This thesis investigates how wave loads act on an OC3 Hywind spar-buoy. The author analyzes the resulting motions of the support platform. This work also contains a detailed presentation of the topic with useful additional information and graphics.
Unlike fossil fuels (like oil, coal and natural gas), wind energy is a renewable energy resource. Since winds at sea are stronger and more consistent than onshore winds, the demand for offshore wind turbines has increased over the last years. As energy can be produced more efficient in deeper water, several floating offshore wind turbine constructions, such as the OC3 Hywind spar-buoy, have been proposed. The design of floating wind turbines depends on the simulation of the system behavior caused by exciting forces.
A general overview of regular and irregular waves as well as hydrostatic and hydrodynamic loads acting on floating structures is given in chapter 2. Furthermore, essential formulations for calculating motions of FOWTs are given at the end of this chapter. Since all simulations carried out in this thesis are based on the OC3 Hywind concept, detailed information about this floating wind turbine model are given in chapter 3. Three different methods are used for the estimation of wave induced loads and motions. Section 4 describes a modified Morison formulation in the time domain which is applied by the commercially available software MATLAB.
On the basis of the diffraction theory the commercial offshore software package SESAM simulates wave excitation forces and responding motions which are presented and discussed in chapter 5. The third method is the open source code FAST that computes wave induced loads and motions based on the first-order potential theory and Kane’s equation of motion. Basic formulations used in FAST and essential hydrodynamic results are shown in chapter 6. The comparisons between the simulations of the three programs are represented and the individual results are analyzed in chapter 7.
I
Abstract
Unlike fossil fuels (e.g. oil, coal and natural gas), wind energy is a renewable energy resource. Since
winds at sea are stronger and more consistent than onshore winds, the demand for offshore wind turbines
has increased over the last years. As energy can be produced more efficient in deeper water, several float-
ing offshore wind turbine constructions, such as the OC3 Hywind spar-buoy, have been proposed. The
design of floating wind turbines depends on the simulation of the system behavior caused by exciting
forces. This thesis deals with the comparison between different methods for calculating wave forces and
resulting platform motions of a floating offshore wind turbine. On the one hand, wave exciting loads
computed with Morison's equation are compared to the hydrodynamic forces simulated by the open
source code FAST on the basis of the diffraction theory. On the other hand, response motions of the float-
ing structure are simulated by the commercial offshore software SESAM in the frequency domain and
compared with the motions calculated by FAST in the time domain.
Keywords: Floating offshore wind turbine, OC3 Hywind, Wave forces, Platform motions
Zusammenfassung
Im Gegensatz zu fossilen Brennstoffen wie Öl, Kohle und Erdgas, gilt Windenergie als eine erneuerbare
Energiequelle. Im Laufe der letzten Jahre ist insbesondere die Nachfrage nach Offshore-
Windenergieanlagen gestiegen, da durch diese zum einen das Leben auf dem Binnenland nicht wesentlich
beeinflusst und zum anderen ein höherer Energieertrag als durch Onshore-Windkraftanlagen erreicht
wird. Insbesondere im Tiefwasserbereich kann Windenergie aufgrund kontinuierlicher Windbedingungen
und hoher durchschnittlicher Windgeschwindigkeiten effizient gewonnen werden. Vor diesem Hinter-
grund wurden bisher unterschiedliche Modelle für schwimmende Windkraftanlagen, wie z.B. die OC3
Hywind Struktur, entwickelt. Im Wesentlichen basiert die Planung von Offshore-Windenergieanlagen auf
Simulationen von Strukturbewegungen verursacht durch einwirkende äußere Kräfte. In der vorliegenden
Arbeit werden verschiedene Methoden zur Berechnung von Wellenkräften und der resultierenden Platt-
formbewegungen vorgestellt und miteinander verglichen. Dabei werden zunächst Wellenkräfte basierend
auf der Morison Gleichung berechnet und mit Simulationen von hydrodynamischen Kräften des Pro-
gramms FAST verglichen. Außerdem wird ein Vergleich zwischen den mithilfe der kommerziellen Soft-
ware SESAM ermittelten Bewegungen im Frequenzbereich und Simulationen der Plattformbewegungen
im Zeitbereich aus FAST gezogen.
Schlagwörter: Schwimmende Windkraftanlagen, OC3 Hywind, Wellenkräfte, Plattformbewegung
II
INHALTSVERZEICHNIS
LIST OF FIGURES ... IV
LIST OF TABLES ... VIII
SYMBOLS AND ABBREVIATIONS ... IX
1
INTRODUCTION ... 1
1.1
Background ... 1
1.2
Outline ... 1
2
STATE OF THE ART ... 3
2.1
Regular Waves ... 5
2.1.1
Description ... 5
2.1.2
Linear Wave Theory ... 7
2.1.2.1
Basic equations ... 8
2.1.2.2
Boundary conditions ... 8
2.1.2.3
Wave kinematics and pressure ... 11
2.1.3
Stretched Airy Theory ... 14
2.2
Irregular Waves... 15
2.2.1
Description in the Frequency Domain ... 17
2.3
Hydrostatics of Floating Structures ... 21
2.3.1
Static Stability ... 21
2.4
Hydrodynamics of Rigid Bodies ... 25
2.4.1
Coordinate Systems ... 25
2.4.2
Diffraction Theory ... 26
2.5
Hydrostatic- and Dynamic Loads on Floating Structures ... 29
2.5.1
Fundamentals ... 29
2.5.2
Forces and Moments ... 30
2.5.3
Radiation and Diffraction Loads ... 31
2.5.4
Wave Excitation Loads ... 34
2.5.5
Hydrostatic Loads ... 36
2.6
Floating Structures in Waves ... 37
2.6.1
Coupled Equations of Motion ... 37
2.6.2
Motions in Regular Waves ... 38
2.6.2.1
Response amplitude operator ... 39
2.6.3
Motions in Irregular Waves ... 39
3
FLOATING WIND TURBINE MODEL... 43
3.1
OC3 Hywind ... 43
3.1.1
Tower and Platform Structural Properties ... 44
3.1.2
Floating Platform Hydrodynamic Properties ... 46
3.1.3
Mooring System Properties ... 47
III
4
MATLAB ... 49
4.1
Morison Forces ... 50
4.1.1
Morison Forces due to Regular Waves ... 51
4.1.2
Morison forces due to Irregular Waves ... 54
5
SESAM ... 60
5.1
GeniE ... 61
5.1.1
The Modelling Process ... 62
5.2
HydroD ... 65
5.2.1
Coordinate System ... 66
5.2.2
Panel Model ... 67
5.2.3
Mass Model ... 69
5.2.4
Analysis Preparation ... 69
5.2.5
Wadam ... 71
5.2.5.1
Global Response Analysis in Wadam ... 71
5.2.6
Postresp ... 73
6
FAST ... 79
6.1
Basic Assumptions ... 80
6.2
Hydrodynamic Module (HydroDyn) ... 81
6.2.1
Diffraction Problem ... 83
6.2.2
Radiation Problem ... 86
6.3
Hydrodynamic Results ... 87
6.3.1
Regular Waves ... 88
6.3.2
Irregular Waves ... 91
7
COMPARISONS OF THE METHODS ... 95
7.1
Comparison of Excitation Loads ... 95
7.2
Comparison of Response Motions ... 97
8
SUMMARY AND CONCLUSION ... 101
9
REFERENCES ... 103
IV
List of Figures
Figure 2.1: Illustration of onshore wind turbines in comparison with offshore wind turbines in
shallow water, transitional water and deep water (Song, et al., 2012) ... 3
Figure 2.2: Depiction of three different concepts of offshore floating wind turbines: a) Spar-buoy,
b) Tension leg platform and c) Barge (Jonkman, 2007) ... 4
Figure 2.3: Definition of the impacts acting on a FOWT in accordance with Elliott et al. (2012) ... 5
Figure 2.4: Illustration of the change of orbits depending on the water depth h according to EAK
(2002) ... 6
Figure 2.5: Depiction of basic parameters of a two-dimensional regular wave (according to Zanke
(2002)) ... 6
Figure 2.6: Horizontal velocity distribution under a wave crest and a wave trough based on the Airy
theory (in accordance with Faltinsen (1999)) ... 13
Figure 2.7: Pressure variation under a wave crest and a wave trough according to Linear Wave
Theory in accordance with Faltinsen (1999) ... 14
Figure 2.8: Comparison between the horizontal fluid velocity according to the linear wave theory
and the horizontal wave velocity based on Wheeler stretching (Skailerud et al.
(2010)) ... 15
Figure 2.9: Simplification of the sea state: a) Superposition of regular waves, b) Short-crested
irregular sea state, c) Long-crested irregular sea state, d) Long-crested regular sea
state (EAK, 2002) ... 16
Figure 2.10: Picture illustrating the relation between the time domain and the frequency domain
analysis of waves in a long-crested swell (according to Faltinsen (1999)) ... 17
Figure 2.11: Illustration of an amplitude spectrum and a phase spectrum of a time series, according
to Oumeraci (1996b) ... 18
Figure 2.12: Illustration of the relation between the measured time domain and the frequency
domain, in accordance with Fröhle (2000) ... 21
Figure 2.13: Illustration of the hydrostatic pressure und -forces acting on floating bodies in
accordance to Clauss et al. (1988)) ... 22
Figure 2.14: Definitions of the buoyancy center and force (a) and of the gravity center and force (b)
in accordance with Taheri (2013) ... 23
Figure 2.15: Definitions of the types of equilibrium: (a) natural equilibrium, (b) stable equilibrium,
(c) unstable equilibrium (in accordance with Clauss, et al. (1988)) ... 24
Figure 2.16: Definitions of the coordinate systems, the wave propagation direction and the modes
of motion of a rigid body according to Faltinsen (1999) ... 26
Figure 2.17: Illustration of the heave motion by superposition of (a) radiation loads and (b) wave
exciting loads (Journée, et al., 2001) ... 27
Figure 2.18: Relationship between acting wave loads on a floating body and the resulting motions
according to Journée et al (2001) ... 28
V
Figure 2.19: Definition of the boundary conditions for a floating structure using in Green's theorem
(Journée, et al. (2001)) ... 33
Figure 2.20: Illustration of a response amplitude operator and the significant frequency areas
regarding uncoupled motional behavior (Journée, et al., 2001) ... 39
Figure 2.21: Illustration of the process of calculating the transfer function of an offshore structure
according to Clauss et al., (1988) ... 41
Figure 3.1: Illustration of the NREL 5-MW offshore wind turbine on the OC3 Hywind platform
(Jonkman, 2010) ... 43
Figure 3.2: Detailed description of the structural properties of the NREL 5-MW offshore wind
turbine and the OC3 Hywind platform ... 44
Figure 3.3: Depiction of the mooring system of the OC3 Hywind spar-buoy (Jonkman, 2010) ... 47
Figure 4.1: Depiction of the modified OC3 Hywind system for a simpler determination of the
Morison forces acting on the floating platform ... 49
Figure 4.2: Illustration of the horizontal velocity u, the acceleration u/t and the Morison force
acting on a unit length dz of a vertical cylinder (Clauss, et al., 1988) ... 51
Figure 4.3: Simulation of regular waves (H = 6 m, T = 10 s) with the appropriate horizontal wave
velocity u and acceleration a
x
(upper figure) and the Morison force F
Total
(lower
figure) at the wave crest/trough (z = ) ... 52
Figure 4.4: Illustration of the horizontal fluid velocity u and acceleration a
x
(upper plot) and the
Morison force F
Total
(lower plot) acting on the platform base (z = -120 m) due to
regular waves (H = 6 m, T = 10 s) ... 53
Figure 4.5: Illustration of the Morison forces acting on the submersed part of the OC3 Hywind
platform due to regular waves (H = 6 m, T = 10 s) ... 54
Figure 4.6: Illustration of the JONSWAP spectrum with a significant wave height of H
s
= 6 m and a
peak period of T
p
= 10 s ... 54
Figure 4.7: Illustration of the division of a wave spectrum into equally spaced frequencies d ... 55
Figure 4.8: Superposition of regular waves for three different frequencies ( = 0.3 rad/s, = 0.7
rad/s, = 1.2 rad/s) and illustration of the resulting fluid velocities u (top) and
accelerations a
x
(bottom) ... 56
Figure 4.9: Demonstration of the surface elevation , the horizontal wave velocity u and
acceleration a
x
due to irregular waves calculated as superposition of three regular
waves ( = 0.3 rad/s, = 0.7 rad/s and = 1.2 rad/s) ... 57
Figure 4.10: Illustration of the horizontal fluid velocity u and acceleration a
x
(top) as well as the
Morison force F
Total
(bottom) at the wave crest and trough (z = ) due to irregular
waves with H
s
= 6 m and T
p
= 10 s ... 58
Figure 4.11: Generation of the horizontal wave velocity u and acceleration a
x
(upper plot) and the
Morison force (lower plot) at the platform base (z = -120 m) due to irregular waves
(H
s
= 6 m, T
p
= 10 s) ... 59
VI
Figure 4.12: Demonstration the Morison force (lower plot) acting on the entire wetted surface of
the structure caused by irregular waves (H
s
= 6 m, T
p
= 10 s) ... 59
Figure 5.1: Illustration of the integrated modules (GeniE, HydroD, DeepC and Postresp) in SESAM
and their dependency on each other ... 61
Figure 5.2: First of three steps for the hydrodynamic analysis of the wind turbine model in SESAM,
depicted in red: Modelling of the floating structure in GeniE ... 62
Figure 5.3: Illustration of the guiding geometry (guiding planes in green and guiding lines in blue),
the modelling process of the tower (a) and the platform (b and c) by using shells
and the final model with continuous surface (d) ... 63
Figure 5.4: Illustration of the finite element model of the wind turbine model with first-order finite
elements (characteristic mesh size of 1 m) ... 63
Figure 5.5: Depiction of the defined wet surface of the wind turbine model (a) and the hydrostatic
pressure acting on the immersed parts of the platform (b) ... 64
Figure 5.6: Presentation of the modelled OC3 Hywind system with the hydrostatic pressure acting
on the platform in a water depth of 320 m ... 64
Figure 5.7: Presentation of the main model types (hydro model, mass model and structural model)
in HydroD used for the hydrodynamic analysis in Wadam (Det Norske Veritas,
2011) ... 65
Figure 5.8: Depiction of several combinations of panel and Morison models (Det Norske Veritas,
2011) ... 66
Figure 5.9: Illustration of a semi-submersible with its center of gravity (COG), global coordinate
system (X
glo
, Z
glo
) and input coordinate system (x
inp
, z
inp
) with ZLOC = -h used in
HydroD (Det Norske Veritas, 2011) ... 67
Figure 5.10: Presentation of a hull form of a crude oil carrier divided in a number of panels
(Journée, et al., 2001) ... 67
Figure 5.11: Illustration of the adjustment or division of elements extending above the still water
level (Det Norske Veritas, 2011) ... 69
Figure 5.12: Generation of the loading condition and illustration of the CoG of the model in red as
well as the water surface at mean sea level in blue ... 70
Figure 5.13: Second of three steps for the hydrodynamic analysis of the wind turbine model in
SESAM, illustrated in green: The calculation of the wave-structure interaction in
Wadam ... 71
Figure 5.14: Description of main wave headings with respect to the input coordinate system of the
offshore floating structure (Det Norske Veritas, 2011) ... 72
Figure 5.15: Third of three steps for the hydrodynamic analysis of the OC3 Hywind system in
SESAM, shown in purple: Evaluation of the hydrodynamic results in Postresp ... 73
Figure 5.16: Presentation of the wave exciting forces X
w
in surge, sway and heave direction caused
by harmonic waves with a unit-amplitude and a wave angle = 0° ... 74
VII
Figure 5.17: Illustration of the wave exciting moments X
w
in roll, pitch and yaw direction due to
harmonic waves and a wave angle of = 0°... 75
Figure 5.18: Transfer functions H
i
( ) of the translational motions in surge, sway and heave
direction due to regular waves with a unit-amplitude and wave heading of = 0° ... 76
Figure 5.19: Transfer functions H
i
() of the rotational motions in roll, pitch and yaw direction
caused by harmonic waves with a unit-amplitude and wave heading of = 0° ... 76
Figure 5.20: Generated JONSWAP spectrum (H
s
= 6 m, T
p
= 10 s) used for the calculation of the
response spectra in six directions of motion in SESAM ... 77
Figure 5.21: Display of the response spectra S
() for the translational motions surge (
1
), sway
(
2
) and heave (
3
) caused by irregular waves (H
s
= 6 m and T
p
= 10 s) with = 0° ... 78
Figure 5.22: Illustration of the response spectra S
() for the rotational motions roll (
4
), pitch (
5
)
and yaw (
6
) caused by irregular waves (H
s
= 6 m and T
p
= 10 s) with a wave
heading of = 0° ... 78
Figure 6.1: Overview of the simulation process of a fully coupled aero-hydro model in FAST ... 80
Figure 6.2: Flowchart of the hydrodynamic analysis in the time domain in FAST according to
Jonkman (2007) ... 83
Figure 6.3: Tower base coordinate system for defining the translations and rotations of the OC3
Hywind platform in FAST (Jonkman, et al., 2005) ... 87
Figure 6.4: Simulation of regular waves (H = 6 m, T = 10 s) with the appropriate horizontal wave
velocity u and acceleration a
x
(upper plot) and the total hydrodynamic force F
Pltf
acting on the platform in x-direction (lower plot) in FAST ... 89
Figure 6.5: FAST-Simulation of the translational motions
1
(surge),
2
(sway) and
3
(heave) of
the floating platform due to regular waves (H = 6 m, T = 10 s) with a wave
direction of = 0° ... 90
Figure 6.6: Depiction of the rotational motions
4
(roll angle),
5
(pitch angle) and
6
(yaw angle)
of the floating platform caused by irregular waves (H
s
= 6 m, T
p
= 10 s) with a
wave direction of = 0° simulated with FAST ... 90
Figure 6.7: Generation of irregular waves (H
s
= 6 m, T
p
= 10 s) with the corresponding horizontal
wave velocity u and acceleration a
x
(upper plot) and the total hydrodynamic force
F
Pltf
acting on the OC3 Hywind platform (lower plot) in FAST ... 91
Figure 6.8: FAST-Simulation of the translational motions
1
(surge direction),
2
(sway direction)
and
3
(heave direction) of the floating platform due to irregular waves (H
s
= 6 m,
T
p
= 10 s) with a wave direction of = 0° ... 92
Figure 6.9: Depiction of the rotational motions
4
(roll angle),
5
(pitch angle) and
6
(yaw angle)
of the floating platform caused by irregular waves (H
s
= 6 m, T
p
= 10 s) with a
wave direction of = 0° simulated with FAST ... 92
Figure 6.10: Response spectra S
() of the translational motions in surge
1
and heave
3
direction
due to irregular waves (H
s
= 6 m, T
p
= 10 s) with a wave heading angle of = 0°... 93
VIII
Figure 6.11: Response spectra S
() of the rotational motion in pitch
5
direction due to irregular
waves (H
s
= 6 m, T
p
= 10 s) with a wave heading angle of = 0° ... 94
Figure 7.1: Comparison of the wave exciting loads F
Waves
according to Morison's equation
calculated with MATLAB and the total hydrodynamic forces F
Pltf
based on the
potential theory determined by FAST caused by regular waves (H = 6 m, T = 10 s) ... 95
Figure 7.2: Comparison of the wave exciting loads F
Waves
according to Morison's equation
calculated with MATLAB and the total hydrodynamic forces F
Pltf
based on the
potential theory determined by FAST due to an irregular sea state (H
s
= 6 m, T
p
=
10 s) ... 96
Figure 7.3: Comparison between the spectrum S
() of the response motions in surge direction
1
calculated by FAST and the response spectrum from SESAM ... 98
Figure 7.4: Comparison between the spectrum S
() of the response motions in heave direction
3
calculated by FAST and the response spectrum from SESAM ... 99
Figure 7.5: Comparison between the spectrum S
() of the response motions in pitch direction
5
calculated by FAST and the response spectrum from SESAM ... 100
List of Tables
Table 3.1: Overview of the essential tower properties of the OC3 Hywind system referred to
Jonkman (2010) ... 45
Table 3.2: Structural properties of the support platform according to Jonkman (2010) ... 45
Table 6.1: List of natural frequencies of the OC3 Hywind spar-buoy calculated in FAST (Roald, et
al., 2013) ... 88
Table 7.1: List of the natural frequencies of the FOWT calculated by FAST and SESAM... 100
IX
Symbols and Abbreviations
Symbol
Notation
Dimension
a
Added-mass coefficient
M
a
x
Horizontal wave acceleration
LT
-2
a
z
Vertical wave acceleration
LT
-2
a
Stretching
Horizontal acceleration according to Wheeler stretching method
LT
-2
A
ij
Impulsive added-mass
M
A
ij
()
(i,j) component of the added-mass matrix
M
b
Damping coefficient
-
b
cr
Critical damping coefficient
-
B
Viscous damping
-
B
ij
()
(i,j) component of the damping matrix
-
c
Restoring spring coefficient
MT
-2
C
a
Added-mass coefficient used in Morison's equation
-
C
D
Drag coefficient used in Morison's equation
-
C
e
External restoring
MT
-2
C
m
Mass coefficient used in Morison's equation
-
C
ij
Linear
Linearized restoring matrix from mooring system
MT
-2
C
ij
()
(i,j) component of the hydrostatic restoring matrix
MT
-2
D
Diameter of a cylinder
L
f
Frequency
T
-1
f
j
Normalized direction cosines (j = 1 - 6)
-
F
B
Buoyancy force
MLT
-2
F
Damping
Linear damping force
MLT
-2
F
Drag
Drag force used in Morison's equation
MLT
-2
F
FK
Froude-Krilov force
MLT
-2
F
G
Force due to gravity and inertia
MLT
-2
F
Hydro
Force due to hydrostatics and dynamics used in FAST
MLT
-2
F
Inertia
Inertia forces used in Morison's equation
MLT
-2
F
Lines
Force due to the mooring system used in FAST
MLT
-2
F
Pltf
Total force acting on the floating platform used in FAST
MLT
-2
F
S
, F
Hydrostatic
Forces due to hydrostatic buoyancy
MLT
-2
F
W
, F
Waves
Wave force acting on structure
MLT
-2
g
Acceleration of gravity
LT
-2
GM
Metacentric height
L
X
GM
Longitudinal metacentric height
L
GM
Transverse metacentric height
L
h
Water depth
L
H
Wave height
L
H
s
Significant wave height
L
I
jk
(j,k) component of the moment of inertia
L
4
k
Wavenumber
L
-1
K
ij
(i,j) component of the radiation-retardation kernel
-
L
Wave length
L
M
Metacenter
L
M
ij
(i,j) component of the mass matrix
M
M
Re
Restoring moment
ML
2
T
-2
m
n
nth moment of a wave spectrum
L
2
T
-n
m
0
Area under the spectral area curve
L
2
p
D
Dynamic pressure
ML
-1
T
-2
r
ij
(i,j) component of the radius of gyration
L
s
a
Wave amplitude of the motion of a rigid body
L
s
j
Motion of a rigid body in i-degree of freedom (i = 1 - 6)
L
S
Water plane area
L
2
S
0
Water plane area of the structure in static condition
L
2
S
(,)
Directional energy spectrum of the incident waves
L
2
T
S
1-Sided
()
One-sided energy spectrum of the incident waves
L
2
T
S
2-Sided
()
Two-sided energy spectrum of the incident waves
L
2
T
S
()
Energy spectrum of the incident waves
L
2
T
S
()
Energy spectrum of the responding waves
L
2
T
t
Time
T
T
Wave period
T
T
1
Mean wave period
T
T
z
Zero-upcrossing wave period
T
T
p
Wave period at spectral peak
T
u
Horizontal wave velocity
LT
-1
u
Stretching
Horizontal velocity according to Wheeler stretching method
LT
-1
U
1
,U
2
Independent and uniformly-distributed random variates
L
w
Horizontal wave acceleration
LT
-2
W()
Fourier transform of a white Gaussian noise
-
x
g
,y
g
,z
g
Coordinates of the center of gravity
L
x
b
,y
b
,z
b
Coordinates of the center of buoyancy
L
XI
x
0
,y
0
,z
0
Coordinates of the earth-bound coordinate system
L
x
s
,y
s
,z
s
Coordinates of the body-fixed coordinate system
L
x
glo
,y
glo
,z
glo
Coordinates of the global coordinate system used in SESAM
L
x
inp
,y
inp
,z
inp
Coordinates of the input coordinate system used in SESAM
L
X
r
Radiation loads
MLT
-2
X
w
Wave excitation loads
MLT
-2
X
S
Hydrostatic loads
MLT
-2
z
Position of the wave crest/trough (Wheeler stretching method)
L
Displayced volume of water
L
3
Phillips constant
-
Ratio between the wave frequency and the natural frequency of the
structure = /
0
-
Peak shape parameter in the JONSWAP spectrum
-
ij
(i,j) component of the Kronecker-Delta function
-
Surface elevation of incident waves
L
a
Amplitude of incident waves
L
k
Motion in k-direction (k = 1 - 6);
1 = surge, 2 = sway, 3 = heave, 4 = roll, 5 = pitch, 6 = yaw
L
Wave direction
-
Density of freshwater
ML
-3
s
Density of saltwater
ML
-3
Scaling factor in the JONSWAP spectrum
-
2-Sided
Variance of the incoming waves
-
Phase angle
-
Velocity potential
L
2
T
-1
r
Velocity potential due to radiation
L
2
T
-1
w
Velocity potential due to incident waves
L
2
T
-1
d
Velocity potential due to diffraction
L
2
T
-1
Angular wave frequency
T
-1
p
Peak of the angular frequency
T
-1
z
Zero-upcrossing angular wave frequency
T
-1
0
Natural frequency of the structure
T
-1
Frequency of incident wave
T
-1
XII
Abbreviation
Meaning
AVCG
Allowable Vertical Center of Gravity
BIEM
Boundary Integral Equation Method
COB
Center of Buoyancy
COG
Center of Gravity
DNV
Det Norske Veritas
DOF
Degree of Freedom
FAST
Fatigue, Aerodynamics, Structures and Turbulence
FEM
Finite Element Method
FFT
Fast Fourier Transformation
FOWT
Floating Offshore Wind Turbine
FPSO
Floating Production Storage and Offloading
HydroD
Hydrodynamic Design
JONSWAP
Joint North Sea Wave Project
MIT
Massachusetts Institute of Technology
MSL
Mean Sea level
NREL
National Renewable Energy Laboratory
Postresp
Postprocessor for Statistical Response Calculations
RAO
Response Amplitude Operator
RGN
Pseudo-Random Number Generator
SESAM
Super Element Structural Analysis Modules
TLP
Tension Leg Platform
WAMIT
Wave Analysis MIT
WGN
White Gaussian Noise
1
1
Introduction
1.1
Background
Currently, most of the energy worldwide is obtained by nonrenewable resources such as coal, oil, natural
gas and nuclear power. Fossil fuels are, however, limited in supply and the increasing consumption is
harmful to the environment. For example, dangerous nuclear waste is constantly produced by obtaining
energy from nuclear power. The safe storage and disposal of this radioactive waste as well as the in-
creased risk from terrorism, radioactive accident and nuclear proliferation pose serious problems
(Jonkman, 2007). Against this background, the demand for renewable energy has increased significantly
in the past years. Onshore wind power has been the fastest growing energy source worldwide for more
than a decade. Due to the limited availability of land and vast shallow-water wind resources at the North
and Baltic seas, Europe is the global leader in the development of offshore wind turbines (Lefebvre, et al.,
2012). A substantial advantage of offshore wind turbines is a higher wind energy production in compari-
son to land-based wind turbines. This occurred as the wind is more consistent and stronger over the sea
because of less turbulence at sea than onshore. As offshore structures are usually produced near the coast-
line, the size of these constructions is not limited by road or rail logistical constraints, which results in
lower transport costs and increasing flexibility in construction.
With regard to offshore wind turbine systems, a distinction is made between bottom-fixed and floating
platforms. Bottom-fixed offshore wind turbines are installed in water depths up to 60 meters. Since the
installation and maintenance of such structures is associated with substantial costs and effort, floating
offshore wind turbines (FOWT) are used in deeper water. Through the positioning of the FOWT in the
open ocean, visual and noise annoyances can be avoided. Besides, in contrast to the bottom-fixed wind
turbines, the floating platforms do not have to be inserted into the seabed. The installation of the wind
turbine system in the ground causes significant habitat disturbance for marine mammals and fish
(Henderson, et al., 2003).
In nature, floating wind turbines are subject to not only aerodynamic loads, but also to hydrodynamics.
Unlike bottom-fixed platforms, the dynamic behavior of FOWT is mainly influenced by structural proper-
ties of the floating system, hydrodynamic processes and further external loads (Butterfield, et al., 2005).
To avoid instabilities of the FOWTs, research on support platform motions is indispensable. Several stud-
ies have been carried out on preliminary design of FOWTs, such as by Bulder (2005), Lee (2005) and
Wayman et al., (2006). However, floating support platforms for wind turbines are still in its research and
development phase. The first full-scale prototype, the so-called OC3 Hywind spar-buoy, has been devel-
oped by the Norwegian oil corporation Statoil and was deployed in 2009 (Lefebvre, et al., 2012).
1.2
Outline
The aim of this thesis is to investigate wave loads acting on an OC3 Hywind spar-buoy and to analyze the
resulting motions of the support platform. A general overview of regular and irregular waves as well as
hydrostatic and hydrodynamic loads acting on floating structures is given in chapter 2. Furthermore, es-
sential formulations for calculating motions of FOWTs are given at the end of this chapter. Since all sim-
ulations carried out in this thesis are based on the OC3 Hywind concept, detailed information about this
floating wind turbine model are given in chapter 3. Three different methods are used for the estimation of
wave induced loads and motions. Section 4 describes a modified Morison formulation in the time domain
which is applied by the commercially available software MATLAB. On the basis of the diffraction theo-
2
ry, the commercial offshore software package SESAM simulates wave excitation forces and responding
motions which are presented and discussed in chapter 5. The third method is the open source code FAST
that computes wave induced loads and motions based on the first-order potential theory and Kane's equa-
tion of motion. Basic formulations used in FAST and essential hydrodynamic results are shown in chapter
6. The comparisons between the simulations of the three programs are represented and the individual
results are analyzed in chapter 7. Finally, a brief summary and conclusion are given in section 8.
3
2
State of the Art
As offshore wind energy is a clean, domestic and renewable resource, academic interest in offshore wind
turbines has attracted considerable attention from industry over the last few years and consequently, a
significant amount of funding has been invested to the development of offshore wind turbines (Roddier,
et al., 2009). Substructures of offshore wind turbines can be categorized depending on the water depth, as
illustrated in Figure 2.1. Wind turbines in shallow water (less than 30 meters) are built on monopoles and
gravity bases which extend to the sea bottom. In a water depth between 30 and 60 meters, also known as
transitional depth, multi-pile structures and frames (e.g. jackets), which reach to the seabed, are used. At a
water depth of more than 60 meters, however, bottom-mounted platforms are not feasible anymore be-
cause the installation and maintenance is associated with higher costs and effort as well as with signifi-
cant habitat disturbance for marine mammals and fish (e.g. dolphins and harbor porpoises). For that rea-
son, prototypes of floating substructures are developed (see Figure 2.1), which proved to be the most eco-
nomical option for generating electricity in the open sea (Song, et al., 2012).
Figure 2.1: Illustration of onshore wind turbines in comparison with offshore wind turbines in shallow water,
transitional water and deep water (Song, et al., 2012)
Until a few years ago, most of the offshore wind turbines in the world were located in shallow water.
However, much of the offshore wind resource potential in China, Japan, Norway, the United States and
other countries is available in deeper water (Jonkman, 2010). Based on significant research and develop-
ment efforts, the world's first full-scale floating platform for offshore wind turbines in deep water has
been installed in the North Sea off the coast of Norway in 2009. The support platform is developed by the
Norwegian oil corporation Statoil as part of the so-called Hywind-project and can be installed in water
depths from 120 meters to 700 meters (AFK, 2009). A more detailed explanation of this project, which is
an essential part of this thesis, is given in chapter 3.
In general, several floating support platform constructions were developed for offshore wind turbines
with modifications to the mooring systems, tanks and ballast options in accordance with the offshore oil
and gas (O&G) configurations. Three different concepts of offshore floating wind turbines are depicted in
Figure 2.2. The spar-buoy construction is moored by taut lines and contains additional ballast to reach
4
static stability by lowering the center of mass below the center of buoyancy. The tension leg platform
(TLP) is stabilized by the mooring line tension (see Figure 2.2b) and the barge achieves stability through
distributed buoyancy and the use of weighted water plane area for righting moment (Jonkman, 2007).
Figure 2.2: Depiction of three different concepts of offshore floating wind turbines: a) Spar-buoy, b) Tension
leg platform and c) Barge (Jonkman, 2007)
One of the fundamental challenges concerning floating structures is the ability to estimate loads and re-
sulting responses of the system. In the offshore environment, floating wind turbines are exposed to sever-
al loads due to wind, waves, currents, sea ice and so on. Figure 2.3 shows the essential impacts on a float-
ing wind turbine. In contrast to a bottom fixed structure, the dynamic behavior of a floating wind turbine
is changed by hydrodynamics, the external loads and the structure itself. So far, the hydrodynamic loads
acting on offshore structures are not well understood (Butterfield, et al., 2005).
This thesis deals with motions of floating bodies caused by wave induced forces. In order to provide an
overview of basic characteristics of incident waves, formulations for regular and irregular waves are
defined in chapter 2.1 and 2.2. These wave induced loads acting on a floating offshore structure may
cause an instabil condition. For this reason, the essential processes and coefficients in the hydrostatic
equilibrium condition are outlined in chapter 2.3. To gain an overview of basic motions of floating
structures, the hydrodynamics of a simple rigid body in an ideal fluid are clarified in chapter 2.4. The
determination of hydrostatic and hydrodynamic loads acting on a FOWT are described in chapter 2.5 and
the calculation of motions of floating systems are presented in chapter 2.6.
Ballast Stabilized
,,Spar-buoy"
Mooring Line Stabilized
Tension Leg Platform
Buoyancy Stabilized
"Barge"
a)
b)
c)
5
Figure 2.3: Definition of the impacts acting on a FOWT in accordance with Elliott et al. (2012)
2.1
Regular Waves
The essential parameters of regular waves are discussed and illustrated in chapter 2.1.1. Then, basic equa-
tions, boundary conditions and wave kinematics according to the linear wave theory are presented in
chapter 2.1.2. Since the linear wave theory neglects the kinematics above mean sea level, the so-called
Wheeler Stretching method is introduced, which enables the calculation of the kinematics up to the free
water surface (see section 2.1.3).
2.1.1
Description
In general, the spatial and temporal development of the sea state can be examined by the linear or non-
linear wave theory. The first-order theory is based on the assumption of small wave steepness, which is
defined as the ratio between wave height H and wavelength L, and considers the movement of the water
particles on orbits. Figure 2.4 depicts the change of orbits depending on the water depth h. For wave
steepness H/L greater than 1/50, waves with a finite amplitude behave nonlinearly. In this case, non-linear
wave theories should be applied (EAK, 2002). This thesis basically deals with the first-order theory.
6
Figure 2.4: Illustration of the change of orbits depending on the water depth h according to EAK (2002)
The basis for this is the mathematical and physical description of gravity waves. Surface waves arise
through vertical movements of the sea surface which can be caused by the tides, volcano eruptions, tsu-
namis, gravity or capillary waves. The two latter are short surface waves. The surface tension of the capil-
lary waves which are considered to be the shortest waves at the water surface has the effect of a restoring
force. In contrast, gravity waves have the restoring influence of gravity or buoyancy in terms of swell and
wind waves. For the derivation of the mathematical and physical description of gravity waves the as-
sumption of a monochromatic wave, which spreads with an amplitude
a
in the direction , is made. The
amplitude
a
describes the crest height above mean sea level and is defined as half of the wave height H
(see Figure 2.5). Thus, the wave height is the difference between wave crest and wave trough. The surface
elevation is directed upward from the undisturbed water surface which is illustrated in Figure 2.5.
Figure 2.5: Depiction of basic parameters of a two-dimensional regular wave (according to Zanke (2002))
Some properties of a two-dimensional regular wave are pictured in Figure 2.5. The illustration shows,
among others, the directions of the wave velocities w and u for a two-dimensional surface wave and the
water depth h. The wavelength L characterizes the range between two adjacent wave crests or troughs and
describes the wavenumber k as follows:
h
H
z
x
(x, t)
L
w
u
a
Transition Area
Shallow Water
L/
2
c = L/T
Deep Water
h 0.05L
7
k =
2
L
(2.1)
The wave period T indicates the time taken for one complete cycle of the wave to pass a reference point
x = (x, y). Equivalently, it describes the time between successive wave crests and wave troughs. The fre-
quency is the inverse of the wave period
f =
1
T
(2.2)
and is related to the angular frequency by
= 2 f
(2.3)
With regard to the undisturbed water surface the equation
= (x, t) = sin kx - t +
(2.4)
describes a spatial and temporal variation of the surface elevation in meters. In equation (2.4) t denotes
the time and the phase angle. The latter represents the displacement of the wave referring to a time t = 0
or to the coordinate origin x = 0 and y = 0 (Mai, et al., 2004).
2.1.2
Linear Wave Theory
In this chapter the linear wave theory which is also called Airy theory (Airy, 1845) or Stokes first-order
theory (Stokes, 1847) is clarified and the assumptions made are discussed. The linear wave theory gives a
reasonable approximation of wave properties. In general, waves can be described more precisely by using
the sum of many successive approximations, where each supplemental wave in the series is a correction
to preceding terms. This can be achieved with higher-order theories, such as finite-amplitude wave
theories, which is explained in detail in the paper of Mei (1991) and also in Dean and Dalrymple (1991).
However, the Airy theory is fundamental for understanding higher-order theories and turns out to be the
least complicated wave theory, which is based on several assumptions, such as the homogeneity and in-
compressibility of an ideal irrotational fluid. Moreover, the specific medium is considered to be inviscid
and the waves are plane or long-crested (two-dimensional). Further, it is assumed that the pressure at the
free surface is uniform and constant and the wave amplitude is small compared to the wave length and the
water depth. The sea bed is considered to be a fixed, horizontal and impermeable boundary, which means
that the vertical velocity at the seabed is zero (Newton, 2009).
8
2.1.2.1
Basic equations
The incompressibility of the sea water requires that the velocity potential
has to satisfy Laplace's
equation:
x
+
y
+
z
= 0
(2.5)
The velocity potential describes the velocity field and is defined as the gradient of a scalar function. To
find the complete mathematical solution for the incompressible, inviscid and irrotational fluid motion, the
Laplace equation has to be solved with relevant boundary conditions on the fluid, as will be discussed in
chapter 2.1.2.2. Thus, the fluid velocity vector V(x, y, z, t) = (u, v, w) at time t at the point x = (x, y, z) in
a Cartesian coordinate system fixed in space can be described depending on the velocity potential:
=
x
+
y
+
z
(2.6)
where i, j and k denote unit vectors along the x-, y- and z-axes, respectively. The fluid is irrotational
when the vorticity vector
= ×
(2.7)
is zero everywhere in the fluid. The pressure p is based on Bernoulli's equation
p +
t
+
1
2
+ gz = C
(2.8)
where g is the gravitational acceleration, the density of the fluid and C an arbitrary function of time. The
z-axis is assumed to be vertical and positive upwards. Equation (2.8) is valid for unsteady, irrotational and
inviscid fluid motion and the assumption is made that the only external force field is gravity. Additional-
ly, it is estimated that the free surface is defined as z = 0 with the fluid domain z < 0.
2.1.2.2
Boundary conditions
In the previous chapters an incompressible fluid and irrotational flow was assumed. As already men-
tioned, all potential flows are based on the Laplace equation that is solved by kinematic and dynamic
boundary conditions. While the kinematic boundary conditions refer to the motions of the water particles,
the dynamic conditions relate to loads acting on the particles. The boundary conditions at the free surface
define that a particle located at the free surface remains in this location (kinematic) and that the pressure
is constant at the water surface (dynamic). The boundary condition at the sea bottom specifies an imper-
meable layer, which means that the vertical velocity at the bottom is zero. Based on the perception of an
ideal fluid (no friction), the boundary condition for the horizontal velocity at the bottom is neglected
(Sorensen, 2006). In the following, the mathematical equations of these boundary conditions are dis-
cussed.
9
Kinematic boundary condition
The body boundary condition on the body surface is valid for fixed structures in a moving fluid
n
= 0
(2.9)
and represents impermeability, which means that no flux at the body surface is present. The body surface
in respect of the normal is characterized by /n. Thus, the positive normal is directed into the fluid
domain. The motion of the structure with velocity u can be considered by
n
=
(2.10)
on the body surface. U also includes translatory and rotational movements for a rigid body.
For a better understanding of the kinematic free surface condition, which will be discussed later, the
meaning of the derivative DF/Dt of a function F(x, y, z, t) should be clarified. It can be written as
DF
Dt
=
F
t
+ F
(2.11)
and represents the rate of change depending on time of the function F by following a fluid particle in
space. The fluid velocity at the point (x, y, z) at time t is defined by V.
It is hypothesized that the free surface is described by the formula
z = (x, y, t)
(2.12)
where represents the wave elevation. Hence, the function F can be expressed by
F(x, y, z, t) = z - (x, y, t) = 0
(2.13)
Assuming the fluid particles which initially are located on the free surface remain on the surface equation
(2.9) is always satisfied, consequently, the substantial derivative DF/Dt is zero. Then, the kinematic free
surface condition reads as follows:
F
t
z - (x, y, t) + z - (x, y, t) = 0
(2.14)
10
and corresponding for z = (x, y, t) it can be written by
t
+
x
x
+
y
y
-
z
= 0
(2.15)
where the fluid velocity V from (2.11) is expressed by the velocity potential (see Eq. (2.6)).
Dynamic free surface condition
The dynamic free surface condition is based on the assumption that the water pressure is identical to the
constant atmospheric pressure p
0
on the free surface. By replacing the constant C (see (2.8)) by p
0
/ the
formula is valid with no fluid motion. Then, the dynamic free surface condition for z = (x, y, t) is
g +
t
+
1
2
x
+
y
+
z
= C
(2.16)
Since (2.15) and (2.16) describe non-linear surface conditions there is no information about the state of
the free surface before the problem is solved. Nevertheless, through a linearization of the free surface
conditions the problem can be simplified and in most cases the information is still sufficient.
Additionally, the assumption that the body has no forward speed and that no current exists is applied.
According to the linear theory it is also considered that the velocity potential is proportional to the wave
amplitude which is valid for small wave amplitude in relation to the representative wavelength and struc-
ture dimension. By applying the Taylor expansion equation (2.15) and (2.16) can be transferred to the
mean free surface position at z = 0. Thus, the kinematic free surface condition reads as
t
=
z
(2.17)
and correspondingly, the dynamic condition is
g +
t
= 0
(2.18)
Equation (2.17) and (2.18) can be combined to
t
+ g
z
= 0
(2.19)
and is also known as
11
- + g
z
= 0
(2.20)
Equation (2.20) describes the harmonic oscillation of the velocity potential with circular frequency .
2.1.2.3
Wave kinematics and pressure
By assuming a free surface of infinite horizontal expansion and a horizontal seabed the sea bottom condi-
tion can be written as a combination of the free surface condition (see (2.20)) and the Laplace equation
(2.5):
z
= 0
(2.21)
Equation (2.21) is valid for z = - h where h denotes the mean water depth. (Faltinsen, 1999)
For a general solution of the Laplace equation (2.5) the velocity potential can be determined based on
the previous boundary conditions:
=
g
Z(kz) cos(t - kx)
(2.22)
The velocity potential of the first-order incident wave describes a preceding wave of the circular frequen-
cy . The depth dependence of the incident wave is represented by the function Z and for infinite water
depth it is defined as
Z(kz) = e
(2.23)
where
k =
g
(2.24)
For a finite water depth h the function is given by
Z(kz) =
cosh k(z + h)
cosh(kh)
(2.25)
where the wavenumber k can be determined by solving the dispersion relation
12
k tanh(kh) =
g
(Wamit, 2000)
(2.26)
The fluid velocity in x- and z-direction can be calculated from (2.6) and is specified by
u =
cosh k(z + h)
sinh(kh)
sin(t - kx)
(2.27)
and the vertical velocity by
w =
cosh k(z + h)
sinh(kh)
cos(t - kx)
(2.28)
Since the acceleration a is defined as the first partial derivative of the velocity u as a function of time, e.g.
a
x
= u/t, the horizontal acceleration for a finite water depth can be expressed by the equation
a =
cosh k(z + h)
sinh(kh)
cos(t - kx)
(2.29)
and equivalent the vertical acceleration by
a = -
sinh k(z + h)
sinh(kh)
sin(t - kx)
(2.30)
For infinite water depth the two-dimensional wave velocity is based on equation (2.6), (2.22) and (2.23)
and can be expressed as
u = e sin(t - kx)
(2.31)
and
w = e cos(t - kx)
(2.32)
Accordingly, the partial derivative of the wave velocity in x- and z-direction for infinite water depth is
defined as
13
a = e cos(t - kx)
(2.33)
and
a = - e sin(t - kx)
(2.34)
The horizontal velocity distribution under a wave crest and a wave trough is illustrated in Figure 2.6
which shows that the fluid velocity under the wave crest is directed in wave propagation direction.
However, the horizontal wave velocity beneath a wave trough is opposite to the flow direction. Further-
more, the figure depicts that the maximum of the horizontal velocity is under either a wave crest or
trough. As formulated in the free surface conditions, cf. equation (2.16) and (2.19), a constant perfor-
mance of the velocity potential and fluid velocity for the mean sea level to the free surface level is as-
sumed. Figure 2.6 displays the consistent velocity distribution under a wave crest above mean sea level.
For calculating the velocity under a wave trough the analytical velocity distribution up to the free surface
level is used. In this context, the assumption is made that the difference between the velocity at a wave
trough and the theoretical velocity at mean sea level is small compared to the velocity itself.
Figure 2.6: Horizontal velocity distribution under a wave crest and a wave trough based on the Airy theory
(in accordance with Faltinsen (1999))
The dynamic pressure, which is already defined in (2.8) by the term /t, can be expressed for finite
water depth by using the depth dependent function Z (cf. equation (2.25)):
p = g
cosh k(z + h)
cosh(kh)
sin(t - kx)
(2.35)
Wave propagation direction
z
x
Horizontal velocity
under a wave crest
Horizontal velocity
under a wave trough
14
and accordingly, based on (2.23) for infinite water depth:
p = g e sin(t - kx)
(2.36)
The pressure variation beneath a wave crest and a wave trough is shown in Figure 2.7. Besides, it is
illustrated that under a wave crest the dynamic pressure is positive and beneath a trough negative. It
should be mentioned that the hydrostatic pressure gz, cf. equation (2.8), should cancel the dynamic
pressure /t|
z=0
at the water surface as defined in the dynamic condition, see Eq. (2.18). This surface
condition is satisfied at the wave crest, as the total pressure above mean sea level in Figure 2.7 implies.
Whereas, a higher-order error remains at the wave trough because the linear wave theory uses the analyti-
cal form of the velocity and pressure distribution (Faltinsen, 1999).
Figure 2.7: Pressure variation under a wave crest and a wave trough according to Linear Wave Theory
in accordance with Faltinsen (1999)
2.1.3
Stretched Airy Theory
As mentioned earlier, the Airy theory only applies up to the mean sea level, which means that the kine-
matics above still water level is excluded. However, ocean waves above mean sea level may have an es-
sential impact on the response of offshore structures and consequently, should be taken into account. For
this reason, several empirical methods, such as Vertical stretching, Wheeler stretching and Extrapolation
stretching, have been developed to improve the linear wave theory (Chakrabati, 2005). One of the most
widely used is the Wheeler stretching method, which modifies the kinematics up to the wave crest or
wave trough, respectively. According to the stretched Airy theory the kinematics applied to the mean sea
level are determined to the instantaneous water surface which can be achieved by computing the coordi-
nate of the free surface by
Wave propagation direction
z
Dynamic pressure
Static pressure
Total pressure
15
z = (z - )
h
h +
(2.37)
(Wheeler, 1980)
Accordingly, the modification of the horizontal velocity, see formula (2.27), can be written as
u
=
cosh k(z + h)
sinh(kh)
sin(t - kx)
(2.38)
and the horizontal acceleration, which is based on (2.29), may be expressed by
a
=
cosh k(z + h)
sinh(kh)
cos(t - kx)
(2.39)
The comparison between the horizontal velocity based on the Airy theory and the modified horizontal
velocity according to the Wheeler stretching, which is displayed in red, is exemplarily represented in
Figure 2.8.
Figure 2.8: Comparison between the horizontal fluid velocity according to the linear wave theory and the
horizontal wave velocity based on Wheeler stretching (Skailerud et al. (2010))
2.2
Irregular Waves
So far, the wave theory described in chapter 2.1 assumes a regular wave field which is a common method
in the design of offshore structures. On this occasion, a regular wave with a corresponding height and
period represents an extreme wave and on this basis, the extreme response of the body can be simply
Wave propagation direction
z
x
Horizontal velocity
(Wheeler stretching)
Horizontal velocity
(Airy theory)
16
analyzed. However, the occurring irregular waves in nature cannot be taken into account. Especially, in
the design and analysis of the response of floating structures the random wave method should be applied
to simulate the natural processes as accurately as possible (Chakrabati, 2005).
As indicated in Figure 2.9a, the irregular surface elevation in nature can be simulated as a linear
superposition of waves with different wave heights, periods and directions which leads to a short-crested
irregular wave field, cf. Figure 2.9b. For facilitating the calculation of the wave parameters the model can
be idealized to a long-crested irregular swell by neglecting the different directions (see Figure 2.9c). On
the basis of the linear wave theory, characteristic wave heights and periods are determined and
consequently, a further simplification to a long-crested regular sea state can be made (Figure 2.9d).
Figure 2.9: Simplification of the sea state: a) Superposition of regular waves, b) Short-crested irregular sea
state, c) Long-crested irregular sea state, d) Long-crested regular sea state (EAK, 2002)
As already mentioned, the random surface elevation of a long-crested irregular sea can be simulated as a
linear superposition of waves (see Figure 2.10) which can be defined by the equation
=
sin t - k x +
(2.40)
where
aj
,
j
, k
j
and
j
denote the wave amplitude, circular frequency, wave number and random phase of
wave component number j. The random phase angles
j
are uniformly distributed in the range of 0 to 2
and are constant with time. For deep water waves
j
and k
j
can be calculated by the dispersion relation,
see equation (2.26). Accordingly, the horizontal fluid velocity u and acceleration a
x
can be written as
a)
b)
c)
d)
17
u =
cosh k (z + h)
sinh k h
sin t - k x +
(2.41)
and
a =
cosh k (z + h)
sinh k h
sin t - k x +
(2.42)
The irregular sea state can be described by two different methods. On the one hand, the description can be
made in the time domain, which will not be further discussed in this thesis, and on the other hand, the
random swell can be elucidated as a function of the frequency (see Chapter 2.2.1). The connection be-
tween the a time domain solution of the waves as defined in (2.40) and the frequency domain analysis by
a wave spectrum S
() is shown in Figure 2.10.
Figure 2.10: Picture illustrating the relation between the time domain and the frequency domain analysis of
waves in a long-crested swell (according to Faltinsen (1999))
2.2.1
Description in the Frequency Domain
By analyzing the sea state in frequency domain, the definition of individual waves is not required, which
is an essential contrast to the description in time domain. The sea state, which is based on a time series,
can be divided into sinusoidal components by using a Fourier analysis (cf. Figure 2.4a&b). Each of these
sinusoidal waves has a different wave height and period that represent a one-dimensional spectrum. By
analyzing the swell in the frequency domain, the directions should be taken into account (directional
wave spectrum). The waves that are determined by the Fourier analysis are divided into frequency groups
(Zanke, 2002). Consequently, certain amplitudes (amplitude spectrum) and phases (phase spectrum) can
be allocated to each frequency, as illustrated in Figure 2.11. (Oumeraci, 1996b)
S
()
t
Frequency domain
Wave spectrum
Time domain
Random wave elevation
Sum
Sum
Sum
Fin de l'extrait de 145 pages
- haut de page
- Citation du texte
- Olga Glöckner (Auteur), 2014, Wind energy. Methods for computation of wave forcing and the resulting motion of a slender offshore floating structure, Munich, GRIN Verlag, https://www.grin.com/document/429615
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X. -
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Téléchargez vos propres textes! Gagnez de l'argent et un iPhone X.