Effect of restricted water on wave making resistance

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i

Effect of Restricted Water on Wave-Making Resistance

by Keizo UENO* member

Tetsuro NAGAMATSU** member

Abstract

There hase been many investigations into the effect of restricted water on wave-making resistance. In numerical calculation of the wave-making resistance, however, ship forms were represented by source distributions obtained for the unbounded water instead of those for the restricted water.

In the present study the authors consider the effect of restricted water to obtain source distributions and calculate the wave-making resistance to compare with the results derived from the source distributions in unbounded water.

Numerical calculations are carried out on a ship in a canal with rectangular cross section. The ship is represented by stepped distribution of sources and sinks on the centre

plane. The side walls aud the bottom of the canal are replaced by infinite series of image sources and sinks. Calculations are made on the following ship form and canal sizes.

Ship form Parabolic water line and frame line L/B=L/T= 10

Canal size WL=O.25, 0.5, 0.75, 1.0, 1.5, 2.0 H/L=O.25, 0.5, 0.75, 1.0

where L=ship length. B=ship breadth, T=draft, W=canal width, H=canal depth From the results of the calculations, the following conclusions are derived.

(1) The source density representing the same ship form increases with the presence of canal walls and decreases with trie presence of canal bottom.

2) For practical purposes, the wave-making resistance in restricted water may be calculated by using the source distribution obtained in unbounded water.

(3) When the canal is wider than 3 2 L and is deeper than 3/4 L,

the efect of restricted water may be neglected in practical application.

* Prof. at the Department of Naval Architecture, Faculty of Engineering, Kyushu University **Experimental Tank, Nagasaki Technical Institute, Mitsubishi Heavy Industries, Ltd.

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1- Introduction

There have been many investigations into the effect of restricted water on wave-making resistance, anti the formulae calculating the wave-making resistance in restricted water for a source distribution have been derived. For numerical calculation of wave-making resistance, however, use was made of the source distribution representing a ship form in the unbounded water. ) (2) ) _{This was mainly} due to the difficulty in carrying out a more rigorous calculation, though it might be expected that the source distribution in unbounded water should he modified when placed itt restricted water to

represent the same ship form for both cases.

Attempts are made in the present work, therefore, to compute the wave-making resistance with an electronic computer, using the source distribution which satisfies the boundary conditions of restricted

water.

The bottom and the side walls of the canal are replaced by infinite series of image sources and sinks. The wave-making resistance is then calculated by the use of the source distribution obtained

for the restricted water as well as the distribution obtained for the unbounded water.

Comparison of the numerical results shows that the wave-making resistance is little affected by the conditions for obtaining the source distribution. For practical purposes, therefore, the source distribution may he replaced by the one obtained for unbounded water.

T

y

//I//I////iI/IlJIIJJ//I/Jí /jJ//////I/III//I//I/I// X

i

il/h/i/I//I//I//I' 1//f//I fl//I/I//I f//I//f//f//Ill

Fig. 1. COORDINATE SYSTEM

-x

2. Formulation of the problem

The problem to be treated here is to calculate the wave-making resistance of a ship set along the centre line of a canal with uniform stream velocity V, width W and depth H.

In formulating the problem Cartesian coor-dinate is adopted as shown in Fig. 1.

I he origin is taken at the midship in the undisturbed water surface with O, in the direction of stream, O normal to O. in the horizontal plane and O vertically upwards.

The velocity potential CI in the canal should satisfy the following conditions

1) The condition of continuity V2 = O

The condittion at the bottom of the canal

at z=H

The condition at the walls of the canal

at y=+

ày

2k t ne velocity potentiat nue to a snip in tfle

O

o

o

o

_o

-canal is obtained in terms of the image system

_O

O

O

_O

_Q

n.i n

by taking the free surface condition into account

and by summing up the contribution of each

O

O

O

source element representing the ship form. Non-

O

O

O

dimensional expressions are introduced with reg- 2W

ard to the half ship length L-2 and the undisturbed _z

-velocity V, _{Fig. 2.} E X y z H L72 ' L 2 ' - L 2

Q

Q

n-2 I.

O

O

2W 5.0 1.F IMAGE SYSTEM (4) (b) VL72'

y

cI'

_{m= -,}

M

k=K0L 2

(1)

- where K0" is related to the Froude number of ship F,, and is considered to be a non-dimensional

parameter which expresses a ship speed.

-K0*=

2,,2 (2

A non-dimentional velocity potential q _{at a point (E. -, ') is given by}

4,r J,m(E'. -'i'. ')I'(E,i7,

; E','ì'.'; m,n)ds

. (3)

where the surface intergral is taken over the plane S on which sources are distributed.

-

i

i

I, (. m

E" 'i'.

" ; m, n)=

m=-'

EÇ--+

r,1,,,,

±1

f del cosh (+d)J(K+ko*se20+z,.csee8}cos/z[K(d+r)J or on--" J - J,, - - cosh(Kd)

{K-x e{K-xp[K(iü,,, -d)] {1H-2cosh(Kd) _{exp[ (2n 1)Kd]}dK}

721mn =(E

E')2 + ( - ')2 +

' 2nd2 72

_(EE')2+

_{(i_'m)2+}

_(+"+2fld)2

m (E-E')cost9+ (rii'm)sin9. 1ì,,= (_ 1)",'+,nl a2

at

ax2 _az

-

ò =0 at z==0 where Ko=g V2

g is the acceleration clue to gravity and is Rayleigh's hypothetical friction coefficient.

As it may be assumed that a ship form can he represented by suitable distribution of sources and

sinks (or doublets), it is sufficient at first to _z

consider the velocity potential of a source element

M in the canal.

_Q

_Q

_Ø

_Q

Q

.i n

Neglecting for the time being the free surface,

f

Q -

O -

- Q - Q

n--i n

the boundary oonclition at the bottom and the 2k - .

-O

O

O

_O

On.on

walls of the canal can he satisfied by infinite series _y

HO

O

of image sources as shown in Fig. 2.

L

4

(E'. '. ') shows the location of source.

Using this ve1ocity potential, non-dimensional velocity components u, ow of the Cartesian coordinates

E, '7, ' due to the disturbance of the source distribution are given by

u= w=- (6)

As the ship surface is symmetrical with regard to the plane 7=0, it is written by

-

I I .

to

7)

where I is the non-dimensional draft.

In this case, the boundary condition to be satisfied on the ship surface is

- []'

+ [W]f(E,t) - (B)

The ship form is replaced for simplicity by the continuous distribution of sources and sinks over the plane q=O. From (3). (7), (8), we get

-

"

₍₉₎

This is the integral equation of the density distribution of sources and sinks m (E',o, ') in which the effect of restricted water is included. Using m (E',o, ') which is obtained by solving (9), the wave-making resistance coefficient in restricted water") is given by

c=

k0*

[A2 +B, +2

(A,+B,)]

(10)

L2V2 n1

where

j°

r ,n(E', o, 1coth[K (d+ (") ]exp[iEVk0*KtanJ.FC,51dE'd'

A0+ iB,=

[1+(?;1)2jcosh2(K0d) _k0*d}i

and K0 is the positive root of the following equation (12) corresponding to a particular value of n.

k0*tanh(Kod) =IÇ

(2r-)2 n=0. 1,2, (12)

When k0*d<1.0. the first root k0 of (12), corresponding to n=0. has to be put zero. This consequently leads to B0=0 and

1 1

fI _m(E,o,')dEd'

As_v 1k0*dL

The solution of the integral equation (9) should he found by numerical methods.

Here the equation (9) is transformed into the simultaneous equations by means of the stepped distribution5 of sources and sinks at the centre plane of ship, i. e. at plane 7=0.

The region 11 is divided into

p equal parts and the region 1'O is divided into q equal parts» source density mii in rectangular part within,

i±EEi,

(i 1,2, ,p, 3

1,2, q) being assumed to be uniform.

It is noticed that the first term and the second term in (4) represent the images of the source with respect to bottom and of the wall, and the third term in (4) represent the perturbation term

surface is assumed to be a rigid wall for simplicity, thus neglecting the third term in (4). J (

i

+

i

) n=- ,=-' 7Ïm 7fh, where (e')2+ ('ml)2+ ( +2nd)2 (14) /2iim,= (E')2+ ('7ml)2+

(+'+2nd)2

From (3) (6) and (13), the induced perturbation velocity components u, y and w at any point

(,

, ) can be obtained as follows.

p q

E E E E

rnÇ di' + 1 4?r

m- n-

_{z=i j=1 JE1}

j

_a 7JJTILn I

pq

E E E E mjjuijmn(j,ei*i, o,

j, m,n) 4,r ii=-" i=i j= p q

Nj,

_a 1 1

E E E

Em1, dOE\ ( + a 4ir ,,,=- n=- i=I j=t ¿ 7ïm,, 711m where Taking

+V(E_)z+ (mI)2 + (+i+2nd)2}

(ml)2 + (-+

2)2

1

pq

Ç-I

E

m=-' o" =I 7=1

5A. p q

Çi

Ii1

a '

i

i

w(E,,)==-- E

_4m

_n

Çl E E

_{1=1 jI}

m%

d\

+

je.

j

7imn 7!Intn

pq

E E

7T n,=-" fl=-' t=1 )=

uijmn(ei,+i,o,j,ji. m,n)

=[u*(ei,ji) _u*(E+i,

)_U*(,j+i)

+u*(,)]

Vijm,,('i.i+i.o,

j,

+i,m,n) u*(1 _f)=lo

+2nd+V(fl2+ (,qJ)2+ (+2nd5

1+2nd

v*(,

_=tan

V(_)2+(_,)2 + (_1+2nd)2

(i

+1+2nd

tan

7=fjj(:oj, oj).

and substituting (15) in (9). the following equation is obtained.

(13)

(15)

,,,.

n=

I

(oi,roi) _{+ oj)}

afu(,i, oi)}

41

E+

7

2

a=V'ntanjj(Kd)

3. Numerical Calculation and Results

In this paper, a ship form represented by the following formula is considered,

77=f(E.') =0.1(1E2) (1_252) (22)

The centre plane =0 on which sources and

y

sinks are distributed was divided into 20 equal parts in E direction and bisected in direction. i.e. p=20 and q=2, as shown in Fig. 3.

Calculations were made on the wave-making resistance coefficient C, and the distribution of source density mj on various canal sections as

follows.

d=0.5, 1.0, 1.5, 2.0

1 =0.5. 1.0. 2.0, 3.0, 4(1

Numerical calculation was carried out by the FACOM 230 electronic computer. In calculating the source density m.j in restricted water, about 100 images were considered in the canal section of d = 0.5. 1 =0.5 and about 50 images in the other canals. 'f he number of images was determined by checking the accuracy of the numerical ruIts for typical canal sizes.

The source density are shown for some canal sections in Fig. 4-7. In theseFigures, j=1 shows the source distribution on the lower half of the centre plane and j=2 shows the upper half. The full line shows the source density obtained in restricted Water and the broken line shows the one obtained

in unbounded water. It is evident from these Figures that the source density increases with the presence of the canal width and decreases with the presence of the canal depth. This tendency agrees with the

case of shallow water° where the canal width tends to infinity:

Nextly, using the source density which was obtained from (19), the wave-making resistance coefficient in restricted water was calculated. Since the ship form (22) is symmetrical with respect to the midship Section, the source density m1 is asymmetric. Therefore A7* in (21) are zero and C is

-

+_.i+i+-j

₂ 'l'his is a simultaneous equation of (x q) unknowns.

Using m1 as solution of this simultaneous equation, the wave-making resistance coefficient in restricted water can be rewritten as follows.

= 4)_- [A0*2 +B2

+ 2j (A,

+ B52) (20)

where

A*

4 sin.h [KC51 cash [K (d + ç) ]sin(a)

cas(aE)

B* -

K7.41+(

)2Jc0sh2(Ksd)

__

sin(aEj) (21)

} 2Et

IOEq Prt.S

expressed by the sum of the infinite series B,,*. In the numerical calculation of C,,,. the number of terms n of the series B,,* was chosen as follows

The results of the calculation are presented in Fig. 8-12 for typical canal sections. The full line shows C,,, in restricted water which was calculated by using the source density obtained in restricted waler and the broken line shows C,,. _{in restricted water which was calculated by using the source} density obtained in unbounded water. _{lt can be seen that the former is slightly less than the latter at} low Froude's numbers hut the former is larger than the latter at high Froude's numbers. However.

when the canal becomes wider, the difference becomes very small.

The discontinuity) in wave-making resistance _{curve in Fig. 10 is due to the solitary wave in} shallow water which appears at 1. 0. _{Fig. 13 shows the effect of the canal width by plotting}

the ratio C,,, C,,,,,. where C,,, being the wave-making resistance coefficient in unbounded water, to

the base of canal width/ship length ratio.

From Fig. 12 and 13, it may be said that the effect of the canal wall on the wave-making resistance is negligible for canals wider than one and half ship length and deeper than three-fourths shiplength

in practical application. 'I'he ratio of the midship section area to this canal section area is about 0.006.

4. Conclusions

Summarizing the results of the calculation. it may be concluded as follws.

The source density representing a ship form increases with the presence of canal walls and decreases with the presence of canal bottom.

Wave-making resistance in restricted water calculated by using the source diatributions for restricted water is slightly less than when calculated by source distributions for unbounded water at low speeds, and this tendency is reversed at high speeds. The difference between them, however,

is very small except for very narrow canal and very high speed range. For practical purposes,

therefore, the source distribution may be replaced by the one obtained in unbounded water.

3 _{When the canal is wider than one and a half ship length and is deeper than three-fourths ship} length, the effect of restricied water on wave-making resistance may he neglected for the ship

form used in the present study.

References

M. Kirsch "Shallow Water and Channel Effects on Wave Resistance" Journal of Ship Research

Vol. 10, 1966.

J. N. Newman and F. A. P. Poole "The Wave Resistance of a Moving Pressure Distribution in a Canal" Schiffstechnik, Vol. 9, 1962.

F,,0.34

n=20

0. 34<Fn< 0.44 n=15

T. Inui and M. Bessho; "Side-Wall Effects on Ship Wave Resistance" Journal. Soc.

N. A.,

Japan. No.92.

J. K. Lunde ; "On the Linearized Theory of Wave Resistance for Displacement Ships in Steady and Acceleration Motion" T.

S. N. A. M. E.

1951.

K. Nakatake and N. Fukuchi ; "On the Source Distribution wihch Represents Ship Form" journal of the Society of Naval Architects of West Japan. No. 34.

S. Morishita ; "On the Source Distribution which Represents Ship Form" Graduate Thesis. Master Course of Naval Architecture. Kyushu University. 1967.

T. Inuj "Wave-Making Resistance in shallow-Sea and Restricted Water, with Special Reference to its Discontinuities" Journal. Soc. N. A. . Japan, No.76.

04

çfr)f)

O)(4

K

K ,V-Yt

tO)t),

tIZt

L LJz5

Source density m1 o)i-l-V'Z,

W/L='/4, H/L=1% 5 canal section O)3-97fliO) image

image X, check

t:5l7

section 0)

canal l97Q

image

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image

t

O)jtD)lJ 0.1% 0) order

5I),

C,,

)j\

aLtcO,

image

to

()

l7kthì K ¿,

weight

It fL Lt:5 *{Ji'

LtO 'J®

Ei

K

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l'llt),

_f&

_{tifi5 clear}

_',

_LatO

tank practicelcr)-,

blockage correction ®

-lmU5

Kreitner

bodily sinkage JjuUit

¿Ji

(I)

2J(.i K

weight C l2 L-thkiC LtC0 71

0)(

tl.

(51I

Lill

source

l}Ut: thin ship

&CtOJYC,

II)t0)7JKi1ffi

0) effect

lit0

(2) ), bodily sinkage L

)

<ftIj -9.-A

0.3 0,2 0.I -OEl FP. In Ret,ted Water - - - In Unbounded Water

Fig. 4. SOURCE STRENGTH

H=L

W=Z L

d =2

A.?. -0.1 --0.1 --0.2 --0.3 0.4-0.3

0.2-

-0.l--

-02-Restric-ted

J&ter

ir'

LJboir.d

LVc..ter

-- o. i

-EP

Fig. 5. SOURCE STRENGTH

d-Z

A -04

--o.'

-0.3 EF CaitaL

s;ze

0.3-g

W*L

0.2-0.

-ì1i * 41

sz

L_.

lo

03-g

0.2-0.I_

O

0.3-

0.2-0. I

-F.P

Fig. 6. SOURCE STRENGTH

-AP. 0. L

0.4-r

Cc.ra1

sze

H= L

03

-W=L

0.2 -

¿=2

0.

--

_--0.4

--0. I

--0.2

--0.3

0.1

-n

Restrictec

VJ&ter n UboLrìcIe. N&ter FR AP.

12

0.4-

0.3-0.2

O. t

--0j

FR

esrictec

\AJdy i1 Unbou.,cJe.cj

Fig. T. SOURCE STRENGTH

size

W+L

¿ 2

size

HL

W=kL

A. P. -0. -0.4 o. --0.1 -0.2 -0.3 0.2 0. I -F.P. A. P.

-1-Fig. 8.

a6xl2 Q4x I0 03x102 02x102 -2 0.1% IO

PrboLc. skip -or

C4L sze

H=L

W*L

Fig. 9.

fi[ j

k R 41 j-14 0.20 0.30 0.40 0.50 0.60

r

0.6k 02

O.5k1Ö2

O.4xI04

3x104-Effect of Restricted Water on Wave-Making Resistance 15

n U160udd Water

0.20 o o 0.40 0.50 0.60

16 L) 02x104-01x104 0.1* IO_2_ 010

J

ParaoLc. sh1p f-ori w;f Ca.r&L size H = 4L

w _L

41. 0.30 PAr&boLc shp ort Cazi&t size H - +1.

W+L

Fig. 11.

wf Ys5'rlo

0.40

n Uoune

Water 0.s0 n strcte Water 0.60 pi 'estrictecL Water , Lh 'orde Water 0.20 030 0.4.0 0.50 0.60 Fn Fig. 12.

J

3,0-o

0.5 .0

I5

2.0

W/L

Fig. 13.

Paybo Lic ship .Çort

i[0

= 0.20

-

Fr0.3O

=0.4O Fi =0.50

L-

3

Cw

2.0

I-0