NUMERICAL COMPUTATIONS. ON INTERACTION ...

NUMERICAL COMPUTATIONS. ON INTERACTION' OF WAVES WITH LARGE SUBMERGED STRUCTURES Hen-Cheng Fan**

Robert R. Hwang*

ABSTRACT Finite-difference techniques based on boundary-fitted coardinates are used to study the interaction of waves with 1arge fixed two-dimensiona1 structures submerged in water of finite depth. Thè physica1 f10w fie1d is transformed to the curvi1inear corrdinate system in which the computationa1 region is rectangu1ars with a fixed square grid regard1ess of the movement of the free surÎace. The free surface of the f10w is obtained in such that the transformation i8 computed simu1taneous1y with the f10w fie1d for each time step. Wave forces and pressure are ca1cu1ated from the ve10city potentia1 of the f1ow. Comparisons of the resu1ts for wave forces of two cases for submerged structures in water of finite depth with solutions obtained by other methods indicate that these finite-difference techniquès can yie1d accurate resu1ts. To demonstrate the usefu1ness of the numerica1 approach ，the prob1ems of two subm~rged horizonmounted cy1inders in different spacings which do not have c1assica1 solutions are

a1so

ana1yzed.

1 .INTRODUCTlm~ The great oi1

dri11ing

has

1ed to

waves

increase

，submerged the

shore

10ading

three

tanks

works

and marine of

dynamic

many marine

，become

the

on submerged

As the

objeçt

genera1

prob1em

presence

of

object

(1)

Professor and

acting

，the

of wave

equation

**

coasta1

prob1em

to

，therefore

decades. wave

that

study

*

stoage

of

bui1t

disposa1 forces

and civi1

a predominant

for

the for

purpose

contaminations

exerted

by water

engineers.

ro1e

in

the

of

Environdesign

of

off-

structures.

incident in

use

on the

structures

has

Wave forces than

the

oi1

attention

on submerged

menta1

in

forces

， (2) ， (3). of

Institute

Chung-Shan

the

on such As the

Department of

Physics

Institute

of

objects s 工ze

has

no effect

has of

the

，Academia

investigated

sma11 compared

come to object

Nava1 Arch 工tecture

of. Science

been

of wave/structure

objects size

is

has

Sinica

and Techno1gy.

-82 一

to

for

the

1ength

interaction on the

incident the

in

to

，Nationa1

of

the

was simp1ified

be known as relation

more

wave.The Morison the

Taiwan

type

wave 1ength

University

，the ，the

increases therefore violated.

inciderit

In

such

account

for

For

parameter

the

the

土ty

of

H/2a

-lem

工n

sea

problem

object

tent

工al

have

the

，et

Green's

andJ

wave is

free

must

surface.

and a indicates

is

to

set

is

and Hirst

complex

the

diffraction

up in

terms

of

a

difference

formula

，which

is

geometries

(15)

have

used

flow

generated

of

on ~he boundary

geometries

conditions

fluid

for

unequal the

into

rectangular approach

at

is

the

on of

the

po-

Bai

(7)

determine

，Zienki-

boundary

work

free

at

the

boundaries those

This

element

for

flow

the

boundary

makes

sur1ace

method

below

systems

Haussling

ar~

particularly

with

unsteady

ap-

and Coleman

water

the

，and

finite-

map arbi-

unsteady

found

To

In another

a free

ac-

，Ni-

grid using

transformations

simplify

the

difficult.

w工th

boundaries.

two-dimensional

in motion

coordinate

that

finite-difference

，numerical (13) ，(14).

the

problem fact

deforming.

marker-and-cell

compute

as

applied

on the

a rectangular the

regions

such

to

，

tech-

equation.

method

by the

surface

cylinder

curved

geometries

for

integration

interaction

conditions

present

to

based surface

Chen and Mei (9)

have

particular

mesh spacing

in

is

the

is

bound-

problems.

free

w工th

solution

used

integral

element

conof

the

equation

boundary

domain.

(11)

the

over

an integral

finite

surface

solving

sources

boundary

the

in

applied

boundary-fitted

time-dependent

the the

for

prob

boundary

，the

approaches

Various

wave/structure

of

have

obtain

and Shepherd

used

this

to

use

complicated

the

，

dynamic

obtainihg

methods

One of

solve

(8)

by a circular

The resulting boundary

to

，

the

a boundary-value

surface

distributing

object.

region

(12)

to

，but

numerical

wave diffraction

representation such

Bird

solution

calculation

chols

the

throughout

the

surface

of

as

free

straightforward

theorem

of

applied

，and

on the

involves

and Hanif

(10)

treat

trary

the

referred

problem

and subjected

developed. It

，

been

solving

a free

proach

as

object

incident interaction

as

interaction

condition

very

been

potential al

in

the

is

boundary

also

velocity

curate

the

wave height

and the

equation

(4 ， 5， 6).

The numerical

of

the

be small

A variety

and using

over have

with

as well

object

to

a large

affect

of wave/structure

H denotes

wave/structure

boundary

and Vongvissesomjai

method

not

size

the

bottom

function

the

th~

problem

neglected

，difficult.

ary-valu~

ewicz

the

by Laplace's

and the

general

niques

does

relative

are

of

and kinematic

Green's

of

upon encountering

object

，the

finite

in which

effects

governed

body

the

potential.

The formulation

~ition

scattered

that

an analysis

dimension

，viscous

theory

土on

effect

characteristic

veloc

wave i8

assumpt

potential surface.

application useful

of

with

water-wave

prob-

lems. This computation

study of

describes unsteady

the

application

potential

flows

of

generated

-83 一

\

boundary-fitted by tþe

coordinates interaction

to

the

of waves

W 工th

1arge

Thé wave tential

fixed

forcesand of

tion

the

obtained

niques

two-dimensiona1

can

pressures

on the

wave f16w. by other

yie1d

structures body

Comparison

methods

accurate

submerged

of

indicates

are

in water

of

from

tþe

ca1cu1ated

the

resu1ts

that

these

for

finite

depth.

ve16city

wave forces

po-

with

solu-

tech 一

finite-difference

resu1ts.

II. FORMULA T10NS OF PROBLEM Consider me!genc~

a circu1ar

h be10w a free

- surface

is

c 土 fied

that

taken

as

surface

the

The y-axis

is

is

sma11 enough

inc ，Ompressib1e.The

roached

1.

depth

d with

The undisturbed

instantaneous

free

surface

sub-

free

wi11

be spe-

viscous

potentia1

gY

is

effects

progressing under

can

in

the

the

H/2a

tha.t

that

can

a ve10city

x-axis.

of

and of

prob1em

We seek

positive

assumptions

be neg1ected

interaction

f10w theory.

中 yy

the

then

be

potentia1

f1uid

app(x ，y ，

，中

+中 T

the

course;

the

the

Y = Y

(6 )土 s the

distant the

，

the

pressure

at

denote

(5)

x = :!:∞

differentiation

c is

the

a11

(x ，y)

in

工on

surface

(6) (7)

，and

condit

body

(4)

= 0

for

on the

condition

radiation

incident

The dynamic

at:t

中r)x

wave and

kinematic

and yoare of

t

dynamic

usua1

+ c(

(中r)t

x ，y ， and t

，

Y

0 on y = -d and on the

incident

a11 time

.

(3)

= 0 on y = Y(x ，t)

中 2)

.X

。。

subscripts

for for

= 0 on y = Y(x ， t)

+主(中 YZ+

L

中=中，

(2)

-Yt

+ c 中x =

中t

the

= 0

中y

V中﹒直=

中。

H/2

two dimensions

satisfies

中 xyx-

tentia1

in

wave/sturcture

中 xx+

ject.

figure

of

upwards.

so that

by way of

t) .which

surface

the

wave o.f amp1i.tude

is

of

shown in

inwater

(1)

formu1ated

is

p.ress

10cated

， t)=o

positive

The incident The problem

ho1d

as

，and

x-axis

is

by y-Y(x

where

cy1 工nder

at

condition

free

ini t iaIcondit

ions

s the

wa~e ce1erity. the

f1uid

at for

of

infinite the

ve10city

Equat

domain.

，and

surface

a boundary (16)

，土

(2)

Equation

Eqs.(3)

a perfect distance

ve1ocity

工on

p

，and

f1uid. from

。一 must

(4)

îs

(5)

ex-

Equation the

potentia1and

obfree

wave.

，

p(x

，y ，t)

，

on the

-84 一

structure

can

be computed

from

the

，

Bernoulli

equation

without

p = -p 中 t in

which

tained

p is

from

the

the

structure.

-

density

中 y2 〉 /2

of

fluid.

the of

per

form

hydrostatic

(申x2+

integration

The forces

and y còmponent

the

the

unit

term

as

(8) The force

dynamic

length

due

pressure

of

the

to

over

object

wave action the

are

is

surface

of

calculated

in

obthe

an x

as

FX=-j-S

P(x

， y' 的

F37.=fs

P(x

， y， t )dx=fs

dy=-fs

PYE dE

(9a)

PXE dE

(9b)

and

THE TRANSFORMATION

回. To simplify physical

region

transformed 2 ，is

to

corriposed

surface

onto

AN and

J1

the

numerical

(Figure

a time'-dependent

the

bottom

As to

.

(~， η) are

in

lines

and/or

scheme

to

For (11)

C.. yy

η+ xx

ll.... =

is

be

yy

P(

the

for

the

upstream

region mapped

which

onto

1H，the

onto onto

，the

problem

，

the

time-dependent and

downstream

，as

shown

upstream

BC and GH.

aIlows

points

in

the

by Thompson

by solving

free

boundaryonto

Theboundaries

et

al

an elliptic

(13)

，the

system

JKLMN

curvilinear

of

the

~， η， t)

form (10)

(11) The source

coordinate

domain

Figure

fluid.

conditions.

it

in

， is

LE ，the

spli t

Q (~ ， η， t)

that

or

lines

on the

functions

to

P and Q are

be attracted

boundaries

to

in

making

of

equations

spe-

specified the

numerical

efficience.

computational transformed

independent

=

is

water

computed

boundary

such

the

boundary

generated

1:;..___ +

appropriate

cified

öf

within

transformations

coordinates

with

cuts

of suitably

The body

downstream

and CDEFG represent

off

computational

o~E rectangles.

AB，the

，and

solution

1 )， cut

purposes to

variables ~~

the to

，the

plane

system

by interchanging

(10)

dependent

yield

- 2 日X~_ + yx__

αy~ë I:;~ -

generating

computational

-"""~llηηE

2ßy~_ -""ð~ll' + yy__ ulln

= -J2(PX~

+ Qx_)

= -J2(PYë ~ '~JI:;η + QYn)

where

一85 一

(13)

(14)

and and

α=

X

+ v

2

η"

ß -

2

n

y=xE2+yi

to

the

free

surface

body

are

and the GH. The

can

following

are

and

Reentrant-type

boundary

physical

plane.

boundaries

Lines

.except.

where

of

results

of

constant

they

the

those

Eqs.(13)

ahd

‘'..~、' 、情

Jι

(14)

(x ， y)-coordinatesof

Figure

2 ，the

lower

boundary

are are

specified

apþlied

the

coordinates are

cuts

boundary

of

given

on AN， JI

on the

on GF; the

stmilarly

the

by solving The

in

boundaries

on CD match

3 displays

shown

conditions

ML and KL，and DE and FE are Figure

、

(15)

conditions:

coordinates

downstream

(x ， y)-coordinates

S

be determined

on AB as

on LE ，the

given

then

boundary

specified

upstream

Ç"n

J = x"y EηnYs

The transformation subject

y ， ..y η J

+

X".X_

，

as

the

，

on HI BC， and

follows: MN and KJ ，

pairs

matched.

of

the

solution.to

extend

intersect

between

the

body

surface.

the

Eqs.(13)

and

(14)

in

the

upstream

and

downstream

j

where

Since

we desire

the

m~sh system

and boundary

to

given

of

must Using

(15)

the

the

，the

rectangulars in

transformed

such

conservative

that form

governing

s and of

plane

the

，

equations

ηare

the

in-

differential

一 (fy ..)_J '~Js， + (fx ..)_J '~"s ， η

some arbitrary

function

(16a) (16b) and J is

the

Jacobian

defined

in

becomes

The transformed On the

fully

Y =主 J [~(fx_)". L '~"n/s

α 中ss

(a)

simple

in

by

fdenotes

，Eq.(2)

computations

be transformed

=主 J [(fy_) L'~Jn/s

where

all

consists

conditions

dependentva:r:iables. operators

perform

free

boundary surface

(Y~)------~--~ =constant

J2 (P中" + Q 中) - '""'s

y 中=一

2自中+ -~"'sηnn

equations

(17)

are

(y=y~):

= [-(y 中L 'Jn"'s

y 中 )/JJ Js"'n"~J

Y ， ../x "s"'s

，.. +'(x

中'''s'''nη

x

中

)/J

at y = y 且 (中)，.. ______~__~=

t /S， η=constant

i

-x~( 中 y"t''''sJn - [(y

中-

S

y， ..)/J "'nJs"~

-中

y J

中

t;ηE

- y~( 中 x Jt''''s''nηE

)2 + (x

-86

一

中-

ηη

x

-中

中)

x， ..)/J

-gY

2 J / ( 2J 2 )

可

(18)

Eq.

at y (b)

(19)

= y且

On the

surface

of

structure

and the

bottom

of

water

(y=n

and y=η'- ) : u

1

(日中 i: ，-

(c)

On the

remote

y 中 η)/(y

= 0

告 J)

(20)

(i: ， =i:， and i:， =i:，.):

boundaries

凡

1

(中 )r

= x~(y "t'Ji:

______~__~

t' i:"η=constant

y

中， "'nηi:

- c(y

中) ，'

y

中，

ηi:

J

中

i:， η

y~(x 中x J t '"' i: ， ηηE

中)

+ F

)/J

(21)

where F"，=

(中)~

+

S1nce

the

flu

1

generate the

'''' 1't

the

c( 中)

工d

reg

工on

is

transforìnation

solution

of

the

not

must

fluid

flow

known in

be solved

problem

，Eqs.(13)

advance at

each

time

in

and

(14)

which

conjunction

with

(17)-(21).

N. NUMERICAL SOLUTION Numerical

computations

lem solution.

The first

transformation. o~ (13)

and

vergence

of

along

Replacing (14) the

lines

of

are

SOR point

and the

transformation These

input

to

，the

tion

equations

numerical

scheme

Given tential Since

is the

vation must

description

also

replaced

flow

be solved

potential elevation followed

i8

in

the

are

combined

and the

potential

by an upHating

region

of

with

on the grid

equations

Once con-

x， y locations

the

，are

then

available

mesh geometry.

pair

coordinate

be sufficient

，Eq.

(主7)

formula in

the

surface

the

interative

coordinates.

the

the

in

1n this transforma-

making

the

P=O and Q=O.

difference

potentialon

prob-

compute

x_ ， y_ ， xr ， and y 尸， ηηιι each gr 工d point are comsize

the

the

to

mesh system

by central

mesh point

on of

工

found

So we set

and the

，defining

appropriate

P and Q of are

a time-dependent

velocity and

the

descript

zero

of

(SOR).

derivatives

flow

s to

工

，differe~ce

formula

achieved

and

procedure

α ，自， y ， and J at

functions

efficiently.

transformation

overrelaxation

spatial

to

a complete

the

and the

is

，scaled

as

the

difference

coefficients

specify

both

c6mputational

by successive iteration

inhomogeneous to

of

the

i:， and η ， the

derivatives

constitute

study

in

by central

solved

constant

puted. as

contain step

{x ， y)-space

A time

at

the

the

，the

advanced

solution advancement

surface

according

point

coordinates

-87 一

for

velocity

and solved

to

SOR.

surface time

ele-

level

for

the

velocity

of

the

surface'

Eqs.(18)

with

po-

with

and

Eqs.(13)

(19) and

is

(14)

and then ) and

an adjustment

of

finite-difference

grid

point

the

initial

the

incident

est

the

convergence

工mates

field

of

the

the

of

surface

boundary

force

elevation for

Y and

from

中

and the

is

with

are

when

reached.

on the

，

for

halted

iteration

，

are

started

are

usually

(17

potential

step

to

criteria

Eq.

structure is

velocity

time

itreration

to

The new

on the

procedure

an advanced

convergent

according

conditions.

acting

The computation

surface

relative

the

the

and the

The iterations

criteria of

below to

simultaneously.

wave.

The magnitude

potential

，flow

distribution

，computed

therefore

the

approximations

order

of

0.001

V .RESULTS AND DlSCUSSION Based

teraction fin

土te

with

merged

the

For dimens

der

[that

to

cy1inder bottom

wave

the

the

cident

the

very per

pressure

that

，as

the

v字 rious

the in

cylinde~

，some

of

，the

spacings

of on

studies this

case

are

wave in-

water

resting

other

results

technique

study.

are

presenteQ

per

unit

To

two sub-

a1so

resu1ts

have

‘

of

wave forces

good.

un 工t

Fu11y

previous1y.

Figs. in

variation

of

submerged

(18)

for

investigated.

in

1ength

that

the

ned

工

the

and the

the of

form

cy1in-

in

the

d/a the

b10ckage Fig.

steps.

the

It

position

do-

equa1

agreeof wave 7 shows

indicates of

the

in-

object.

--

wave

that

cy1inder.

time upon

on the

and ver-

ratio

shows

Some stud

interaction

a semic1osed-form

es

工

for

has solution

a fu11y

been

sub-

solved

for

，

com-

computationa1

a greater

on the

resting

show the

hOrizonta1

，it

different

一88 一

•

(17)

dependent of

(b)

depth-to~radius

resu1ts

cy1inder.

with

region

cy1inde~

and

dimension1ess

forces

= 4 at

a ha1f 4 (a)

studies

o~ d/a

main1y

to

obta

ka

1arger'

d/a

is

physica1

other

horizonta1

encountered

Ogi1vie

for

for

Figs.

6 show the

with

re1ative

4-7.

the

5 and

and hence

distribution

cy1inder

in

A 10wer va1ue

1ength

pressure

一- Resu1ts

bottom.

functions

comparison

wave profi1e

merged

in

norma1ized

system Figs.

forces

In

is

calidate

，numerica1

ana1ysis

given

coordinate

4.0.

energy

to

(circular)

numerica1

mounted

﹒ The

on the are

respective1y.

ment

the

studying

submerged

cyclinder

工fference

，the horizonta1 force f_ = (F_)___/(pgaa_) and the vertica1 x xmax ~ 0 (Fv)m~x/(pgaan)] are p10tted against the dimension1ess wave number ymax" ~ 0 various va1ues of the depth-to-radius ratio ，d/a.

putationa1

tica1

of

p10ts

half

circular

be used of

for

finite-d

=

ocean

main

can

out

the

structures

a submerged

cy1inders

工on1ess

Ha1f the

which

carried

of

is

fvy

，for

of

usefu1ness

been

system

two-dimensional

submerged

convenience

of

force

cases

horizonta1

have

f 土xed

a fully

made before

illustrate

ka

For

and

coordinate

solutions

large

depth.

bottom

been

boundary-fitted

，numer 土cal

approach

the

on the

a fu11y

sub-

rnerged tion

cylinder

of

infinitely

a ka and kh are

and Shepherd ifa11y

in

in

(11)

the

d 旬出 on the the

present

for

d/a

depth

incident nwnerical

for

d/a

noted as

that

the

h/a

the

this

of

s1stern fect and

in

of fy

Fig.

the

the

，are

results

of

cylinder

is

will

reduce

inder

，the

crease

of

cylinders

cylinder

the

versus

upstrearn

reduction

water

at

effect

工nder

of

the

water

of

Ogilvie

，and

B工rd

and Shepherd

results

are

seen

to

the

fx

and

fy

experirnental

the

nurner-

these

cornparison

curves

agal

，where

工s

01

be the

plotted results

nurnerical

ka for

It

of

can

in

ka

叫

﹒ The

arree-

good.

It

is

ka rnay be interpreted

vertical

工te

tirne

that

the at

It

steps.

is

is

for

are

giveri

叫

coordinate

To illustrate

with

increase srnall

of

case

the

Fig.

12.

eff

individual

cy]inder

To the

in

the

of

spacing

downstream

proportional

in

observed

the

ka.

inversely

can be seen

wave force

water

cor恥tatio

results

，the dirnens 工onless wave forces. ，2.o and 3.0.FM-11Show-tie

specially

wave force

depth

and the

The cornparison

spacing.

一- Nurnerical

S.

a fin

surface

s/a=1.0

note

，fx'

the

cylinder

a spacing

different

cylinder.

rnade.

in

two cylinders

wave force

on the

，Bird

approach cyl

8 shows

ka decreases

subrnerged the

region

p1omd

the

as

the

a func 一

Fig.

and the

rnounted

10 shows

of

also

w工th

results

cylinders

physical

the

kd 川

as

elernent

a sub~erged

wave forces

irnproves

spacj.ng

the

as

that

cornparing

wave forces

wave steepness.

horizont~l

10-13.

with

results

note

cmenslonjess

agreernent

dirnensionless

a pair

with

the

on a boundary

be negligible.

can

= 4 in

which

water-Slnce

experirnental

Two subrnerged

Figs.

It

the

= 6 and

between

of

solution

= 6 and ka = 2.

in

waveinteraction

wave will

sme-Fig-9Shows

rnent

the

water

﹒ Based

preserited

solved

finite

deep

to

cyl-

the

Theeffect

inof

two

insignificant.

"VI . CONClUSIONS A description systern.

has

been

structures.

nwnerical to

The ，governing

/transformed

in

us 色r entensive aries.

of

presented

using control

Wàve pressure

the

rnethod

study

the

equation bound

over

based

and.the

于ry-fitted

the

de~ign

and wave forces

cOmparewell to the previous solutions inder and a horizontal cylinder.

on the

boundary-fitted

wave interaction boundary

coordinate of

the

obtained for

-89 一

with conditions approach

cornputational frorn this

a bottorn-seated

coordinate

large

subrnerged

have which

been

allows

rnesh and the study

are

horizontal

found half

the boundto cyl-

REFERENCES 1.

，H. ，and ，"J. the

Beckmann Pipe1ines

，M.

Thibodeaux Waterways

H.

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Offshore

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88

pp.125-138. 2.

Grace Sphere

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95

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291-317. 3.

，C.

Garrison F1ow ，" J.

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Bodi.es

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Axisyrnmetric

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6.

Dimensiona1

circu1ar

Objects

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K.

Finite

Water

23 ，No. 1 ，1979 ，pp. 32-42. ，K. J. ， "Diffusion of Ob1ique Waves by Mech. ，Vo1. 68 ，Part 3 ，1975 ，pp. 513-535. Von'gvissessomjai ，S. ，and Hanif ，M. ， "Wave

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Depths

Waves by Two-

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Ship

Research

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Cy1inder

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F1uid

Vo1. 7.

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9.

Bai

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E1ement

Forceson

Eng.

pp.

219-235.

Chen

，H.

Harbor sium 10.

Zienk

Solution

，H.

F10ws

13.

，B. in

12

D.

the

the

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Vicinity

，1973 ，pp.

Thompson rious

the

，J.

F.

1st

Co11ege

at

Boundary 1976 ，pp.

-

Fitted

Shepherd

Intern.

Conf.

J.

Cy1inders

on Environ.

，London ，Eng1and ，Ju1y ，1979 ， and Wave Forces

the

，"Intern.

on E1liptica1

10th

Nava1

in

a Man-Made

Hydrodynamics

of

the

Finite

of

Numerica1

E1ement Methods

Sympo-

Method in

and

Engi-

355-376.

，R. ， "Wave Interaction Port ，Coasta1 and

Waterways.

with

Large

Ocean

Div.

Submerged

，Vol.

108

，

146-162. Hirt of

，C.

W. ， "Ca1culating

Submerged

Three-'-Dimensina1

and Exposed

，"J.

Structures

Free

Surface

，

Comp. Phys.

234-246.

，et.

F10wRegimes

Forces

，"0sci11ations

Procedures

，and

，"J.

WW2 ，May ，1982 Nicho1s

Vo1.

C.

，"Presented

，1977 ，pp.

11

W. K.

Structures

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Open Sea

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neering Bird

the

，"Proc.

，MIT ，Cambridge. ，Mass. ，June ，1974. 工ewicz ，O. C. ，et al. ， "The Coup1ing

Boundary

11.

， and

S.

in

Methods

Structures

an Infinite

al.

， "Solutions

on Fie1ds Coordinate

of

Contatining Systems."

421-450.

-90 一

the

Navier-Stokes

any Number of Lecture

Notes

Equations Arbitary in

Physics

in

Va-

Bodies

Using

，Vo1.

59

，

14.

Hwang ，R. R. Circular 1986

15.

，et.

Cyli.nder

，pp.

Incompressible tute

of

Viscous

Flow past

，Vol.

Engineers

9 ，No.

a 6，

617-631.

，H.

Haussling

al. ， "Time-Dependent ，"J. the Chinese Insti

Accelerated

J.

and Coleman

Circular

Cylinder

，R. M. ， "Nonlinear Water ，"J. Fluid Mech. ，vol.

Waves Generated 92 ，part

by an

4 ，1979

，pp.

767~781. 16.

Chapman

，D.

Barotropic pp 17.

C.

，"Numerical

Coastal

Treatment Model

of

，"J.

Cross-shelf

Phys.

Open Boundaries

，Vol.

Oceanogr.

15 ，No.

in

a

8 ，1975

﹒1060-1075. ，R.

Naftzger mensional

A. and Chakrabart

Circular

23 ，No. 1 ， 1979 18.

Ocean

Ogilvie

，t.

Under

a Free

F'.

Obstacles

，pp.

，"First

Surface

，S. in

K.

， "Scattering

Finite

Water

Depths

of

Waves by Two-Di-

， "J.

Ship

Research

32-42. and Second-Order

，"J.

Flu

土d

Mech.

-91

Forces

，Vol.

一

on a Cylinder

16 ，1963.

，pp.

Submerged 451-472.

，Vol.

，

.-

Wove

SWL

凡

一

RS」 n

d

11

凡

Fig 1. Definition sketch of the physical problem

8

A

N

仁一

D

'"L K

III

-

-

~

C

t

，，

F

J

r"7

G

f

If

Fig 2. The computational region and coordinate system.

Fig 3. The computational coordinate system in physical space.

~92

一

(a)

The computational in

:1ηL: 單 4.

coordinate

system

regio 耳﹒

physical

Bη

團

ηE

C

F (b)

Fig

the

EηEηb

The computational

The~application

of the

，

to. bottom-seated

half

domain

boundary-fitted

coord

土nate

system

cylinder.

。 ω

O

N

F

電

關叫;

。 D

-

H

.

o o

'、 。

'"o.

O

可

-

'"

- ASYHPTOTIC SOLUTIONS φPI1ESEln: N 【)II-LINEAA B. C

o

"--'

。

-.ASYHPTOTIC SOLUTIONS PAESENT:NON-LINEAA B.C

>-'"。

d/D=2.0

>-

L

.. 。 0

g

-

BIRO F.E.M SOL EXPëflIMENT AESULTS PAESi:NT: r~ON-L1NEAfI PAESENT: NON-L1NEAA

。

.6

g

f

N

o

01

'"

g

N

I +

。 0.00 Fig

Id

s

/-、

HZ

L

SWL

6f

o o 開

o

.

\y~

﹒

B. C. H-O. 1 B. B H/L 1/62 1.50

1.00 K

0.50

9. Dimensionless

Fig

10. The

in physical

BIAO F.E.M SOL EXPEAIMENT AESULTS PAESENT: NON-LINEAR PAESENT: NON-LINëAA

。 。 +

o

0.50

forces

WATER WAVE口 N 8 口OY A PAIR OF CYLIUOER NON-LINUR C口NOITIOfl .150 6.2

日

，

a c H-O.I a. C H/L-l/i62

O

2.00

maximum

FLOW TYPE : 80 口Y FORl1 su 肉 HCE COIt W~γE HEIGHT W~γE LENGTH:

-

3

on a submerged

"FROUOE NO 口 EPTH "8001 OEPTH 個 'UI"FORH VEL "W~'IE PERIOO

‘醫WATER

computational

: :

time

horizontal

.993 2.500 1.500 .000 6.283

coordinate

space at different

-95 一

: :

1.00 K

system steps.

1.50

2.00

cylinder

. . z

z

零

" 5-0.5 5-1.0 5-1.5

‘

自

4 X

E

區。i

司，

，

.

E

E

" 莖

jdoi

眉

h

L

l!l 5-0.5 " 5-1.0 X 5-1.5

軍

m

E

Z

豆"

囡

，‘ 〉

s

. -

u. 穹

z

2

m

宮

"

E h1D-2.0 dl口-3.0 H/L-l/62 +1 阻 1V1DUAI.CYL1間回 Xl!l" A PAIR OF 1AI CYl.I間自

. ....

宜

。

z ..圓

0.00

0

1.20

K

1."

0•00

2.00

0

h1D-2.0 d1D-3.0 H丸 -1/62 +1 阻 IVIDUAI. CYL1阻回 Xl!l'" A PA1R OF ωCVL1 間自

。.....8。

1.20

1..8

K

。

2.00

Fig 11. Dimensionless maximum forces of cylinder.A for two submerged horizon-mountedcylinders

at different spacings

z

零

"

回

.. ..

。 +

5-0.5

5-1. 0

(!) 5-1.5

E

idoi

司，

+ 5-0.5 (!) 5-1.0 5-1.5

。

idd

. 自

h

z

E

"

g

E

E

L

i， Ð

咧

M

M

芷瑋

u.g 司，

z

宮

"

"

. -

s E

E 1IID-2.0

dlD 3.0

z

目

H/L-V62 + 1N11IV1DUAI. CYL1阻四 +(!)~ A P.u 胸前個 l CYU

e

閉目

E

-..曲..咽

....

1."

..

-..叩..

E 1.霆，

。

h/D-2.0 d/D-3. H/L-l/62 + lNIIlV1DU且 CYL1N11 個 +(!)~ A PAIR.OF lBl CYL1閉目

2.00

K

..曲

1.20

1.&Q

2.00

K

Fig 12. Dimensionless maximum forces of cylinder B for two submerged horizon-mounted cylinders at different spacings

-96 一