New concept

LNG marine terminal

Hydrodynamics of a LNG carrier
Behind a detached Breakwater

Thesis report
Wieke Wuisman
Delft, February 2005
I

Preface
The final section of the Masters study program at Delft University of Technology
consists of an individual thesis on a subject related to the students specialization. This
report is an overview of the results of an investigation on the thesis subject: New
concept LNG Marine terminal; Hydrodynamics of a LNG carrier behind a detached
breakwater. The research objective is the development of a simulation model that
predicts the response of a moored LNG carrier in an exposed LNG marine terminal, as
a function of the breakwater configuration and coast.
During this investigation a graduation committee, consisting of the following persons,
supervised all stages of the thesis work, for which I am very grateful; Prof.ir. H.
Ligteringen (head of the committee), Dr.ir. A.J.H.M. Reniers, Ir. L. Groenewegen and
Ir. W. van der Molen.
Especially Wim van der Molen I would like to thank for his daily supervision and
support during the past months. Due to his clear explanations on the subject and our
coorporation, this thesis has been a very constructive and educational process.
I also would like to thank Prof.dr.ir. J.A. Pinkster for the DELFRAC simulation results.
Besides the graduation committee I would like to thank all those who have donated
their time, energy and have given me advice, my family, and friends.
Delft, 24th of February 2005
Wieke Wuisman

II

Abstract

As the worldwide gas market continues to grow and environmental concerns with
respect to in-port unloading of gas have increased, there has been a boom of interest
in new liquefied natural gas (LNG) import terminals in the past five years. For these
terminals, which are more and more located in areas with hostile sea conditions,
dedicated provisions are required to create sufficient shelter for the carriers. Proposals
have been made to construct a marginal low crested breakwater parallel to the coast
protecting a ship moored at a jetty close to the shore. For an optimal economic design
of such an LNG marine terminal, the dimensions and orientation of the detached
breakwater have to be optimized as a function of the weather related downtime of the
moored LNG carrier. Doing so requires adequate simulation tools. However, for the
combination of wave and ship motion, a link between an efficient wave simulation tool
and a program for ship response calculations is not available at present.
The research on ship behaviour has resulted in the development of various so-called
six degrees of freedom (SDF) computer programs. These programs solve the
equations of motion of a moored vessel for all six degrees of freedom. As a
consequence of the non-linear characteristics of the mooring system the equation of
motion is solved in the time domain. The wave force time series are calculated from a
homogeneous wave field of irregular, long-crested waves. In case of an open jetty
configuration these assumptions are valid. However, considering a carrier behind a
detached breakwater, the wave field is not homogeneous, but the wave height varies
over the ship length. Consequently the influence of the detached breakwater on the
ship motions must be considered. In addition, the reflection of the waves at the coast
also has to be taken into account.
This thesis describes a methodology to predict the hydrodynamics of a moored LNG
carrier behind a detached breakwater. A rapid assessment tool has been developed in
order to assess the optimum breakwater dimensions in the preliminary design stage of
an LNG marine terminal. In particular the effects of the breakwater dimensions on the
hydrodynamic behaviour of the moored LNG carrier are considered. The
computational approach for the calculation of ship motions from a given offshore wave
field is described. In addition results are presented for different terminal layouts.

Contents
LIST OF FIGURES ..................................................................................... I
LIST OF TABLES .................................................................................... III
LIST OF SYMBOLS ................................................................................. IV
1

INTRODUCTION ................................................................................. 1
1.1
1.2

SCOPE OF WORK.............................................................................. 5
2.1
2.2
2.3
2.4

PROBLEM CONTEXT ...............................................................................5

RESEARCH OBJECTIVE ...........................................................................5
PROBLEM APPROACH.............................................................................5
REPORT SET-UP ....................................................................................7

WAVE DESCRIPTION ........................................................................ 9

3.1
3.2

LNG TRANSPORT ..................................................................................1

NEW CONCEPT ......................................................................................2

SPECTRA ..............................................................................................9
SURFACE ELEVATION .............................................................................9

WAVE TRANSFORMATION PHENOMENA..................................... 11

4.1
4.2

INTRODUCTION ....................................................................................11
DIFFRACTION ......................................................................................12

4.2.1

Diffraction: definition of the phenomenon....................................................... 12

4.2.2
4.2.3

4.3

Diffraction solution of Penny and Price .......................................................... 13

Transformation of wave parameters by diffraction ......................................... 15
REFLECTION .......................................................................................17

4.3.1
4.3.2

Reflection: definition of the phenomenon....................................................... 17

Reflection from beaches ............................................................................... 18

4.3.3

4.4

Transformation of wave parameters by reflection .......................................... 19

TRANSMISSION....................................................................................21

4.4.1

Transmission: definition of the phenomenon.................................................. 21

4.4.2

Transmission studies .................................................................................... 22

4.4.3
4.4.4

Spectral change by wave transmission.......................................................... 26

Implementation in WSP................................................................................. 26

4.4.5

Transformation of wave parameters by transmission ..................................... 27

LNG CARRIER DYNAMICS.............................................................. 29

5.1

EQUATIONS OF MOTION ........................................................................29

5.1.1

Solving the equations of motion by SDF simulation ....................................... 31

HYDRODYNAMIC LOADS ............................................................... 35

6.1
6.2

INTRODUCTION ....................................................................................35
STRIP THEORY METHOD ......................................................................36

6.2.1

6.3

6.3.1

First order wave forces.................................................................................. 39

6.3.2

Second order wave forces............................................................................. 39

6.4
7

Relative motion approach.............................................................................. 37

W AVE FORCES ....................................................................................38

TOTAL WAVE FORCE ............................................................................44

WAVE SIMULATION PROGRAM ..................................................... 45

7.1

MODEL SET-UP OF THE W AVE SIMULATION PROGRAM ............................45

7.1.1
7.1.2

Requirements of WSP................................................................................... 45
Schematization of the model ......................................................................... 46

7.1.3

7.2

7.2.1

Introduction................................................................................................... 50

7.2.2

Input Files ..................................................................................................... 50

7.2.3
7.2.4

Simulation Module ........................................................................................ 51

Output data................................................................................................... 52

7.2.5

Application of WSP calculations in TERMSIM ............................................... 53

7.2.6

Ship response assessment tool..................................................................... 53

PROGRAM VALIDATION ................................................................. 54

8.1
8.2

INTRODUCTION ....................................................................................54
VALIDATION TRANSFORMATION OF WAVE PARAMETERS ...........................55

8.2.1
8.2.2

Diffraction ..................................................................................................... 55
Reflection ..................................................................................................... 57

8.2.3

Transmission ................................................................................................ 61

8.3

VALIDATION OF WAVE FORCES ..............................................................64

8.3.1

Introduction................................................................................................... 64

8.3.2

Vessel characteristics ................................................................................... 64

8.3.3
8.3.4

First order Transfer Functions ....................................................................... 65

Comparison of the wave force spectra .......................................................... 67

8.3.5

Conclusion validation of wave forces............................................................. 70

8.4

Coordinate system conventions .................................................................... 47

COMPUTER SIMULATION PROGRAM .......................................................49

APPLICATION OF EXTERNAL WAVE FORCES IN TERMSIM .......................71

8.4.1

Introduction................................................................................................... 71

8.4.2

Starting-up time ............................................................................................ 71

8.4.3
8.4.4

Duration........................................................................................................ 71
Simulation runs ............................................................................................. 72

8.4.5

Conclusions external input of wave forces in TERMSIM ................................ 73

INFLUENCE OF BREAKWATER DIMENSIONS ON SHIP MOTIONS75

9.1

INTRODUCTION ....................................................................................75

II

9.2

SIMULATION RUNS ...............................................................................75

10 CONCLUSIONS AND RECOMMENDATIONS ................................. 79

10.1

DEVELOPMENT OF THE W AVE SIMULATION PROGRAM .........................79

10.1.1

Near-shore wave parameters ........................................................................ 79

10.1.2

Wave forces.................................................................................................. 80

10.1.3
10.1.4

Application WSP in TERMSIM ...................................................................... 81

Main conclusions and recommendations development of WSP ..................... 82
SHIP MOTION DEPENDENCY ON THE BREAKWATER CONFIGURATION AND

10.2

COASTAL CHARACTERISTICS ..........................................................................82

III

List of Figures
Figure 1-1: World LNG trade development ........................................................................... 1
Figure 1-2: Schematisation of a LNG marine terminal concept [not to scale]......................... 2
Figure 1-3: Aspects of a LNG marine terminal ...................................................................... 3
Figure 2-1: Wave Simulation Program.................................................................................. 6
Figure 2-2: Wave force determination .................................................................................. 6
Figure 2-3: Optimization of the breakwater configuration ...................................................... 7
Figure 3-1: Frequency-directional wave component ............................................................. 9
Figure 4-1: Diffraction around a semi-infinite rigid breakwater............................................. 12
Figure 4-2: Derivation of direction ...................................................................................... 16
Figure 4-3: Reflected wave propagation [ref 4] ................................................................... 17
Figure 4-4: Reflected wave parameters.............................................................................. 19
Figure 4-5: Parameters influencing transmission ................................................................ 21
Figure 4-6: Influence Rc/Hs on transmission coefficient Kt ................................................... 22
Figure 5-1: Six degrees of freedom .................................................................................... 29
Figure 5-2: Flow diagram SDF-program TERMSIM ............................................................ 33
Figure 6-1: Wave phenomena and resulting wave forces ................................................... 35
Figure 6-2: Segmented floating body.................................................................................. 36
Figure 6-3: Wave set down ................................................................................................ 41
*

Figure 6-4: Determination added mass coefficient using ................................................ 43

Figure 7-1: Model set-up Wave Simulation Program........................................................... 45
Figure 7-3: Schematization diffraction, reflection and transmission ..................................... 47
Figure 7-4: Earth bound coordinate system ........................................................................ 48
Figure 7-5: Ship-bound coordinate system ......................................................................... 48
Figure 7-6: Breakwater bound coordinate system............................................................... 49
Figure 7-7: Integration coordinate systems......................................................................... 49
Figure 7-8: Flow scheme WSP........................................................................................... 50
Figure 7-9: Flow scheme computer program WSP ............................................................. 51
Figure 7-10: Flow diagram modified TERMSIM .................................................................. 53
Figure 8-1: Validation process............................................................................................ 54
Figure 8-2: Diffraction 90o after Wiegel
Figure 8-3: Diffraction 90o WSP ....................... 55
o

Figure 8-4: Diffraction 45 WSP

Figure 8-5: Diffraction 135 WSP .............. 55

o

Figure 8-6: Diffraction around a detached breakwater, wave incidence 45 and 135 .......... 56
Figure 8-7: Reflection at the coast, semi-infinite breakwater, wave incidence 45o ............... 57
o

Figure 8-8: Reflection at the coast, detached breakwater, wave incidence 45 ................... 57
Figure 8-9: Project site and output locations breakwater in Limbe....................................... 59
Figure 8-10: Diffraction patterns run 02, wave incidence o = 90o, peak period Tp = 10 s .... 59
Figure 8-11: Parameters transmission................................................................................ 61

Figure 8-12: Validation Rc/Hs distribution d'Angremond formula for varying permeability
coefficients ................................................................................................................. 62
Figure 8-13: Kt vs incoming wave direction o.................................................................... 62
o

Figure 8-14: Comparison surge, heave and pitch with FTF, wave incidence o = 0 . ........... 66
o

Figure 8-15: Comparison surge, sway and yaw with FTF, wave incidence o = 45 ............. 66
Figure 8-16: Comparison surge, sway and yaw with FTF, wave incidence o = 90o............. 66
o

Figure 8-17: Wave force spectra, surge, sway, roll and yaw, o = 45 , Tp = 13 s ................. 69
Figure 8-18: Validation scheme of external input wave forces in TERMSIM ........................ 72
Figure 8-19: Re-introduction wave force time series in TERMSIM, for surge and sway ....... 73
Figure 9-1: First order transfer functions, 0 = 45; (- - -) no breakwater, (___) Lb = 800 m ... 76
Figure 9-2 Ship motion spectra, 0 = 45; (- - -) Lb = 600 m, (

___

) Lb = 800 m, ( ) Lb = 1000

m ................................................................................................................................ 76
Figure 9-3: Significant motions for increasing breakwater length Lb, 0 = 45 ...................... 77
Figure 9-4: Significant motions for increasing breakwater crest height Rc, 0 = 90 ............. 77
Figure 9-5: Significant motions for different shore types; (___) 0 = 45, (- - -) 0 = 90 ......... 78
Figure 10-1: Wave Simulation Program.............................................................................. 79
Figure 10-2: Wave force determination............................................................................... 80
Figure 10-3: Optimization of the breakwater configuration .................................................. 82

II

List of tables
Table 1: Simulation runs calibration.................................................................................... 58
Table 2: Results run 01, model tests Delft Hydraulics, ANSYS and WSP............................ 60
Table 3: Results run 02, model tests Delft Hydraulics, ANSYS and WSP............................ 60
Table 4: Comparison of ANSYS and WSP to Delft Hydraulic model tests ........................... 60
Table 5: Carachteristics LNG carrier .................................................................................. 64
Table 6: Simulation runs validation wave forces ................................................................. 68
o

Table 7: Difference spectral parameters TERMSIM and WSP, o = 45 , Tp = 13 s .............. 69

Table 8: Simulation runs external input wave forces in TERMSIM ...................................... 73

III

List of symbols
Roman symbols

a
a

Added inertia matrix

Coefficient of reflection coefficient formula

[-]

b
b
c

Damping matrix
[-]
[-]

Coefficient of reflection coefficient formula

Permeability parameter

Hydrostatic resoring coefficient

2

g
h

[m/s ]
[m]

Acceleration of gravity
Water depth

i
i

[-]

Imaginary unit =

[-]

i-th frequency component

[-]

j-th directional component

k
j

[-]
[-]

Mode of motion [1..6]

Mode of motion coupled with mode of motion k

[m ]

-1

m
mo

Wave number
Added mass matrix
Zeroth moment spectrum
First moment spectrum

m1
r
sop

[m]
[1/s2]

Radius
Wavwe steepness

[s]

Time

[m]

Spectral width
Displacement or rotation in the j-mode in ship-bound coordinate

xj

system

xo
xs

[m]
[m]

earth-fixed x coordinate
ship-bound x coordinate

yo

[m]

earth-fixed y coordinate

ys
yrefl

[m]
[m]

ship-bound y coordinate
Mirrored y-coordinate with respect to coast

A
A

[-]
[m2/s]

Scaling parameter
Complex amplitude velocity potential

[m]

Breadth of ship

B
bG

[m]
[m]

Cross-sectional area of breakwater

Distance sectional buoyancy point CoG

[-]

Fresnel integral

[m]

Depth of ship

IV

D50

Nominal diameter

F
Ftotal

[kN]
[kN]

Force
Total wave force

(1)

[kN]

First order wave force

(2)

F
Fdrift

[kN]
[kN]

Second order wave force

Wave drift force

FFK

[kN]

Froude-Krilov force

F
H

[-]
[m]

Complex x,y dependent function of velocity potential

Wave height

Hs

[m]

Significant wave height

Hd
Hi

[m]
[m]

Diffracted wave height

Incoming wave height

Hr

[m]

Reflected wave height

I
Kd

[-]

Moment of inertia
Diffraction coefficient

KG

[-]

Distance keel CoG

Kr
Kt

[-]
[-]

Reflection coefficient
Transmission coefficient

LGB

[m]

Longitudinal centre of gravity relative to station 10

Lpp
M

[m]

Length between perpendiculars

Inertia matrix

[-]

N of frequency intervals

N
P

[-]
[-]

N of directional intervals
Left breakwater-end [earth-fixed coordinate system]

o
o

In-phase part of second order transfer function

Q
Q

[-]

oG

Right breakwater-end [earth-fixed coordinate system]

Out-of-phase part of second order transfer function

[m]

Height CoG above free surface

R
Rc

[m]

Retardation function
Crest freeboard relative to SWL

[m ]

Submerged body surface

S
SGS

[-]

Fresnel integral
Gaussian variance density

SJS

Jonswap variance density

SPM
T

[m]

Pierson-Moskowitz variance density

Draft of ship

[s]

Wave period

Tz
Tp

[s]
[s]

Mean zero wave-crossing wave period

Peak wave period

Tm

[s]

Wave period transmitted waves

Xj

[m]

Displacement or rotation in the j-mode in earth-fixed coordinate system

Greek symbols

(1)
(2)

ij ,0
ij , D
ij , R
ij ,T

s
Fk
o
b
ij , D
ij , R
ij ,T

Fk

[rad]

Normalised velocity potential

Random phase angle

Velocity potential

First order velocity potential

[m /s]

Second order velocity potential

[rad]
[rad]

Phase angle
Phase angle incoming wave component ij

[rad]

Phase angle diffracted wave component ij

[rad]

Phase angle reflected wave component ij

[rad]

Phase angle transmitted wave component ij

[-]

Peak enhancement factor Jonswap spectrum

[m /s]
[rad]
[m /s]
[m /s]

[m]

Wave length

[deg]

Ship-bound rotation

[kN]

Mean of wave force

[deg]

Incident wave direction, earth-fixed coordinate system

[rad]

Breakwater angle with respect to xo -axis

[rad]

Wave direction of diffracted wave component ij, polar coordinate

[rad]

system
Wave direction of diffracted wave component ij, polar coordinate
system

[rad]

Wave direction of transmitted wave component ij, polar coordinate

system
3

[kg/m ]

Density of water

[-]
[kN]

Numerical parameter Jonswap spectrum

Standard deviation of wave force

[s]

Time lag

[rad/s]
[rad/s]

Wave angular frequency

Peak wave angular frequency

[rad/s]

Mean frequency

[rad/s]

Wave angular frequency, bound second order waves

Irribarren parameter

(1)

Beach slope

[m /s]

Normalised velocity potential which is independt of the vessel

movement

[m]

Wave surface elevation

[m]

First order surface elevation

VI

 (2)
a ,ij ,0
ij , D
ij , R

ij ,T
*
&wk
&&*

[m]

Second order surface elevation

[m]

Off-shore wave amplitude of wave component ij

[m]

Diffracted water surface elevation at point (xo,yo) of wave

[m]

component ij
Reflected water surface elevation at point (xo,yo) of wave
component ij
Transmitted water surface elevation at point (xo,yo) of wave
component ij

[m]
[m/s]
2

Equivalent velocity of the water particles in direction k

[m/s ]

Equivalent acceleration of the water particles in direction k

x
y
xb

[m]

Small partition in x direction

[m]

Small partition in y direction

[m]

Longitudinal partition of ship length

[rad/s]

Small frequency interval partition

wk

[]

Small directional partition

Volume displacement

Abbreviations
AP
AQWA

Aft perpendicular
Atkins Quantitative Wave Analysis

BAS

Beweging Afgemeerde Schepen

CoG
FFT

Centre of Gravity
Fast Fourier Transform

FK

Froude-Krilov

FP
FTF

Forward perpendicular
First order Transfer Function

JONSWAP

JOint North Sea WAve Project

LAO
LNG

Length Over All

Liquid Natural Gas

LNGC

LNG Carrier

MARIN
MatLab

MAritime Research Institute Netherlands

Matrix Laboratory

MSL

Mean Sea water Level

PM
QTF

Pierson-Moskowitz
Quadratic Transfer Function

RAO

Response Amplitude Operator

SDF

Six Degrees of Freedom

VII

SWL

Sea Water Level

TERMSIM
WSP

TERminal SIMulation
Wave Simulation Program

VIII

Introduction

Introduction
1.1

LNG transport

Liquefied Natural Gas (LNG) is the liquid form of natural gas. It is normal gas, cooled to
o

approximately -160 C, in order to store it as a boiling liquid in insulated tanks. From the point
of view of the major electric utilities, electricity generation is increasingly dependent on gas
as a flexible contributor to the merit order, particularly given the environmental desire to
reduce dependence on nuclear and oil, and to phase out coal. Furthermore, LNG is one of
the cleanest and most efficient forms of energy available. LNG appears to be a feasible
alternative for meeting the increasing demand for gas.
Figure 1-1 gives the worldwide LNG trade development from 1970 to 2003 [ref 1]. The total
capacity of the world's liquefaction plants in 2003 was 155 billion m3 per year. This was a
10% increase over the past year according to 2002.
WORLD LNG TRADE, 1970-2003
180
160

LNG, billion cu m

140
120
100
80
60
40
20
1970

1980

1990

1992

1994

1996

1998

2000

2002

Figure 1-1: World LNG trade development

As the worldwide gas market continues to grow in order to supply domestic/industrial users
and in many cases new power generation projects, there has been a recent renewal of
interest in new LNG import terminals. Historically, the LNG terminals have been located in
sheltered sites, without significant influence of swell and wind induced waves, or at exposed
sites with relatively calm sea conditions. The availability and discovery of gas fields in areas,
where (naturally) sheltered sites are not often available, ask for the development of new
concepts for sites in increasingly hostile marine environments.

Introduction

1.2

New concept

In LNG terminals, exposed to the open sea, dedicated provisions are required to create
sufficient shelter for LNG carriers. Proposals have been made to construct a marginal low
crested detached breakwater parallel to the coast to protect a ship moored at a jetty closer to
the shore. Figure 1-2 presents a schematization of this new concept. The figure is not drawn
to scale.

Reflection

Diffraction
Transmission

Breakwater

LNG Carrier

Reflective coast

Figure 1-2: Schematisation of a LNG marine terminal concept [not to scale]

Generally, due to the high cost of LNG ships, the cost of storage and the impact of a shortfall
in delivery, any investment in port development to eliminate marine/wheather related
downtime was justified without (any) question. However, as terminal exposure and the cost
of breakwater structures increase, these old assumptions need to be challenged in order to
reduce capital expenditure and to increase the economic viability of a project.
The weather related downtime is defined as the ratio of period of delay of marine operations
due to adverse weather conditions. It depends on the ships response and thus on the near
shore wave parameters. These wave parameters are influenced by the phenomena of wave
diffraction and transmission, which on their turn depend on the breakwaters dimensions.
Furthermore, the wave field at the jetty is influenced by reflection of the incoming wave field
by the coast.
The relation between the different aspects is clarified in Figure 1-3 on the next page.

Introduction

Wave f ield
outside breakw ater

Breakw ater
dimensions

Diffract ion
Transmission
Wave field
at jetty

Coast al
characteristics

Ship
response

Weather related
Dow nt ime

Ref lect ion

Figure 1-3: Aspects of a LNG marine terminal

A more thorough knowledge of the influence of diffraction, transmission and reflection on the
weather related downtime would most likely result in options for further optimization of the
concept.
For an optimal economic design of the LNG (un)loading facility, the dimensions of the
detached breakwater need to be optimised as a function of the weather related downtime of
the moored LNG-carrier.

Scope of work

Scope of work
2.1

Problem context

Studies are carried out for a new concept for a LNG marine terminal, consisting of a
detached breakwater protecting a LNG jetty. Due to the presence of the detached
breakwater and a reflective coast, ocean waves experience changes in height, direction and
phase. The wave transformation process comprises diffraction, transmission and reflection.
For an optimal economic design of a LNG marine terminal, the dimensions of the detached
breakwater have to be optimised as a function of the weather related downtime of the
moored LNG-carrier. Doing so requires adequate simulation tools. A survey of the currently
available simulation tools for predicting wave the wave field at a LNG jetty and the ships
response can be found in Appendix A. From this overview, the conclusion can be drawn that
a link between an efficient wave simulation tool and a program for ship response
calculations, for the preliminary design stage of a LNG marine terminal, is not available at
present.

2.2

Research objective

The research objective is the development of a model that predicts the response of a
moored LNG carrier in an exposed LNG marine terminal, as a function of the breakwater
configuration and coast.

2.3

Problem approach

Based on the above-formulated objective, the procedure of research breaks in two parts:
Part A: Development of an efficient wave - and wave force simulation tool
Wave Simulation Program (WSP)
An efficient and simplified model to calculate the effects of site-specific (protective)
structures (i.e. breakwater, coast) on the wave field, as a supplement to the ship motion
simulation programs, is not available. Without such a wave field simulation tool, the
influence of the breakwater dimensions on diffraction, transmission and reflection, and
subsequently on the ships response and the weather related downtime, cannot be predicted
accurately.

Scope of work

For a better understanding in the dynamic behaviour of a moored LNG carrier behind a
detached breakwater, an efficient wave field simulation program needs to be developed. This
new simulation tool needs to provide more accurate input of wave characteristics along the
ships hull for a convenient ship response simulation program and must be well applicable in
the preliminary design stage of a LNG mooring facility.

WAVE SIMULATION PROGRAM

Wave field
outside breakw ater

Breakw ater
dimensions

Diff raction
Transmission
Wave field
at jetty

Coastal
characteristics

Ship
response

Weather related
Dow ntime

Reflection

Figure 2-1: Wave Simulation Program

Wave force calculation

As the near shore wave field parameters are strongly spatially varying, and the LNG carriers
length is significant with respect to the wave length, the wave parameters and the resulting
wave forces have to be introduced along the ships hull.
Therefore a solution for introducing the strongly varying wave parameters and the resulting
wave forces along the ships hull, is required

WAVE FORCE SIMULATION

ALONG SHIP' S HULL
Wave field
outside breakw ater

Breakw ater
dimensions

Diffraction
Transmission
Wave field
at jetty

Coastal
characteristics

Reflection

Figure 2-2: Wave force determination

Wave
forces

Ship
response

Weather related
Dow ntime

Scope of work

Part B: Investigation to the influence of site-specific features on the ship

response.
Ship motion dependency to the breakwater configuration
Ocean wave conditions, breakwater configuration (especially the main parameters length
and height), and coastal features have influences on the ships response, the mooring line
and fender forces and eventually on the downtime of the moored LNG carrier. Influences of
these site-specific features have to be examined in order to get more insight in the
development of the new LNG marine terminal concept.

OPTIMISING BREAKWATER CONFIGURATION

Wave field
outside breakw ater

Breakw ater
dimensions

Weather related
Dow ntime

Diffraction
Transmission
Coastal
characteristics

Wave field
at jetty

Wave
forces

Ship
response

Reflection

Figure 2-3: Optimization of the breakwater configuration

2.4

Report set-up

This report is set-up in the following way; the development of the WSP is based on theory of
waves and wave forces given in literature. In chapter three (wave description), four (wave
transformation phenomena), the theory, used during the development of the program, is
described. The dynamics of a LNG carrier can be computed using the equation of motion. In
order to get more insight in this topic, the theory of ship motions is discussed in chapter five.
In chapter six the determination pf the hydrodynamic loads is explained. The model set-up of
the Wave Simulation Program (WSP) and the structure of the computer simulation program
are presented in chapter seven. In chapter eight the program is validated. Applications of the
Wave Simulation Program and the influence of diffraction, transmission and reflection on the
ship response are evaluated in chapter nine. Finally, the conclusions and recommendations
of this thesis project are presented in chapter ten.

Wave description

Wave description
3.1

Spectra

In order to describe the wave climate at a specific location, it is usually considered to be

stationary within periods of three hours. This implies that within such a period of time, the
statistical properties of the wave climate are assumed to be constant. The wave
characteristics within such durations are called sea states and can be described statistically
by a variance density spectrum, commonly referred to as a wave spectrum
A directional spectrum describes the distribution of wave energy in the frequency domain as
well as in the direction (angle ). It is generally expressed as

S ( , ) = S ( ) D ( )

(1)

in which S(,) is the directional wave spectrum, S() denotes the frequency spectrum and
D(I) is the directional spreading function, i.e. the frequency spectrum represents the
absolute value of the wave energy density and the spreading function represents the relative
magnitude of directional spreading of wave energy. The frequency distributions and
directional spectra that have been used in the model can be found in Appendix B.

3.2

Surface elevation

Changing the spectrum from its continuous form to a discrete form transforms the spectra
into wave time series. Doing so requires a small bandwidth for the frequency spectrum
and for the directional spectrum. This is illustrated in Figure 3-1. It shows a 2-dimensional
frequency spectrum. A series of bands with the same amount of wave energy (gray area) is
created.

Figure 3-1: Frequency-directional wave component

Wave description

Irregular waves can be described in the time domain and in the spatial domain as a
summation of regular waves with random phase, also known as a Fourier series:
M

(1) , (t ) = a ,ij e

i (ij t ki x cos j ki y sin j + ij )

(2)

i =1 j =1

in which the frequency range of the spectrum is divided into segments from i = 1 to M and
the directional range is divided in parts form j = 1 to N, and:

(1)

= First order surface elevation [m]

a,ij
ij

= Amplitude of wave ij [m]

= Angular frequency of wave component ij [rad/s]

ki

= Wave number [m ]

j
t

= Wave direction [rad]

= Time [s]

i,j

= Random phase angle of wave component ij [rad].

-1

For each angular frequency and direction segment, a wave amplitude and a phase are
required. The phase is to be chosen randomly, according to the random-phase/amplitude
model. This phase angle is uniformly distributed in the range from 0 to 2.
The amplitude can be derived from the discrete wave spectrum with use of the following
equation:

1
S (i , j ) i j = (1)2a ,ij
2

(3)

(1) a ,ij =

(4)

or

2 S (i , j ) i j

= Small frequency interval partition [ rad/s]

= Small directional interval partition [o]

The wave number can be determined iteratively by evaluating the dispersion relationship:

kij =

with:

ij2

(5)

g tanh kij h

= Water depth [m]

= Acceleration of gravity [m/s ]

10

Wave transformation phenomena

Wave transformation phenomena

4.1

Introduction

Ocean waves experience, as they hit the detached breakwater and coastline, changes in
height and direction. The wave transformation process modeled here comprises diffraction,
reflection and transmission. This section contains theoretical background on diffraction,
reflection and transmission and their individual influence on the wave parameters, surface
elevation and direction.
In this thesis the transformation processes are approached as linear problems. This implies
that the following assumptions have to be made like in all other linear wave theories;
Water is an ideal fluid; i.e. non-viscous and incompressible
Waves are of small amplitude and can be described by linear wave theory
Flow is irrotational and conforms to a potential function, which satisfies the
Laplace-equation
Water depth shoreward of the breakwater is constant, i.e. no refraction
Following the separate determination of the diffraction, reflection and transmission patterns,
the wave characteristics at location (x0,y0) at time t will be derived by the superpositioning of
the three individual wave components:
M

Diffraction

, , D ( x, y, t ) = ij , D ( x, y ) e

i (ij + ij )

(6)

i =1 j =1

Reflection

, , R ( x, y, t ) = ij , R ( x, y ) e

i (ij + ij )

(7)

i =1 j =1
M

Transmission

, ,T ( x, y, t ) = ij ,T ( x, y ) e

i (ij + ij )

(8)

i =1 j =1

, ( x, y, t ) = ij ( x, y ) e

i (ij + ij )

(9)

i =1 j =1

Of each wave component, and for the three phenomena, the following parameters have to
be derived;
(x,y)
= Amplitude surface elevation at point x,y [m]
(x,y)
= Phase at point x,y [rad/s]

= Direction at point x,y [rad]

11

Wave transformation phenomena

4.2

Diffraction

4.2.1

Diffraction: definition of the phenomenon

When a wave train meets an obstacle such as a breakwater or an offshore platform it may
be reflected backward, but the wave crests can also bend around the obstacle and thus
penetrate into the lee zone of the obstacle. This phenomenon is called diffraction. The
degree of diffraction depends on the ratio of the characteristic lateral dimension of the
obstacle (i.e. the length of a detached breakwater) to the wavelength [ref.1].
Figure 4-1 shows a long-crested monochromatic wave approaching a semi-infinite
breakwater in a region where the water depth is constant (i.e. no wave refraction or
shoaling). The portion of the wave that passes the breakwater tip will diffract into the
breakwater lee. The diffracted wave crests will essentially form concentric circular arcs with
the wave height decreasing along the crest of each wave.

Figure 4-1: Diffraction around a semi-infinite rigid breakwater

The diffraction coefficient can be defined by:
Kd = Hd/Hi
where Hd is the diffracted wave height at a point in the lee of the breakwater and Hi is the
incident wave height at the breakwater tip. If r is the radial distance from the breakwater tip
to the point where Kd is to be determined and is the angle between the breakwater-end and
this radial, then
Kd = fcn(r/,,0 )

12

Wave transformation phenomena

where 0 defines the incident wave direction and is the wave length. Consequently, for a
given location in the lee of the breakwater, the diffraction coefficient is a function of the
incident wave period and direction of approach. So, for a spectrum of incident waves, each
frequency component in the wave spectrum would have a different diffraction coefficient for a
given location in the breakwater's lee.

4.2.2

Diffraction solution of Penny and Price

The first to solve a general diffraction problem was Sommerfeld (1896) for the diffraction of
light passing the edge of a semi-infinite screen. Penny and Price [ref. 2] showed that the
same solution applies to the diffraction of linear surface waves on water of constant depth
that propagate past the end of a semi-infinite, vertical-faced, rigid, impermeable barrier with
negligible thickness.
Thus, the diffraction coefficients in the structure lee include the effects of the diffracted
incident wave and the much smaller diffracted wave that reflects completely from the
structure. Wiegel [ref. 3] made a summary of the Penny and Price findings and has listed the
results in tables (Kd = fcn(r/, ,0 ) for selected values of r/, and 0. The results of the
method of Penny and Price will be explained in this section. The complete derivation of the
solution can be found in Appendix C.
Surface elevation progressive waves
Separation of variables is a convenient method that can be used when solving linear partial
differential equations like the velocity potential (see Appendix C). Following this method, the
water surface elevation can be expressed as

(1) =

i A it
e cosh kh F ( x, y )
g

(10)

in which,

(1)

= First order surface elevation [m]

= complex number [-]

= Angular frequency of wave component [rad/s]

A
g

= Complex amplitude of the velocity potential with unit [m /s].

= Acceleration of gravity [m/s2].

= Wave number [m ]

h
t

= Water depth [m]

= Time [s]

= Complex function depending on x and y

-1

13

Wave transformation phenomena

Diffraction coefficient
The diffraction coefficient Kd is defined as the ratio of the wave height in the area affected by
diffraction to the wave height in the area unaffected by diffraction. It is given by the modulus
for the diffracted wave

Kd ( x, y ) = F ( x, y )

(11)

The phase difference is given by the argument of F(x,y)

( x, y ) = arg ( F ( x, y ) )

(12)

For the general case of waves approaching the breakwater under any angle 0, which has to
be used during the development of the Wave Simulation Program, Penny and Price derived
an equation to express F in polar coordinates, thus

Kd (r , ) = F (r , )

(13)

where F(r,) is still a complex function defined as

F ( r , ) = E + iF

(14)

The values of E and F can be determined using Fresnel integrals C and S. Appendix D
provides background on these integrals. Wiegel has worked out the solution. The equation of
E and F become:

U1 cos[kr cos( 0 )] + U 2 cos[kr cos( + 0 )]

E=

W1 sin[kr cos ( 0 )] W2 sin[kr cos( + 0 )]

(15)

W1 cos[kr cos( 0 )] + W2 cos[kr cos( 0 )]

F =

+U1 sin[kr cos( 0 )] + U 2 sin[kr cos( 0 )]

(16)

in which;
U1

= 1 (1 + C + S )
2

W1

= 1 (S + C)
2

U2

= 1 (1 C S )
2

W2

= 1 (S + C)
2

= Fresnel integral given by equation D.1 (see Appendix D)

= Fresnel integral given by equation D.2 (see Appendix D)

14

Wave transformation phenomena

4.2.3

-1

= Wave number [m ]

= Incident wave direction [rad]

= Angle with respect to breakwater-end [rad]

= Radius with respect to breakwater-end [m]

Transformation of wave parameters by diffraction

Amplitude
The diffracted wave amplitude at location (x0,y0) thus becomes:
M

ij , D ( x, y ) = K d ( x, y ) a ,ij ,0

(17)

i =1 j =1

ij ,D(x,y,t)

= Diffracted water surface elevation at point x,y [m]

Kd(x,y)

= Diffraction coefficient at point x,y [-]

a,ij,0

= Off-shore wave amplitude [m]

Phase
The phase of each monochromatic wave in the diffraction field will be represented by the
summation of the argument of the real and imaginary part of the diffracted wave and the
random phase angle :

ij , D ( x, y ) = arg ( F ( x, y ) )

(18)

Direction
The direction of the diffracted wave crests can be obtained by the argument of the phasedifferences in both x0 and y0-direction.

ij , y

ij , x

ij , D ( x, y ) = a tan

(19)

with:

ij , y = ij , x , y +y ij , x , y

y = very small y-directional interval

ij , x = ij , x +x , y ij , x , y

x = very small x-directional interval

15

Wave transformation phenomena

ij,y

ij

ij,x

Figure 4-2: Derivation of direction

Surface elevation
The resulting surface elevation at point (x0,y0) and time t is:
M

, , D ( x, y, t ) = ij , D ( x, y ) e

i (ij , D + ij )

i =1 j =1

16

(20)

Wave transformation phenomena

4.3

Reflection

4.3.1

Reflection: definition of the phenomenon

Water waves may be either partially or totally reflected from both natural and manmade
barriers. When a wave interferes with a vertical, impermeable, rigid surface-piercing wall,
essentially all of the wave energy will reflect from the wall. On the other hand, when a wave
propagates over a small bottom slope, only a very small portion of the energy will be
reflected. Consideration of wave reflection may often be as important as diffraction in the
design of coastal structures or harbor development. Figure 4-3 illustrates a wave reflected by
a barrier. It shows that the incident waves bend while they approach the coast. This indicates
a decreasing water depth in coastal direction. Nevertheless, the water depth considered in
the model is constant.

Figure 4-3: Reflected wave propagation [ref 4]

The degree of wave reflection is defined by the reflection coefficient
Kr = Hr/Hi

(21)

where Hr and Hi are respectively, the reflected and incident wave heights.
Reflection coefficients for most structures are usually estimated by means of laboratory
model tests. Approximate values of reflection coefficients as reported in various sources are
listed in Table E.1 of Appendix E.

17

Wave transformation phenomena

Theoretical analysis found that, the reflection coefficient for a surface-piercing sloped plane
not only depends on the slope angle, surface roughness, and porosity, but it also on the
incident wave steepness Hi/. Consequently, for a given slope roughness and porosity, the
wave reflection will depend on a parameter known as the surf similarity number or Iribarren
number [ref. 1]:

4.3.2

tan

(22)

( H / )2

Reflection from beaches

Sloping sand beach

For a sloping sand beach the following formula can be used:

K r = 0.1 2 [ref 1]

(23)

Rocky coast
When the coast is rocky, it can be considered as a rigid structure. Therefore Kr should be
obtained by using a formula for rigid structures, in stead of a formula for sloping beaches.
Laboratory investigations (Seelig and Ahrens 1981; Seelig 1983; Allsop and Hettiarachchi
1988) [ref. 5] indicate that the reflection coefficients for most structure forms can be given by
the following equation:

Kr =

a 2
b + 2

(24)

where the values of coefficients a and b depend primarily on the structure geometry and to a
smaller extent on whether waves are monochromatic or irregular. The Iribarren number
employs the structure slope and the wave height at the toe of the structure. Values of a and
b are tabulated in Table E.2 of Appendix E. Generally the following values can be used for a
rocky coast: a = 0.6 and b = 6.6.

18

Wave transformation phenomena

4.3.3

Transformation of wave parameters by reflection

Amplitude
The reflected surface elevation equals the mirrored diffracted surface elevation with respect
to the reflecting barrier; multiplied by the reflection coefficient Kr, see Figure 4-3.
M

ij , R ( x, y ) = K r ij , R ( x, yrefl )

(25)

i =1 j =1

Phase
The phase of the reflected waves has the same value; as if they would propagate without
reflection, i.e. the same phase as the incident non-reflecting waves, see

ij , R ( x, y ) = ij , D ( x, yrefl )

(26)

Direction
The direction of the reflected wave crests equals the mirrored direction of the incoming wave
crests, see Figure 4-4.

ij , R ( x, y ) = 2 ij , D ( x, yrefl )

(27)

" Diffracted w aves"

y ref l
IyI

IyI

1
Barrier

2
Reflected waves:
Surf ace elevation(2) = KrSurface elevat ion(1)
Phase (2) = Phase(1)
Direction (2) = 2 -Direction(1)

Breakw at er

Figure 4-4: Reflected wave parameters

19

Wave transformation phenomena

Surface elevation
The resulting surface elevation at point (xo,yo) and time t is:
M

, , R ( x, y, t ) = K r ij , R ( x, yrefl ) e

i (ij ,R + ij )

i =1 j =1

in which:

,,R(x,y,t)

= Reflected water surface elevation at point x,y [m]

Kr

= Reflection coefficient [-]

.ij,D(x,yrefl)

= "Diffracted" surface elevation at point x,yrefl [m]

= Mirrored y-coordinate with respect to the coast:

yrefl

yrefl = y + 2IyI

20

(28)

Wave transformation phenomena

4.4

Transmission

4.4.1

Transmission: definition of the phenomenon

When waves interact with a structure, a portion of their energy will be dissipated, a portion
will be reflected and, depending on the geometry of the structure, a portion of the energy
may be transmitted past the structure. If the crest of the structure is submerged, the wave
will simply transmit over the structure. However, if the crest of the structure is above the
waterline, the wave may generate a flow of water over the structure, which, in turn,
regenerates waves in the lee of the structure. Also, if the structure is sufficiently permeable,
wave energy may transmit through the structure. When designing structures that protect the
interior of a harbor from wave attack, as little wave transmission as possible should be
allowed, when optimizing the cost versus performance of the structure.
The degree of wave transmission that occurs is commonly defined by a wave transmission
coefficient
Kt = Ht/Hi

(29)

where Ht and Hi are respectively, the transmitted and incident wave heights. When
considering irregular waves, the transmission coefficient might be defined as the ratio of the
transmitted and incident significant wave heights or some other indication of the incident and
transmitted wave energy levels. Figure 4-5 shows the parameters affecting wave
transmission

K REFLECTION

LEE SIDE

SEA SIDE

Ko OVERTOPPING
HI INCIDENCE

Rc

HT TRANSMISSION

D
h

Kt THROUGH THE
BREAKWATER

Figure 4-5: Parameters influencing transmission

21

SWL

Wave transformation phenomena

4.4.2

Transmission studies

Extensive studies with 2-D models have been performed in order to investigate wave
transmission at structures. These studies resulted in various predictive formulae and are
dependent on the following input data: configuration and properties, water level, and wave
condition. These formulae have been empirically derived from data collected from different
laboratories without the certainty of equal analysis procedures; while most predictive
equations of wave transmission perform quite well for the limited conditions in which they
were tested, the question rises whether they are generally applicable. Recently different
comparison studies have been carried out, leading to recommendations for the most
appropriate predictive formulae for wave transmission modeling.
This section gives an overview of the present formulae. Appendix F contains some
evaluation studies on this subject.
Empirical derived transmission formulae
Many authors have investigated the effects of wave transmission. This has resulted in the
diagram presented in Figure 4-6 [ref 6]. Note that the transmission coefficient can never be
smaller than 0 or larger than 1. In practice, limits of about 0.1 and 0.9 are found.

Figure 4-6: Influence Rc/Hs on transmission coefficient Kt

The following transmission studies and formulae are all empirically derived by hydraulic
model tests on permeable breakwaters. In chronological order:
Van der Meer (1990)

22

Wave transformation phenomena

Van der Meer [ref. 7] has analysed several hydraulic model tests, carried out by Seelig
(1980), Powell and Allsop (1985), Deamrich and Kahle (1985), Ahrens (1987) and van der
Meer (1988),
From this analysis a simple prediction formula has been derived in which the transmission
coefficient decreases with relative crest board Rc/Hs.. Crest width effect is not directly taken
into account.

R
K t = 0.46 0.3 c
Hs
Rc
< 1.53
H si

(30)

in which:
= Crest freeboard relative to SWL [m]
= Incident significant wave height [m]

Rc
Hs
Daemen (1991)

Daemen [ref. 8] re-analysed the same data set as van der Meer, excluding the data of
Ahrens (1987), because hydraulic response of reef structures and conventional breakwaters
deviates to a high extent. Daemen introduced a different dimensionless freeboard, including
the permeability of the armour layer:

R
KT = a c + b
D50

(31)

H
a = 0.031 si 0.24
D50
1.84

B
H
b = 5.42 sop + 0.0323 si 0.0017

D50
D50
1<

H si
<6
D50

0.01 < s op < 0.05

where:
sop

= Wave steepness = 2 Hs/D50 Tp

Lp
Dn50

= Local wave length [m]

= Nominal diameter armour stone

23

+ 0.51

Wave transformation phenomena

DAngremond et al. (1996)

Further analysis has been performed by dAngremond et al. [ref. 9]. Model test results have
been filtered, deleting tests with :
High steepness,

s op 0.6

Breaking waves,

H si / h 0.54
Rc / H si < 2.5
Rc / H si > 2.5

Highly submerged structures,

Highly emerged structures,
Resulting in:

Kt = a

Rc
+b
H si

(32)

a = 0.4

B
b=

Hs

0,31

(1 e0.5 ) c

where:
Rc

= Crest freeboard relative to SWL [m]

= Crest width [m]

= Permeability parameter
= Irribaren-parameter = (tan()/Hs/)0.5

Seabrook and Hall (1998) [ref. 10]

0.65 Rc 1.09 H
B Rc
Rc H s
H
B
0.0067
K t = 1 e s + 0.0047

D50
B D50

B Rc
7.08
L D50
H R
0 si c 2.14
B D50
0

24

(33)

Wave transformation phenomena

where:
Rc

= Crest freeboard relative to SWL [m]

= Cross-sectional area of structure [m]

= Wave length [m]

= Nominal diameter armour stone

Dn50

Ahrens (2001)
Dominant mode approach
Dominant modes of wave transmission can be identified as a function of the relative
freeboard. If the structure is submerged (i.e. Rc/Hsi < 0), the dominant mode will be wave
transmission over the crest. For surface-piercing structures (i.e. 0 < Rc/Hsi < ), the dominant
mode will be by wave run-up and overtopping, where is a specific threshold that defines
high structure. If the structure is high, defined as Rc/Hsi < , transmission is primarily through
the breakwater.
Ahrens [ref. 11] developed a parameterization and prediction equation that comprehends
each of the fundamental modes. The transmission over and through the structure are
computed using empirically derived formulas are combined into an overall transmission
coefficient, Kt, using

Kt =

( Kt )over + ( Kt )thru

(34)

From this approach, an empirical general predictive procedure that produces logical trends in
the transmission coefficient over a broad range of configurations and conditions can be
developed.
Analytical derived transmission formulae
In the article Interaction between porous media and wave motion, [ref 12], Chwang and
Chan review the use of Darcys law for analyzing waves moving past a porous structure. It
summarizes a large amount of literature on the analytical study of free-surface wave motion
past porous structures. The ratio between the reflected and transmitted wave energy forms
an important subject in these studies. Yu and Chwang provided plots showing the variation
of reflection and transmission coefficients as a function of the dimensionless thickness of the
structure.
Dalrymple et al., [ref 13], studied the reflection and transmission of a wave train at an oblique
angle of wave incidence by an infinitely long porous structure of thickness b. This provides
the basis for treating an incident directional spectrum

25

Wave transformation phenomena

4.4.3

Spectral change by wave transmission

Due to transmission at the breakwater the wave spectrum is changed not only with respect
to the total energy, but also with respect to the spectral shape. The loss of total energy
results in the decrease of significant wave height, while the spectral shape change results in
lower mean wave periods. Whithin the scope of this subject only little research is available:
Lower mean wave period
In 1998 Mai et al. introduces the ratio of mean period of the transmitted waves Tm,T to the
incident waves Tm in his study on study on wave transmission at summer dikes. According to
Mai, Ohle, Daemrich and Zimmerman (2002) [ref. 14] non-linear wave transformation over
submerged breakwaters causes the change in wave spectrum, by the effect of transfer of
energy from spectral peak to the higher harmonics. The incoming spectrum is changed into a
double peak wave spectrum and subsequently causes the reduction of the mean wave
period. However, the spectral peak remains nearly constant (van der Meer et al. 2000) and
the influence of wave transmission on the spectral shape diminishes for relative high
freeboards.
Overtopping
Overtopping generates so-called new waves with higher frequencies in the breakwater's lee.
Because of the random character of the sea waves, and the discontinuous wave crests, the
waves generated by overtopping are located at various positions along the breakwater,
depending on the layout of a particular breakwater. According to Goda [ref. 15], analysis on
such individual situations is not feasible. Therefore it is necessary to assume that transmitted
waves propagate in a pattern similar to that of the incident waves.

4.4.4

Implementation in WSP

Due to lack of detailed research and literature, which is available on the subject of spectral
shape change due to wave transmission, it is assumed that the spectral shape of the
transmitted waves is similar to the spectral shape of the offshore wave field.
The Wave Simulation Model as described here, the user may choose either a constant value
of the transmission coefficient Kt or allow the model to calculate values based on wave and
structure characteristics. It could be interesting to integrate the empirical and analytical
derived transmission formulae; the empirical formulae provide more information about the
overtopping over the structure and breakwater characteristics. On the other hand, the
analytical solutions provide the possibility to include oblique wave incidence and dependency
on wavelength.
As such integration requires more detailed study, only the empirical formulae have been
included in the WSP. For each structure the user specifies geometric properties (crest height
and width, and median rock size) and can select between calculation methods of van der
26

Wave transformation phenomena

Meer (1990), Daemen (1991), d'Angremond et al. (1996), Seabrook and Hall (1998), and
Ahrens (2001).
The evaluation studies of Wamsley and Ahrens (2003, ref 16), Wamsley, Hanson, Kraus
(2002, ref 17) and Calabrese, Vicinanza, Buccino (2003, ref 18), provide guidance for
selecting a calculation method for a given application. A survey of these evaluation studies
is given in Appendix F.

4.4.5

Transformation of wave parameters by transmission

Amplitude
The transmitted wave amplitude at location (xo,yo) becomes:
M

ij ,T ( x, y ) = Kt ( x, y ) a ,ij ,0

(35)

i =1 j =1

in which:

ij,T(x,y)

= Transmitted water surface elevation at point (x,y)[m]

Kt(x,y)

= Transmission coefficient [-]

a,ij,0

= Off-shore wave amplitude [m]

Phase
Assuming that the phase of the incoming wave components will not change by wave
transmission, the transmitted phase is defined similar to the incoming phase:

ij ,0 = ij ,T

ij,0
ij,T

(36)

= Phase incoming wave component [rad/s]

= Phase transmitted wave component [rad/s]

Direction
According to the assumption that transmitted waves propagate in a pattern similar to that of
the incident waves, the direction will not be altered:

ij ,0 = ij ,T
ij,0
ij,T

(37)
o

= Direction incoming wave component [ ]

o

= Direction transmitted wave component [ ]

Surface elevation
The transmitted water surface elevation thus becomes:

27

Wave transformation phenomena

, ,T ( x, y, t ) = ij ,T e

i (ij ,T + ij )

i =1 j =1

28

(38)

LNG carrier dynamics

LNG carrier dynamics

This chapter presents the theory of the response of a moored LNG carrier at a jetty.

5.1

Equations of motion

Ships floating on water have six degrees of freedom, i.e. surge (translation in x-direction),
sway (translation in y-direction), heave (translation in z-direction), roll (rotation around xaxis), pitch (rotation around y-axis) and yaw (rotation around z-axis). The motions of a
floating body under actions of winds and waves are analyzed by solving the equations of
these six modes.

Figure 5-1: Six degrees of freedom

Frequency domain
A moored floating body can be described as a linear mass-damper-spring system. The six
modes of motions are coupled and can be estimated by solving the equations of motion. For
forces that are linear and independent of the body motions xj and its time derivatives, the
equation of motion is:
6

{( M
j =1

kj

+ akj ) &&
x j + bkj x& j + Ckj x j = Fk

29

k = 1, 2,...6

(39)

LNG carrier dynamics

with:
k

= 1 6 six modes of motion respectively

j
xj

= mode of motion coupled with mode of motion k

= motion in j-mode

Mkj

= Inertia matrix; the mass of the floating body in the direction of

akj

k when the body makes a motion in the mode j

= Added mass matrix; represents the coefficients of the
component of fluid resistance proportional to the acceleration

bkj

in the direction of k when the floating body moves in the mode j

= Wave damping matrix; coefficients of the component of fluid
resistance proportional to the velocity

Ckj
Fk

= static restoring matrix

= External force in the k-mode

The masses in these equations of motion consist of:

solid masses or solid mass moments of inertia of the ship (Mkj ) and
added masses or added mass moments of inertia caused by the disturbed
water, called hydrodynamic masses or mass moments of inertia ( akj ).
An oscillating ship generates waves, since energy will be radiated from the ship. The
hydrodynamic damping terms ( bkj ) account for this.
In the linear approach solving the equations of motion for every frequency and then
integrating the results, can solve them in the frequency domain.
Time domain
However, in many cases the external forces Fk are not linear, and the mooring system shows
non-linear behavior, which violates this linear assumption. Examples are non-linear viscous
damping, forces and moments due to currents and wind, second order wave drift forces and
non-linear characteristics of fender and mooring-line. If the system is non-linear, then the
superposition principle - the foundation of the frequency domain approach - is no longer
valid. Instead, the direct solution of the equations of motion as functions of time needs to be
reverted. These equations of motion result from Newton's second law.
The hydro mechanical reaction forces and moments, due to time varying ship motions, can
be described using the classic formulation given by Cummins [ref 19]. In order to describe
this non-linear behavior, of the impulse response theory can be used, which in it turn
describes the fluid reactive forces. The equation of motion, to be solved in the time domain,
is denoted as:

30

LNG carrier dynamics

&&
( M kj + mkj ) X j (t ) + Rkj ( t ) X& j ( ) d + Ckj X j (t ) = Fk (t )

j =1

(40)

in which:
k

= 1 6 six modes of motion respectively; surge, sway, heave,

roll, pitch yaw

= 1 6 mode of motion coupled with mode k

Xj
Mkj

= motion in j-mode
= Inertia matrix

mkj

= Added mass matrix

Rkj
Cjk

= matrix of retardation functions

= static restoring matrix

Fk

= In time varying external force in the k-mode

As it can be convenient to keep the left hand side of the equation motion linear, all the nonlinear effects can be transferred to the opposite side, where they all form a part of the
external force F(t)
The derivation of this equation is given in Appendix G. The only basic assumption in the
approach is the separate treatment of the hydrodynamic reactive forces and all other
external loads.

5.1.1

Solving the equations of motion by SDF simulation

In order to solve equation (40), a time-domain analysis has to be carried out, based on the
input of time-varying external forces Fk(t). TERMSIM, developed by the Maritime Research
Institute Netherlands (MARIN), is a simulation program, used to calculate vessel motions
and forces in mooring lines and fenders. It prepares a time series of external forces for each
mode of freedom by numerical simulation techniques based on the spectra of forces and
moments exerted on the vessel by wind, waves and currents.
Usually many coefficients from equation (40) can be neglected. The equations of motion for
a moored LNG-carrier to a jetty is given by the following set of equations, in which the
mooring forces consist of both mooring line and fender (equations 41 47):
6

mx&1 + a1k X&&k + R1k (t ) X& k ( ) d + b11 X& 1 = F1wind + F1current + F1wave + F1moor
k =1

k =1

31

LNG carrier dynamics

mx&2 + a2 k X&&k + R2 k (t ) X& k ( ) d + b22 X& 2 = F2wind + F2current + F2wave + F2moor

k =1

k =1
6 t

mX& 3 + a1k X&&k +

k =1

3k

(t ) X& k ( ) d + c33 X 3 + c35 X 5 = F3wave + F3moor

k =1

I 44 X& 4 + a4 k X&&k +
k =1
6

I 55 X& 5 + a5k X&&k +

6

4k

5k

(t ) X& k ( ) d + c44 X 4 = F4wave + F4moor

(t ) X& k ( ) d + c53 X 3 + c55 X 5 = F5wave + F5moor

k =1
6 t

I 66 X& 6 + a6 k X&&k +
k =1

k =1
t

k =1

6k

(t ) X& k ( ) d + b66 X 6 = F6wind + F6current + F6wave + F6moor

k =1

in which:
16
m

= Respectively surge, sway, heave, roll, pitch yaw (k-direction)

= Mass displacement of the ship

Ikk

= Moment of inertia in the k-mode

akj
bkk

= Frequency dependent hydrodynamic mass coefficient

= Additional viscous damping coefficient

ckj

= Hydrostatic restoring coefficient

= Time lag
= Wind force in k direction (k = 1,2,6)

Fjwind
current

Fj

= Current force in k direction (k = 1,2,6)

wave
Fj
moor
Fj

= Wave drift force in k direction (k = 1,6)

= Mooring force due to fenders and mooring legs in k direction (k = 1,6)

The hydrodynamic loads in this thesis only consist of the wave forces; the forces due to
current and wind loads are not considered.
Figure 5-2 presents the calculation procedure that has to be evaluated for each time-step.
The limitations, in terms of the integration of site-specific wave parameters, of such a ship
motion simulation program are discussed in section A-2 of Appendix A. The gray outlined
area, namely the computer module which generates the input wave parameters and the
eventual wave forces, has to be re-designed.

32

LNG carrier dynamics

Viscous
damping

Viscous damping forces

Linear
Filter A

Fluid reaction forces

Time-step

Velocities

integration of
Linear
Filter B

First order wave forces

Linear
Filter C

Low frequency drift forces

Wind and
current force
calculation

Wind and current forces

Position and orientation

differential

Wave height record

Wind and current data

six coupled

equations of
motion

Mooring forces

Figure 5-2: Flow diagram SDF-program TERMSIM

33

Calculation
Of restoring
Forces of the
mooring
system

Mooring forces

Hydrodynamic loads

Hydrodynamic loads
6.1

Introduction

In a sea state, as described in chapter 3, different wave phenomena occur, resulting in

different kinds of wave forces affecting the carriers movements. Taking only the first order
wave forces in consideration will not be sufficient, regarding a moored ships movement. The
second order wave forces also influence the ship response, especially the low-frequency drift
forces, which may be close to the natural frequency of the ship. However, the contribution of
the difference second order potential must be taken into account in order to complete the
low-frequency wave forces. Thus the time-varying external wave forces are divided in three
categories: first order wave forces, second order waves forces due to wave set-down and
the low-frequency drift forces, which are related to the phenomena of wave grouping. This
section describes the above-mentioned division and the derivation of the wave forces acting
on the ship's hull. The time series of the first - and second order wave forces are prepared
for each mode of freedom based on Strip Theory Method calculations. The wave drift force is
determined using the Newman's approximations; the varying drift forces are calculated using
mean drift force transfer functions in regular waves. These methods will be explained in
section 6.3.2.

WAVE PHENOMENON

RESULTING WAVE FORCE

STRIP THEORY METHOD

First order wave
phenomenon

First order wave forces

Wave set-down

Second order wave

phenomena

Second order wave forces:

Wave setdown

NEWMAN
Wave grouping

Figure 6-1: Wave phenomena and resulting wave forces

To start, the strip theory method will be discussed.

35

Second order wave forces:

Low frequency drift forces

Hydrodynamic loads

6.2

Strip Theory Method

The strip theory solves the three-dimensional problem of the hydro mechanical and exciting
wave forces and moments on the ship by integrating the two-dimensional potential solutions
over the ships length. The strip theory considers a ship to be made of a finite number of
transverse two-dimensional slices, which are rigidly connected to each other. The total wave
force acting on each cross-section is determined by evaluating the six forces, which originate
from the six modes of motion. Integrating these total wave forces over the length of the
floating body, leads to the total wave force acting on the vessels hull.

Fk = Fk' dxb

k = 1, 2...6

(48)

Each of the slices will have a shape which closely resembles the segment of the ship which
it represents, and is treated hydro dynamically as if it is a segment of an infinitely long
floating cylinder, see Figure 6-2.

Figure 6-2: Segmented floating body

Two different loads act on the floating body: the waves, produced by the oscillating ship
(hydro mechanical loads) and the diffracted waves (wave loads). These are assumed to
travel perpendicular to the middle line plane (parallel to the ys,zs plane) of the ship. This
means that the fore and aft side of the body, do not produce waves in the x-direction. For the
zero speed case, interactions between the cross sections are ignored as well.
The separate treatment of each cross section simplifies the coupling to a wave model. At
each component of the cross section, the incident wave field is introduced as a summation of
individual linear free surface waves. This implies that differentiation of amplitude, direction
and phase-angle along the length of the ship is possible, causing a high applicability at
terminals located in a strongly varying wave field. The wave parameters of the wave field
behind the breakwater are introduced at each cross-section, at the "aft-side" of the section.

36

Hydrodynamic loads

6.2.1

Relative motion approach

According to the classical strip theory, the exciting wave moments for the six modes on a
restrained cross-section of a ship in waves, are based on the relative motion principle. The
exciting wave loads were found from the loads in undisturbed waves the so-called FroudeKrilov forces or moments completed with diffraction terms, accounting for the presence of
the ship in these waves. The principle states that the force generated by diffraction of a
floating body in waves equals an equivalent oscillation of the body in still water: the
equivalent accelerations and velocities in the undisturbed wave are used to determine the
additional wave loads due to diffraction of the waves. They are considered as potential mass
and damping terms, as applied for the hydro mechanical loads.
The forces are obtained by evaluating the following equations:
Surge

&&w*1 + n11 &w*1

Fw1 = F1FK + m11

(49)

Sway

&&w* 2 + n22
&w* 2
Fw2 = F2FK + m22

(50)

Heave

&&w* 3 + n33
&w* 3
Fw3 = F3FK + m33

(51)

Roll

1 + m42
&&w* 2 + n42
&w* 2 + X w 2 Os G
Fw4 = FFK

(52)

Pitch

Fw5 = Fw1 bG X w 3 xb

(53)

Yaw

Fw6 = Fw2 xb

(54)

where:

&wk

= Equivalent velocity of the water particles in direction k [m/s ]

&&wk

= Equivalent acceleration of the water particles in direction k [m/s ]

OsG

= Height centre of gravity above free surface [m]

bG

= Distance between sectional buoyancy point and CoG [m]

FFKj

= Froude-Krilov force [N]

In beam waves the strip theory method could lead to erroneous results. In the strip theory
method, the floating body is assumed to be a cylinder in still water. Considering a restrained
body, the total force is assumed to be the sum of the Froude-Krilov force FFK and the force
*
exerting on a cylinder in still water F ( wk
) , which is oscillating in the opposite direction of

the particle motions. In case of heave these particle motions

*
are averaged over width of
wk

the carrier. For long waves this approach is plausible, whereas in short wave the velocities

37

Hydrodynamic loads

vary over the width of the ship. The same yields for the sway motion: particle motions are
averaged over the height of the carrier. In practice the velocities will decrease with
increasing depth.
In order to overcome these errors the radiated wave approach could be used. This method
overcomes the above-mentioned problems for beam waves. However, due to the
assumptions made in this approach, it leads to inaccurate results for oblique waves. In
consideration of uniformity, the radiated approach is not included in the model.
Appendix H lists the detailed formulae of the Strip Theory Method, the relative motion
approach and the radiated wave approach. It also gives an illustration of the ship-related
notations used in the strip theory method. For further literature background reference is
made to the theoretical manual of Seaway. [ref. 20]

6.3

Wave forces

Wave loads on a moored ship, as well as the ship response, can be split into several
components. The first order wave forces have the same frequencies as the waves and the
amplitudes are linearly proportional to the wave amplitudes. Forces with both higher and
lower frequencies than the wave frequencies are second order wave forces, which are
proportional to the square of the wave amplitudes. Low-frequency second order wave forces
have frequencies that correspond to the wave group frequencies, present in an irregular
wave field. The forces, consisting of a time varying and a non-zero mean component are
called wave drift forces.
A moored floating structure can be considered as a nonlinear system. Analysis of the
horizontal motions of a moored ship in a seaway show that the response of the structure on
the irregular waves include three important components [ref. 21]:
First order:
1. An oscillating displacement of the ship at frequencies corresponding to those of the
waves, caused by first order wave forces.

Second order:
2. A mean displacement of the structure, resulting from a constant load component,
caused by the mean wave drift force.
3. An oscillating displacement of the structure at frequencies much lower than those of
the irregular waves, caused by low-frequency wave drift forces. These are caused
by non-linear (second order) wave potential effects, also referred to as wave setdown.

38

Hydrodynamic loads

Although the second order forces are much smaller then the first order forces, they are of
importance due to the long periods of the slowly varying part.

6.3.1

First order wave forces

The contribution of the first order wave forces ( Fk(1) ) to the total wave force can be
determined using the first order wave parameters and the theory from section 6.2 and
Appendix H. The time-averaged value of this wave load and the resulting motion component
are zero.

6.3.2

Second order wave forces

The ship motions excited by the second order wave forces, the slow-drift motions, are
resonance oscillations excited by non-linear interaction effects between waves and the body
motion. Slow-drift motions are of equal importance as the linear first-order motions in design
of mooring systems for large volume structures. Generally, a moored ship has a low natural
frequency in its horizontal modes of motion as well as very little damping at such
frequencies. Resonance can lead to very large motions, which could dominate the ship's
dynamic displacement. For a moored structure, slow-drift resonance oscillations occur in
surge, sway and yaw.
The second order wave forces are generally referred to as wave drift forces. The total drift
force consists of a slowly varying drift force (the low-frequency wave drift force) around a
mean value (the mean wave drift force).
Low-frequency wave drift forces
In a sea state, the wave amplitude provides information about the slowly varying wave
envelope of an irregular wave train. The wave envelope is an imaginary curve joining
successive wave crests (or troughs); the entire water surface motion takes place within the
area enclosed by these two curves. This requires a spectral analysis of the square of this
wave envelope. In other words: the spectral density of the square of the wave amplitude
gives information about the mean period and the magnitude of the slowly varying wave drift
force.
Based on the phenomenon on wave grouping, a general formula of the slow-drift excitation
loads, can be derived in terms of the second order transfer functions [ref. 22], (Faltinsen):
N

Fkdrift = i j ( Pij + iQij ) e

i (i j ) t + (i j )

j =1 k =1

39

(55)

Hydrodynamic loads

in which Pij and Qij can be interpreted as second-order quadratic transfer functions for the
difference frequency loads:
Pij =in-phase part of second-order transfer function P i , j and Qij =out-of-phase part of

second-order transfer function P i , j .

Newman [ref. 23] stated that the quadratic transfer function (QTF) is equal to the average of
the mean wave drift forces. His approximation implies that an off-diagonal QTF term formed
by two frequencies can be approximated by the main diagonal term at the average frequency
of the two:

P ( i , i ) + P ( j , j )

P ( i , j ) =

(56)

Q ( i , j ) = 0

Resulting from his findings the formula of the low-frequency wave drift force can be reduced
to:
N N
Pii + Pjj
Fkdrift = i j
j =1 k =1
2

i( (i j ) t + (i j ) )
e

(57)

in which the QTF's Pii and Pjj represent the mean drift forces in a regular wave field, i.e.
without second order waves.
In shallow water the second order wave effects become more important, because the
second order potential to the varying drift forces becomes significant. The contribution of the
second order potential to the quadratic transfer functions is zero at the main diagonal and
non-zero for the off diagonal terms and thus neglected in Newmans approximation: The
QTF's are obtained without the contribution of the second order potential.
The contribution of the second order potential to the total wave force will not be introduced
by means of the quadratic transfer functions. Its contribution will be considered as a linear
second order wave acting on the hull of the vessel.
Contribution of the second order potential
Especially when the water depth is finite, the contribution of the second order potential is
considerable. The contribution of the second order potential to the wave forces is determined
by the phenomenon of wave set down.

40

Hydrodynamic loads

In shallow water, irregular incoming waves exhibit the wave set-down phenomenon. This
non-linear effect appears as long waves bound to the incoming short waves. Set-down wave
elevations are related to second order pressures in the wave field, which in shallow water, is
dominated by second order potential effects. The phase of this long wave - relative to the
wave group - is such that it has a trough where the wave group attains its maximum wave
elevation and a crest where it attains its minimum elevations.
An example of wave set-down can be seen in Figure 6-3.

Regular
w ave group

Wave set-dow n

Figure 6-3: Wave set down

Based on an idealized sea state consisting of two wave components of frequencies i and

j, and directions k and l , the second order potential in a bi-chromatic wave group is:

(2) = i(1) (1)

j A ijkl e

i ( ki cos k k j cosl ) x + ( ki sin k k j sin l ) y (i j ) t )

(58)

and has to satisfy the free surface boundary condition, which is:

1
g z(2) + tt(2) = 2 (1) t(1) + t(1) zz(1) + ttz(1) .
g

A formulation of

(59)

A ijkl can be derived by satisfying the free surface condition (Huijsmans [ref.

24] )

A ijkl =

B ij + C ij
1
g2
2
2 (i j ) ( ki k j ) g tanh h ( ki k j )

with:

41

(60)

Hydrodynamic loads

B ij =

k 2j
ki2

i cosh 2 ki h j cosh 2 k j h

C ijkl =

(61)

2ki k j (i j ) ( cos ( k l ) + tanh ki h tanh k j h )

(62)

i j

Subsequently the low frequency components on the free surface determine the total wave
set-down surface elevation:

1
g

(2) = t(2)

1
1
(1) (1) + (1)tz(1)
2g
g

(63)

As the contribution of the quadratic terms of the first order quantities 1 (1) (1) + 1 (1) (1)
tz
2g

already have been taken into account in the calculation of the low-frequency drift forces by
the quadratic transfer function Pij, only the contribution of the second order potential is
considered. As a consequence the second order free surface elevation is denoted as:
2

1
g

(2) = a(1),ik a(1), jl A ijkl e

i =1 j =1 k =1 l =1

i ( ki cos k k j cos l ) x + ( ki sin k k j sin l ) y (i j ) t )

(64)

Dispersion relation
The incoming wave that results from the low frequency second order potential have a wave
number equal to

(2)
(1)
(1)
k (2) = ki(1) k (1)
and a wave frequency of = i j . These waves
j

do not satisfy the dispersion equation. If the incoming waves have a frequency of

(2) = i(1) (1)

j

then the diffracted waves have the same frequency. The wave number

follows from the relationship:

j ) = kg tanh kh
2

(65)

In order to simplify the situation, the diffracted waves are assumed to have the same wave
number

k (2) = ki(1) k (1)

as the incoming waves. Problems refer to a situation where the
j

wave exciting force on the body due to a wave, which has a velocity potential given by
equation (58), has to be determined. Diffracted waves have the same wave number as
(1)

incoming waves. This is solved by considering the ordinary first order wave exciting force F
on the body in a regular wave with wave number equal to

k (2) = ki(1) k (1)

j in an ordinary

gravity field. For such a case the associated wave frequency will be in accordance with the
dispersion relationship equation. The frequency of the second order wave can be made

42

Hydrodynamic loads

equal to the frequency

(2) = i(1) (1)

j of the second order waves by selecting a different

value for the acceleration of gravity:

*
ij

g =

(i j )2
(ki k j ) tanh(ki k j )h

[ref 21]

(66)

Actually, this is no more than just a trick in order to fit the second order frequency to the
second order wave number in order to fulfill the dispersion relationship.
The adaptation of the acceleration gravity implies different corresponding hydrodynamic
coefficients for the bound long waves. Regarding the model presented in this thesis it is not
useful to determine these new hydrodynamic coefficients, as the waves due to wave set
down will be regarded as free long waves while passing the breakwater. However, in order to
compare the results of the WSP model with TERMSIM simulations without the presence of a
*
breakwater, the parameter is introduced:

*(2) = k (2) g tanh k (2) h

(67)

* is used to determine the hydrodynamic coefficients: in the same matrices but with different
frequency.

Figure 6-4: Determination added mass coefficient using *

The solutions for

*(2)
(2) = i(1) (1)
j and

are both represented in the model. The first

has to be used with the presence of a breakwater, because the set-down waves are then
regarded as long free waves.

*(2)

can only be used incase of bound waves, i.e. without the

presence of the breakwater.

43

Hydrodynamic loads

For usage in relatively shallow water the wave set-down can be considered as a long wave
and subsequently the pressure is hydrostatic and the vertical particle velocities can be
neglected.
The contribution of the second order waves to the total wave force is therefore introduced as
a linear wave exerting on the ship. The wave forces resulting from the contribution of the
(2)

second order potential ( Fk

) is treated equal to the first order wave forces, i.e. using the

strip theory method. These are referred to as "second order waves" in the model.
Since the wavelength of the second order waves due to the phenomenon of wave set down
is considerably long, and the breakwater has less influence on reducing the energy of lowfrequency waves, it is likely that contribution of the second order waves on the moored ship
response will be significant.

6.4

Total wave force

The total wave force exerting on the ship's hull is the summation of the different wave force
contributions:

Fktotal = Fk(1) + Fk(2) + Fkdrift ,

with k = 1, 2...6

(68)

An explanation of the method presented in this section is also described by W. van der Mole,
[ref. 25]

44

Wave Simulation Program

Wave Simulation Program

7.1

Model set-up of the Wave Simulation Program

Simulation of the new LNG marine terminal concept requires a model set-up that functions
as a stepping-stone. The general requirements, project site schematization, co-ordinate
system conventions and of the model are presented in this chapter. The model's set-up is
given in Figure 7-1:

WAVE SIMULATION PROGRAM

Offshore wave
patemeters

Diffraction
Transmission
Reflection

Nearshore wave
paremeters

SDF MODEL

Wave forces on
ship's hull

Ship motions

Figure 7-1: Model set-up Wave Simulation Program

7.1.1

Requirements of WSP

As a supplement on the currently available ship motion simulation models the Wave
Simulation Program must be able to calculate:
Directional spreading of the waves

Introducing spectra, defined in both frequency- and directional domain

Modified near shore wave parameters, under influence of a breakwater

Introducing wave parameters which are changed under influence of a

breakwater (diffraction, transmission), and/or coast (reflection)
Ship-length varying input wave parameters

Introducing time-series of wave forces which originate from calculations by

the Strip Theory Method
In addition the program must be:
Well applicable in the preliminary design stage of a LNG mooring facility
Easy to incorporate in a current ship motion simulation program

45

Wave Simulation Program

7.1.2

Schematization of the model

Schematization of the project site

The first step is to schematize the project site. Figure 7.2 presents the project site
schematization.

OFF SHORE

Dif f ract ion

Transmission

NEA R SHORE

SHIP' S HULL

NEA R SHORE

Ref lect ion

Q
Breakw at er

LNG Carrier

Of f shore w ave f ield

M odif ied w ave f ield

Signif icant w ave height

Hs
Peak Period
Tp
Dominant w ave direct ion 0
0
Phase

Surf ace elevat ion

Frequency
W av e direct ion
Phase

W ave f orces
Six Degrees of
Freedom

Carrier response

Ref lect ive coast

M odif ied w ave f ield

Surf ace elevat ion
Frequency
W ave direct ion
Phase

Figure 7-2: Project site schematization

As can be seen in Figure 7.2, the different areas of the model are defined as follows
Offshore area

The area at the seaside of the breakwater

The offshore wave field is defined by the offshore wave parameters

Near shore area

The area between LNG carrier and breakwater and the LNG carrier
and the coastline
The modified wave field is defined by the near shore wave
parameters

Ship's hull

The hull of the LNG carrier

The wave forces are defined by the near shore wave parameters

Furthermore, point P and Q denote the breakwater-ends. Here they will be referred to as:
Point P = right breakwater-end
Point Q = left breakwater-end

46

Wave Simulation Program

Schematization of the wave transformation phenomena

Figure 7-3 shows the schematization of the wave transformation phenomena. Diffraction,
reflection and transmission are treated separately, without influence on each other.

DIFFRACTION

REFLECTION

TRANSMISSION

No influence
reflection

No influence
reflection
No influence
diifraction

No influence
re-reflection
Impermeable
breakwater

Impermeable breakwater

No influence
transmission

Permeable
breakwater

No influence
transmission

Figure 7-3: Schematization diffraction, reflection and transmission

Diffraction
The incoming waves bend around the two breakwater-ends. Given that the waves, coming
from breakwater-end P and Q, have different wave characteristics (amplitude , phase and
direction ), they are treated separately at the hull of the LNGC. The breakwater is assumed
to be impermeable.
Reflection
In practice the reflected waves at the coast will be re-reflected at the breakwater. Given that
these re-reflected waves would not influence the ship motions substantially, due to
dissipation in the breakwater, this will not be included in the model. Since the reflection
coefficient is dependent on the Irribarren number, it also depends on the wavelength. Longer
waves have a higher coefficient. With the aim of integrating this aspect in the model, the
reflection coefficient is chosen to be dependent on the wavelength.
Transmission
As has been shown previously, the transmission formulae do not provide the possibility to
include an incident direction spectrum. So the following assumption has been made to
include the effect of oblique wave incidence in the transmission:

K t , = K t sin (0 )

(69)

The transmission formulae are empirically derived for short waves. Therefore only
transmission of the first order waves will be taken into account.

7.1.3

Coordinate system conventions

47

Wave Simulation Program

Earth bound coordinate system S(x0, y0, z0)

A right-handed Cartesian co-ordinate system S(x0, y0, z0) is set in space. The (x0, y0)-plane
lies in the still water surface. The x0-axis is set parallel to the shoreline, with the positive x0direction pointing to the right. The y0-axis is directed perpendicular to the x0-axis, pointing in
landward direction and z0 is directed upwards.
y 0 -axis

x 0 -axis

shore line

S
- angle of w ave
attack

Figure 7-4: Earth bound coordinate system

Ship-bound coordinate system CoG(xs,ys,zs)

With respect to the ship another right-handed Cartesian co-ordinate CoG(xs,yb,zb) system is
fixed in space. This sytem is connected to the ship with its origin at the ship's center of
gravity, CoG. The directions of the positive axes are: xs in the longitudinal forward direction,
ys in the lateral direction and zs upwards. If the ship is floating upright in still water, the (xs,

ys)-plane is parallel to the still water surface.

270o

ys-axis

s
0o

CoG

xs-axis

90o
zs-axis

ys-axis

CoG

zs-axis

CoG

xs-axis

Figure 7-5: Ship-bound coordinate system

48

180o

Wave Simulation Program

Breakwater-bound coordinate systems PP(r,,z), PQ(r,,z)

In order to transform the off-shore wave field in near-shore wave parameters, two different
co-ordinate systems are set with respect to the breakwater. These co-ordinate systems are
polar with the origins Pp and PQ lying at the breakwater ends. r is the radial distance from the
breakwater tip to the point where the modified wave characteristics are to be determined,
and is the angle between the breakwater-end and this radial, z is pointing upwards. The (r,
)-plane lies in the still water surface.

rQ

PQ

rP
b
P
0

PP
Incoming w ave

Figure 7-6: Breakwater bound coordinate system

Integration of the three coordinate systems is shown in Figure 7-7:

z0

zb

yb
y0
CoG
S

xb

PP,Q

x0
Figure 7-7: Integration coordinate systems

7.2

Computer simulation program

49

Wave Simulation Program

7.2.1

Introduction

A computer simulation program has been developed to perform the calculations on the wave
field behind the breakwater and the wave forces acting on the ship's hull. The program runs

under MatLab , which itself is a computer program that is considerably adapted for making
calculations with large matrices.
The set-up of the model has been explained in section 7.1. In this section, the structure of
the computer program will be clarified. First the required input data for the program is
treated, followed by the different calculation procedures that eventually lead to a modified
wave field and wave forces. An output procedure arranges these results in order to produce
usable output data. Subsequently the output is integrated in a TERMSIM simulation. Finally
the results can be assessed using the assessment tool. This basic procedure of the program
is illustrated in Figure 7-8.

Input
Files
Section 7.2.2

WSP
Simulation
Module
Section 7.2.3

Output
Data

Application
in TERMSIM

Ship response
assessment

Section 7.2.4

Section 7.2.5

Section 7.2.6

Figure 7-8: Flow scheme WSP

7.2.2

Input Files

The input data for the calculations in the simulation module are to be entered in six different
input files in MatLab, called M-files. These six M-files are:
Input_menu
i.e.: in the menu file, different options of simulation modes must be chosen. The
WSP allows different types of simulations. These are specified in Appendix I.
Input_wave_parameters
Input_breakwater_parameters
Input_ship_parameters
Input_coastal_parameters
Input_time
The exact content of the input files is described in Appendix I (User manual), and will not be
discussed in this section. Note that all input parameters in the five input files are to be
entered in SI-units.

50

Wave Simulation Program

7.2.3

Simulation Module

The simulation module of the program uses the input parameters to make all necessary
calculations in order to determine the modified wave field parameters behind the breakwater
and the wave forces on the ship's hull. This is done using a sequence of programs in
MatLab. The main procedure in the simulation module is illustrated in Figure 7-9 where each
cell represents a separate calculation module.

start

OFF SHORE

NEAR SHORE

SHIP'S HULL

Hull_form
Coordinates
FIRST ORDER WAVE PARAMETERS

MODIFIED FIRST - AND SECOND ORDER

WAVE PARAMETERS

Gauss_spectrum
Diffraction

Jonswap_spectrum

First order

FIRST- AND SECOND

ORDER WAVE FORCES

hf_strip

Second order

drift

Pierson_Moskowitz_spectrum

SECOND ORDER WAVE PARAMETERS

Reflection

First order

hf_strip2

Second order

Set_down

Transmission

First order

Output
time_series
"shipfor.dat"

TERMSIM

Figure 7-9: Flow scheme computer program WSP

The main procedure runs by the M-File 'start'. The procedure within each separate program
is described in Appendix I (User manual). In this section, the task of each program shown in
Figure 7-9 is only briefly described.

51

Wave Simulation Program

The subprograms in the main simulation procedure are:

Start

Executes the main procedure of simulation

Hullform

Determines the hull-shape of the LNG carrier and reads the

hydro-dynamic file

Coordinates

Sets the local co-ordinates in the earth-fixed co-ordinate

system

Offshore:
Pierson_Moskowitz

Calculation of wave components by the evaluating the

_spectrum
Gauss_spectrum

Pierson-Moskowitz spectrum
Calculation of wave components by the evaluating the Gauss
spectrum

Jonswap_spectrum

Calculation of wave components by the evaluating the

JONSWAP spectrum

Set-down

Computes the set-down wave parameters

Nearshore:
Diffraction

Calculation of the diffraction coefficients and pattern using the

Reflection

diffraction theory as described in section 4.2

Calculation of the reflection coefficients and pattern using the
reflection theory as described in section 4.3

Transmission

Determines the transmission coefficient,

transmission theory as described in section 4.4

using

the

Ship's hull:
Hf-strip

Calculation of first order wave forces, using the strip theory

method, described in section 6.2 and Appendix H
o

Hf-strip2 n 1
o

Hf-strip2 n 2

Calculation of second order wave forces, when considered as

free waves, see section 6.3.2
Calculation of second order wave forces, when considered as
bound long waves, also see section 6.3.2

Drift

7.2.4

Calculation of wave drift forces, theory is given in section

6.3.2

Output data

The Wave Simulation Program has been designed to provide site specific input data for the
ship motion simulation program. For introduction of the WSP-generated wave forces in
TERMSIM, the time series have to be set in a specific format. As seen in Figure 7-9 this is
done in the output file "shipfor.dat". This file is introduced by saving it in the current directory
of TERMSIM. So it can be considered as the link between the two simulation programs.
Appendix H shows an example of a shipfor.dat file. As WSP generates a large amount of
output parameters, the results are arranged in different types of format for validation and

52

Wave Simulation Program

evaluation purposes. These files, for instance contour plots of the diffracted and reflected
wave fields or tables with relevant wave parameters, can be found in the current directory in
which WSP is run. Appendix I gives a complete summary and examples of the output files
generated by the WSP. Figure on the next presents the flow scheme of the parameters
calculated by WSP.

7.2.5

Application of WSP calculations in TERMSIM

Figure 7-10 represents the flow diagram of the modified TERMSIM program. The numerical
solution of the equation of motion given by equation (40) in section 5.1 is evaluated in the
time domain for all Six Degrees of Freedom. The input of the wave data is replaced by WSP
calculations by saving the "shipfor.dat" file in the current directory in which TERMSIM is run.

Viscous
damping

Viscous damping forces

Linear
Filter A

Fluid reaction forces

Time-step

Velocities

integration of
Strip Theory
method
Wave height record
Newman

Wind and current data

Wind and
current force
calculation

First order wave forces

six coupled

Second order wave forces

differential

Mean
Low frequency drift forces

equations of
motion

Wind and current forces

Position and orientation

Calculation
Of restoring
Forces of the
mooring
system

Mooring forces

Mooring forces

Figure 7-10: Flow diagram modified TERMSIM

A detailed description of TERMSIM and its hydrodynamic theory, is not discussed here, but
can be found in Appendix J.

7.2.6

Ship response assessment tool

Using the assessment tool, provides knowledge of the ship response to the wave field. It
provides the analysis of the spectra of motions, mooring line and fender forces. In addition it
offers the possibility to investigate the relative increase or decrease of the motions and
mooring forces as a function of varying input parameters.

53

Program validation

8 Program validation

8.1

Introduction

The best way to validate the WSP program is to compare the output of the program with the
results of a laboratory model test. As such a test is not available for the project specified in
this research, i.e. a moored LNG carrier sheltered by a detached breakwater, other
possibilities of validation have been used. This section outlines the validation process of the
Wave Simulation Program and its application in TERMSIM. The main issues in this respect
are: the validation of the modified near-shore wave parameters, validation of the wave forces
generated by WSP and validation of a correct application of the WSP generated wave force
time series in TERMSIM.
Firstly the near-shore wave parameters will be examined. Diffraction and reflection patterns
of the modified wave field provide information about the near shore wave parameters. These
can be compared with corresponding diagrams given in literature. Secondly the wave forces
generated by the WSP are checked. A situation is considered without the presence of a
breakwater. The results of the TERMSIM wave force calculations must be in line with the
results obtained by WSP. Furthermore, the wave forces must equal the first order transfer
functions provided by the hydrodynamic file. Thirdly it has been evaluated whether
TERMSIM reads in the wave force time series correctly. Lastly the results with the presence
of a breakwater are validated. Comparing the results with DELFRAC simulations, where
diffraction around the breakwater is considered, provides this validation. The validation
process is schematised in Figure 8-1. All simulation runs, which have been performed for
this validation, are tabulated in Appendix K.
VALIDATION

Wave parameters
1. Diffraction
2. Reflection
3. Transmission

Wave forces
1. First order Transfer
Functions
2. Wave force spectra

Application in
TERMSIM
Motion spectra

Figure 8-1: Validation process

54

Comparison with
DELFRAC
Motion spectra

Program validation

8.2

Validation transformation of wave parameters

The validation of diffraction and reflection in fact is a validation nd verification of the

program. It is set-up as follows: first a comparison of the diffraction coefficients is made with
a corresponding diagram given in literature. Then the program is tested to calculate the
diffraction and reflection patterns for different angles of wave incidence, both for semi-infinite
and detached breakwaters.

8.2.1

Diffraction

Semi-infinite breakwater
For the validation of the diffracted wave parameters it is useful to compare the wave field
patterns to diffraction diagrams given in literature, for instance by Wiegel [ref 31]. However,
the output provided by the WSP program is set in a Cartesian coordinate system, whereas
the diagrams provided by Wiegel are drawn in polar coordinates. One diagram is available
for lateral coordinates, and it is shown in Figure 8-2. It represents the diffraction coefficients
o

for an incoming wave field with wave incidence of an angle of 90 and a semi-infinite
impermeable rigid breakwater. Figure 8-3 shows the results the WSP pattern under the
same circumstances

Figure 8-3: Diffraction 90 WSP

Figure 8-5: Diffraction 135 WSP

Figure 8-2: Diffraction 90 after Wiegel

Figure 8-4: Diffraction 45 WSP

55

Program validation

Figure 8-4 and Figure 8-5 show the diffraction coefficients for wave incidence o= 45 and
o

135 respectively. These are enclosed in order to show the patterns in case of oblique wave
incidence.
An interesting feature demonstrated by Figures 8-2 8-5, is that for any approach angle, the
value of the diffraction coefficient along a line in the lee of the breakwater that extends from
the breakwater tip in the direction of the approaching wave is approximately 0.5. Note also
that for a given location in the lee of the breakwater, a one-dimensional spectrum of waves
that comes from the same direction will undergo a greater decrease in height (energy
density) for successively higher frequency waves in the spectrum. Thus the diffracted
spectrum will have a shift in energy density towards the lower frequency portion of the
spectrum.

Detached breakwater
Figure 8-6 represents the diffraction patterns of an impermeable rigid detached breakwater
with a total length of L = 700 m (about two times the length of a LNG-carrier). The
wavelength of the monochromatic wave is 30 m. The patterns are shown for two angles of
wave incidence o = 45o and 135o.

wave incidence o = 45

wave incidence o = 135

Figure 8-6: Diffraction around a detached breakwater, wave incidence 45 and 135

The diffraction coefficients coming from the breakwater tips show values higher than 1,
indicating an increase of the surface elevation. The figures also illustrate the strongly varying
wave field.

56

Program validation

8.2.2

Reflection

Semi-infinite breakwater
The reflected wave equals the diffracted wave; mirrored at the coast and multiplied by the
reflection coefficient, see Figure 4-3 in section 4.3. The wave field pattern obtained by a
o

reflection calculation for a semi-infinite breakwater, under wave incidence of o = 45 and

o

135 are presented in Figure 8-7. The contours shown in the reflection plots have a reflection
coefficient of Kr = 1.
0

wave incidence o = 45

Figure 8-7: Reflection at the coast, semi-infinite breakwater, wave incidence 45

Detached breakwater
illustrates the reflection at the coast when in case of a single detached breakwater for wave
incidence o = 45o.
0
wave incidence o = 45

Figure 8-8: Reflection at the coast, detached breakwater, wave incidence 45o

57

Program validation

Incase of oblique wave incidence, the plots show a reflected relative wave height of 1.1 to
1.2 over the whole area. It can be concluded that the influence of reflection could be very
important for the response of the moored LNGC in case of a high reflection coefficient
defined by coastal characteristics.

Comparison hydraulic model tests

For the construction of the breakwater at Limbe in Cameroon, a wave diffraction analysis has
been carried out. Details about this study can be found in Project construction of breakwater

for Limbe shipyard, [ref 26]. During a 3D model test, wave measurements have been made
at various locations in the basin. These measurements have been used to calibrate the
numerical model ANSYS.

ANSYS
The calculations have been made using the finite element model ANSYS version 6.1. In this
2-D model the effect of wave reflection against structures, refraction due to local bathymetry
and diffraction around objects are included. The model does not include energy dissipation
due to wave breaking. Output of the program is given in relative wave heights.

Simulation runs
The offshore wave height is always equal to 1 m, whereas the wave period is the actual
wave period. Two ANSYS calibration runs were carried out for the calibration of the model
with the physical model tests of Delft Hydraulics. Therefore these runs do not include the
bathymetry and the reflecting wall, which has been assumed to be fully absorbing.

Table 1: Simulation runs calibration

Run

Water level

Wave height Hs

Wave period Tp

Wave direction

01
02

MSL: +2.4-0 m
MSL: +1.50 m

1m
1m

14 s
10 s

65
90o

Figure 8-9 shows the project site and the output locations (1,2,3) of the physical model tests.

58

Program validation

REFLECTING
WALL
Kr = 0

BREAKWATER

Figure 8-9: Project site and output locations breakwater in Limbe

Figure 8-10 demonstrates the diffraction patterns of simulation run 02. The left figure shows
the results of WSP, the figure at the right side shows the results of ANSYS.

y-axis [m]

Diffraction pattern, Limbe breakwater (ANSYS)

x-axis [m]

Figure 8-10: Diffraction patterns run 02, wave incidence o = 90o, peak period Tp = 10 s

Comparison with model test

Since the numerical model tests are based on monochromatic waves, the wave pattern is
predicted accurately in terms of nodes and anti-nodes. Therefore the numerical results have
been obtained for areas of 50 m x 50 m around the output locations. Because the wave
heights vary within these areas, the minimum, mean and maximum wave heights are
determined. The comparison has been made based on both Hm0 and H1/3. The
measurements of Delft Hydraulics of these different wave heights show small deviations.
Table 2 and Table 3 show the comparison of the ANSYS and WSP results with the
measured relative wave heights.

59

Program validation

Table 2: Results run 01, model tests Delft Hydraulics, ANSYS and WSP

Run 01
Tp =14 s
Location

o = 65

Measured
Hm0
H1/3

ANSYS results
Min
Mean

Max

WSP results
Min
Mean

Max

0.6

0.5

0.4

0.6

0.8

0.5

0.6

0.7

1.0

1.0

0.7

0.9

1.0

0.8

0.9

1.0

0.5

0.5

0.4

0.5

0.6

0.5

0.5

0.5

Table 3: Results run 02, model tests Delft Hydraulics, ANSYS and WSP

Run 02
Tp =10 s
Location

o = 90

Measured
Hm0

ANSYS results
H1/3

WSP results

Min

Mean

Max

Min

Mean

Max

0.8

1.1

1.0

0.9

0.5

0.7

1.0

0.6

1.5

1.5

1.3

1.5

1.8

1.4

1.6

1.7

0.6

0.6

0.5

0.7

0.8

0.6

0.7

0.9

The comparison of ANSYS and WSP to the measured data is given in Table 4.

Table 4: Comparison of ANSYS and WSP to Delft Hydraulic model tests

Run 01
Tp =14 s

Run 02
o = 65

Tp =10 s
WSP

o = 90

Loca

ANSYS

tion

to Hm0

100 %

111 %

105 %

107 %

70 %

78 %

80 %

85 %

87 %

89 %

92 %

95 %

102 %

102 %

103 %

103 %

93 %

97 %

100 %

100 %

109 %

109 %

111 %

111%

to H1/3

ANSYS

to Hm0

to H1/3

to Hm0

WSP
to H1/3

to Hm0

to H1/3

The comparison between the model tests and the simulation programs shows that WSP
underestimates the wave heights at location 2 of run 01 and location 1 of run 02. On the
other hand, the underestimation is smaller than the results of ANSYS.
The difference between the simulation programs and the model test measurements could be
explained by the fact that the simulations are based on monochromatic waves, whereas the
wave heights of the hydraulic model tests are based on random waves. Overall, it can be
concluded that WSP provides good assessment of the wave heights and generally the
simulations lead to conservative results.

60

Program validation

8.2.3

Transmission

In the input menu of the Wave Simulation Program the user may choose either a constant
value of the transmission coefficient Kt or allow the model to calculate values based on wave
and structure characteristics. For the latter case, the program provides different transmission
formula. Based on the characteristics of the terminal location and the breakwater, a suitable
formula can be chosen. The LNG marine terminal needs a submerged breakwater as a
consequence of visibility and safety considerations. As a result of the application for rubble
mound and solid structures, with a relative low crest freeboard, the dAngremond formula
(equation 32) is very applicable. For that reason the modelling of the dAngremond formula
will be validated in this section.

Validation of the dAngremond formula

Figure 4-6 in section 4.4.2 shows the dependency of the transmission coefficient to the
dimensionless parameter Rc/Hs. Considering the dAngremond formula, the parameters
shown in Figure 8-11 influence the transmission coefficient:
B
Rc

Hs

SWL

Figure 8-11: Parameters transmission

Hs
Rc

= Incoming wave height [m]

= Crest freeboard relative to SWL [m]

= Crest width [m]

= Permeability parameter [-]

= Irribaren-parameter = (tan()/Hs/)0.5

Dependency on dimensionless parameter Rc/Hs

Several runs have been carried out with varying permeability coefficients c [0.64-0.8], in
order to determine whether the WSP simulations generate the same distribution as the
empirical model tests show. In theory the transmission coefficient cannot exceed the range
[0,1]. In practice, limits of about 0.1 and 0.9 are found. The limits in WSP are set on [0.05,
0.95]. This is illustrated in Figure 8.12

61

Program validation

transmission coefficient Kt [-]

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-2

-1.5

-1

-0.5

0
0.5
Rc/Hs [-]

1.5

Figure 8-12: Validation Rc/Hs distribution d'Angremond formula for varying permeability
coefficients

According to Figure 8-12 it follows that the transmission coefficient, calculated using the
dAngremond formula, approximates the empirical distribution over Rc/Hs.

Dependency on incoming wave direction o

Figure 8-13 illustrates the distribution of the transmission coefficient over the incoming wave
direction for different relative crest heights. Decreasing the incoming wave angle o with
respect to the breakwater leads to lower values of the transmission coefficient Kt. It is given
by the equation (69), see section 7.1.2:

K t , = K t sin (0 )

Figure 8-13: Kt vs incoming wave direction o

62

Program validation

Dependency on peak period Tp and breakwater width B

In Appendix L the dependency of the transmission coefficient on the breakwater width and its
distribution over the frequency is shown. From this it can be concluded that the transmission
coefficient can be considered independent of the wave period and frequency,. In practice the
frequency distribution of the transmission coefficient over the frequencies probably is
significant. But this needs further research and is not included in the program.

63

Program validation

8.3

Validation of wave forces

8.3.1

Introduction

This section presents the validation of the wave forces calculated by WSP. The strip theory
method, previously described in section 6.2 has been used for the calculation of the wave
forces exerting on the ship.
The First order Transfer Functions provided by the hydrodynamic file of the LNG carrier and
the first order wave forces have been compared. Furthermore, a spectral analysis of the
WSP wave force time series in comparison with the results of TERMSIM has been
performed.

8.3.2

Vessel characteristics

A 276.15 m long LNG carrier was considered during the validation process. The main
characteristics and stability data are presented in Table 5.

Table 5: Carachteristics LNG carrier

Symbol

Unit

Hydfile 14 m

Type of vessel

Gas carrier

Type of tanks

Spherical tanks
3

Capacity

130.000

Length between perpendiculars

Breadth

Lpp
B

m
m

276.15
41.15

Draft

11

Displacement volume
COG above keel

COG forward of station 10

GK

m
m

96361
14.30

LCG

Projected side area above waterline

Projected front area above waterline

AL
AT

m
m2

7555
1545

Loading condition in % of max. draft

100

Water depth

14

The general arrangements of the carrier and the mooring configuration can be found in
Appendix L.
The strip theory method calculations require a hull-shape file, which provides information
about the carrier's hull shape at the cross-section of every strip. Since such a file was not

64

Program validation

available for the carrier considered here, a hull-shape file has been made using the crosssectional figure of a similar LNG-carrier. The hull shape thus obtained has been scaled to fit
the LNG carrier used in the validation of the program. The hull-shape file, as well as the
figure of the cross-sections, are enclosed in Appendix L. In the Appendix also plots of the
added mass and damping coefficients are given.

8.3.3

First order Transfer Functions

The hydrodynamic file provides us information about the wave force First order Transfer
Functions (FTF). These are transfer functions between height and phase of the waves and
the amplitude and phase of the wave forces.

FTF =

(69)

Fk =
= j =i

ij ,k e

i (ij ,k + ij )

(70)

ij , k

For every combination of frequency and incoming wave direction the FTF's are specified.
The MARIN program DIFFRAC can determine these first order transfer functions. The
results are part of the hydrodynamic file. The wave forces obtained in this way have to equal
the wave forces derived by the Strip Theory Method.
Generally, the first order wave forces show a good agreement with the First order Transfer
Functions. Especially the forces in the most important directions with regard to the mooring
line forces, i.e. surge, sway and yaw approximate the values provided by the hydrodynamic
file. Apparently, the deviations mainly are located nearby the peaks and troughs. In the
higher frequency regions the strip theory method tends to deviate more from the transfer
functions.
Figures 8-14 8-16 show the results of the comparison between the transfer functions given
in the hydrodynamic file and the wave forces calculated using the strip theory method. The
most important modes affecting the mooring line forces, in the directions o = 0o, 15o, and
o

90 , are shown in the figures. The plots of the other modes and wave directions are shown in
Appendix N.

65

Program validation

Figure 8-14: Comparison surge, heave and pitch with FTF, wave incidence o = 0 .
o

With regard to wave incidence o = 0 , the plots of surge, heave and yaw are shown. The
wave forces of the three other modes are negligible (see Appendix M). The surge motion is
the most important motion, and shows close correlation to the FTF. The oscillation of the
strip theory method is stronger over all frequencies.

Figure 8-15: Comparison surge, sway and yaw with FTF, wave incidence o = 45o
Regarding the comparison for wave incidence o = 45o, the sway and yaw motion are almost
similar to the FTFs. Again, the stronger oscillation can be noticed in the plot of the surge
motion.

Figure 8-16: Comparison surge, sway and yaw with FTF, wave incidence o = 90

66

Program validation

Figure 8-16 demonstrates that the wave forces, calculated for a wave incidence of 90o,
correspond less with the transfer functions. The representation of the difference in surge
direction, which is less important in this direction of wave incidence, is a little misleading. The
difference between the transfer functions and wave forces is probably the result of the
difference between the calculation methods used by WSP (strip theory method) and
TERMSIM (panel-method). The strip theory method used by WSP considers the LNG carrier
to be a long slender cylinder. The surge motion is determined by defining an equivalent
longitudinal cross section, which is swaying. In case of beam waves the method assumes
the surge motion to be zero. Regarding the panel-method, the panels are not only orientated
in the x-direction, but also in the y-direction. For that reason it is likely that some surge
motion occurs.
The sway and yaw are the most important motions in beam waves. As can be seen in Figure
8-16 the sway motion approximates the transfer function closely. However, the graph
presenting the yaw motion shows less correspondence.
Theoretically, the yaw motion is likely to approach zero in beam waves. As the carriers
shape is not symmetrical (i.e. the front and aft side of the vessel differ) some yaw motion is
still noticeable. The yaw motion is one order smaller than the yaw motions in other wave
directions, so indeed the yaw motion decreases. In section 6.2.1 it has been pointed out that
the strip theory method could lead to erroneous results in beam waves, as the total wave
force has to be completed by a diffraction part. This is probably the reason that the first order
wave force of the yaw motion differs from the First order Transfer function.
Furthermore, the application for beam-waves in the overall model will be of less importance,
o

as the waves are not likely to approach the moored LNG carrier under an angle of 90 when
the breakwater is present.

8.3.4

Comparison of the wave force spectra

To form a better understanding of the wave forces calculated by WSP, a spectral analysis
has been carried out. Using spectral parameters, the wave force spectra of TERMSIM and
WSP have been compared for three angles of wave incidence (o = 0o, 45o and 90o). From
section 8.3.3 it follows that the deviation of the wave forces to the First order Transfer
Functions is frequency dependent. Hence the wave forces are compared for different wave
periods (Tp = 8 s, 13 s, 25 s) as well. The results of wave direction o = 45o and peak period

Tp = 13 s are presented in this section. Plots and tables of the other wave directions and
periods can be found in Appendix 0.

Parameters spectral analysis

67

Program validation

A series of characteristic numbers, called the spectral moments, is related to the spectrum.
These numbers, denoted by mr, r=0,1... are defined as,

mr = r S ( ) d

(71)

where S() is the power density function of the wave forces. The spectral parameters are the
combination of spectral moments. The mean frequency and spectral width are important
parameters, because the distribution of the spectral density over the frequencies has a high
influence on the ship response. They are defined as:

Mean frequency:

m1
m0

(72)

Spectral width:

v2 =

m0 m1
1
m12

(73)

The spectral width is the measure of spectrum concentration around the mean frequency.
2

The spectrum is extremely narrow when v but is wide when the spectral width
increases.

Simulations runs
An overview of the simulation runs for the comparison of the wave force spectra is given in
Table 6. All runs have been performed both by TERMSIM and WSP. The Pierson-Moskowitz
distribution has been used to describe the offshore wave field. The duration t of all
simulations is three hours or 10.800 s.
For the calculation of the second order wave forces due to wave set-down the two solutions
mentioned in section 6.3.2 are included in the program. Regarding the absence of a
breakwater, the second order waves can be assumed as bound long waves, thus the second
solution is used.

Table 6: Simulation runs validation wave forces

o

nd

N run

Tp [s]

Hs [m]

o [ ]

h [m]

Spectrum

13

2.5

14

PM

bound

13

2.5

45

14

PM

bound

3
4

13
8

2.5
2.5

90
45

14
14

PM
PM

bound
bound

20

2.5

45

14

PM

bound

Spectral analysis

68

2 order waves

Program validation

Figure 8-17 shows the wave force spectra of wave incidence O = 45o

Figure 8-17: Wave force spectra, surge, sway, roll and yaw, o = 45 , Tp = 13 s
The spectral parameters of these spectra of wave forces are tabulated in Appendix N in
Table N.4. Table 7 shows the difference in terms of percentage between TERMSIM and
WSP
o

Table 7: Difference spectral parameters TERMSIM and WSP, o = 45 , Tp = 13 s

Difference TERMSIM - WSP

o

o = 45
Tp = 13s

Zero

First moment

Mean

Spectral

moment

m1

freq.

width

( )

(v )

mo

FSURGE

- 25.0 %

- 23.2 %

2.6 %

- 5.4 %

FSWAY
FHEAVE

3.0 %

- 3.5 %

- 6.4 %

13.8 %

100 %

7.1 %

- 50 %

300 %

FROLL

9.5 %

12.4 %

2.8 %

- 6.6 %

FPITCH
FYAW

- 10.6 %

- 18.3 %

8.1 %

15.7 %

- 2.6 %

- 3.1 %

- 0.4 %

- 8.7 %

69

Program validation

Comparing the spectra in Figure 8-17 with the First order Transfer functions, it is remarkable
that a small deviation from the FTF leads to higher deviations in the spectra. Furthermore it
is surprising to see that the sway motion spectrum shows the highest difference, whereas
the comparison with the FTFs showed very close resemblance. Nonetheless, the mean
frequency and spectral width are almost for every motion within a margin of 15 %
Heave is the only exception. It strongly differs from the TERMSIM calculations. Once more,
this can be explained by the difference between the calculation methods used by TERMSIM
and WSP. TERMSIM considers a floating vessel, which is lifted by the waves. As a result the
wave forces acting on the hull are small regarding the heave motion. The strip theory method
considers the vessel to be a restrained cylinder with an initial position, which is fixed in space
(see section 6.2.1). The oscillating motion is calculated by means of the equivalent particle
motions. Hence the resulting wave forces on the floating body are much higher. However,
the effect of this relatively high wave force is reduced by the hydrostatic restoring component
of the equation of motion of heave (see equation 43). Therefore the response will not differ
from the TERMSIM calculations to a large extend. Besides that, the heave motion is of minor
importance concerning the mooring line forces.
The spectra of motion of the other runs demonstrate that the mean frequency

and the

spectral width v are generally within a margin of 20-30 % in comparison with TERMSIM. A
higher peak period Tp leads to closer correspondence, whereas a small peak shows
divergent spectral shapes.

8.3.5

Conclusion validation of wave forces

The first order wave forces calculated by WSP show close correlation to the First order
Transfer Functions provided by the hydrodynamic file. Difference has been noticed for the
higher frequencies and beam waves. This problem could be solved by using the radiated
approach of the strip theory method for a wave incidence o of 90o (or 270o). In consideration
of uniformity, only the relative motion approach is used in the Wave Simulation Program.
The wave force spectra show good agreement. The spectral parameters are generally within
a margin of 20-30% in comparison with TERMSIM. The higher the peak period, the lower the
spectral resemblance. According to the comparison with the First order Transfer functions,
this is not surprising.
Due to the peak in the low frequency region, the heave motion forms the exception. As a
consequence of the different calculation methods of WSP and TERMSIM, this difference
occurs. However, it will not exert high influence on the mooring line forces, since the heave
motion is of minor importance in this respect.

70

Program validation

8.4

Application of external wave forces in TERMSIM

8.4.1

Introduction

The link between the Wave Transformation Program and TERMSIM consists of the
shipfor.dat file, which is generated by WSP. When this file is stored in the current directory in
which TERMSIM is run, TERMSIM will read the input of the wave force time series
automatically. Initially, the shipfor.dat file option has been developed by MARIN to provide
the possibility to include the influence of a passing ship. In this section the application of the
external wave force time series in TERMSIM is investigated.

8.4.2

Starting-up time

The TERMSIM simulation includes a start-up time possibility to damp out the initial ship
motions. The actual simulation starts after this starting time, which is often set on 1800
seconds (30 minutes). The data provided by WSP is without the starting-up time. The
question is whether the starting-up time influences the results when the wave force time
series are given by the shipfor.dat file. Figure O-1 in Appendix O demonstrates the influence
of the starting-up time on a simulation of 10.800 seconds, with the wave forces time series
generated by WSP. It can be concluded that the motions, although to a small extension, are
influenced by the starting time. Similar results have been found for other simulation durations
(run 5 10; 625 s, 1250 s, 2500 s and 7200 s). Therefore the starting-up time has to be
included in all simulation runs, and is set on 1800 seconds.

8.4.3

Duration

Three hours (10800 seconds) is a widely accepted duration for model tests in an irregular
sea state. Usually the time step needs to be at least ten times smaller than the smallest
natural period of the system that is simulated. Here we can assume that the time step must
be ten times smaller than the peak period of the wave spectrum. However, the maximum
number of data records in the shipfor.dat file, which can be read by TERMSIM, is 5000. This
implies that, for a duration of three hours, the time-step will increase to a value of t = 2.16
s. This value could be too high. It has been investigated for a peak period of Tp = 13 s (p =
0.48) whether this time-step leads to erroneous results.
The time step of the wave force time series generated by TERMSIM is constant, namely t =
0.5 s. For a simulation time of 10.800 seconds and peak periods typical for an irregular sea
state, this time-step is small enough. For this reason, the wave force time series and spectra
obtained from TERMSIM have been compared with the time series and spectra generated

71

Program validation

by WSP. It can be concluded that a time-step, t = 2.16 s, seems to be small enough

considering a peak period Tp = 13 s. However, to be sure the time-step will not influence the
results, a simulation time of two and a half hour (9000 s) has been used. Considering a peak
period of Tp = 13 s, the time step reduces to t = 1.8 s. If higher frequencies are analysed, it
is recommended that the time-step and subsequently the duration of the WSP simulation
would be decreased within the same ratio.

8.4.4

Simulation runs

In order to verify whether TERMSIM reads in the shipfor.dat file correctly, TERMSIM has
been run reading its own generated wave forces from such a file. The results have been
compared. The procedure is clarified in Figure 8-18.

Wave forces
TERMSIM run 1

TERMSIM run 1

Motions
TERMSIM run 1

Internal calculation
wave forces

comparison

Wave f orces
generated by
TERMSIM run 1

Shipf or.dat file

Wave forces
TERMSIM run 2
External input
wave forces

TERMSIM run 2
Motions
TERMSIM run 2

Wave forces
generated by
TERMSIM run 1

Figure 8-18: Validation scheme of external input wave forces in TERMSIM

If the external input of wave forces is read in correctly the results of the wave forces and
motions for TERMSIM run 1 and TERMSIM run 2 must be equal. Table 8 lists the simulation
runs which have been carried out, all have been carried out for TERMSIM 1 (internal
calculation wave forces) and TERMSIM 2 (external input wave forces).
Originally the option for external wave force input has been developed for a passing ship.
The wave forces generated by passing ships unequivocally have a smaller wave height,
period and duration time. By means of the simulation runs, the influence of these aspects

72

Program validation

has to be investigated. For that reason, the runs are carried out with varying incident wave
direction o, peak wave period Tp, significant wave height Hs and simulation time t.

Table 8: Simulation runs external input wave forces in TERMSIM

o

Simulation

run

Program

11

TERMSIM 1-2

13

12

TERMSIM 1-2

13
14

nd

Hs

Duration

Spec-

2 order

[m]

[]

[m]

[s]

trum

waves

2.5

14

9000

PM

bound

13

2.5

45

14

9000

PM

bound

TERMSIM 1-2
TERMSIM 1-2

13
8

2.5
2.5

90
45

14
14

9000
9000

PM
PM

bound
bound

15

TERMSIM 1-2

20

2.5

45

14

9000

PM

bound

16
17

TERMSIM 1-2
TERMSIM 1-2

13
13

0.5
1

45
45

14
14

9000
9000

PM
PM

bound
bound

18

TERMSIM 1-2

13

3.5

45

14

9000

PM

bound

19
20

TERMSIM 1-2
TERMSIM 1-2

13
13

2.5
2.5

45
45

14
14

7200
2500

PM
PM

bound
bound

21

TERMSIM 1-2

13

2.5

45

14

1250

PM

bound

22
23

TERMSIM 1-2
TERMSIM 1-2

13
8

2.5
1

45
45

14
14

625
625

PM
PM

bound
bound

Tp [s]

Poco, the post-processor of TERMSIM, provides time series and spectra for analysis of the
results.
Figure 8-19, illisutrates that the external wave forces are correctly introduced in
TERMSIM, for wave incidence o = 90o and peak period Tp = 13 s.

Figure 8-19: Re-introduction wave force time series in TERMSIM, for surge, sway and yaw

8.4.5

Conclusions external input of wave forces in TERMSIM

Based on the analysis of the application of the wave force time series in a TERMSIM
simulation, the following conclusions can be drawn:

73

Program validation

The duration of the simulation depends on the peak period Tp of the wave spectrum. For a
peak period of 13 s a duration time t of 9000 s seems reasonable. If the peak period
decreases it is recommended to decrease the simulation time with the same ratio. The
starting-up time is set on 1800 s and has to be included when the wave forces are derived
from an external file.
Introducing wave force time series in a TERMSIM run leads to similar results, which
indicates a correct application of external wave forces in TERMSIM.

74

Influence of breakwater dimensions on ship motions

Influence of breakwater dimensions on ship motions

9.1

Introduction

The application of the Wave Simulation Program in the preliminary design stage of a LNG
terminal will be explained in this chapter. With the use of several simulation runs the
influence of the breakwater dimensions on the ship motions and wave forces will be
investigated.

LNG carrier
The same LNG carrier as in the validation process has been considered. The main
characteristics and stability data are presented in Table 5 in section 8.3.2. The general
arrangements of the carrier can be found in Appendix L.

Duration
The duration of the diffraction simulations is 9000 seconds. The starting-up time is set on
half an hour or 1800 seconds.

9.2

Simulation runs

Several simulation runs have been carried out for a 130,000 m3 LNG carrier moored at a
jetty behind a breakwater. The rubble-mound breakwater, with length Lb = 800 m and crest
height Rc = 2 m, and the jetty are situated at 600 m and 350 m from the coast respectively.
The water depth is assumed constant, h = 14 m. For the determination of wave reflection a
bottom slope of 1:40 is assumed in the surf zone. A typical mooring arrangement is used.
The ship is equipped with 16 steel wire ropes with 11 m polypropylene tails.
The results presented in this section are for the terminal dimensions as described above and
for different breakwater lengths, crest heights and bed slopes of the coast to investigate the
influence of these parameters on the moored ship behaviour. The results are given for the
horizontal motions of the carrier, surge, sway and yaw. These degrees of freedom are
chosen because they are critical for the safe mooring of the ship. The ship is able to move
rather freely in heave, roll and pitch. The mooring system prevents the ship from large
horizontal motions. If these motions still become large, line-breaking accidents can occur.

75

Influence of breakwater dimensions on ship motions

Figure 9-1: First order transfer functions, 0 = 45; (- - -) no breakwater, (

___

) Lb = 800 m

First order Transfer Functions (FTF) are given in Fig. 9-1 to provide more insight in the
dependence of the ship behaviour on the wave frequency. The FTF is defined as the ratio
between the offshore wave amplitude and the wave force on the ship. Fig. 2 shows the
reduction due to the presence of an 800 m long impermeable breakwater for an incident
wave direction 0 = 45. Only diffraction is considered here to get a better insight in this
contribution. The effects due to reflection and transmission are neglected. For low-frequency
waves there is only minor reduction, whereas for higher frequencies the reduction is about
90% for sway and yaw. A remarkable fact is the frequency shift comparing the graphs with
and without the breakwater. This shift is due to the change of wave direction in the diffracted
wave pattern behind the breakwater.
The results for ship motions presented in this section are for irregular waves. The offshore
wave field is defined by a Pierson-Moskowitz wave spectrum, with a significant wave height
Hs = 2.5 m, peak period Tp = 13 s and the calculations have been carried for two different
wave directions 0 = 45 and 90, where the latter corresponds to waves perpendicular to the
breakwater. The vessel motion spectra in Fig. 9-2 and the significant values of the motion
amplitudes in Fig. 9-3 are for a wave direction 0 = 45 and different breakwater lengths. The
surge motions are dominated by low-frequency behaviour, mainly due to diffracted bound
waves. Higher frequencies have no effects on the motions, because the mooring system is
very soft in surge. Evidently, a very long breakwater would be necessary to protect the
terminal against these long waves. The sway and yaw motions are influenced by both first
and second order wave forcing. The breakwater is much more effective for these motions.

Figure 9-2 Ship motion spectra, 0 = 45; (- - -) Lb = 600 m, (___) Lb = 800 m, ( ) Lb = 1000 m

76

Influence of breakwater dimensions on ship motions

Figure 9-3: Significant motions for increasing breakwater length Lb, 0 = 45

The investigation of the influence of the crest height and thus the effect of overtopping has
been carried out for normally incident waves, see Fig. 9-4. The influence of transmission is in
the range of the first order wave frequencies, as the empirical transmission formulae are
formulated for first order waves only. Surge motions are low for these transverse waves.
Regarding the results for sway and yaw the influence of transmission is large for low crest
elevations. In these cases with dominant transmission the model gives an approximation of
the ship motions and one should be aware of the necessary rude assumptions on
transmitted waves.

Figure 9-4: Significant motions for increasing breakwater crest height Rc, 0 = 90
Regarding the reflection at the shore the (second order) low-frequency waves have higher
reflection coefficients, and thus dominate the ship motions due to reflection, only in case of a
rocky coast there is also considerable short-wave reflection. Results for different bed slopes
and for a rocky coast are given in Fig. 9-5. The large difference between a rocky coast and a
mild-slope beach shows that reflection can play an important role. In case of oblique waves,
the waves reflected from the coast are hardly reduced due to the presence of the breakwater
and in case of normally incident waves, where the ship is perfectly sheltered against
diffracted waves, the reflected wave energy is dominant for rocky coasts.

77

Influence of breakwater dimensions on ship motions

Figure 9-5: Significant motions for different shore types; (

78

___

) 0 = 45, (- - -) 0 = 90

Conclusions and recommendations

10 Conclusions and Recommendations

The conclusions and recommendations will be given in two categories:
The first category is the development of the Wave Simulation Program (WSP). It is subdivided in the near shore wave parameters, the wave forces and its application in TERMSIM.
The second category is the ship motion dependency on the breakwater configuration. For
each subject conclusions and recommendations are given.

10.1 Development of the Wave Simulation Program

10.1.1 Near-shore wave parameters

WAVE SIMULATION PROGRAM

Wave field
outside breakw ater

Breakw ater
dimensions

Diffraction
Transmission
Wave field
at jetty

Coastal
characteristics

Ship
response

Weather related
Dow ntime

Reflection

Figure 10-1: Wave Simulation Program

Conclusions
Comparison of WSP with hydraulic model tests by Delft Hydraulics and the computer
program ANSYS, indicates that the Wave Simulation Program correctly calculates
the diffracted and reflected waves behind a breakwater.
Literature on transmission gives empirical derived transmission formulae, which
determine the transmission coefficient as a factor of the incoming wave height and
breakwater parameters. No practical theory was found on the subject of spectral
change behind a detached breakwater due to transmission, and this has not been
included in WSP. For the calculation of the transmission coefficient, the
dAngremond formula has been found to be applicable considering a detached
rubble-mound breakwater protecting a LNG marine terminal.
Considering the dAngremond formula, the wave period has a negligible influence on
the transmission coefficient. It is not taken into account in WSP.

79

Conclusions and recommendations

Recommendations
Transformation of the near-shore wave parameters has only been validated by
separate treatment of diffraction, reflection and transmission. Comparison with a
hydraulic model test that includes a combination of the three wave transformation
phenomena is recommended.
A more accurate implementation of transmission in the program requires better
understanding of the influence of transmission on the spectral shape of the wave
field. Integrating theoretical and empirical knowledge on transmission could be a
solution. Influence of incoming and transmitted wave direction needs further
research as well.
The computer program provides two options for the calculation of second order
waves: one considers the second order wave to be a bound long wave; the other
considers it to be a free wave. The difference between these calculation methods
has not clearly been investigated, but is recommended.
In practice combinations of sea and swell can occur. It is recommend that the WSP
is adjusted for this possibility.

10.1.2 Wave forces

WAVE FORCE SIMULATION

ALONG SHIP' S HULL
Wave field
outside breakw ater

Breakw ater
dimensions

Diffraction
Transmission
Wave field
at jetty

Coastal
characteristics

Wave
forces

Ship
response

Weather related
Dow ntime

Reflection

Figure 10-2: Wave force determination

Conclusions
The relative motion approach of the strip theory method proved to be a good method
to calculate the wave forces:

The wave forces calculated with the strip theory method show close
correlation to the First order Transfer function, given in the hydrodynamic file
of a moored LNG carrier. Difference has been noticed in case of beam
waves. This deviation is generally acknowledged and can be solved

80

Conclusions and recommendations

introducing the radiated motion approach. In consideration of uniformity of

the computer program, this approach is not included in WSP.

Differences between the spectra (spectral parameters) of wave forces of

WSP and TERMSIM are within reasonable limits (0% - ca. 25%). The higher
the peak period, the better the correspondence. In contrast to TERMSIM,
heave shows a high peak in the lowest frequency part of the spectrum. This
can be explained by the different calculation methods used by TERMSIM
(panel-method) and WSP (strip theory method).

10.1.3 Application WSP in TERMSIM

Viscous
damping

Viscous damping forces

Linear
Filter A

Fluid reaction forces

Time-step

Velocities

integration of
Strip Theory
method
Wave height record
Newman

Wind and current data

Wind and
current force
calculation

First order wave forces

six coupled

Second order wave forces

differential

Mean
Low frequency drift forces

equations of
motion

Wind and current forces

Position and orientation

Calculation
Of restoring
Forces of the
mooring
system

Mooring forces

Mooring forces

Conclusions
The external input of wave forces are correctly introduced in TERMSIM.

Recommendations
5000 is the maximum number of wave force time series components that can be
read in by TERMSIM. For an irregular sea state this could lead to either
unacceptably high time-steps or short simulation times. For that reason it is
recommended to increase the maximum number of components.

81

Conclusions and recommendations

10.1.4 Main conclusions and recommendations development of WSP

Main conclusions
The making and testing of the computer simulation program WSP, in which all the calculation
procedures were incorporated, lead to the following main conclusion:
A computer model has been developed to calculate the motions and subsequent
mooring forces of a ship moored behind a breakwater. Analytical formulations for
diffraction of first- and second order waves by a breakwater are combined with
empirical formulations for wave transmission over and through the breakwater and
reflection at the shore. Quick computations can be carried out to investigate the
influence of varying breakwater dimensions and terminal layouts. This is especially
useful in the preliminary design stage of a LNG terminal. The calculations give good
approximations of the behaviour of the ship, although model testing remains
necessary for particularly the difficult modelling of wave overtopping. Hence, the
numerical results can be used as a preparation to the model tests.

Main recommendations
More extensive testing of WSP is advised to validate its use for design purposes.
Particularly comparison with a model test is recommended.

10.2 Ship motion dependency on the breakwater configuration

and coastal characteristics

OPTIMISING BREAKWATER CONFIGURATION

Wave field
outside breakw ater

Breakw ater
dimensions

Weather related
Dow ntime

Diffraction
Transmission
Coastal
characteristics

Wave field
at jetty

Reflection

Figure 10-3: Optimization of the breakwater configuration

82

Wave
forces

Ship
response

Conclusions and recommendations

Conclusions

The results presented in this paper show that a detached breakwater of

approximately three times the vessel length efficiently protects the berth against
diffracted short waves. However, the protection against long waves is less effective
and in case of steep coasts the influence of wave reflection at the shore is dominant.
Marginal overtopping over the breakwater can be allowed, although the lack of
knowledge about the deformation of the wave spectrum behind the breakwater has
to be kept in mind.

Recommendations
Evidently it is recommended to analyze the influence of the breakwater dimensions
on the basis of motions, mooring line and fender forces. A ship response
assessment tool has been designed for this purpose.
Some aspects influencing the ship response need further research
- Directional wave spreading

Choice of transmission formula

Combination of wave-, wind- and current forces

83

References

References

[1]

Battjes, J.A. (2001). Korte golven. Faculty of Civil Engineering and Geosciences, D
Technical University

[2]

Penny, W. G., and Price (1944). Diffraction of Waves by Breakwaters" Journal of

fluid mechanics, Civil engineering, TU Delft

[3]

Wiegel, R.L. (1964), "Oceanographical Engineering", Prentice-Hall Inc.,

Englewood Cliffs, New Yersey., Library Delta Marine Consultants, Gouda

[4]

http://users.coastal.ufl.edu/~mcdougal/CEM/Part_II_Coastal_Hydrodynamics/II3_Estimation_of_Nearshore_Waves.pdf

[5]

Seelig, W.N., Ahrens, J.P. (1981) "Estimation of wave reflection and energy
dissipation coefficients for beaches, revetments and breakwaters", US. Army
Corps of Enhineers, Technical paper 81-1, pp. 40

[6]

dAngremond, K. and F.C. van Roode (2001) "Breakwaters and closure dams"
Delft, Delft University Press. Ill.

[7]

Van der Meer, J.W. (1990), "Data on wave transmission due to overtopping",
Technical report, Delft Hydraulics, report n. H986

[8]

Daemen, I.F.R., (1991), "Wave transmission at low-crested structures", Msc

thesis, Delft University of Technology, Delft Hydraulics report n. H462

[9]

D'Angremond, K. et al (1996), " Wave transmission at low-crested structures",

Proc. of Int. Conf. of Coastal Engineering, ASCE pp. 2418-2426

[10]

Seabrook, S.R. and Hall, K.R. (1998), "Wave transmission at submerged rubblemound breakwaters", Proc. of Int. Conf. of Coastal Engineering, ASCE pp.20002013

[11]

Ahrens, J.P. (2001) "Wave transmission over and through rubble-mound

breakwaters",

[12]

A. T. Chwang and A. T. Chan (1998), "Interaction between porous media and

wave motion", Annu. Rev. Fluid Mech. 1998. 30:5384

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[14]

Mai, S., Ohle, N., Daemrich, K.-F., Zimmerman, C., (2002) "Numerical simulation
of transmission at submerges breakwaters compared to physical modelling",
Conf. Physics of Estuaries and Coastal seas (PECS), Hamburg

[15]

Goda Y. (2000), "Random seas and design of maritime structures"., World

Scientific Publishing Co. Pte. Ltd

[16]

Calabrese, M., Vicinanze, D., Buccino, M., (2003) "Low-crested and sub-merged
breakwaters in presence of broken waves", University of Naples

[17]

Wamsley, T., Hanson, H., Kraus, N.C., (2002), "Wave transmission at detached
breakwater for shoreline response modelling", US Army Corps of engineers

[18]

Wamsley, T., Ahrens, J.P., (2003), "Computation of wave transmission

coefficients at detached breakwaters for shoreline reponse modelling", Proc.
Coastal structures 2003, Portland, USA

[19]

W.E, Cummins, "the Impulse Response Functions and ship motions", D.T.M.B
Report 1661, Washington DC, 1962

[20]

Journee, J.M.J., (2001), "Theoretical manual of Seaway", Delft University of

Technology, report n. 1216a

[21]

Journee, J.M.J., Massie, W.W. (1999), "Offshore hydromechanics", lecture notes,

Delft University of Technology

[22]

Faltinsen, O., (1990) "Sea loads on ships and offshore structures". Cambridge,
University Press

[23]

Newman, J.N. (1974) "Second order, slowly varying forces on vesseld in irregular
waves". Int. symp. on the Dynamics of Marine Vehicles and structures in waves.
pp. 193-197

[24]

Huijsmans R.H.M. (2002), "Second order Wave elevation in Irregular Short

Crested seas", technical report, MARIN

[25]

Van der Molen, W. "A hybrid strip theory method to calculate first and second
order wave forces

[26]

Interbeton, Delta Marine Constultans (2004), "Project construction of breakwater

for Limbe Shipyard, Projectnr: 501220, Doc.no: LB-BW-CR-1005

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[27]

Maritime Research Institute Netherlands (MARIN), (1998), "TERMSIM II, Users's

Guide"

[28]

Pinkster, J.A. (1995), "Hydrodynamic interaction effects in waves', Proceedings of

the fifth Int. Offshore and Polar Engineering Conference, The Hague, The
Netherlands.

[29]

Iperen, van E.J., Battjes, J.A., Forristall G.Z., Pinkster J.A. (2004), "Amplification
of waves by a concrete gravity sub-structure:linear diffraction analysis and
estimating the extreme crest height", Proveedings of OMAE04, Vancouver,
Canada

Software
DIFFRAC, Maritime Research Institute Netherlands
DELFRAC, Pinkster (1993), Delft Technical University
MATLAB, Matrix Laboratory, The Mathworks
TERMSIM II, Terminal Simulation, Maritime Research Institute Netherlands

Websites
[i]

www.shipmotions.nl

[ii]

www.marin.nl

[iii]

www.ct.tudelft.nl

[iiii]

www.coastal.ufl.edu

86