On the properties of vertical wave oscillation in a Moonpool

University of Strathclyde

Department of Ship and Marine Technology Glasgow. July, 1982

77:127:23112 trznn:TET

LEtoretoriurn vizor

Saheepshydromechanica

Archief Mekeftveg 2, 2828 CD De!ft

i$LOth-?8--F*: ois

-181X$ ON THE PROPEFiTIES OF

VERTTCAL WATER OSCULATION IN A MOONPCOL

by

B. S. Lee, B.S.E., M.Sc

A thesis submitted for the degree of Doctor of Philosophy

IP

SUMMARY

This project is concerned with the water oscillation in a Moonpool which is employed for a range of offshore support taskt. The problem is am important one, as not enough attention has been given to

its

optitUt design despite its Wide popUlarity among the offshore

support system operators. The fundamental aspects of the

ahipmoonpool .system is investigated and the emphasis

is

placed on the practical application

of

the results. The research is performed

in two

parts.

In the

first part the moonpool Is idenlised to a two-dimensional system so that it

becomes a narrow duct formed betWeen two

infinitely

long Obstacles in

the wave field separated by a Small distance.. Initially the barriers are fixed in space with a train of simple harmonic waves incident upon them in the beaMwise direction. The problem is treated as a singular perturbation problem and is solved by the Method of matChed asymptotic expansions. The reaponte amplitude operator of the water column oscillation is cdtputed and the results are compared to the experimental results. The barriers are then allowed to, heave only and the fixed problem solution is

extended ta incorporate the heave motion. The results

are also

compared to experimental data.

This

study enables a key parameter

governing the behaviour of water oscillation in a two-dimensional

moonpool to be identified.

The second

part deals

with a three-dimensional coupled system of

ship and moonpool. The ship is allowed to heave only and a train of simple harmonic waves is incident along the length of the Ship. The coupled

problem

is formulated by Using a.two degrees of. freedom

system and the

hydrodynamic coefficients are obtained through experimental approach.

A practical design calculation procedure is put forward and its

ii

ACKNOWLELGEMENTS

The project was ,initiated

and

supervised. by Professor C.KUO of

Strathclyde University and.' an indebted

to him for

his

patient

gUidance and constant

encourage bent

during the research.

The help and advice of Dr. J.

Martin.

Of 'Edinburgh Univeraity, Mr + D.

Vas4a,lOs and

Dr. P. Sayer of

Strathclyde

UniVersity aid PrOfessor Ursell

of

Mandhester University are gratefUlly adknioWledged.

The computer program used for the solution of the various bounaarY

value

probTems discussed

in

this thesis is a modified version

of a

program originally

Written

by Mr. Vassalos.

T would. alSo like

to

offer my thanks to W. Wett, G. Vitt

and T.

Chapter

SUMMARY

ACKNINLEIXEYETIS _ii

PART A ABOUT THE PROJECT AND KEY FINDINGS

INTRODUCTION

AIMS OF THE PROJECT ₅

LITERATURE REVIEW

PROJECT STRATEGY ₁₀

KEY FINDINGS AND_ :DISCUSSIONS ₁₂

AREAS OF FUTURE _RESEARCH ₁₄

MAIN CONCLUSIONS ₁₆

PART B ON THE PROPERTIES OF WATER OSCILLATTON

IN '1W0-DIMENSIONAL DUCTS _-18

INTRODUCTION ₁₉

BASIC APPROACH TO THE PROBLEM ₂₁

WATER COLUMN _{OSCILLATION BETWEEN TWO FIXED}

BARRIERS. WITH THICKNESS ₂₄

3.1 Fbrmulation of

the

Problem ₂₄

I.

iv

3.3

Solution of the Inner Problem

26

3.4

Solution by Matching 28

3.5

Estimation of Water Column Response

29

3.6

Experimental Verification

31

WATER COLUMN OSCILLATION BEWEEN TWO HEAVING .BARRIERS

WITH THICKNESS

34

4.1

Formulation of the Problem .

34

4.2

The Solution of the Inner Regian-

35

4.3 The

Solution of the Outer Problem

36

4.4

Matching

39

4.5

Heave Problem 40

4.6

Estimation of Water Column Response

42

4.7

Experimental Verification

44

APPLICATION OF THE FINDINGS

45

rucussIoNs

48

CONCLUSIONS

51

LIST OF SYMBOLS FOR PART B 75*

PART C ON THE THREE-DIMENSIONAL MCONPOOL PROBLEM

77

INTRODUCTION

78

ON THE PROPERTIES OF WATER OSCILLATION IN A FIXED TUBE

79

2.1 System Representation with a Mechanical

Oscillator Model

79

2.2

Description of Experiment 83

85

3.1 Introduction ₈₅

3.2 Formulation of the Problem ₈₆

3.3 SOlution Of the

_{Coupled Equations}

₈₈ StrateV for Experimental Studies ₉₁

3.5

Description of Experiments _'92

3.6

Methods of Analysis

95

'3.7

Experimental Results Obtained

₉₇

3.8

Interpretation of the Experimental .Results and Comnents

99

PRACTICAL DESIGN CALCULATION PROCEDURE AND

EXAMPLE COMPUTATIONS ₁₀₂

4.1 Introduction ₁₀₂

4.2 Design

_{Computational}

_{Procedure 102} 4.3 Worked Examples and. Comparison with Experiments 104

.DISCUSSIONS ₁₀₇

CONCLUSIONS ₁₁₀

LIST OF SYMBOLS FOR PART C

148

REFERENCES

150

APPENDIX I

THE SOLUTION OF FIXED INNER.PROBLEM.BY

SCHWARZ-CHRISTOFFELTRANSPORMATION ₁₅₄

APPENDIX II

FORMULATION OF BOUNDARY VALUE PROBLEM FOR

Vi

APPENDIX III

DESCRIPTIONS OF EXPERIMENTAL FACILITIES USED DI THE STUDY

APPENDIX IV

ANALYSIS OF FEM OSCILLATION 9:Esir RESULTS

APPENDDC

RESULTS OF FORCED OSCILLATION TESTS ON

WATER COLUMN IN :VERTICAL TUBES

APPENDIX VI

CCMPUTATION OF HEAVE EXC ITATION PUCE AND WATER COLUMN EXCITATION FORCE

169

17

INTRODUCTTON

With the :rapid expansion of Man's exploitation of ocean resources

in

the last two decades, the

demand for

the 'effective meanS of

launching and retrieving equipment, personnel

and

materials

into/fram the

sea has

increased tremendously. Despite the huge inVestment in developing more COSt effective and reliable sUpport

systems, at leatt one hurdle remains unresolved in launch/retrieval:.

activities,' i.e. the hatard of air/sea interface. Although various ideas of alleviating this problem have been put forward and in spite of a few claims as to the contrary, the ability to pass through t*6

interface succesSfUily

is

still at the .mercy of weather donditiOirsli

This difficulty is

expected to become greater as

the offshore -activities advance into more hazardous environment.

The major hazard Of the air/Sea

interface in launoh/retnieVal operations lies in the forces exerted by the environmental element8 and experienced by, the objects launched/recovered at they pass frOM

one medium into another. These forces can

be broadly divided into

two dompOnents: vertical

and

horizontal. The vertical force it mainly

due

to the relative vertical motion between the objects

and

the water

and can lead to a dangeroub situation, particularly wneh cotpled with horizontal forces Of

Which the

likely causes

are

current, wind and 'theorbital motion

or

the water particles induCed

by waves.

With this

problem in mind

several Methods of launching And

retrieving subsea Units, especially manned and =conned Submersibles

and diVing bells, have been developed and Some have been put

to

practical use with Varying degrees of success. One of these SysteMs Which has Shown considerable merits and consequently won reasonable

popularity

is

the mompool system.

A moonpool is essentially a vertical well of various cross sectional shapes usUally situated at the centre Of floatation

or

a ship

with

its lower end open to the sea. At present

only

relatively small arid

'compact units, such as diving bells, remote control vehicles (RCV's) and one man atmospheric diving suits, are launched and recovered

through the moonpool, but there is a sign that the system will be widely used to handle manned submersibles and heavy equipment as

well in the future.

The relative advantages of using the moonpool system for these

purposes are threefOlds:

it provides good protection from the

horizontal force

elements; .

by positioning the,moonpool at the centre of floatation of the ship at the normal working draught, the adverse effects of the ship's angular motions can be. minimised;_

the Characteristics of the water oscillation ,inside the

moonpool is such that the higher frequency components of the wave are 'filtered out.'

For the

summary of various

diving support systems and the

discussions of their relative advantages and drawbacks, see Kuo [1].

Unfortunately, however, launch/retrieval operations are often hampered by 'unpredictable' large amplitude oscillations

of the

water column inside the moonpool. One other

aspect of the

'unpredictability' of the moonpool behaviour can be illustrated by a-very puzzled diving support operator who observed the prevalent wave period of six seconds and the resultant

water oscillation in a

moonpool which appeared to be of .about.ten seconds period. This puzzlement is due to the

lack of understanding of the

basic properties of the moonpool water column oscillation. The consequence of this present ignorance is that moonpools are usually designed simply as vertical wells of a size suitable for the subsea units to

be launched/retrieved. Various damping devices are added to them almost as an afterthought in an attempt to decrease the water column

oscillation.

4

The vertical water oscillation in the moonpool is important only

in

the context- of the forces imparted by the oscillation on the. ObjectS

to be launched or retrieved. Once the characteristics of the

water-oaciilation are

known,

however, these forces am be :calculated by using e.g. Mbrrizonis equation (see Madsen Fbr.this reason the research Work described

in this

thesis- is devoted to -the Investigation of the Vertical water oscillation problem alone.

The behaviour of amconpool.is of a very oamplicated. nature. Apart

from the size and geometrical shape of the

moonpoal,

there are the

lmillmoonpoO1

interaction and the effects Of various damping devices

to consider.. It is well.-known, however, that

the more

parameter

there are, - the more complex the problem becomes. It is, therefore,

,,necessary to simplify' the probleft

and

.start with as few parameters

as possible. The Choice of these parameters is a difficult one

due

to lack of reliable data and for this reason a theoretical approach to the problem using

a

simplified geometry was adapted as a starting

point.

In this approach the main features of the

moonpool

are retained but two-dimensionalised

in order

to accommodate the linearized

theoretical hydrodynamics more easily.

Damping

devices are ignbred and main emphasis is placed on determining the characteristia.

behaviour of such a system. This part of the work is described

in

Part B of the thesis.

Useful though this exercise is; it only provides qualitative

information Which could not be readily used for design purposes. For

this reason the problem is tackled once more using an entirelly

different approach with particular emphasis on practical_

application. This

entails

formulating the problem in terms of

4

two-degrees of freedam-imedhanical, oscillator system.

Again

the

effects of damping devices are ignored and stoothwalled circular

cylinders are taken to represent the moonpool. This ,apprbach

is

discussed in Part C.

2. AS OF THE PROJECT

The main .aim of the project. is to develop a. mathematical model describing the shiloHmoonpool coupled system, the ktOwledge of Which wiil..provide the basic, understanding of the

behaviour

of the water

column .oscillation in a moonpoOl and

an effective meansof

predicting

the moonp001 properties at

the

design stage. More

specifically the aims are;

(i) To examine the

method of

solution to

the problem using

theoretical hydrodynamics and to verify the theory by

experiments;

1. To investigate

the

effects of ship heave motion on

the

..-behaviour of vertical water oscillation in.

_{the moonp001;}

To establish a practical

_method

of

_evaluating the

3. LrEERATURE REVIEW

Although the mconpool system has been in practical applicatidn forra

number of years,

it is 'only

during

the last 5 years that tlyie

characteriptics.

of moonpoOls

have been studied tseribusIV. Consequently there hat been Very little work attempting todescribe

. I

the vertical motion. Of the water column

in

the moonpool, t4P

information of Which is 'essential

for assessing the fore

experienced by subsea units laundhea/recovered throUgh the moonpoOl air/Sea

intetface..-.

The papers Which have been-published to date on the subject are reviewed below.

FUkuda [2] investigated the effects of water column oscillation

in a

-:...moonpool on Ship Motions with particular emphasis on the increase

of

ahip's:fOrward motion resistance. It was shown that,' When a model with a moonpool was towed in calm water, the resistance increased by up to 7% compared to a model without one. The work has provided good

iiisi,gbt into the importance of the water oscillation in Ship motion

study, but the behaviour of the water column was

studied with a

number of uniform forward velocities of the Ship and not in the wave

field. Therefore', there it no direct application to the present

problem, except the fact that an empirical formula for the Water

column added Mass was derived through a series of transient tests.-This formula was found fairly consistent with the result of ttie

present work described in Part C.

Madsen [31carried out a research specifically directed at modelling the water column motion in the mconpool mathematically so that .the

forces experienced by the diving bells passing through the- moonpoOl.

air/tea interface

could be

predicted. The equation of motion was derived by using the principle of equilibrium of forces. The added

mast of the water column was taken from Fukudes [2]

suggestion While the

damping

coefficient was determined by assuming a flOW

pattern which- is not fully justifiable. This work Was the first attempt to give a full description to the water motion including

the

I

effect of

a

diving bell blocking the flow path. However, there Are

the following weak features:

Froude-Kryloff force alone is considered as external excitation force and other factors,

such as

wave diffraction, are not taken into account.

The numerical values

of the water column hydrodynamic

coefficients are selected in an incorrect

manner.

(iii) It is believed by

the present author

that the various 'damping devices'

mentioned in this

paper will have a

significant effect not only on damping but on added mass as well, but this point is ignored in his formulation.

(iv) A significant weakness is that the cross-coupling effects

between

ship

and moonpool are not considered.

Knott and MacKley [14] performed experimental studies an the water

oscillation' in half immersed vertical tubes fixed in _ space. Their

'main interest was to study the eddy' motion neat the lower exit of the tube die to the water oscillation. A series of. free-oscillation tests was performed and

the decaying water Oscillation was

formulated by using Bernoulli's equation. The equation thus derived

is very similar to that describing

the

motion of a single degree of freedan mechanical oscillator with non-linear damping. The decaying motion of the Water column was then simulated on a computer by this equation using various numerical Values of damping coefficient until a satisfactory matching between the prediction and experimental

observation Was achieved. The empirical formula for damping

coefficient thus

derived

proved to be very close to the result of

present study. Unfortunately, however, only the isolated tubes were treated and the coupled

systen of ship-moonpool was not included

in

the study.

Flower and Aljaff [15] Showed that KnOtt and MacKley's experimental data could be analysed by a method based an a first order averaging

method due

to Kryloff and HogpliUboff (see Minorsky [20]). This is a very Useful technique in analysing

the

free Oscillation test data of

8

Although these

researchers

have contributed to the general. knowledge

of the moonpool

probleth

considerably, it is felt that the basic understanding of the physical phenomenon is still lacking.

A good

starting point would be to investigate the problem using theoretical hydrodynamic methods

but

without invOlVing hydrodynamic coefficients

as such. It is .natUral; therefore,, to examine the theoretical

studies which are relevant to the problem of vertical water column oscillation in

a

moonpool.

Newman [6] solved the problem of a closely spaced double barrier with infinitesimal thickness in waves. The flow field. was diVided into inner and Outer regions and examinedseparatay. The inner

'region covers the flow field near the lower exit of the Slit formed

1

.)between the two barriers and the channel-like flow near

the

free-surfacelmaide the slit. The flow

in

thiS regal is solved

'using- Sdhwar2-Christoffel tranSformatiOn,

but the

solution its

incomplete since one constant is still undetermined. In the. outer

region,

which covers the flow field fartemoved from the slit, the two barriers are assumed to be collapsed into one. The flow into and but of the lower entrance is represented by a source sing0Iarity of an Undetermined strength. The velocity potential in this region is obtained by summing incident wave potential, diffraction potential and the potential due to the source. The solution of the two regions are

then matched to

each other in the overlap domain to give the expressions for two unknown constants. This completes tha solution

for the singular perturbation problem. and the water column

oscillation in the

slit can be estimated from linearised free-surface condition.. NUmerical studies based on this solution have shown that the characteristic features of the

water

column oscillation are very simil.Arto those of the heave motion response of a free floating thin solid Column in waves.

Another

important

conclusion of this work is that the most influential parameter

governing the behaviour of watwer oscillation is the ratio

of

barrier Separation distance to the depth of immersion.

-tUbe. In the absence of prOper. evaluation of damping. terms the result

is

removed from the reality, but it is

meaningful

In a sense that this approach gives the maximum rate of energy -extraction of this particular device, and this was shown to be 0.5.

The basic geometrical assumptions of these authors

ate

that the

barrier thickness is.infinitesiTAT and that the gap between the two barriers is much Smaller than the depth of immersion. In terms of

shipmoonpoOl

system the barriers correspond to the

ship hull

and the gap between them to the Size of

mconpool,

e.g. its diameter. The latter assumption may be valid for a large number Of cases, but the .zero thiakness.assumptign is not, since the ratio of ship breadth to , moonpool diameter is

usually

larger than 5. Their. work has provided

'Valuable insight into this complex problem And it is now &Own in

this thesis how the Method can be Modified to Suit. the present

problem of double

barrier with a

ship-like section,

which is a

similar but

a

more generalised problem of the double barrier with

zero thickenss. The results obtained by Newman are particularly

In View of the critical review, it

is

clear that the ship-moonpoOl probleM'will

have

to be tackled in two ways as follows:

Theoretical hydrodynamic method can be used to inhance the fUndamental understanding of the system through studying a two-dimensionalised model. The technique developed by

Newman [6] is followed closely, but the barriers are allowed

to have an arbitrary Ship-like

section. The barriers are assumed to be fixed in space initially and the solution is extended to incorporate the heave motion of the

barriei's at a later stage. This approach is described in

Part EL

Since the assumptions inVolVed

in the theoretical

analysis

would be

too

restrictive

for the

results to

be 1284b1

directly for design Calculation purposes, a more practical

approach is required. This is achieved by formulating

the

Shipmoonpool system as

a

coupled mechanical oscillator

with the

help

of insight gained during the theoretical

study. The

numerical values of hydtodytamic coefficients Including coupling terms are then obtained by experimental methods of both free- and forced-oscillation tests. The atO

of providing a guide for designers

is achieved by

suggesting a design caldblation procedure. This part of the

.Study is presented in Part. C.

Finally, the whole project strategy is summarised in Fig.A.1. The, items inside double rectangles are

_{the work carried out in}

_the

present study. The investigation into the fundamental aspects of the Problem is

described on the

left, While the approach aimed at practical application is shown on the

melt.

It is worth noting that the theoretical line of research concentrates On the closely Spaced double barrier problem and not the widely spaced One. This point is discussed fUrther in Chapter A.6. The insight gained through the

10

work

on double. barrier with shiplike section

is

then utilised in fotmUlating the problem as a coupled mechanical oscillator. After solving the coupled equations with the hydrodynamic coefficients obtained, by experimental methods, the practical approach results in proposing a design calculation procedure.

12

.5, KETFINDINGS AND BRIEF DISCUSSIONS.

Following the Strategy shown in the previous Chapter, the properties Of Vertical Water oscillation in a moonpool were investigated in two ways. Here the main features and the key findings of the research

work described in Parts B and 0 are highlighted and a few salient

points are briefly discussed.

In the

theoretical study the work by Newman

[6]

with two closely

spaced and infinitely thin barriers was generalised to

deal With

barriers with arbitrary thickness,

Although in this

stUdY

tie

. barriers were assumed to be symmetrical About the Vertical axis

and

have

a

Ship-like section.

The result from the fixed barrier case

showed a similar

trend to the

I

-thin barrier prObleM, but distinctly diffetent.in details. The two problemS use different transformation and the flow lines indicate

that

they are dissimilar physical phenomena. The thickness effecti,

therefore, cannot be ignored.

The numerical studies

based on the

theoretical solution have produced A

set of graphs with water

column response amplitude operator plotted against wave number. The response characteristics

are

seen to be similar in many ways to those of

a

damped Medhanicai oscillator. In conjunction with experiMental observation,

this

can

be viewed as a theoretical

badk. Up to the current practice

of

treating the Vertical water oscillation in a tube as a

spring pasS

system. Another contribution of the theoretical exercise is that it clearly shows the composition of the excitation forces on the watelr

column. This knowledge was made use of in the study presented

in

Part C.

However, the theory has its

restrictions mainly due to

the'

assumptions usually Associated with linearized hydrodynamics and ttl,

geometrical assumption of small duCt width to depth of .iMtersia4,

because of its two-dimensionality, but it can be a .starting point to tackle the three-dimensional problet or it Can be Used to check the viability of other theoretical methods Which

May.

be developed to

cope

with

the three-dimensionality.

Following

the line of approach aiMed at providing a practical design tool, the ,problem of shipmoonpool was reformUlated.aS.a coupled

Mechanical oscillator in Part C.. Four cross coupling coefficients. were introduced to link the ship's heave motion to the water column oseillatidn And vice versa. The numerical values of these. cOUplihg

coefficients as wU as

the hydrodynaMic coefficients

of the

individual

single degree of

freedom

systems were obtaned by

experimental Methods._ _Eased on this formulation, a .design calculation procedure was put forward and a-set of worked examples

even.

Some of

the experimental results were considered to be unreliable due to inadequate experimental facilities, but the values obtained

were found

to be fairly consistent With the.

findings reported

by

other researchers. It is encouraging, but more data are required tO

firmly

establish these results.

The calculation procedure suggested.

in

this theSia is ample

and.

eaaly adaptable. Once the hydrodynamic coefficients are known, the Simultaneous equations can. be solved without

any

difficulties, and the while process can then be computer programed.

The forMUlation

however,

may

require further

refinement

depending

on how well it agrees with further experiments. The .major

contributions of this part Of research work, therefore, cart be

Summarised as having

suggested

a simple but versatile approach to the compleX. engineering problem and having demonstrated the

use of

experimental techniques to obtainhydrodynamic coefficients of the ship-mconpool coupled system.

AREAS OF FUTURE

RESEARCH-The areas of fututre research

in

this field of practical importance are discussed below.

(a) Solution for Wide Duct Problem

The present theoretical analysis described in Part B oniy

deals with very narrow ducts, but there are Many cases Where

this assumption is not Valid. More importantly it is

-difficult at present to identify the region of width to depth

ratio Where it becomes unacceptable. It: is, therefore, -necessary to tackle the problem from

the. other

eXtremity,

i.e. a wide duct problem.

The

result can be brought side by side

with the

present method and compared to each other neat

1

the

intermediate region. This work would complete the

picture,

So

to speak.

-. () Influence Of MbOnpool On Horizontal ship motioh

It is known that the water in the tOonpoOl dOes oscillate or sloSh in all

SiX

directions, i.e. pp and down, bad k and forth and side to side. The initial work by FUkuda [2] and the practical experience

of diving

Support vessel operatorS

I tUggeSt that the Water oscillation Mentioned above increases

the

resistance of the Ship's fotWard MOVement and has adverse:

effects on coUrse keeping abiltiy. This is an important poin! to consider when designing a vessel with a toonp601,-but

the

Work in this

field is far from complete yet, as no attempt has been made to take into account the effects of surface

waves..

(C)

COnfirmation of Hydrodynamid Coefficients

14

The hydrodynaMic coefficients including the coUP11;4g

4ria not totally conclusive, mainly due to lack of data. More

eXperitental and theoretical

studies

are required to

determine these all iMpottant quantities.

Inclusion of Other Modes of Motion

Although heave motion of. the ship is believed to be. the most important mode influencing the vertical water oscillation

in

the mconpool, the effect of other mode8 of motion need be

investigated at least to confirm that

4.t is

so.

Effects of Various Damping Devices and MoOnpool Geometry

Many

damping devices are being eMployect

in

an :attempt_ to reduce

the

water oscillation in the moonpool at present, e.g.

injection of high pressure air under

the water surface, Installing various forms of baffles,- etc. Their effectiveness, however,

is

not at

all

clear and the many contradicting claiMs confuse the issue considerably. The saMe

can be

said abOut the geometrical arrangements of the

mconpool itself. Some claim the best configuration

tO

be a .trapezdidal. one, while others favour a rectangular,

triangular or circular moOnpool.

As

far as

the vertical water oscillation is concerned, the

effectiveness of these devices and the

geometrical

configurations

can be

investigated by .the experimental

methods

described In

Part

C of

this thesis.

When the

hydrodynam3.c coefficients are obtained through systematic

16

7. MAIN CONCLUSIONS

The main conclusions of this research work are as follows:

To understand the

fundamentals

of the water column oscillation inside a moonpool theoretically, it is esential to idealise it as a two-dimensional system represented by two surface piercing parallel barriers in waves and the results of this treatment showed that the most important parameter is the ratio of the

gap

between the two barriers to the depth of immersion..

By idealising the moonpool as a fixed vertical. tube, it. is possible to employ the knowledge of the spring-mass system to

obtain approximate response characteristics of the vertical oscillation of the water column.

The coupled system of amoonpool on a vessel can be treated

as a two degrees of freedom system involving the water column oscillation and the vessel heave motion.

The hydrodynamic coefficients contained

in the

coupled

equations have been obtained by experimental methods, but additional experimental data are required for design purposes.

Any effective procedure for designing moonpools an vessels

must incorporate factors concerning the Characteristics of water column oscillation in the moonpool and vessel motions

FUNDAMENTAL ASPECTS

CLOSELY SPACED DOUBLE BARRIER WITH ZERO THICKNESS

No.

MATCHED ASYMPTOTIC EXPANSIONS

CLOSELY SPACED FIXED BARRIERS WITH

SHIP-UKE SECTION

THEORY; EXPOIMENT

1

HEAVING BARRIERS WITH SHIP-UKE SECTION

-THEORY, EXPERIMENT

17

FIGA.1 SUMMARY OF APPROACH

EVERIMENT ESTIMATION OF HYDRODYNAMIC COUTICIENTS: MOONPOOL; SHIP; COUPLING TERMS

APPLICATION

SOWTION OF COUPLED EQUATIONS

PROPOSING

DESIGN CALCULATION PROCEDURE SINGLE DEGREE OF FREEDOM

FORMULATION

MOONPOOL : NON-LINEAR OSCILLATOR SHIP: UNEAR OSCILLATOR

TWO DEGREES OF FREEDOM

-FORMULATION

PART B ON 'ME PROPERTIES OF WRIER OSCILLATION

IN TWO-DIMENSIONAL DUCTS

1. INTRODUCTION

It is now known

that

the water Oscillation

in

a moonpool has a peak response phenomenon Which suggests its semblance tO

a siMple

mechanical oscillator system. Fat' such a system it is essential

to

know the

peak response frequency either to detune it from the operational environment as

in

the case of the mbonpool, or to tune Into it as

in the

case of a certain type of wave energy device. Therefore, it is highly desirable to predict the

behaviour of the

water colt= oscillation

In

a

moonpool at the design stage. This requirement for prediction

has

led a few authors (e.g. Madaen

[3]

and AUghes [T]) to represent the system

in

workable fOrtulae. It is

felt, however, that the

approaches used by

theft are rather

over-intuitive and the fUndamental understanding of the systemis still lacking.

Accordingly the linear potential theory is utilised to study the

basic behaviour pattern

of

the ship-moOnpoOl system.

In

1974

Newman

[8]

solved the problem of water oscillation between two

closely

spaced barriers

in waves using

the method of matched

Asymptotic expansions si*elar to Tuck [5], by treating the problem

as a singular perturbation problem. Evan

[8]

extended the solution

to

the

three-dimensional case, i.e. the problem of water oscillation

in

a vertical tube using the 'pipe-end correction factor

given

by Levine and Sdhwinger [9].

References

[6]

and [8] use the basic assumptions that the thickness of the barriers is infinitesiral

and

the

gap

between them is email

compared to the depth of lmmePsion. The latter is acceptable for a large number of cases

but the

former is not for the mcohpool

problem. Accordingly

an

approach simi_lar to these authors is adopted

to investigate the water oscillation between two thick barriers of ship-like section. The barriers are assumed to be fixed in space initially (fixed case) and at

a later

stage

the

heave motion.

restraint is released so that the body, formed by a rigid connection of the two barriers undergoes pure heavng motion (heaving case).

20

Expansion to three-dimensional model using this approadh

in

a,

similar manner as

[8], however, is not Valid

in

this case since there will be considerable croaa-flOwt at the lower exit of the

tube.

The basic

approach and the construction of a tWo=dimensiOrial Model

is discussed in Chapter 2. After Saving the problem of the fixed barriers case

in

Chapter

3,

the solution is developeOrfUrther to incorporate the heave motion of the body in Chapter 4. PVadtideil

applications of the theory thus developed are discussed

in chaptee.

5.

2. BASIC APPROACH TO THE PROBLEM

The actual moonpool problem can be illustrated by the diagram shown in Fig.B.1(a),

and as

it stands it is a daunting task to tackle theoretically. Some simplifications and idealisations, therefore, are necessary and the steps involved are shown in Fig.B.1.

The. problema considered in this Part are represented by Fig.B.1(.d) and (c). As mentioned above, Newman

[6]

solved the

problem

of two

barriers shown in Fig.B.2. This

can

be either a two-ditensionalised representation of

a

vertidal tube or a model simplified one step

further

frdit Fig.8.1(d) by removing

the

thickness of

the

barriers.

We Can therefore regard the problem in Fig.B.2 as

a

special case of

Noting its similarity to -the present. problem, we adopt a

SlmilAr

approach to that of

[6]

and [8] and the method of tackling the fiXed barriers case.(Fig.B.1(d)) is summarised below:

(i) The flow field i8 divided into two regions, i.e. the inner region comprising

or

the local flow near the lower exit Of the moonpool and the Channel-like flow near the free surface within the moOnpool; and the outer region covering

the wave

field exterior to the barrierse

ii) The outer problem i8 Staved With

the

assumption

that the two barriers are

eSsentiAlly

.collapsed into one

and

consequently

the outer

contour of the barriers represents a solid boundary Of the Ship Section. The MBSS flow into and out

of the moonP061

i8

then represented by A source

singularity placed Where the exit of the MoOnpdol

S.

Then the velocity potential

in this

region

is

the linear

addition of the incident wave,

potential,

diffraction potential

and

the potential due to the SoUrce. The litret of this compound velocity potential near the exit of the MoonpOol is taken as the inner expansion of the outer

solution.

(iii)

The inner problem is solved with the chwarz,JChriatoffel

transformation and the asymptotic expression of this

Implicit solution for the region near the lower exit of the duct is taken as the outer eXpansiOn of the

inner

_solution.

.(iv) The inner expansion Of the outer solution is _{then matched} - to the outer expansion of the inner solution.

(V) With the solution obtained in (iv) the

behaviour of the

water column oscillation

in

the duct

is

evaluated in the form of. the response amplitude operators, which are defined at the ratios of the oscillation amplitude to the _incident wave amplitude.

1

The heaving barrier problem is shown in Fig.B.1(c), and essentially it is a similar probleth as the fixed barrier case,

_{but it}

_differs

I

from the

latter in that the heaving Motion Of the ship is expected to be influenced. by the source singularity placed at the exit of he

duct, and that the water oscillation is affected by the heave

potential.

Therefore, the approach employed for the fixed case is

modified to stilt the present problem as follows:

IntrodUce an

unknown

complex heave amplitude to

deal with

the influence of the heave

potential.

The inner soldtion- will now be

the one

obtained

in the

fixed da8a plus another term to satisfy the body boundary cOndition dUe to the heave motion of the hull.

The

velocity potential of the outer solution is the linear summation of incident waVe potential, diffradtion potential,

heave potential

And source potential.

22

solutions In the

overlap domain and

the heave motion equation of the body.

With these modifications the .behaviour of the water column oscillation relative to a fixed datum and the

heaving

barriers can be estimated. We begin, however, by tackling the fixed barrier

24

WATER COLUMN OSCILLATION BETWEEN TWO FIXED BARRIERS WITH THICKNESS

3.1 FOrmulation of the Problem

The

geometry

of the two-dimensional moonpool model is

identical

t0 that of an infinitely

long

duct

fOrmed by

two barriers Of

a

Ship-like tectiOn. We, therefore, represent the prOblem as a doub. 0 barrier problet.

A cartesian coordinate system (x,y) is Chosen with the

origin

at

the

undisturbed free

surface On the

centreline of

the

dUct With y JoOsitive vertically

doWnwarda.

Tha barriers are.hIed in space

and

the width of the duct is 2b, While the depth of immersion is a

(See

AlUA3.3)..

The

depth

Of Water is infinite and plane progressive waves of frequency (P/21t

are

normally

incident

fPam x = 140 upon the

two-dimensional Ship hull. The Usual astumptiOns of lineariaed ideal_ fluid and ittotational flow are made and the time,-dependence of tbe velocity potential is expressed by

(xiY it)

=- eec.4x,y)

eg'xt]

The spatial-.dependence 4'(x y) satisfies the following

set Of

conditions.:

v*=

0 for y> 0 external to the body-Winds*.

K+

_+a+/a7

=-0 on y = O.

alb/CRI = 0 an the by boundary.

- 0 as y

where i =

K = ci/g

g = gravitational acceleration R = reflectidn coefficient T = transmission coefficient

normal vector positive outward from the body. A solution

or

this boundary Value problem is

sought

with the

assumptions

on the

geometry Of the problem, i.e.

b/a = e << 1 and Ka = 0(1),

Which means

that the width of the duct is small relative to the draught of the ship a and wave length .= 27r/K.

The flow field is now divided into two regions.

The

outer region for (Kx,Ky) = 0(1) and the inner region for x/b = 0(1), 0 < y "<d; a are

considered separately.

3.2 Solution of

the outer

Problem

The outer problem can be best illustrated by an analogy to a giant observer far away from the eXit of the duct (see

Tack.

[5]). Fbr him

the two barriers will look effectively collapsed into one and will then represent a Ship section with

a

solid boundary.. The

outgoing

ftow

of the duct is represented by a source

singularity

placed

Where

the

exit

of

the

duct is Situated as shown in Fig.B.4. This treatment is somewhat similar to the approach adopted by

Taylor [10].

The .body is fiXed

in

space, and the time-independent part of the Velocity potential can be expressed by

cp(x,y) = t+430+ 2mG (5. 1)

where4k=

incident wave potential

!there

26

cPt, = diffraction potential

m = sOurce

strength

to be determined later G = Green's.fUnction.

Let

0, =

ctti+4s,, and r =

Cia +

(y=a)43ra Green's

function

G(x,y;a) for deep Water is giVen by Wehausen and Laitone [111 in

the

form NA0HougsAgx , G(x,Y;a) =

_44t.'"°

X4+(1%:/;°: 1 1-

e

, 4.2

r

Jtv-d<

-k (y+°)

y+a)

-se. COS

k

X ,

enotes Cauchy PrinciPal value integral.

If r-r0, then X 0 and y a, and the asymptotic expression of as r becoires Ocx,y ; a 11%."0 m

-3--

"wa 0

-K

KA

.ie

0((r/04)

)

ak

For

any

given

wave

frequency

and body

contour

regarded

as a constant and We let

Do=

I'm

n-ipo 4).

The inner expansion of the outer solution,

then, is

11'11

cx y) =

D. +

_27c

(L)

ftze±-

A

ime2K4

0 ((

ria)11Z )

via

.

r-ve

This

completes

the solution

of the outer region. with the source strengthm Still. undetermined. .

3.3

The Solution of the Inner Problem

The geometry of the inner region is shown in Fig.B.5(a) and, .since, the flow will be syMmetridal about the y-agiS, we need consider onlil

(32)

be

the

eight

half of the 2-plane with the coordinate system as shown

in

Fig.B.5(b).

In order to find the exact solution of

the flow

in this

region bounded by the solid walls of Fig.B.5(b), the technique of

Schwarz-Christoffel transformation is utilised. The flow is

described by the implicit solution of the form (see Appendix I).

z =

b[Z

cexp cow +p3 + Pf'e

4 ( dm/

_+fS

)

-2

OcjitCexPCom+P)

+I j vai (E3. 3)

where j.

rlf.

and does not interact with i

CI and

p

are constants real relative to

w. is the complex potential of the form (

4, +

Sufficiently far above the lower exit the asymptotic expression of

equation (B.3) is

x + gay)

i4r ci)

By redefining

p

2 the term (2 2log2) can be absorbed by

p

and

separating the real

and

imaginary parts,

7-=

(Y-a)

C5-4)

On the free surface within the

duct

equation (B.4) should satisfy the free surface condition, i.e.

+ a+,/ay

= 0 on y-=

o.

Substituting (.B.4)

into (B.5)

gives

p =

+

it

with d'0.

In the

overlap dathain b << r <<

a,

where r =

Ix + j(ay)1._,

asymptotic : expression of equation (B.3).

is

+ 2 2log2)

(B. S)

28

x

14) [exP( clw +p)

+

nY2.

Hence

zvi; + 0 ((

b/r)2

This completes the solution for the inner region With cL still undetermined.

3.4 Solution 1:m Matching

Now the

inner expansion of the outer solution is matched to the outer expension of the inner

solution in the overlap &main tO

obtain

the two

unknowns cl and m. The outer limit of the inner

'solution is given by (3.7) and the inner limit of the outer solution by equation (B.2). The outer limit of the inner solution, then it

expressed: by

the

outer solution perturbation quantity r/a after Van Dyke [4] as

a?

jaa

-ar

'"93

with the order term omitted.

Equation (B.8) can now be readily matched to equation (B.2) to give

m

/

211c

= 2 /d

or

= LC/

m

and _CB-9.) otr, _2kck

a

clic

k

ly Do

f

di?

(8. °)

From (B.6), (B.9) and (B.10) ;ea

M =

[

-a

log

aft

+ ;71h5 (a -

1/K) +

otk

+ ie

-2"

_{CB. 10.}

With

p

from (B.6), Cl from (B.9) and M frOM (S.11) the solution is

at

Using this solution, the response Characteristics of the water coll.= oscillation in the duct is examined in the next section.

3.5

Estimation of Water Column Response

Our Main interest here is the amplitude of water column oscillation and accordingly the Phase angle part of the transfer function is Ignored. Since we are dealing with a linearised prOblem, this can best be approached by estimating the ratio of response amplitude to the incident wave amplitude at various wave, frequencies.

From the free surface condition

K

+ ativa

y = 0 at y 0

Therefore

=

(x,o)

Thus the amplitude of the wave is

A -/c-41;

14Vx,o)

The velocity potential near y = 0 inside the duct

is

given by

equation (B.4)

...yc -a)

_b

As8Uting a flat water. surface in the duct, the amplitude of the

water column oscillation is Obtained in similar manner as

A =

_14,(0,0)1

Therefore the response amplitude operator M is

4 (0,0)1

M

If the

amplitude of the incident wave potential at y = 0 is unity, i.e. cl)x,(x,0) e-iKx

= 14)(0,0)

I

_{= rett)}

la%(5

We now consider eqUation (B.11) in order to evaluate

DO

213:10()+

(1I-TD)

e-Ky

e-iKx

but we are interested

in the

region

x

-0 And y -*a. Therefore

.ri.oie-14c4

OD is evaluated numerically from the relationShip

_ 4 *111on

Zs

-an

- 04

where

Se denotes the body Silt:face.

. As shown in Fig.B.6, this constitutes a boundary value probIet,

and

it is solved by the source distribution method closely following the work of Frank [12].

A computer program

MS written

to evaluate the response amplitude

operator of the water oscillation and

a

numerical Study

was

perforted

for

two typical section shape8 shOWn in Fig.B.7..lhe twO sections shown as Fig.B.7(b) and (c)

represent the

same section shApe but of different draughts. Fbr the reasons discussed below this rectangular section with

rounded bilge was

chosen for the expetitental study.

30

I

irI

'I

= (8.12)

oticb _4)(b

the

form

of R.A.O. of the water oscillation plotted against Ka. For the purpose of comparison the Characteristics of the Water oscillation between two barriers of zero thickness are

given in

Fig.B.1I.

These were

evaluated using the

solution given in

Newman [6].

Experimental Verification

The two-dimensional fixed mconpool problem has been theoretically

investigated and as -a result the vertical water .column oscillation

in Atwo-dimensional moonpool.can be predicted for a ship fixed in

space. The qualitatiVe agreement with Newman's resUlts, as can be

seen from

Figs

8.10 and B.11, gives some credibility to the present solution.

It was decided, however, to perform a series of experiments aimed at achieving the following objectives:

validate the theoretical results;

find the limitations of the theoretical analysis.

(a) Description of the Model

The basic configuration of the model used in this experiment

is Shown in Fig.B.12. Due to the cross-sectional shape of the mpdel the diffraction of the incident waves was expected to be considerable. However, the

inner

region geometry of the

.theoretical ahalysis Was assumed to

have. a 110 bottom

extenoarig to a reasonable distance,

and this

is

the main

reason for Selecting such a 'body Shape. The experienceof experiments with fixed vertical tUbes has .shown that an accurately identifiable

peak

does not occur below the depth of immersion

of

approximately 15 am. Therefore, it was decided that the draught of the 'Model should be at least 20 an for

this experiment.

towing

tank

/ma

the

main

dimensions

32

25 mL x 1.5 mW x 0.8 mD and the model is _{to be placed across} the tank _{to maintain the wo-dimensionality. Thus the length} of the model was made to be 1.3 m which leaves _{a 10 am gap} each side between the tank wall and the model.

Since the result of the theoretical study suggested the ratio b/a to be the most important

_{parameter governing the peak}

response frequency, the width of the duct was varied as shown In Table B.1. The effect of the

depth of immersion was

investigated by testing for two draughts of 20 an and 25 an.

The

immersed cross sectional shapes when normalised relative to the draughts, are identical to those given in Fig.B.7(b) and (c).

(b) Instruments Used and the Experimental Procedure

Once the duct was set to a desired width and the model placed

and fixed

to two cross bars across the tank at the first draught (20 am), the simple harmonic long

crested waves of

frequency ranging from

0.4

to 1.2 HZ were generated by the wave maker.

In less than one minute after

starting a wave of a given.

frequency standing waves were quickly established between the model and the wavemaker due to the large reflection of the incident wave.

For this

reason the input wave height was

measured as near to the hull as practicality would allow.

_The

resulting water oscillation and

the

input wave profile were measured and recorded on a three channel pen recorder. The wave in the tank was allowed to die down and the water surface calm again before the next frequency wave was generated. The model was then adjusted to the second draught (25 am) and the

whole process repeated before the duct was adjusted to a new width.

For the description of towing

tank

and instrumentation system see Appendix III.

(c) Results Obtained and Comparisons with the Theory

The results of the experiment are presented in Figs B.13 and

B.14. By comparing these to Figs B.9 and .B.10, the trend of

the

peak

response frequency obtained from the experiment is

seen to agree with those from the theoretical analysis. The

peak

response magnification factor, however, dhows an opposite trend, i.e. the theory predicts it would decrease as b/a increases, but the experimental results show otherwise, albeit In a much smaller scale. The discrepancy is due to following factors:

experimental error arising from severe distortion of

waves caused by a relatively large model Obstructing the

now field;

the theory considers

only wave

radiation losses as damping but the

peak

response magnification factor is extremely sensitive to damping value. It is worth noting that the theory predicts a virtually infinite response

for small b/a ratios, but it is

hirely

unlikely in

reality.

It is very difficult to determine in what manner these factors

would affect the result, and therefore the information concerning the magnitude of peak response should be treated with caution. Finally, we can conclude from these results that

the b/a ratio is amain parameter influencing the response Characteristics of the water column.

34

4. WATER COLUMN OSCILLATION BETWEEN TWO HEAVING BARRIERS WTEH THICKNESS

4.1 F6rmUlation of the Problem

Following the conventions Of the previous chapter, a cartesian

coordinate system is chosen with its origin at the undisturbed free surface on the centreline of the duct

with y positive vertically

downwards. The two barriers are rigidly connected to each other and are allowed to heave only. The width of the duct is 2b while the depth of immerSion is a (see Fig.B.15).

It it assumed that the

depth of water

is infinite and plane progressive waves of frequency (.0/27C are normally incident from

= +co

upon the tWo-diMenSiOnal Ship hull. The usual assumptions

of

lineariSed

ideal

fluid

and

irrotational flow are made and the

time-del6endence is assumed to be harmonic SO that, for

instance,

the Velocity

potential

is written. as

) =

ReCCx,y)e'

3.

The spatial-dependence 1?(x,y) satisfies the following

set of

conditions:

174+= 0 for y > 0 externel to the body boandary.

10-1, a+/1Y =.0 on

(14) a+Ain =

-icoAdose on the body boundary.

-.0 as y

(v) 40(;

y) e-KY (6-1Kx' + Rei" ), x 4.1ms.

ct(x,y)

,

-ca.

g = gravitational acceleration R = reflection coefficient T = transmission coefficient A = heave amplitude of the body

e

= the angle between the vertical axis and the normal outward vector of the body surface R, T and A can be complex to represent

phase relationships.

A solution of this

boundary value problem is

sought

under the

further assumptions on the geometry of the problem that:

b/a

= 6

<< 1 and Ka = 0(1),

,which means that the width of the duct is small relative to the draught of the barriers a and the wave length ).= 2n/K.

The

flow field is now divided into two regions to be considered . separately: the outer region for (Kx,Ky) = 0(1) and the inner region for x/b = 0(1), 0 < y a.

4.2 The Solution of the Inner Region

The geometry of the inner region is Shown in Fig.B.16(a) with the

cooranate system as shown in Fig03.16(b). We consider the right half of the z-plane, since the flow

is

symmetrical about y-axis.

Fbr

the

problem with fixed barriers, the line y = a, x > b is part of

a

Streamline and

thue-acOiy_=

0 an y = a, X > b. However, this line is no longer

a

streamline

in

the heaving case. Suppose the unknown complex heave amplitude of the barriers is A, then ..4) must satisfy

= 4)1- iwAY,

+,satlzfies the Same

prOblem

as

4, in

the fixed barrier case

The

asymptotiC

ekpressions of 47, are found in the previous chapter

and

in Appendix I:

4)1 al

JE-(y-a) _

36

near y = 0 for

Ix( <

b

where d and p are constants

r =, .

'Hence near y = 0,

for

[xl < b, we have

IC4)

34) /ay = 0 on y = O.

Substituting (B.13) into (B.14) gives

p =

- a) + icaccA/K

The outer Limit of the inner solution is

(1)

log tk- ,

-

iwAy.

4.3 The Solution of the Outer Problem

This mutt satisfy the free surface condition within the duct, i.e.

(8.15)

(5.16)

This cpmpletes the solution for the inner region with pt,

p

and

A, still Undetermined, although equation (B.15) gives one telationShip between them.

ft

log for b<<r<<a, y a,

In the outer region, as 6 .0, the two barriers are effectively collapsed into one and the barriers then represent a. ship section with a solid boundary. The flow into and out of the duct is then represented by a source, placed at (0,a), of strengthmt Which is to be determined. This is shown in Fig.B.17.

The Section is allowed to heave only, and the spatial dependence of the potential is expressed by

(x,Y) = +

A tit 4- m t's

(B. 17)

Where indident wave potential

46-

diffraction potential

A.= unknown heave .amplitude of the barriers $6= heave potential

for

Unit aMPlitude of heave

= unknown source strength

4k= unit

source Potential

The

incident waVe

potential is given in the form

and the four potentials of the

ral.t.

of equation (B.17) Should satisfy the free Surface condition and Laplace't equation.

cp., $4

and +5 should also satisfy the radiation condition. In addition to these conditionS the following bOdy bOtIndary conditions apply:

(a) Diffraction Potential

(Ii)

The diffraction potential

is

the correction to the incident potential required if the ship were fixed in space and the source absent (i.e. the duct sealed). It must satisfy'

341)/an =

-a4/an

on Se (8.19)

where is the northal Vector positive outwards

from the

body.

Heave Potential

(+0)

The heave potential represents the flow generated by the unit amplitude heave motion of the barriers with the source absent

(i.e. the

duct

'Sealed). It Must satitfy

-icoAdose on Ss

where 9 is as given in §4.1.

Source Potential

The sourdepotential represents the flow due to

a

unit source

at'

(0,e) in the

presence Of the

ship fixed at its

mean

position. We

require

that

ciog P, as

e

+

(y-a)Z 0 with y a.

The potential in y> 0 of a source at (0,a) of unit strength satisfying the free strface condition throughout y- 0

Laplace'S equation and the radiation condition is

just the

well-knOwel Green's function (see Wehausen and Laitone [11]):

G(x,y;4) = log

x:4-- 1=11._2.ele"4"Y440,

'-X cy+a

)

Z

Je A

-K

obakx dk 38. = G +51

-W7+4)

e

Kx

We write

Cs. al)

and compute the correction +51

in order

to satisfy the

aVan

= 0 on

Ss.

Hence fsomist satisfy the boundary condition

a40/..a n = 0 - a

G/

a

n on Sp.

c5.23)

The explicit form of (B.23) is given in

Appendix II.

These

three boundary value problems, as suMmatised in Figs B.18.-20, can be solved by various numerical methods and here- we follow

Ftank's close fit source distribution method [12].

Fbr the purpose of matching we require the leading asymptotic forms of the four potentials as r 0, and thus we obtain

(x f.-xiog( ) - dic

-+ Um (

cfxj_ + 4) + A cis, + m4,51)

rwe

+. 0((r/0), as r 0

which is the inner expansion of the outer solution,.

4.4 Matching

We now Consider the .matching between the inner expansion of the

outer solution and the outer expansion of the inner solution in the overlap daMain in order to obtain two 'equations for the unknown

quantities.

In the overlap domaih b<<r<<a, the asymptotic expression for the

velocity potential in the inner region is given by equation (B.16):

log()

-

Jo) Ay.

Cr

CB . 24)

dp 14-*

Introducing the outer perturbation quantity

r/a and noting

that

40

r log e-Tx +

With the order term omitted.

Equation

(B.25) is now matched to

equation (B.24). By matdhing

0(log r) terMs

_2 en

air

and by

matching 0(1) terms we

obtain

2

az

log -6-

a- -

icaAa

.-ak4

= De

+

14

A +ti curt 13 ia)Aa where D, ( 42z. p )

We thus bye equations

(B.15),

(5.26)

and (B.27)

for the

four

unknowns.

4.5

Heave Problem

One further-eqUatiOn is required for a

closed

System and this comes':

from the equation of motion for the heaving of the barriers:

(4811iMe+

AN.*

+ Fs + Fo + mF5 .

Solving this

equation

for A gives

A = CI

+ Cm

where

C, = -(P1

+Pp)/(46+

FM -

Fi

Ca =

-Fs

AdMi+

FM 130)

Ma = gess of the barriers

FH

,

etc. = force due to

etc.

Re = heave restoration.

(9. 25)

(6. 29)

(8.26)

-

ime-2"

_{34;110 451} (13.2-7)

_

The forces

FH ,

etc.

can be estimated

by using

the linearized

Bernoulli's equation,

thus

F(4),t)

=r1;1(X,5t;t)ftdS6 Se aE(X/Y;

y

i

55s

at

_ Ase,

p

where p = mass density of the fluid

5

= normal outward vector on the body surface P(x,y;t) = time-dependent pressure at (x-,Y) St = body contour.

Hence the vertical component of the complex-force amplitude is

F(ca) =

f p(x

,y) cos dS

= - 463 fi(x ,y)

wee dSe

.(B-31)

'where p(x,y) = spatialrdependent pressure at (x,y)

The velocity potentials are given by

Saving

the boundary value:

problems of

the

previous section, and therefore equation (B.29) can be evaluated to give a relationship between A and m.

We thus have four equations (B.15), (B. 26), (B.27) and (B.29) for

the four unknowns d, p, A and t. Solving

in

particular for m, we

have

(

S. 30)

m = CD0 + Di -

-a)]

x

EakiogV -(A

-

1A)c2)(iE-

a) - Da + J.J7S2±Ldk +. le-2" lim

r

7C A-14 0-i+0

;!

0 where Di = C,lizi+pi Da = CalAmeism

(a. 32-)

This completes

the

solution and We can proceed to calculate the response of the water column oscillation in the duct.

42

1L6 Estimation of Water Column Response

With the complete solution

obtained

above .the behaviour of

the.

Water

column oscillation,

in the form

of a response amplitude operator,

iS

now inVettigated.

FrOt the viewpoint

of moonpo01 applications the vertical Water oacillatiOn relative to

the ship is of more interest

tO us, but it LS- Convenient first tO consider

the oscillation

relative to the fixed Coordinate system.

P. the

free surface condition

K

4): ÷ tiVay =

0 on y 0,

We have

Ott/ay )yzio =

-K cf,L(x,0).

Thus the

amplitude of the incident

wave is

Aw

=1A144x,0)1

.

The water column potential near y = 0

is given by

equation (B.13) as

/c

P

dip

(Sra)

4-i4

Ay.

The amplitude of the

Water

column oscillation

is

Obtained

In

similar

manner as

Jo

,o

1 .

Therefore, the response amplitude operator

for

the

absolute oscillation

113(9,0.)1 1 Ick(xi0) .

MA = I(0,0)1

=

-= 17;A

ic"/Ki

(5.33)

With m and A given by (B..32) and (B.29) respectively.

Now we turn to the water oscillation relative to the

heaving

barriers. The heave amplitude A is now known and the time-dependence of the heave motion can be described in the form

ytt

= Ae-t

.,7he velocity potential of the water column near y p is given by

(B.13)

from which we derive the position of the free surface

in

the duct at time t to be

tnt

i

+ A \

b ch,)

)

The Water oscillation relative to the heave motion of the barriers -is thUS

_im

yo

-e

.

The response amplitude operator of the relative water oscillation

14k, then is

m I / 41,1

A computer program was written to compute the R.A.O. of the water column oscillation based an

the

solutions Obtained above,

With the

aid of the

program a numerical study

was

performed for the three

cross sectional

Shapes of the

barriers as described

in h3.5 and

Shown In Fig.B.7.

The latter two sections

are

the shape of

an

identical model used for the experimental study described in the next section with different draughts and normalised relative to the draughts... The resUlts of the numerical .study are given in Figa B.21

'

44

23.

The

_{R.A.O.'s-of the Water oscillation relative to the fixed} coordinate systel are plotted

against

Ka and shown as

graphs (a)

and

the

same

relative to the heaving barriers are presented as

graphs (b).

_{FU.B.24 shows the influence of the water oscillation in}

the _{duct on the heave motion of the barriers for the case of the}

semi-circUlar cylinder.

4.7 Ekperimental Verification

(a) Description of.Experitent

The results of

the

_{theoretical analysis on the} heaving moonpool

problem

_{discussed above are}

series

of experiments Wing the same model

.3.6. The mpdel, however, was allOWed to pure resulting motion was

measured With an

arrangement

of

the instrumentation system

Appendix III.

two-dimensional Checked

by

a

as described

in

heave and

the

For

the

eftployed see

The model heave guide medhanism is shown in Fig.B.-25 and the

experimental setup

is

Similar to that shown in Photos B.1

_and

The rest of the instrumentation system is exactly,

identical to that used for the experiments

with the fixed

barriers. The combinations of the parameters tested are Shown, In Table B.2.

(b) Results

Obtained and Comparison

with

Theory

The experimental results are summarised in Figs B.26 and B.2 .

It may be

noticed

that these graphs do not show a well-defined pattern. This was due to the severe distortion of the waves near the Model caused by the

_heaving

motion of a relatively

large

model

in

a small

tank, thus introducing appreciable

experimental error.

those fram the fixed barrier experiment, we can conclude that bOth

dapping

and added mass are dependent upon b/a ratio.