7/1.2

Fig: 12 Ship trajectories as a parameter of rudder

con-trOl.

of ship 1 and ship 2 were chosen to be 4.5kn and 30kn respectivelyr and k and k2 4M. Loci of the ship i that started from x/L2=0.0 and

the ship 2 that started from x/L2=2.0 are

shown in the figures. The initial separation

distance between ships was chosen, to be

77.5m(S/Lz=0.5). This figure shows that as

Li/Lz.is increased from 0.8 to 1.4, the

possibil-ity of collision is larger.

Fig.11 shows the ship trajectories as a

parameter of difference of length between ships, in the case that ship i passes in 6.Okn. This figure shows that as the relative speed is increased from 1.5 to 20, the possibility of collision is smaller in various ship length. The

reason of this results is caused that as the

relative speed becomes smaller, both ships take

the effect of mutual interaction in a long time.

Fig.12 shows the ship trajectories as a

.parameter of rudder control. The relative ship length L1/L2 was chosen to be 1.2 and water depth 17.40m(h/do= 2.0). In the case of '1No Control", both ships collide with the channel

wall of right hand side.

By simulation study in varioUs ship speed,

ship length, äte? depth àîìd sêpâiation

dis-tance of ships, we sought the control limit for

keeping her course for the safety of navigation.

Fig.13 shows the limit line on ship 2. Here, we express the limit line in the minimum separa-tion distance between ships. The figure shows that if ship 2 is passed by ship 1 at smaller separation distance than this limit line, ship 2 falls in unsteerable situation. As the length of

ship 1 becomes larger, the relative speed

smaller and water depth SñIáller. the minimum

separation distance between ships becomes longer and the possibility of falling in uncon-trollable situation becomes higher.

5. Conclusion

Hydrodynamic behaviors .of two' 'ships

dur-ing meetdur-ing and passdur-ing in narrow water

'channel were examined by using slender body theory, and ship motions with rudder control during passing in water channel Were

discus-sed from viewpoint of safety of navigation. The

major conclusion from the present results can be summarized as follows.

(1) The. results obtained by using the present

method agreed well with the experimental

SpIL2 1.0 0.5 o o LO-Sp/L2 0.5 o o Sp/L. LO-0.5 O 1WU2l2 -- -- ö15

- 8-35'

L,/L2,1j. 74

//L/Li0

1.5 _{HId 2.0} L,1LaI.2 _L/L1.4 1.5 L,/Ll.2

//

L,/L2l.0/ U,/U21.5 H/d1 2.0 U,/U22.0 8=15 2.0 - Hid, (C)

Fig. 13 Minimum lateral distance between ships in.

dicating that the rudder force and moment are much smaller than the hydrodynamic

interac-tions during passing.

two ships.

During passing, the interaction forces,

such as lateral force and yaw moment between two ships; are affected by the distance between ship and channel wall, and water depth.

By simulation study of ship motions with

rüdder control during passing in water channel, the possibility of collision or

ramming of ship to be caused by hydrody

namic interaction was cleared.

In passing condition, the ship that is

passed by another ship takes greater

effect of the hydrodynamic interaction between ships in water channel than the ship that passes another ship.

When the size of ship that passes another ship is larger, the relative speed smaller,

the separatiön distance between ships

smaller and water depth smaller, the ship that is passed by another ship must be controlled with cautiom

Some guides on two way traffic in water channel were obtained.

The compUtations were carried out using by

FACOM M-382 of th Computer Center in Kyushu University. This research was sup-ported by the Grant.in-Aid for Scientific Re-search of the Ministly of Education.

References

J. N. Newman: Lateral motion of a slender

body between two parallel walls. Jour. of Fluid Mechanics, vol. 39, 1969.

R. W. Yeung: On the interaction of slender

ships in shallow water. Jour. of Fluid

Mechanics, vol. 85, 1978.

R. W. Yeung, W. T. Tan: Hydrodynamic interactions of ships with fixed obstacles. Jour. of Ship Research, vol. 24, 1979. W. T. Tan: Unsteady hydrodynamic

in-teraction of ships in the proximity of fixed

objects. Master's theses, Department of

Ocean Engineering, M. I. T., 1979.

L. I. Sedov: Two dimensional problems in hydrodynamic and aerodynamic. John

Wil-ly and Sons, N. Y., 1965.

P. J. Taylor: The blockage coefficient for flow about on arbitrary body immersed in

a channel. Jour. of Ship Research, vol. 17,

1973.

K. Kijima: Model experiments on hydrody-namic interaction between two ships.. Ship

Performance Committee of West-Japan

12 Katsuro KIJIMA, Hironori YASUKAWA

LS o

Society of Naval Architects, SP8S-30, 1981. (in Japanese)

Appendix

The Green functions defined by eqs.(15), (1) and (17) can be obtained by conformal transformation using the following mapping

function.

' =e'

(36)

where z and ' are complex variables

repre-senting point on the physical and mapped

plane. z is defined as z=x+iy. The complex potentials for a source and a vortex located in

the e-plane, satisfying the rigid boundary

condition, are the following form,

f1n( o)+1n( c)

₍₃

fY i in

(

- 'o)+i in (

-where o denotes the location of the source or

vortex and o is the complex conjugate of o.

The Green Functions can be obtained by taking the real part of eq.(37). By taking the

Manoeuvrability of Ships in Narrow Waterway 13

complex derivative of eq.(37), the following

expressions for the complex velocities are

obtained.

i

i

+

-dz

o

az

df" r j = I

+

- j az dz L

-where the complex velocities are defined as

iv.

The normal velocities are then given by the

following, aG

=

_{Im Iwe° I}

1 (39)

= Im [w(7ere

I

ay

where Im denotes the imaginary part and Th is

the angular displacement of the oxy coordinate

system relative to the Oxy system. When ship travels in the positive direction of x-axis, O,

equals O and in the negative direction, O