Prediction of WaveâStructure Interaction by Advanced

Prediction of Wave–Structure Interaction by Advanced Wave Field Forecast A deterministic, linear approach for the prediction of natural sea states and wave induced vessel/structure motions from surface elevation snapshots of the surrounding wave field

vorgelegt von Diplom-Ingenieur Sascha Kosleck aus Berlin

Von der Fakultät V – Verkehrs- und Maschinensysteme der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften Dr.-Ing. genehmigte Dissertation

Promotionsausschuss: Vorsitzender: Prof. Dr.-Ing. Andrés Cura Hochbaum Berichter: Prof. Dr.-Ing. Günther F. Clauss Berichter: Prof. Dr.-Ing. Paul Uwe Thamsen Tag der wissenschaftlichen Aussprache: 21.02.2013

Berlin 2013 D 83

Acknowledgements This thesis has been inspired by my research work for the joint project ‘LaSSe – Loads on Ships in Seaway’ (original German title: ‘LaSSe – Lasten auf Schiffe im Seegang’) at the Ocean Engineering Department of the Technical University Berlin. My gratitude goes to everyone who encouraged me to work on this topic and supported me in doing so ever since. First of all, I would like to thank Prof. Dr.-Ing. Günther F. Clauss for his never ending enthusiasm for this work and his faith in me. His experience in ocean engineering and his encouraging advice have been motivation throughout the years. Most important, he gave me the freedom to accomplish this work. Many thanks also go to my second promoter Prof. Dr.-Ing. Paul Uwe Thamsen who took interest in my work and the time to support me despite all other duties as well as to the chairman of the doctoral committee, Prof. Dr.-Ing. Andrés Cura Hochbaum. I would like to thank my colleagues from the Ocean Engineering Department for the always lively, fun-loving and entertaining atmosphere. I wouldn’t have missed one day! Special thanks go to my former workmates Janou Hennig and Christian Schmittner, who opened up the door and encouraged me to enter the ocean engineering world many years ago. Special thanks also go to my roommate Florian Sprenger, whom I worked with on so many different, always challenging projects. We shared pain and glory – but more than this, I found a friend. I’d never hesitate to work with you again. Beyond that I’d like to thank my dear colleague Daniel Testa for the countless discussions we had on this topic and for the advice he gave and Marco Klein, who was always willing to share his knowledge, his project equipment and a good laugh to help me through this work. I am especially indebted to Kornelia Tietze. She supported me in all administrative issues – always in a good mood, she is irreplaceable. Furthermore I would like to thank the rest of the team, the students, the technical staff and all my former workmates for their constant support and the good time we had. But my greatest debt of gratitude I owe to my beloved wife Tina and my two kids Maja and Ylvi. Tina, you always encouraged me to go on with my work and supported me in compensating for all the many hours I couldn’t share in the daily family life or have just been plain absent minded. Maja and Ylvi, you never forfeit in reminding me – although you may not know yet – that there are much more important things in life than a PhD. I love you for that! Sascha Kosleck, 2013

Abstract The presented work focuses on the possibility of forecasting natural sea states as well as wave induced vessel/structure motions based on information gathered from surface elevation snapshots of the surrounding free water surface. Objective target of this thesis is the development of a linear, deterministic approach for the just-in-time prediction of an ocean wave field. The knowledge of the precise evolution of the free water surface is used to forecast encountering wave trains at predefined coordinates in space, identified by special points of interest, such as the position of a moored structure or a cruising vessel. Knowing the motion characteristics (transfer functions) of the structure/ vessel, its motions can be predicted by frequency domain analysis. Predicting the sea state as well as the motion behaviour of a structure/vessel prospectively enables the development of a comprehensive decision support system, giving the possibility to identify critical or uncritical situations, respectively, in advance, securing ship and offshore operations. For the investigation of the predictability, unidirectional sea states (with only one direction of wave propagation) with different characteristics are generated in a seakeeping basin. Simultaneous measurements at 451 different positions enable not only the artificial generation of surface elevation snapshots but also the comparison of predictions and measurements at an almost arbitrary number of different positions. It is shown, that frequency domain analyses and the characterization of the sea state using a linear approach, enable the development of procedures for the prediction of natural sea states in time and space, inside a well defined range of validity. By forecasting the wave train to be encountered by a ship or offshore structure, the wave induced motion behaviour can subsequently be derived – given that the frequency domain transfer functions of the system are known. For cruising vessels, the dependency of the transfer functions on the cruising speed of the vessel has be paid special attention to. To validate the procedures developed within this thesis, additional test runs at model scale are conducted, where the vessel is exposed to the sea states already investigated and the motion behaviour of the system is recorded. Data input for the prediction of vessel motions are snapshots of the free water surface. Thus, not only the forecast of the encountering wave train can be compared to measurements taken at the vessel position but also the predicted motion behaviour can be validated. This work clearly shows, that the developed linear methods for the prediction of the motion behaviour of a stationary or cruising vessel/structure deliver excellent results. In the course of this work, the so far unidirectional methods are, in a subsequent step, enhanced for the prediction of multidirectional, short crested seas – with an arbitrary number of directions of wave propagation. At the same time the procedures for the forecast of wave induced motions are

adapted, in order to be able to predict the motion behaviour in short crested seas as well. The respective methods are exemplified using full scale measurements. Data input for the forecast of natural, short crested seas and subsequent wave induced vessel motions is delivered by a WaMoS II® wave monitoring system, registering and analysing snapshots of the free water surface within radar range of a platform or cruising vessel.

Zusammenfassung Die vorliegende Arbeit untersucht die Möglichkeit der Vorhersage von natürlichen Seegängen sowie die Vorhersage des Bewegungsverhaltens von Schiffen oder Strukturen auf der Basis von Momentaufnahmen der bewegten, freien Wasseroberfläche. Ziel ist die Entwicklung eines deterministischen, vollständig auf linearer Theorie basierenden Verfahrens, mit dem auf offener See die Entwicklung eines Wellenfeldes in Echtzeit vorhergesagt werden kann. Die Kenntnis über die präzise Entwicklung der freien Wasseroberfläche wird genutzt, um gezielt Zeitreihen für Begegnungsseegänge an definierten Koordinaten im Raum vorauszuberechnen. Diese Koordinaten kennzeichnen Orte von speziellem Interesse, wie z.B. die Position einer fest verankerten Struktur oder eines fahrenden Schiffs. Ist die Bewegungscharakteristik eines Schiffs/einer Struktur, in Form von Übertragungsfunktionen für die 6 Freiheitsgrade, bekannt, so kann durch eine Frequenzbereichsanalyse des zu erwartenden Begegnungsseegangs das Bewegungsverhalten vorhergesagt werden. Die Vorhersage von Seegang und Bewegungsverhalten soll zukünftig, in Form eines Entscheidungshilfeprogramms, die Möglichkeit bieten, Gefahrenpotentiale zu erkennen, um Schiffs- und Offshoreoperationen sicherer und effizienter zu gestalten. Für die Untersuchung der Vorhersagemöglichkeit werden in einem ersten Schritt unidirektionale Seegänge (Seegänge mit nur einer Wellenausbreitungsrichtung) mit verschiedener Charakteristik im Wellenkanal simuliert. Die simultane Registrierung des Seegangs an 451 verschiedenen Positionen ermöglicht sowohl das synthetische Generieren von Momentaufnahmen der freien Wasseroberfläche, als auch den Vergleich von Vorhersagen und Messungen an fast beliebig vielen verschiedenen Positionen. Die Arbeit zeigt, dass sich durch die Analyse im Frequenzbereich und die Beschreibung der natürlichen Seegänge mittels des gewählten linearen Ansatzes, Verfahren entwickeln lassen, mit denen die aufgemessenen Seegänge, innerhalb eines klar definierten Gültigkeitsbereichs in Raum und Zeit, schnell und zuverlässig vorhergesagt werden können. Durch die Vorhersage eines zu erwartenden Begegnungsseegangs am Ort einer Struktur/eines Schiffs lässt sich in weiterer Abfolge ebenfalls das Bewegungsverhalten vorhersagen, Voraussetzung ist die vorab erfolgte Untersuchung der frequenzabhängigen Bewegungscharakteristik (Übertragungsfunktionen) des zu untersuchenden Systems. Für die Betrachtung fahrender Schiffe ist insbesondere zu berücksichtigen, dass sich die Bewegungscharakteristik mit zunehmender Fahrtgeschwindigkeit deutlich ändert. Um die innerhalb dieser Arbeit entwickelten Bewegungsvorhersagemethoden zu validieren, werden zusätzliche Versuche im Wellenkanal durchgeführt. Dabei wird entweder das fest verankerte oder fahrende Schiff den bereits untersuchten Seegängen ausgesetzt und das Bewegungsver-

halten aufgemessen. Als Basis für die Vorhersage des Bewegungsverhaltens dienen wiederum Momentaufnahmen der freien Wasseroberfläche. Somit lässt sich sowohl die Vorhersage des Begegnungswellenzuges, durch Messungen am Ort des fest verankerten/fahrenden Schiffs, als auch das vorausberechnete Bewegungsverhalten des Schiffs validieren. Die Arbeit zeigt deutlich, dass die angewendeten linearen Methoden das Bewegungsverhalten der Struktur/des Schiffs, innerhalb der Zeiträume in denen die Seegangsvorhersage verlässliche Ergebnisse liefert, ausgezeichnet abbilden. Die für unidirektionale Seegänge entwickelten Methoden wurden im weiteren Verlauf der Arbeit für die Vorhersage multidirektionaler, kurzkämmiger Seegänge mit beliebig vielen Wellenausbreitungsrichtungen angepasst. Gleichsam werden auch die Methoden zur Bewegungsvorhersage dementsprechend weiterentwickelt. Die jeweiligen Verfahren werden beispielhaft für in der Realität aufgemessene Seegänge durchgerechnet. Als Eingangsgröße für die Vorhersage eines natürlichen, kurzkämmigen Seegangs dienen mittels eines Seegangsradars (WaMoS II®) ermittelte Momentaufnahmen des Umgebungswellenfeldes im Radarbereich einer fest installierten Offshoreplattform, bzw. eines fahrenden Schiffs.

Contents Acknowledgements

v

Abstract

vii

Zusammenfassung

ix

Contents

xi

1

Introduction and Executive summary

1

2

State of the art and relevant theoretical basics

7

2.1

1-dimensional Fourier Transform

8

2.2

Linear wave theory (Airy)

10

2.3

Prediction horizon / Prediction error

11

2.4

Forecast of natural sea states

12

2.5

Forecast of wave induced motions

14

2.6

Multidirectional sea states

19

2-D wave forecast for unidirectional long crested sea states

21

General approach

21

3 3.1

xii

CONTENTS

3.2

Predictability of 2D waves

23

3.3

Different numerical methods

33

3.4

Investigated sea states

37

3.5

Preliminary comments

40

3.6

Forecast for sea state II – snapshot range 100% (2450m)

42

3.7

Summary and Comparison of results

46

3-D wave forecast for multidirectional short crested sea states

51

4.1

General approach

51

4.2

Different methods

53

4.3

Predictability for multidirectional waves

58

4.4

Investigated sea state

61

4.5

Forecast for a fixed reference frame

64

4.6

Forecast for a moving reference frame

69

4.7

Summary

71

Motion forecast for unidirectional long crested sea states

73

5.1

General approach

73

5.2

Predictability

80

5.3

Preliminary Comments

80

5.4

Motion forecast for a stationary vessel (Fn = 0)

81

5.5

Motion forecast for a cruising vessel (Fn ≠ 0)

84

5.6

Summary

91

4

5

CONTENTS

xiii

Motion forecast for multidirectional short crested sea states

93

6.1

General approach

93

6.2

Definition of the coordinate system

95

6.3

Investigated sea state

100

6.4

Motion forecast for a stationary vessel (Fn=0)

102

6.5

Motion forecast for a cruising vessel (Fr ≠ 0)

107

6.6

Summary

113

Conclusions and Outlook

115

7.1

Conclusions

115

7.2

Outlook

118

6

7

Nomenclature

121

Bibliography

125

List of figures

127

List of tables

139

2-D wave forecast – supplementary test runs with reduced snapshot ranges

A1

A.1

Forecast for sea state I – snapshot range 100% (2450m)

A1

A.2

Forecast for sea state III – snapshot range 100% (2450m)

A5

A.3


A9

A.4

Forecast for sea state I – Snapshot range 50% (1225m)

A14

A.5


A18

A.6


A22

A

xiv

A.7

Forecast for sea state III – Snapshot range 75% (1834m)

A27

A.8


A31

A.9

Summary

A36

2-D motion forecast – supplementary test runs with differing cruising velocity B1

B

C

CONTENTS

B.1


B1

B.2


B4

B.3

Summary Overview of forecast process steps

B22 C1

Chapter 1

Introduction and Executive summary “The basic law of the seaway is the apparent lack of any law.” – Lord Rayleigh

Despite all scientific and technologic progress, even a hundred years after Lord Rayleigh’s statement, the seaway remains the fascinating, often enough dreadful unknown. It’s sheer complexity and apparently chaotic nature make it almost impossible to foresee. This uncertainty implies an enormous handicap to ocean engineers, as it complicates the exact planning and execution of all kinds of offshore operations. Although the limitations an operation is bound to are exactly known, their exceedance can only be described by a probabilistic approach, resulting in the overall downtime of the operation. Nevertheless, the downtime can not guarantee, that a certain offshore operation can be conducted within a fixed schedule. Often enough ‘waiting on weather’ causes delays and enormous expenditure. Therefore, even a short forecast of the near future would enable the estimation of loads, forces and motions to be expected. The prediction of the sea state by means of a just-in-time decision support system allows, that even in apparently unclear conditions, offshore operations can be conducted safely. Based on that idea, this work presents a holistic, deterministic, linear approach for the just-in-time prediction of unidirectional and multidirectional natural wave fields and subsequent wave induced vessel/structure motions including the following key steps: a) Sea state registration • For the first time surface elevation snapshots of the free water surface are used to determine the sea state characteristic, as new technics allow for the instant registration of the sea state in the far field of the vessel, using on-board radar. • Multidirectional, natural sea state is decomposed into unidirectional subsystems, allowing for a prediction method using linear wave theory. Ö Ö The approach allows to take into account directional spreading with an arbitrary number of wave components and wave encounter angles. b) Sea state prediction • The sea state can be predicted from close vicinity to the far field (radar range) of the vessel/ structure, from present to the near future (up to 2.5min).

2

CHAPTER 1

INTRODUCTION AND EXECUTIVE SUMMARY

• Encountering wave trains for the prediction of motions of a stationary or cruising vessel/structure can be predicted. Ö Ö The overall encountering wave train is decomposed into sector wise encountering trains with different wave encounter angles with respect to the vessel/structure fixed coordinate system. c) Motion prediction • The vessel/structure motions are predicted in 6 degrees of freedom using vessel transfer functions derived by potential theory. Ö Ö As the vessel response is highly dependent on the wave encounter angle, the responses are derived sector wise using the sector wise calculated encountering wave trains. Ö Ö As the vessel responses are also highly dependent on the actual cruising speed, all transfer functions are modified by adapting the velocity dependent hydrodynamic masses and damping coefficients. Ö Ö The overall vessel responses are calculated by superimposing the sector wise derived responses, for each degree of freedom respectively. d) Estimation of the predictability in time and space • The predictability of the sea state and with it the predictability of vessel/structure motions in time and space is estimated by a prediction error that examines the propagation of the energy contained within the sea state. e) Optimisation of the procedure for just-in-time applications • All methods and procedures are optimised in order to meet the requirements of a just-in-time decision support system, which means: optimum accuracy at a minimum calculation time. It is shown, that these surface elevation snapshots of the surrounding wave field (contour of the free water surface) can provide the basis for successful instantaneous predictions of sea state and vessel/structure motions, allowing for a look into the future and the identification of critical or uncritical operational slots. Note, that in this thesis radar registrations (full scale – future application) are substituted by wave probe registrations taken at model scale of 1:70. Continuous registrations at 451 different positions, covering a length 3150m (full scale), enable the generation of snapshots of variable range as well as the comparison of predictions and the evolution of the ‘real’ sea state. Fig. 1.1 (1) presents a surface elevation snapshot, i.e. the wave elevation z(x) at a certain instant of time. Based on this snapshot of a natural, unidirectional, long crested sea state, its evolution can be predicted. The exact forecast of the time dependent encountering wave train at the position of a vessel delivers the key for a successful subsequent prediction of vessel motions. Fig. 1.1 (6) illustrates the forecast of the free water surface elevation at the position of the vessel. Fig. 1.1 (11) presents the subsequently derived vessel motions (heave and pitch). The prediction is compared to measurements (blue lines) and the time frames for acceptable forecasts are highlighted. Thus Fig. 1.1 shows, that both, sea state and motions, can be predicted very well into the near future. Within this thesis it is also shown that the time frames for acceptable wave and vessel motion forecasts directly correlate to the snapshot range. As only a few seconds are required to calculate the prediction of sea state and motions, this approach provides an excellent basis for future just-in-time decision support systems.

3 x=0

+

snapshot range 2450m (full range) snapshot range 1834m (75% of full range) snapshot range 1225m (50% of full range)

Direction of wave propagation

sea state II (HS=9.1m) snapshot range 2450m (100%)

−2000

−1500

−1000 x[m]

of the snapshot

ζ ( x ) = Σζ a e

−500

of the snapshot)

(vessel is not cruising -> Fn=0) dispersion relation

2

Fourier wave spectrum (equidistant samples)

k1

ζa =

ω = kg tanh (kd )

k=2π/L

harmonic wave components

1

0 x=0m (trailing edge500

−ikx

wave number

signal decomposition via FFT

space domain (wave components )

124m

3

N

4

Fourier wave spectrum (non-equidistant samples)

Fwave (k )

k2

N is the number of wave components

F (ω) ⋅2

5

wave amplitude spectrum (non-equidistant samples)

Fwave (ω)

phase frequency domain angle

ζ(x) [m]

4 2 0 −2 −4 −2500 leading edge

position of the CG of the vessel (124m beyond trailing edge of the snapshot)

ζ a (ω )

kn space ->

IFFT-method (section 4.3.2) n

ζ ( x = const ., t ) = ∑ ζ a ,i e

superposition (section 4.3.1)

−i (ωi t +kx ,i x +ϕi )

i =1

encountering wave train at the position of the CG of the vessel (x=124m) - prediction vs. measurement

6

measurement prediction (snapshot range 100%)

0

50

ζ (t ) = Σζ a e

frequency domain

7

150

t [s]

8

response amplitude operator (RAO) sa ,i iεn H i ,l (ω) = e ζ a ,i

Fourier wave spectrum

Fwave (ω)

200

9

6 DOF

250

Fresponse (ω)

=


IFFT n

sl , j (t ) = ∑ sa ,i ,l , j e 5

3

s [m]

6 DOF

superposition −i (ω i , j t +ϕi , j )

i =1

vessel response for heave (s3) and pitch (s5) motion - prediction vs. measurment

heave

11

0 measurement prediction (snapshot range 100%)

−5 10 s 5 [°]

10

Fourier response spectrum

phase angle

time domain (wave components )

component waves

time ->

100

−i ωt

phase angle frequency domain

ζ(t) [m]

4 2 0 −2 −4

0

−10

pitch 0

50

100

t [s]

150

200

250

Fig. 1.1: Example for the deterministic, linear prediction of a natural, unidirectional ocean wave field and resulting wave induced vessel motions

4

CHAPTER 1

INTRODUCTION AND EXECUTIVE SUMMARY

The basic forecast procedure The first step for the prediction of the sea state at a certain position is the registration of the surrounding wave field. This is achieved by analysing radar information, commonly available on the majority of ocean going vessels or platforms. These surface elevation snapshot in space domain z(x) are transferred into frequency domain, i.e. the snapshot z(x) is decomposed into a well defined number of wave components, using a Fourier transform algorithm. As is noted above, in this thesis the radar snapshots for the prediction of unidirectional sea states are substituted by measurements conducted in a wave tank. Fig. 1.1 (1) visualises a snapshot of the free water surface with a full scale range of 2450m. The registration in space is decomposed into harmonic waves, each defined by its amplitude za, angular frequency w=2p . f and phase. All components form Fourier, the amplitude spectrum za(w) or the sea state spectrum S(w)= za(w)²/(2 . Dw) – which can be converted into each other (see Fig. 1.2).

ζa =

F (ω) ⋅2 N

[m]

sea state spectrum / energy density spectrum Szz(ω) =

ζ a2 [m²s] 2∆ω

∞

HS = 4

∫S

zz ,n

(ω ) d ω

0

phase angle

frequency domain phase angle

frequency domain phase angle

Fζ (ω) [m]

frequency domain

amplitude spectrum


Fig. 1.2: Fourier, amplitude and sea state spectrum (energy density spectrum)

Note, that only the magnitudes of the spectra are transferred. The phase information stays constant. Due to their harmonic shape, the propagation of each component can easily be predicted in time and space. Separately forecasting all wave components and subsequently superimposing them results in the prediction of the surface elevation. Following this procedure, the evolution of the sea state can be calculated – based on a linear, deterministic approach. Fig. 1.1 (1-6) gives an overview of the described process steps. The approach allows for all types of forecasts, e.g. the prediction of time dependent wave trains (at a fixed z(t,x=const.) or moving position z(t,x)) or the prediction of the surface elevation for a fixed instance in time z(x,t=const.). Fig. 1.1 (6) presents the prediction over time at a fixed position – here the position of the vessel the radar is installed on – which is compared to measurements. The predicted encountering wave train is used for calculating the motion behaviour of the vessel. Therefore the time dependent information z(t) is again transferred into frequency domain, resulting in a spectrum of the encountering wave train. With the spectrum calculated, the frequency dependent excitation of the vessel is known. Together with the frequency dependent transfer functions of the vessel (RAOs) the responses can be calculated. The resulting response spectra consist of harmonic components of the vessel response. Superimposing these in time yields the time dependent responses of the vessel si(t). Fig. 1.1 (7-10) gives an overview of the described process. Fig. 1.1 (11) shows the forecast for heave and pitch motion, based on the prediction of the encountering wave train, as given in (6). All predictions are compared to measurements (blue lines) and the time frames for good to excellent forecasts are highlighted. The example given in Fig. 1.1 shows that the deterministic, linear approach enables the prediction of irregular ocean waves. For the example presented in Fig. 1.1 – a snapshot range of 2450m=1.3nm and a vessel position at x=124m (beyond the trailing edge of the snapshot) – the predictable time frame turns out to be ~160s (~2.5min).

5

Throughout this thesis it is shown, that different natural, irregular sea states and subsequent vessel motions can be predicted up to 2.5min into the future depending on the characteristics of the respective sea state.

Structure of the thesis The thesis opens up with an overview of the current state of the art in sea state prediction and a brief introduction to the relevant theoretical principles this work is based on. The introduction is followed by two main parts – the prediction of sea states and the prediction of vessel/structure motions. Each of these two parts is subdivided into the prediction in unidirectional, long crested wave fields on one hand and the forecast in multidirectional, short crested wave fields on the other. The following summary gives a short overview on the topics investigated within this work: • 2-D wave forecast for unidirectional, long crested sea states The investigation of the predictability of natural, unidirectional sea states is presented in chapter 3. The procedures are exemplified using three different sea states. A detailed example for one of the three sea states is given in chapter 3. Further examples for all sea states can be found in Appendix A. The radar based snapshots are substituted by snapshots generated from measurements, taken at a model scale of 1:70. • 3-D wave forecast for multidirectional, short crested sea states The development of methods for the prediction of multidirectional, short crested seas care presented in chapter 4. These enhanced procedures, allow for taking into account an arbitrary number of directions of wave propagation. • Motion forecast for unidirectional, long crested sea states Chapter 5 explains the forecast of vessel motions based on a previously predicted encountering wave train. The approach is exemplified for a stationary vessel (v=0kn) as well as for a cruising vessel (v=6kn). Further examples with cruising speeds of v=6kn, v=8kn and v=15kn are summarized in Appendix B (the vessel is alternatively exposed to head seas as well as following seas). Depending on the cruising speed, the shape of the sea state spectrum, as encountered by the vessel, changes (Doppler-shift). The energy of the sea state, however, remains constant:

∫S

v ≠0

we

(we ) d we = ∫ Sv=0 (w) d w

(1.1)

w

Similarly the RAOs of the vessel change depending on the cruising speed. Both effects have to be considered when predicting responses for cruising vessels. • Motion forecast for multidirectional, short crested sea states The key to a successful prediction of vessel motions, induced by a multidirectional sea state is the segmentation of the ship fixed x,y-plane (compass rose) into an arbitrary number of sectors (like pieces of a cake). Based on that, the overall encountering wave train is decomposed into sector wise encountering wave trains, with different directions of wave propagation. Subsequently the vessel/structure responses are also derived sector wise. As each sector wise subsystem is treated linear, the overall response in 6 DOF (degrees of freedom) is calculated superimposing the sector wise computed responses. Thus, the dependency of the system response to the wave encounter angle is accounted for. The workflow and all procedures are presented in chapter 6.

The examples given within this thesis show, that reliable predictions of irregular, natural sea states can be achieved by a deterministic, linear approach. The time frames and spatial ranges for reliable predictions depend on the position in time and space a forecast is conducted for. Furthermore they are dependent on the characteristic of the sea state. The coherence between sea state spectrum and predictable range (in time and space) is approximated by the ‘prediction horizon’, which is elaborately described in chapter 3. The prediction horizon is calculated and illustrated for every forecast exemplified within this work. This enables not only the forecast of sea states and wave induced motions, but also the identification of the time frames for acceptable predictions. Thus, critical or uncritical situations can be recognized in advance and comprehensive decision support can be given. The thesis closes with comprehensive conclusions and an outlook to future work.

Chapter 2

State of the art and relevant theoretical basics

The basis for a deterministic forecast of ocean waves and wave induced vessel motions is the identification of the exact sea state parameters to be found at present. Therefore the wave field within a given coordinate system is recorded either by spatially fixed time domain registrations, e.g. wave buoy measurements or by taking snapshots of the free water surface at a certain instance in time. 3D

3D

z

ζ(x,y)

y

y ζ(x,y)

z

x

wave on of directi ation g propa

x

d pr irect op io ag n o at f io w n av e

=

n1

z ζ(x,y)

y

ζ(x,y)

+

2D

y z

x

z

ζ(x,y)

n1

z=0 x

d

domain Ω z=-d

z y

ζ(x,y)

+

ζ(x,y)

n2

z=0

z

x

z ζ(x,y)

y

x

domain Ω

x

decomposition of an irregular multidirectional sea state into unidirectional irregular sea sates with different directions of propagation

Fig. 2.1: Multidirectional and unidirectional sea states

n3

z=0

z

n3

ζ(x,y)

z

y ζ(x,y)

d

z=-d

n2

direction of wave propagation


ζ(x,y) ve wa of on on cti ati re g di ropa p

dir e pro ction pag of atio wav e n

z

y

domain Ω

x

d

z=-d

reduction of the irregular, unidirectional 3D-wave field into an irregular, unidirectional 2D-wave registration

8

CHAPTER 2

STATE OF THE ART AND RELEVANT THEORETICAL BASICS

Given a 2-dimensional time or space dependent wave registration – z(t) or z(x) – the sea state can be analysed by either linear or non-linear methods of higher order. For a linear approach the wave field is decomposed into harmonic wave components using for example a Fast Fourier Transform (FFT), determining the amplitude, frequency and phase information for each underlying wave component. In order to analyse a multidirectional sea state with respect to linear wave theory it primarily has to be decomposed into 2-dimensional time or space dependent wave registrations – z(t) or z(x) – as is illustrated in Fig. 2.1. The following section (2.1) introduces the frequency domain analysis using a 1-dimensional Fourier transform.

2.1

1-dimensional Fourier Transform

A time domain wave registration z(t) or registration of body responses sj(t) can be represented in frequency domain by its Fourier transform ∞

F (ω) = ∫ ζ (t )e −iωt dt

(2.1)

−∞

with w=2pf being the angular wave frequency, t the time and i = −1 the imaginary unit. The Fourier transform can also be written as (w ) F (w) = F (w) ei argF phase

(2.2)

absolute value information ( amplitude )

The inverse operation, the so-called inverse Fourier transform yields the original time domain registration. ∞

1 ζ (t ) = F (ω) e −iωt d ω ∫ 2π −∞

(2.3)

Within this work not time but space dependent registrations (snapshots of the water surface elevation over the spatial domain) are used for developing the respective applications. The Fourier transform of these registrations turns out as ∞

F (k ) = ∫ z ( x )e −ikx dx

(2.4)

−∞

or (k ) F (k ) = F (k ) eiargF phase

(2.5)

absolute value information ( amplitude )

and in its inverse form as ∞

z ( x ) = ∫ F (k ) e −ikx dk

(2.6)

−∞

For a discrete finite registration with N being the number of samples, a sample rate of Dt the resolution in frequency domain is Dw=2p/(N Dt). In space domain the resolution is Dk=2p/(N Dx). The Fourier transform can then be written as the following summation

1-DIMENSIONAL FOURIER TRANSFORM

N −1

9

F (r ∆ω) = ∆t ∑ ζ (δ∆t ) e

−i 2 πr δ N

for

r = 0,...,

for

r = 0,...,

δ =0

N

(2.7)

2

for transferring time domain registrations, and N −1  ∆x  −i 2 πr δ N = ∆ F r x ζ (δ∆x ) e  ∑  L  δ =0

N

(2.8)

2

for the transformation of space domain registrations. The summation can be completed by the Fast Fourier Transform (FFT). Its inverse form, the Inverse Fast Fourier Transform (IFFT) can be written as ζ (δ∆t ) =

N 2

∆ω i 2 πr δ N F (r ∆ω) e ∑ 2 π r =0

for

r = 0,..., N − 1

for

r = 0,..., N − 1

(2.9)

for the transformation into time domain, and N 2

ζ (δ∆x ) = N ∆x ∑ F (r ∆x ) e

i 2 πr δ N

(2.10)

r =0

for the transformation into space domain. The outcome of the Fourier analysis of a time domain registration is the Fourier wave spectrum F (ω) = FFT (ζ (t ))

(2.11)

With w=2p/T being the angular frequency and T being the wave period. The magnitudes of the corresponding amplitude spectrum follow from Eq. (2.12) ζ a ,i =

F (ωi ) ⋅2 N

(2.12)

Further on the magnitudes of the sea state spectrum or energy density spectrum can be derived using Eq. (2.13) S (ωi ) =

ζ a2,i 2∆ω

(2.13)

In case of a space domain registration z(x) the Fourier analysis yields the following F (k ) = FFT (z ( x ))

(2.14)

With k=2p/L being the wave number and L being the wave length. The magnitudes of the corresponding amplitude spectrum follow from Eq. (2.15) z a ,i =

F (ki ) ⋅2 N

The magnitudes of the sea state spectrum follow from Eq. (2.16)

(2.15)

10

CHAPTER 2


S (ki ) =

z a2,i 2∆k

(2.16)

Note, that only the magnitudes of the spectrum are transferred. The phase information of each wave component stays constant. As with the linear approach the random sea state is seen as the superposition of harmonic wave components, the sea state spectrum S(w) describes the distribution of the component wise wave energy as part of the overall energy contained within the sea state. ∞

∞

E ∼ ∫ S (w ) d w

or

0

E ∼ ∫ S (k ) dk

(2.17)

0

Given that the direction of wave travel is known, the propagation of the wave field can be calculated by linear wave theory, assuming that all waves are travelling in the same direction (2D-problem).

2.2

Linear wave theory (Airy)

The linear or Airy-wave theory is a solution of the Laplace equation, assuming uniform, mean water depth, impermeable ocean bottom and inviscid, incompressible and irrotational flow. Although Airy-wave theory considers small amplitude waves it is often applied in ocean and coastal engineering, giving a description of the wave kinematics and dynamics of high enough accuracy for many purposes. Using linear wave theory the celerity of each wave component is only dependent on its wave length. This correlation is described by the dispersion relation. w = kg tanh (kd )

d =water depth

(2.18)

with k=

2p L

(2.19)

being the corresponding wave number. Hence, the celerity for linear waves is ci =

g tanh (ki d ) ki

(2.20)

Note, that whereas the phase of the wave travels with the wave celerity, as given in Eq. (2.20), the energy propagates with the group velocity c gr ,i =

ci 2

  1 + 2kd   sinh (2kd )   

(2.21)

The water wave problem is considered of harmonic 1st order. As the celerity of each wave component stays constant and wave/wave interaction is neglected the corresponding free surface elevation for each wave component and thus its propagation is obtained as ζi ( x , t ) = ζ a ,i cos (ω (ki ) t + ki x + ϕi )

(2.22)

The irregular overall free water surface elevation results from a superposition of the element wise calculated surfaces

PREDICTION HORIZON / PREDICTION ERROR

11

N

z ( x , t ) = ∑ zi ( x , t )

(2.23)

i =1

Considering a higher order approach the celerity of the wave is not only dependent on its wave length but also on its amplitude. In addition to that the wave components interact. The equations describing non-linear waves include terms of higher order and the shape of the wave changes – higher, steeper wave crests and lower, smoother wave troughs. The superposition does not generally apply as additional interaction terms complicate the calculation of an irregular wave field. Thus, with higher order the computational effort considerably increases to a level that is not acceptable or at least questionable for instantaneous prediction. At the same time, the prediction error does not necessarily decrease by the same rate. Based on the work of Morris [13], Wu [23] introduced the approach of reconstructing/forecasting the evolution of the sea state based on time domain registrations taken from wave buoy measurements. He successfully proved that a sea state can be satisfyingly predicted taking into account non-linearities in wave kinematics using a low-order Stokes solution and a high-order-spectral (HOS) non-linear wave model. Morris also introduced an approach to estimate the error of prediction.

2.3

Prediction horizon / Prediction error

The prediction of a wave field is mathematically not limited if no restrictions are put to the spatial or time domain. However, in practice the forecast is bound to limitations. The validity of the forecast is therefore determined by the wave components that limit the spectral range of the sea state. Hence, the longest as well as the shortest wave components within the spectrum define what was introduced by Morris [13] and furthermore described by Naaijen ([14] and [15]), Wu [23] and Blondel-Couprie [2] as the predictable area/zone and is referred to within this work as the prediction horizon. Within the referenced publications the prediction horizon was exemplified for forecasts computed from time domain registrations as is illustrated in Fig. 2.2. The parallelogram describes the area in which either prediction or reconstruction of the sea state is possible. The theoretical prediction error within that area is defined as e=0. Assuming that the spectral information (wave components) is derived at t1 (with the time domain registration covering a length of Dt=t1-t0), it is self-evident that for instances in time where t0 ≤ ti ≤ t1 only reconstruction is possible (blank area), whereas for all instances t1 0.2, with D being the structural diameter), it can be observed that this structure significantly influences the incident wave field, causing scattering and radiation. Scattering is the alteration of the incident wave field caused by the body, i.e.

FORECAST OF WAVE INDUCED MOTIONS

15

reflection and deflection. Radiation, on the other hand, is the generation of waves due to body motions which, due to their nature, radiate away from the body. Both effects together are called diffraction. Within this work, the potential theory code WAMIT [22] is used to calculate vessel motions and to determine the transfer functions, also called RAOs (Response Amplitude Operators).

2.5.1.1

Boundary condition on the wetted body surface

With a body floating in waves, a new boundary value problem along the wetted body surface arises, which has to be solved. This boundary condition ensures that no fluid particle can pass through the wetted body surface SB and therefore turns out as ∂Φ sT n = ∂n

(2.28)

Based on potential theory, the overall wave field is in the following described by the superposition of the a) incident wave field b) radiation wave field c) scattering wave field Hence, each effect can be described by an individual velocity potential Φ=

Φ 0

+

n

∑Φ j =1

j

+

incident potential

Φn+1

(2.29)

scattering potential

radiation potential

where n is defined by the number of degrees of freedom of the body, since motions in each degree of freedom create their own radiation wave field, hence velocity potential. As for a freely moving three dimensional body, the number of degrees of freedom (DOF) is n=6. Eq. (2.29) turns out as Φ=

Φ 0

incident potential

+

6

∑Φ

j

+

1 j=

Φ7

(2.30)

scattering potential

radiation potential

where the total potential F is defined as a superposition of various wave systems and their potentials, respectively, assuming a linear problem (Newman [17]). Surly each velocity potential F0...7 and thus the total potential has to satisfy the Laplace equation ∂ 2Φ ∂ 2Φ ∂ 2Φ + + = ∇∇Φ = ∆Φ = 0 ∂x 2 ∂y 2 ∂z 2

(2.31)

Furthermore the following boundary conditions have to be taken into account for deriving F: • generalised, linearised free surface boundary condition ∂Φ ω 2 − Φ=0 ∂t g

(2.32)

16

CHAPTER 2


• bottom boundary condition w=

∂Φ =0 ∂z

for z = −d

(2.33)

as well as the • boundary condition on the wetted body surface (Eq. (2.28)) and the • Sommerfeld radiation condition (see section 2.5.1.2)

2.5.1.2

Sommerfeld radiation condition

The Sommerfeld radiation condition ensures that for F1 to F7 the waves travel away from the body generating them, decaying towards infinity.   ∂Φj − ik Φj  = 0 lim R   ∂R 

for j = 1,..., 7

R→∞

2.5.1.3

(2.34)

Solution to the boundary value problem

Looking again at Eq. (2.30), the potentials of the incident wave field (the undisturbed wave field) F0 can be derived from linear wave theory according to Clauss [10] Φ0 =

ζ a g cosh  k ( z + d ) cos θ ω sinh (kd )

(2.35)

The six radiational velocity potentials can be transferred into six centre of gravity velocities s j and six local body potentials fj , respectively. Φ j = s j y j

for j = 1,..., 6

(2.36)

Hence, j denotes the six degrees of freedom of the body in the following order surge (1) sway ( 2) (3 ) heave

roll ( 4) pitch (5) yaw (6 )

translations

rotations

The boundary problem is solved adopting Green’s second identity

∫∫∫ Ω

 Φ( ∆G ) − G ( ∆Φ) dV =  

∫∫ SB

 ∂G ∂Φ  Φ  ∂n − G ∂n dSB

(2.37)

where G ( x , ξ ) is the Green function. Applying this, the differential equation over the fluid domain W can be transferred into integrals over the wetted body surface SB. The integral equations for the six radiation potentials F1,..., F6 are

FORECAST OF WAVE INDUCED MOTIONS

2πψ j ( x ) + ∫∫ ψ j ( ξ )

17

∂G ( x , ξ ) ∂n

SB

for

dSB = ∫∫ G ( x , ξ )

∂ψ j ( ξ )

SB

∂n

dSB

(2.38)

j = 1, ..., 6

For the scattering potential F7 the solution turns out to be 2πΦ7 ( x ) + ∫∫ Φ7 ( ξ )

∂G ( x , ξ ) ∂n

SB

dSB =

∫∫ G ( x , ξ ) SB

∂Φ7 ( ξ ) ∂n

dSB

(2.39)

The incident potential F0 can be combined with the scattering potential F7 . Thus, the integral equation for the total potential FD= F0+ F7 comes out as 2πΦD ( x ) + ∫∫ ΦD ( ξ )

∂G ( x , ξ ) ∂n

SB

dSB = 4πΦ0 ( x )

(2.40)

Discretising the body surface into a finite number of panels, the integrals shown above can be solved numerically, considering a constant potential on each panel.

2.5.2

Hydrodynamic Forces and Motions

Physically, the total dynamic force acting on a floating body arises from the integration of the dynamic pressure over the wetted body surface SB. Fdyn =

∫∫

pdyn n dSB

(2.41)

SB

According to the linearised Bernoulli equation ∂Φ + p + ρ gz = p0 ∂t ∂Φ −ρ − ρ gz = p − p0 ∂t hydrostatic pressure

ρ

with z = ζ ( x , t )

(2.42)

dyn . pressure

the hydrodynamic pressure turns out as pdyn = −r

∂Φ ∂t

From Eq. (2.30) and Eq. (2.36) follows 6   ∂Φ ∂Φ pdyn = −ρ  0 + 7 + ∑ s j ψ j    ∂t ∂t j =1

Combining Eq. (2.43) and Eq. (2.41) yields

(2.43)

18

CHAPTER 2


 ∂Φ ∂Φ  Fdyn = − ρ ∫∫  0 + 7 n dSB − ρ ∫∫ s j ψ j n dSB  ∂t ∂t  SB SB

(2.44)

dynamic internal force

dynamic exitation force

The dynamic internal force depends on the added mass A as well as on the potential damping B. Additionally, a static internal force has to be considered, which pays attention to the hydrostatic restoring coefficient C, due to a change in buoyancy.   Fint ,stat = C s = −r g ∫∫ n z dSB

(2.45)

SB

Finally, the equation of motion describes the equilibrium of forces based on Newton’s second law. Generally, this equation is written in the form (2.46)

( M + A ) s + B s + C s = FE Thus, the body motion for every degree of freedom can be derived from 6

∑ −w (M 2

j =1

ij

+ Aij ) − i wBij + C ij  s j = FE , j 

(2.47)

Note that the added mass (A), potential damping (B) and restoring force coefficients (C) as well as masses and moments of inertia (M) come in matrix form  A11   0   A31 A =   0 A  51  0 

0 A22 0 A42 0 A62

0  0  0 C =  0 0   0

2.5.3

A13 0 A33 0 A53 0

0 A24 0 A44 0 A64

A15 0 A35 0 A55 0

0 0 0 0 0 0 0 0 0 C 33 0 C 35 0 0 C 44 0 0 C 53 0 C 55 0 0 0 0

0   A26   0  A46  0  A66  0  0  0 0 0 0

 B11  0   B31 B =  0 B  51 0 

0 B22 0 B42 0 B62

B13 0 B33 0 B53 0

0 B24 0 B44 0 B64

B15 0 B35 0 B55 0

0   B26   0  B46  0  B66 

 m 0 0 0 mzG −myG     0 m 0 −mzG 0 mxG      0 0 m my − mx 0 G G  M =  −mzG myG I xx − I xy − I xz   0  mz I yy 0 −mxG − I yx − I z   G −my mxG I zz  0 − I zx − I zy  G

Transfer functions

Transfer functions (Response Amplitude Operators – RAOs) are widely used, in order to describe the linear motion behaviour of a body exposed to a linear sea state. Assuming linear behaviour the time dependent exiting wave train can be transferred into frequency domain, decomposing it into complex components, each describing a harmonic wave. From the complex value the wave amplitude za(wj), the angular wave frequency wj and the phase shift jj can be derived.

MULTIDIRECTIONAL SEA STATES

19

Due to the linearity of the system, the response of a body floating in waves is again of harmonic form, having the same frequency wj , but different amplitude sa and phase shift jj . The ratio of response sa(wj) to excitation za(wj) gives the transfer function H (ω j ) =

s a (ω j )

ζ a (ω j )

e iϕ

for

j = 1,..., n

(2.48)

The transfer function is either written in a form of absolute values and phase shifts or as form of complex components. Fig. 2.5 illustrates the correlation between wave excitation (cause) and response (effect) in a linear system.

2.6

Multidirectional sea states

Clauss, Kosleck et al. ([5] and [8]) enhanced the approach introduced in section 2.5 (see also [4]) in order to be able to predict the evolution of multidirectional wave fields and the subsequently derived vessel motions. The presented procedure consists of three main steps: • Forecast of the short crested, multidirectional sea state either in the near or in the far field of a floating or fixed structure from present to the near future using radar images. • Forecast of encountering wave trains from present to the near future considering the multidirectionality of the sea state by a sector-wise, calculation of encountering wave trains within the ship/structure -fixed coordinate system. • Time-domain forecast of ship motions for six degrees of freedom as a superposition of the sector wise determined vessel responses (according to the sector-wise calculation of encountering wave trains). As the sector wise vessel encountering wave trains as and the vessel transfer functions (RAOs) are dependent on the wave encounter frequency, hence the cruising speed, procedures for the correct adaption of the vessel RAOs were introduced as well. Based on the work referenced above this thesis focuses on a deterministic, linear approach for the prediction of unidirectional and multidirectional natural wave fields and resulting wave induced vessel motions. Basis for all forecasts are surface elevation snapshots of the surrounding wave field from close vicinity to the far field. A variety of examples for sea states with different characteristics are comprehensively investigated and general conclusions for the prediction of natural sea states and wave induced vessel motions are drawn.

time domain (time series -> ζ(t))

20

CHAPTER 2


input signal (cause / wave excitation)

output signal (effect / response)

transfer system

a.pp.

Lpp/2

f.pp.

z

(0,0,0)

x

ζ (t ) = ∑ ζ a ,n e −iωnt

s (t ) = ∑ sa ,n e

−i (ωn t +εn )

ζ1 / ω1

s1 / ω1

ζ2 /ω2

s2 /ω2 .... .... sn / ω3

ζn (t ) = ζ a ,n e −iωnt time ->

response amplitude operator (RAO)

frequency domain

H (ω ) =

sa ,n iεn e ζ a ,n

phase angle

phase angle

Fζ (ω) [m]

Fourier response spectrum Fs (ω)

=

amplitude response spectrum

F (ω) ⋅2

sa =

[m]

squared magnitude of RAO

N

Szz(ω) =

ζ a2 [m²s] 2∆ω

H (ω )

∫S

zz ,n

2

Sss (ω) =

=

∞

(ω ) d ω

0

s a2 2d ω ∞

2 sS = 4

∫S 0

Fig. 2.5: Response of vessels/offshore structures in random seas

phase angle

HS = 4

response spectrum

SS ,n

(ω ) d ω

frequency domain

sea state spectrum / energy density spectrum

phase angle

frequency domain

Fs (ω) ⋅2

phase angle

N

phase angle

frequency domain

amplitude spectrum

phase angle

−i (ωn t +εn )

time ->


ζa =

sn (t ) = sa ,n e

frequency domain

.... .... ζn / ω3

time domain (response components )

response components

frequency domain

component waves

phase angle



Chapter 3

2-D wave forecast for unidirectional long crested sea states

3.1

General approach

At open sea, the phenomenon of 2D unidirectional, long crested waves all having the same direction of propagation is only a very rare and special event, e.g. during swell conditions. Nevertheless, for the understanding of wave travel and the propagation of ocean wave fields in time and space, the assumption of 2D unidirectional wave travel provides an excellent basis, as these conditions are comparatively easy to investigate at model scale in nearly every modern wave flume/tank, such as the seakeeping basin at the Technical University Berlin. The fundamental basis for predicting ocean wave fields and wave induced motions is a comprehensive knowledge of the actual sea state condition. Therefore, in situ measurements as data base for predictions are inevitable. There are mainly two different ways of collecting information on the present ocean wave field: a) Registration of the water surface elevation from close vicinity to the far field at certain instances in time –> snapshots z(x,y,t=const.) of the water surface elevation b) Time domain registrations of the water surface elevation at fixed positions in space –> e.g. wave gauge measurements z(x=const.,y=const.,t) This work focuses on the prediction from snapshots of the water surface. The accuracy of the wave information that is obtained via surface elevation snapshots is considerably dependent on the snapshot range and the spatial resolution chosen and is given in space domain. The snapshot is analysed in frequency domain resulting in a spectrum consisting of harmonic wave components defined by amplitude, frequency and phase information. Based on linear wave theory, the propagation of every wave component is calculated in time and space. Fig. 3.1 shows the procedure schematically. The following sections give a comprehensive overview of the linear methods used for the deterministic prediction of unidirectional, natural sea states. Section 3.4 introduces the investigated sea states, followed by some preliminary comments on the conducted wave forecasts (section 3.5) and a comprehensive prediction example in section 3.6.

CHAPTER 3

2-D WAVE FORECAST FOR UNIDIRECTIONAL LONG CRESTED SEA STATES

i 1 L )x


wave number

dispersion

k=2π/L

w = kg tanh (kd )


Fwave (1 L)

ζa =

F (ω) ⋅2 N


Fwave (k )

Fwave (ω)

n

ζ j ( x , y , t ) = ∑ ζ a ,i , j e i =1

j= 1,...,m

(with m=number of sectors) (with n=number of wave components)


3) superposition of sector wise derived responses to overall response of vessel/structure (in 6DOF)

H i ,l , j (ω) =

sa ,i , j ζ a ,i , j

e iϕn

phase angle

l = 1,..., 6 DOF j = direction

time domain (wave components ) frequency domain

2) sector wise calculation of responses in 6 DOF

6 DOF

decomposition of overall encountering wave train into sector wise encountering wave trains

ζ (t ) = ∑ ζ j

Σ

j =1

time ->


Fj ( ω )


6 DOF

frequency domain


6 DOF

phase angle

Fs ,l , i , j (ω) l = 1,..., 6 DOF j = direction

m

s j (t ) = ∑ sl , j j =1

6 DOF

j= 1,...,m

(with m=number of sectors) (with n=number of wave components) n

sl , j (t ) = ∑ sa ,i ,l , j e i =1

−i (ωi , j t +ϕi , j )

time domain (time series)


IFFT output signal (effect / response)

predicted encountering wave train at CG

= 6 DOF

phase angle

frequency domain

= Fourier response spectrum

superposition method

ζ(x,y,ti)


6 DOF

m

Σ


6 DOF



2D-FFT method

predicted surface elevation snapshot

phase angle

Fj (ω)

ζ(x,y,t1)

component waves

1) sector wise calculation of encountering wave trains


signal decomposition via 3D-FFT

snapshot of the free water surface


ζa(kx,ky)

m

frequency domain

time ->

ζ(x,y,t1)

m

ζ1,j / ω1,j ζ2,j /ω2,j .... ζ (t ) = ζ e −i(ωi , j t +kx , j x +k y , j y +ϕi ) a ,i , j ζ3,j / ω3,j i , j

phase angle

frequency domain


component waves

frequency domain

−i (ωi , j t +kx ,i , j x +k y ,i , j y +ϕi )


phase angle


with t=const.


space domain ζ(x,y,ti)

phase angle

ζ a (ω )




frequency domain

space domain ζ(x)

IFFT-method (section 4.3.2)




ζ ( x ) = ∑ ζae (

array of subsequently taken snapshots


space domain ζ(x,y)


space domain ζ(x,y)

3D WAVE FIELDS multidirectional sea states

phase angle

frequency domain

space domain ζ(x)

2D WAVE FIELDS unidirectional sea states

frequency domain

22

Fig. 3.1: Schematic flow chart for the prediction of 2D unidirectional long crested waves from surface elevation snapshots (highlighted area) (for enlarged overview see Fig. C.1)

PREDICTABILITY OF 2D WAVES

23

Further forecasts for varying sea states and snapshot ranges can be found in Appendix A, sections A.1 to A.8. Section 3.7 summarises the results.

3.2

Predictability of 2D waves

This section aims at the investigation of the validity of a forecast based on linear methods. Therefore, following from the wave spectrum, the predictable range in time and space is deduced and the prediction horizon or predictable zone is calculated.

3.2.1

Definition of the coordinate system z,ζ direction of wave travel

ζa,i

Hi

x,kx

L

sea bottom

z=-d

Fig. 3.2: 2D - coordinate system

Note, that for all examples calculated during the prediction of unidirectional sea states the origin of the y-position (x=0m) is by definition located at the trailing edge of each snapshot.

3.2.2

Determination of sea state parameters from surface elevation snapshots

In order to derive the sea state parameters enclosed in the space domain registration, the snapshot is analysed in frequency domain by means of FFT. Contrary to the spectral information obtained from a time domain registration the spectrum derived from a space domain snapshot is not given over the angular wave frequency w but over the wave number k, where the increment is ∆k =

2p Lmax

with

Lmax = SL

(3.1)

with SL being the snapshot range and thus the length of the longest wave within the spectrum. The shortest wave within the spectrum is defined by 1 N = → Lmin 2 ⋅ SL 1 N +1 = → 2 ⋅ SL Lmin

2 ⋅ SL N 2 ⋅ SL Lmin = N +1 Lmin =

for ( N = even)

(3.2)

for ( N = uneven)

whit N being the number of samples within the snapshot. Hence, the corresponding minimum and maximum wave numbers are

24

CHAPTER 3


2p 2p = Lmax SL N p = ∆k = N 2 SL ( N + 1) p = ∆k = ( N + 1) 2 SL

kmin = ∆k = kmax kmax

for ( N = even)

(3.3)

for ( N = uneven)

From the FFT follows the Fourier spectrum. The magnitudes of the amplitude spectrum are deduced using Eq. (3.4), whereas the magnitudes of the sea state spectrum are subsequently be derived as described by Eq. (3.5). z a ,i (ki ) = F (ki ) ⋅

2 N

(3.4)

z a2,i (ki ) S (ki ) = 2 ⋅ ∆k

(3.5)

ζ(x) [m]

Fig. 3.3 exemplifies the procedure. Note, that only the absolute values of the spectra change, the phase information stays constant. In order to calculate the wave propagation according to linear theory the angular frequencies w of each wave component are calculated using the dispersion relation (Eq. (2.18)). Because of the nonlinearity of the dispersion relation, the sampling points of the converted spectrum, given over the angular frequency, are no longer equidistant (see Fig. 3.4). That limits the use of IFFT for re-conversion of the spectral information into time domain, since equidistant sampling points are inevitable. Hence, only space domain registrations z(x, t=const.) can directly be calculated by applying IFFT to the spectrum given over the wave number k. snapshot x [m]

FFT ζa(k)

|F(k)|

S(k)

Fourier spectrum

amplitude spectrum

sea state spectrum

k

k

k

phase angle

ζ a ,i (ki ) = F (ki ) ⋅

2 N

S (ki ) =

ζ a2,i (ki ) 2 ⋅ ∆k

Fig. 3.3: Snapshot of the water surface (space domain) and correlated spectra (frequency domain)


ζa

25

amplitude spectrum over wave number

k ζa

amplitude spectrum over angular wave frequency

ω Fig. 3.4: Spectra with equidistant and non equidistant samples

3.2.3

Characteristics and resolution of the derived sea state parameters

Given a snapshot as shown in Fig. 3.3 with a spatial dimension SL and a resolution of N sampling points, the spectral analysis via FFT results in a spectrum with N/2 samples, for N being even or (N+1)/2 samples, for N being uneven. The longer SL the longer Lmax and the smaller the spectral increment. Simultaneously, for constant values of N, the shortest wave becomes longer. With that, the length of the longest wave can only be influenced by the snapshot length, whereas the shortest wave is not only influenced by the snapshot length but also by the number of samples within the snapshot. Each of the derived wave components is defined by its wave number/frequency, its magnitude and phase information, which can be converted into complex numbers.

26

CHAPTER 3

3.2.4


Prediction horizon tan τ=cgr,max

x [m]

(group velocity of the longest wave component within the derived spectrum)

prediction from a snapshot maximum period of time for prediction

maximum period of time for reconstruction

α

xe no prediction possible dge

t-1

area where reconstruction is possible

t0

PAST

hor izon “

ta,2

period prediction at position x1 no reconstruction possible

„pr edic e of edg

ling

no prediction possible

tp,1(x1)=te,1(x1)- ta,1(x1)

α

tion

area where prediction is possible

ta,1

(group velocity of the shortest wave component within the derived spectrum)

o

trai

no reconstruction possible

snapshot covers a length of ∆x=x1-x0

direction of wave travel

lea

τ x-2

e ing

d

x1 x0

o

cti

di pre f„

“

on

riz

o nh

maximum spatial range for prediction

x2

x-1

tan α=cgr,max

τ

te,1

time [s]

tend te,2

tp,2(x2)=te,2(x2)- ta,2(x2) period of prediction at position x2

FUTURE

Fig. 3.5: Prediction horizon for a forecast based on a snapshot of the water surface (space domain registration)

The prediction horizon for a forecast based on data taken from a snapshot of the surface elevation is basically the same as that for a forecast based on a time domain registration (see section 2.3). Note, that the area in which reconstruction is possible notably changes. The exemplification is given in Fig. 3.5, where the striped triangle shows the area for prediction whereas the blank area shows the reconstruction zone. Again, the group velocity of the longest wave cgr,max as well as that of the shortest wave cgr,min define the slopes that delimit the prediction horizon (leading and trailing edge of the prediction horizon). The longest wave length Lmax is correlated to the smallest angular frequency wmin and the smallest wave number kmin but the highest group velocity and vice versa.


27


x [m]

fastest (longest) wave τ

tan τ= cgr,max α tan α= cgr,min

xe

x0

slowest (shortest) wave

x-2

time [s]

t1

ti

wave spectrum, consisting of n harmonic wave components, is derived (Lmin,cgr,min - green; Lmax,cgr,max - blue)

x-2

PAST

harmonic wave components travel at different speeds (Lmin,ωmax - green; Lmax,ωmin - blue)

te fastest (longest) wave component has passed slowest (shortest) wave component - no prediction beyond this point (Lmin,ωmax - green; Lmax,ωmin - blue)

FUTURE

Fig. 3.6: Wave travel through prediction horizon based on dispersion relation

Fig. 3.6 reveals the physical phenomenon on which the limitations of the prediction/reconstruction horizon are based. At t1, all wave components derived from the snapshot are completely overlapping. As waves due to dispersion relation are propagating at different velocities (longer waves are faster than shorter waves), the longer waves within the spectrum start overtaking as is pointed out for ti > t1. At that instant in time, a valid prediction is only possible for the area where all wave components are still partly overlapping. At te the longest wave has completely overtaken all other components, so that theoretically no valid prediction is possible beyond that point in space and time. For the determination of the prediction horizon the propagation of the energy contained within each wave component is investigated. Hence, the focus is put on the group velocity (see Eq. (2.21)), being

28

CHAPTER 3


half the wave celerity. The trailing edge as well as the leading edge of the prediction horizon are thus defined by the group velocities of the longest as well as of the shortest wave component, respectively. From Eq. (3.1) and Eq. (2.18) follows ωmin = kmin g tanh [ kmin d ] =

 2π  2π g tanh  d   SL  SL

(3.6)

The corresponding group velocities are derived using

c gr ,max

c = max 2

  1 + 2kmin d  = c max  sinh(2kmin d )  2  

      4pd 1 +   4pd      SL ⋅ sinh   SL   

(3.7)

with g

c max =

kmin

tanh [ kmin d ] =

 2 pd  gSL tanh    SL  2p

(3.8)

For L=Lmin follow kmax =

2p Np = Lmin SL

ωmax = kmax g tanh [ kmax d ] =

(3.9)

Nπ  Nπ g tanh  d  SL  SL

(3.10)

and c c gr ,min = min 2

  1 + 2kmax d  = c min  sinh(2kmax d )  2  

  Np 2 d    1 + SL   N p   sinh(2 d )  SL 

(3.11)

with g

c min =

kmax

tanh(kmax d ) =

gSL Np tanh( d) Np SL

(3.12)

Depending on a chosen position in space xi ,the time frame for a valid prediction can be calculated by identifying xi

= x1 + c gr ,min ⋅ t a ,i ( xi ) − t1 

t a ,i ( xi ) =

and

xi − x1 + t1 c gr ,min

(3.13)


29

xi

= x0 + c gr ,max ⋅ t e ,i ( xi ) − t1 

t e ,i ( xi ) =

xi − x 0 + t1 c gr ,max

(3.14)

The uppermost corner of the prediction horizon is defined by tmax and xmax with x1 + c gr ,min (t max − t1 )

= x0 + c gr ,max (t max − t1 )

x1 − x0 = (c gr ,max − c gr ,miin )(t max − t1 ) t max

=

( x1 − x0 )

(c gr ,max − c gr ,min )

− t1

(3.15)

and x max = c gr ,max ⋅ (t max − t1 )

(3.16)

Vice versa the valid range for prediction for a given time step ti is calculated as follows x a ,i (ti ) = c gr ,max ⋅ (ti − t1 ) + x0

(3.17)

xe ,i (ti ) = c gr ,min ⋅ (ti − t1 ) + x1

(3.18)

For predictions beyond te , new information has to be gathered in terms of succeeding snapshots as is exemplified in Fig. 3.7, focusing on the prediction horizon only. Depending on the position in space xi , the period between two succeeding snapshots should preferably equal the period for a valid prediction at that certain location Dt(xi)=te,i(xi)-ta,i(xi).

30

CHAPTER 3


continuous prediction from a snapshot

tan α=cgr,min

tan τ=cgr,max

x [m]

(group velocity of the longest wave component within the derived spectrum)

τ


α

x-1 t

0



x0 snapshot covers a length of ∆x=x1-x0


xi

t1

t3

∆t(x1) ∆t(x1)

time [s] ta,1

te,1=ta,2 te,2=ta,3

te,3

∆t(x1) ∆t(x1) ∆t(x1)

Fig. 3.7: Overlapping prediction horizons for a continuous forecast

3.2.5

Enlarged prediction horizon

As described in section 3.2.4, the prediction horizon is limited by the snapshot length as well as by the group velocities of the bounding wave components. That basically means, that a forecast can no longer be precise when the information included in only one wave component is missing. However a sea state forecast beyond these theoretical limitations will still deliver reasonable results for periods with ti Lmin ). Subsequently the group velocity of the longest wave is reduced (cgr,max > cgr,max,5% ) whereas that of the shortest wave is increased (cgr,min cgr,max,en

c gr,min,en > cgr,min

spectrum, as derived from snapshot

ω


Fig. 3.9: Enlarged prediction horizon

Prediction error The method described in section 3.2.5 has its limitations as more and more information gets lost with increasing distance beyond the original prediction horizon. Thus, at some stage, the prediction becomes increasingly imprecise. As Wu [23] pointed out, one possibility to realise the error is to define it over the loss of energy that goes along with a reduced sea state spectrum. At a given position ti,out,xi,out outside the original prediction horizon – so that ta,i,out (Eq. (3.13)) equals te,i,out (Eq. (3.14)) – ti,out,xi,out describe the uppermost corner of one specifically enlarged prediction horizon. The corresponding bounding group velocities cgr,min,out and cgr,max,out and thus the wave frequencies wmin,out and wmax,out define the prediction error e, ranging from 0 to 1.

DIFFERENT NUMERICAL METHODS

33

ωmax,out    ∫ S (ω) d ω   ω out ε (ti ,out , xi ,out ) = 1 − min,  ωmax   S ω d ω ( ) ∫ωmin  

(3.20)

Note, that within the original prediction horizon the error is e=0. Depending on the shape of the spectrum, hence the energy distribution, the distribution of the error outside the original prediction horizon is changing. Fig. 3.10 illustrates the distribution based on the spectrum as shown in Fig. 3.8. energy error distribution (ε=1=100%)

1

0.4 0.3

0.4

0.9

0.7

0

0.2 0.4 0

0.6

0.4

0.1

0.5

0.3

0.3

0.4

0.3 0.2

0.4 0.1 0 .2

00..42

0.3

−200

0.1 .3 0 2 0. .4 0

−100

0

ε=0

0.1

error in energy [%]

0.8

.3

0.3

.2 0.4 0 0.1

0.2 0.1

0 t [s]

100

200 −200

−100

0 t [s]

100

200

0

Fig. 3.10: Distribution of the prediction error as defined in Eq. (3.20)

3.2.6

Resolution of the prediction

Within the given boundaries ta,i(xi), te,i(xi) the resolution of the prediction is not bound to any limitation, neither in space nor in time and can thus be arbitrarily chosen.

3.3

Different numerical methods

In the following two different methods – the 2D-Superposition method and the IFFT method – are introduced, whose objectives are • the forecast of time domain registrations – either for a fixed position in space, e.g. the encountering wave train at the CG of a moored vessel, or a transient position, e.g. the CG of a cruising vessel as well as • the forecast of the surface elevation (snapshot) in a given space domain – for a single or for subsequently following time steps. Regardless of which method is used, the procedure starts with a frequency domain analysis of the surface elevation snapshot. This analysis results in a number of harmonic wave components. So far the two methods do not differ. As they have pros and cons each method has its special area of application as is described in the following sections.

34

3.3.1

CHAPTER 3


2D-Superposition method

Analysing the surface elevation snapshot using FFT results in complex Fourier coefficients ci – one for each spectral component. ci = ai + bi i = ri e iji

(3.21)

These complex numbers are transferred into a Fourier magnitude Fi = ri = ai2 + bi2

(3.22)

b  ji = arctan  i   ai 

(3.23)

and a phase information

according to the correlation shown in Fig. 3.11. Subsequently the wave amplitude is derived from the Fourier magnitude using Eq. (3.4). In addition the angular wave frequency w (Eq. (2.18)) and the wave number k (Eq. (2.19)) are calculated for each component. Each wave component is now identified by its amplitude za,i, its phase information ji, its angular frequency wi and its wave number ki.

Im +

c=a+bi=reiϕ a r ϕ

+

C

−

bi Re

− Fig. 3.11: Complex numbers

The amplitudes as well as the angular frequencies and corresponding wave numbers of every single wave component are not subject to change and remain constant, whereas the overall phase information q (see Eq. (3.24)) changes depending on the time (shift in time t) and position (shift in space x) a forecast (or reconstruction) is requested for. θ = −ωi ∆tm + ki ∆ x (∆tm ) + ϕi

(3.24)

According to linear wave theory the surface elevation of each harmonic wave component is described by Eq. (2.22). For a fixed position in space, a prediction of a time domain wave registration follows from the superposition of all wave components and can be calculated as follows

DIFFERENT NUMERICAL METHODS

35

ζ (∆tm ,∆ x = const .) = ∑ ζ a ,i cos (−ωi ∆tm + ki ∆ x + ϕi ) i

(3.25)

θ

where Dx=const. defines the distance between the fixed position and the point of origin of the surface registration (snapshot), with Dtm being the discrete time variable. Following from Eq. (3.25), a time domain wave registration in a moving reference frame, such as e.g. the CG of a cruising vessel, is defined as ζ (∆tm ,∆ xu (∆tm )) = ∑ ζ a ,i cos (−ωi ∆tm + ki ∆ xu (∆tm ) + ϕi ) i

(3.26)

θ

where Dxn(Dtm) now describes the time-dependent distance between the ship and the point of origin of the surface registration (snapshot) and Dtm is the discrete time variable. For predictions in a spatial domain, i.e. t=const., Eq. (3.25) is modified into ζ (∆t = const .,∆ xu ) = ∑ ζ a ,i cos (−ωi ∆t + ki ∆ xu + ϕi ) i

(3.27)

θ

With an increasing number of components in the wave spectrum and an increasing number of time steps a wave registration shall be predicted for, the 2D-Superposition method becomes more and more time consuming. Therefore, the use of the 2D-Superposition method is only expedient for the prediction of spatially fixed wave registrations or for encountering wave trains (prediction of the wave registration in a moving reference frame) but is not applicable for the prediction of the evolution of widespread wave fields.

3.3.2

IFFT method

The IFFT method also starts with the frequency domain analysis of the space domain snapshot by FFT resulting in a spectrum with the same complex Fourier coefficients as resolved with the 2D-Superposition method (see section 3.3.1). In contrast to the 2D-Superposition method, no further conversion into amplitude and phase information is necessary as the complex spectrum can be transferred directly back into time domain by the use of IFFT (Inverse Fast Fourier Transformation). As explained in section 3.2.2, the spectrum derived from a space domain snapshot z(x) is given over the wave number k, hence the IFFT of that spectrum will again result in a space domain snapshot z(x) with the same range as the input snapshot – no matter what modifications are made to the phase information of each wave component (shift in time or position shift in space). In order to directly obtain time domain registrations z(t) via IFFT the spectrum would have to be transformed in a way which assures that the wave components are no longer given over the wave number but over the angular wave frequency w. As described above, that transformation – due to the non-linearity of the dispersion relation (Eq. (2.18)) –inevitably changes the spectrum in a way that its sampling points are no longer equidistant. Hence, this new spectrum can no longer be used as an input for the IFFT, as the IFFT requests equidistant sampling points.

36

CHAPTER 3


ζ(x)

Fi

Fourier spectrum

FFT

snapshot

ϕi

angular wave frequency

ω0 ω1

ω

k0 k1

ωi

ki

ωi

ω0 ω1

k

wave number

κi

k0 k1

ωi

ki

element wise operation ∆tm

∆x0 ∆x1

time shift ∆t0

F0 F1

ω0∆tm

∆t ∆t0 1

ω ∆t ω0 ∆tm 1 m

∆xu

=

+

-

original phase information

k1∆x0 k0∆x0

ωi ∆tm

ω0 ∆t1 ω0 ∆t0 Fi

x - shift ∆xu

∆tm

=

Fi

∆x0 ∆x1

∆x0

∆t ∆t0 1

ki ∆x0 ki ∆x0 ki ∆x1

k0∆x0 k0∆x1

F0 F1

Fi

ω0 ∆t0 ω1 ∆t0

k0∆xu

ωi ∆t0

k0∆xu k1∆xu

ϕi

+

ki ∆xu

ωi ∆tm ωi ∆t1 ωi ∆t0

ϕ0 ϕ1

ϕi

ϕ0 ϕ1

ki ∆xu

ϕi

element wise operation

r

θnew=−ω ∆t + k ∆x + ϕ i

absolute value

j

Dim 3 (shift in time) r eiθ(∆x0,t1)

r eiθi =

i

l

i

new phase information

r eiθ(∆x0,t0) r eiθ(∆x1,t0)

Dim 1 (shift in space)

Dim 1 Dim 2 Dim 3

r eiθ(∆x0,tm) r eiθ(∆x1,tm)

r eiθ(∆xu,tm) r eiθ(∆xu,t1) r eiθ(∆xu,t0) Dim 2

ζ(x+∆x0 ,t+∆t1)

IFFT

Dim2

ζ(x+∆x0 ,t+∆t0) ζ(x+∆x1 ,t+∆t0)

(r eiθi)

(u . m times)

u = number spatial shifts n = number of spectral values m = number time shifts

ζ(x+∆x0 ,t+∆tm) ζ(x+∆x1 ,t+∆tm)

ζ(x+∆xu ,t+∆tm) ζ(x+∆xu ,t+∆t1) ζ(x+∆xu ,t+∆t1)

x = spatial dimension of snapshot t = instant in time at which snapshot was taken = reconstruction of original snapshot

Fig. 3.12: IFFT method for the prediction of unidirectional wave fields

Consequently, a forecast of a time domain registration via IFFT can only be achieved by calculating a series of subsequently following snapshots that include the position in space one is interested in – with the new phase information q. Fig. 3.12 illustrates the procedure. The size of the matrix reiqi is determined by the number of values within the spectrum n (Dim 2), the number of spatial shifts u (Dim 1) and the number of time shifts m (Dim 3), which the forecasts are to be calculated for (see also Fig. 3.12). The IFFT has to be executed u by m times and requires no loop over the spectral

INVESTIGATED SEA STATES

37

values n as does the 2D-Superposition method. Henceforward, the IFFT method is quick and supplementary delivers the evolution of the surrounding wave field.

3.4

Investigated sea states

Within the experimental frame of this thesis, three different tailor made, deterministic sea states with the following parameters have been investigated at a model scale of 1:70 and a model water depth of 1m: sea state I): moderate irregular sea state – HS,m=0.05m (full scale HS,f=3.5m) – Jonswap spectrum with g=3.3 – signal duration is t1,m=360s (full scale t1,f=3012s)

I model scale full scale

ω PEAK

ω MEAN

T PEAK

T MEAN

HS

γ

[rad/s] 6.0 0.72

[rad/s] 7.7 0.92

[s] 1.05 8.73

[s] 0.82 6.86

[m] 0.05 3.5

[-] 3.3 3.3

Tab. 3.1: Parameters of sea state I

sea state II): high irregular sea state – HS,m=0.13m (full scale HS,f=8.5m) – Jonswap spectrum with g=3.3 – signal duration is t1,m=275s (full scale t1,f=2301s)

II model scale full scale

ω PEAK

ω MEAN

T PEAK

T MEAN

HS

γ

[rad/s] 2.5 0.30

[rad/s] 3.2 0.39

[s] 2.51 20.94

[s] 1.95 16.29

[m] 0.12 8.5

[-] 3.3 3.3

Tab. 3.2: Parameters of sea state II

sea state III): high irregular sea state – HS,m=0.16m (full scale HS,f=11.2m) – HMAX,m=0.34m (full scale HMAX,f=23.8m) -> HMAX/HS=2.13 – signal duration is t1,m=275s (full scale t1,f=2301s) ω PEAK

ω MEAN

T PEAK

T MEAN

HS

γ

[rad/s] model scale full scale -

[rad/s] 3.9 0.47

[s] -

[s] 1.61 13.37

[m] 0.16 11.2

[-] -

III

Tab. 3.3: Parameters of sea state III

Fig. 3.14 shows the spectral distributions (wave amplitude spectra) of the investigated sea states. Note, that full scale values are given in brackets.

38

CHAPTER 3


8

ζ [m]

4

sea state I sea state II sea state III

0 −4 −8 −2500

−2000

−1500

−1000

x[m]

−500

0

500

Fig. 3.13: Sea state examples

ζa [m]

sea state I

sea state II

1

0.25

0.1

sea state III

S [m²s]

0.5

5

0

25

1

ω [rad/s]

measured spectrum

2

0

Fig. 3.14: Spectra of investigated sea states

3.4.1

50

1

ω [rad/s]

smoothed spectrum

2

0

1

ω [rad/s]

2

theoretical spectrum

Model test setup

All model tests have been conducted at the seakeeping basin of the Technical University Berlin, implying that the waves propagate within the wave tank identically as in reality but in a scaled manner. This is indeed the case, as the wave steepness kza (wave number k=2p/L , wave amplitude za) which characterizes wave propagation, is equal at full and model scale. The scaling is performed according to Froude’s law, i.e. the geometric properties wave length, wave height and water depth are scaled linearly and time according to the square root of scale. Data acquisition and control of the wave maker is performed by a single computer system, ensuring time synchronization of steering, measuring and the repeatability of the experiment. For measuring the wave elevations, resistant type wave gauges are used. The motions of the models are registered by an optical tracking system that is synchronized by the computer system. With a length of 120m, a width of 8m and a maximum water depth of 1.1m, the wave tank features an electrically driven 3-flap wave maker that can operate in piston- or flap-type mode. The wave maker is able to generate deterministic, tailor made, regular and irregular sea states with wave

INVESTIGATED SEA STATES

39

heights up to H~0.4m for regular waves and HS~0.25m for irregular seas. The angular wave frequency range is wMIN–>0 up to wMAX=12rad/s. The basin is equipped with a slope (beach) in order to minimize reflections.

3.4.2

Generation of surface elevation snapshots during model test piston type wave maker

controlling and measuring computer

wave gauges

15.0 m

1.0 m SWL

slope

incoming wave

s

45.0 m

120.0 m 8.0

m

Fig. 3.15: Model test setup (source: [4])

As no radar or photometric system is installed at the test facility the required snapshots used as input for the forecast are calculated from 451 single wave registrations taken at spacings of x=0.1m along the wave tank (the distance to the wave maker ranges from 15m to 60m). This procedure allows to register even the shortest waves with respect to the Nyquist -Shannon-Theorem (see following section). As it is not feasible to register 451 wave gauges simultaneously the measurements have to be repeated reproducing the chosen wave sequence until every gauge position has been considered. As the registrations yield the time dependent wave elevation at a certain position, the associated snapshots are found by evaluating these registrations simultaneously at chosen instants t0...t1...ti. Fig. 3.16 illustrates this process. The frequency of the wave measurements is 200Hz, thus the minimum time step size between two consecutive snapshots is t=0.005s. space domain registrations (snapshots)

0.05

ζ [m]


0 −0.05

12 10

40 35

8 30 6

25 20

t [s]

4

15 10

x [m]

time domain registrations

2

5 0

Fig. 3.16: Generation of snapshots (source: [21])

The snapshot range can be variably chosen up to a maximum range of 45m (full scale 3150m = 1.7nm). This allows for the investigation of the influence of the snapshot range on the time frames for acceptable forecasts and precision.

40

CHAPTER 3


Sample rate based on Nyquist-Frequency In order to be able to correctly reconstruct the surface elevation along the wave channel, the spatial sample rate Dx (distance between two adjacent wave registrations) has to satisfy the NyquistShannon-Theorem. The required sample rate in time domain is defined as f sample > 2 f max ⇒

1 f

with

T=

follows

Tmin > Tsample 2

(3.28)

and

f max =

1 Tmin

In space domain, the wave length L is analogue to the period T in time domain. From that follows

⇒

Lmin > Lsample 2 Lsample = Dx

with

(3.29)

Lmin > Dxmax 2

follows

The minimum wave length throughout all tests is defined by the limitation of the wave maker wMAX=12rad/s, hence the maximum spatial distance between two adjacent wave registrations shall not exceed the following 2π = 12 rad / s Tmin 2π = = 0.524 s ωmax

⇒

ωmax =

⇒

Tmin

⇒

Lmin =

⇒

∆xmax

2 gTmin = 0.429m 2π 0.429 < m = 0.215m 2

(3.30)

Eq. (3.30) shows that a chosen spatial sample rate of x=0.1m meets the conditions defined by the Nyquist-Shannon-Theorem as x=0.1m frequency domain -> time domain) and superposition over all sectors to derive over all vessel responses in 6 DOF (number of sectors in this example is m=12)

76

Fig. 5.5: wave coordinate system (top), ship fixed coordinate system (middle) and alignment of both systems (bottom)

77

Fig. 5.6: Impression of the LNG-carrier model (scale 1:70)

78

Fig. 5.7: Heave RAOs for the stationary vessel (full blue line: wave encounter angle 0°,

dashed black line: wave encounter angle 90°, dottet red line: wave encounter angle 180°)

79

Fig. 5.9: Heave RAOs for the stationary vessel (all encounter angles)

79

Fig. 5.8: Pitch RAOs for the stationary vessel (full blue line: wave encounter angle 0°, dashed black line: wave encounter angle 90°, dotted red line: wave encounter angle 180°)

79

Fig. 5.10: Pitch RAOs for the stationary vessel (all encounter angles)

80

Fig. 5.11: Surface elevation snapshot (sea state II, v=0kn, full spectrum)

81

Fig. 5.12: Prediction horizon (sea state II, v=0kn, full spectrum) – dotted line indicates vessel position

82

Fig. 5.13: Surface elevation snapshot (sea state II, v=0kn, full spectrum) – (measurements

[blue line], prediction [red line])

82

Fig. 5.14: Frequency domain analysis (heave motion) – (sea state II, v=0kn, full spectrum)

83

Fig. 5.16: Time domain registration (heave motion) – (sea state II, v=0kn, full spectrum) – (measurements [blue line, prediction [red line])

83

Fig. 5.15: Frequency domain analysis (pitch motion) – (sea state II, v=0kn, full spectrum)

83

Fig. 5.17: Time domain registration (pitch motion) – (sea state II, v=0kn, full spectrum) – (measurements [blue line, prediction [red line])

84

Fig. 5.18: Change of spectra in dependency of the vessel cruising speed

85

Fig. 5.19: Added mass matrix (highlighted coefficients are changed according to cruising velocity)

Fig. 5.20: Potential damping matrix (highlighted coefficients are changed according to cruis-

86

x

LIST OF FIGURES

ing velocity)

131

86

Fig. 5.21: Heave RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=6kn [right hand side])

88

Fig. 5.22: Pitch RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=6kn [right hand side])

88

Fig. 5.23: Surface elevation snapshot (sea state II, v=6kn, full spectrum)

89

Fig. 5.24: Prediction horizon (sea state II, v=6kn, full spectrum)

89

Fig. 5.25: Surface elevation snapshot – (sea state II, v=6kn, full spectrum) – (measurements [blue line], prediction [red line])

90

Fig. 5.26: Frequency domain analysis (heave motion) – (sea state II, v=6kn, full spectrum)

90

Fig. 5.27: Frequency domain analysis (pitch motion) – (sea state II, v=6kn, full spectrum)

90

Fig. 5.28: Time domain registration (heave motion) – (sea state II, v=6kn, full spectrum) – (measurements [blue line], prediction [red line])

91

Fig. 5.29: Time domain registration (pitch motion) – (sea state II, v=6kn, full spectrum) – (measurements [blue line], prediction [red line])

91

Fig. 6.1: Schematic flow chart for the calculation of wave induced motions in time domain for multidirectional sea states (highlighted area) (for enlarged overview see Fig. C.1)

94

Fig. 6.2: Global coordinate system for the prediction of vessel/structure responses in multidirectional sea states

95

Fig. 6.3: Ship fixed coordinate system

95

Fig. 6.4: Alignment of wave coordinate system and ship fixed coordinate system

96

Fig. 6.5: Surge RAOs for the stationary LNG-carrier at v=0kn (all encounter angles)

97

Fig. 6.6: Sway RAOs for the stationary LNG-carrier at v=0kn (all encounter angles)

97

Fig. 6.7: Heave RAOs for the stationary LNG-carrier at v=0kn (all encounter angles)

98

Fig. 6.8: Roll RAOs for the stationary LNG-carrier at v=0kn (all encounter angles)

98

Fig. 6.9: Pitch RAOs for the stationary LNG-carrier at v=0kn (all encounter angles)

99

Fig. 6.10: Yaw RAOs for the stationary LNG-carrier at v=0kn (all encounter angles)

99

132

x

LIST OF FIGURES

Fig. 6.11: Samples of snapshots taken by radar and post processed by the WaMoS II® -system

101

Fig. 6.12: Distribution of the wave encounter angles measured by the WaMoS II® -system

102

Fig. 6.13: Example for a stationary vessel (constant course heading m=180°) and the time dependent change of the wave encounter angles

103

Fig. 6.14: Predictable zone for motion predictions with a stationary vessel

104

Fig. 6.15: Dependency of the predicted overall response on the number of sectors chosen

105

Fig. 6.16: Overall results for the stationary vessel

106

Fig. 6.17: Example for a cruising vessel and the time dependent change of the wave encounter angles

107

Fig. 6.18: Surge RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=15kn [right hand side])

109

Fig. 6.19: Sway RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=15kn [right hand side])

109

Fig. 6.20: Heave RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=15kn [right hand side])

110

Fig. 6.21: Roll RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=15kn [right hand side])

110

Fig. 6.22: Pitch RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=15kn [right hand side])

111

Fig. 6.23: Yaw RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=15kn [right hand side])

111

Fig. 6.24: Predictable zone for motion predictions with a cruising vessel on a predefined course 112 Fig. 6.25: Comparison on the influence of the velocity dependent RAOs on the predicted overall responses

113

Fig. A.1: Surface elevation snapshot and reconstruction based on reduced spectrum (sea state I, range 100%)

A1

Fig. A.2: Amplitude spectrum za(1/L) [top], original/full sea state spectrum S(w) [middle], full vs. reduced sea state spectrum S(w) [bottom] (note that the dashed vertical lines indicate the boundaries of the reduced spectrum)

Fig. A.3: Distribution of the prediction error and positions for forecasts in time and space

A2

x

LIST OF FIGURES

(sea state I, range 100%) - note that both diagrams show the same error distribution

133

A3

Fig. A.4: Time domain forecasts at fixed positions in space (sea state I, range 100%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: black dashed lines, measurements: blue lines]

A4

Fig. A.5: Space domain forecasts for a range from x=-2450m to x=700m (sea state I, range 100%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: dashed black lines, measurements: blue lines]

A5

Fig. A.6: Surface elevation snapshot and reconstruction based on reduced spectrum (sea state III, range 100%)

A6


A6

Fig. A.8: Distribution of the prediction error and positions for forecasts in time and space (sea state III, range 100%) - note that both diagrams show the same error distribution

A7

Fig. A.9: Time domain forecasts at fixed positions in space (sea state III, range 100%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: black dashed lines, measurements: blue lines]

A8

Fig. A.10: Space domain forecast for a range from x=-2450m to x=700m (sea state III, range 100%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: dashed black lines, measurements: blue lines]

A9


A10


A10

Fig. A.13: Distribution of the prediction error and positions for forecasts in time and space (sea state I, range 75%) - note that both diagrams show the same error distribution

A11


A12

Fig. A.15: Space domain forecast for a range from x=-2450m to x=700m (sea state I, range 100%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: dashed black lines, measurements: blue lines]

A13


A14

134

x

LIST OF FIGURES


A15


A15


A16


A17

Fig. A.21: Surface elevation snapshot and reconstruction based on reduced spectrum (sea state II, range 75%)

A18


A19

Fig. A.23: Distribution of the prediction error and positions for forecasts in time and space (sea state II, range 75%) - note that both diagrams show the same error distribution

A20

Fig. A.24: Time domain forecasts at fixed positions in space (sea state II, range 75%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: black dashed lines, measurements: blue lines]

A21


A22


A23


A23


A24


Fig. A.30: Space domain forecast for a range from x=-2450m to x=700m (sea state II, range

A25

x

LIST OF FIGURES

135

50%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: dashed black lines, measurements: blue lines]

A26


A27


A28


A28


A30


A31


A32


A32


A33


A34


A35

Fig. A.41: Comparison of the time domain forecast for sea state I and all three snapshot ranges (SL=2450m (100%), SL=1834m (75%) and SL=1225m (50%)

A36

Fig. A.42: Comparison of the time domain forecast for sea state II and all three snapshot ranges (SL=2450m (100%), SL=1834m (75%) and SL=1225m (50%)

A36

Fig. A.43: Comparison of the time domain forecast for sea state III and all three snapshot ranges (SL=2450m (100%), SL=1834m (75%) and SL=1225m (50%)

A37

136

x

LIST OF FIGURES

Fig. B.1: Surface elevation snapshot (sea state III, v=0kn, full spectrum)

B1

Fig. B.2: Prediction horizon (sea state III, v=0kn, full spectrum)

B2

Fig. B.3: Surface elevation snapshot (sea state III, v=0kn, full spectrum) – (measurements

[blue line], prediction [red line])

B2

Fig. B.4: Frequency domain analysis (heave motion) – (sea state III, v=0kn, full spectrum)

B3

Fig. B.6: Time domain registration (heave motion) – (sea state III, v=0kn, full spectrum) – (measurements [blue line], prediction [red line])

B3

Fig. B.7: Time domain registration (pitch motion) – (sea state III, v=0kn, full spectrum) – (measurements [blue line], prediction [red line])

B3

Fig. B.5: Frequency domain analysis (pitch motion) – (sea state III, v=0kn, full spectrum)

B3

Fig. B.8: Heave RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=6kn [right hand side])

B4

Fig. B.9: Pitch RAOs for the cruising vessel (all encounter angles) – (v=0kn [left hand side], v=6kn [right hand side])

B5


B5


B6


B6


B7

Fig. B.14: Surface elevation snapshot (sea state II, v=8kn, full spectrum)

B7

Fig. B.15: Prediction horizon (sea state II, v=8kn, full spectrum)

B8

Fig. B.16: Surface elevation snapshot (sea state II, v=8kn, full spectrum) – (measurements [blue line], prediction [red line])

B8

Fig. B.17: Frequency domain analysis (heave motion) – (sea state II, v=8kn, full spectrum)

B9

Fig. B.19: Time domain registration (heave motion) – (sea state II, v=8kn, full spectrum) – (measurements [blue line], prediction [red line])

B9

x

LIST OF FIGURES

Fig. B.18: Frequency domain analysis (pitch motion) – (sea state II, v=8kn, full spectrum)

137

B9

Fig. B.20: Time domain registration (pitch motion) – (sea state II, v=8kn, full spectrum) – (measurements [blue line, prediction [red line])

B10


B10


B10


B11


B12


B12


B12

Fig. B.27: Time domain registration (pitch motion) – (sea state II, v=15kn, full spectrum) – (measurements [blue line], prediction [red line])

B13


B13


B14

Fig. B.30: Surface elevation snapshot (sea state III, v=6kn, full spectrum) – (measurements [blue line], prediction [red line])

B14


B15


B15


B15


B16


B16


B17


B17


B18

138

x

LIST OF FIGURES


B18


B18


B19


B19


B20

Fig. B.44: Surface elevation snapshot (sea state III, v=15kn, full spectrum) – (measurements [blue line, prediction [red line])

B20

Fig. B.45: Frequency domain analysis (heave motion) – (sea state III, v=15kn, full spectrum) B21 Fig. B.47: Time domain registration (heave motion) – (sea state III, v=15kn, full spectrum) – (measurements [blue line], prediction [red line])

B21


B21


B22

Fig. C.1: Overview of process steps for the deterministic, linear approach to the prediction of ocean waves and wave induced vessel/structure motions

C1

List of tables Tab. 3.1: Parameters of sea state I

37

Tab. 3.2: Parameters of sea state II

37

Tab. 3.3: Parameters of sea state III

37

Tab. 3.4: Overview of conducted 2-D wave forecasts

41

Tab. 3.5: Selected parameters of sea state II, represented by the snapshot illustrated in Fig. 3.17 (range 100%, full and reduced spectrum)

43

Tab. 3.6: Summary of results for sea state I (time domain predictions [left hand side]; space domain predictions right hand side])

47

Tab. 3.7: Summary of results for sea state II (time domain predictions [left hand side]; space domain predictions right hand side])

47

Tab. 3.8: Summary of results for sea state III (time domain predictions [left hand side]; space domain predictions right hand side])

47

Tab. 4.1: Tabular spectrum as provided by the WaMoS II® system

55

Tab. 4.2: Parameter of WaMoS II® -analysis

62

Tab. 4.3: Sea state parameter

62

Tab. 5.1: LNG-carrier – main dimensions

78

Tab. 5.2: Overview of conducted 2-D wave forecasts

81

Tab. 6.1: LNG-carrier – main dimensions

96

140

xi

LIST OF TABLES

Tab. 6.2: Parameters of WaMoS II® -analysis

100

Tab. A.1: Selected parameters of sea state I , represented by the snapshot illustrated in Fig. A.1 (range 100%, reduced spectrum)

A2

Tab. A.2: Selected parameters of sea state III, represented by the snapshot illustrated in Fig. A.6 (range 100%, full and reduced spectrum)

A6

Tab. A.3: Selected parameters of sea state I, represented by the snapshot illustrated in Fig. A.11 (range 75%, full and reduced spectrum)

A10


A15

Tab. A.5: Selected parameters of sea state II, represented by the snapshot illustrated in Fig. A.21 (range 75%, full and reduced spectrum)

A19


A23


A28


A32

Appendix A

2-D wave forecast – supplementary test runs with reduced snapshot ranges

A.1


Data base for the forecast of sea state I (with a snapshot range of 100% (2450m in full scale)) is a snapshot of the water surface elevation taken at t=150s (model scale) after the beginning of the measurements/wave generation. At that instant in time, the sea state is developed over the whole snapshot range.

Surface elevation snapshot Fig. A.1 (blue line) shows the surface elevation snapshot based on which the sea state is analysed in frequency domain, resulting in the spectra as shown in Fig. A.2. Fig. A.1 (red line) illustrates the reconstruction of the snapshot using the reduced spectrum as is given in Fig. A.2 (bottom diagram). The changes evoked by the reduction of the spectrum turn out to be marginal. leading edge of snapshot

direction of wave travel measurements for validation available (3150m)

trailing edge of snapshot

ζ(x) [m]

snapshot range 100% (2450m) 4 2 0 −2 −4 −2500

−2000

−1500

−1000

−500 0 500 x[m] snapshot (sea state: 1, snapshot range: 100%, reduced spectrum) original snapshot (blue line) / reconstruction of snapshot with reduced spectrum (red line)


A2

APPENDIX A 2-D WAVE FORECAST – SUPPLEMENTARY TEST RUNS WITH REDUCED SNAPSHOT RANGES

Sea state spectrum The reduced spectrum is highlighted by the red area (see Fig. A.2, (bottom diagram)). Instead of the original (full spectrum), the reduced spectrum consists of only 100 wave components. spectra of snapshot (sea state: 1, snapshot range: 100%, reduced spectrum) amplitude spectrum ζa(k)

0.8

amplitude spectrum (absolute values)

ζa [m]

0.6 0.4 0.2 0

0.06

0.13

S( ω) [m2s]

0.25 k [1/m]

0.31

0.38

0.44

0.50

full sea state spectrum S( ω)

8

sea state spectrum (absolute values) smoothed distribution (see Fig. 3.14)

6 4 2 0

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2

full vs. reduced sea state spectrum S( ω)

10

original sea state spectrum (absolute values) reduced sea state spectrum (absolute values)

2

S( ω) [m s]

0.19

5

0

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

sea state: water depth: snapshot range (100%): reduced spectrum: ω_{start,enlarged}:

1 70 m 2450 m yes 0.42 rad/s

ω_{end,enlarged}:

1.65 rad/s

minimum prediction error ε reduction of spectrum (by total energy): reduction of spectrum (by number of wave components):

10.7 % 1.15 % 43 %

∆t_{max,enlarged}:

227.9 s

∆x_{max,enlarged}:

679.1 m

cgr_{max}: cgr_{max,enlarged}: cgr_{min}: cgr_{min,enlarged}:

25.8 13.7 2.3 3

m/s m/s m/s m/s

2


Tab. A.1: Selected parameters of sea state I , represented by the snapshot illustrated in Fig. A.1 (range 100%, reduced spectrum)

Although the overall energy is reduced by only 1.15%, the inherent, i.e. the minimum prediction, error is 10.7%. For this example, the number of wave components can be reduced by 43% for this example. Except for A0(w0=0), being the absolute element or offset of the spectrum, the reduced spectrum now ranges from wmin=0.42 to wmax=1.65rad/s.

Prediction horizon The distribution of the energy error e based on Eq. (3.20) is shown in Fig. A.3. The theoretical horizon for correct predictions is e(x(t) )=0), see Fig. A.3. The inherent error for the reduced spectrum is e=0.1 (11%), bound by the minimum and maximum group velocities cgr,min,enl=3m/s and cgr,max,enl=13.7m/s. Hence, the upper right corner of the prediction horizon is identified by tmax,enl=227.9s after the snapshot has been taken and xmax=679.1m beyond the trailing edge of the snapshot. All values are summarised in Fig. A.3.

FORECAST FOR SEA STATE I – SNAPSHOT RANGE 100% (2450M)

A3

fixed time steps and energy error distribution (ε=1=100%) (sea state: 1, snapshot range: 100%)

fixed positions and energy error distribution (ε=1=100%) (sea state: 1, snapshot range: 100%)

t=69.6s x=245m

t=0s

0.3

0.4 0.1

.4

0.3

−3000 0.4

−200

−100

0.03.2 .4 00.1 0

0.1

0.03.2

0 0.1 0.2

0 t [s]

0.3

0.03.2 0.4

.4 0.1

x [m]

0.03.2

x=-609m

−1000

t=34.8s

.4 00.1 0

x=119m x=0m

0

−2000

x=679m

0.1

1000

x=-2450m

100

0.3

200

0.4

−200

−100

0 0.1 0.2

t=227.9s t=104.5s

0 t [s]

100

200


Time domain forecast for a fixed position in space Based on the snapshot (Fig. A.1) and the subsequently computed spectra (full and reduced) (Fig. A.2), time domain forecasts for fixed positions in space have been calculated and validated with measurement at six positions (x=-2450m, -609m, 0m, 119m, 245m, 679m). The forecast for x=2450m(see Fig. A.4) shows no compliance with measurements as consequently follows from the fact, that no information is available on waves passing that position for instances t >0s. This can also be educed from the distribution of the prediction error, as the prediction error e instantly turns 1 for t>0s. The other five examples also show that the time frame for excellent predictions can be deduced from the prediction error distribution (see Fig. A.4). Focusing on the comparison of time domain forecasts for a fixed position in space based on the full spectrum against those derived from the reduced spectrum (Fig. A.4 (diagrams 1 to 5 (from top))), it can be pointed out that for sea state I the difference are marginal. No comparison between full and reduced spectrum is available for x=679m. However the comparison between the prediction deduced using the reduced spectrum and measurements taken at x=679m (Fig. A.4 (diagram 6 (from top)) reveal good agreement from t~80s to t~180s. In summary, that the time domain forecasts, calculated using the reduced spectrum, show only insignificant changes compared to those deduced using the full spectrum. Hence, the comparison shows excellent to good similarities up to e(x(t))~0.1. Thus the reduction of the spectrum focusing on the energy conservation is reasonable and does not deteriorate the results.

A4


ζ [m]

Forecast at fixed positions (sea state: 1, snapshot range: 100%, reduced spectrum and full spectrum) position: x = −2450 [m] = at beginning of snapshot 4 2 0 −2 −4

0

50

100

ζ [m]

position: 4 2 0 −2 −4

0

50

ζ [m] ζ [m] ζ [m] ζ [m]

4 2 0 −2 −4

4 2 0 −2 −4

4 2 0 −2 −4

0

50

0

50

0

50

0

50

200

250

200

250

200

250

150 x = 245 [m] = xmax

200

250

150 x = 679 [m] = xmax,enlarged

200

250

200

250

100 position:

4 2 0 −2 −4

150

x = −609 [m] = at 3/4 of snapshot length

150 x = 0 [m] = at end of snapshot

100 position:

100 position:

100 position:

100

x = 119 [m] =

t [s]

150 x /2 max

150


Space domain forecast Fig. A.5 presents the comparison for space domain forecasts at five instances in time, t=0s (moment at which snapshot has been taken), t=34.8s=1/3tmax, t=69.6s=2/3tmax, t=104.5s=tmax and t=227.9s=tmax,enl. Again the comparison between forecasts using the reduced spectrum and those derived from the full spectrum show no changes. Furthermore the prediction for t=227.9s still shows similarities, albeit for a close range from x~-450m until x~-450m only.

FORECAST FOR SEA STATE III – SNAPSHOT RANGE 100% (2450M)

A5

ζ [m]

Forecast of snapshots (sea state: 1, snapshot range: 100%, reduced spectrum) time: t = 0 [s] 4 2 0 −2 −4 −2500

−2000

−1500

−1000 x [m]

ζ [m]

time: 4 2 0 −2 −4 −2500

−2000

−1500

−2000

−1500

−2000

−1500

−2000

ζ [m] ζ [m] ζ [m]

−500

−1500

0

500

0

500

0

500

0

500

t = 69.6 [s] = 2/3 tmax

−500

t = 104.5 [s] = t

max

−1000 ∆x [m] time:

4 2 0 −2 −4 −2500

500

max

−1000 x [m] time:

4 2 0 −2 −4 −2500

0

t = 34.8 [s] = 1/3 t

−1000 x [m] time:

4 2 0 −2 −4 −2500

−500

−500

t = 227.9 [s] = tmax,enlarged

−1000 x [m]

−500

Fig. A.5: Space domain forecasts for a range from x=-2450m to x=700m (sea state I, range 100%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: dashed black lines, measurements: blue lines]

Discussion and general remarks In summary, the reduced spectrum delivers excellent results, although the inherent error is e(x(t))=0.11. No deteriorations could be detected compared to the prediction using the full spectrum. Thus, if necessary – e.g. for the reduction of computation time – the size of the spectrum can be downsized using the presented procedure.

A.2


Within this and the following two sections sea state III is investigated, again starting with a snapshot range of 100% (2450m in full scale). This time, the snapshot is taken at t=190s (model scale) after the beginning of the measurements/wave generation.

A6


A.2.1

Forecast based on full as well as reduced spectrum

Surface elevation snapshot Fig. A.6 shows the surface elevation snapshot based on which the sea state is analysed in frequency domain (blue line), resulting in the spectra as shown in Fig. A.7. Additionally, a reconstruction of the snapshot based on the reduced spectrum (see also Fig. A.6 bottom diagram) is presented. leading edge of snapshot



ζ(x) [m]

snapshot range 100% (2450m) 8 4 0 −4 −8 −2500

−2000

−1500

−1000

0 500 snapshot (sea state: 3, snapshot range: 100%) original snapshot (blue line) / reconstruction of snapshot with reduced spectrum (red line) x[m]

−500


Sea state spectrum (full and reduced) spectra of snapshot (sea state: 3, snapshot range: 100%, reduced spectrum) amplitude spectrum ζa(k)

ζa [m]

3


2 1 0

0

0.06

0.13

S( ω) [m2s]

0.25 k [1/m]

0.31

0.38

0.44

0.50

full sea state spectrum S( ω) sea state spectrum (absolute values) smoothed distribution (see Fig. 3.14)

ω_{start}:

0.07 rad/s

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


104.5 s

∆x_{max}:

244.6 m

cgr_{max}: cgr_{min}: reduced spectrum:

100 50 0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


25.8 m/s 2.3 m/s 9.2 %

reduction of spectrum (by total energy): reduction of spectrum (by number of wave components):

0.85 % 75 %

ω_{start,enlarged}:

0.13 rad/s

ω_{end,enlarged}:


150

2.1 rad/s

∆t_{max}:

minimum prediction error ε

50

200 S( ω) [m2s]

0.19

100

0

3 70 m 2450 m

ω_{end}:

150

0

sea state: water depth: snapshot range (100%): full spectrum:

1.06 rad/s


122.4 s


565.1 m

cgr_{max,enlarged}: cgr_{min,enlarged}:

24.6 m/s 4.6 m/s


For the reduced spectrum the overall energy is decreased by 0.85%. Thus the inherent, i.e. the minimum prediction error is 9.2%. The number of wave components is reduced by 75%. Except for A0(w0=0), being the offset of the spectrum, the full spectrum ranges from wmin=0.07 to wmax=2.1rad/s (see Tab. A.2). The reduced spectrum is highlighted by the red area (see Fig. A.7, (bottom diagram)),


A7

having a frequency range from wmin=0.13 to wmax=1.06rad/s (see also Tab. A.2). The full spectrum consists of 176 wave components, whereas the reduced spectrum is downsized to 45 wave components.

Prediction horizon The distribution of the energy error e is shown in Fig. A.8. The horizon for e(x(t) )=0 is defined by the minimum and maximum group velocities cgr,min=2.3m/s and cgr,max=25.8m/s. Hence, the upper right corner of the prediction horizon is identified by tmax=104.5s after the snapshot has been taken and xmax=244.6m beyond the trailing edge of the snapshot. For the reduced spectrum the values turn out to be cgr,min,enl=4.6m/s, cgr,max,enl=24.6m/s, tmax,enl=122.4s, and xmax,enl=565.1m. All values are summarised in Tab. A.2. fixed time steps and energy error distribution (ε=1=100%) (sea state: 3, snapshot range: 100%)

0.2

0.1

x=567m 0.4

t=0s

0

0

3 0.

2 0.

0.1

0.2 0. 4

0.2 0. 4

3 0. 4

0

0.1

0.

0. 01

0.3

0.

2 0.

−200

0

0.3

x=-2450m 0.2

0.2

0.3

−3000

0.4

x=-609m

4

0

0.

3

0.1

x [m]

−1000

−2000

0.1

t=34.8s

3

x=119m x=0m

0

0.3

0.1

x=245m

0.1

1000

0.2

t=69.6s 0.1


4 0.2 0.

−100

0 t [s]

100

200 −200

01

0

t=104.5s

4 0.2 0.

−100

t=122.4s

0 t [s]

100

200


Time domain forecast for a fixed position in space Based on the snapshot (Fig. A.6) and the subsequently computed spectra (Fig. A.7), time domain forecasts for fixed positions in space have been calculated and validated with measurement. Altogether, six positions (x=-2450m, -609m, 0m, 119m, 245m, 567m) have been investigated, at which x=-2450m and x=0m define front and trailing edge of the snapshot. The forecast for x=-2450m (Fig. A.9) delivers no compliance with measurements, as consequently follows from the fact that no information is available on waves passing that position for instances t>0s. This can also be educed from the distribution of the prediction error, as the prediction error e instantly turns 1 for t>0s.

A8


ζ [m]

position: 8 4 0 −4 −8

0

50

100

ζ [m]

position: 8 4 0 −4 −8

0

50

Forecast at fixed positions (sea state: 3, snapshot range: 100%) x = −2450 [m] = at beginning of snapshot

ζ [m] ζ [m] ζ [m] ζ [m]

8 4 0 −4 −8

8 4 0 −4 −8

8 4 0 −4 −8

0

0

50

50

0

50

0

50

200

250

200

250

150 x = 119 [m] = x /2

200

250

150 x = 245 [m] = x

200

250


200

250

200

250

100 position:

8 4 0 −4 −8

150



100 position:

100 position:

100 position:

100

max

max

t [s]

150


At x=-609m, good agreement between prediction and measurements can be fund until t~115s, where the prediction error is already e(x(t))~0.25. Comparisons between predictions, based on full and reduced spectrum, show no significant difference. For x=0m (trailing edge of the snapshot), the prediction shows excellent compliance with measurements within a time frame from t=0s until t~130s. For x=119m, as identified by the distribution of the prediction error, excellent prediction can be found from t~10s up to t~135s. For x=245m, the time domain forecast shows good comparison for a time frame from t~15s until t=140s. For x=567m, it can be pointed out, that the comparison still shows good agreement in between t=50s and 155s. Focusing on the comparison of time domain forecasts based on the full spectrum with those derived from the reduced spectrum (Fig. A.9 (diagrams 1 to 5 (from top))), it is shown that also for sea state III the difference are marginal. For x=567m, no comparison between full and reduced spectrum is available.

Space domain forecast For the comparison of surface elevation snapshots the complete range of x=3150m in which measurements have been taken is available for validation. Fig. A.10 presents the comparison at five instances in time (t=0s, t=34.8s=1/3tmax, t=69.6s=2/3tmax, t=104.5s=tmax and t=122.4s=tmax,enl). The distribution of the prediction error is given in Fig. A.8.


A9

For t=0s, the top diagram of Fig. A.10 shows absolute agreement for a range from x=-2450m to x=0m. With growing instances in time (t >0s), the comparison presents excellent compliance within a decreasing range – for t=34.8s from x~-1700m until x~400m, for t=69.6s from x~-1200m to x~700m, for t=104.5s from x~-600m up to x=600m and for t=122.4s from x~-350m to x~700m, as is supported by the prediction horizon Fig. A.8. Forecast of snapshots (sea state: 3, snapshot range: 100%)

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−1000 x [m]

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−2000

−1500

−2000

−1500

−2000

−1500

ζ [m] ζ [m] ζ [m]

500

−500

0

500

−500

0

500

0

500

0

500

t = 104.5 [s] = tmax

−1000 x [m] time:

8 4 0 −4 −8 −2500

0

t = 69.6 [s] = 2/3 tmax

−1000 x [m] time:

8 4 0 −4 −8 −2500

−500

t = 34.8 [s] = 1/3 tmax

−1000 x [m] time:

8 4 0 −4 −8 −2500

t = 0 [s]

−500


−1000 x [m]

−500


Discussion and general remarks In summary, all forecasts show excellent to good results until e(x(t))~0.1. Thus, for calculations based upon the full range snapshot, the time t for predictions is approximately 105 to 130s. t is depending on the actual position x(t).

A.3


Within this section the length of the snapshot is reduced to 75% (1834m in full scale) in order to investigate the influence of the snapshot range on the precision of the forecast. The snapshot is again taken at t=150s (model scale) after the beginning of the measurements/wave generation, thus the snapshot is the same as in section 3.6 albeit 25% - starting at the leading edge – have been cut off.

A10


A.3.1


Surface elevation snapshot leading edge of snapshot



ζ(x) [m]

snapshot range 75% (1834m) 4 2 0 −2 −4 −2500

−2000

−1500

−1000

−500 0 500 snapshot (sea state: 1, snapshot range: 75%) x[m] original snapshot (blue line) / reconstruction of snapshot with reduced spectrum (red line)


Fig. A.11 shows the surface elevation snapshot, based on which the sea state is analysed in frequency domain (blue line), resulting in the spectra as shown in Fig. A.12. Additionally, a reconstruction of the snapshot based on the reduced spectrum (see also Fig. A.12 (bottom diagram)) is presented.

Sea state spectrum (full and reduced) spectra of snapshot (sea state: 1, snapshot range: 75%, reduced spectrum)


1 70 m 1834 m

0.4

ω_{start}:

0.09 rad/s

0.2

ω_{end}:

2.09 rad/s

∆t_{max}:

79.3 s

∆x_{max}:

185.7 m

amplitude spectrum ζa(k)

0.8


ζa [m]

0.6

0

0

0.06

0.13

S( ω) [m2s]

0.25 k [1/m]

0.31

0.38

0.44

0.50

full sea state spectrum S(ω)


25.5 m/s 2.3 m/s

2


10.7 %

0


1.15 % 48 %


0.51 rad/s

6


4

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


10

ω_{end,enlarged}:


1.6 rad/s


236.7 s

5


727.8 m

0


2

S( ω) [m s]

0.19

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


10.8 m/s 3.1 m/s


The limitations for the reduction of the spectrum are the same as for the spectrum derived from a snapshot with a range of 100%:


A11

• the energy loss in the low band (low frequencies) of the reduced spectrum shall not exceed 0.5% of the overall energy contained in the sea state • the energy loss in the high band (high frequencies) of the reduced spectrum shall not exceed 0.75% of the overall energy contained in the sea state Hence again, a maximum energy loss of 1.25% is permitted, resulting in an inherent prediction error of e=0.11 (11%). The overall energy for this case is reduced by 1.15% and the inherent, i.e. the minimum prediction error is 10.7%. The number of wave components is reduced by 48%. Except for A0(w0=0), being the absolute element of the spectrum, the full spectrum ranges from wmin=0.09 to wmax=2.09rad/s (see Tab. A.3). The reduced spectrum, as highlighted by the red area (see Fig. A.12, (bottom diagram)), has a frequency range from wmin=0.51 to wmax=1.60rad/s (see also Tab. A.3). The full spectrum consists of 132 wave components, whereas the reduced spectrum is downsized to 68 wave components.

Prediction horizon fixed time steps and energy error distribution (ε=1=100%) (sea state: 1, snapshot range: 75%) t=52.8s


1000

x=189m x=91m x=0m 4

0.2

0.3

0.1

t=0s 2

0.

2

0.

4

.3 −2000 0

0.1

.43 00.

0.2

0.1

.3 −3000 00.4

t=236.7s

00..43

−200

−100

0 t [s]

100

.43 00.

3

0.

x=-1834m

0.2

1

0.

2 0.

0

2 0.

0

1 0.

0

x=-609m

0

−1000

0.1

0.

0.1

0.

0.2

0.3

0.1

0 x [m]

t=24.6s

x=728m

200

t=79.3s −200

−100

0 t [s]

100

200


The distribution of the energy error e is shown in Fig. A.13. The horizon for e(x(t))=0 is defined by the minimum and maximum group velocities cgr,min=2.3m/s and cgr,max=25.5m/s. Hence, the upper right corner of the prediction horizon is identified by tmax=79.3s after the snapshot has been taken and xmax=185.7m beyond the trailing edge of the snapshot. For the reduced spectrum the values turn out to be cgr,min,enl=3.1m/s, cgr,max,enl=10.8m/s, tmax,enl=236.7s and xmax,enl=727.8m. All values are summarised in Tab. A.3. The inherent error for the reduced spectrum is e=0.11.

Time domain forecast for a fixed position in space Based on the snapshot (Fig. A.11) and the subsequently computed spectra (Fig. A.12) time domain forecasts for fixed positions in space have been calculated and validated with measurement. Six positions (x=-1834m, -609m, 0m, 91m, 189m, 728m) have been investigated at which x=-1834m and x=0m define leading and trailing edge of the snapshot. The forecast for x=-1834m (Fig. A.14) delivers no compliance with measurements as consequently follows from the fact, that no information is available on waves passing that position for instances t >0s. It can also be deduced from the distribution of the prediction error, as the prediction error e instantly turns 1 for t >0s.

A12


The other five examples also show that the time frame for excellent predictions can be deduced from the prediction error distribution (see Fig. A.13). For the reduced snapshot range the forecast at x=-609m shows the same unexpectedly good comparison as for the full range. A good agreement between prediction and measurements can be found until t~165s, where the prediction error already rises to e(x(t))~0.25. Comparisons between predictions based on full and reduced spectrum show no significant difference. For x=0m (trailing edge of the snapshot) the prediction shows excellent compliance with measurements within a time frame from t=0s until t~75s. For x=91m, excellent predictions can be identified from t~15s up to t~110s. For x=189m, the time domain forecast shows excellent comparison for a time frame from t~30s until t=125s. For x=728m, it can be pointed out, that the comparison no longer shows any agreement, if at all between t=130 and 140s.

ζ [m]

position: 4 2 0 −2 −4

0

50

100

ζ [m]

position: 4 2 0 −2 −4

0

50


100

ζ [m]

ζ [m]

ζ [m]

ζ [m]

position: 4 4 0 −2 −4

4 2 0 −2 −4

4 2 0 −2 −4

4 2 0 −2 −4

0

50

0

50

0

50

0

50

150

200

250

200

250



100 position:

150 x = 91 [m] = x /2

200

250

100 position:

150 x = 189 [m] = xmax

200

250


200

250

200

250

100 position:

100

max

t [s]

150


Focusing on the comparison of time domain forecasts based on the full spectrum with those derived from the reduced spectrum (Fig. A.14 (diagrams 1 to 5 (from top))), it can be pointed out, that the differences for sea state I are marginal. For x=728m, no comparison between full and reduced spectrum is available. The reduction of the snapshot range does not reduce the accuracy of the forecast but the time frame in which the agreement between forecast and measurement is good to excellent. The time frames are cutback by approximately 20%.


A13

Space domain forecast The complete range of x=3150m, in which measurements have been taken, is available for validation of space domain forecasts. Fig. A.15 presents the comparison at five instances in time (t=0s, t=26.4s=1/3tmax, t=52.8s=2/3tmax, t=79.3s=tmax and t=236.7s=tmax,enl). The distribution of the prediction error is given in Fig. A.13. Forecast of snapshots (sea state: 1, snapshot range: 75%)

ζ [m]

time: 4 2 0 −2 −4 −2500

−2000

−1500

−1000 x [m]

ζ [m]

time: 4 2 0 −2 −4 −2500

−2000

−2000

−2000

−1500

−1000 x [m]

−1000 x [m]

−2000

−1500

ζ [m] ζ [m] ζ [m]

0

500

−500

0

500

−500

0

500

0

500

−500

t = 79.3 [s] = tmax

−1000 x [m] time:

4 2 0 −2 −4 −2500

500

t = 52.8 [s] = 2/3 tmax

−1500

−1500

0

max

time: 4 2 0 −2 −4 −2500

−500

t = 26.4 [s] = 1/3 t

time: 4 2 0 −2 −4 −2500

t = 0 [s]


−1000 x [m]

−500


For t=0s, the top diagram of Fig. A.15 shows absolute agreement for a range from x=-1834m to x=0m. Measurements and prediction are congruent, as for t=0s the prediction is only a reconstruction of the snapshot itself. With growing instances in time (t >0s), the comparison presents excellent compliance within a decreasing range – for t=26.4s from x~-1600m until x~100m, for t=52.8s from x~-1500m to x~300m and for t=79.3s from x~-1250m up to x=550m. The prediction at t=236.7s shows hardly any similarity with measurements any longer as is supported by Fig. A.13. Compared to predictions based on the snapshot with full range not only the time frames, as before for time domain predictions, are reduced. Also the spatial ranges for good predictions are downsized by about 15 to 20%.

A14


Discussion and general remarks In summary, all forecasts show excellent to good results up to e(x(t))~0.1, sometimes even up to e(x(t))~0.25. Thus, the time t for predictions – approximately (~2 to 2.5min) for calculations based upon the full range snapshot – is cut down by around 20% for snapshots reduced to 75% of the full length, turning out to be (~1.5min). t is certainly depending on the actual position x(t). The spatial range for prediction is coevally downsized by around 15 to 20%.

A.4

Forecast for sea state I – Snapshot range 50% (1225m)

Within the following section, the snapshot range is further reduced to 50% (1225m in full scale). The snapshot is again taken at t=150s (model scale) after the beginning of the measurements/ wave generation.

A.4.1


Surface elevation snapshot Fig. A.16 shows the surface elevation snapshot, based on which the sea state is analysed in frequency domain (blue line), resulting in the spectra as shown in Fig. A.17. A reconstruction of the snapshot based on the reduced spectrum (see also Fig. A.17 (bottom diagram)) is also presented. leading edge of snapshot direction of wave travel measurements for validation available (3150m)


ζ(x) [m]

snapshot range 50% (1225m) 4 3 0 −3 −4 −2500

−2000

−1500

−1000

x[m]

−500

0

500

snapshot (sea state: 1, snapshot range: 50%, reduced spectrum) original snapshot (blue line) / reconstruction of snapshot with reduced spectrum (red line)


Sea state spectrum (full and reduced) The limitations for the reduction of the spectrum are the same as for the spectrum derived from a snapshot with a range of 100 and 75%. The overall energy is reduced by 1.08% and the inherent, i.e. the minimum prediction error is 10.4%. Thus, the number of wave components can be reduced by 41%. Except for A0(w0=0), being the absolute element of the spectrum, the full spectrum ranges from wmin=0.13 to wmax=2.09rad/s (see Tab. A.4). The reduced spectrum, as highlighted by the red area (see Fig. A.17, (bottom diagram)), has a frequency range from wmin=0.25 to wmax=1.63rad/s (see also Tab. A.4). The full spectrum consists of 88 wave components, whereas the reduced spectrum is downsized to 51 wave components.

FORECAST FOR SEA STATE I – SNAPSHOT RANGE 50% (1225M) spectra of snapshot (sea state: 1, snapshot range: 50%, reduced spectrum)


a

ζ [m]

0.6

1 70 m 1225 m

0.4

ω_{start}:

0.13 rad/s

0.2

ω_{end}:

2.09 rad/s

0

0

0.06

0.13

0.19

S( ω) [m2s]

0.25 k [1/m]

0.31

0.38

0.44

∆t_{max}:

0.50

sea state spectrum (absolute values) smoothed (ideal) distribution

6 4 2 0

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


10

55 s

∆x_{max}:


8


129.2 m


24.6 m/s 2.4 m/s


10.4 %


1.08 % 41 %


0.25 rad/s

ω_{end,enlarged}:

1.63 rad/s


68.2 s

5


205.5 m

0


2

S( ω) [m s]



0.8

A15

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


21 m/s 3 m/s


Prediction horizon fixed time steps and energy error distribution (ε=1=100%) (sea state: 1, snapshot range: 50%)


t=36.6s 1000

t=18.3s x=63m x=0m

−1000

0.4 03

−2000

.1 0.3

0.2

0.1

0.4

.3.4 0.200

0

x [m]

0

0

2

0.

t=0s

x=-609m

x=-1250m

0.20.3 0.4

0.4 03

0.1

0.4

2

0.

.3.4 0.200

0

0.20.3 0.4

t=68.2s

−3000 −200

.1 0.3

0.2 0

x=126m

x=203m

t=55s −100

0 t [s]

100

200 −200

−100

0 t [s]

100

200


The distribution of the energy error e is shown in Fig. A.18. The horizon for e(x(t))=0 is defined by the minimum and maximum group velocities cgr,min=2.4m/s and cgr,max=24.6m/s. The upper right corner of the prediction horizon is identified by tmax=55s after the snapshot has been taken and xmax=129.2m beyond the trailing edge of the snapshot. The values turn out to be cgr,min,enl=3m/s,

A16


cgr,max,enl=21m/s, tmax,enl=68.2s and xmax,enl=205.5m for the reduced spectrum. All values are summarised in Tab. A.4.

Time domain forecast for a fixed position in space Based on the snapshot (Fig. A.16) and the subsequently computed spectra (Fig. A.17), time domain forecasts for fixed positions in space have been calculated and validated with measurement. Six positions (x=-1225m, -609m, 0m, 63m, 126m, 203m) have been investigated at which x=-1225m and x=0m define leading and trailing edge of the snapshot. Again, the forecast at the leading edge of the snapshot (x=-1225m (Fig. A.19)) delivers no compliance with measurements, as consequently follows from the fact that no information is available on waves passing that position for instances t >0s.

ζ [m]

position: 4 2 0 −2 −4

0

50

100

ζ [m]

position: 4 2 0 −2 −4

0

50


100

ζ [m]

ζ [m]

ζ [m]

ζ [m]

position: 4 2 0 −2 −4

4 2 0 −2 −4

4 2 0 −2 −4

4 2 0 −2 −4

0

50

0

50

0

50

0

50

150

200

250

200

250



100 position:

150 x = 63 [m] = x /2

200

250

100 position:

150 x = 126 [m] = xmax

200

250


200

250

200

250

100 position:

100

max

t [s]

150


The other five examples show that the time frame for excellent predictions can be deduced from the prediction error distribution (see Fig. A.18). For x=-609m an excellent agreement between prediction and measurements can be found until t~90s, where the prediction error turns out to be already e(x(t))~0.25 to 0.3. Comparisons between predictions based on full and reduced spectrum show no significant difference. For x=0m (trailing edge of the snapshot) the prediction shows excellent compliance with measurements within a time frame from t=0s until t~55s and still good agreement until t~80s. For x=63m, one can identify excellent prediction from t~5s up to t~75s. For x=126m, the time


A17

domain forecast shows excellent to good comparison for a time frame from t~20s until t=120s. For x=203m, it can be pointed out that the comparison shows excellent agreement between t=30 and 105s. Focusing on the comparison of time domain forecasts based on the full spectrum against those derived from the reduced spectrum (Fig. A.19 (diagrams 1 to 5 (from top))) the differences are marginal. For x=203m no comparison between full and reduced spectrum is available. In summary, the time domain forecasts calculated using the reduced spectrum show only insignificant changes compared to those deduced using the full spectrum. It can also be concluded, that the reduction of the snapshot range by 50% reduces the time frames in which the agreement between forecast and measurement is good to excellent. Compared to the full spectral range, the time frames are cutback by approximately 40 to 50%.

Space domain forecast

ζ [m]

Forecast of snapshots (sea state: 1, snapshot range: 50%) time: t = 0 [s] 8 4 0 −4 −8 −2500

−2000

−1500

−1000 x [m]

ζ [m]

time: 4 2 0 −2 −4 −2500

−2000

−1500

−2000

−1000 x [m]

−1500

−2000

−2000

−1500

ζ [m] ζ [m] ζ [m]

500

−500

0

500

−500

0

500

0

500

max

t = 55 [s] = tmax

−1000 x [m] time:

4 2 0 −2 −4 −2500

0

−500

−1000 x [m]

−1500

500

t = 36.6 [s] = 2/3 t

time: 4 2 0 −2 −4 −2500

0

t = 18.3 [s] = 1/3 tmax

time: 4 2 0 −2 −4 −2500

−500

t = 68.2 [s] = t

max,enlarged

−1000 x [m]

−500


Measurements within the complete range of x=3150m are available for validation of space domain forecasts. Fig. A.20 presents the comparison at five instances in time (t=0s, t=18.3s=1/3tmax, t=36.6s=2/3tmax, t=55s=tmax and t=68.2s=tmax,enl). The distribution of the prediction error is given in Fig. A.18.

A18


For t=0s the top diagram of Fig. A.20 shows, self-evidently, absolute agreement for a range from x-1225m to x=0m. With growing instances in time (t >0s), the comparison presents excellent to good compliance within a decreasing range – for t=18.3s from x~-1100m until x~150m, for t=36.6s from x~-1000m to x~250m and for t=55s from x~-800m up to x~400m. The prediction at t=68.2s shows good similarity with measurements between x~-700m and x~400m. Thus the spatial ranges for good predictions are downsized by about 50%.

Discussion and general remarks In summary, all forecasts show excellent to good results up to e(x(t))~0.1, sometimes even up to e(x(t))~0.25. Thus, the time t for predictions – approximately (~2 to 2.5min) for calculations based upon the full range snapshot – is cut down by around 40 to 50% for the reduced spectrum, turning out to be ~75s. t is certainly depending on the actual position x(t). The spatial range for prediction is coevally downsized by around 50% for the snapshot range of 50%.

A.5


Within this section the snapshot range is reduced to 75% (1834m in full scale) in order to investigate the influence of the snapshot range on the precision of the forecast. The snapshot is again taken at t=120s (model scale) after the beginning of the measurements/wave generation, thus the snapshot is the same as in section 3.6 albeit 25% – starting at the leading edge – have been cut off.

A.5.1


Surface elevation snapshot Fig. A.21 shows the surface elevation snapshot based on which the sea state is analysed in frequency domain (blue line), resulting in the spectra as shown in Fig. A.22. Additionally a reconstruction of the snapshot based on the reduced spectrum (see also Fig. A.21 (bottom diagram)) is presented. leading edge of snapshot



ζ(x) [m]

snapshot range 75% (1834m) 8 4 0 −4 −8 −2500

−2000

−1500

−1000

−500 0 500 x[m] snapshot (sea state: 2, snapshot range: 75%, reduced spectrum) original snapshot (blue line) / reconstruction of snapshot with reduced spectrum (red line)


Sea state spectrum (full and reduced) The limitations for the reduction of the spectrum are the same as for the spectrum derived from a snapshot with a range of 100%. For the reduced spectrum the overall energy is decreased by 0.8% and consequently the inherent, i.e. the minimum prediction error is 8.9%. The number of wave components is reduced by 84%. Except for A0(w0=0), being the offset of the spectrum, the full spectrum ranges from wmin=0.09

FORECAST FOR SEA STATE II – SNAPSHOT RANGE 75% (1834M)

A19

to wmax=2.09rad/s (see Tab. A.5). The reduced spectrum, as highlighted by the red area (see Fig. A.22, [bottom diagram]), has a frequency range from wmin=0.09 to wmax=0.84rad/s (see also Tab. A.5). Whereas the full spectrum consists of 132 wave components, the reduced spectrum is downsized to 22 wave components. spectra of snapshot (sea state: 2, snapshot range: 75%, reduced spectrum) 1.5


1

a

ζ [m]



0.5 0

0

0.06

0.13

S( ω) [m2s]

0.38

0.44

0.50


10 5 0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


20 2

0.31

15

0

S( ω) [m s]

0.25 k [1/m]


20

50

0.19


15 10 5 0

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


2 70 m 1834 m

ω_{start}:

0.09 rad/s

ω_{end}:

2.09 rad/s

∆t_{max}:

79.3 s

∆x_{max}:

185.7 m


25.5 m/s 2.3 m/s


8.9 %


0.8 % 84 %


0.09 rad/s

ω_{end,enlarged}:

0.84 rad/s


93.5 s


547.1 m


25.5 m/s 5.9 m/s


Prediction horizon The distribution of the energy error e is shown in Fig. A.23. The horizon for e(x(t))=0 is defined by the minimum and maximum group velocities cgr,min=2.3m/s and cgr,max=25.5m/s. The upper right corner of the prediction horizon is identified by tmax=79.3s and xmax=185.7m. For the reduced spectrum, the values turn out to be cgr,min,enl=5.9m/s, cgr,max,enl=25.5m/s, tmax,enl=93.5s and xmax,enl=547.1m. All these values are summarised in Tab. A.5.

A20

APPENDIX A 2-D WAVE FORECAST – SUPPLEMENTARY TEST RUNS WITH REDUCED SNAPSHOT RANGES fixed positions and energy error distribution (ε=1=100%) (sea state: 2, snapshot range: 75%)

fixed time steps and energy error distribution (ε=1=100%) (sea state: 2, snapshot range: 75%) t=52.8s

0.2

0.4

0.4

−100

0 t [s]

100

200 −200

0.40 1e .3 −

00.2.3

0.4

−100

0.1

0.1

0.40

0 .2

0

x=-1834m

.3

.3 00.2

007 0 .2 0 .1

007 0 .2 0 .1

0.40 1e .3 −

x=-609m

0.1

0.40

.4 0 .1 0

0

0.3

0.3

0.2

0.1

t=0s

0.20.3

0.20.3

0.4

.3 00.2

0 .2

0

−2000

−200

0

00.2.3

−1000

−3000

. 0 .1 0

0.1 0.4

x=91m x=0m

0.1 0.4

x [m]

0

x=546m x=189m 4

0.4

t=24.6s

0 .4 0 .3

1000

0 .4 0 .3

0.4

.3

t=93.5s t=79.3s 0 t [s]

100

200


Time domain forecast for a fixed position in space Based on the snapshot (Fig. A.21) and the subsequently computed spectra (Fig. A.22) time domain forecasts at six fixed positions in space have been calculated and validated with measurement (x=-1834m, -609m, 0m, 91m, 189m, 546m). x=-1834m and x=0m define front and trailing edge of the snapshot. Again, no compliance with measurements can be found for x=-1834m (Fig. A.24), due to the fact, that no information is available on waves passing that position for instances t >0s. This can also be educed from the distribution of the prediction error, as the prediction error e instantly turns 1 for t >0s. The other examples show that the time frame for excellent predictions can be deduced from the prediction error distribution (see Fig. A.23). For x=-609m good agreement between prediction and measurements can be fund until t~90s, where the prediction error is already e(x(t))~0.3. Comparisons between predictions, based on full and reduced spectrum, show no significant difference. For x=0m (trailing edge of the snapshot), the prediction shows good compliance with measurements within a time frame from t=0s until t~95s. For x=91m, one can identify excellent agreement from t~10s up to t~100s. For x=189m, the time domain forecast shows excellent comparison for a time frame from t~15s until t=105s. For x=546m, it can be pointed out that the comparison shows good agreement in between t=35 and 115s. Focusing on the comparison of time domain forecasts based on the full spectrum, with those derived from the reduced spectrum (Fig. A.24 [diagrams 1 to 5 (from top]), it can be pointed out that the differences are marginal. For x=546m, no comparison between full and reduced spectrum is available. In summary, it can be pointed out that the time domain forecasts calculated using the reduced spectrum show, if at all, only insignificant changes compared to those deduced using the full spectrum. It can also be concluded that the reduction of the snapshot range reduces the accuracy of the forecast, by means of a reduced time frame in which the agreement between forecast and measurement is good to excellent. The time frames are cutback by approximately 30%.


ζ [m]

position: 8 4 0 −4 −8

0

50

ζ [m]

8 4 0 −4 −8

0

50


100 position:

ζ [m] ζ [m] ζ [m] ζ [m]

8 4 0 −4 −8

8 4 0 −4 −8

8 4 0 −4 −8

0

0

50

50

0

50

0

50

150

200

250

200

250


100 position:

8 4 0 −4 −8

A21


100 position:

150 x = 91 [m] = x /2

200

250

100 position:

150 x = 189 [m] = x

200

250


200

250

200

250

100 position:

100

max

max

t [s]

150


Space domain forecast Fig. A.25 presents the comparison of space domain snapshots at five instances in time (t=0s, t=26.4s=1/3tmax, t=52.8s=2/3tmax, t=79.3s=tmax and t=93.5s=tmax,enl). The distribution of the prediction error is given in Fig. A.23. For t=0s, the top diagram of Fig. A.25 shows absolute agreement for a range from x=-1834m to x=0m. Measurements and prediction are congruent, as for t=0s the prediction is only a reconstruction of the snapshot itself. With growing instances in time (t >0s), the comparison presents excellent compliance within a decreasing range – for t=26.4s from x~-1200m until x~100m, for t=52.8s from x~-500m to x~700m, for t=79.3s from x~-100m up to x=700m and for t=93.5s from x~0m to x~700m. Thus, compared to predictions based on the snapshot with full range not only the time frames, as before for time domain predictions, are reduced. Also the spatial ranges for good predictions are downsized by about 35 to 40%.

A22

APPENDIX A 2-D WAVE FORECAST – SUPPLEMENTARY TEST RUNS WITH REDUCED SNAPSHOT RANGES Forecast of snapshots (sea state: 2, snapshot range: 75%)

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−1000 x [m]

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−2000

−1500

−2000

−1500

−2000

−1500

ζ [m] ζ [m] ζ [m]

500

0

500

−500

0

500

−500

0

500

0

500

max

−500

t = 52.8 [s] = 2/3 tmax

t = 79.3 [s] = tmax

−1000 x [m] time:

8 4 0 −4 −8 −2500

0

t = 26.4 [s] = 1/3 t

−1000 x [m] time:

8 4 0 −4 −8 −2500

−500

−1000 x [m] time:

8 4 0 −4 −8 −2500

t = 0 [s]


−1000 x [m]

−500


Discussion and general remarks In summary, all forecasts show excellent to good results up to e(x(t))~0.1, often even up to e(x(t))~0.3. Thus, the time t for predictions – approximately ~2.5min for calculations based upon the full range snapshot for sea state II– is cut down by around 30% for the reduced spectrum, turning out to be ~1.5min. t is certainly depending on the actual position x(t). The spatial range for predictionis coevally downsized by around 35 to 40% for the snapshot range of 75%.

A.6


Within this section the snapshot range is further reduced to 50% (1225m in full scale). The snapshot is again taken at t=120s (model scale) after the beginning of the measurements/wave generation. 50% of the full snapshot, starting at the leading edge have been cut off.


A.6.1

A23


Surface elevation snapshot Fig. A.26 shows the surface elevation snapshot, based on which the sea state is analysed in frequency domain (blue line), resulting in the spectra as shown in Fig. A.27. Additionally, a reconstruction of the snapshot based on the reduced spectrum (see also Fig. A.27 (bottom diagram)) is presented. leading edge of snapshot direction of wave travel measurements for validation available (3150m)


ζ(x) [m]

snapshot range 50% (1225m) 8 4 0 −4 −8 −2500

−2000

−1500

−1000

−500 0 500 x[m] snapshot (sea state: 2, snapshot range: 50%) original snapshot (blue line) / reconstruction of snapshot with reduced spectrum (red line)


Sea state spectrum (full and reduced) spectra of snapshot (sea state: 2, snapshot range: 50%, reduced spectrum) 2


a

ζ [m]

1.5 1

ω_{start}:

0.13 rad/s

ω_{end}:

2.09 rad/s

0.06

0.13

S( ω) [m2s]

0.19

0.25 k [1/m]

0.31

0.38

0.44

0.50

∆t_{max}: ∆x_{max}:


15


10


55 s 129.2 m 24.6 m/s 2.4 m/s

5


0


0.73 % 89 %


0.13 rad/s

ω_{end,enlarged}:

0.71 rad/s

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2

full vs. reduced sea state spectrum S(ω)

20 2

2 70 m 1225 m

0.5 0

S( ω) [m s]




15 10 5 0

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


8.5 %


69.5 s


486.8 m


24.6 m/s 7 m/s


The overall energy is again reduced by 0.73% and the inherent, i.e. the minimum prediction error is 8.5%. The number of wave components is reduced by 89%. Except from A0(w0=0),the full spectrum ranges from wmin=0.13 to wmax=2.09rad/s (see Tab. A.6). The reduced spectrum as highlighted by the red area (see Fig. A.27, (bottom diagram))), with a frequency range from wmin=0.13 to

A24


wmax=0.71rad/s (see also Tab. A.6). The full spectrum consists of 88 wave components, whereas the reduced spectrum is downsized to 11 wave components only. The limitations for the reduction of the spectrum are the same as for the spectrum derived from a snapshot with a range of 100%.

Prediction horizon fixed positions and energy error distribution (ε=1=100%) (sea state: 2, snapshot range: 50%)

0 0.1

0.1

0.2

x=-1250m

0.4

0 0.1

t=69.5s

−3000 −200

0

0.4

0.2

x=-609m

0.3

0.3

−2000

0.4

0.3

t=0s

0.4

0.2

0.2

−1000

t=18.3s

0.4

0.3

0

0.1

x=126m x=63m x=0m

0.4

x [m]

0

x=490m 0.3

0.3

1000


t=55s −100

0 t [s]

100

200 −200

−100

0 t [s]

100

200


The distribution of the energy error e is shown in Fig. A.28. The horizon for e(x(t))=0 is defined by the minimum and maximum group velocities cgr,min=2.4m/s and cgr,max=24.6m/s. The upper right corner of the prediction horizon is identified by tmax=55s and xmax=129.2m. For the reduced spectrum the values turn out to be cgr,min,enl=7m/s, cgr,max,enl=24.6m/s, tmax,enl=69.5s and xmax,enl=486.8m. All values are summarised in Tab. A.6.

Time domain forecast for a fixed position in space Based on the snapshot (Fig. A.29) and the subsequently computed spectra (Fig. A.27), time domain forecasts for fixed positions in space have been calculated and validated with measurement. Six positions (x=-1225m, -609m, 0m, 63m, 126m, 490m) have been investigated at which x=-1225m and x=0m define leading and trailing edge of the snapshot. The forecast for x=-1225m (Fig. A.29) delivers no compliance with measurements, as consequently follows from the fact, that no information is available on waves passing that position for instances t >0s. This can also be educed from the distribution of the prediction error, as the prediction error e instantly turns 1 for t >0s.


ζ [m]

position: 8 4 0 −4 −8

0

50

ζ [m]

8 4 0 −4 −8

0

50


100 position:

ζ [m] ζ [m] ζ [m] ζ [m]

8 4 0 −4 −8

8 4 0 −4 −8

8 4 0 −4 −8

0

0

50

50

0

50

0

50

150

200

250

200

250


100 position:

8 4 0 −4 −8

A25


100 position:

150 x = 63 [m] = x /2

200

250

100 position:

150 x = 126 [m] = x

200

250


200

250

200

250

100 position:

100

max

max

t [s]

150


The other examples show that the time frame for excellent predictions can be deduced from the prediction error distribution (see Fig. A.28). For the reduced snapshot range the forecast at x=609m shows good agreement between prediction and measurements from t=0s until t~65s, where the prediction error is already e(x(t))~0.3. Comparisons between predictions based on full and reduced spectrum show no significant difference. For x=0m (trailing edge of the snapshot) the prediction shows excellent compliance with measurements within a time frame from t=0s until t~95s. For x=63m, excellent to good prediction can be identified from t~10s up to t~100s. For x=126m, the time domain forecast shows excellent to good comparison for a time frame from t~25s until t=100s and for x=490m, good comparison can be identified in between t=50 and 125s. Focusing on the comparison of time domain forecasts based on the full spectrum with those derived from the reduced spectrum (Fig. A.29 (diagrams 1 to 5 (from top))) the differences are marginal. For x=490m, no comparison between full and reduced spectrum is available. In summary, the time domain forecasts calculated using the reduced spectrum show only insignificant changes compared to those deduced using the full spectrum. It can also be concluded that the reduction of the snapshot range reduces the time frames, in which the agreement between forecast and measurement is good to excellent. The time frames are cutback by approximately 35%.

A26


Space domain forecast Fig. A.30 presents the comparison of surface elevation snapshots and space domain forecasts at five instances in time (t=0s, t=18.3s=1/3tmax, t=36.6s=2/3tmax, t=55s=tmax and t=69.5s=tmax,enl). The distribution of the prediction error is given in Fig. A.28. For t=0s, the top diagram of Fig. A.30 shows absolute agreement for a range from x=-1225m to x=0m. Measurements and prediction are congruent, as for t=0s the prediction is only a reconstruction of the snapshot. With growing instances in time (t >0s), the comparison presents excellent compliance within a decreasing range – for t=18.3s from x~-1100m until x~50m, for t=36.6s from x~-900m to x~250m, for t=55s from x~-450m up to x=500m and for t=69.5s from x~-400m up to x=600m, as is supported by Fig. A.28. Thus, it can be revealed that, compared to predictions based on the snapshot with full range, not only the time frames, as before for time domain predictions, are reduced. Also the spatial ranges for good predictions are downsized by about 10 to 30%. Forecast of snapshots (sea state: 2, snapshot range: 50%)

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−1000 x [m]

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−2000

−1000 x [m]

−1500

−2000

−1000 x [m]

−1500

−2000

−1500

ζ [m] ζ [m] ζ [m]

500

0

500

−500

0

500

−500

0

500

0

500

−500

t = 55 [s] = tmax

−1000 x [m] time:

8 4 0 −4 −8 −2500

0

t = 36.6 [s] = 2/3 tmax

time: 8 4 0 −4 −8 −2500

−500

t = 18.3 [s] = 1/3 tmax

time: 8 4 0 −4 −8 −2500

t = 0 [s]


−1000 x [m]

−500

Fig. A.30: Space domain forecast for a range from x=-2450m to x=700m (sea state II, range 50%, full and reduced spectrum) [prediction based on reduced spectrum: red lines, prediction based on full spectrum: dashed black lines, measurements: blue lines]


A27

Discussion and general remarks In summary, all forecasts for sea state II show excellent to good results. Thus, the time t for predictions – approximately ~2.5min for calculations based upon the full range snapshot – is cut down by around 30 to 50% for the reduced snapshot, turning out to be ~70 to 90s (~1 to 1.5min). t is certainly depending on the actual position x(t). The spatial range for prediction is coevally downsized by up to 50% for the snapshot range of 50%.

A.7

Forecast for sea state III – Snapshot range 75% (1834m)

Within this section the snapshot range is reduced to 75% of its original length (1834m in full scale), in order to investigate the influence of the snapshot range on the precision of the forecast. The snapshot is taken at t=190s (model scale) after the beginning of the measurements/wave generation.

A.7.1


Surface elevation snapshot Fig. A.31 shows the surface elevation snapshot based on which the sea state is analysed in frequency domain (blue line), resulting in the spectra as shown in Fig. A.32. Additionally, a reconstruction of the snapshot, based on the reduced spectrum (see also Fig. A.31 (bottom diagram)), is presented. leading edge of snapshot



ζ(x) [m]

snapshot range 75% (1834m) 8 4 0 −4 −8 −2500

−2000

−1500

−1000

−500 0 500 snapshot (sea state: 3, snapshot range: 75%) x[m] original snapshot (blue line) / reconstruction of snapshot with reduced spectrum (red line)


The limitations for the reduction of the spectrum are the same as for the spectrum derived from a snapshot with a range of 100%. The overall energy is reduced by 0.8% and the inherent prediction error turns out to be 8.9%. The number of wave components can be reduced by 87%. Except from A0(w0=0), the full spectrum ranges from wmin=0.09 to wmax=2.09rad/s (see Tab. A.7). The reduced spectrum, as highlighted by the red area (see Fig. A.32, (bottom diagram)), has a frequency range from wmin=0.09 to wmax=0.75rad/s (see also Tab. A.7). The full spectrum consists of 133 wave components, whereas the reduced spectrum is downsized to 18 wave components.

A28


Sea state spectrum (full and reduced) spectra of snapshot (sea state: 3, snapshot range: 75%, reduced spectrum) amplitude spectrum ζa(k)

ζa [m]

3


2

S( ω) [m2s]

3 70 m 1834 m

ω_{start}:

0.09 rad/s

1

ω_{end}:

2.09 rad/s

0

∆t_{max}:

79.3 s

∆x_{max}:

185.7 m

0

0.06

0.13

0.19

0.25 k [1/m]

0.31

0.38

0.44

0.50


100


sea state spectrum (absolute values) smoothed distribution (see Fig. 3.14) 50

0

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


100 S( ω) [m2s]



25.5 m/s 2.3 m/s


8.9 %


0.8 % 87 %


0.09 rad/s

ω_{end,enlarged}:

0.75 rad/s


96.8 s

50


632.1 m

0


0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


25.5 m/s 6.5 m/s


Prediction horizon fixed positions and energy error distribution (ε=1=100%) (sea state: 3, snapshot range: 75%)

−2000

4

0.

0.1

0

4

0.3

−200

2

0.1

0

0

3

0.

x=-1834m

4

0.

0.1

0

4 0.

0.3

0.2

0.2 0.4

−100

0.3

0 t [s]

100

4

4

0.3

4

0.

0.3

0

0.3

4 0.

0.2

0.2

−3000

0.2 0.1

0.3

0.

x=-609m

0.1

2

3

0.

0.1 0.

x [m]

−1000

t=0s

4

0. 0.2

4

0.

0

0

0.2 0.1

0.3

0.4

t=24.6s

4 0.

0.3

0.2

x=189m x=91m x=0m

t=52.8s

0.3

0.4

x=630m

fixed time steps and energy error distribution (ε=1=100%) (sea state: 3, snapshot range: 75%)

0.1 0.

1000

4 0.

200 −200

0.4

−100

t=96.8s t=79.3s 0 t [s]

100

200


The distribution of the energy error e is shown in Fig. A.33. The horizon for e(x(t))=0 is defined by the minimum and maximum group velocities cgr,min=2.3m/s and cgr,max=25.5m/s. The upper right corner of the prediction horizon is identified by tmax=79.3s and xmax=185.7m. For the reduced spectrum,


A29

the values turn out to be cgr,min,enl=6.5m/s, cgr,max,enl=25.5m/s, tmax,enl=96.8s and xmax,enl=632.1m. All values are summarised in Tab. A.7.

Time domain forecast for a fixed position in space Based on the snapshot (Fig. A.31) and the subsequently computed spectra (Fig. A.32), time domain forecasts for fixed positions in space have been calculated and validated with measurement. Six positions (x=-1834m, -609m, 0m, 91m, 189m, 630m) have been investigated, at which x=-1834m and x=0m define leading and trailing edge of the snapshot. The forecast for x=-1834m (Fig. A.34) delivers no compliance with measurements, as consequently follows from the fact, that no information is available on waves passing that position for instances t >0s. This can also be educed from the distribution of the prediction error, as the prediction error e instantly turns 1 for t >0s. The other five examples show that the time frame for excellent predictions can be deduced from the prediction error distribution (see Fig. A.33). For the reduced snapshot range the forecast at x=-609m shows good agreement between prediction and measurements from t=0s until t~70s, where the prediction error is already e(x(t))~0.25. Comparisons between predictions based on full and reduced spectrum show no significant difference. For x=0m (trailing edge of the snapshot), the prediction shows excellent compliance with measurements within a time frame from t=0s until t~105s. For x=91m excellent prediction is identified from t~20s up to t~110s, whereas for x=189m, the time domain forecast shows excellent comparison for a time frame from t~20s until t~120s. Finally for x=630m the comparison shows good agreement in between t=65 and 150s. Focusing on the comparison of time domain forecasts based on the full spectrum with those derived from the reduced spectrum (Fig. A.34 [diagrams 1 to 5 (from top]), it can be pointed out that for sea state I the difference are marginal. For x=630m, no comparison between full and reduced spectrum is available. In summary, it can be pointed out that the time domain forecasts calculated using the reduced spectrum show, if at all, only insignificant changes compared to those deduced using the full spectrum. It can also be concluded, that the reduction of the snapshot range reduces the accuracy of the forecast by means of a reduced time frame, in which the agreement between forecast and measurement is good to excellent. The time frames are cutback by approximately 30%.

A30


ζ [m]

position: 8 4 0 −4 −8

0

50

100

ζ [m]

position: 8 4 0 −4 −8

0

50


100

ζ [m]

ζ [m]

ζ [m]

ζ [m]

position: 8 4 0 −4 −8

8 4 0 −4 −8

8 4 0 −4 −8

8 4 0 −4 −8

0

0

50

50

0

50

0

50

150

200

250

200

250



100 position:

150 x = 91 [m] = x /2

200

250

100 position:

150 x = 189 [m] = x

200

250


200

250

200

250

100 position:

100

max

max

t [s]

150


Space domain forecast The comparison of surface elevation snapshots over a range of x=3150m is available for validation. Fig. A.35 presents the comparison at five instances in time (t=0s, t=26.4s=1/3tmax, t=52.8s=2/3tmax, t=79.3s=tmax and t=236.7s=tmax,enl). The distribution of the prediction error is given in Fig. A.33. For t=0s, the top diagram of Fig. A.35 shows absolute agreement for a range from x=-1834m to x=0m. Measurements and prediction are congruent, as for t=0s the prediction is only a reconstruction of the snapshot itself. With growing instances in time (t >0s), the comparison presents excellent compliance within a decreasing range – for t=26.4s from x~-1200m until x~200m, for t=52.8s from x~-1050m to x~500m, for t=79.3s from x~-700m up to x=500m and for t=96.8s from x~-300m to x~600m as is supported by Fig. A.33. Thus compared to predictions based on the snapshot with full range not only the frames for time domain predictions, are reduced. Also the spatial ranges for good predictions are downsized by up to 40%.


A31

Forecast of snapshots (sea state: 3, snapshot range: 75%)

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−1000 x [m]

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−2000

−1500

−2000

−1500

−2000

−1500

ζ [m] ζ [m] ζ [m]

500

0

500

−500

0

500

−500

0

500

0

500

max

−500

t = 52.8 [s] = 2/3 tmax

t = 79.3 [s] = tmax

−1000 x [m] time:

8 4 0 −4 −8 −2500

0

t = 26.4 [s] = 1/3 t

−1000 x [m] time:

8 4 0 −4 −8 −2500

−500

−1000 x [m] time:

8 4 0 −4 −8 −2500

t = 0 [s]


−1000 x [m]

−500


Discussion and general remarks In summary, it can be concluded that all forecasts show excellent to good results until e(x(t))~0.1. Thus, the time t for predictions – approximately 110s for calculations based upon the full range snapshot – is cut down by around 20% for the reduced spectrum, turning out to be 90s. t is certainly depending on the actual position x(t). The spatial range for prediction, on the other hand, is coevally downsized by around 25% for the snapshot range of 75%.

A.8


At last, the snapshot range is reduced to 50% (1225m in full scale). The snapshot is again taken at t=190s (model scale) after the beginning of the measurements/wave generation. Compared to the full snapshot 50% – starting at the leading edge – of the snapshot have been cut off.

A32


A.8.1


A.8.1.1

Surface elevation snapshot

Fig. A.36 shows the surface elevation snapshot based on which the sea state is analysed in frequency domain (blue line), resulting in the spectra as shown in Fig. A.37. Additionally, a reconstruction of the snapshot based on the reduced spectrum (see also Fig. A.36 (bottom diagram)) is presented. leading edge of snapshot direction of wave travel measurements for validation available (3150m)


ζ(x) [m]

snapshot range 50% (1225m) 8 4 0 −4 −8 −2500

−2000

−1500

−1000

−500

0 500 snapshot (sea state: 3, snapshot range: 50%) original snapshot (blue line) / reconstruction of snapshot with reduced spectrum (red line) x[m]


Sea state spectrum (full and reduced) spectra of snapshot (sea state: 3, snapshot range: 50%, reduced spectrum) amplitude spectrum ζ (k) a

ζa [m]

3


2 1 0

0.06

0.13

S( ω) [m2s]

0.25 k [1/m]

0.31

0.38

0.44

0.50


40

ω_{start}:

0.13 rad/s

ω_{end}:

2.09 rad/s

∆t_{max}:

cgr_{max}: cgr_{min}: reduced spectrum: minimum prediction error ε

20 0

3 70 m 1225 m

∆x_{max}:


60

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2


100 S( ω) [m2s]

0.19



55 s 129.2 m 24.6 m/s 2.4 m/s 8.8 %


0.77 % 70 %


0.13 rad/s

ω_{end,enlarged}:

1.14 rad/s


60.2 s

50


259 m

0


24.6 m/s 4.3 m/s

0

0.2

0.4

0.6

0.8

1 1.2 ω [rad/s]

1.4

1.6

1.8

2



The overall energy is reduced by 0.77% and the inherent, i.e. the minimum prediction error is 8.8%. The number of wave components is reduced by 70%. Except for A0(w0=0), the full spectrum ranges from wmin=0.13 to wmax=2.09rad/s (see Tab. A.8). The reduced spectrum is highlighted by the red area (see Fig. A.37, (bottom diagram)), with a frequency range from wmin=0.13 to wmax=1.14rad/s


A33

(see also Tab. A.8). The full spectrum consists of 88 wave components, whereas the reduced spectrum is downsized to 27 wave components. The limitations for the reduction of the spectrum are the same as for the spectrum derived from a snapshot with a range of 100%.

Prediction horizon The distribution of the energy error e is shown in Fig. A.38. The horizon for e(x(t))=0 is defined by the minimum and maximum group velocities cgr,min=2.3m/s and cgr,max=25.5m/s. Hence, the upper right corner of the prediction horizon is identified by tmax=79.3s, after the snapshot has been taken, and xmax=185.7m beyond the trailing edge of the snapshot. For the reduced spectrum the values turn out to be cgr,min,enl=3.1m/s, cgr,max,enl=10.8m/s, tmax,enl=236.7s and xmax,enl=727.8m. All values are summarised in Tab. A.8.


fixed positions and energy error distribution (ε=1=100%) (sea state: 3, snapshot range: 50%) 0.4

−2000

−200

0.4

0.3

0.3

0.01.3

0.1

0.4

0.4

0.1

0.2 0.3

4

0.

0.4

t=60.2s 0.4

−3000

0

x=-1250m

4

0.

0.2

0.2 0.3

0

x=-609m 0

0.3

t=0s

0

0.01.3

−1000 0.2

x [m]

x=63m x=126m x=0m

x=259m

0.2

0.3

0.2

0

0.4

t=18.3s

0.4

1000

t=55s −100

0 t [s]

100

200

−200

−100

0 t [s]

100

200


Time domain forecast for a fixed position in space Based on the snapshot (Fig. A.36) and the subsequently computed spectra (Fig. A.37) time domain forecasts for fixed positions in space have been calculated and validated with measurement. Six positions (x=-1225m, -609m, 0m, 63m, 126m, 259m) have been investigated, at which x=-1834m and x=0m define front and trailing edge of the snapshot. The forecast for x=-1225m (Fig. A.39) delivers no compliance with measurements, as consequently follows from the fact, that no information is available on waves passing that position for instances t >0s. This can also be educed from the distribution of the prediction error, as the prediction error e instantly turns 1 for t >0s. The other five examples also show that the time frame for excellent predictions can again clearly be deduced from the prediction error distribution (see Fig. A.38). The forecast at x=-609m shows good agreement between prediction and measurements can from t=0s until t~40s, where the prediction

A34


error is already e(x(t))~0.25. Comparisons between predictions based on full and reduced spectrum show no significant difference. For x=0m (trailing edge of the snapshot), the prediction shows excellent compliance with measurements within a time frame from t=0s until t~75s. For x=63m good prediction can be identified from t~15s up to t~90s. For x=126m, the time domain forecast shows good comparison for a time frame from t~20s until t=75s and for x=259m, from t~10s until t~90s. Focusing on the comparison of time domain forecasts based on the full spectrum with those derived from the reduced spectrum (Fig. A.39 (diagrams 1 to 5 (from top))) the differences are marginal. For x=259m no comparison between full and reduced spectrum is available.

ζ [m]

position: 8 4 0 −4 −8

0

50

100

ζ [m]

position: 8 4 0 −4 −8

0

50


100

ζ [m]

ζ [m]

ζ [m]

ζ [m]

position: 8 4 0 −4 −8

8 4 0 −4 −8

8 4 0 −4 −8

8 4 0 −4 −8

150

200

250

200

250



0

50

100 position:

150 x = 63 [m] = xmax/2

200

250

0

50

100 position:

150 x = 126 [m] = xmax

200

250

0

50


200

250

0

50

200

250

100 position:

100

t [s]

150


In summary, the time domain forecasts calculated using the reduced spectrum show only insignificant changes compared to those deduced using the full spectrum. The reduction of the snapshot range reduces the time frames, in which the agreement between forecast and measurement is good to excellent. Compared to the full snapshot range the time frames are cutback by an average of about 55%.

Space domain forecast This part of the validation shows the comparison of surface elevation snapshots. The complete range of x=3150m in which measurements have been taken is available for validation. Fig. A.40 presents the comparison at five instances in time (t=0s, t=18.3s=1/3tmax, t=36.6s=2/3tmax, t=55s=tmax and t=60.2s=tmax,enl). The distribution of the prediction error is given in Fig. A.38.


A35

For t=0s, the top diagram of Fig. A.40 shows absolute agreement for a range from x=-1225m to x=0m. Measurements and prediction are congruent, as for t=0s the prediction is only a reconstruction of the snapshot itself. With growing instances in time (t >0s), the comparison presents excellent compliance within a decreasing range – for t=18.3s from x~-800m until x~400m, for t=36.6s from x~-500m to x~700m, for t=55s from x~-350m up to x=700m and for t=236.7s sfrom x~-250m up to x=700m as is supported by Fig. A.38. Forecast of snapshots (sea state: 3, snapshot range: 50%)

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−1000 x [m]

ζ [m]

time: 8 4 0 −4 −8 −2500

−2000

−1500

−2000

−1000 x [m]

−1500

−2000

−2000

−1500

ζ [m] ζ [m] ζ [m]

0

500

−500

0

500

−500

0

500

0

500

−500

t = 55 [s] = tmax

−1000 x [m] time:

8 4 0 −4 −8 −2500

500

t = 36.6 [s] = 2/3 tmax

−1000 x [m]

−1500

0

max

time: 8 4 0 −4 −8 −2500

−500

t = 18.3 [s] = 1/3 t

time: 8 4 0 −4 −8 −2500

t = 0 [s]


−1000 x [m]

−500


Thus, it can be revealed that, compared to predictions based on the snapshot with full range, not only the time frames, as before for time domain predictions, are reduced. Also the spatial ranges for good predictions are downsized by about 50%.

Discussion and general remarks In summary, it can be concluded that all forecasts show excellent to good results for e(x(t))≤0.1. The time t for predictions – approximately ~2min for calculations based upon the full range snapshot – is cut down by an average of 55% for the reduced spectrum, turning out to be ~1min. t is certainly depending on the actual position x(t). The spatial range for prediction is coevally downsized by around 50% for a snapshot range of only 50% of the original size.

A36


A.9

Summary

Fig. A.41 to Fig. A.43 summarize the results shown in section A1 to A6, comparing the predictions for all snapshot ranges (100%, 75% and 50%) tos the measurements taken. The reduction of the time frame for reliable prediction with decreased snapshot range can clearly be deduced for all three sea states. Forecast at fixed positions (sea state: 1, snapshot ranges: 100%, 75% and 50%, full spectrum)

ζ [m]

position: 4 2 0 −2 −4

0

50


100

ζ [m]

position: 4 2 0 −2 −4

0

50

150

200

250

200

250

x = 0 [m] = at trailing edge of snapshot

100

150 t [s]

(blue line) measurements

(red line) snapshot range 2450m (100%)

(green line) snapshot range 1834m (75%)

(black line) snapshot range 1225m (50%)

Fig. A.41: Comparison of the time domain forecast for sea state I and all three snapshot ranges ( SL=2450m (100%), SL=1834m (75%) and SL=1225m (50%)

Forecast at fixed positions (sea state: 2, snapshot ranges: 100%, 75% and 50%, full spectrum)

ζ [m]

position: 8 4 0 −4 −8

0

50


100

ζ [m]

position: 8 4 0 −4 −8

0

50

150

200

250

200

250

x = 0 [m] = at end of snapshot

100

150 t [s]





Fig. A.42: Comparison of the time domain forecast for sea state II and all three snapshot ranges ( SL=2450m (100%), SL=1834m (75%) and SL=1225m (50%)

SUMMARY

A37 Forecast at fixed postions (sea state: 3, snapshot ranges: 100%, 75% and 50%, full spectrum)

ζ [m]

position: 8 4 0 −4 −8

0

50


100

ζ [m]

position: 8 4 0 −4 −8

0

50

150

200

250

200

250

x = 0 [m] = at end of snapshot

100

150 t [s]





Fig. A.43: Comparison of the time domain forecast for sea state III and all three snapshot ranges (SL=2450m (100%), SL=1834m (75%) and SL=1225m (50%)

Appendix B

2-D motion forecast – supplementary test runs with differing cruising velocity

B.1


B.1.1

Motion forecast for sea state III (wave heading 180°)

In the following the LNG-carrier is surrounded by sea state III. The vessel is again fixed at a distance of 124m (full scale) behind the trailing edge of the snapshot. The overall analysed time frame covers a total of 250s (full scale).3

Surface elevation snapshot Fig. B.1 shows the surface elevation snapshot of sea state III, based on which the sea state is analysed in frequency domain using the 2D-Wave forecast procedure. leading edge of snapshot



ζ(x) [m]

snapshot range 100% (2450m) 8 4 0 −4 −8 −2500

position of vessel

−2000

−1500

−1000

x[m]

−500 0 500 snapshot (sea state: 3, snapshot range: 100%, full spectrum)


Prediction horizon Fig. B.2 illustrates the prediction horizon for the examined scenario. The prediction horizon indicates excellent to good sea state and motion prediction for a time frame from 15 up to 150s. 3 The

snapshot is taken at 210s (model scale) after the beginning of the measurements/wave generation.

B2

APPENDIX B 2-D MOTION FORECAST – SUPPLEMENTARY TEST RUNS WITH DIFFERING CRUISING VELOCITY energy error distribution (ε=1=100%) (sea state: 3, snapshot range: 100%) 0.2 0.4

0.2

1000 vessel start position

0.1

0.1

0.3

0.3

0

−200

0.1

0

0.3 0.2

−3000

0.4

−2000

vessel end position

0.2 0.4

0.2 0.1

−1000 0.3

x [m]

0

0.4

01

0

4

0.2 0.

−100

0 t [s]

100

200


Encountering wave train at CG of the vessel/structure Fig. B.3 (top diagram) gives the position of the CG of the vessel for the investigated period of time. The lower diagram of Fig. 5.13 shows good comparison between predicted encountering wave train (red line) and measurements (blue line) for a time frame from 15 to 160s (highlighted red area). transient postion for a cruising velocity of v=0kn (sea state: 3, snapshot range: 100%, full spectrum)

∆x [m]

1000 0 −1000 −2000 0

50

100

∆t [s]

150

200

250

ζ [m]

transient postion for a cruising velocity of v=0kn (sea state: 3, snapshot range: 100%, full spectrum) 8 4 0 −4 −8

0

50

100

∆t [s]

150

200

250


Amplitude spectrum of encountering wave train, RAOs and vessel response spectra Fig. B.4 shows the results of the frequency domain analysis focusing on the heave motion (s3), whereas Fig. B.5 gives the results for pitch motion (s5).

MOTION FORECAST FOR A STATIONARY VESSEL (FN = 0)

B3

4

4 amplitude spectrum of the encountering wave train (v=0kn) (sea state: 3, snapshot range: 100%, full spectrum)

2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

2

0.6

0.2

0.4

0.6

0.4 0.2

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2

pitch RAO (v=0kn) (sea state: 3, snapshot range: 100%, full spectrum)

1

heave RAO (v=0kn) (sea state: 3, snapshot range: 100%, full spectrum)

0.8

0 180 −180 0

s5a /ζa [°/m]

s3a /ζa [m/m] phase [°]

1 ω [rad/s]

phase [°]

0.2

1

0.8 0.6 0.4 0.2

0 180 −180 0

0.2

0.4

0.6

2

0.8

1 ω [rad/s]

1.2

1.4

1.6

1.8

2

0 180 −180 0

s5a[°]

1

0.2

0.4

0.6

0.8

1 ω [rad/s]

1.2

1.4

1.6

1.8

2


phase [°]

0.5 0 180 −180 0

0.2

0.4

0.6

3

amplitude spectrum of heave vessel response (v=0kn) (sea state: 3, snapshot range: 100%, full spectrum)

1.5 s3a[m]

2 1

0 180 −180 0

phase [°]

phase [°]

1

phase [°]

amplitude spectrum of the encountering wave train (v=0kn) (sea state: 3, snapshot range: 100%, full spectrum)

3 ζa [m]

ζa [m]

3

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2

amplitude spectrum of pitch vessel response (v=0kn) (sea state: 3, snapshot range: 100%, full spectrum)

2 1

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


Results and validation for time the domain prediction vessel heave response for a cruising velocity of v=0kn (sea state: 3, snapshot range: 100%, full spectrum)

s3 [m]

5 0 −5

0

50

100

∆ t [s] predictable range as indicated by sea state forecast predictable range as indicated by motion forecast

150

200

250


vessel pitch response for a cruising velocity of v=0kn (sea state: 3, snapshot range: 100%, full spectrum)

s5 [°]

10 0 −10

0

50

100

∆ t [s] predictable range as indicated by sea state forecast predictable range as indicated by motion forecast

150

200

250


Transferring the frequency domain response spectra for heave and pitch motion, as shown above (Fig. B.4 and Fig. B.5) into time domain results in registration for heave and pitch responses. Fig. B.6 compares heave prediction (red line) and measurements (blue line). The comparison turns out to be

B4

APPENDIX B 2-D MOTION FORECAST – SUPPLEMENTARY TEST RUNS WITH DIFFERING CRUISING VELOCITY

good within a time frame of 5 to 155s. Fig. B.7 illustrates the results for the pitch motion, where the time frame turns out to be slightly smaller than for the heave motion (10 to 155s).

B.2


B.2.1

Change of the RAOs of the vessel in dependency of the cruising speed

As in the following the vessel motion is predicted for three different cruising velocities (v=6kn [following seas at a cruising velocity of 6kn], v=8kn [head seas at a cruising velocity of 8kn] and v=15kn [head seas at a cruising velocity of 15kn]), the RAOs for these velocities have been pre-calculated using the a adaption of damping and added mass coefficients, as has been introduced in section 5.5.2. Fig. B.8 to Fig. B.13 show heave and pitch RAOs – in dependency of the wave encounter angle – for v=6kn, v=8kn and v=15kn in comparison to those derived for v=0kn.


MOTION FORECAST FOR A CRUISING VESSEL (FN ≠ 0)

B5



B6





B7


B.2.2

Motion forecast for Sea State II (wave heading 180°, vcruise = 8 kn)

Next the motion forecast procedure will be exercised for the cruising LNG-carrier surrounded by sea state II. The cruising speed is v=8kn, the cruising direction is against the direction of wave propagation. The overall analysed time frame covers a total of 250s (full scale). 4

Surface elevation snapshot Fig. B.14 shows the surface elevation snapshot of sea state II, based on which the sea state is analysed in frequency domain using the 2D-Wave forecast method. leading edge of snapshot



ζ(x) [m]

snapshot range 100% (2450m) 8 4 0 −4 −8 −2500

vessel start position

vessel end position −2000

−1500

−1000

x[m]



4 The


B8


Prediction horizon Fig. B.15 illustrates the prediction horizon for the examined scenario. The black dashed line shows the CG-position of the vessel. The prediction horizon indicates excellent to good sea state and subsequent motion predictions for a time frame from 0 up to 75s, where the prediction error is e~0.3. energy error distribution (ε=1=100%) (sea state: 2, snapshot range: 100%)

1000

−200

0.2 0. 0 4 .3 0.3


vessel end position

0.3

0.4 0.1 0 .2

0 .42

−2000

−3000

0 .1

0.10.4 0.2

−1000

0.3

x [m]

.2 0.4 0 0.1 0

0.3

0

0.3

0.

4

0.4

0

0.3 0.2.4 0 −100

0.1

0 t [s]

100

200


Encountering wave train at CG of the vessel/structure Fig. B.16 gives the position of the CG of the vessel for the investigated period of time (top diagram) as well as the corresponding predicted encountering wave train (red line) compared to measurements (blue line) (see bottom diagram). The comparison shows excellent to good results for a time frame from 0 to 75s (highlighted red area). transient postion for a cruising velocity of v=8kn (sea state: 2, snapshot range: 100%, full spectrum)

΢x [m]

1000 0 −1000 −2000 0

50

100

΢t [s]

150

200

250

΢[m]


0

50

100

΢t [s]

150

200

250



B9

Amplitude spectrum of encountering wave train, RAOs and vessel response spectra Fig. B.17 shows the results of the frequency domain analysis focusing on the heave motion (s3), whereas Fig. B.18 gives the results for pitch motion (s5). 1.5

1

ζa [m]


0.5

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2

phase [°]

phase [°]

ζa [m]

1.5

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

s5a /ζa [°/m]

2

phase [°]

0.5

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2

0.5

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

0.5

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2

1.5

s5a[°]

1


1

0 180 −180 0


2


phase [°]


1

0 180 −180 0

s3a[m]

0.2

1.5 heave RAO (v=8kn) (sea state: 2, snapshot range: 100%, full spectrum)

1.5

phase [°]

0.5

0 180 −180 0

1.5


1


1 0.5

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


Results and validation for time the domain prediction Transferring the frequency domain response spectra for heave and pitch motion into time domain results in the time domain registration for heave and pitch. Fig. B.19 compares the predicted heave response (red line) to measurements (blue line). The comparison turns out to be good within the determined time frame of 0 to 70s with a slight offset. Fig. B.20 illustrates the comparison for the pitch motion. Good results are pointed out even until 90s. vessel heave response for a cruising velocity of v=8kn (sea state: 2, snapshot range: 100%, full spectrum)

s3 [m]

5 0 −5

0

50

100

΢t [s] predictable range as indicated by sea state forecast predictable range as indicated by motion forecast

150

200

250


B10

APPENDIX B 2-D MOTION FORECAST – SUPPLEMENTARY TEST RUNS WITH DIFFERING CRUISING VELOCITY vessel pitch response for a cruising velocity of v=8kn (sea state: 2, snapshot range: 100%, full spectrum)

s5 [°]

5 0 −5

0

50

100

150


200

250

Fig. B.20: Time domain registration (pitch motion) – (sea state II, v=8kn, full spectrum) – (measurements [blue line, prediction [red line])

B.2.3

Motion forecast for Sea State II (wave heading 180°, vcruise = 15 kn)

As the last example for sea state II, the forecast procedure will be tested for a cruising speed of v=15kn and head waves. The overall analysed time frame covers again a total of 250s (full scale). 5

Surface elevation snapshot Fig. B.21 shows the surface elevation snapshot of sea state II, based on which the sea state is analysed in frequency domain using the 2D-Wave forecast method. leading edge of snapshot



ζ(x) [m]

snapshot range 100% (2450m) 8 4 0 −4 −8 −2500

vessel start position vessel end position −2000

−1500

−1000


x[m]


Prediction horizon

1000

0.1 0. 4

energy error distribution (ε=1=100%) (sea state: 2, snapshot range: 100%) 0.3 0.2 0.4 vessel start position

2

3 0.

0

0

0.

2

0.10 .

−1000

2

−2000 0

3

3

0 0.1 .2 0.

−200

0.

0.

0.

4

−3000

vessel end position

4 0.

0.1

0

x [m]

3

0.

4 0.

0.1 −100

0 t [s]

100

200

Fig. B.22: Prediction horizon (sea state II, v=15kn, full spectrum) 5 The



B11

Fig. B.22 gives the resulting prediction horizon. The CG-position of the vessel can be identified by the black dashed line. The prediction horizon indicates excellent to good sea state and subsequently calculated motion predictions for a time frame from 30 up to 110s.

Encountering wave train at CG of the vessel/structure Fig. B.23 (top and bottom diagram) give the position of the CG of the vessel for the investigated period of time as well as the corresponding predicted encountering wave train (red line) compared to measurements (blue line), showing excellent to good results for a time frame from 30 to 125s (highlighted red area). transient postion for a cruising velocity of v=15kn (sea state: 2, snapshot range: 100%, full spectrum)

΢x [m]

1000 0 −1000 −2000 0

50

100

΢t [s]

150

200

250

΢[m]


0

50

100

΢t [s]

150

200

250



B12

APPENDIX B 2-D MOTION FORECAST – SUPPLEMENTARY TEST RUNS WITH DIFFERING CRUISING VELOCITY 3

2

0.2

0.4

0.6

1 ω [rad/s]

1.2

1.4

1.6

1.8

2

0.5

0 180 −180 0

0.2

0.4

0.6

1 ω [rad/s]

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1 ω [rad/s]

1.2

1.4

1.6

1.8


0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


0.5

0 180 −180 0

0.2

0.4

0.6

3

1

0 180 −180 0

0.4

1


2

0.2

1.5

s5a[°]

s3a[m]

3

0.8

1

2


1


2

0 180 −180 0

s5a /ζa [°/m]


0.8

phase [°]

0 180 −180 0

phase [°]

1

1.5

phase [°]

ζa [m]


2

phase [°]

phase [°]

ζa [m]

3

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


2 1

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


Results and validation for time the domain prediction Transferring the frequency domain response spectra for heave and pitch motion, as shown above (Fig. B.24 and Fig. B.25), into time domain via IFFT, results in the time domain registration for heave and pitch, respectively. Fig. B.26 compares the predicted heave response (red line) to measurements (blue line). The comparison turns out to be good from 20 to 125s. Fig. B.27 illustrates the results for the pitch motion. Again the time frame, indicated by an sea state forecast, turns out to be a good approximation, as the pitch motion shows good comparison from 25 to 130s. vessel heave response for a cruising velocity of v=15kn (sea state: 2, snapshot range: 100%, full spectrum)

s3 [m]

10 0 −10

0

50 100 ΢t [s] predictable range as indicated by sea state forecast predictable range as indicated by motion forecast

150

200

250


MOTION FORECAST FOR A CRUISING VESSEL (FN ≠ 0) vessel pitch response for a cruising velocity of v=15kn (sea state: 2, snapshot range: 100%, full spectrum)

10 s5 [°]

B13

0 −10

0

50

100


150

200

250

Fig. B.27: Time domain registration (pitch motion) – (sea state II, v=15kn, full spectrum) – (measurements [blue line], prediction [red line])

Discussion For sea states II and the cruising vessel – either in following seas (cruising velocity of v=6kn) or in head seas (cruising velocities of v=8kn and v=15kn) – the prediction for heave and pitch motion of the vessel shows good results, with slight deviations compared to the measurements taken. This shows, that vessel motions, even in high irregular sea states can be forecasted, presupposing a qualitatively good wave forecast as well as the correct representation of the motion behaviour of the vessel, i.e. the RAOs are adopted to the present cruising velocity. The time frames, as found reliable by the forecast of the encountering wave train, predefine the time frames for trustworthy motion predictions.

B.2.4

Motion forecast for Sea State III (wave heading 0°, vcruise = 6 kn)

In the following sections the motion forecast procedure will be exercised for the cruising LNGcarrier surrounded by sea state III, at three different cruising velocities. During this test run the cruising speed is v=6kn and the cruising direction goes along with the direction of wave propagation (following seas). The overall analysed time frame covers a total of 250s (full scale). 6

Surface elevation snapshot and underlying sea state spectrum Fig. B.28 shows the surface elevation snapshot of sea state III, based on which the sea state is analysed in frequency domain using the 2D-Wave forecast method. leading edge of snapshot



ζ(x) [m]

snapshot range 100% (2450m) 8 4 0 −4 −8 −2500

vessel start position −2000

−1500

vessel end position −1000

x[m]

−500 0 500 snapshot (sea state: 3, snapshot range: 100%, full spectrum )


6 The


B14


Prediction horizon Fig. B.29 illustrates the prediction horizon for the examined scenario. The black dashed line shows the CG-position of the vessel. The prediction horizon indicates excellent to good sea state and subsequent motion predictions for a time frame from 0 up to 75s. energy error distribution (ε=1=100%) (sea state: 3, snapshot range: 100%)

0.4

0.1

0.1

0.3 0

0.2

0.1

0

0. 3

0.3

0.2

−3000


0. 4

−2000

vessel end position

0.2 0. 4

−1000

0.1

x [m]

0. 3

0

0.4

0.2

0.2

1000

−200 0 1

0.2 0.4 −100

0

0 t [s]

100

200


Encountering wave train at CG of the vessel/structure Fig. B.30 (top diagram) gives the position of the CG of the vessel for the investigated period of time. The bottom diagram of Fig. B.30 shows the corresponding predicted encountering wave train (red line) compared to measurements (blue line). The comparison shows excellent to good results for the in beforehand indicated time frame from 0 to 80s (highlighted red area). transient postion for a cruising velocity of v=6kn (sea state: 3, snapshot range: 100%, full spectrum)

΢x [m]

1000 0 −1000 −2000 0

50

100

΢t [s]

150

200

250

΢[m]


0

50

100

΢t [s]

150

200

250



B15

Amplitude spectrum of encountering wave train, RAOs and vessel response spectra 3

2

ζa [m]


1

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

s5a /ζa [°/m]

2

0

0.2

0.4

0.6

2

0.5

1.2

1.4

1.6

1.8

2

0.2

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


1.5 s5a[°]

1

1

ω [rad/s]

0.4

0 180 −180


0.8


0.6

phase [°]

0.5

1 0.5

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


phase [°]


1

0 180 −180 0

s3a[m]

0

0.8 heave RAO (v=6kn) (sea state: 3, snapshot range: 100%, full spectrum)

1.5

phase [°]

1

0 180 −180

1.5


2

phase [°]

phase [°]

ζa [m]

3

0 180 −180

0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


Fig. B.31 shows the results of the frequency domain analysis focusing on the heave motion (s3). Fig. B.32 gives the results for pitch motion (s5). As a matter of fact the amplitude spectra of the encountering wave train for both analyses is the same.

Results and validation for time the domain prediction Transferring the frequency domain response spectra for heave and pitch motion, as shown above (Fig. B.31 and Fig. B.32), into time domain via IFFT, results in the time domain registration for heave and pitch, respectively. Fig. B.33 compares the predicted heave response (red line) to the measured heave response (blue line). Fig. B.34 illustrates the results for the pitch motion. Both comparisons, heave and pitch, show good agreement within the determined time frame of 0 to 75s, as has been identified by the sea state forecast (see Fig. B.30). vessel heave response for a cruising velocity of v=6kn (sea state: 3, snapshot range: 100%, full spectrum)

s3 [m]

5 0 −5

0

50

100


150

200

250


B16


s5 [°]

10 0 −10

0

50

100


150

200

250


B.2.5


Within the next run the procedure is tested at a cruising speed of v=8kn, the cruising direction is against the direction of wave propagation. The overall analysed time frame covers a total of 250s (full scale).7

Surface elevation snapshot Fig. B.35 shows the surface elevation snapshot of sea state III, based on which the sea state is analysed in frequency domain using the 2D-Wave forecast method. leading edge of snapshot



ζ(x) [m]

snapshot range 100% (2450m) 8 4 0 −4 −8 −2500

vessel end position vessel start position −2000

−1500

−1000

x[m]



Prediction horizon Fig. B.36 illustrates the prediction horizon with the black dashed line showing the CG-position of the vessel. The prediction horizon indicates excellent to good sea state and subsequent motion predictions for a time frame from 35 up to 140s, where the prediction error is already e~0.3.

7 The



B17

energy error distribution (ε=1=100%) (sea state: 3, snapshot range: 100%)


0.1

0.1

0.3 0

0.2

0.1

0

0. 3

0.3

0.2

−3000

0. 4

−2000

vessel end position

0.2 0. 4

−1000

0.1

x [m]

0. 3

0

0.4

0.4

0.2

0.2

1000

−200 0 1

0.2 0.4 −100

0

0 t [s]

100

200


Encountering wave train at CG of the vessel/structure Fig. B.37 gives the position of the CG of the vessel for the investigated period of time (top diagram)as well as the corresponding predicted encountering wave train (red line) compared to measurements (blue line), (see bottom diagram). The comparison shows excellent to good results for a time frame from 35 to 145s (highlighted red area), slightly more as has been indicated by the prediction horizon. transient postion for a cruising velocity of v=8kn (sea state: 3, snapshot range: 100%, full spectrum)

΢x [m]

1000 0 −1000 −2000 0

50

100

΢t [s]

150

200

250

΢[m]


0

50

100

΢t [s]

150

200

250



B18


ζa [m]

2

0.2

0.4

0.6

phase [°]

1 ω [rad/s]

1.2

1.4

1.6

1.8

2

0.5

0 180 −180 0

0.2

0.4

0.6

2

0.8

1 ω [rad/s]

1.2

1.4

1.6

1.8

2

0.4

0.6

1

ω [rad/s]

1.2

1.4

1.6

1.8

2

0.5

0 180 −180 0

0.2

0.4

0.6

s5a[°]

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


3

1

0.8


4

2 1

0 180 −180 0

0.2

0.4

0.6

0.8

1 ω [rad/s]

1.2

1.4

1.6

1.8

2


phase [°]

0.5 phase [°]

0.2

1


1.5

1

1.5


1


2

0 180 −180 0

s5a /ζa [°/m]

s3a /ζa [m/m]

1.5

0.8

phase [°]

1

0 180 −180 0

s3a[m]

3


phase [°]

phase [°]

ζa [m]

3

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


Results and validation for time the domain prediction Transferring the frequency domain response spectra for heave and pitch motion, as shown above (Fig. B.38 and Fig. B.39), into time domain via IFFT (Inverse Fast Fourier Transformation), results in the time domain registration for heave and pitch, respectively. Fig. B.40 compares the predicted heave response (red line) to measurements (blue line). The comparison turns out to be good within the determined time frame of 15 to 140s, which varies from the time frame indicated by the prediction horizon as well as by the comparison of measured and predicted sea state. Fig. B.41 illustrates the results for the pitch motion. Again the time frame slightly differs (25 to 145s). The overall comparison turns out to be good. vessel heave response for a cruising velocity of v=8kn (sea state: 3, snapshot range: 100%, full spectrum)

s3 [m]

10 0 −10

0

50

100


150

200

250


MOTION FORECAST FOR A CRUISING VESSEL (FN ≠ 0) vessel pitch response for a cruising velocity of v=8kn (sea state: 3, snapshot range: 100%, full spectrum)

20 s5 [°]

B19

0 −20

0

50

100


150

200

250


B.2.6


The last test run deals with LNG-carrier cruising at v=15kn surrounded by sea state III. The cruising direction is once more against the direction of wave propagation. The overall analysed time frame covers a total of 250s (full scale).8

Surface elevation snapshot Fig. B.42 shows the surface elevation snapshot of sea state III, based on which the sea state is analysed in frequency domain using the 2D-Wave forecast method. leading edge of snapshot



ζ(x) [m]

snapshot range 100% (2450m) 8 4 0 −4 vessel end position −8 −2500 −2000


−1500

−1000

x[m]



Prediction horizon Fig. B.43 illustrates the prediction horizon. The prediction horizon indicates excellent to good prediction for a time frame from 5 up to 105s, where the prediction error is e~0.3.

8 The


B20

APPENDIX B 2-D MOTION FORECAST – SUPPLEMENTARY TEST RUNS WITH DIFFERING CRUISING VELOCITY energy error distribution (ε=1=100%) (sea state: 3, snapshot range: 100%)

0.4

1000

0. 3

0.02.1

0.3

0. 3

0

0.4

0.4

0

−1000

0.2 0.1

x [m]

0.2


0

−2000

0 −3000

0.3

−200

−100

.3 00.4

vessel end position

00..21

0 t [s]

100

200


Encountering wave train at CG of the vessel/structure Fig. B.44 (top diagram) gives the position of the CG of the vessel for the investigated period of time. The lower diagram of Fig. B.44 shows the corresponding predicted encountering wave train (red line) compared to measurements (blue line). The comparison shows excellent to good results for a time frame from 5 to 105s (highlighted red area), same as has been indicated by the prediction horizon. transient postion for a cruising velocity of v=15kn (sea state: 3, snapshot range: 100%, full spectrum)

΢x [m]

1000 0 −1000 −2000 0

50

100

΢t [s]

150

200

250

΢[m]


0

50

100

΢t [s]

150

200

250

Fig. B.44: Surface elevation snapshot (sea state III, v=15kn, full spectrum) – (measurements [blue line, prediction [red line])

Amplitude spectrum of encountering wave train, RAOs and vessel response spectra Fig. B.4 shows the results of the frequency domain analysis focusing on the heave motion (s3). Fig. B.5 gives the results for pitch motion (s5).


B21 3

2

0.2

0.4

0.6

1 ω [rad/s]

1.2

1.4

1.6

1.8

2

0.5

0 180 −180 0

0.2

0.4

0.6

1 ω [rad/s]

1.2

1.4

1.6

1.8

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2

2


1 0.5

0 180 −180 0

0.2

0.4

0.6

5


2

0.2

1.5

1

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


4 s5a[°]

s3a[m]

3

0.8

1

2


1


2

0 180 −180 0

s5a /ζa [°/m]


0.8

phase [°]

0 180 −180 0

phase [°]

1

1.5

phase [°]

ζa [m]


3 2 1

0 180 −180 0

0.2

0.4

0.6

0.8

1 ω [rad/s]

1.2

1.4

1.6

1.8

2


phase [°]

phase [°]

ζa [m]

3

0 180 −180 0

0.2

0.4

0.6

0.8

1

ω [rad/s]

1.2

1.4

1.6

1.8

2


Results and validation for time the domain prediction Fig. B.47 and Fig. B.48 show the heave and pitch response spectra transferred to time domain. Comparing the predicted heave response (Fig. B.47, red line) to measurements (blue line) gives good results from 0 to 100s. The comparison for the pitch motion (Fig. B.48) turns out to be good for a time frame of 0 to 105s. Thus, both time frames very slightly differ from the time frame indicated by the sea state forecast comparison (see Fig. B.44). vessel heave response for a cruising velocity of v=15kn (sea state: 3, snapshot range: 100%, full spectrum)

s3 [m]

10 0 −10

0

50

100


150

200

250


B22


s5 [°]

20 0 −20

0

50

100


150

200

250


Discussion The forecast of vessel motions, induced by sea state III, deliver even better results as the forecast with sea state II. For all cruising velocities v=6kn (with following seas) as well as v=8kn and v=15kn (head seas), respectively, the predictions for heave and pitch motion of the vessel show good to excellent results, compared to the measurements taken. The time frames as found reliable by the forecast of the encountering wave train predefine the time frames for trustworthy motion predictions.

B.3

Summary

The forecast of motions, as developed and investigated within this chapter clearly shows that the motion behaviour of either a stationary or a cruising vessel can be predicted throughout a large variety of cruising speeds and different sea states – for head as well as following seas. All examples demonstrate, that the methods developed provide good to excellent predictability – based on a reliable forecast of the sea state conditions. The next chapter will now cope – as a logic consequence – with the adaptation of these methods for the even more challenging forecast of motions in multidirectional short crested seas.

space domain ζ(x) frequency domain

−i (ωi , j t +kx ,i , j x +k y ,i , j y +ϕi )

ζ a (ω )


Fwave (ω)

6 DOF

6 DOF

ζ a ,i , j

sa ,i , j

e iϕn

i =1

sl , j (t ) = ∑ sa ,i ,l , j e

n

−i (ω i , j t +ϕi , j )


j= 1,...,m

IFFT

l = 1,..., 6 DOF j = direction

Fs ,l , i , j (ω)

j = direction

l = 1,..., 6 DOF

H i ,l , j (ω) =

3) superposition of sector wise derived responses to overall response of vessel/structure (in 6DOF)

2) sector wise calculation of responses in 6 DOF

1) sector wise calculation of encountering wave trains

ζ1,j / ω1,j ζ2,j /ω2,j .... ζ (t ) = ζ e −i(ωi , j t +kx , j x +k y , j y +ϕi ) a ,i , j ζ3,j / ω3,j i , j



j= 1,...,m

i =1

n

with t=const.


Fwave (k )

N

F (ω) ⋅2


ζa =

array of subsequently taken snapshots output signal (effect / response)


=


Fj ( ω )


time ->

component waves


ζ(x,y,ti)

6 DOF

6 DOF

6 DOF

m

Σ

Σ

m

decomposition of overall encountering wave train into sector wise encountering wave trains


ζ(x,y,t1)


ζ(x,y,t1)



j =1

m

s j (t ) = ∑ sl , j

j =1

m

ζ (t ) = ∑ ζ j


superposition method

2D-FFT method

signal decomposition via 3D-FFT

ζa(kx,ky)


6 DOF


frequency domain

Fig. C.1: Overview of process steps for the deterministic, linear approach to the prediction of ocean waves and wave induced vessel/structure motions



=


Fj (ω)


time ->

component waves

dispersion

w = kg tanh (kd )


k=2π/L

wave number

ζ j ( x , y , t ) = ∑ ζ a ,i , j e

6 DOF



IFFT-method (section 4.3.2)

Fwave (1 L)


i 1 L )x

ζ ( x ) = ∑ ζae (


space domain ζ(x)


frequency domain

phase angle

phase angle time domain (time series) time domain (wave components ) frequency domain phase angle frequency domain phase angle frequency domain phase angle time domain (time series)

time domain (time series) time domain (wave components ) frequency domain phase angle frequency domain phase angle frequency domain phase angle time domain (time series)

space domain ζ(x,y) space domain ζ(x,y) space domain ζ(x,y,ti)

3D WAVE FIELDS multidirectional sea states


C

2D WAVE FIELDS unidirectional sea states

Appendix Overview of forecast process steps

Prediction of WaveâStructure Interaction by Advanced

Prediction of WaveâStructure Interaction by Advanced

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