OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Design Methods for Offshore Wind
Turbines at Exposed Sites
(OWTES)
Hydrodynamic Loading on Offshore Wind Turbines
OWTES Task 4.2 OWEC Tools Task B.1 - B.2
Dr. A. R. Henderson (TUDelft, editor) March 2003
This contribution to the OWTES project has been carried out under contract
JOR3-CT98-0284 awarded by the European Union. This work has been co-financed
by NOVEM under contract 224.750-9854.
Duwind 2003.025
Delft University of Technology, Section Wind Energy
Stevinweg 1, 2628 CN, Delft, The Netherlands
Phone +31 15 278 5170, Fax +31 15 278 5347
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Duwind 2003.025 Page i Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Summary
The calculation and determination of design wave loads on offshore structures is a complex
undertaking involving different wave models, load-calculation methods and probability
analyses. It is however of vital importance if a cost-effective and durable structure is to be
designed. Both the extreme and fatigue load cases need to be considered and the actual
approach will depend both upon which of these two cases is being considered and upon the
type of support-structure. The key to the problem is to determine the nature of the waves:
their distribution and their hydrodynamic properties.
In the shallow seas that are the favoured locations for offshore wind farms, the limited water
depths can result in highly non-linear waves. The determination of the design wave loads will
involve selection of appropriate models of wave kinematics as well as force and structural
dynamics models. Each selection will involve a compromise between accuracy and usability
(speed, ease of use and simplicity of evaluation). Currently, the offshore oil and gas industry
is focusing on ever deeper waters as much of the available hydrocarbon resource in accessible
medium depth waters is already being exploited. In contrast, offshore wind energy is being
developed in shallower waters, often at sites exposed to extreme weather such as the
European North Sea. The approach needed here will be subtly different but equally
demanding, with economic aspects being more prominent. Electricity is a low-value
commodity in a highly competitive market and the costs of generation using offshore wind
farms are approaching the costs of conventional generation. It is vitally important that
inappropriate and excessively conservative design approaches do not sabotage this trend.
Because of the range of aspects that need to be considered, different approaches have been
developed to analyse different situations, for example, the Morison method for slender
structures (such as monopiles) and diffraction theory for massive structures (such as gravity
base support-structures). Substantial design experience has been built up in the offshore
industry, albeit in situations that can be different from the relatively shallow waters that are
the preferred locations for offshore windfarms.
This report focuses on the following aspects of this design problem:
• appropriate wave kinematics models, wave load models and structural dynamics
models for the water depths, sea climates and wind-turbine support-structures
applicable for offshore windfarm projects,
• extreme hydrodynamic loading, including breaking waves, using non-linear wave
theories and appropriate corrections,
Duwind 2003.025 Page ii Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
• stochastic wave loading using linear wave theories with empirical corrections,
• consideration of both slender and massive structures.
Depending on the type and size of support structure and turbine selected, waves will be at
least a significant source and can be the dominant cause of fatigue and extreme loads on parts
of the support structure. Hence the selection of an appropriate method of determining the
resulting loads can have an important effect on the cost of that structure and its ability to
withstand environmental and operating loads.
Duwind 2003.025 Page iii Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Foreword
Results are reported for Task 4.2 of the project ‘OWTES’ commissioned by the European
Union and for Task B.1 - B.2 of the project ‘OWEC Tools’ commissioned by NOVEM. The
work can be characterised as an analysis of hydrodynamics loading of support structures of
offshore wind turbines. The main part of the report has been written by the editor with the
other parties making important contributions in their fields of expertise.
• TU Delft
Andrew Henderson (editor),
Michiel Zaaijer
Po Wen Cheng
Jasper Barreveld
• Garrad Hassan
Tim Camp
• John Brown Hydrocarbons Limited
Bernie Smith
John Harrison
The report has been published by TUDelft, Section Wind Energy.
Parts of the report are based on work previously published at conferences including [36], [37],
[38], [39], [40] and [41].
The work reported here forms part of the European Commission funded research project
entitled 'Design Methods for Offshore Wind Turbines at Exposed Sites’ (OWTES) and is
being undertaken by AMEC Border Wind, Delft University of Technology, Germanischer
Lloyd WindEnergie, PowerGen Renewables Developments and Vestas Wind Systems under
the leadership of Garrad Hassan and Partners.
Delft, December 2002
Duwind 2003.025 Page iv Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Duwind 2003.025 Page v Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
List of Symbols
Alocal area above which the local ice pressure appears [m2]
CD drag coefficient [-]
CM inertia coefficient [-]
D diameter (of circular gravity base structure or pile) [m]
Dr soil relative density [-]
E modulus of elasticity [N/m2]
EP pile equivalent modulus of elasticity [N/m2]
F force [N]
G shear modulus of elasticity [N/m2]
Hb breaking wave height, [m]
HS significant wave height, [m]
I second moment of inertia of pile cross-section [m4]
K0 coefficient of earth pressure at rest [-]
LC critical pile length [m]
M moment on pile head [Nm]
Nq bearing capacity coefficient [-]
U windspeed [m/s]
R cylinder radius, [m]
TZ zero-crossing period [s]
V volume [m3]
Xp distance from point of initial breaking to point of maximum wave load [m]
c wave celerity [m/s]
cS time varying impact force function [-]
cu undrained shear strength [N/m2]
dw depth of water [m]
e ice thickness [m]
f frequency [Hz]
fm characteristic frequency [Hz]
k wave number [1/m]
q gravitational constant [9.81 m/s2]
h0 deep-water wave height [m]
k stiffness matrix
Duwind 2003.025 Page vi Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
m rate of change of soil shear modulus with depth [-]
qc cone penetration test tip resistance [N/m2]
r0 pile outer radius [m]
rf characteristics bending strength of ice [Pa]
rlocal local ice pressure [Pa]
ru characteristics compression strength of ice [Pa]
sq shape correction factor for bearing capacity [-]
u horizontal translation of pile head [m]
z vertical axis (depth below the seabed, above surface) [m]
Ir Irribarren number, ( )0
0
.tanh βλ
=I r [-]
KC Keulegan-Carpenter number, UTD
=KC [-]
Re Reynolds number, ULν
=Re [-]
S Strouhal number, fDU
=S [-]
β sea-bed slope, [-]
ϕ friction angle [°]
ηb crest elevation, [m]
λ curling factor (0.5 recommended by 104), [-]
λ0 deep-water wave length, [m]
� Poisson’s ratio [-]
� kinematic viscosity [m2/s]
ρ density, [kg/m3]
0vσ vertical effective soil pressure [N/m2]
θ rotation of pile head around horizontal axis [rad]
µ coefficient of friction [-]
ω angular velocity [rad/s]
Duwind 2003.025 Page vii Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Table of Contents
SUMMARY............................................................................................................................................. I FOREWORD ....................................................................................................................................... III LIST OF SYMBOLS............................................................................................................................. V TABLE OF CONTENTS ................................................................................................................... VII 1 INTRODUCTION ......................................................................................................................... 1 2 DESCRIPTION OF THEORY INCLUDING LITERATURE REVIEW ................................ 4
2.1 Sea Climate Modelling.......................................................................................................... 4 2.1.1 Defining the Sea Climate .................................................................................................... 5 2.1.2 Selecting Fatigue Cases.................................................................................................... 11 2.1.3 Selecting the Extreme Case............................................................................................... 11 2.1.4 Presence and Description of Breaking Waves.................................................................. 13
2.2 Wave Kinematics Modelling .............................................................................................. 16 2.2.1 Regular Linear Wave Modelling (incl. Empirical Corrections) ....................................... 22 2.2.2 Regular Non-Linear Wave Modelling............................................................................... 24 2.2.3 Stochastic Seas ................................................................................................................. 26 2.2.4 Breaking Wave Height Modelling..................................................................................... 26 2.2.5 Breaking Wave Kinematics Modelling.............................................................................. 28 2.2.6 Effect of Current ............................................................................................................... 28
2.3 Wave Loads Calculation..................................................................................................... 29 2.3.1 Morison Method ............................................................................................................... 31 2.3.2 Diffraction Method ........................................................................................................... 37 2.3.3 Froude-Krylov Method ..................................................................................................... 38 2.3.4 Breaking Wave Loads ....................................................................................................... 40 2.3.5 Effect of Structural Dynamics........................................................................................... 42
2.4 Ice Loads .............................................................................................................................. 43 2.5 Design Practices................................................................................................................... 44
3 DESCRIPTION OF BLYTH WINDFARM .............................................................................. 46 3.1 The Windfarm and Windturbines ..................................................................................... 47
3.1.1 Description of the Site ...................................................................................................... 47 3.1.2 Description of Windturbine .............................................................................................. 50 3.1.3 Wave Climate at Blyth ...................................................................................................... 51
3.2 Measurement Systems ........................................................................................................ 52 4 HYDRODYNAMIC LOADING – EXAMINATION OF THEORY....................................... 56
4.1 Slender Support Structures................................................................................................ 56 4.1.1 Structural Dynamics ......................................................................................................... 67 4.1.2 Current ............................................................................................................................. 70
4.2 Compact Support Structures ............................................................................................. 71 4.2.1 Determining the Wave Loads for GBS Structures ............................................................ 74 4.2.2 Evaluating the Uncertainties ............................................................................................ 76 4.2.3 Effect of Surface................................................................................................................ 90 4.2.4 Comparison of Methods.................................................................................................... 91
5 HYDRODYNAMIC LOADING – EVALUATION OF MEASUREMENTS ........................ 93 5.1 Individual Waves ................................................................................................................ 93 5.2 Campaign (30 minute sea state) ......................................................................................... 95 5.3 Probabilistic Approach....................................................................................................... 97 5.4 Alternative Approaches .................................................................................................... 100
6 DISCUSSION AND CONCLUSIONS ..................................................................................... 102 7 RECOMMENDATIONS........................................................................................................... 105 REFERENCES ................................................................................................................................... 106 APPENDIX A: CALIBRATION OF STRAIN GAUGES........................................................... 115 APPENDIX B: ANALYSIS OF 30 MINUTE CAMPAIGN DURING STORM....................... 118
Duwind 2003.025 Page viii Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
App B.1 Introduction and Overview of Data........................................................................ 118 App B.2 Identification of Individual Waves.......................................................................... 121 App B.3 Statistical and Spectral Properties of the Sea Surface .......................................... 122 App B.4 Properties of Individual Waves............................................................................... 124
APPENDIX C: PRELIMINARY DESIGN OF GRAVITY BASE STRUCTURES................. 127 App C.1 Resistance factors ..................................................................................................... 127 App C.2 Load and resistance factor analysis ........................................................................ 128 App C.3 Effects of hydrodynamic modelling on design solution......................................... 130
APPENDIX D: DEANS STREAM THEORY.............................................................................. 133 App D.1 Velocities ................................................................................................................... 134 App D.2 Accelerations............................................................................................................. 135 App D.3 Pressure ..................................................................................................................... 138 App D.4 Description of Methods for Solving for Regular Waves ....................................... 143 App D.5 For Irregular Wave Profiles .................................................................................... 146
App D.5.1 Wave Kinematics.................................................................................................... 146 App D.5.2 Finding the Solution............................................................................................... 149
Duwind 2003.025 Page 1 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
1 Introduction
The calculation and determination of design wave loads on offshore structures is a complex
undertaking involving different wave models, load-calculation methods and probability
analyses. It is however of vital importance if a cost-effective and durable structure is to be
designed. Both the extreme and fatigue load cases need to be considered and the actual
approach will depend both upon which of these two cases is being considered and upon the
type of support-structure. The key to the problem is to determine the nature of the waves:
their distribution and their hydrodynamic properties.
Offshore windturbines will generally be located in what is termed intermediate water depths1,
a region of the seas where until now relatively few structures have been built. The offshore
oil and gas industry has significant experience building structures for deeper waters, while
coastal engineering have naturally focused on the shallowest water regions. Both these
branches of engineering have also had different primary design objectives than the offshore
wind energy industry. In their cases, cost has had a lower priority compared with aspects
such as time-scale, reliability and safety. Therefore, although much very useful knowledge
can be gained by studying both the offshore and coastal engineering experiences, their design
methods may need to be modified if an excessively expensive or possibly even unexpectedly
unreliable structure is to be avoided.
The procedures necessary to calculate the critical wave loading, for either the fatigue or
extreme cases, can be divided into three stages:
(i) determining the design wave or wave climate
(ii) selecting an appropriate wave load calculation procedure
(iii) determining the effect on the structure
This report focuses on all three stages, as each is of equal importance for achieving an
appropriately design solution. Also, each stage cannot be considered in isolation, as they are
interrelated: for instance, the design wave can depend on the structural response when a larger
wave at a frequency away from the structure's natural frequency can be less critical than a
smaller wave close to the natural frequency.
1 intermediate refers to the ratio between the wave length and the water depth
Duwind 2003.025 Page 2 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Hence an important aspect in the prediction of extreme- and fatigue loading of the support
structure of an offshore wind energy converter (OWEC) can be its dynamic response. The
predictability of this dynamic response differs in some important aspects from that of
platforms for the offshore oil industry and of onshore wind energy converters. The natural
frequency of an OWEC can be wedged between different excitation frequencies, whereas the
natural frequency of a fixed platform for the offshore oil industry is usually designed to be
well above the wave excitation frequencies. The geometry and dimensions of offshore
foundations differ from typical onshore solutions, resulting particularly in an expected larger
influence of soil characteristics for the slender monopile foundation.
It should also be made clear that there are many uncertainties in this process of calculating the
wave loading. Through experience, the offshore industry has developed integrated methods
where uncertainties in one stage may be covered by additional safety margins in another. The
question arises as to whether these methods are appropriate for the intermediate water depths
in which offshore wind farms will be built, since the extent of these uncertainties may vary
greatly with water depth. At the other extreme of water depth, in shallow waters, the coastal
engineering community has developed different design methods because of the different
problems they face. For example, at these water depths, waves are generally more non-linear,
breaking waves are more common and the structures, such as breakwaters, piers, sea
defences, are more compact.
At this moment, the size of the offshore windenergy market does not warrant intensive
research on developing new and bespoke methods, and hence judgment of appropriateness
and applicability of existing methods, which can easily be a very subjective process, is
needed.
Points of concern with the application of the existing offshore engineering methods include:
• increased uncertainties in the wave modelling, because of increased non-linearities
• increased occurrence and importance of breaking waves
• inappropriate safety margins
Regarding safety margins, the offshore oil and gas industry deals with an exceedingly high
value and potentially explosive and environmentally toxic material, hence failure has a
disproportionately high cost on the operators' safety, the environment and profitability.
Likewise, the failure of a coastal sea defence structure can have catastrophic consequences.
Offshore wind farms, by contrast, generate a low-value, non-toxic commodity and should
Duwind 2003.025 Page 3 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
always be unmanned during storms and hence lower design margins will deliver both higher
levels of safety to operators as well as improved economic viability.
Although existing offshore design methods can undoubtedly result in a durable structure,
there may be excessive cost penalties. On the other hand, non-linear and breaking waves
experienced in shallow waters may mean that the design methods based on experience in
deeper water are unconservative. The coastal engineering branch does have substantial
experience in designing in shallow water conditions, albeit again to much more stringent
durability criteria than are appropriate here.
For the evaluation of proposed engineering models, the report draws on data collected at the
Blyth offshore windfarm, where one turbine is comprehensively instrumented and a winter-
season of heavy seas, including breaking waves, has been recorded.
Duwind 2003.025 Page 4 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
2 Description of Theory including Literature Review
When considering the hydrodynamic loading aspects of the design process, two cases need to
be considered, each of which can require a different approach:
(i) the fatigue cases
(ii) the extreme case
Defining the design procedure consists of selecting an appropriate wave or wave climate,
kinematic model, loading model and structural model. Only a limited number of
permutations of these models are possible (an example being that diffraction cannot be
applied to high order wave theories) and the more sophisticated or accurate the model, the
greater the demands on the engineers and the computers time.
In addition to describing the main points of the background theory, this section also provides
a brief review of literature relevant for hydrodynamic design aspects of offshore windturbine
support structures, hence it includes literature that defines the sea-climate, the waves and the
resulting loadings and response of the structure. As mentioned in the introduction, an
integrated approach is necessary, with all aspects of the design procedure having to be
examined with care: the weakest-link principal applies since the appropriateness of the
resulting design can be no better than that of the most uncertain or conservative aspect of the
whole procedure.
This chapter is divided into five sections investigating different aspects of the design process:
(i) sea climate
(ii) wave kinematics
(iii) wave loading
(iv) ice loading
(v) design practices
2.1 Sea Climate Modelling
The first question facing the offshore windturbine designer is to define the external loads and
within this report the hydrodynamic wave forces are of greatest interest. When the
windturbine is located in a sea, it will encounter a minimum of twenty-years of waves of
varying forms. How can this be distilled into a limited number of cases that can be dealt with
in a timely and cost-effective manner and yet represent the full-life experience of the
structure?
Duwind 2003.025 Page 5 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
2.1.1 Defining the Sea Climate
The lifetime wave-environment, or sea climate (termed long term distribution), can be
considered to consist of a number of sea states (termed short-term distribution; it is assumed
that the sea will remain statistically identical throughout a single sea state). Each sea state in
turn consists of a number of waves and possibly other parameters, such as a current and a
tide-level, Figure 1. The focus of research is to define the sea climate and the sea states in as
few parameters as possible while maintaining as much of the important information as
possible.
Wave 1 T; H
deterministic
Wave nT; H
Wave 2T; H
Sea State 1 TZ; HS
stochastic
Wave Climate p(TZ; HS)
Probability
Sea State n TZ; HS
Figure 1: Relationship between Individual Waves, Sea States and the Sea Climate
Individual Waves, Sea States and Wave Climates
The waves of the ocean environment can be viewed at three levels, Figure 1: at the lowest
level are (i) individual waves, ranging in size from the smallest ripple to the extreme or freak
wave. These individual waves combine to make a (ii) sea state, which is generally assumed
to be stationary over three hours (as there is relatively little energy at this frequency; this is
analogous with the 10 minute spectral gap in the wind). Sea states are usually defined in terms
of stochastic spectra parameterised by the spectral wave height and period. The entire
collection of sea states as the (iii) wave climate, typically represented as a sea-state scatter
probability table. These show the probability of occurrence of the sea states of each
combination of wave height and period. A climate can also be defined in terms of the
probability of individual waves rather than three hour sea states.
At the most detailed level, a sea state could be described using the time-history of the surface
elevation, for example recording the level of the surface every half-second for 3 hours
(i.e. 21,600 numbers), however this is not particularly efficient to store, analyse or
communicate, hence attempts continue to define the sea state with fewer parameters, with the
Duwind 2003.025 Page 6 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
extreme being just one number: a characteristic wave-height (usually the significant wave
height), with the assumption that the characteristic period is related using a pre-determined
function, such as through the windspeed in Eq. 1 and Eq. 2 [71].
2.0212.0 UH S = Eq. 1
���
����
�=gU
TZπ2
81.0 Eq. 2
The sea can then be described using a spectral function, such as the Pierson-Moskowitz [74],
Eq. 3, or its more general derivative, the JONSWAP [80] [34] [35], Eq. 4. Because of their
relative simplicity and robustness, these formulations have found widespread use, however
there are criticisms including that its ease of use encourages inappropriate use (for example
for seas with a significant swell2 component when a two peaked spectra would be more
appropriate) and that the measured tails of severe storms show a f -4 slope rather than the f -5
slope in the equation [45] but challenged by [98].
( )( ) �
�
�
�
��
�
�
���
����
�−=−4
54
2
.45
exp.2 m
PM
ff
f
AgfS
πηη Eq. 3
( )( )
( )���
�
� −−
��
�
�
�
���
����
�−=22
2
..2exp4
54
2
.45
exp..2
. m
m
f
ff
mHass
ff
f
gafS σ
ηη γπ
Eq. 4
Where A, a, σ = constants γ = amplitude
Taking a different approach, attempts have been made to determine the joint probability of
individual wave periods and amplitudes, such as Eq. 5 [62] within the sea-state.
( )
2
2
11
12
22
2 1, e 1
4
TR
Rp R T
T
ν
νπν
� � �− �� � �− +� � �� �� � �� �� �
� �� � � �= +� � � �� �� �� �� �
Eq. 5
Where 02
Rm
ρ=
ρ = amplitude
Tττ
= 0
1
22 mmππτ
σ= =
2 because of their long periods, swell waves can be very persistent and travel great distances; for example, the Dutch north sea
coast experiences both locally generated wind waves as well as swell waves from the Atlantic Ocean
Duwind 2003.025 Page 7 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
And the spectral bandwidth, ν, follows the definition in Eq. 6 [60]:
0 22
1
1m mm
ν = − Eq. 6
This may be useful in determining appropriate combinations of the wave height and periods
of extreme waves though [95] suggests that Eq. 5 over predicts the probability of large wave
heights and that Eq. 7 provides a better estimate of the probability of above-average sized
waves. (It is not clear how this complements the assertion made by other authors that use of
linear theory underestimates the highest waves but it may be that linear based theories
overestimate the higher half but underestimate a small extreme proportion, i.e. figure in [79]).
( )( )
( )
22|
|
12 2 2 1
2
|
1, 1 e
42 4 1
rrp
rr r
τ ξ
τ ξ
τ µξσ
τ ξ
ξξ τξπ σ
� �� �−� �� �− +� �� �+� �� �� �
� �−= +� �+ � �
Eq. 7
Where 0
Hm
ξ = r � 0.6-0.7 for ν � 0.4-0.6 [95]
It should be stated that there is no definitive agreement yet over which model should be used,
or indeed over what the parameters should be (alternatives include wave height and wave
steepness and wave height and crest front velocity) and other candidates are briefly reported
on in [54]: five-parameter bivariate log-normal, bivariate Weibull, Eq. 8 (variables are wave
height and steepness), and the Plackett distribution [6]. An attempt has been made to fit the
measured distribution from six North Sea storms to Eq. 8 but in spite of the parameters being
shown to vary systematically with the severity of the storm, the authors felt the search was not
complete [54].
( )1 1
, ; , , , e eh sh s
p s h
γ βγ β
η αγ β
γ βα β γ ηη α
� � � �− −− −� � � �� � � �= Eq. 8
Where sλη τ=
Likewise, a definition of a suitable model for the probability distribution of sea states within a
sea-climate has also been attempted, with the above equations potentially applicable. The
Plackett distribution [6] has been applied over a wide range of conditions of European seas in
[75] and [78] evaluated a number of models against data from two sites and recommended a
relatively simple model (by one of the co-authors [101]) using wave steepness, Eq. 9, as the
Duwind 2003.025 Page 8 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
second parameter. The use of wave steepness instead of wave period has the advantage in
that this parameter is uncorrelated (or perhaps less closely correlated) with the wave height.
2
2
SP
P
Hs
gTπ
=� �� �
Eq. 9
Figure 2 illustrates the difference between the distribution of the individual waves (left chart)
and the sea states (right chart) for the Blyth design sea conditions using logarithmic scales.
Note that this data is for a nearby deepwater location. Often when selecting data for the
design of offshore windfarms, only deepwater data will be available for a suitably close
location. It is then necessary to estimate the true sea climate at the shallower windfarm site
from this, preferably using a coastal model such as SWAN [93].
log Data WaveDistT 100
0 5 10 150
5
10
15
43.5
3
3
2.5
2.5
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
T
0 5 10 150
5
10
15
1.51
0.50.5
0
0
0
0.5
0.5 1
1
1 1.5
1.5 1.5
1.5
2
2
2
2 2.5
2.5
2.5
2.5
2.5
Figure 2: Individual Wave (left) and Sea State (right) Distributions (log scale)
Determining the Local Wave Conditions
Unless the offshore windfarm developer has installed a wave measurement system at the site,
the wave data is likely to be from another nearby location, with differing immediate water
depth and local bathymetry. Hence the wave-data needs to be adjusted to take account of
these different conditions. Two forms of data are likely:
• measured data, for example from a waverider buoy,
• a hind-cast database; where a wave record has been estimated from regional weather
records such as the NESS database [73].
Wave Period [s]
Wav
e H
eigh
t [m
]
Zero-Crossing Wave Period [s]
Sig.
Wav
e H
eigh
t [m
]
Duwind 2003.025 Page 9 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Extrapolation to the actual site can be done using a coastal model, such as Swan [93], or a
simple analytical model, such as proposed by Tucker [98], which calculates a revised
significant wave height and wave period for the assumed JONSWAP spectrum, taking into
account the reduction in water depth but not the horizontal distance.
Sea Surface Elevation Distribution
In theory, the sea surface should have a Gaussian distribution, Eq. 10, though for steep seas in
shallow waters, this is not the case, see Figure 143 (page124).
( )���
�
�−= 2
2
.2exp.
.21
ηη ση
σπηp Eq. 10
Spectral Definition of Sea State
It is convenient to define sea state parameters directly from the spectrum. This section lists a
number of definitions which are based on the spectral moments being weighted by the angular
velocity, ω, Eq. 11. An alternative, which can lead to confusion, is to define the spectral
moment using the frequency, f; it is still then possible to define the parameters through the
spectral moments but each expression may need to be adjusted by factors of 2π.
( ) ωωω ηη dSm kk ..
0 ∞
= Eq. 11
For narrow-banded spectra, the following holds for the r.m.s. wave height:
0.22
mH
arms == σ Eq. 12
While the significant wave height can be defined as:
( ) 003
1 .4 mHHH mS =≈= Eq. 13
The characteristic time period of these spectra can be defined in numerous ways, which are
listed below.
Eq. 14 gives the peak period, or modal period (noting that this is not the same as peak
frequency, or model frequency):
4
20
1mm
fT
m
== Eq. 14
Duwind 2003.025 Page 10 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
as mean/average wave period, there being two alternatives, Eq. 15 and Eq. 16:
( )1
01 ..2
mm
T π= Eq. 15
1
21 m
mT = Eq. 16
zero-crossing period:
( )2
0..2mm
TZ π= Eq. 17
mean (energy) period:
0
1
mm
Te−= Eq. 18
average period between crests:
4
2
mm
Tcrest = Eq. 19
And mean period of average 1/3 highest waves
04 .5
4 TTS = Eq. 20
In addition to Eq. 6 above (for which broad band spectra � ∞) , The spectral bandwidth can
be defined through Eq. 21 [16] (for which broad band spectra � 1) and Eq. 22 where it is
defined through the irregularity factor, Eq. 23:
40
2240
..
mmmmm −
=ε Eq. 21
2
2 11 ���
����
�−=−=
Z
Crest
TTγε Eq. 22
And the irregularity factor though:
40
22
.mmm
T
T
zero
peak ==γ Eq. 23
Duwind 2003.025 Page 11 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
2.1.2 Selecting Fatigue Cases
For the case of fatigue, a range of waves need to be considered, as smaller, more probable
waves make a significant contribution to the overall fatigue damage. The individual wave
height and wave period cases can be lumped together into groups to reduce calculation time
with a limited and acceptable reduction of accuracy [52], these groups being focused on
regions of:
• maximum fatigue damage,
• the turbine not operating (i.e. low and high wind speeds and failures),
• and sea state frequencies near to the natural frequency of the structure.
The recommended wave model and load calculation method depends on the context, i.e.
whether the fatigue or the extreme situation is of interest and what type of structure is under
investigation.
2.1.3 Selecting the Extreme Case
Many different procedures have been developed to determine the design extreme wave. Some
approaches are based on an elaboration of the following procedure:
• use statistical analysis of the extremes of the sea states to find the significant wave
height of the 1-in-50 years sea state,
• multiply by a factor (typically between 1.8 and 2) for the height of the extreme
individual wave within that sea state.
Evidence from field measurements does show that the extreme wave may be higher than
probability distributions based on a Rayleigh distribution, Eq. 24, (i.e. linear theory) would
suggest (i.e. [79] shows divergence for waves higher than twice the significant wave height
from the Danish sector of the North Sea). There are several approaches being taken in an
attempt to provide a theoretical justification, including effects of bathymetry or variable
currents (though extreme waves occur in very deep water and where strong currents are not
present as well), non-linearities in the waves (possibly the most likely explanation) or that the
Fourier components coincide (though this would seem to contradict the assertion that the
actual probability is higher than the theoretical probability) [51]. A second problem with
linearly-generated stochastic seas is that the wave-frontal steepness may not be modelled
correctly [54]; this asymmetry has been seen at Blyth and occurs prior to wave breaking.
Duwind 2003.025 Page 12 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Hence it may not be adequate, regarding the selection of an appropriate extreme wave height,
to run sufficient linear simulations of the design sea state.
( )��
���
��
���
���
����
�−−=<
2
2exp1SH
hhHP Eq. 24
A simpler method suggested by [61] is to use a modified Rayleigh formula, Eq. 25, for the
extreme waves since they are within broad band locally generated seas (rather than the narrow
band seas that dominate fatigue). [15] suggested a factor of 0.9 for K.
( )��
���
��
���
���
����
�−−=<
2
.2exp1
SHKh
hHP Eq. 25
However, for dynamic structures, stochastic analysis is necessary and this design wave would
need to be embedded into a stochastic simulation. From the probability distribution of the
structural response of multiple runs, an estimate of the extreme design loads can be made.
Since the non-linear nature of the problem makes a definitive analytical solution impossible,
there are other similarly defendable approaches. For example, the sea state wave height
depends upon the probability distribution model and the plotting position chosen.
The main difficulty regarding designing for the extreme loading case is that the necessary
wave model (high order stream function) cannot be combined with a stochastic or frequency-
domain response calculation. Even using time-domain modelling techniques, a compromise
needs to be made between modelling the individual wave accurately (i.e. using high order
stream function) when only regular waves can be utilised or modelling the stochastic process
correctly, in which case at most second order non-linear seas can be modelled. This second
approach has been chosen by the offshore industry since wave non-linearity is less of a
feature in the deeper waters, for example the NewWave method [96].
Regarding the selection of the wind conditions corresponding with the extreme wave
condition, Turkstra’s rule can be applied, which states that the maximum value of the
principal load should be combined with the stochastic value of the secondary loads, however
studies have shown that this can lead to an underestimate of the response for non-linear
systems [69].
The probability distribution of the wave-heights can be assumed to follow a Rayleigh
distribution at the first instance. However, the wave heights of the maximum values follow
different distribution patterns. [97] evaluates the family of Fisher-Tippett distribution
Duwind 2003.025 Page 13 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
functions, recommending that type I (also called Gumbel), Eq. 27 may give a poor fit for the
most extreme values, type 2 (also called Frechet), Eq. 28, is an improvement but that the best
fits for shallow water sites would be the type 3, Eq. 29.
( )���
���
��
�
� −−−=β
αxxF expexp Eq. 26
( )��
���
��
���
���
����
�−=−α
βx
xF exp Eq. 27
( ) expx
F xβθ
α� −� �= − �� �� � �� �
Eq. 28
For a comprehensive analysis of designing offshore windturbines for the extreme load case,
the reader is advised to obtain the recent Ph.D. thesis by Cheng [20].
2.1.4 Presence and Description of Breaking Waves
In the shallow seas that are the favoured locations for offshore wind farms, breaking wave
loads will be a more important design phenomenon than in the deeper waters utilised by the
offshore oil and gas industry. As waves approach shallower waters, they become steeper and
the probability of breaking increases significantly. Wave steepness is one important parameter
for determining the likelihood and type of breaking and this is dependent on both the wave-
height to wave-length ratio and the wave-height to water-depth ratio. Both of these ratios
increase as waves progress into shallower waters. The most severe breaking wave loads are
caused by so-called ‘plunging breakers’ which result from a rapid reduction in water depth as
the waves approach the shore.
Recently, there has been renewed interest in researching breaking waves, particularly in the
coastal engineering community but also to a lesser extent in the offshore engineering
community, where it is a less important phenomena. Attention is currently focusing on
modelling the kinematics using CFD and on further experiments of breaking waves,
particularly on the simplest case of a slender cylinder.
The shallow water depth at Blyth means that wave breaking is a recorded phenomena there
and hence a review has been made of various approaches in modelling this aspect with the
focus being on:
Duwind 2003.025 Page 14 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
• wave breaking modelling, including determination of the height, the mode of
breaking and the resulting forces
• evaluation of these models and recommendations for appropriate analyses for this and
other offshore wind farms
As waves move into shallower waters, where offshore windfarms are likely to be located,
waves are generally modified as follows:
• as the waves begin to feel the sea bottom, their length decreases and the height will
decrease and then increase a little; the period remains constant (1st phase in Figure 3),
• the waves are likely to break when their height exceeds the local breaking height; this
breaking height reduces with water depth and is also strongly influenced by the
seabed slope; breaking causes the wave energy to be partly dissipated and partly
redistributed over a wider range of frequencies (2nd phase in Figure 3).
0 2 4 6 8 10
2
0
2
4
6
8
Wave Motion Direction [not to scale]
Ver
tical
Dir
ectio
n [n
ot to
sca
le]
Figure 3: Effect of Shallow Water on Waves
Some breaking occurs throughout the passage from deepwater to the shoreline but it becomes
increasingly frequent as the depth reduces, until the surf zone is reached, where breaking
occurs continuously.
The objective of this part of the design process is to determine the structural response due to
breaking waves. This procedure can be divided into three stages:
(i) determining the wave climate at the location
(ii) selecting an appropriate wave model
1st 2nd
Duwind 2003.025 Page 15 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
(iii) determining the loading and response of the structure, of particularly important for
compliant structures such as monopiles, see section 2.3.4 Breaking Wave Loads
Focusing on the second stage, selecting appropriate models, this can be broken down further
into the following steps:
(i) determining the nature of the breaking wave process
- the form, see below
- the height, see section 2.2.4 Breaking Wave Height Modelling
- the probability of occurrence, see below
(ii) determining the breaking wave kinematics, see section 2.2.5 Breaking Wave
Kinematics Modelling
Breaking Wave Forms
The breaking wave process can be classified into three forms, given below in order of
increasing wave steepness and increasing seabed slope:
(i) spilling, where the crest of the wave becomes covered with white foam, are also
known as 'white capping' (if ζ � 0.5, Eq. 29)
(ii) plunging, where there top of the wave curls over and can project a flat wall of water
onto the structure (if 0.5 < ζ < 3, Eq. 29)
(iii) surging, where the whole wave crest becomes turbulent (if ζ � 3, Eq. 29)
The form of the breaking process depends on both the initial wave steepness and the slope of
the sea bed. [8] provides a guide and indicates that plunging breakers occur for medium-steep
waves and are more likely as the sea-bed slope becomes steeper. [29] suggest the use of a
non-dimensional slope to determine the form of breaking, Eq. 29, the conditions for which
type of breaker is likely to occur are given in the list above.
( )βλς tan.0
0
h= Eq. 29
Wave loads due to spilling breakers will not differ significantly from the non breaking case,
hence non-linear wave models can be used to model them. Surging waves are a relatively rare
occurrence and again the resulting loads are similar to the extreme non-breaking case. The
projection of a vertical wall of water onto the structure by plunging breakers results in very
Duwind 2003.025 Page 16 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
high impact loads and consequent "ringing" in a compliant structure and may well be the
design extreme case, hence the interest in understanding this breaking form in particular.
Plunging breakers impose the maximum structural loads a distance downwave from the initial
point of breaking. A simple rule of thumb is 5 times the wave height, however more accurate
models, [89] and [108], suggest that this distance is shorter for steeper slopes. The first
reference, [89], suggests using the following relationship to determine the distance:
.95.3 25.0−= NrIrb
P
hX Eq. 30
Probability of Breaking
With a regular deterministic wave, the probability of wave breaking can be determined quite
accurately; for example, in laboratory experiments, the wave generator can be programmed to
produce waves that break everytime at the same location in the wave tank. With a stochastic
sea, breaking probability is more difficult to determine and the models proposed are perhaps
too approximate and complicated to be worth applying in engineering. Since wave breaking
is relevant for the extreme case and the maximum wave height is limited by the expressions,
using the limiting wave should be acceptable for most design cases.
2.2 Wave Kinematics Modelling
A number of wave kinematics models are reviewed within this section, including several
whose complexity made them impractical for implementation within this project. However
brief details are included for the information of the reader. Regarding implementation and
evaluation, the focus was on widely available and used wave models so that any conclusion
can be implemented by designers should they so wish. Hence the following have been
considered:
(i) Airy; or linear wave theory was the first to be proposed [1]; its relative simplicity is
both its strength and weakness, in that what it lacks in accuracy itself, it can compensate
by being more easy to integrate with other aspects of the wave-load calculation process,
such as stochastic waves, diffraction etc. Its primary weakness is that the kinematics are
assumed only to reach to the mean water level, hence wave peaks and troughs are cut-off
but this is important for the highest waves only (i.e. extreme case) and hence Airy theory
may be suitable for the vast majority of the waves at any site (i.e. the fatigue cases)
(ii) Chakrabarti Stretching [17] is an example of the flexibility of the Airy wave in that the
kinematics as calculated at the mean-water level are moved up to the true surface and the
Duwind 2003.025 Page 17 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
distribution down to the seabed is stretched appropriately; this gives a reasonable
prediction of what happens in reality and this approach has sufficient flexibility to be
applied to non-linear seas for example
(iii) Wheeler Stretching [106] is very similar to Chakrabarti stretching (i.e. see Figure 12
on page 25) and predates it; it is more widely used;
(iv) Constant Crest, assumed that the kinematics in the wave crest are as calculated at the
mean water level; troughs are removed
(v) Extrapolated Crest; the kinematics above the mean surface are calculated applying the
true vertical co-ordinate directly within the airy theory solution
(vi) Delta stretching; the partial derivative of the kinematic property at the surface applies
up to the free surface
(vii) Stokes second order wave theory [92] was proposed a few years after the simpler Airy
theory in 1880 and is appropriate for low-amplitude waves, hence over time higher orders
of this theory have been developed, including third order and fifth order; the fifth order is
the most widely applicable form of the theory to have been in recent use; the mathematics
to derive higher orders becomes progressively more convoluted hence the fifth order form
was developed a century after the second order; this theory was widely used before
increased computation power allowed the more flexible and widely applicable Deans
theory to be utilised,
(viii) Boussinesq, a solitary wave theory originally developed in 1872 [12], is currently
receiving much attention and more accurate formulations are being proposed [64]; it is
not yet clear whether this will lead to widespread acceptance by design rather than
research engineers but it has appeared in at least one design guidelines,
(ix) Cnoidal wave theory [47] should also be mentioned for the sake of completeness; at its
limits, it reduces to both linear theory and solitary wave theory,
(x) Deans Stream Function [23]; has largely superseded all other wave-theories for regular
waves where Airy is insufficient.
Duwind 2003.025 Page 18 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0 50 100 150 200 2505
0
5
10
horizontal [m]
vert
ical
[m]
LinearStokes 2nd OrderStokes 5th orderStream Function
Figure 4: Wave Surface Profile
Figure 4 shows the wave profile for the more important of the above wave models for an
extreme wave3 for which seventh order non-linear stream theory would be recommended
(Figure 6); note that all derivatives of the linear wave theory (i.e. Wheeler, Chakrabarti,
constant and extrapolated crest) assume the same sinusoidal profile. It can be seen that,
relative to linear theory, the non-linear theories predicts that waves have:
• sharper crest and flatter trough profiles,
• higher crest and trough elevations.
These wave characteristics are also clearly visible in stochastic seas as the surface recording
from Blyth in Figure 5 clearly shows.
- 2
- 1
0
1
2
3
4
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0T i m e [ s ]
Wav
e Su
rfac
e [m
]
Figure 5: Recorded Surface Elevation of a Heavy Seas at Blyth
A weakness of low order theories when applied to extreme waves is apparent in Figure 4 in
that the profile of the second order Stokes theory includes an erroneous higher order harmonic
and in that the exact wave length for the fifth order Stokes theory could not be calculated
because the algorithm used here failed to converge. Note that for highly linear waves (low
wave height and deep water compared with the wave length), the profiles will be identical and
3 for 10m 15s wave in 21m water
Duwind 2003.025 Page 19 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
wave tank experiments have shown that for small waves in deep water, all wave theories are
adequate.
Figure 6 shows copies of wave-model guideline charts from two sources, Dean (left chart
[24]) who developed the original stream function theory and from the API (right chart [5])
which is widely used within the offshore engineering industry. Although Stokes theory is
applicable for many situations, it does not cover as wide a range as the stream function theory,
hence for convenience it may be advisable to reduce the number of wave theories in the
design process to two: linear theory for the lowest waves, since it is significantly faster than
the alternatives, and stream function theory for all other cases. The criteria for
recommendation of a particular model is divergence between the predicted kinematics
(typically 1%) between the recommended wave model and its more accurate higher order
neighbour, it being assumed that the highest order stream function generates the correct
solution.
Figure 6: Recommended Wave Theory (sources Dean [24] and API [5])
There have been many attempts to improve the understanding of wave kinematics, breaking
wave heights and wave loads by undertaking experiments in wave tanks or in the open sea
and those that are referred to within the section include:
o Marine Research Facility of the Chicago Bridge and Iron Technical Services,
Plainfield, Illinois wave tank tests [18] and [50]; the tank is 250 ft (76m) long, 33ft
(10m) wide and 18 ft (5.5m) deep; the caisson model has a diameter varying between
53mm and 46mm.
Duwind 2003.025 Page 20 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
o British Maritime Technology compliant test-tower sea tests at Christchurch Bay [13]
which were accompanied by regular and random wave tests in the De Voorst flume in
the Netherlands (1987); the cylinder has a diameter of 480mm and is located in 5m
water depth.
o Coastal Research Centre (Forschungszentrum Küste), Hannover, Germany has
recently undertaken experiments to determine breaking wave loads on both vertical
[104] and inclined [105] piles; the wave tank has dimensions of 309m length, 5m
width and 7 m height; the cylinder diameter was 0.7m.
o Norwegian University of Science and Technology (NTSU) have undertaken a number
of scaled experiments (1:100) on small fixed structures (cylinder-dimensions of
around 1.5 m) located on shallow shoals (between 5 m and 7.5 m) surrounded by
deeper water, [53] and [33].
Today, waves kinematics are generally modelled using two different methods: (i) linear or
Airy theory, possibly incorporating vertical extrapolation, such as Wheeler stretching, to take
account of the wave height, and (ii) stream function theory, a numerical approach which can
be extended to any order. In the past, Stokes theory has been used widely; however for many
users it has now been superseded by the stream function theory, since its analytical nature
means that extending to each subsequent order becomes progressively more difficult. A
number of other theories are of interest to researchers and include solitary wave, cnoidal and
Boussinesq theories, however these are generally not incorporated into commercial software
and hence are not considered here. The choice of actual wave model to be used depends upon
the relationship between the wave height-and-period and the water-depth-and-wave-period.
Waves with low steepness (i.e. where the height to length ratio is low) in deep waters can be
modelled accurately with linear wave theory; in other cases, stream function theory of an
appropriate order is recommended, see Figure 7.
Duwind 2003.025 Page 21 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Figure 7: Recommended Wave Model [8]
This figure also shows which wave model is recommended for traditional offshore
engineering versus offshore windfarm engineering and for fatigue and extreme cases
respectively. For the case of traditional offshore engineering, the fatigue region is still
sufficiently close to the linear zone that a modified linear method, i.e. Wheeler or Chakrabarti
stretching, is acceptable, while stream function theory is recommended for the extreme case.
However, for a dynamically complaint structure, a stochastic analysis is needed; hence a
compromise of the linear wave model with extrapolation is used. As yet, it is not yet
practicable to model stochastic waves with higher order models, except in research to second
order. In contrast, if the windfarm is located in intermediate water depth, with the waves
being generated in a deepwater region, then there is a strong possibility that the extreme wave
will be the breaking case. In addition, the dominant fatigue wave region is further from the
linear zone. Hence the uncertainties will be increased compared with the typical offshore
case.
The extent of the potential difference between linear, extended linear and stream function is
illustrated within section 4.1 Slender Support Structures.
Windfarm Extreme & Fatigue
Deep Water Extreme
& Fatigue
Duwind 2003.025 Page 22 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
The wave loads generated by breaking waves are extremely variable; they depend on both the
nature of the breaking (spilling, plunging or surging) and the phase of the process in which
contact is made with the structure. The worst case is a plunging breaker striking the structure
just after the crest has broken, as the wave projects a flat wall of water onto the turbine.
2.2.1 Regular Linear Wave Modelling (incl. Empirical Corrections)
Figure 8 displays the calculated horizontal velocity under the wave crest for an extreme
wave4. The linear model together with a number of extrapolation methods are shown and
these can be evaluated against the stream function (7th order) solution, which is assumed to be
correct. It can be seen that all linear-based models underestimate both the maximum
velocities and the crest elevation, which will result in an underestimate of the calculated wave
loads.
D w
0 1 2 3 4 5 6 730
20
10
0
10
Maximum Horizontal Velocity [m/s]
Dis
tanc
e ab
ove/
belo
w M
WL
[m]
Linear WaveChakrabarti StretchingWheeler StretchingExtrapolatedConstant CrestStream Function
Figure 8: Variation of Horizontal Velocity at Crest with Depth
The following set of charts show the calculated velocities throughout the wave for four wave
models: linear (Figure 9 left), with Wheeler stretching (Figure 9 left), with Constant Crest
(Figure 10 left), and with Extrapolated Crest (Figure 10 right). Note that all stretching
techniques violate the local mass conservation and linear dynamic free surface boundary
conditions.
4 for 10m 15s wave in 21m water, as previously shown in Figure 4
Duwind 2003.025 Page 23 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
VelM
0 100 200
20
10
0
3
3
3
3
2.5
2.5
2.5
2.5
2
2
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0.50
0
0
0
0
0.5
0.5
0.5
0.5
1
1
1
1 1.5
1.5
1.5
1.5
1.5 2
2
2
2
2
2.5
2.5 2.5
2.5
2.5
3
3
3
3 3.5
0 100 200
20
10
0
3
3
3
3
2.5
2.5
2.5
2.5
2
2
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0
00
0
0
0.5
0.5
0.5
0.5
1
1
1
1
1
1.5
1.5
1.5
1.5
2
2
2 2
2 2.5
2.5
2.5
2.5
3
3
3
3 3.5 3.5
Figure 9: Variation of Horizontal Velocity – Airy (left) and Wheeler (right)
VelM
0 100 200
20
10
0
3
3
3
3
2.5
2.5
2.5
2.5
2
2
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0.5
0.5 0.5
0.5
1
1 1
1
1 1.5
1.5 1.5
1.5 2
2
2
2
2.5
2.5
2.5
2.5
3
3
3
3
0 100 200
20
10
0
3
3
3
3
2.5
2.5
2.5
2.5
2.5
2.5
2
2
2
2
2
2
1.5
1.5
1.5
1.5
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0.5
0.5 0.5
0.5
1
1 1
1
1 1.5
1.5 1.5
1.5 2
2
2
2
2.5
2.5
2.5
2.5
3
3
3
3
Figure 10: Variation of Horizontal Velocity – Constant (left) and Extrapolated (right)
Acc M
0 100 200
20
10
01.5
1
11
1
10.5
0.5
0.50.5
0.5
00
0
0
0.5
0.5
0.5
1
1 1
1
1
1.5
0 100 200
20
10
0 1.5
1
1
11
1
1
0.5
0.5
0.5
0.5
0
0
0
0.5
0.5
1
1 1
1
1
1.5
Figure 11: Variation of Horizontal Acceleration – Airy (left) and Wheeler (right)
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Duwind 2003.025 Page 24 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
2.2.2 Regular Non-Linear Wave Modelling
In the past, the most popular non-linear wave theory has been Stokes Second and Fifth Order,
since they are explicit (i.e. the solutions can be obtained directly by taking the parameters that
define the wave: height, period, water depth etc., and performing a clear string of calculations
until all parameters have been calculated). Deans stream function was originally developed in
1965 [23], just as computing was becoming more widely available in selected research
facilities. Its solution requires the solution of a set of implicit (i.e. there is no direct line from
wave details to the solution) and iterative calculation-procedures are needed. These are
detailed in Appendix D: fore regular waves, with some suggestions how the irregular case
could be solved.
Chakrabarti [18] investigated the use of irregular stream function theory, where the stream
function parameters are modified to match the measured profile. In theory this should
improve the accuracy of the predicted loads however there remained a significant amount of
scatter for both breaking and non-breaking waves. An attempt was made to implement the
irregular form of the Deans stream function within this research project, unfortunately without
success within the time available and against the other priorities within this task. Since the
use of the irregular stream functions seemed to leave a large amount of uncertainty even
within a laboratory environment, since it is unlikely that such tools will become available to
industry within the time-frame of interest here, and since it was not within the terms of
reference for this project this omission cannot be held to be a major loss.
Figure 12 shows the variation of the maximum horizontal velocity (i.e. at the crest) with water
depth for the different wave kinematics models. A more detailed presentation of these
kinematics for the Stokes second-order and the stream function are shown in Figure 13. The
wave form is too extreme to be correctly modelled using either of the Stokes theory orders, as
noted above Figure 4 (page 18).
Duwind 2003.025 Page 25 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
D w
0 2 4 6 8 1030
20
10
0
10
Maximum Horizontal Velocity [m/s]
Dis
tanc
e ab
ove/
belo
w M
WL
[m]
Linear WaveStokes 2nd OrderStokes 5th OrderStream Function
Figure 12: Variation of Horizontal Velocity at Crest with Depth
VelM
0 100 200
20
10
0
77
6
6
5
5
5
5
4
4
4
4
3
3
3
3
3
2
2
2
2
2
1
1
1
1
1
1
0
000
0
0
1
1 1
1
1
2
2
2
2
2
2
2
2
3 3
X = along wave / Y = above MWL
0 100 200
20
10
0
4
4
3
3
3
3
3
3
2.5
2.5
2.5
2.5
2
2
2
2
1.5
1.5
1.5
1.5
1.5
1
1
1
1
1
1
0.5
0.5
0.5
0.5
0.5 0
00
0
0
0
0.5
0.5 0.5
0.5
0.5
1
1 1
1
1
1.5
1.5 1.5
1.5
1.5
X = along wave / Y = above MWL
Figure 13: Variation of Horizontal Velocity – Stokes 2nd (left) and Stream Funct. (right)
Acc M
0 100 200
20
10
0
2.5
2.5
2
2
2
2
2
1.5
1.5
1
1
11
1
1
0.5
0.5
0.5
0
0
0
0
0
0
0
0.5 1
1.5
1.5
2
2
2
2
2
2.5 2.5
2.5
2.5
X = along wave / Y = above MWL
0 100 200
20
10
02
1.5
1.5
1
1
11
1
1
0.5
0.5
0.5
0
0
0
0.5
0.5 1
1 1
1
1.5
2
2
X = along wave / Y = above MWL
Figure 14: Variation of Horizontal Acceleration – Stokes 2nd (left) and Stream F. (right)
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Along wave [m]
Dis
tanc
e fr
om M
WL
[m]
Duwind 2003.025 Page 26 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
2.2.3 Stochastic Seas
Effort is now being directed to generating non-linear stochastic wave sequences, examples
include Clauss [21] whose method inserts a predefined three-wave sequence within a
stochastic sea and the hybrid wave model (HWM) developed by Zhang et al at Texas A & M
University and is described in [110] and evaluated against laboratory measurements within
[90] and against the response of a offshore jacket structure [22]. Although the HWM is more
accurate at predicting wave kinematics and the structural response, the additional effort
compared with using Wheeler stretching is considerable (regarding both code development
and analysis time). Approaches such as these two make intensive demands on both researcher
and computation time and hence there has not been an opportunity to investigate them within
this project.
Linear [99] but also second order [91] [48] simulation can give incorrect under predictions of
the extent of wavegrouping, which can have important effects on wave loads. [91] reports
that narrow-band seas in a deepwater laboratory diverged from the second-order case after
about 10-15 wavelengths of propagation with the result that kurtosis, and extreme wave and
crest heights were under-predicted by theory.
However, as stated also elsewhere within this report, that approaches are being developed
with deepwater locations in mind, where waves are less severely non-linear.
2.2.4 Breaking Wave Height Modelling
Many attempts have been made to develop formulae that predict the wave breaking height as
a function of water depth and in some cases sea bed slope. One of the earliest was by
McGowan in 1894 [66], who recommended the relatively simple expression, Eq. 31, which is
still in user today; its simplicity and hence ease and reliability of use making up for its lack of
sophistication.
wb dh .78.0= Eq. 31
Kriebel [50] has investigated the distribution of breaking wave heights within a wave-tank
and found that the Miche [67] form, Eq. 32, was an upper bound for distribution of breaking
wave heights observed, the mean being around 20% lower. The conclusion for the other two
forms: Battjes & Janssen [10] Eq. 33 and Goda [32] Eq. 34 were similar. In all cases, the
lowest recorded heights of breaking waves were a little more than half the formula-value.
Duwind 2003.025 Page 27 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
( )wbbb dkh tanh142.0 λ= Eq. 32
( )wbbb dkh 9.0tanh14.0 λ= Eq. 33
��
�
�
��
�
�
��
���
��
���
���
�
�+−−= 3
4
00 151
.5.1exp1.17.0 β
λπ
λ wb
dh Eq. 34
Rattanapitikon and Shibayama [76] list 24 models and evaluate these against a comprehensive
list of laboratory (only) wave data from other experiments. It recommends that any of three
models, including Eq. 35 [31] (also recommended by Barltrop [8] and is an earlier version of
Eq. 34), are universally applicable and several others are acceptable for the range of sea-bed
slopes likely to be found around offshore windfarms. This and other expressions suggests that
the breaking wave height increases over sloped regions, as shown in Figure 15. Wave heights
up to twice the water depth are possible over very steep slopes.
��
�
�
��
�
�
��
���
��
���
���
�
�+−−= 3
4
00 151
.5.1exp1.17.0 β
λπλ w
bd
h Eq. 35
2002040608020
15
10
5
0
5
10
15
20
25
Distance from Turbine [m]
Wat
er D
epth
[m]
sea bed level (w.r.t. mean water level, incl tides)breaking wave heightbreaking wave actionwind-turbine
Figure 15: Plunging Breaking Waves5 [89]
However, it should be noted that this expression is not accurate for very steep seabeds (> 7%)
and that two of the simplest and oldest expressions, Eq. 31, McGowan [66] and Eq. 32, Miche
[67], are rated among the best by both this and two other evaluation undertaken by Anastasiou
[2] and Kriebel [50]. These three evaluation-papers do not agree with each other on all
matters; for example [2] evaluated 19 expressions against their own data taken over 1D and
2D laboratory bathymetric profiles and found that the Goda expression, Eq. 34, to be
5 in the SE direction, at high tide level (=LAT + 6m)
Duwind 2003.025 Page 28 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
significantly poorer than the other two. The paper mentioned in the previous paragraph,
Kriebel [50], on the other hand, found the Goda expression, Eq. 34, to be even marginally
better than the other two
Care should be taken when using laboratory corroborated coefficients to determine the
breaking weave height since these will generally be from long-crested waves data; short-
crested seas include waves with significantly higher maximum wave crests; [86] report that
the mean breaking-wave crest elevation doubles from the long-crested case to a sea with a 70
degree spreading angle.
2.2.5 Breaking Wave Kinematics Modelling
Determining the kinematics of the plunging breakers requires CFD and this is already being
used with success within the research community, for example, reference [27] shows
remarkably good correlation between the measured and modelled wave kinematics for a
plunging breaker. Generally no attempt is made to calculate the kinematics of breaking
waves accurately except for research purposes because of the high degree of randomness,
instead the simplified approaches listed below (and described further in section 2.3.4
Breaking Wave Loads) are applied; advances in computer power means that limited numbers
of simple cases can be examined in detail, i.e. [27].
o Calculating the kinematics using regular wave theory and the highest non-breaking
wave, waves tend to break when the surpass the breaking wave height and hence are
not able to become significantly larger than this value (an exception is over sloping
sea beds,,, when the breaking wave height limit itself can change rapidly)
o Applying simple empirically based formulae using basic wave parameters
2.2.6 Effect of Current
If a uniform current is present, then the calculation of the wavelength needs to be applied in
the coordinate system moving with the current rather than in the fixed frame of reference in
which the surface elevation readings are taken; for example for linear waves the dispersion
relationship becomes Eq. 36.
( ) ( )kdgkkU C tanh2 =−ω Eq. 36
Regarding the implications for breaking waves, these are that for a favourable current, the
breaking wave height increases with [50] ascertaining that increase in the order of 5%
Duwind 2003.025 Page 29 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
occurred for Eq. 32, Eq. 33 and Eq. 34 with an in-line current (scaled-down but equivalent to
1.65 m/s).
2.3 Wave Loads Calculation
There are two main methods currently extensively used, among the wider possible range,
listed in Table I:
• Morison's method, usually in the time domain, used for slender structures, such as
monopiles and tripods,
• Diffraction theory, used for massive structures, such as gravity base supports
In addition the pressure integration (or Froude-Krylov) method offers the advantage of being
able to model massive and complex structural geometries with any wave model [17], however
diffraction has to be estimated in a similar manner as for Morison's method with the more
complicated geometries make this harder to achieve. In the situations where this method
could offer the most beneficial results, i.e. gravity base structures in shallow water, the wind
loads on the turbine do dominate the design process [87], however this method has already
been used for the design of gravity base structures for offshore windfarms.
In the longer term, CFD offers promising benefits of being able to model all aspects, though
at undoubted penalties of time and clarity.
Table I: Wave load calculation Methods
Mor
ison
Diff
ract
ion
Frou
de-K
rylo
v /
Pres
sure
In
tegr
atio
n
CFD
Time / Frequency Domain TD FD FD TD TD
Forces
inertia √ √ √ √ √ Transverse
drag √ √ 6 X X √
Lateral (drag) √ √ 6 X X √
Pressure X 1 X 1 √ √ √
Geometry
Diffraction X 2 X 2 √ X √
Duwind 2003.025 Page 30 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Mor
ison
Diff
ract
ion
Frou
de-K
rylo
v /
Pres
sure
In
tegr
atio
n
CFD
1D √ √ X √ √ Surface Effects 3
3D X X X √ √
Massive Structures X X √ √ √
Wave Model
Non-linear wave models & extrapol. √ X X √ √
Stochastic (Linear) √ √ √ X 5 X 5
Applicability (* = poor / *** = good)
Commercial Availability *** *** *** ** *
Ease of Use *** ** ** ** *
Calculation Speed ** *** *** * *
1 = can be modelled relatively easily by adding an extra term 2 = can be modelled using MacCamy-Fuchs [63] correction for simple shapes 3 = non-linear surface effects between the structure and the wave-field:
1D = in vertical direction only (i.e. wave height considered only at the vertical-axis of the structure) 3D = full geometric field (i.e. wave height at each surface element of the structure)
4 = must be estimated 5 = high demands on computation power 6 = linearised
Both diffraction and Froude-Krylov methods suffer from substantial weaknesses when
calculating the wave loads on massive structures. However, the weaknesses of each method
are different from each other, see Table I, and hence the obvious approach is to use both
methods together: diffraction theory to estimate the effect of the structure on the flow field
and Froude-Krylov to calculate the wave loads using non-linear waves. Since GBS structures
tend to be fairly simple, i.e. consisting of a round base, a tower section and possible an ice-
cone at the water surface, in many cases, it should also be possible to estimate the diffraction
coefficients by comparing with other similar structures.
Duwind 2003.025 Page 31 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
2.3.1 Morison Method
In 1950, Morison [68] developed a simple expression, Eq. 37, for calculating the normal loads
on fixed piles in shallow water, Figure 16. The method combines a theoretical approach and
rationalisation with the flexibility of empirically adjustable drag and inertia coefficients.
Added mass, Froude-Krylov and drag forces can be modelled depending on the coefficients
and terms included.
Waveloading
Figure 16: Morison Equation - Normal Loads
1. . . . . . . .
2M DdF C dV U C dS U Uρ ρ= +� Eq. 37
where dF = total wave force,
dS = frontal area, U = fluid velocity,
U* = fluid acceleration.
To apply the equation, an expression for the fluid velocity and acceleration is required. This
is derived from the wave potential flow model selected, for example, Airy linear wave theory.
Because of these important advantages, over the intermediate decades researchers have
attempted to stretch the equation to cover other applications, which now include fully-
submerged structures, floating structures and different geometries. The critical aspect of this
work is to develop simple yet widely applicable methods of determining the value of the
coefficients CM and CD. In summary, that work has shown that the value of the two
coefficients, CM and CD, depends primarily on the Keulegan-Carpenter number, K and
secondarily on the Reynolds number, Re.
Regarding this choice of inertia and drag coefficients, Table II lists selected recommended
values obtained from wave and open sea experiments and from design recommendations. It is
clear that there is significant scatter, not only due to the different situation being examined
(i.e. smooth or rough cylinders, low or high K). Basically it is exceeding difficult to separate
the inertia and drag coefficients from measurement data. [7] provides a more comprehensive
summary and concludes that for high K, CD converges to the steady flow value, Figure 18. It
Duwind 2003.025 Page 32 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
also states that it is acceptable to ignore the effect of K, i.e. to select a constant value for the
coefficients; see Figure 22 (page 36) for an evaluation of the effect of the values
recommended by DNV on an offshore windturbine support structure geometry which come to
a different conclusion, probably because drag forces are more important in this case.
A common mistake made by designers is to use the smooth cylinder coefficients, this being
unconservative for the drag coefficient. [7] suggests that this is acceptable for offshore
engineering because of conservative assumptions elsewhere:
(i) Use of independently determined extreme values for wave height, wave period,
current and mean water level, hence ignoring correlation
(ii) Use of regular wave theories
(iii) Use of uni-directional seas
(iv) Ignoring shielding
However, the situation for offshore windenergy design engineer is different:
a.) the conservativism of independently determined extreme values, (i), is being
challenged by use of correlated design values [20]
b.) it is unclear why the use of would be conservative for offshore windfarms
c.) shielding, (iv), is currently not relevant being applicable for jacket structures; it
will become important when tripod support structures come into use
Of the four conservative assumption listed, two are probably applicable here: the affect of
using regular wave theories, (ii), instead of stochastic case, since research suggests that inertia
coefficients are slightly higher in regular waves than in real stochastic seas, and the uni-
directional seas, (iii), is definitely applicable here. In addition, there are other unconservative
assumptions:
d.) use of linear-extrapolated stochastic wave theory for the extreme design case; the
shallow water depth means that non-linear wave theory is required,
e.) ignoring the effect of access structures; this is unlikely to affect the inertia forces
but the drag forces could be increased depending on the size and location of the
ladder and J-tubes (i.e. see recommendations in API [5]).
Duwind 2003.025 Page 33 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Table II: Comparison of Recommended Values for CM and CD
Wave Only Wave Plus Current Source Details
CM CD CM CD Ref
API Recommendations for Design 1.7 1.05 [5]
Chakrabarti Wave-tank tests; 46/53 mm diameter 1.46 Drag
dominated 17 [18]
Christchurch Bay
Offshore tests; 480 mm diameter8 1.65-1.9 0.75-
0.95 [13] [70]
City University
Horizontal cylinder; wave tank; 210 mm &
500 mm diameter 1.2 0.6 –
1.2 1.2 0.6 – 1.2 [19]
Delta wave flume
Roughened cylinders; 216 mm & 513 mm
diameter 2 1.7 � 1.8 (?) � 1.5 (?) [107]
DNV Recommendations for Design 1.8 1.2 [25]
Looking at the API [5] recommendations in greater detail, the drag and inertia coefficients
advised can be accurately expressed as functions of the relative surface roughness, the
Reynolds number, the Keulegan Carpenter number, the current/wave velocity ratio and the
member orientation. The advised CDS, which is defined as the steady flow drag coefficient, CD
and CD values can be seen in Figure 17, Figure 18 (left) and Figure 19 (left) as functions of
the roughness (e), and Keulegan-Carpenter number (K), respectively.
The DNV recommendations [25] are slightly simpler in that no intermediate step involving
CDS is required; the corresponding charts can be seen adjacent to the API recommendations in
Figure 18 (right) and Figure 19 (right).
6 Keulegan-Carpenter values of 10-30 7 Keulegan-Carpenter values of 15-40 8 Keulegan-Carpenter values of 2.5-8.5
Duwind 2003.025 Page 34 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Figure 17: CDS (source API [5])
Figure 18: CD (source API [5] and DNV [25])
Figure 19: CM (source API [5] and DNV [25])
The tendency towards cheaper softer support structures means that the structural motion
response may need to be considered; [70] suggests that the usual coefficients can continue to
be used if the relative velocity is determined. An exception was if significant lateral motion
Duwind 2003.025 Page 35 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
occurred, termed lock-in [83], when there was an increase in the drag coefficient of up to
25%, however this should not occur for offshore windturbine structures.
The charts below show the calculated wave loads for the previously defined geometry for a
selection of wave kinematic models; Figure 20 shows a time-series of the calculated shear
force (left) and overturning moment (right).
0 5 10 151000
500
0
500
1000
1500
Linear WaveChakrabarti StretchingWheeler StretchingExtrapolatedConstant CrestStream Function
Time [s]
Shea
r For
ce [k
N]
0 5 10 1520
10
0
10
20
30
Linear WaveChakrabarti StretchingWheeler StretchingExtrapolatedConstant CrestStream Function
Time [s]
Ove
rtur
ning
Mom
ent [
MN
m]
Figure 20: Shear Force (left) and Overturning Moments (right)
Effect of Surface Roughness
Figure 21 and Figure 22 illustrate how the forces will change with age and increases in the
surface roughness of the cylinder. A roughness of 1/100 is equivalent to 3.5 cm on the 3.5 m
diameter Blyth monopile, which results in an increase in the maximum overturning moment
by 20%. Since the minimum value becomes smaller, the range only increases by a little over
10%.
Duwind 2003.025 Page 36 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Effect of Cylinder Smoothness
-15
-10
-5
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Time [s]
Ove
rtur
ning
Mom
ent
[MN
m]
smooth roughness 1/1000
roughness 1/100 roughness 1/20
Effect of Cylinder Roughness
0
0.5
1
1.5
2
2.5
smooth 1/1000 1/100 1/20
Cylinder Roughness
Iner
tia &
Dra
g C
oeffi
cien
t
Inertia Coefficient Drag Coefficient
Figure 21: Effect of Cylinder Roughness
Effect of Increasing Cylidner Roughness [i.e. age]
-30%
-20%
-10%
0%
10%
20%
30%
40%
1/1000 1/100 1/20
Cylinder Roughness
% C
hang
e in
Ove
rtur
ning
M
omen
t
minimum value maximum value range
Effect of Cylinder Roughness
-20%
-10%
0%
10%
20%
30%
40%
50%
60%
1/1000 1/100 1/20
Cylinder Roughness
% C
hang
e in
Iner
tia &
Dra
g C
oeffi
cien
t
Inertia Coefficient Drag Coefficient
Figure 22: Change of Properties with Increasing Roughness
Effect of Current
If a current as well as the waves is present, then the procedure needs to be modified as
follows::
[1] Apply Doppler shift to find the apparent wave period
[2] Use linear theory or an appropriate order of the Deans stream function to calculate the
wave kinematics
[3] Superimpose current profile
[4] Use recommended values of CM and CD within Morison to calculate the loads
Effect of Inclination of the Structural Member
If the cylinders are inclined, for example as on a tripod structure, the wave loads will
generally be decreased for members inclined away from the waves and increased for members
inclined into the waves [3] and the drag coefficient is more sensitive to the angle than the
inertia coefficient [17]. Sarpkaya [85] reported some additional phenomena where the wave
Duwind 2003.025 Page 37 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
loading dipped at certain Keulegan-Carpenter numbers, however this may be due to the way
in which the water kinematics where created (as oscillating water within a tube).
If the structural member is horizontal, laboratory experiments [19] suggests that the inertia
coefficient is reduced to around 1.2 while the drag coefficient remained within the normal
band of variability at 0.6 and 1.2 for smooth and rough cylinders respectively but that
variability did increase, hence the conservative assumption of using values from the vertical
case may be appropriate.
Effect of Drag
If the wave height is varied, Figure 23 shows how the difference between the wave models
and the exclusion of drag varies with wave height, when the range of the overturning moment
loads are calculated with the Morison method. It can be seen that at low wave heights, such
as those that determine the fatigue loads, the difference is smaller and hence linear models
may be acceptable. For the higher waves and this geometry, the choice of wave model is
important but drag could be ignored. 8s wave in 6m water
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1 2 3 4Wave Height [m]
Ove
r-tu
rnin
g M
omen
t [M
Nm
]
Deans[with drag]
Deans [nodrag]
Wheeler[with drag]
Wheeler[no drag]
Airy [withdrag]
Airy [nodrag]
Figure 23: Effect of Wave Model and Drag Resulting OTM at Mudline
2.3.2 Diffraction Method
Diffraction is the process whereby the effect of the structure on the incoming wave field is
determined. Of course, all structures have an impact on the wave field, including monopiles,
however the diffraction effects of simple geometries such as monopiles are easy to determine
(being unity for a circular cylinder for example) and the difficulties arise when complex
geometries, such as gravity base structures, are analysed.
Duwind 2003.025 Page 38 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
The focus of the analysis is to calculate the added mass and possibly damping due to the
presence of the water. Part of the wave force on the structure can be found by integrating the
pressures over its surface (the Froude-Krylov force) and the additional force (due to
diffraction) is quoted as a factor of this. As stated above, for a simple circular cylinder, this
value equals unity, resulting in an inertia coefficient of two. For other geometries, the inertia
coefficient is generally less than two, being at the limit one for a flat plat side-on to the flow.
Diffraction has been used for the analysis of candidate gravity base structures in section 4.2
Compact Support Structures. For further background information of the technique, the reader
is directed to offshore engineering books such as [17].
2.3.3 Froude-Krylov Method
Froude-Krylov or pressure integration theory involves determining the pressure due to the
undisturbed fluid at the surface of the structure. The effect of the structure on the fluid needs
to be estimated to find the total loads and since the structure may be complicated, this is more
difficult than for the Morison method.
The Froude-Krylov method can also be applied to simple geometries and in the following
section, an investigation is made into what the effect of considering the three-dimensional
geometry would be on the wave loads on a simple monopile. Figure 25 illustrates the
influence of the geometry on the overturning moment range by comparing the Morison
method against the panel method. The Froude-Krylov method is a type of panel method,
hence the wave loads are calculated by integrating the pressure around the wetted surface, in
comparison with the Morison method, which, at its simplest, calculates the wave field along
the tower axis and applies this to the entire volume. This suggests that the application of the
Morison method together with Deans wave theory may also underestimate the wave loads in
some cases, due to the imbalance in wetted surface caused by the steep surface slope of
extreme waves, Figure 24.
Duwind 2003.025 Page 39 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
D w
4 2 0 2 40
5
10
X-axis
Z-a
xis
imbalance in wetted surface
water surface
Tower (panel mesh
shown)
Figure 24: Effect of 3D Geometry
In Figure 25 it can be seen that there is a significant difference of up to 50% between the
application of Morison (with Deans stream function) and the Panel method (also with Deans
stream function). Note that it is not possible to calculate drag forces directly with the panel
method, hence drag forces are excluded from the other curves but Figure 23 showed that drag
was relatively unimportant. Figure 25 also shows the panel and drag forces calculated using
Wheeler (which also includes surface effects) and Airy (which does not) wave theory. It can
be seen that for linear Airy theory, the Morison and Froude-Krylov approaches predict the
same wave forces.
The conclusions from this chart are that even if stream function theory could be combined
with diffraction effects, lack of inclusion of the three-dimensional surface effects would mean
that significant errors remained.
Duwind 2003.025 Page 40 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
8s wave in 6m water
0
0.5
1
1.5
2
2.5
0 1 2 3 4Wave Height [m]
OTM
Wav
e Lo
ad [M
Nm
]
Deans[panel]
Deans[Morison /no drag]Wheeler[panel]
Wheeler[Morison /no drag]Airy [panel]
Airy[Morison /no drag]
Figure 25: Effect of Geometry on Resulting OTM at Mudline
2.3.4 Breaking Wave Loads
Guidance for determining breaking wave loads are not common in offshore engineering; for
example the API recommendations [5] do not give any specific recommendations, probably
partly because there is incomplete consensus that the modelling this aspect accurately is
important. The following contrasting conclusions from recent experiments illustrate this:
• Chakrabarti [18]9 did not find a significant difference for breaking wave loads
compared with the non-breaking cases of similar heights, however the high-frequency
components of the response data were filtered out prior to analysis,
• Kriebel [49], Basco [9] and Kjeldsen, [46] (via [9]), found that the force due to
breaking waves was up to twice the non-breaking case,
• Apelt [4] measured a ratio of up to eight-to-ten between the base overturning
moments of the two cases,
• and Peregrine [72] predict local accelerations within the crest of a plunging breaker of
up to eight times gravity using numerical modelling (accelerations within a Deans
stream function wave would typically reach half of gravity).
Hence, some form of analysis is recommended. The worst case of a plunging breaker could
be modelled as an impulse as per Eq. 38:
9 results from the experiments at Marine Research Facility of the Chicago Bridge and Iron Technical Services, Plainfield, Illinois
Duwind 2003.025 Page 41 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
bSI CCRF ηλρ ..... 2= Eq. 38
Two forms of CS, the time varying impact force function, were examined in [104]:
• that suggested by von Kármán [102], Eq. 39, which predicts a sawtooth variation of
the force with time; with an initial peak coefficient of π falling linearly until the
submersion of half the cylinder10. The coefficient value of π was derived
theoretically and has been corroborated by experimental work by Sarpkaya [82]
(= 3.17±0.05) though other experiments suggest larger values and scatter, with
numbers up to 10 published (i.e. [81]).
2
0
11
2I S S
tF C Dlv
tρ
� �= −� �
� � Eq. 39
• and that by Wagner [103] (a shorter more complex variation, with a peak coefficient
of 2π).
The conclusion in [104] was that Wagner's method was the more accurate (and [53] appears
to agree though no mention is made of Wagner’s method), however von Kármán’s is
currently more widely used. These formulations refer to the total loads on the cylinder; local
pressures can be much higher at up to 32ρv2, i.e. more than ten times higher but are also much
shorter at less than 1% of the wave period [111].
Note that this force should be added to the Morison force.
If the cylinders are inclined, for example as on a tripod structure, the wave loads will
generally be decreased, since they are inclined away from the waves; in the event that a wave
is able to break on an element inclined into the wave, the loads are significantly increased
[105] and [94].
Recent studies by the NTSU have examined breaking wave loads on vertical cylinders ([42]
and [53]) and tripod structures ([33]) and have proposed a new curling factor, λH-T, dependent
on the length of the pile (as opposed to the wave height). Experimentally-derived values are a
little higher than for the λ defined in the traditional manner, at up to around 0.6 for large
waves, probably because of the extreme conditions at the site (the seabed could become dry).
Of general interest, it was found that the overall loads were higher relatively speaking on the
monopile than the tripod because the breaking waves do not impact all members
10 Note that this formulation was developed for predicting forces on the floats of a seaplane as it came in to land.
Duwind 2003.025 Page 42 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
simultaneously on the second structure. Note that the conditions examined in this series of
experiments are unlike those faced by windturbines in that the steep seabed slopes mean that
wave heights can be up to three times the water local depth and that the conclusions from [42]
and [53] differed even though they examined similar structures probably because of differing
local bathymetric detail.
2.3.5 Effect of Structural Dynamics
The field tests carried out using a flexible column at Christchurch Bay [13] suggested that the
usual CM coefficients could be used if the relative motion was taken into account (and that and
CD should be increased; however since the experiments were undertaken solely in an inertia-
dominated regime, this recommendation carries additional uncertainty). Alternatively, the
values of both CM and CD coefficients should be slightly enhanced.
There is however a basic difficulty in including the effects of structural dynamics, since a
properly designed structure is one where its natural frequencies do not correspond with the
frequency-range of the wave energy spectra. Hence dynamic amplification will often be
triggered by harmonics of the waves rather than the fundamental frequencies themselves. On
the other hand, a properly analysis of the structural response requires the application of a
stochastic sea in the time domain and as yet this can only be done for linear waves. Hence the
vitally important harmonics are not included.
One approach is to use regular non-linear waves instead, however this is unsatisfactory since
the structural response seen is then primarily a function of the ratio between the wave
frequency and the structures natural frequency: if this is a ratio, large dynamic amplification
will occur; if not, the response will appear quasi-static. Figure 26 shows this in that the
maximum and minimum values of the applied wave load curve (thick line) and the calculated
internal bending moment curve (thin line) are very similar, i.e. the there is no dynamic
amplification. This approach is able to model the effect of the harmonics in a simple manner
but because for this example the main wave force is applied when the natural response cycle
is in an opposing phase, the amplitude of the response cycles do not intensify.
Duwind 2003.025 Page 43 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
16 20 24 28 32 Time [s]
Bend
ing
Mom
ent [
kNm
]wave loads
structuraldynamics
Figure 26: Effect of Structural Dynamics with No Dynamic Amplification
2.4 Ice Loads
Ice loads are a design issue in the more sheltered and less saline Baltic Sea; however in the
North Sea, it is not an issue even at the most northern potential offshore windfarm sites.
According to the IEC recommendations 61400-03 [43], the local static ice pressure can be
determined by Eq. 40 (and cannot exceed 20MPa):
ulocal
local rAe
r
5.02
15���
�
�+���
����
�= Eq. 40
The following two tables give ice properties for interior Danish Waters, which are susceptible
to ice. Table III lists some basic permanent properties of ice while Table IV illustrates how
the variable properties of the ice alter with increasing return period of winter severity.
Table III: Properties of Ice [43]
Symbol Property Value
ρi Density 900 kg/m3
γi Gravity 8.84 kN/ m3
E Modulus of Elasticity 2 GPa
ν Poisson’s ratio 0.33
Duwind 2003.025 Page 44 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Symbol Property Value
Ice-ice 0.1
Ice-concrete 0.2 µ Coefficient of Friction
Ice-steel 0.1
Table IV: Design Values for Interior Danish Waters [43]
Annual Risk of Deviation 0.2 0.1 0.02 0.01 8 ×××× 10-4 10-4
Recurrence Period years 5 10 50 100 1250 10,000
Kmax - °°°°C 24
hrs 170 245 410 480 744 960
ru MPa 1.0 1.5 1.9 2.0 2.4 2.6
rf MPa 0.25 0.39 0.50 0.53 0.64 0.69
e m 0.33 0.42 0.57 0.63 0.80 0.91
2.5 Design Practices
Regarding advise for design procedures, design procedures specified by certification
authorities for traditional fixed offshore structures can be followed, however these may not
always be directly applicable or even suitable for offshore windturbines for the reasons
explained above:
(i) API [5]
(ii) DNV [25]
(iii) Lloyd's Register [55] [56] [57] [58]
(iv) Germanischer Lloyd, specifically for offshore windenergy [30]
Alternatively, design guidelines from other sources could be used, though an eye should be
kept on the guidelines published by the certification authority to be used to ensure a smooth
process. Examples for offshore engineering include
(i) UK Department of Energy Guidance Notes
(ii) Barltop and Adams [8] unfortunately out of print
While for windenergy, there are publications from:
(i) DNV/Risø’s windenergy guidelines include sections on offshore windenergy [26]
Duwind 2003.025 Page 45 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
(ii) IEC [43].
Duwind 2003.025 Page 46 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
3 Description of Blyth Windfarm
Blyth offshore wind farm is situated off the Northumberland coast, on the north-east of
England. It comprises two Vestas V66 2MW wind turbines, situated approximately 1km
offshore. The turbines were installed between August and October 2000 in rock-socket
foundations on a submerged rock known as the ‘North Spit’. Both turbines are sited in a
mean water depth of approximately 9m. Figure 27 indicates the locations of all offshore
windfarms as of the end of 2002, with the sizes of the markers indicating the generating
capacity of the windfarms. It can be seen that Horns Rev, on the Danish west coast,
overshadows the other windfarms and in fact has more generating capacity than all the others
together. Blyth offshore windfarm itself is not significant as far as generating capacity is
concerned, however it was the first to be installed in the North Sea, two years prior to Horns
Rev.
�
�
�
Yttre Stengrund
Utgrunden
Bockstigen-Valor
Vindeby
Middelgrunden�
�
�
Samsø
�
Tuno Knob
Dronten
Lely
�Horns Rev
�
�
Blyth�
Figure 27: Location of Previously Built Windfarms
Of the two turbines installed at Blyth it was decided to instrument the southern-most turbine.
This turbine is positioned at the top of a steeply shelving region of the sea bed which was
considered would increase the likelihood of breaking waves at this turbine. Breaking waves
were indeed experienced at the site during installation of the turbine and subsequently.
The offshore wind turbines installed at Blyth are the first turbines to experience the full force
of North Sea wave conditions. They therefore provide an ideal opportunity to study wave
loading at full scale in an aggressive environment.
Size of Dot represents Windfarm Capacity Size of Font represents Windturbine Size Small Font = older, smaller machines Large Font = recent MW machines
Duwind 2003.025 Page 47 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
The measurement programme at Blyth and supporting theoretical studies are being performed
in a collaborative project sponsored by the European Commission (JOR3-CT98-0284), the
UK Department of Trade and Industry and Novem, The Netherlands agency for energy and
the environment. The project, named ‘Offshore Wind Turbines at Exposed Sites’ (OWTES),
is being undertaken collaboratively by Delft University of Technology, Germanischer Lloyd
WindEnergie, Vestas Wind Systems, AMEC Wind and Powergen Renewables Developments
under the leadership of Garrad Hassan and Partners. This report forms part of the deliverables
of this project.
3.1 The Windfarm and Windturbines
3.1.1 Description of the Site
Figure 28 shows the local bathymetry surrounding the windfarm site plus some details of the
local shoreline. There is a second windfarm built along the breakwater near the centre at the
bottom of this figure.
Duwind 2003.025 Page 48 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Figure 28: Sea Chart of Area Surrounding Windfarm Site
Figure 29 and Figure 30 provide more details of the bathymetry at the windfarm based on a
survey undertaken at the site. In Figure 29, the wind-turbines are shown as spikes and the
unmeasured region is shown at -20m; note that the horizontal scale is heavily compressed
making the slopes appear steeper than they really are and that the unmeasured region is shown
at -20m, hence the apparent cliffs. The shallow spit surrounded by deeper water upon which
the two offshore windturbines at Blyth are constructed can be clearly seen; the mean water
depth at the instrumented southern turbine varies between 6 m (LAT, Lowest Astronomical
Tide) and 11m (MHWS, Mean High Water Springs), while the water depth is 5m deeper in
the south-east direction at a distance of 50-100 metres.
Duwind 2003.025 Page 49 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0
20
40
0 20 40
20
10
0
Figure 29: Bathymetry at Blyth
Figure 30 gives a more conventional plan view of the same information; the axes are
'northing' and 'easting' in m with the grid spacing being 100 m.
4.323 105 4.324 105 4.325 105 4.327 105 4.328 105 4.329 1055.822 105
5.823 105
5.824 105
5.825 105
5.826 105
5.827 105
7 7
7
7
8 8
8
8
8
9
9
9
9
9
9
9
9
9
9
9
9
9 9
9 9
9
10
10
10
10
10
10
10
10
10 10 10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
11
11
11
11
11
11
11
11
12
12
12
12
12
12
12
13
13
13
13
13 14
14
14
14
14
14
14
15
15
15
15
16
16
16
17
17
17
18
18
18
19
19
19
20
20
20
Figure 30: Bathymetry at Blyth
East - West [x10m]
North - South [x10m]
Southern Windturbine
Southern Windturbine
Duwind 2003.025 Page 50 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
3.1.2 Description of Windturbine
Figure 31 shows the layout of the windturbine. Elevations are given with respect to the LAT
(Lowest Astronomical Tide) and it can be seen that this is 5.8 m above the mudline (in this
case rocks) and that the 50 year tide is 6.1 m above this. The entrance platform is 11.7 m
above LAT, giving at least 5 m clearance above the water level. The nacelle hubline is 62 m
above LAT and the blade tip when vertically upwards is 95 m above.
Figure 31: Dimensions of Windturbine
Duwind 2003.025 Page 51 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
3.1.3 Wave Climate at Blyth
The wave-height exceedance (in hours) at Blyth for the prevailing offshore directions are
shown in Figure 32, this information being taken from a nearby deeper water location. It can
be seen that for a substantial proportion of the year, the significant wave height is higher than
the local breaking height. When we consider that the highest probable wave in a 3 hour sea
state is 1.86 times the significant wave height, we can conclude that breaking on or before the
spit will occur frequently. Note that some of the energy will have been dissipated prior to the
wave reaching the spit, [98] suggests a simple method of adjusting the wave height and period
to take account of this. Alternatively, [93] is more advanced but neither approach will be able
to properly account for the influence of bathymetry in the immediate vicinity.
LAT MWHS
0 5 10 150.01
0.1
1
10
100
1 103
from NNW-Nfrom NNE-NEfrom ENE-Efrom ESE-SE
Significant Wave Height [m]
Prob
able
Exc
eeda
nce
[hrs
/yea
r]
Figure 32: Expected Sea State Distribution at Blyth
Examining the depth variation up-wave from the turbine in the SE direction (where the edge
of the spit is closest), Eq. 30 predicts that the wave-breaking triggered by the shelf (40-60m
from the turbine) would fortunately cause the worst potential wave loads before the actual
turbine location, see 'breaking wave action' curve in Figure 15. Applying Eq. 29 suggests that
plunging breakers will form mainly at the steep part of the slope between 60 m and 40 m from
the turbine.
Duwind 2003.025 Page 52 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
3.2 Measurement Systems
The measurement system installed at Blyth comprises three main elements:
(i) measurement of the turbine structural loading,
(ii) measurement of the sea-state, and
(iii) measurement of wind conditions at an onshore meteorological mast close to the
turbine.
The instrumentation comprising each of these sub-systems is described briefly below. The
turbine loading is measured using a large array of strain-gauges which have been applied to
every major structural element of the turbine. On the tower and pile foundation, strain-gauges
are used to measure bending moments in two dimensions at eight vertical stations. Blade
loads are measured as flapwise and edgewise bending moments at the blade roots. The low-
speed shaft of the turbine is also instrumented to measure torque and bending moments in two
orthogonal directions. In addition, signals related to the control and operational status of the
turbine are recorded, including blade pitch angles, the speed and position of the rotor, nacelle
orientation, brake status and generated power. The locations of the strain gauges on the
turbine and support structure are shown schematically in Figure 33. A large number of
measurement stations were instrumented on the pile to provide redundancy, because it was
believed that the strain-gauges might not have a long life in the salt water environment. In
fact, after seven months, only one gauge had failed, resulting in excellent strain gauge
coverage on the pile. The tower and pile strain gauges were calibrated during summer 2002
however this was too late to include in the work described within this report, hence an
alternative method of calibration was used, as described in Appendix A: Calibration of Strain
Gauges.
Duwind 2003.025 Page 53 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Figure 33: locations of Strain Gauges
The wave and current climate is recorded using instruments mounted both above and below
water level. A Saab WaveRadar unit is mounted on the turbine walkway to measure the
instantaneous water level at the turbine base, including time-history profiles of passing waves.
Simultaneously, instruments mounted on the sea bed approximately 40m from the foundation
record statistics describing the wave climate and the current profile. These instruments
include a wave and tide recorder (Coastal Leasing Microspec) and an acoustic doppler current
profiler (Nortek ADCP). The sea-state instrumentation is shown schematically in Figure 34
Saab Waveradar
Coastal LeasingMicrospec
Video Camera(mounted on 2nd turbine,looks at painted lines on tower)
Sea level
Sea bed
Nortek ADCP
Power & data cables
~30m
Figure 34: Sea Climate Monitoring Instrumentation
Wind conditions are measured using anemometers and wind vanes mounted on an onshore
meteorological mast and on the turbine nacelle. Although the meteorological mast for the
Duwind 2003.025 Page 54 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
project would be ideally located offshore, close to the monitored turbine, the large cost of
such an installation was beyond the budget of the OWTES project. The mast has therefore
been positioned on the coast, approximately 1km from the southern turbine. The mast features
anemometers at heights of 10m, 20m, 30m and 40m above ground level and instruments to
measure atmospheric pressure, temperature and precipitation. This data was also not
available at the time of undertaking the main work described within this report; however that
was of secondary importance as the focus here is on the wave loading rather than the wind
loading.
Measurements have been collected from the southern Blyth wind turbine since November
2001 and the main data is recorded in two forms:
• as 10 minute summary values (mean, minimum, maximum and standard deviation of
each channel),
• as 30 minute campaign datasets, with 40Hz data from all channels.
47 channels of data are recorded and the datasets utilised in this report include the:
• water surface elevation,
• the strain gauge readings in the tower and pile,
• the accelerometer readings in the nacelle.
A full list of the measurements made in the time domain is given in Table V.
Table V: Description of Measurements
Channel Description Unit Channel Description Unit
1 time [s] 25 Gen Star
2 Blade B root flap BM [V] 26 Gen Delta
3 Blade B root edge BM [V] 27 turbine OK
4 LS Shaft x BM [V] 28 Tower Top tors. [V]
5 LS Shaft y BM [V] 29 Tower Top x BM [V]
6 LS Shaft torque [V] 30 Tower Top y BM [V]
7 blade B pitch [deg] 31 Tower Base y BM [V]
8 Blade A root flap BM [V] 32 Tower Base x BM [V]
9 Blade A root edge BM [V] 33 Grid Connection
10 Blade C root flap BM [V] 34 Pile mwl torsion [V]
11 Blade C root edge BM [V] 35 Pile mwl x BM [V]
Duwind 2003.025 Page 55 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Channel Description Unit Channel Description Unit
12 blade A pitch [deg] 36 Pile mwl y BM [V]
13 blade C pitch [deg] 37 Pile depth 1 x BM [V]
14 power [kW] 38 Pile depth 1 y BM [V]
15 react. power [kW] 39 Pile depth 2 x BM [V]
16 Nac. orient. [deg] 40 Pile depth 2 y BM [V]
17 Nac. wind speed [m/s] 41 Water Elevation [m]
18 Nac. wind dir. [deg] 42 Pile ml x BM [V]
19 Nac. Acc. f-a [m/s2] 43 Pile ml y BM [V]
20 Nac. Acc. s-s centr. [m/s2] 44 Pile Found. 1 x BM [V]
21 Nac. Acc. s-s front [m/s2] 45 Pile Found. 1 y BM [V]
22 Rotor Az. [deg] 46 Pile Found. 2 torsion BM [V]
23 Rotor Speed [rpm] 47 Pile Found. 2 x BM [V]
24 Rotor pls / rev 48 Pile Found. 2 y BM [V]
The current profile measurements are recorded separately and given as ten-minute averages
only. The windspeed measurements from the main mast were not available when writing this
report, however the windspeed measurements made with the nacelle-anemometer were
included within the main group listed in Table V above. This data has to be used with care as
the measured windspeed is susceptible to turbulence around the nacelle and will measure the
wake windspeed rather than the free windspeed when the turbine is operating. Since the focus
of this work is hydrodynamic loading, an accurate windspeed recording was not vital. In fact,
only datasets where the turbine was off where used, since the wind loads on the rotor tended
to dominate the strain gauge measurements. With the turbine off, only the longer period,
turbulence induced wind loads were present, which could be filtered out or treated in a similar
manner as all the many other uncertainties. Further information on the instrumentation can be
found in [14].
Duwind 2003.025 Page 56 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
4 Hydrodynamic Loading – Examination of Theory
This chapter examines the application of hydrodynamic loading. The first section examines
slender support structures but in a brief manner, since the evaluation of theory against the
measured wave loads at Blyth are of greatest interest for this type of structure in this report.
The second section investigates the hydrodynamic loading on gravity base structures in much
greater detail, since no similar measurement-data is available to this project.
4.1 Slender Support Structures
In this section, the wave loads calculated using different kinematic models is examined. In
each case, Morison theory is used, since this is most appropriate for the geometry under
consideration, Figure 36. A matrix of waves was considered with the wave period ranging
between 5 s and 15 s at 1 s intervals and the wave height ranging between 1 m and 7 m at 1 m
intervals. In Figure 35, waves of 8 m height are also shown and it can be seen that they are
very close to the breaking limit and hence could not be calculated using stream function
theory. The structure was assumed to be a 4 m diameter monopile, with high roughness due
to age, hence CM=1.79 and CD=1.55, as recommended by DNV [25]. The water depth was
taken as 21 m.
Figure 35 shows the recommended wave theory for the above matrix of waves; it is based on
Figure 7 (page 5). It can be seen that for even the smallest 1 m waves, non-linear theory is
recommended, however the differences between the wave kinematic models for the smallest
waves are relatively minor.
1 10 3 0.01 0.1 1
1 10 5
1 10 4
1 10 3
0.01
0.1
d/gT2
H/g
T2
Airy LimitBreaking WaveAiryDean Stream 3rd OrderDean Stream 5th OrderDean Stream 7th OrderDean Stream 9th OrderDean Stream 11th OrderPost Breaking Wave Height
Figure 35: Recommended Wave Theory
Increasing wave height
Increasing wave period
Duwind 2003.025 Page 57 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Figure 36 plots the matrix of waves on the load-model recommendation chart, for example
given in [17]. It can be seen that for all waves, diffraction effects are important and that both
drag and inertia forces need to be considered. As a guide, the bottom-right corner is the
diffraction region, the left side suggests the use of the Morison formula, with the bottom-left
side being inertia dominated and drag being ever more important higher up in the same left-
hand side region. The drag dominated region is the top-left, which in this case is beyond the
wave breaking limit.
0.01 0.1 1 100.01
0.1
1
10
100
pi.D / wave-length
Keu
lega
n-C
arpe
nter
Num
ber
Airy LimitBreaking WaveBreaking WavesDiffractionInertia DominatedInertia & DragDrag Dominated
Figure 36: Recommended Wave Load Model
The following set of charts show the calculated maximum values of the wave-induced shear
force (for selected wave kinematic models) and overturning moment (for all the kinematic
models under consideration). Figure 37 shows the wave loads calculated from Airy linear
theory. Figure 38 shows the overturning moment only from the Wheeler and Chakrabarti
stretching correction models.
Increasing wave period
Increasing wave height
Duwind 2003.025 Page 58 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
GrDataT
6 8 10 12 14
5
10
1 1061 106
1 106
5 105
5 105
5 105
5 105
GrDataT
6 8 10 12 14
5
10 1 107
1 107
1 107
5 106
5 106
5 106
Figure 37: Airy Shear Force (left) and Overturning Moment (right)
GrDataT
6 8 10 12 14
5
10
2 1072 107
1.5 1071.5 107
1 107
1 107
1 107
5 106
5 106
5 106
GrData
T
6 8 10 12 14
5
101.5 107
1.5 107
1 107
1 107
1 107
5 106
5 106
5 106
Figure 38: Overturning Moment Wheeler (left) and Chakrabarti (right) Stretching
Figure 39 shows the calculated overturning moment for extrapolated and constant crest
corrections and Figure 40 for Stokes second and fifth order theory. It can be seen that for
high amplitude waves, both theories have difficulties in predicting the wave kinematics, with
the second order theory generating excessively high values and the fifth order algorithm used
here being unable to reach a solution. These waves are all beyond the region of applicability
for this theory, as given in Figure 7.
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
] Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Duwind 2003.025 Page 59 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
GrDataT
6 8 10 12 14
5
10
3 107
2.5 107
2 107
2 1071.5 107
1.5 1071 107
1 107
1 107
5 106
5 106
5 106
GrDataT
6 8 10 12 14
5
10
2.5 1072 107
2 1071.5 107
1.5 107
1 107
1 107
1 107
1 107
5 106
5 106
5 106
Figure 39: Overturning Moment Extrapolated (left) and Constant (right) Crest
GrDataT
6 8 10 12 14
5
10
1 108
5 1075 107
GrDataT
6 8 10 12 14
5
10 1 106
1 1061 106
5 105
5 105
5 105
5 105
5 105
5 105
Figure 40: Overturning Moment Stokes 2nd11 (left) and 5th(right)
Figure 41 shows the shear force and overturning moment wave loads calculated using stream
function theory. It can be seen that generally the use of this theory gives higher values for the
loads.
11 The Stokes 2nd chart uses a different scale to the other charts
Wave Period [s]
Wav
e H
eigh
t [m
] Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves No Solution
Found
Duwind 2003.025 Page 60 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
GrDataT
6 8 10 12 14
5
10
2 1062 106
1.5 1061.5 106
1 1061 106
1 106
5 105
5 105
5 105
5 105
GrDataT
6 8 10 12 14
5
10
4.5 1074 1074 10
7
3.5 107
3 1073 10
72.5 10
7
2 1072 107
1.5 107
1.5 107
1 1071 107
1 107
5 106
5 1065 106
Figure 41: Stream Function Shear Force (left) and Overturning Moment (right)
The fact that when using the stream function, the wave loads seem relatively independent on
the wave period may be related to the fact that the crest-section of the wave profile does not
change shape much, with just the relatively flat trough expanding as the wave length
increases. This can be observed in Figure 42, where it can be seen that the slope of the waves
increases with a shorter wave period for the linear theory, while it is similar for the same three
cases shown for the stream function. Note that for the initial determination of the kinematic
distribution within the wave, the linear theory and hence also its derivatives assume a flat
surface profile. The correction formulae are only able to extrapolate kinematics from this
flat-surface field.
4 2 0 2 4
5
0
5
8s period11s period14s periodDean Stream 3rd Order
Time [s]
Surf
ace
Ele
vatio
n [m
]
4 2 0 2 4
0
5
8s period11s period14s periodDean Stream 3rd Order
Time [s]
Surf
ace
Ele
vatio
n [m
]
Figure 42: Evolution of Surface Profile for Airy (left) and Stream Function (right)
The following set of charts show the percentage difference of the above theories against the
stream function theory, which is assumed to be accurate within its intrinsic limitations
(i.e. models regular waves only). It can be seen that all theories are reasonable accurate for
Wave Period [s]
Wav
e H
eigh
t [m
] Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Duwind 2003.025 Page 61 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
small wave heights of up to perhaps 2 m but beyond the errors increase. It should be noted
that these charts reflect a particular case and that other situations will result in different
conclusions however the fact that linear or low-order wave theories under predict the wave
kinematics and hence the resulting wave loads will be true for all slender structures in the
shallow waters inevitably selected for the construction of offshore windfarms.
For linear theory, the overturning moment is less than 25% of the true value determined using
stream function theory, Figure 43, while the Stokes second order generates conservative
values. This chart is also of interest for stochastic sea simulations, since we can conclude that
both first and second-order random seas will contain large inaccuracies for the higher waves
for the conditions of interest to the offshore windenergy industry.
Gr ZeroT
6 8 10 12 14
5
10 0
5
5
5
10
10
10
10
15
15
15
20
20
20
25
25
25
30
30
35
35
35
40
40
40
45
45
50
50
55
55
55
60
60
60
65
65
65
70
70 75
Gr Zero
T
6 8 10 12 14
5
10
250200150
100100
50
50
50
50
Figure 43: Error in Overturning Moment: Airy (left) & Stokes 2nd Order (right)
The following charts show the errors associated with the use of Wheeler and Chakrabarti
stretching, Figure 44, and extrapolated and constant crest, Figure 45. Although the errors are
reduced compared with linear theory (Figure 43, left), they remain considerable and
excessive. It is also of interest that the extrapolated crest theory (Figure 45, left) can
overestimate the wave loads in some cases. This problem has been noted in literature and is
why this theory is only rarely used; it is likely that this decision reflects the errors encountered
in deeper waters since for this geometry, the theory is generally more accurate than any of the
other linear-derivative theories.
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
] Breaking
Waves
Duwind 2003.025 Page 62 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Gr ZeroT
6 8 10 12 14
5
10 0
5
5
5
5
10
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
35
35
35
40
40
40
45
45
45
50
50
50
55
55
Gr ZeroT
6 8 10 12 14
5
10 0
5
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
35
35
35
40
40
40
45
45
45
50
50
50
55
55 55
60
60
Figure 44: Error in Overturning Moment: Wheeler (left) & Chakrabarti (right)
Gr ZeroT
6 8 10 12 14
5
10 105
0
0
0
5
5
10
10
15
15
15
20
20
20
25
25
25
30
30
30
35
35
40
40
45
45 50
Gr ZeroT
6 8 10 12 14
5
10
0
0
0
5
5
5
10
10
10
15
15
20
20
20
25
25
25
30
30
30
35
35
35
40
40
40
45
45 50 55
Figure 45: Error in Overturning Moment: Extrapolated (left) & Constant (right) Crest
If we examine the case of shallower waters with 6 m depth and a matrix of waves with wave
heights up to 4 m and wave periods of 5 to 15 s, the position of the plots on the wave theory
selection chart shifts to the left, Figure 46, with higher order wave theories recommended
compared with the previous case, see Figure 35. It can be seen in Figure 46 that the most
extreme waves require eleventh order theory.
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
] Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Wave Period [s]
Wav
e H
eigh
t [m
]
Breaking
Waves
Duwind 2003.025 Page 63 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
1 10 3 0.01 0.1 1
1 10 5
1 104
1 10 3
0.01
0.1
d/gT2
H/g
T2
Airy LimitBreaking WaveAiryDean Stream 3rd OrderDean Stream 5th OrderDean Stream 7th OrderDean Stream 9th OrderDean Stream 11th OrderPost Breaking Wave Height
Figure 46: Recommended Wave Theory
Turning to the selection of wave load model, the position of the plots on the selection chart,
Figure 47, is slightly shifted to the right compared with the previous case, Figure 36, but the
breaking wave limit line is also moved, reflecting the different breaking conditions in shallow
water (note that this line is permanent in the above wave theory recommendation chart). As
before, diffraction is not considered important (reflecting the relatively small diameter of the
monopile) and the calculation of both inertia and drag forces is shown to be necessary.
0.01 0.1 1 100.01
0.1
1
10
pi.D / wave-length
Keu
lega
n-C
arpe
nter
Num
ber
Airy LimitBreaking WaveBreaking WavesDiffractionInertia DominatedInertia & DragDrag Dominated
Figure 47: Recommended Wave Load Model
The following set of charts shows the shows the calculated overturning moment at the
monopile base calculated using Airy (Figure 48 left chart), Wheeler Stretching (Figure 48
right chart), Chakrabarti Stretching (Figure 49, left chart), Extrapolated Crest (Figure 49, right
Increasing wave height
Increasing wave period
Increasing wave period
Increasing wave height
Duwind 2003.025 Page 64 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
chart), Constant Crest (Figure 50, left chart) and Stream Function (Figure 50, right chart)
theories.
T
6 8 10 12 14
1
2
3
4
7 105
6 105
5 105
5 105
4 105
4 105
4 105
3 105
3 105
3 1052 105
2 105
GrData
T
6 8 10 12 14
1
2
3
4
7 105
6 105
6 105
5 105
5 105
4 105
4 105
4 105
3 105
3 105
3 105
2 105
2 105
Figure 48: Calculated Overturning Moment Airy (left) & Wheeler (right)
Gr T
6 8 10 12 14
1
2
3
4
7 105
6 105
6 105
5 105
5 105
4 105
4 105
4 105
3 105
3 105
3 105
2 105
2 105
Gr T
6 8 10 12 14
1
2
3
4
8 105
7 1056 10
5
6 105
5 105
5 105
5 105 4 105
4 105
3 105
3 105
3 1052 105
2 105
Figure 49: Calculated OTM Chakrabarti (left) Extrapolated Crest (right)
Wave Period [s]
Wav
e H
eigh
t [m
]
Wave Period [s] W
ave
Hei
ght [
m]
Wave Period [s]
Wav
e H
eigh
t [m
]
Wave Period [s]
Wav
e H
eigh
t [m
]
Duwind 2003.025 Page 65 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
GrDataT
6 8 10 12 14
1
2
3
4
8 105
7 1056 10
5
6 105
5 105
5 105
5 105
4 105
4 105
4 105
3 105
3 105
3 105
2 105
2 105
GrDataT
6 8 10 12 14
1
2
3
4
1.5 106
1 1061 106
5 1055 1055 10
5
Figure 50: Calculated OTM Constant Crest (left) & Stream Function (right)
Making the assumption again that the solution derived using stream function is the most
accurate, we can evaluate the other wave theories against this, as shown in Figure 51 for Airy
and Wheeler stretching and in Figure 52 for the extrapolated and constant crest models. The
errors are larger than in the corresponding charts for the deeper water depth above (Figure 43
to Figure 45 starting on page 61) and in the extreme, Airy predicts wave loads that are only
one-fifth of the correct solution (i.e. a 15 s 4 m wave in Figure 51). As before, the accuracy
of the calculated loads using linear theory for even small waves (which dominate the fatigue
damage) is poor.
T
6 8 10 12 14
1
2
3
4
5 10 15
20
20
25
25
30
30
30
35
35
35
40
40
40
45
45
45
45
50
50
50
55
55
55
60
60
65
65
70
70 75
T
6 8 10 12 14
1
2
3
4
5 10
15
15
20
20
25
25
25
30
30
30
35
35
35
35
40
40
40
45
45
45
50
50
50
55
55
55
60
60 65
Figure 51: Error in Calculated OTM Airy (left) & Wheeler (right)
Wave Period [s]
Wav
e H
eigh
t [m
]
Wave Period [s]
Wav
e H
eigh
t [m
]
Wave Period [s]
Wav
e H
eigh
t [m
]
Wave Period [s]
Wav
e H
eigh
t [m
]
Duwind 2003.025 Page 66 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
T
6 8 10 12 14
1
2
3
4
5 10
15
15
20
20
25
25
25
30
30
30
35
35
35
40
40
40
45
45
45
50
50
50
55
55
60
60 65
T
6 8 10 12 14
1
2
3
4
5 10
15
15
20
20
25
25
25
30
30
30
30
35
35
35
40
40
40
45
45
45
50
50
50
55
55
60
60 65
Figure 52: Error in Calculated OTM Extrapolated (left) & Constant (right) Crest
The effect of this on the fatigue damage calculated for the two situations above is examined in
a preliminary manner below.
FatDamT
0 5 10 150
2
4
6
0.4
0.4
0.35
0.35
0.3
0.3
0.3
0.250.2
0.2
0.2
0.15
0.15
0.15
0.15
0.1
0.1
0.1
0.1
0.05
0.05
0.05
0.05
0.05
FatDam nT
0 5 10 150
2
4
6
0.80.80.7
0.65
0.65
0.6
0.6
0.550.5
0.5
0.50.4
0.4
0.4
0.4
0.4
0.4
0.35
0.3
0.3
0.3
0.3
0.25
0.25
0.2
0.2
0.2
0.2
0.2
0.15
0.15
0.15
0.15
0.1
0.1
0.10.1
0.1
0.1 0.05
0.05
0.050.05
0.05
0.05
Figure 53: Total Fatigue Damage using Wheeler (left) and Stream Function (right)
Turning to the second shallow water (6 m depth) case first, if the total fatigue damage is
summed, use of the Wheeler wave model results in a lifetime fatigue of 43% of the more
accurate stream function value. Use of linear theory gives a result of 35%. However, the
corresponding figures for the deeper water (21 m depth) case is 97% and 95%, suggesting that
selection of an appropriate wave model for fatigue calculations is only important for the
shallowest waters.
Hence, if for fatigue calculations, the hydrodynamic loads on the support structure at Blyth
where calculated using stochastic linear-extrapolated wave theory (Wheeler for example), the
Wave Period [s]
Wav
e H
eigh
t [m
]
Wave Period [s]
Wav
e H
eigh
t [m
]
Wave Period [s]
Wav
e H
eigh
t [m
]
Wave Period [s]
Wav
e H
eigh
t [m
]
Duwind 2003.025 Page 67 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
effect of these loads on the total fatigue damage would be underestimated by a factor of
around 2.3. This does not matter if the fatigue damage is dominated by rotor loads, however
it should also be noted that this error only reflect the inaccuracy of not using stream function
theory and not the inaccuracies in ignoring surface effects for example.
4.1.1 Structural Dynamics
Slender structures, such as monopiles, are compliant and hence move dynamically, either to
shed part of the wave loads or to amplify them, depending on how close are the wave and
structural resonant frequencies. This is shown as the peak in Figure 54. Thre results from
three models are shown here: (i) the commercial package SESAM (Det Norske Veritas),
(ii) DUWECS (Section Windenergy’s in-house integrated wind turbine model) and
(iii) Bladed for Windows (Garrad Hassan's commercial wind turbine modelling suite). The
response spectra due to a severe sea state is shown in Figure 55
0.E+00
1.E+07
2.E+07
3.E+07
4.E+07
5.E+07
6.E+07
7.E+07
8.E+07
0 0.5 1 1.5 2 2.5 3Frequency (rad/s)
TRF
(Nm
/m) DUWECS
SESAMBladed
Note Bladed results are for Wheeler stretching
Figure 54: Response Transfer Function
Duwind 2003.025 Page 68 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0.E+00
5.E+13
1.E+14
2.E+14
2.E+14
3.E+14
3.E+14
4.E+14
0 0.5 1 1.5 2 2.5 3Frequency (rad/s)
SM
M (
N2m
2s
/ rad
)
DUWECSSESAMBladed
Figure 55: Response Spectra for a Severe Sea State12
Examining the response to an regular wave train in the time domain, Figure 26 (page 43)
shows a comparison of the imposed wave overturning moment at the mud-line and the
resulting moment taking into account the structural dynamics, calculated using BLADED
[11]. For this wave period, the range is similar, however the compliance of the tower results
in additional dynamic response at higher frequencies.
However, with closer investigation, it is clear that this approach suffers important
weaknesses, with the impossibilities of modelling non-linear waves within stochastic seas.
The alternative of using regular waves in the analysis also suffers from important weakness in
that the calculated response depends on the ratio between the wave’s and the structure’s
natural response periods. The peaked surface elevation of the non-linear waves shown in
Figure 4 (page 18) results in harmonic loads at integer multiples of the wave frequency, a
feature that is less pronounced, if present at all, in stochastic linear waves. When harmonics of
the wave frequency coincide with the structural natural frequency, resonance of the structure
results in amplification of the response, Figure 56. The presented amplification is valid for an
infinite sequence of regular and identical periodic waves. For a single incoming wave, the
relation between the phase of the wave and the initial structural motion determines the
effective amplification, as is shown below in Figure 112 (page 94) and Figure 113 (page 95)
from the measurements taken at the Blyth turbine.
12 Hs = 6.25 m and Tz = 7.5
Duwind 2003.025 Page 69 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0.5
0.75
1
1.25
1.5
1.75
2
2.25
6 6.5 7 7.5 8 8.5 9Wave Period [s]
Dyn
amic
Am
plifi
catio
nof
the
OTM
ExtremeWaves
LowAmplitudeWaves
Figure 56: Dynamic Amplification of Non-Linear Waves
Figure 56 shows that there is significant dynamic amplification at wave periods of around
three (� 6.4 s) and four (� 8.5 s) times the tower’s first natural period and that this is only
apparent for the extreme wave (the undulations in the small wave curve are due to insufficient
run times in the simulations) and also that no cancellation can be identified at the mid-natural
period point (i.e. 3 ½ × natural period) but instead being present immediately prior to the
multiple value (i.e. 2.9 and 3.9 × natural period). Examining the dynamic amplification in
terms of kinematics, if the ratio is an integer, the structure will be moving in phase with the
wave when each rising crest reaches the monopile and severe dynamic amplifications will be
stimulated, while if it is an integer plus a half, it is expected that the structure would be
moving into the wave when it impacts.
Relatively little focus has been made on the structural dynamics modelling within this report
because of the perceived need to obtain a good understanding of the underlying wave loads
first and of the inherent difficulties of obtaining a good match between measurements and the
models without resorting to flattering logarithmic scales. The reason for this difficulty is
almost certainly one of the core problems identified: the inability that the available structural
dynamics models have of modelling stochastic non-linear seas.
Recently, the deepwater offshore design industry has turned to probability-analysis methods,
where instead of matching individual waves to their resultant structural response, the
probability distribution of the waves is matched to the probability distribution of the response.
This approach has become popular because of its relative accuracy, albeit in deeper waters
where non-linear waves are less of a problem. Cheng has applied this approach to the design
process for offshore windturbines (see [20] for a detailed description) but the non-linear seas
at Blyth makes the matching process much more difficult, as described below in section 5.4
Probabilistic Approach [40].
Duwind 2003.025 Page 70 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
4.1.2 Current
The current data was gathered in a different manner to the wave data in that it was recorded as
10 minute averages and was measured a short distance away from the windturbine. When
analysing the data from measurement campaigns taken during the storm event, the possibility
of including the effect of the current on the hydrodynamic loadings was investigated, however
as Figure 57 shows, the current speed varied greatly with time during this storm making it
impossible to estimate what the current would be at any single wave.
0
0.1
0.2
0.3
0.4
0.5
0:00 3:00 6:00 9:00 12:00 15:00 18:00 21:00 0:009th November 2001
Cu
rren
t sp
eed
(m
/sec
)
Figure 57: Depth Averaged Measured Current
Hence, the investigation into the affect of the current on the hydrodynamic loads had to focus
on what magnitude could potentially be expected. Figure 58 shows the time trace for three
case of current (inline, none and counter) on an extreme wave at Blyth. It can be seen that the
extreme loads are shifted significantly but that the range within each wave does not alter as
much. This is also shown in Figure 59, where, for example, it can be seen that for current up
to 0.5 m/s (the highest that was recorded), the maximum values vary between +12% and –8%,
the minimum values vary between –10% and + 8% (depending on the current direction) while
the range varies between +3% to –2%. Since the strain gauge calibrations were not available
during the main work within this report, only the measured stress range could be considered
anyway (i.e. any peaks were identified with respect to a local mean value, rather than an
absolute zero value). Hence currents of even as high as 0.5 m/s would only cause the
measured strains to vary by up to 3%, which is significantly less than the scatter due to other
wave phenomena (i.e. non-linear, stochastic and breaking waves). Hence the exclusion of the
current in the analysis of the data is acceptable.
Duwind 2003.025 Page 71 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0 2 4 6 8 10 121.2 10
6
8 105
4 105
0
4 105
8 105
1.2 106
1.6 106
2 106
inline currentno current
Figure 58: Effect of Current on Profile
Current Min Current Max
1 0.5 0 0.5 110
7.5
5
2.5
0
2.5
5
7.5
10
maximum positive valuerange (/2)maximum negative value (-ve)
Current (+ve = wave direction) [m/s]
Cha
nge
in M
udlin
e B
endi
ng M
omen
t [%
]
Figure 59: Effect of Current on Max
Values
It is generally known that the current velocity will vary with depth but the current direction
can do so also. Figure 60 shows the directional distribution at the surface and near the sea
bed during a complete month (November ’01) and it can be seen that there is a small
difference. The surface current is affected by wind drag forces whilst the seabed current will
be influenced by geometry to a greater extent.
0 45 90 135 180 225 270 315 3600
0.2
0.4
0.6
at surfacenear sea bed
Current Direciton (from) [deg]
Prob
abili
ty D
isitr
butio
n
T
Figure 60: Directional Distribution of Current
4.2 Compact Support Structures
Gravity base foundations have been a popular foundation concept for offshore wind turbines
in sheltered and ice-infested waters. As wind turbines increase in size and move to deeper and
more exposed sites, the hydrodynamic loading on the large gravity base structures becomes an
ever more important contribution. The determination of hydrodynamic loads on a gravity base
structure is more complicated than on slender monopiles, due to the irregular geometry and
the complicated effect that the structure has on the wave field (termed diffraction).
Furthermore, the design of gravity base structures with a large contribution of hydrodynamic
Duwind 2003.025 Page 72 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
loading will be more susceptible to inaccuracies in the determination of the hydrodynamic
loading. The hydrodynamic loading on the gravity base structure itself is part of the design
loop with respect to its load bearing function. On the other hand, the geo-technical design of a
pile foundation can be performed directly using the hydrodynamic loading on the structure
above the seabed as a priori knowledge [109].
The traditional approach to wave load calculation for gravity base structures in the offshore
industry has been to use diffraction analysis. In the deep waters, in which such structures are
located, the wave height is relatively low compared with the water depth. Therefore the use
of linear wave theory (also called Airy theory), upon which the most commonly implemented
form of diffraction theory is based, is applicable. On the other hand, offshore windfarms are
located in much shallower seas, where highly non-linear waves are a more frequent
phenomenon. Breaking waves also become more frequent and the balance of the type of
breaking shifts from the relatively benign spilling breaker (which can be modelled using the
appropriate non-breaking method) to include cases of the more severe plunging breakers
(which impose significantly higher loads than the non-breaking case). A comprehensive
evaluation of the impacts-forces due to breaking waves is beyond the scope of this work,
however the effect of non-linearities in the wave field (i.e. the asymmetry in the vertical
direction) and the structural geometry (i.e. the departure from simple cylindrical columns) is
examined. Of particular interest is the effect of the ice-cone, which is located at the water
surface, where wave kinematics and hence wave load effects are greatest. The hydrodynamic
loads are also assessed in the perspective of all loads on the offshore wind-turbine, since
although the non-linearities in shallow waters are higher, the hydrodynamic loads themselves
will be lower, because of the shallowness.
As set out previously, the modelling of wave loadings can be divided into three stages:
(i) selection of the wave model, (ii) selection of the load-calculation model and
(iii) examining the response of the structure (i.e. structural dynamics). For a rigid structure,
such as a GBS, the small magnitude of the structural dynamics do not affect the wave loads
and hence this stage can be ignored. The choice of wave models for engineering application
has converged on to two main theories: (i) the linear Airy model, with or without
modifications and (ii) the non-linear stream function model. Regarding the load calculation
method: three approaches are applicable according to the type of structure: (i) Morrison
theory, a slender body theory and (ii) Diffraction theory and (iii) Froude-Krylov method, both
for massive structures, such as the gravity base structures (GBS) being considered within this
report. Of the second and third methods, both have particular weaknesses, in the case of
Duwind 2003.025 Page 73 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
diffraction it is in the application of non-linear wave models and in the case of Froude-Krylov
it is in the modelling of the effect of the structure on the flow field (i.e. diffraction). In the
longer term, computational fluid dynamics (CFD) promises the benefits of being able to
model all aspects of interest, though the current penalties of complexity for the user and
computational costs necessary will need to be reduced.
Figure 61 and Figure 62 show the two candidate GBS support structures analysed in this
section; the first is suitable for shallow waters and similar to that used at Middelgrunden,
while the second is the optimised design from the Opti-OWECS research project [28], which
assumed a deeper water depth of 15m. Note also that the shallow water design includes an
ice-cone; ice is a frequent phenomenon in the Baltic Sea but does not occur in the North Sea
at the latitudes of interest. Table VI gives the design conditions used in this section; the
values for the deep-water concept are again taken from the Opti-OWECS study [28]. Of the
following example calculations in this section, most are for the deeper-water concept, to tie in
with the subsequent geotechnical analysis; exceptions are when the analysis of the shallow-
water concept leads to different conclusions, for example due to the presence of the ice-cone.
Figure 61: Shallow Water (6m) Gravity
Base Support Structure
Figure 62: Deep Water (15m) Gravity
Base Support Structure
Table VI: Design Conditions
Parameter Shallow Water Concept Deep Water Concept
Water Depth 6 m 15 m
Return Period 50 years
Height 4 m 9.7 m Extreme
Wave Period 8 s 11 s
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4.2.1 Determining the Wave Loads for GBS Structures
Accepting that the diffraction and Froude-Krylov methods offer the most appropriate paths
for calculating wave loads on GBS structures, the relative importance of the different sources
of error needs to be evaluated. Starting with diffraction, Table I (page 29) shows that this is
unable to include the effect of (i) viscous effects (transverse and lateral drag), (ii) surface
effects (1D and 3D) and (iii) non-linear waves. Likewise, the Froude-Krylov method is also
unable to include transverse drag, (i) viscous effects plus (iv) diffraction effects.
In the early stages of the design process, it may be necessary to use the Morison method to
determine the wave loads, for example in the procedures described in Appendix
C: Preliminary Design of Gravity Base Structures. It is not viable to integrate diffraction
analysis into this procedure because of its complexity and time-consuming nature (ignoring
for the moment that fact that it contains it’s own inherent inaccuracies), hence the Morison
formula has been used.
In the following sub-sections, an attempt is made to identify and quantify the uncertainties in
the procedures for the two types of GBS structures selected above and identified in Figure 61
and Figure 62 (page 73).
Table I (page 29) identifies the main weaknesses of the three models, some of which are
strongly related to each other. For the diffraction model, these are that it:
• Does not calculate drag loads (transverse or lateral)
• Ignores surface effects (i.e. the effect of the sea surface rising and falling around the
column) hence also the full effects of complex geometries at the water surface, see
Figure 63
• Cannot model non-linear waves
Figure 63 illustrates potential impact of the surface effects for a design similar to that used at
the Middelgrunden windfarm. The wave is as defined in Table VI (page 73), a 8s 4m high
wave in 6m water depth. The wavelength is 57.49 m, around 20% of the diameter of the base
slab. It can be seen that the wetted surface is significantly greater on the left than on the right,
which will lead to a higher horizontal force than would be predicted by linear theory.
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D w
15 10 5 0 5 10 150
5
10
15Profile of Support Structure with Wave
in wave direction [m]
in v
ertic
al d
irect
ion
[m]
Figure 63: Profile of a Gravity base Structure (design similar to Middelgrunden)
Considering the Froude-Krylov approach, the main weaknesses identified are:
• Does not calculate drag loads (transverse or lateral)
• Does not calculate diffraction effects
In addition, utilising non-linear wave theories is very demanding on computational resources,
that being a disadvantage in industry rather than in research.
Finally turning to the Morison method, the major deficiencies are that it:
• Ignores surface effects (i.e. the effect of the sea surface rising and falling around the
column)
• Ignores three dimensional effects of loads on the column, (i.e. more complicated
variations of the wave field through the column’s volume; only the first differential is
calculated at the centre-line is considered)
• and two dimensional effects of end-loads (i.e. i.e. more complicated variations of the
wave field over the base-slab surface; again only the first differential is calculated at
the centre-line is considered)
• cannot model complicated geometries.
In addition, in its usual form, it excludes pressure effects, such as those acting on the ends
In the following sections, the importance of these weaknesses are examined, the approach
being as follows. The effect of:
• drag loads can be examined using Morison method by switching the drag coefficient
off
• surface effects can be examined using the Morison method by comparing simple
Airy and Wheeler models
Duwind 2003.025 Page 76 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
• non-linear wave theory can be examined using either Froude-Krylov or Morison
methods
• diffraction can be examined by comparing the diffraction and Froude-Krylov
approaches, with a simple coefficient used within the Froude-Krylov model
The importance of complex geometry (i.e. the ice-cone) is difficult to evaluate in a simple
manner as it has several impacts, which couple with each other.
The importance of the omission of lateral drag loads on the base slab is examined using first
principals.
4.2.2 Evaluating the Uncertainties
4.2.2.1 Effect of Drag
It is possible to examine the effect of drag (i) by examining the Keulegan-Carpenter number,
K, see the List of Symbols (page v) and (ii) utilising the Morison equation.
The Keulegan-Carpenter number depends on the peak fluid velocity, U, the wave period, T,
and the structure diameter, D and, for this deepwater GBS structure under this design wave,
will be approximately 2.5 and 15 for the base and tower respectively. Values of K below 5
mean that the flow is inertia dominated while above 25 indicates flow is drag dominated;
hence drag forces may have some importance for the load calculations on the tower but can be
ignored for the base. Shear loads occur predominantly on the base-slab, hence it can be
concluded that drag can be ignored. The same conclusion applies to the drag force on the
upper surface of the base, which can be shown to be insignificant [65].
4.2.2.2 Effect of Diffraction (Froude-Krylov)
Diffraction is the effect that the structure has on the fluid flow field. It involves the
calculation of the added mass, which can be envisaged as the mass of the fluid within the
surrounding flow field that is entrained to move or remain stationary with the structure. This
is illustrated in Figure 64. From analytical theory, it can be shown that the added mass of a
circular cylinder lying perpendicular to the flow field is equal to the mass of the fluid
displaced by the element, Figure 64 (left), hence CM has a value of 2 (i.e. one for the fluid
displaced by the element and one for the added mass). Note that there is a no-flow boundary
condition at the symmetrical horizontal axis, hence the solution is also applicable for a half-
cylinder lying on the sea-bottom.
Duwind 2003.025 Page 77 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Element
Added Mass
Element
Added Mass
Figure 64: Illustration of Added Mass
Similarly, the added mass of a sphere is 0.5; hence the CM value used for the ice-cones is 1.5.
For a rectangular cross-section, the added mass depends on the ratio of the width (i.e. the
dimensions projected into the flow) to the depth (i.e. the dimension in parallel with the flow).
Regarding the dimension projected into the flow, the significant one is the shorter of the two,
hence for a monopile, it is the diameter rather than the length. (This can be thought of as
around which side will the flow be diverted.) The base slab of a GBS can be modelled as
rectangle, with the important dimension being the slab height, since the wave particles will
generally flow over the slab rather than around it. Figure 64 (right) illustrates this, with the
slab being modelled as a rectangle of twice the height of the slab placed symmetrically about
the ground level. The added mass will be less than illustrated and the following sections
investigate though comparison of results between Froude-Krylov and diffraction, what the
value should be. For simple geometries, such as the deepwater structure, the relationship
between the geometry and the added mass is fairly straight-forward and it is possible to
develop a simple function relating the added inertia coefficient, for the base slab, to the slab
height, B and diameter, D, such as Eq. 41. In the following section, an attempt is made to
find a suitable value for k.
DB
kCm += 1 Eq. 41
Inclined GBS Base Slab
The base slab used in the Middelgrunden configuration has a sloping upper surface, Figure 71
(page 79), which makes the determination of the diffraction effects more difficult. The height
of this sloping section is equal to the height of the vertical part of the slab, while in
comparison, the upper surface of the slab used in the deepwater configuration is level, Figure
72 (page 79). We can try to guess why this shape was chosen from the results of the analyses
that follow, as it seems that this form reduces both horizontal and vertical loads.
Using a small wave height, the wave forces calculated using diffraction and Froude-Krylov
(with added mass coefficients of unity and linear wave theory) methods are compared for a
matrix of wave periods (5s to 15s) and water depths (5m to 9m). The calculated surge and
heave forces are shown in Figure 65 and Figure 66 respectively.
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0
20
40
60
80
100
120
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
5m Diffraction 6m Diffraction 7m Diffraction
9m Diffraction 5m Froude-Krylov 6m Froude-Krylov
7m Froude-Krylov 9m Froude-Krylov Figure 65: Surge Force
0
200
400
600
800
1000
1200
1400
5 7 9 11 13 15Wave Period [s]
Hea
ve F
orce
[kN
]
5m Diffraction 6m Diffraction 7m Diffraction
9m Diffraction 5m Froude-Krylov 6m Froude-Krylov
7m Froude-Krylov 9m Froude-Krylov Figure 66: Heave Force
It can be seen that for this case that using the Froude-Krylov method with no added mass
gives a very good estimation of the surge but a less close prediction of the heave forces; in
other words there is very little net diffraction in the surge direction and that the pressure
forces on the upper surface of the slab are reduced. Figure 67 shows the ratios of the
calculated forces for the two methods, which would be the added mass coefficients
recommended for use with Froude-Krylov, which are 1.025 and 0.85 for the horizontal and
vertical directions respectively (read at the design wave period of 8s, see Table VI, page 73).
We know what the diffraction effects will be for the vertical column (CM = 2) and the ice-
cone (CM = 1.5), however any interaction between the components is ignored here. Figure 68
shows the calculated overturning moment for the same cases; the difference varies between
5% and 10%.
0.80
0.85
0.90
0.95
1.00
1.05
1.10
5 7 9 11 13 15Wave Period [s]
Sur
ge &
Hea
ve C
oeff
icie
nts
5m Surge 6m Surge 7m Surge 9m Surge
5m Heave 6m Heave 7m Heave 9m Heave
Figure 67: Coefficients
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
5 7 9 11 13 15Wave Period [s]
OTM
[MN
m]
5m Diffraction 6m Diffraction 7m Diffraction
9m Diffraction 5m Froude-Krylov 6m Froude-Krylov
7m Froude-Krylov 9m Froude-Krylov Figure 68: Overturning Moment
However, if the above recommended values (CM(H) = 1.025 and CM(V) = 0.85) are applied
uniformly across the base slab, the surge, Figure 69, and heave forces may be correctly
predicated but the error in overturning moment actually increases, Figure 70. This is not
Duwind 2003.025 Page 79 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
necessarily critical though, since the overturning moment is not required in the design
procedure used here.
0
20
40
60
80
100
120
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
5m Diffraction 6m Diffraction 7m Diffraction
9m Diffraction 5m Froude-Krylov 6m Froude-Krylov
7m Froude-Krylov 9m Froude-Krylov Figure 69: Surge Force (revised CM)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
5 7 9 11 13 15Wave Period [s]
OTM
[MN
m]
5m Diffraction 6m Diffraction 7m Diffraction
9m Diffraction 5m Froude-Krylov 6m Froude-Krylov
7m Froude-Krylov 9m Froude-Krylov Figure 70: OTM (revised CM)
Alternatively, two values for the inertia coefficient should be used, one at the lower vertical
section of the slab and a second value at the higher inclined section, however the associated
added complexity made this not worthwhile pursuing here.
0 2 4 6
0
50
0.2
0.4
0.6
0.8
1
Figure 71: Inclined Base Slab
0 5 10
10
0
100
1
2
3
4
Figure 72: Rectangular Base Slab
Rectangular GBS Base Slab
Two configurations of the deepwater GBS base slab were used, with 2m and 4m, Figure 72,
slab heights. Starting with the 2m base slab, Figure 73 and Figure 74 compare the surge and
heave forces respectively for a matrix of wave periods and water depths. As previously, it can
be seen the heave force predicted by the Froude-Krylov method is very close, since there is
little diffraction in that direction but that in the surge direction, some correction is necessary.
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0
100
200
300
400
500
600
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 73: Surge Force (2m)
0500
1000150020002500
300035004000
45005000
5 7 9 11 13 15Wave Period [s]
Hea
ve F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 74: Heave Force (2m)
The recommended inertia coefficients are shown in Figure 75. It can be seen that there is
some variation with wave period and to a lesser extent with water depth. To avoid the
complexity of a wave-period and water-depth coefficient, the values at the design wave period
(11s, see Table VI, page 73) could be chosen, since the water depth effects are smaller. This
would suggest a value of 1.2 for the inertia coefficient, lower than the value for the longest
waves. Figure 76 shows the calculated OTM, where the error reaches around 10%.
0.95
1.00
1.05
1.10
1.15
1.20
1.25
5 7 9 11 13 15Wave Period [s]
Iner
tia C
oeff
icie
nts
12m Surge 15m Surge 18m Surge 21m Surge
12m Heave 15m Heave 18m Heave 21m Heave
Figure 75: Inertia Coefficients (2m)
0
2
4
6
8
10
12
5 7 9 11 13 15Wave Period [s]
OTM
[MN
m]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 76: OTM (2m)
The next set of four charts shown the same information for the deeper 4m base slab, with
Figure 77 and Figure 78 comparing the surge and heave forces for diffraction and Froude-
Krylov as before.
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0
200
400
600
800
1000
1200
1400
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 77: Surge Force (4m)
0500
1000150020002500
300035004000
45005000
5 7 9 11 13 15Wave Period [s]
Hea
ve F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 78: Heave Force
Following the same process as above, Figure 79 shows the calculated inertia coefficients for
use with the Froude-Krylov theory with the recommended value for the design wave
condition being 1.35. The error in the calculated OTM is higher than for the shallower base
slab, at up to around 20%, Figure 80.
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
5 7 9 11 13 15Wave Period [s]
Iner
tia C
oeff
icie
nts
12m Surge 15m Surge 18m Surge 21m Surge
12m Heave 15m Heave 18m Heave 21m Heave
Figure 79: Inertia Coefficients
0
2
4
6
8
10
12
5 7 9 11 13 15Wave Period [s]
OTM
[MN
m]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 80: OTM
The recommended force coefficients are collated in to the following Table VII.
Table VII: base Slab Force Coefficients for Froude-Krylov Method
Slab Dimensions Force Coefficients
Height Diameter Horizontal Vertical
Inclined Slab 0.5m + 0.5m 12.5m 1.025 0.85
2m Slab 2m 25m 1.2 1
4m Slab 4m 25m 1.35 1
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Using the simplified rule for selection of the force coefficients recommended below, Table X
(page 92), the wave loads on the complete GBS structure using diffraction and Froude-Krylov
can be compared. Here on the deepwater concept is considered however this is equally
applicable to the Middelgrunden type structure.
0 5 10
10
0
100
10
20
30
Figure 81: Deepwater GBS with Ice-Cone
0 5 10
10
0
100
10
20
30
Figure 82: Deepwater GBS w/o Ice-Cone
Looking at the structure with an ice-cone, Figure 81 (shows mesh used in panel method
analysis; note the drawing includes some additional superfluous lines), Figure 83 and Figure
84 show that the surge and heave forces are generally similar, with an average error of 5%
and 9% respectively. Base slab force coefficients of 1.33 and 1 were used for the horizontal
and vertical directions.
0
200
400
600
800
1000
1200
1400
1600
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 83: Comparison of Surge Force
0500
1000150020002500
300035004000
45005000
5 7 9 11 13 15Wave Period [s]
Hea
ve F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 84: Comparison of Heave Force
Figure 85 compares the calculated loads for each case, where it can be seen that with the
selected coefficients, the longer period waves are more accurately calculated. Figure 86
shows the calculated overturning moments and it can be seen that the errors are large, on
average being almost 50%.
Duwind 2003.025 Page 83 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0.80
0.85
0.90
0.95
1.00
1.05
1.10
5 7 9 11 13 15Wave Period [s]
Com
pari
son
of F
roud
e-K
rylo
v an
d D
iffra
ctio
n
12m Surge 15m Surge 18m Surge 21m Surge
12m Heave 15m Heave 18m Heave 21m Heave
Figure 85: Error in Froude-Krylov
01
23
456
78
910
5 7 9 11 13 15Wave Period [s]
OTM
[MN
m]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 86: OTM
Next examining the structure without an ice-cone, Figure 82, Figure 87 and Figure 88
compare the calculated surge and heave forces for diffraction and Froude-Krylov. It can be
seen that the agreement is generally good.
0
200
400
600
800
1000
1200
1400
1600
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 87: Surge Force
0500
1000150020002500
300035004000
45005000
5 7 9 11 13 15Wave Period [s]
Hea
ve F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 88: Heave Force
The ratio between the two methods is shown in Figure 89; with the average error being again
5% and 9% for the surge and heave forces respectively. The shape of these curves depends of
course on the force coefficients chosen in the Froude-Krylov analysis. Better chosen
coefficients (i.e. depth and wave-period varying) would improve the overall accuracy
somewhat; however if it is assumed that exact values are not known and a simplified
relationship such as shown in Table X (page 92) has to be used, accuracy similar to this would
result. Figure 90 shows the calculated overturning moments, for which the difference is much
larger at around 47%. This is primarily because the surge and heave forces act against each
other (i.e. see the cancellation point for the 21 m deep curve at around 6s wave period) and
the overturning moment is the result of the difference.
Duwind 2003.025 Page 84 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
0.80
0.85
0.90
0.95
1.00
1.05
1.10
5 7 9 11 13 15Wave Period [s]
Com
pari
son
of F
roud
e-K
rylo
v an
d D
iffra
ctio
n
12m Surge 15m Surge 18m Surge 21m Surge
12m Heave 15m Heave 18m Heave 21m Heave
Figure 89: Error in Froude-Krylov
01
23
45
67
89
10
5 7 9 11 13 15Wave Period [s]
OTM
[MN
m]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Froude-Krylov 15m Froude-Krylov
18m Froude-Krylov 21m Froude-Krylov Figure 90: OTM
4.2.2.3 Effect of Diffraction (Morison)
It may be beneficial to use the Morison method to calculate the wave loads on simple gravity
base structures during early conceptual evaluation stages, when the time needed and
complexity of diffraction or Froude-Krylov analysis makes them not viable for incorporation
into design tools. In this section, only the deepwater concept is considered, since a concept
optimisation model has been developed within the section.
Starting with the case of the 2m base slab, Figure 91 and Figure 92 show a comparison of the
surge and heave forces respectively calculated using both diffraction and Morison (inertia
coefficients of unity) for a matrix of wave periods and water depths. The diffraction results
are as shown previously in the evaluation of the Froude-Krylov theory above, (Figure 73 and
Figure 74 being the equivalent charts.)
0
100
200
300
400
500
600
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Morison 15m Morison
18m Morison 21m Morison Figure 91: Surge Force
0500
1000150020002500
300035004000
45005000
5 7 9 11 13 15Wave Period [s]
Hea
ve F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Morison 15m Morison
18m Morison 21m Morison Figure 92: Heave Force
Figure 93 shows the ration between the results, which can be used as inertia and pressure
coefficients in the Morison equation. The value for the horizontal inertia coefficient at the
Duwind 2003.025 Page 85 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
design wave is around 1.14, while the pressure coefficient I below unity in this case. Figure
94 shows the calculated overturning moments. It can be seen that the Morison methods
severely underestimates the overturning moment, primarily because it assumes that the entire
vertical pressure force acts at the vertical central axis and hence there is no moment arm.
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
5 7 9 11 13 15Wave Period [s]
Iner
tia C
oeff
icie
nts
12m Surge 15m Surge 18m Surge 21m Surge
12m Heave 15m Heave 18m Heave 21m Heave
Figure 93: Inertia Coefficients (Morison)
0
2
4
6
8
10
12
5 7 9 11 13 15Wave Period [s]
OTM
[MN
m]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Morison 15m Morison
18m Morison 21m Morison Figure 94: OTM
Turning to the case of the 4m base slab, Figure 95 shows the surge force and Figure 96 the
recommended force coefficients. For the higher slab, the recommended horizontal inertia
coefficient has now increased to around 1.27, while the pressure coefficient is again a little
below unity for some cases.
0
200
400
600
800
1000
1200
1400
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Morison 15m Morison
18m Morison 21m Morison Figure 95: Surge Force
0.80
0.90
1.00
1.10
1.20
1.30
1.40
5 7 9 11 13 15Wave Period [s]
Iner
tia C
oeff
icie
nts
12m Surge 15m Surge 18m Surge 21m Surge
12m Heave 15m Heave 18m Heave 21m Heave
Figure 96: Inertia Coefficients (Morison)
The recommended force coefficients are collated in to the following Table VIII.
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Table VIII: base Slab Force Coefficients for Morison Method
Slab Dimensions Force Coefficients
Height Diameter Horizontal Vertical
2m Slab 2m 25m 1.2 0.9
4m Slab 4m 25m 1.27 0.95
Using the simplified rule for selection of the force coefficients recommended below, Table X
(page 92), the wave loads on the complete GBS structure using diffraction and Morison can
be compared.
Looking at the structure with an ice-cone, Figure 81, Figure 97 and Figure 98 compare the
calculated surge and heave loads (the equivalent charts for the Froude-Krylov method are
Figure 83 and Figure 84, page 82). It can be seen that while the surge forces are calculated
with reasonable accuracy, in particular for the longer wave periods, the heave forces are
extremely conservative. Regarding the surge force, a constant value of 2 for the main column
was used, i.e. the MacCamy Fuchs correction was ignored,; its inclusion would reduce the
error. Regarding the heave force, the error is because the base-slab pressure is calculated at
the centre of the slab (i.e. the vertical axis) and it is assumed that the same force is applied
over the entire slab surface. In reality, as the wave passes over the slab, each segments of the
slab experience a different pressure, depending on which phase of the wave is passing directly
above. This lack of correlation means that the actual wave force is lower, significantly so in
the case of short waves.
0
200
400
600
800
1000
1200
1400
1600
1800
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Morison 15m Morison
18m Morison 21m Morison Figure 97: Comparison of Surge Force
0
10000
20000
30000
40000
50000
60000
70000
5 7 9 11 13 15Wave Period [s]
Hea
ve F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Morison 15m Morison
18m Morison 21m Morison Figure 98: Comparison of Heave Force
Figure 99 shows the difference between the two methods, the average error being 9% and
69% for surge and heave forces respectively. Figure 100 shows the calculated OTM, the
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average error being74%, however as stated above, the assumption that the heave force is
correlated across thee whole base slab means there is no moment arm.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
5 7 9 11 13 15Wave Period [s]
Com
pari
son
of M
oris
on a
nd
Diff
ract
ion
12m Surge 15m Surge 18m Surge 21m Surge
12m Heave 15m Heave 18m Heave 21m Heave
Figure 99: Error in Morison
0
1
2
3
4
5
6
7
8
9
10
5 7 9 11 13 15Wave Period [s]
Hea
ve F
orce
[kN
]
12m Diffraction 15m Diffraction 18m Diffraction
21m Diffraction 12m Morison 15m Morison
18m Morison 21m Morison Figure 100: OTM
The removal of the ice-cone, Figure 82, makes little difference as can be seen in Figure 101
(surge force) and Figure 102 (comparison) respectively.
0
200
400
600
800
1000
1200
1400
1600
5 7 9 11 13 15Wave Period [s]
Sur
ge F
orce
[kN
]
12m Diffraction 12m Diffraction 15m Diffraction
18m Diffraction 12m Froude-Krylov 15m Morison
18m Morison 21m Morison Figure 101: Surge Force
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
5 7 9 11 13 15Wave Period [s]
Com
pari
son
of M
oris
on a
nd
Diff
ract
ion
12m Surge 12m Surge 15m Surge 18m Surge
12m Heave 12m Heave 15m Heave 18m Heave
Figure 102: Error in Morison
In conclusion, the Morison method appears to miscalculate the surge wave loads (a slight
underestimate in the longer wave periods using this base-slab surge coefficient); the heave
force must be calculated using by dividing the base slab into segments lying parallel with the
wave crest.
4.2.2.4 Effect of Wave Model
Two of the analyses methods, Morison and Froude-Krylov, are able to incorporate both finite
wave heights and non-linear waves, whilst diffraction is not able to do either. This section
examines the importance of this aspect. Several phenomena generate any differences:
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- firstly the non-uniform profile of the structure at the sea-surface if an ice-cone is
utilised (likely to be necessary at all Baltic sites),
- secondly the increased submerged length of the structure within the wave crest,
- and thirdly the higher wave kinematics (and different distribution with time) due to
non-linear waves.
Using the deepwater GBS as the more detailed example, Figure 103 shows the wave loads
due to a small (2m) wave on the GBS structure without the ice-cone, the force coefficients
having been chosen for there to be good agreement for all three wave models showsn: airy,
wheeler stretching and stream function (Deans). As mentioned previously, in contrast the
overturning moment will show significant differences, Figure 104, in this case of around 35%.
0
200
400
600
800
1000
1200
1400
0 5 10 15 20Wave Period [s]
Sur
ge F
orce
[kN
]
DiffractionAiry 2mWheeler 2mDeans 2m
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 5 10 15 20Wave Period [s]
Hea
ve F
orce
[kN
]
DiffractionAiry 2mWheeler 2mDeans 2m
Figure 103: Surge (left) and Heave (right) Forces on Deepwater GBS (at 15m depth)
0
1
2
3
4
5
6
7
0 5 10 15 20Wave Period [s]
OTM
[M
Nm
] DiffractionAiry 2mWheeler 2mDeans 2m
Figure 104: OTM
0
1000
2000
3000
4000
5000
6000
0 5 10 15 20Wave Period [s]
Sur
ge F
orce
[kN
]
DiffractionAiry 8mWheeler 8mDeans 8m
Figure 105: Extreme Wave Surge Force
If the wave height is increased to an extreme value (8m) in this case, Figure 105 shows that
although the stream function now predicts a reduced wave force, the dominance of the base
structure on the wave load means that surface effects are not very important. Similarly, the
heave and OTM loads, Figure 106, are also scaled linearly, with the stream function reporting
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lower loads as before. Note that no results are shown for periods less than 7s, this being the
breaking limit.
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5 10 15 20Wave Period [s]
Hea
ve F
orce
[kN
]
DiffractionAiry 2mWheeler 2mDeans 2m
0
5
10
15
20
25
30
0 5 10 15 20Wave Period [s]
OTM
[M
Nm
] DiffractionAiry 2mWheeler 2mDeans 2m
Figure 106: Extreme wave Heave (left) and OTM (right)
It appears that the use of a more accurately modelled wave frequently results in lower wave
loads on the structure. If the ice-cone is added, it can be seen that the above conclusion
continues to apply, Figure 107, though the trend suggests that in some cases, the linear wave
model may become unconservative (surge force from 15s wave in Figure 107, left). The
average difference between the stream function and the diffraction result is 10%, 20% and
35% for the calculated surge, heave and OTM wave loads.
0
1000
2000
3000
4000
5000
6000
0 5 10 15 20Wave Period [s]
Sur
ge F
orce
[kN
]
DiffractionAiry 8mWheeler 8mDeans 8m
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5 10 15 20Wave Period [s]
Hea
ve F
orce
[kN
]
DiffractionAiry 2mWheeler 2mDeans 2m
Figure 107: Extreme Case with ice-Cone - Heave (left) and Surge (right)
For the shallower water GBS concept based on the Middelgrunden design, this difference
increases significantly, to a factor of 2 or 3 (200% or 300%) in some cases.
Up to this point, extreme values have been considered; however a GBS structure may fail
though the combination of heave, which reduces the apparent weight of the structure, and
surge, which then moves it. However, linear wave theory generates leads to conservative
conclusion as can be seen from Figure 108 and Figure 109. If the more accurate stream
function theory is used, both the maximum surge and heave forces are reduced and the
maximum surge force does not occur when there are significant heave forces. Obviously the
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wave load time-trace varies with the GBS structure and the wave conditions and the peak
surge force may not always be reduces, however assuming that the GBS structure continues to
be at least based on this form, then the peak heave load will be reduced and the peak surge
load will occur at a less critical time. Note that when non-linear theory is used, the erratic
shape of the time-trace means that the time-resolution needs to be higher to ensure that peak
loads are detected.
-6000
-4000
-2000
0
2000
4000
6000
0 2 4 6 8 10 Time [s]
Sur
ge F
orce
[kN
]
Airy
Wheeler Stretch.
Chakrabarti Stretch.
Extrapolated Crest
Constant Crest
Stream Function
Figure 108: Surge Force
-20000
-15000
-10000
-5000
0
5000
10000
15000
20000
25000
0 2 4 6 8 10 Time [s]
Hea
ve F
orce
[kN
]
Airy
Wheeler Stretch.
Chakrabarti Stretch.
Extrapolated Crest
Constant Crest
Stream Function
Figure 109 Heave Force
4.2.3 Effect of Surface
The importance of surface effects depends on the size of the structure at the water surface,
i.e. the presence and dimensions of the ice-cone. Even with an ice-cone, for the deepwater
GBS the loads calculated using the unmodified Airy & Wheeler methods give similar results:.
This is not the case for the shallow water GBS, however it is difficult to separate the affects
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
from those due to the use of linear vs. non-linear waves hence it can be considered to be
included in that aspect.
4.2.4 Comparison of Methods
In attempting to evaluate the uncertainties associated with the different wave theories, a
certain amount of selection must occur for the cases on which to base conclusions. In Table
IX, the extreme load case for the deepwater GBS structure has been used where possible and
the error values are based on the analyses in the previous section 4.2.2 Evaluating the
Uncertainties. See Table I (page 29) for an overview of the strengths and weaknesses of the
different models. The conclusion is that Morison is the least appropriate, as would be
expected, but it is sufficient for initial concept evaluation and preliminary optimisation if the
identified steps are followed. For a later stage, the choice is between diffraction and Froude-
Krylov method, with both having important omissions in their scope. For deepwater
structures, the weakness in the diffraction analysis become smaller, while for shallow waters,
where waves become less linear, the weaknesses in the Froude-Krylov theory become
smaller, assuming that care has been taken is selecting appropriate force coefficients. Since
GBS foundations have and will continue to generally be built in shallow waters, the Froude-
Krylov method will be needed in the later stages of the design process.
Table IX: Evaluation of Errors in Wave-load Methods
Error Morison Diffraction Froude-Krylov
Transverse Drag 5% - X X
Lateral Drag 0 X X X
Pressure 250% X 1 - -
10% X Diffraction 2
20% X -
Surface Effects 5% X X -
Non-Linear Waves 10% - X -
Total Error 30% 20% 15%
1 = for the total error, it is assumed that the more accurate method of calculating the pressure forces (as
described above) is used 2 = the diffraction error depends on the wave load method and is higher with Morison than with the
Froude-Krylov method
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As mentioned in the previous paragraph, the main weakness of the Froude-Krylov design
method is the selection of appropriate force coefficients. Table X attempts to provide
preliminary guidance though a separate diffraction analysis would also always be
recommended.
Table X: Inertia Coefficients for Use with Froude-Krylov Method
Froude-Krylov Morison
Horizontal Vertical Horizontal Vertical
Inclined Slab 1 0.85 - -
Rectangular Slab 1 1.75
BD
+ 1 1 2BD
+ 1
Column 2 - 2 or see [63] -
Ice-Cone 1.5 1 1.5 1
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5 Hydrodynamic Loading – Evaluation of Measurements
This chapter evaluates wave load theory against measurements taken at the instrumented
offshore wind-turbine at Blyth. The campaign datasets used within this chapter are
summarised in Table XI below.
Table XI: Campaign Datasets Details
Ref. Date & Time
Wave Height
Hs m
Tide Level
m (cf. LAT)
Wind Speed
m/s Turbine Status
0B0 19 Oct 01 22:30 1.21 1.19 9.82 Off
158 9 Nov 01 02:33 4.63 1.53 13.92 Off
159 9 Nov 01 04:07 4.9 1.62 13.64 Off
5.1 Individual Waves
A high wave selected from the dataset is examined in Figure 110 (surface elevation) and
Figure 111 (pile mudline bending moment). It can be seen that for this example the stream
function theory predicts the same crest elevation but that, as would be expected, linear theory
does not. This is also reflected in the bending moment traces in Figure 111, which show that
all theories underestimate the maximum bending moment but that the stream function is
closest.
Note that in the following Figure 111 measured internal bending stresses are being compared
against theoretical external wave loads, i.e. the dynamics are not taken into account in the
theoretical traces. Inclusion of dynamics in the theoretical trace will change the profile (by
adding high frequency oscillations due to modal response) but would probably not change the
maximum value that much for this example; where dynamics is of particular importance is
when the structure is already oscillating when the wave impacts on it. This is clearly shown
in Figure 112 and Figure 113, which illustrate amplification and cancellation respectively.
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0 1 2 3 4 5 6 7 84
2
0
2
4
Time [s]
Surf
ace
Ele
vatio
n [m
]
Wave Profile [m]LinearStream Function Theory
Figure 110: Wave Profile for a Selected Wave: Measurement vs. Theory
0 1 2 3 4 5 6 7 82 10
6
1 106
0
1 106
2 106
3 106
4 106
5 106
Time [s]
Ben
ding
Mom
ent [
Nm
]
rms (X&Y) Strain gauge Pile MudlineAiry LinearWheeler StretchingChakrabarti StretchingConstant CrestExtrapolatedStream Function
Figure 111: Pile Mudline Bending Moment for a Selected Wave:
Measurement vs. Theory
5 0 5 10 15 205 106
0
5 106
1 107
Time [s]
Ben
ding
Mom
ent [
Nm
]
rms (X&Y) Strain gauge Pile MudlineAiry LinearWheeler StretchingStream Function
Figure 112: Effect of Dynamics on Pile Mudline Bending Moment (Amplification)
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In all figures in this chapter, the wave height is defined as the difference between the crest and
the mean of the previous and following troughs. This provides a generally good match for the
stream function wave, see in Figure 110, a particularly close example.
10 5 0 5 10 15 202 106
1 106
0
1 106
2 106
3 106
Time [s]
Ben
ding
Mom
ent [
Nm
]
rms (X&Y) Strain gauge Pile MudlineAiry LinearWheeler StretchingStream Function
Figure 113: Effect of Dynamics on Pile Mudline Bending Moment (Cancellation)
5.2 Campaign (30 minute sea state)
0 1 106 2 106 3 106 4 106 5 106 6 106 7 106 8 1060
1 106
2 106
3 106
4 106 Identical Crest Elevations
Measured Maximum Bending Moment [MNm]
Cal
cula
ted
Max
Ben
ding
Mom
ent [
MN
m]
Figure 114: Maximum Bending Moment - Calculated verses Measured
If a comparison is made between the measurements and theory for all the waves in the
campaign (30 minutes), Figure 114, it can be seen that in general all theories underestimate
the maximum bending moment, with linear and stretched-linear theories being lower than
stream function. There is a large amount of scatter, due to several reasons (identified as
regions in Figure 114):
a. ringing induced by the previous wave resulting in an apparent underestimation by theory
of the loads,
region (a)
region (c)
region (b)
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b. dynamic amplification resulting in high measured loads in comparison with theory,
Figure 112, and underestimation by theory
c. dynamic cancellation resulting in low measured loads in comparison with the theory,
Figure 113.
5.3 Multiple Campaigns
The following charts refer to data collected from around one hundred campaigns. Since the
period covered is less than one year and campaigns are a biased sample (they were selected
depending on how interesting” they were, rather than how “representative”, hence extreme
seas are over-represented), they cannot be considered to represent typical turbine life.
0 1 2 3 4 5 61 10 3
0.01
0.1
1
10
100
Wave Height [m]
Fatig
ue D
amag
e [a
rbitr
ary
units
]
0 5 10 15 20
1 10 3
0.01
0.1
1
10
100
Wind Speed [m/s]
Fatig
ue D
amag
e [a
rbitr
ary
units
]
Figure 115: Campaign Tower Mudline Fatigue Damage
Regarding the stochastic or probabilistic approach, this appears to underestimate the structural
response since it does not model the highest waves accurately (exclude harmonics) and does
not take account of impact loading [40]. These excluded aspects may cause damage
disproportionate to their size because they potentially act near to the structure’s natural
frequencies resulting in both extreme but also fatigue loading being underestimated. Together
with other aspects identified within this paper, this may lead to the hydrodynamic loading
being relatively more important in the overall design of the support structure than expected.
Figure 115 shows the estimated fatigue damage derived from a selection of measurements
campaigns (both 4 and 30 minute), plotted against significant wave height (left) and wind
speed (right). The turbine status (off, switching or on) is identified by the marking used in the
charts. From the left chart, it can be seen that when the turbine is off, fatigue damage
correlates well with significant wave height, while from the right chart, it can be seen that
when the turbine is on, fatigue damage correlates well with windspeed. Apart from the worst
turbine ONturbine SWITCHINGturbine OFF
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
fatigue occurring when the turbine is being switched on and off, wind as well as wave loads
appear to be important. The conclusions are similar if the extreme loads are examined, Figure
116.
0 1 2 3 4 5 60
5
10
15
20
Wave Height [m]
Ext
rem
e M
udlin
e O
TM
[MN
m]
0 5 10 15 20
0
5
10
15
20
Wind Speed [m/s]
Ext
rem
e M
udlin
e O
TM
[MN
m]
Figure 116: Campaign Tower Mudline Extreme Loads
5.4 Probabilistic Approach
Stream function models the waves in a deterministic way with non-linear properties but the
stochastic nature of the waves is not taken into account [20]. On the other hand, the linear
wave theory can produce stochastic waves but with linear properties. For this reason two
random non-linear wave models are applied to calculate the response of the wind turbine.
The first wave model [44] is based on a Volterra series which is composed of first order linear
waves and second order waves with frequency interactions. The second wave model [77] is
based on Stokes type waves based on a wave potential. Figure 117 shows the simulated wave
profiles with different wave models. The asymmetry of the wave profiles for non-linear waves
can be observed. This asymmetry is more pronounced for larger waves.
turbine ONturbine SWITCHINGturbine OFF
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0 20 40 60 80 100 120-4
-2
0
2
4
6
8
time (s)
wave elevation (m)
linear wave wave model 1 wave model 2
Figure 117 Simulated non-linear wave profiles
The kinematics of the water particles are calculated accordingly. The influence of the current
has not been considered. The response of the turbine is obtained with a simple modal model
with the first two modes. From the response time series peaks are selected and fitted to a 3
parameter Weibull distribution. The integrated bending moment of the external wave forces is
also analysed. The 99 percentiles of the bending moment peak distribution at the mudline are
compared. The non-linear wave model 1 yields the highest estimate followed by the wave
model 2. This is due to the higher water particle velocity and acceleration. The linear wave
theory gives a considerably lower estimate of the 99 percentile of the peak bending moment
distribution. Table XII shows the 99 percentile of the static as well as the dynamic bending
moments at the mudline. In this table the 99 percentile of dynamic bending moment using the
linear wave theory is used as a reference value.
It can be seen that the dynamics of the structure plays an important role. The dynamic
response is considerably higher than the integrated bending from the external wave forces. A
direct comparison with the measured bending moment in a similar sea state reveals that the
estimated 99 percentile of the peak bending moment using the wave model 1 lies about 50%
below that of the measured peak bending moment. This, however, has to be put into
perspective due to the simple response model used here and the presence of the uncertainty in
the calibration factors. Nevertheless, one can say that the linear wave theory can
underestimate the extreme bending moment significantly.
Table XII: 99 percentile of the bending moment with different wave models at mudline
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Wave Model 99% dynamic 99% static
Linear 1 0.64
Wave model 1 1.87 0.92
Wave model 2 1.43 1.07
Measured 2.75
If the probabilistic approach is applied to non-linear wave theories (ignoring dynamics), it can
be seen in Figure 118 that the calculated loads are again significantly lower than the measured
values. The 99th percentile values are again found and displayed in Table XIII, together with
the same normalising form as in Table XII. Interestingly, the most severe stochastic dynamic
model gives a similar result to the most severe regular static (1.87 vs. 1.74), which suggests
that both dynamic effects and non-linear wave effects are of similar importance..
0 1 106 2 106 3 106 4 106 5 106 6 106 7 106 8 1060
0.2
0.4
0.6
0.8
1
rms (X&Y) Strain gauge Pile MudlineAiry LinearWheeler StretchingChakrabarti StretchingConstant CrestExtrapolatedStream Function
Identical Wave Heights
Bending Moment [Nm]
Cum
ulat
ive
Prob
abili
ty [%
]
0 1 106 2 106 3 106 4 106 5 106 6 106 7 106 8 1060
0.2
0.4
0.6
0.8
1
rms (X&Y) Strain gauge Pile MudlineAiry LinearWheeler StretchingChakrabarti StretchingConstant CrestExtrapolatedStream Function
Identical Crest Elevation
Bending Moment [Nm]
Cum
ulat
ive
Prob
abili
ty [%
]
Figure 118: Peak Value Distribution; wave characterised by height (left) or crest (right)
Table XIII: 99 percentile of the bending moment with different regular waves
Max Bending Moment [MNm] Normalised
Airy 1.301 0.58
Wheeler Stretching 2.6 1.16
Chakrabarti Stretching 2.276 1.01
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Max Bending Moment [MNm] Normalised
Extrapolated Crest 2.925 1.30
Constant Crest 2.764 1.23
Stream Function 3.901 1.74
Measured 6.179 2.75
5.5 Alternative Approaches
In the longer term, CFD should provide designers with tools capable of including any and all
non-linearities and considering the current state-of-the art in the research field together with
the speed that computer power continues to advance, even a pessimist should admit that a
decade will be long enough to develop the necessary hardware and software tools. Until then
a combination of regular non-linear waves and linear stochastic seas will have to be used.
An alternative approach that may offer promise would be to change what is being examined.
The initial approach was to generate individual waves and examine the structural response.
Probabilistic methods step back and consider the complete distributions. In both cases, the
process is circular: starting with the waves, the loads are determined followed by the motion;
now the affect of this motion on the loads can be examined and the process converge to a
solution. However, there is a flaw in that the initial wave loads cannot be predicted correctly
because of non-linearities.
A solution could be to generate the motion independently of the wave and then apply a single
regular extreme wave and to do this for a variety of conditions (initial displacement and
motion of the structure and wave height and period) and from the determined database of
loads to find extreme and fatigue values from the probability distribution of the initial
conditions.
Initial examinations of the Blyth data are promising. It would be expected that the phase of
the structure’s motion at the time of wave impact would be uncorrelated (i.e. whether the
turbine is moving into or out of the wave when the front slope reaches it), since it should be a
random fraction of natural period response cycles since the previous wave. And this does
seem to be the case: Figure 119 shows that there is a fairly uniform distribution of the first
mode oscillations. It would be expected that the direction of this motion would concentrate
around a single direction, the wave direction: Figure 120 shows this is the case, with the first
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mode oscillations moving mainly in a direction between 45° and 90° (and a smaller group
concentrated between –135° to – 90°, that being the same direction).
180 135 90 45 0 45 90 135 1800
0.05
0.1
0.15
Pre-Wave Motion Cycle Phase
Prob
abili
ty D
istr
ibut
ion
Figure 119: Phase of Structural Motion
Immediately Prior to Wave Impact
180 135 90 45 0 45 90 135 1800
0.05
0.1
0.15
0.2
Pre-Wave Motion Cycle Direction [deg]
Prob
abili
ty D
istr
ibut
ion
Figure 120: Direction of Struct. Motion
Immediately Prior to Wave Impact
Figure 121 shows the distribution of the range of internal bending moment measured during
this pre-wave period, hence it indicates the amplitude of the motion that the structure is
undergoing. It can be seen that the highest values are significant, equal to the 99th percentile
of the calculated static loads (Table XIII, page 99). This would be expected since these
reflect measurements taken a few seconds after wave impacts, during which time the ringing
oscillations have generally not decayed significantly.
0 1 106 2 106 3 106 4 1060.01
0.1
1
Pre-Wave IBM Range
Prob
abili
ty D
istr
ibut
ion
Figure 121: Amplitude of IBM Cycle
It would now be expected that there should be a clear correlation between the phase of the
structural motion at the point of wave impact and the dynamic amplification factor (i.e. the
ratio between the measured strain and the theoretical strain for the measured wave. That has
been more difficult to identify, probably because the dynamic amplification factor depends on
other factors as well and because field data inevitably has a certain amount of noise. This
could be the subject of future work.
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6 Discussion and Conclusions
This report focuses on understanding sand predicting the hydrodynamic loads and hence the
structural response of offshore windturbine support-structures. The principal problem
identified is the less linear nature of waves in shallow water, compared with the deeper waters
that have been the focus of the methods currently being applied within this new industry.
This affects the analysis of slender and compact structures in different ways:
(i) slender structures respond dynamically to the loads, however no design approach
is currently able to include this structural response together with non-linear waves
of an appropriately high order; the difficulty is creating a model of the stochastic
sea using non-linear theory beyond the second order; (on a positive note,
diffraction effects are predictable for slender structures),
(ii) compact structures exhibit little structural response however non-of the available
design methods are able to include both diffraction effects and non-linear waves
simultaneously
A long term solution to both these dilemmas will be CFD however we await further
development of theory as well as necessary increases in computer power, both of which
should be available in around a decade. This means that any investment in improved
techniques may well have a short shelf-life.
Slender Support Structures
Currently, the design process for slender offshore windturbine support structure takes two
approaches: regular non-linear waves and stochastic linear seas. The problem with the regular
non-linear wave approach is that although dynamic amplification can be modelled, what is
actually being examined is how the structural response depends on the ratio of the wave
period to windturbine natural period. If this ratio is an integer, the structure will be moving in
the right direction each time a simulated wave reaches the tower and the motion response will
be amplified. If the ratio were any number plus one half, the tower would be moving into the
wave when the next wave arrives and motion would be suppressed or even stopped altogether.
Regarding the stochastic or probabilistic approach, this appears to seriously underestimate the
structural response because it cannot model the highest waves accurately; it does include all
existing harmonics in such extreme waves and generally does not take account of impact
loading. The harmonics may cause damage disproportionate to their size because of potential
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correlation with structural natural frequencies. Hence, both extreme but also even fatigue
loading will be underestimated.
The measured wave loads were significantly higher than those predicted from the best
available theory (stream function), and it is concluded that this is due to several factors, in
order of importance:
• impact loading due to steep and breaking waves,
• amplification effects of structural dynamic response,
• surface-effects of steep waves (imbalance of submergence leading to an additional net
force),
• additional drag due to landing and J-tube, see Figure 122.
Figure 122: Landing and J-Tube at Blyth Offshore Windturbine
Breaking Waves
Offshore wind farms will generally be built on large and relatively flat and shallow regions of
sea-bed, for economic reasons. The main part of such regions are unlikely to be susceptible to
breaking waves but if the windfarm stretches to the edge of the sub-sea plateau or a few
windturbines are built on a narrow reef or spit, such as at Blyth, those windturbine could be
vulnerable to breaking waves. In that case, the effect of these wave loads should be
examined, because even though the amplitude of the forces and moments are likely to be less
Duwind 2003.025 Page 104 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
than that due to the turbine thrust, the breaking wave force will be impulsive, which can
trigger ringing and dynamic amplification. Breaking waves continue to be recorded at Blyth.
Compact Support Structures
Examination of the results from the above hydrodynamic analyses suggests that diffraction is
necessary to determine the added mass coefficient of the support structure, in particular of the
base. For simple structures, such as the deepwater GBS examined here, a simple relationship
can be determined using a handful of diffraction analysis. The loads should then be checked
using the Froude-Krylov method, to allow the implementation of non-linear wave theory,
utilising the added mass coefficients calculated using diffraction analysis previously. For the
deepwater geometry examined here, it was found that linear theory was conservative, since
using linear theory gives both a higher maximum lifting force, and a higher base shear (surge)
force at that critical moment in the phase of the wave.
Due to the variation of the combined loading on the structure the sliding and bearing capacity
of the gravity base foundation varies as well when the wave passes. As a consequence, the
variation of the utilisations is erratic and doesn’t resemble the shape of the variation of any of
the loadings. Therefore, the highest utilisation is only found when sufficient phases of the
wave are tested. Two design studies were compared and an underestimation of the GBS mass
in one of the designs by nearly 30% could be explained from the difference in tested wave
phases (4 and 24, respectively).
The correction of the bearing capacity due to the inclination of the combined loading is a
dominant factor in the bearing utilisation. Therefore, hydrodynamic load calculations must
not only lead to correct prediction of load amplitudes, but the shape of the variation during
passing of the wave must also be accurate. Although not investigated numerically, the results
point in the direction that both modelling of non-linearity and of (linear) diffraction are
important to obtain a safe lightweight design solution. In early phases of the design emphasis
of the hydrodynamic modelling should be on the variation of the load contributions during
passing of the wave. Sensitivity to variation in shape and amplitude of the load contributions
can reveal the necessity to apply diffraction or non-linear theory for the particular design
conditions.
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7 Recommendations
The main focus for future work on hydrodynamic loading of offshore windturbines should be:
• To examine how stochastic non-linear wave loading of a higher order than at present
can be applied to slender support structures
• To examine how both diffraction and non-linear surface effects can be applied to
compact support structures
In the meantime, for slender support structures, it is recommended that analyses involving a
combination of linear stochastic seas and non-linear regular waves are performed, as is the
current practise.
For compact structures, it is recommended that diffraction analyses be performed as well as
checks using the Froude-Krylov (or pressure integration) method. During conceptual
evaluation, Morison may be used but care must be taken how the forces on the base slab are
determined.
The measured wave loads at Blyth are higher than any of the applied theories predict. It is
expected that this is partly due to breaking wave impact loads but the fact that waves in the
non-breaking seas will be of an extreme form, which cannot be adequately modelled, must
also be important.
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Appendix A: Calibration of Strain Gauges
This appendix presents a description of how the strain gauges in the tower and pile were
calibrated against the periodic first mode oscillations during short periods (of a few seconds)
of relative calm in between the waves that were driving the loads. This method depends on
the fact that extreme non-linear waves have a particular shape: they consist of short and sharp
peaks in between long regions of relatively smooth water where wave loadings are minimal.
At the time of writing, it had not been possible to calibrate the strain gauges; hence the initial
focus of the data-analysis was on performing a preliminary calibration. Examining the data
for the nacelle accelerometers and the strain gauges, it could be seen that at certain times,
when the turbine was turned off, there was good correlation between the signals, Figure 123.
0.3 0.2 0.1 0 0.10.2
0.1
0
0.1
0.2
Acceleration at Nacelle [m/s2]start of cycleStrain Gauge Pile MWL [mV]start of cycle
Nacelle Acceleration & Strain Gauge Outp
X axis
Y a
xis
Figure 123: Correlation of Nacelle Accelerations and Strain Gauge Readings
When all such cycles within a 30 minute campaign are evaluated, and the most appropriate
selected (i.e. with little higher order noise) a calibration of the strain gauge voltage reading
against the nacelle acceleration can be made Figure 124. Making an assumption for the mode
shape allows an estimation of the calibration against the moments at the strain gauge to be
made in both X and Y axes, Figure 125.
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RecValue
0 2 10 4 4 10 4 6 10 4 8 10 4 0.001 0.0012 0.0014 0.0016
0
0.1
0.2
0.3
Estimated Strain Gauge Calibration
Prob
abili
ty
Figure 124: Preliminary Calibration of the Strain Gauges
1 2 3 4 5 61 108
1 109
1 1010
1 1011 Recommended Strain Gauge Calibratation
Gauge Location
Stra
in G
auge
Cal
ibra
tion
[Nm
vs
V]
Tower Top
Tower Base
Pile MWL
Pile dep 1
Pile dep 2
Mud-line
X
Y
Figure 125: Calibration Factors (Moment per Volt output) for Tower and Pile Strain
Gauges (shows + std. dev.)
These calibration factors were in good agreement with the externally calibrated values: in
particular, the differences for the mudline strain gauges were significantly less than the
accepted uncertainty of the external calibration process itself (a few percent), see Table XIV.
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Table XIV: Evaluation of Strain Gauge Calibration Method
Calibration Factor[Nm per V] Axis
Calculated [40] Externally Measured
X 1.27 × 1010 1.251 × 1010
Y 1.21 × 1010 1.208 × 1010
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Appendix B: Analysis of 30 Minute Campaign during Storm
This appendix reports in detail on the analysis of a single 30 minute campaign recorded
during the intense storm of November 8th – 9th, 2001. The key sea-state details are
summarised Table XI (page 93) and Table XV:
Table XV: Description of Sea State for X0158
Charts
30 Minute Detail
Significant Wave Height 4.63 m Figure 126 Figure 130
Mean Wind-Speed (recorded at nacelle) 13.92 m/s Figure 127 Figure 129
Turbine Status Off - -
Mean Water Level 1.53 Figure 128 -
App B.1 Introduction and Overview of Data
Within this appendix, of the total of 47 data streams, only the three which are directly relevant
and of most interest are examined, these being the sea surface and the mudline strain gauge
readings. The first group of the following charts display the full 30 minute time-trace for the
sea surface, Figure 126, the wind, Figure 127, and the moving mean water level, Figure 128,
and a second pair showing a shorter period of the sea surface, Figure 130, and wind, Figure
129, in greater detail. Time [s]
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 18002
0
2
4
6
8
Time [s]
Surf
ace
Ele
vatio
n [m
]
Figure 126: 30 Minute Surface Elevation Record
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 18000
10
20
30
40
Time [s]
Sele
cted
Dat
a
Figure 127: 30 Minute Windspeed Record
0 200 400 600 800 1000 1200 1400 1600 18001.4
1.5
1.6
1.7
Time [s]
Loc
al M
ean
Wat
er L
evel
[m]
Figure 128: Mean Water Level During Campaign
80 85 90 95 100 105 110 115 120 125 1302
0
2
4
6
8
Time [s] Figure 129: Detail of Surface Elevation (Waves)
80 85 90 95 100 105 110 115 120 125 1300
5
10
15
Time [s] Figure 130: Detail of Windspeed Record
Short time-traces (corresponding to the two charts) for two of the wind-turbine performance
response channels are shown below, for nacelle acceleration, Figure 131, and strain gauge
reading, Figure 132, respectively.
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80 85 90 95 100 105 110 115 120 125 1301
0.5
0
0.5
1
Time [s] Figure 131: Detail of Fore-aft Acceleration
80 85 90 95 100 105 110 115 120 125 1302
0
2
4
6
Time [s] Figure 132: Detail of Mudline Strain Gauge Reading
Recordings of the current are also available and the mean value and direction during the 30
minute campaign is shown in Figure 13313.
0
30
60
90
120
150
180
210
240
270
300
330
0.6
0.4
0.2
0
Mean Current Speed & Direction
0 0.1 0.2 0.3 0.4 0.5 0.60
1
2
3
4
5
6
Current Profile at Start TimeCurrent profile at End TimeMaximum CUrrentMinimum Current
Current Speed [m/s]
Dis
tanc
e ab
ove
Seab
ed [m
]
Figure 133: Current Direction and Profile
IF the variation of the mean strain values, given in terms of static internal bending moment in
Figure 134, is examined, it can be seen that there is significant variation through the campaign
cycle. This could be due to the current speed and direction, Figure 133, the windspeed,
Figure 127 (directional information was not available because of a fault in that part of the
13 Note that the compass directions are correct but not than the values on the circular axis. The blue point is the first reading.
N
W
S
E
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measurement system) or averaging bias due to several large waves occurring together, Figure
126.
0 200 400 600 800 1000 1200 1400 1600 18000.4
0.2
0
0.2
0.4
X strain gaugeY strain gauge
Time [s]
Loc
al M
ean
Stra
in R
eadi
ng [m
]
Figure 134: Variation of Mean Strain Value
App B.2 Identification of Individual Waves
Using appropriate criteria, individual waves can be identified from the campaign time-trace;
in this case, the zero-up crossings were located, with the overall campaign mean water level
being using as the trigger. (An improvement could be to use a moving mean water level at
locations, such as Blyth, were tides are relatively high, see Figure 128, page 119). The zero-
up crossing was chosen as the criteria as it generated the most sensible divisions of the sea
surface in to individual waves. Alternatives could be peaks or troughs (but an individual
wave could have several of each) or the down-crossing. Once the waves had been identified,
the very highest and lowest points within the waves could be defined as the crest and trough.
Extremely short waves could then also be removed of; i.e. two upward crossings very close
together. A sequential record of each individual wave height and wave mean water level is
shown in Figure 135 and the same waves ordered in terms of period is Figure 136.
0 200 400 600 800 1000 1200 1400 1600 18002
0
2
4
6
Mean Water Level [m]Wave Min / Max [m]
Time of Start of Wave [s]
Mea
n W
ater
Lev
el [m
]
Figure 135: Sequential Record of Individual Waves
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0 2 4 6 8 10 12 14 16 18 202
0
2
4
6
Mean Water Level [m]Wave Min / Max [m]
Wave Period [s]
Mea
n W
ater
Lev
el [m
]
Figure 136: Height and Period of Individual Waves
The information is shown in a different from in the following two charts, in terms of crest
elevation, Figure 137, and wave height, Figure 138. Crest elevation is thought of as being a
more reliable wave parameter than height in defining the wave (i.e. predicting what the wave
loads will be).
0 5 10 15 200
2
4
6All Waves
Wave Period [s]
Max
Sur
face
Ele
vatio
n [s
]
Figure 137: Crest / Period Distribution
0 5 10 15 200
2
4
6
8All Waves
Wave Period [s]
Wav
e H
eigh
t [s]
Figure 138: Height / Period Distribution
App B.3 Statistical and Spectral Properties of the Sea Surface
The statistical properties of the sea surface are shown in the following set of charts, with the
distribution of the crest height and wave period shown Figure 139 and Figure 140.
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Min Crest.η
0 2 4 60
20
40
60Wave Crest Elevation Distribution
Wave Crest Elevation [m cf current mean]
Prob
abili
ty
Figure 139: Crest Height Distribution
Min CyclePeriod
0 5 10 150
5
10
15
20Wave Period Distribution
Wave Period [s]
Prob
abili
ty
Figure 140: Wave Period Distribution
Figure 141 and Figure 142 show the two dimensional distribution of individual waves within
this selected 30 minute sea, in terms of crest and wave height.
5 10 150
2
4
6Joint PD - Period [X] / Crest [Y]
Figure 141: Crest & Period Distribution
5 10 150
2
4
6
8Joint PD - Period [X] / Height [Y]
Figure 142: Height & Period Distribution
It should be noted that the surface elevation is clearly non-gaussian, as indicated in Figure 143
below. The statistical parameters are listed in Table XVI.
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Table XVI: Moments of the
Probability Distribution
Order Name Sea Gaussian
0th - 1 1
1st - -0.004 0
2nd - 1.34 -
3rd 14 skewness 1.195 0
4th 14 kurtosis 4.482 3
Water MWL
5 0 5 100
20
40
60Surface Elevation Distribution
Surface Elevation [m]
Prob
abili
ty
Figure 143: Surface Elevation Distribution
Turning to the frequency domain record of the surface elevation, Figure 144 shows the
power-spectral density of several windows of the same sea-state. It can be seen that it differs
from the JONSWAP spectrum normally used in deeper waters at both low and high
frequencies.
Peak WF Harm 1st
0 0.1 0.20.1
1
10
100
measured campaignJONSWAP spectrum
Frequency [Hz]
PSD
[m2/
s]
non-linear effects
low frequency changes in
MWL
Figure 144: Comparison of Measured Spectrum with equivalent JONSWAP Spectrum
App B.4 Properties of Individual Waves
Stochastic sea simulation necessitates the application of linear wave theory, however Figure
145 and Figure 146 gives an indication of how inappropriate that is. In Figure 146, the
recommended wave theory is shown for each individual wave and it can be seen that all
except one require non-linear theory. Figure 145 shows the distribution and that a significant
14 normalised with the standard deviation [σn/2]
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proportion require high order theory, 9th being the most frequent order overall and that a
number of waves are beyond the breaking wave limit (shown as order = –1). This does not
necessarily mean those waves were breaking, as it could have been a temporary phenomenon
or due to how the wave period was determined.
1 0 1 2 3 4 5 6 7 8 9 10 110
20
40
Deans Stream Function Order
Prob
abili
ty D
istri
butio
n [%
]
Figure 145: Distribution of Wave Non-Linear Order
1 103
0.01 0.1 1
1 10 5
1 104
1 103
0.01
0.1
Airy Limit
d/gT2
H/g
T2
Airy LimitBreaking WaveAiryDean Stream 3rd OrderDean Stream 5th OrderDean Stream 7th OrderDean Stream 9th OrderDean Stream 11th OrderPost Breaking Wave Height
Figure 146: Recommended Wave Models
Likewise Figure 147 shows the recommended wave load model, following [17]. It can be
seen that for most waves, inertia and drag are important and that apart from two waves,
diffraction effects are straightforward (i.e. CM � 2).
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0.01 0.1 1 100.01
0.1
1
10
pi.D / wave-length
Keu
lega
n-C
arpe
nter
Num
ber
Airy LimitBreaking WaveBreaking WavesDiffractionInertia DominatedInertia & DragDrag Dominated
Figure 147: Recommended Waveload Model
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Appendix C: Preliminary Design of Gravity Base Structures
App C.1 Resistance factors
The gravity base structure (GBS) must provide sufficient resistance against sliding and
sufficient vertical bearing capacity. Evidently, the hydrodynamic heave force may not lift the
entire structure, in which case sliding resistance reduces to zero. The structure may also not
be tilted by the overturning moment of the wind loads and hydrodynamic pitch, but before
that point is reached the vertical bearing capacity will be reduced to zero. This will be shown
below, when the calculation of bearing resistance is explained. Sliding resistance and bearing
resistance are determined according to [5] and [100], assuming cohesionless, sandy soil.
When the horizontal force on the structure equals FX and the vertical force equals FX and the
z-axis is taken positive upward, the criterion for the sliding resistance becomes
( ) ( )ϕtan⋅−⋅Φ≤ zSSx FF (1)
with: SSΦ = resistance factor for sliding, taken equal to 0.8,
ϕ = friction angle of the soil, taken equal to 30º.
This equation is based on Coulomb’s relation for frictional material, which gives a linear
relation between the normal force and the shear force. Equation 1 can be interpreted as a
criterion for the maximum inclination of the combined forces with respect to the vertical.
Bearing capacity is calculated according to the theory developed by Prandtl, Terzaghi and
Brinch Hansen. Only the contribution of the soil weight is taken into account, because this is
commonly the largest contribution to bearing capacity for an offshore GBS. The bearing
capacity is corrected for inclined loading and overturning moment according to
( ) 2
41
'21
GBSGBSSBz DDNmsiF πγγγγγ ⋅⋅Φ≤− (2)
with: SBΦ = resistance factor for bearing, taken equal to 0.67,
γi = inclination factor,
γs = shape factor, taken equal to 0.7 for a circular gravity base,
γm = reduction factor of effective area,
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γN = empirical dimensionless function of ϕ ,
'γ = submerged unit weight of the soil, taken equal to 9000 N/m3
GBSD = diameter of the gravity base.
Equation 2 appears to give an upper limit for the downward vertical force, but the inclination
factor and reduction factor of the effective area are functions of FX that lead to lower limits.
The inclination and reduction of the effective area are illustrated in Figure 148 below.
Horizontal force at effective height
Vertical force
Combined force
Intersection point
Inclination
Top view of contact surface:
Reduced bearingarea in grey
Figure 148: Inclination and reduced bearing area of combined loading
The intersection of the combined force with the soil-structure contact plane is the centre of the
reduced area. The gravity base would be tilted when the intersection reaches the edge of the
contact surface, but as can be seen the effective bearing area is then already reduced to zero.
Thus, the soil would fail near the rim of the base well before the structure topples from a
geometrical point of view. The inclination factor reduces from 1 to zero between vertical
loading and the inclination at which sliding occurs. Because the correction factors reduce to
zero at tilting and sliding the bearing criterion is an all-embracing criterion.
App C.2 Load and resistance factor analysis
A case study is performed to assess the various parameters that affect the gravity base
stability. A simple circular gravity base is designed for a 3 MW turbine in the conditions
listed in Table VI. An operational load case has also been considered, but this appeared to be
less demanding. A previous design study for the same location and turbine resulted in a
gravity base with a 25 m diameter and a mass of 3000 t [28]. In that study linear diffraction
theory was applied to calculate the wave loading for 4 phases of the wave period. Weak to
firm soil conditions were assumed, with soil shear strength of 50 kPa. The current case study
assumes cohesionless soil, with a friction angle of 30º. The current design is made using
linear wave theory without diffraction to calculate hydrodynamic loading at 50 phases of the
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wave period. The pressure distribution according to Bernoulli’s equation is integrated over the
horizontal surface of the gravity base and horizontal loading is calculated with Morison’s
equation with Cm = 2.0 and Cd = 0.7. Under these conditions a gravity base with the same
25 m diameter is designed. The minimum required mass in this case equals 4200 t, with a
height of 2.15 m. The different mass of the previous study and this design will be addressed
later. This section analyses the loading, capacity and stability of the reference design during
different phases of the extreme wave, in order to get insight in the design drivers. The various
relevant parameters are combined in the plots of Figure 149. The horizontal axis is positive in
downwind direction and the vertical axis is positive upward. Pitch is related to the centre of
the gravity base at the contact surface and positive for the wind force. The utilisation is the
ratio between the loading and the capacity.
-1E+7
0E+0
1E+7
(N)
0E+0
2E+7
4E+7(N)
-1E+8
0E+0
1E+8
(N)
0E+0
6E+8(N)
-4E+7
0E+0
4E+7
(Nm)
0E+0
5E-1
1E+0(m)
0
1
0
1
0
1
0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00
Load contributions
Horizontal Vertical Overturning moment
Utilisations
Sliding Bearing Tilting stability
Wind
Surge
Total
Weight tower+nacelle
Weight GBS
BouyancyHeave
Total
Weight nacelle
Wind
Pitch due to heave
Pitch due to surge
Total
Abs(Load)
Capacity UtilisationIntersection
Utilisation
-Load
Effective area correctionInclination correction
Capacity
Utilisation
Figure 149: Loads and utilisations of ∅∅∅∅ 25 m x 2.15 m GBS (linear wave model with
Morison equation and integrated horizontal surface pressure)
(Horizontal axis is wave phase)
During the extreme wave the horizontal wind loading on the idle turbine is insignificant. The
submerged weight and hydrodynamic heave force are of the same order of magnitude and
therefore nearly cancel at wave phase 0.5. Pitch due to heave and surge partially cancel and
are slightly larger than pitch due to wind loading. Both capacity against sliding and bearing
capacity vary enormously during passing of the wave. The heave reduces the normal force at
the contact surface and the effect on sliding capacity is clearly visible. The inclination
correction, which depends on surge and heave, clearly dominates the variation of the bearing
capacity. The effective area correction is nearly equal to 1, because the offset of the total force
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OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
from the centre of the contact surface is small as can be seen in the lower-right plot. The
offsets between the extremes of loading and capacity result in a maximum utilisation of
sliding and bearing capacity just before and just after maximum heave occurs. The maximum
bearing utilisation at wave phase 0.4 is not detected in the previous design study, because of
the low number of wave phases, and this is probably the most important reason why the
previous study resulted in a lighter GBS.
App C.3 Effects of hydrodynamic modelling on design solution
Design optimisation
Often the design of a GBS will be optimised toward smallest dimensions and lowest weight.
To determine how the optimum solution is affected by hydrodynamic modelling the stability
of gravity bases with a range of diameters and heights is tested. The result for the
hydrodynamic model used for the design in the previous section is shown in Figure 150
(however, here 24 phases of the wave are used, instead of 50). Gravity bases with dimensions
within the shaded area are stable, while other dimensions might fail. For different failure
mechanisms the boundaries are given, with the instable area directed away from the shaded
area. As stated before, the bearing criterion is all-embracing and hence the shaded area is on
the stable-side of all boundaries. The bearing capacity without surge and pitch is the fictitious
capacity that is obtained when the inclination correction and effective area correction are
omitted in Equation 2. The lightest stable structure according to this model is found in the
lower-left corner of the shaded area.
0 10 20 30 40 50GBS diameter (m)
0
5
10
GB
S he
ight
(m)
Stability boundaries
Bearing
Sliding
Tilting
Bearing (no surge and pitch)
Lifting
StableInstable
Figure 150: Dimensions of stable and instable gravity bases with stability boundaries
The lower boundary of the stable area is close to the boundary below which the structure is
lifted. Therefore, a good prediction of heave forces is a first essential step toward finding this
lower boundary. The sliding boundary is even closer to the stable area, indicating that this
criterion incorporates the most important effect of the relation between surge and heave on the
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lower stability boundary. The upper bound and particularly the related left boundary of the
stable area are farther away from the other stability boundaries. This demonstrates the
importance of the correction factors for the combined loading. Since the correction factors
depend on the relative magnitudes of the loading contributions, proper determination of these
contributors during several stages of the passing of the wave is crucial to find the correct
lightest possible structure.
Influence of modelling on optimisation
The main question of the design study is how the area with stable gravity bases changes when
different hydrodynamic models are used, particularly in the region of low diameters. Figure
151 plots the stability area for several alternatives to the reference of the previous section.
0 10 20 30 40 50GBS diameter (m)
0
5
10
GB
S he
ight
(m)
Reference
Linearised pressure
4 wave phases
Cd and Cm 10% reduced
Instable Stable
Figure 151: Stability boundaries for several alternative hydrodynamic models
When the pressure on the horizontal surface of the GBS is linearised around the centre an
analytic solution of the heave and pitch can be obtained, which is convenient in early design
phases. This model has only a small conservative difference with the reference model, since
the wavelength of 189 m is much larger than the analysed diameters. Only for larger
diameters the deviation is visible. Reduction of the number of wave phases to 4 lowers the
lower boundary approximately to the boundary where the structure is lifted by the heave.
Simultaneously, smaller diameters appear to be possible. This approach clearly
underestimates the required minimum diameter and height. The underestimation of the
minimum height is nearly the same as the difference observed between the two 25 m diameter
designs mentioned earlier. Reduction of Cm and Cd also results in a significant reduction of
minimum required diameter and height, even though this doesn’t affect the heave force. The
effect is caused by the change in correction factors for the bearing capacity.
Diffraction and non-linear wave theory are not implemented in the used design tool.
However, when linear diffraction is considered, surge, heave and pitch amplitudes will be
Duwind 2003.025 Page 132 Hydrodynamic Loading on Offshore Wind Turbines
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affected differently, but their variation in time will remain the same. Therefore, the effect on
the stability diagram will be similar to that of changing Cm and Cd, although numerically
somewhat different. Since diffraction will particularly result in reduced surge, omission of a
diffraction model is likely to result in a conservative design solution. The use of non-linear
wave theory will not only change the amplitudes of the loading, but also the shape of the
variation in time. Because the instantaneous relation between the load contributions is so
important for the capacity correction factors, this may have a larger effect on the stability
diagram. For the water depth and extreme wave height of this study non-linearity can be
significant. It is not known a priori whether omission of non-linearity will result in a
conservative or underestimated design.
Duwind 2003.025 Page 133 Hydrodynamic Loading on Offshore Wind Turbines
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Appendix D: Deans Stream Theory
This section describes how to apply Deans stream function for regular waves and an
introduction to the case for irregular waves. If an attempt is going to be made to write code to
apply this theory, it is strongly recommended to read a range of appropriate literature, not
limited to the original source [23], for example [17], [84] [88]
The basic stream function equation is:
( ) { }( ) ( )�=
++=N
nn xkndyknXycyx
1
..cos...sinh..,ψ Eq. 1
yds += Eq. 2
Note that, unlike the linear Airy wave theory, the axis system moves relative to the wave and
hence the argument of the cosine term is (n.k.x) and not (n.k.x - ω.t), see Figure 152. This
allows easier (maybe possible at all) manipulation of the Eq.s.
In this section, [x,y] represents co-ordinates in the local axes system and [x ,̀y] is used for the
global axes. Note that since there is no relative motion in the y-axis, no new nomenclature is
necessary. The relationship between the axes systems is as Eq. 3.
tcxx .' +=
cuu +='
dtdu
dtdu =`
Eq. 3
Therefore Eq. 1 is rewritten as Eq. 4 according to this terminology of global axes:
( ) ( ) { }( )�=
+=N
nn ctxnknksXyx
1'cossinh,'ψ Eq. 4
wave axes[x’, y’]
global axes[x, y]
wave velocity[c, 0]
Figure 152 – Global and Wave Axes
dtdy y
u∂∂
tu
∂∂
x
y
Figure 153 – Acceleration within a Fluid
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The boundary conditions are that both the stream function, Eq. 1, and the total pressure, Eq. 5,
are constant at the wave surface:
��
�
�
��
�
���
���
�+���
����
�+=
22
.21
dxd
dyd
gzQ
ψψ Eq. 5
Note that the dynamic pressure is calculated with the relative velocity and not the global
velocity. No doubt this is because the wave profile should remain permanent in these wave
axes, though should questions be asked whether this is a weakness?
App D.1 Velocities
The horizontal and vertical velocities can be found from the stream function as:
dyd
uψ−= and
dxd
vψ+= Eq. 6
However, the signs can be reversed, resulting in an alternative notation, such as in 15
dyd
uψ+= and
dxd
vψ−=
In that case, the sign of the wave celerity expression is also reversed and the stream function,
Eq. 4, will be:
( ) ( ) ( )�=
+−=N
nn nkxnksXcyyx
1cos.sinh,ψ
Applying the first convention, Eq. 6, gives:
( ) ( )�=
−−=+−==−N
nn nkxnksnXkcucu
dyd
1cos.cosh.'
ψ
( ) ( )�=
−==N
nn nkxnksnXkv
dxd
1sin.sinh
ψ
Eq.s 7
kTc
ωλ == Eq. 8
15 Developments in stream-function wave theory, Chaplin, CoastEng, V3-p179, 1980 (P745 To WTheory)
Duwind 2003.025 Page 135 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Hence the expression for the horizontal velocity in the global axes is:
( ) ( )�=
−=N
nn xknsknXnku
1'..cos...cosh...' Eq. 9
App D.2 Accelerations
The general expression for the horizontal acceleration is:
dtdy
yu
dtdx
xu
tu
dtdu
..∂∂+
∂∂+
∂∂= Eq. 10
where
• du/dt is the called the substantial acceleration, the total acceleration experienced by the
fluid particle,
• �u/�t is the local or temporal acceleration, the acceleration experienced if the particle
remained at motionless
• and �u/�x × dx/dt and �u/�y × dy/dt are the convective acceleration, the acceleration due
to the change in the particle’s position.
This can be performed either in the global axes or in the wave axes, since there is no relative
acceleration between them, the acceleration values will be identical. The second order terms
are ignored for Linear theory, probably because second order terms are ignored in the
derivation of the potential flow function.
If Eq. 10 is applied in the global axes [x,y], it becomes:
dtdy
yu
dtdx
xu
tu
dtdu
.''
.''''
∂∂+
∂∂+
∂∂= Eq. 11
The terms are (with the expressions shown in the wave axes [x ,̀y ]̀ but the differentiation
must be applied in the global axes [x,y]):
( ) { }( )�=
−=∂∂ N
nn xknsknXnk
tu
1
2 '..sin...cosh...' ω
( ) { }( )�=
=∂∂ N
nn xknsknXnk
xu
1
22 '..sin...cosh..''
( ) ( )�=
−=∂∂ N
nn xknsknXnk
yu
1
22 '..cos...sinh..'
Eq.s 12
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and Eq. 11 becomes:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )��
��
�
==
==
=
−−
−+
−=
N
nn
N
nn
N
nn
N
nn
N
nn
xknsknXnkxknsknXnk
xknsknXnkxknsknXnk
xknsknXnkdtdu
11
22
11
22
1
2
'..sin...sinh....'..cos...sinh..
'..cos...cosh....'..sin...cosh..
'..sin...cosh...' ω
Eq. 13
If Eq. 10 is applied in the wave axes, the terms become:
0=∂∂
tu
( ) ( )�=
=N
nn nkxnksXnk
dxdu
1
22 sin.cosh
( ) ( )�=
−=N
nn nkxnksXnk
dydu
1
22 cossinh
Eq.s 14
Note that the top Eq.s 14 is now zero because of the choice of axes (it is the differential of the
upper Eq.s 7, which is not a function of time).
Eq. 10 can thus be written as follows, with the function being identical to Eq. 13 following a
substitution and rearrangement:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )���
����
�−−
���
����
���
�
�+−+
=
��
��
==
==
N
nn
N
nn
N
nn
N
nn
nkxnksnXknkxnksXnk
nkxnksnXkcnkxnksXnk
dtdu
11
22
11
22
sinsinh.cossinh
coscoshsincosh
0
Eq. 15
The general expression for the vertical acceleration is:
dtdy
yv
dtdx
xv
tv
dtdv
..∂∂+
∂∂+
∂∂= Eq. 16
and if this is applied in the wave axes, then:
dtdy
yv
dtdx
xv
tv
dtdv
.'
.' ∂
∂+∂∂+
∂∂= Eq. 17
Duwind 2003.025 Page 137 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
and the terms become:
( ) ( )�=
=∂∂ N
nn xknsknXnk
tv
1
2 ..cos...sinh...ω
( ) ( )�=
−=∂∂ N
nn xknsknXnk
xv
1
22 ..cos...sinh...'
( ) ( )�=
−=∂∂ N
nn xknsknXnk
yv
1
22 ..sin...cosh...
Eq.s 18
Note that the differentiation is performed in the global axes, [x',y], and then transformed back
into the local axes, [x,y].
and hence Eq. 16 becomes:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )���
����
�−−+
���
����
�−−+
=
��
��
�
==
==
=
N
nn
N
nn
N
nn
N
nn
N
nn
xknsknXnkxknsknXnk
xknsknXnkxknsknXnk
xknsknXnkdtdv
11
22
11
22
1
2
..sin...sinh......sin...cosh...
..cos...cosh......cos...sinh...
..cos...sinh...ω
Eq. 19
Likewise, to check this expression, the differentiation can again also be performed in the local
axes [x,y], Eq. 16 gives:
0'=
∂∂
tv
( ) ( )�=
−=∂∂ N
nn xknsknXnk
xv
1
22 ..cos...sinh...''
( ) ( )�=
−=∂∂ N
nn xknsknXnk
yv
1
22 ..sin...cosh...`'
Eq.s 20
hence Eq. 17 becomes:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2 2
1 1
2 2
1 1
'0 . . .sinh . . .cos . . '. . . .cosh . . .cos . . '
. . .cosh . . .sin . . '. . . .sinh . . .sin . . '
N N
n nn n
N N
n nn n
dvk n X n k s n k x c k n X n k s n k x
dt
k n X n k s n k x k n X n k s n k x
= =
= =
� �� �= + − − +� �� �� �� �
� �+ − −� �� �
� �
� �
Eq. 21
Duwind 2003.025 Page 138 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
App D.3 Pressure
Starting with the fundamental Eq.s relating pressure with force and force with acceleration:
APF .=
amF .= Eq.s 22
where:
xxP
P δ.∂∂=
xAm δρ ..= Eq.s 23
Hence (the negative sign represents the fact that a negative pressure gradient generates a
positive acceleration):
AxxP
axA ..... δδρ∂∂−= Eq.s 24
dxaP x .. −= ρ and dyaP y .. −= ρ Eq.s 25
Considering the case for first order potential flow theory first, by way of illumination:
���
���
∂∂=
xdtd
a xφ and
���
���
∂∂=
ydtd
a yφ Eq.s 26
which both can be substituted into Eq.s 25 to give:
tP
∂∂−= φρ. Eq. 27
For the case of the Deans stream function, the situation is more complex, as second order
terms must be considered, see Eq. 13 and Eq. 19:
���
���
∂∂−=
ydtd
a xϕ and
���
���
∂∂=
xdtd
a yϕ Eq.s 28
Hence resolving for the x-direction, Eq.s 25 becomes:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
dx
xknsknXnxknsknXnk
xknsknXnxknsknXnk
xknsknXnk
P
N
nn
N
nn
N
nn
N
nn
N
nn
.
..sin...sinh.....cos...sinh..
..cos...cosh.....sin...cosh..
..sin...cosh...
.
11
23
11
23
1
2
��
��
�
����
�
����
�
�
����
�
����
�
�
+
−
−
−=
==
==
=ω
ρ Eq. 29
Duwind 2003.025 Page 139 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
This expression is very difficult to integrate, as the second two terms are products of series.
The first term is straight forward to integrate:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) �������
�
�
�
���
���
�
���
���
�
+
−+
−=
��
��
�
==
==
=
dxxknsknXnxknsknXn
xknsknXnxknsknXnk
xknsknXn
P
N
nn
N
nn
N
nn
N
nn
N
nn
.'..sin...sinh...'..cos...sinh..
'..cos...cosh...'..sin...cosh..
'..cos...cosh...
.
11
2
11
2
3
1
ω
ρ Eq. 30
An attempt is made to integrate the second and third term using the integration by parts rule,
the general Eq. for which is:
dxdxdu
vvudxdxdv
u ..... −= Eq. 31
If we apply this to the second term:
( ) ( )
( ) ( )�
�
=
=
=
=
N
nn
N
nn
xknsknXnkdxdu
xknsknXnu
1
3
1
2
'..cos...cosh...
'..sin...cosh..
( ) ( )
( ) ( )�
�
=
=
=
=
N
nn
N
nn
xknsknXndxdv
xknsknXk
v
1
1
'..cos...cosh..
'..sin...cosh.1
Applying the theory a second time to the second term of Eq. 31
( )dxdxvdx
uddxv
dydu
dxvdxdu
....... 2
2
−= Eq. 32
does not bring us any closer to a solution, as the factors for the expressions are diverging:
( ) ( )�=
−=N
nn xknsknXnk
dxud
1
422
2
'..sin...cosh... ( ) ( )� =
−=N
nn xknsknX
nkdxv
12 '..cos...cosh..
11.
However if we apply Eq. 31 the other way round:
( ) ( )
( ) ( )�
�
=
=
=
=
N
nn
N
nn
xknsknXnkdxdu
xknsknXnu
1
2
1
'..sin...cosh...
'..cos...cosh..
( ) ( )
( ) ( )�
�
=
=
=
−=
N
nn
N
nn
xknsknXndxdv
xknsknXnk
v
1
2
1
'..sin...cosh..
'..cos...cosh..1
Duwind 2003.025 Page 140 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
We get a situation where the two integration terms are identical:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) dxxknsknXnk
xknsknXnk
xknsknXnk
xknnsknXn
dxxknsknXnxknsknXn
N
nn
N
nn
N
nn
N
nn
N
nn
N
nn
.'..cos...cosh..1
.'..sin...cosh...
'..cos...cosh..1
.'..cos...cosh..
.'..cos...cosh...'..sin...cosh..
11
2
11
11
2
��
��
��
���
���−
���
���−=
���
���
==
==
==
Eq. 33
and hence we have the solution:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
2
'..cos...cosh..1
.'..cos...cosh..
.'..cos...cosh...'..sin...cosh..
11
11
2
���
���
−=
���
���
��
��
==
==
N
nn
N
nn
N
nn
N
nn
xknsknXnk
xknsknXn
dxxknsknXnxknsknXn
Eq. 34
Applying Eq. 31 in the same manner to the third term, the substitutions are:
( ) ( )
( ) ( ).'..cos...sinh...
'..sin...sinh..
1
2
1
�
�
=
=
=
=
N
nn
N
nn
xknsknhXnkdxdu
xknsknXnu
( ) ( )
( ) ( )�
�
=
=
=
=
N
nn
N
nn
xknsknXndxdv
xknsknXnk
v
1
2
1
'..cos...sinh..
'..sin...sinh..1
making the substitutions:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) dxxknsknhXnxknsknXn
xknsknXnk
xknsknXn
dxxknsknXnxknsknXn
N
nn
N
nn
N
nn
N
nn
N
nn
N
nn
.'..cos...sinh...'..sin...sinh..
'..sin...sinh..1
.'..sin...sinh..
.'..cos...sinh...'..sin...sinh..
1
2
1
11
1
2
1
��
��
��
==
==
==
−
= Eq. 35
Allows the integral of this term to be found as well:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )2
'..sin...sinh..1
.'..sin...sinh..
.'..cos...sinh...'..sin...sinh..
11
1
2
1
��
��
==
==
=
N
nn
N
nn
N
nn
N
nn
xknsknXnk
xknsknXn
dxxknsknXnxknsknXn
Eq. 36
Duwind 2003.025 Page 141 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Substituting Eq. 34 and Eq. 36 into Eq. 30:
( ) ( )
( ) ( ) ( ) ( )�����
�
�
�
��
���
��
���
��
���
�+��
���
�+=
��
�
==
=
2
1
2
1
2
1
'..sin...sinh..'..cos...cosh...2
'..cos...cosh...
.N
nn
N
nn
N
nn
xknsknXnxknsknXnk
xknsknXn
P
ω
ρ Eq. 37
This result can be checked by resolving for the y-direction; Eq.s 25 becomes:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
dy
xknsknXnkxknsknXnk
xknsknXnkxknsknXnk
xknsknXnk
P
N
nn
N
nn
N
nn
N
nn
N
nn
.
'..sin...sinh....'..sin...cosh...
'..cos...cosh....'..cos...sinh...
'..cos...sinh...
.
11
22
11
22
1
2
��
��
�
���
�
���
�
�
���
�
���
�
�
−−
−−−=
==
==
=ω
ρ Eq. 38
Again the first term is straightforward to integrate but the second and third terms require
integrating by parts:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) �������
�
�
�
���
���
�
���
���
�
+
++
−=
��
��
�
==
==
=
dyxknsknXnxknsknXn
xknsknXnxknsknXnk
xknsknXn
P
N
nn
N
nn
N
nn
N
nn
N
nn
.'..sin...sinh...'..sin...cosh..
'..cos...cosh...'..cos...sinh..
'..cos...cosh...
.
11
2
11
2
3
1
ω
ρ Eq. 39
Applying Eq. 31 to the second term, the substitutions are:
( ) ( )
( ) ( )�
�
=
=
=
=
N
nn
N
nn
xknsknXnkdydu
xknsknXnu
1
2
1
'..cos...sinh...
'..cos...cosh..
( ) ( )
( ) ( )�
�
=
=
=
=
N
nn
N
nn
xknsknXndxdv
xknsknXnk
v
1
2
1
'..cos...sinh..
'..cos...cosh..1
making the substitutions results in the two integral terms being equal:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) dxxknsknXnk
xknsknXnk
xknsknXnk
xknsknXn
dxxknsknXnxknsknXn
N
nn
N
nn
N
nn
N
nn
N
nn
N
nn
.'..cos...cosh..1
.'..cos...sinh...
'..cos...cosh..1
.'..cos...cosh..
.'..cos...sinh...'..cos...cosh..
11
2
11
1
2
1
��
��
��
���
���−
���
���=
���
���
==
==
==
Eq. 40
Duwind 2003.025 Page 142 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
and hence we have the solution:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )2
'..cos...cosh..1
.'..cos...cosh..
.'..cos...sinh...'..cos...cosh..
11
1
2
1
��
��
==
==
=
���
���
N
nn
N
nn
N
nn
N
nn
xknsknXnk
xknsknXn
dxxknsknXnxknsknXn
Eq. 41
Applying Eq. 31 to the third term, the substitutions are:
( ) ( )
( ) ( )�
�
=
=
=
=
N
nn
N
nn
xknsknXnkdydu
xknsknXnu
1
2
1
'..sin...cosh...
'..sin...sinh..
( ) ( )
( ) ( )�
�
=
=
=
=
N
nn
N
nn
xknsknXndxdv
xknsknXnk
v
1
2
1
'..sin...cosh..
'..sin...sinh..1
making the substitutions results in the two integral terms being equal:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) dxxknsknXnk
xknsknXnk
xknsknXnk
xknsknXn
dxxknsknhXnxknsknXn
N
nn
N
nn
N
nn
N
nn
N
nn
N
nn
.'..sin...sinh..1
.'..sin...cosh...
'..sin...sinh..1
.'..sin...sinh..
.'..sin...cosh...'..sin...sinh..
11
2
11
1
2
1
��
��
��
���
���−
���
���=
���
���
==
==
==
Eq. 42
and hence we have the solution:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )2
'..sin...sinh..1
.'..sin...sinh..
.'..sin...cosh...'..sin...sinh..
11
1
2
1
��
��
==
==
=
���
���
N
nn
N
nn
N
nn
N
nn
xknsknXnk
xknsknXn
dxxknsknhXnxknsknXn
Eq. 43
Substituting Eq. 41 and Eq. 43 into Eq. 39 gives Eq. 44, which is identical to Eq. 37:
( ) ( )
( ) ( ) ( ) ( )
�
�
��
���
��
���
�+��
���
�+−=
��
�
==
=
2
1
2
1
2
1
'..sin...sinh..'..cos...cosh...2
'..cos...cosh...
.N
nn
N
nn
N
nn
xknsknXnxknsknXnk
xknsknXn
P
ω
ρ
Eq. 44
Duwind 2003.025 Page 143 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
App D.4 Description of Methods for Solving for Regular Waves
Given wave height, period and water depth H, T, d
Find wave length, Dean function parameters, vertical
displacement of profile
λ, Xn=1…N, ηx=0 or ψη
It can be seen that both T, Eq. 8, and d, Eq. 2, are present in the Eq.s, but H is only
represented implicitly, in Eq. 45.
Hence the conditions for a successful solution are:
• correct wave height, where η(x) is given in Eq. 48:
( ) ��
���
�−=2
0ληηH Eq. 45
• zero mean surface elevation:
( ) 0.2
0= dxx
λη Eq. 46
• constant pressure along surface:16
( ) ( ) ( )��
�
�
��
�
���
���
�+���
����
�+=
22,,
.21
,dx
xddy
xdg
xQηψηψηη Eq. 47
to find the surface elevation for a given surface-stream-function value, rearrange Eq. 1 and
find the root, η(x):
( ) ( ) ( ){ }[ ] [ ]���
��� ++−= �
=
N
nn xknxdknXxcrootx
1
..cos...sinh.. ηηψη η Eq. 48
The Bernoulli pressure is defined as (see Eq. 47):
( ) ( ) ( )��
�
�
��
�
���
���
�+���
����
�+=22
,,.
21
,dx
xddy
xdg
xQηψηψηη Eq. 49
The mean error in the Bernoulli pressure is:
( ) ( ){ }�=
−=I
i
ji
j QQI
E1
2.
1 Eq. 50
16 Note that the dynamic term of the Bernoulli pressure is calculated from the absolute horizontal velocity, i.e. see Chakrabarti,
equation 3.113.
Duwind 2003.025 Page 144 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
The error in the Bernoulli pressure for the next set of parameters can be estimated to be:
( ) ( )( )
� �= =
+
��
���
��
���
−���
����
�′∆
∂∂
+=I
i
N
nn
n
jij
ij QX
XQ
QI
E1
2
1
1 ..1 Eq. 51
The vertical and horizontal velocity components are:
( ) ( )�=
−−=+−==−N
nn xknsknXnkcucu
dyd
1
'..cos...cosh...`ψ
( ) ( )�=
−==N
nn xknsknXnkv
dxd
1
'..sin...sinh...ψ
Eq. 52
Substituting into the expression for the Bernoulli pressure Eq. 49:
( )( )( ) ( )
( )( ) ( ) ������
�
�
������
�
�
���
����
�+−+
���
����
�+−−
+=
�
�
=
=2
1
2
1
sinsinh
coscosh
21
,N
nn
N
nn
nkxdnknkX
nkxdnknkXc
gxQ
η
ηηη Eq. 53
The negative signs can be removed since:
( ) ( )22 xdxd
xdxd =− Eq. 54
Differentiating with respect to the Deans stream function parameter:
( )( ) ( ) ( )( ) ( )
( )( ) ( ) ( )( ) ( ) �����
�
�
�����
�
�
���
����
�+++
���
����
�+++
=∂∂
�
�
=
=
N
nn
N
nn
nkxdnknkXkxdk
nkxdnknkXckxdk
gk
XQ
1
1
sinsinh'sinsinh
coscosh'coscosh
ηη
ηη
nnn
nnn
n Eq. 55
Differentiating with respect to the wave number:
( )( ) ( )( ) ( )( ) ( )
( )( ) ( )( )( ) ( )
( )( ) ( )( ) ( )( ) ( )
( )( ) ( )( )( ) ( )
���������
�
�
���������
�
�
���
����
�+
����
�
�
����
�
�
++
+++
+
���
����
�++
����
�
�
����
�
�
+−
+++
+
=∂∂
��
��
==
==
N
nn
N
nn
n
n
N
nn
N
nn
n
n
nkxdnknkX
nkxdnkxXn
nkxdnkXdn
nkxdnknX
nkxdnknkXc
nkxdnkxXn
nkxdnkXdn
nkxdnknX
gkQ
11 2
2
11 2
2
sinsinh.
cossinh
sincosh
sinsinh
coscosh.
sincosh
cossinh
coscosh
.1
η
η
ηη
η
η
η
ηη
η
Eq. 56
Duwind 2003.025 Page 145 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
To minimise the error in the Bernoulli expression, Eq. 51, the differential is set to zero:
( ) ( )( )
( )0...2.
1
1 1
1
=���
�
�
��
���
��
���
−���
����
�′∆
∂∂
+∂∂
=′∆∂
∂� �= =
+ I
i
N
nn
n
jij
i
ji
j
QXXQ
QXQ
IXE
nn
Eq. 57
for each order of the Deans stream function:
[ ] 0.. 1
,1,
,1,
,11,11
,,,1 =����
�
�
����
�
�
−��
�
�
∆∆
���
�
�
�
∂∂∂∂∂∂∂∂∂∂∂∂
+���
�
�
�
∂∂∂∂∂∂ QX
X
Q
Q
Q
QQQN
NII
Nii
N
I
inInin Eq. 58
for all orders of the Deans stream function:
��
�
�=
����
�
�
����
�
�
−��
�
�
∆∆
���
�
�
�
∂∂∂∂∂∂∂∂∂∂∂∂
+���
�
�
�
��
�
�
∂∂∂∂∂∂∂∂∂∂∂∂
0
0.. 1
,1,
,1,
,11,11
,,,1
1,1,1,1 QX
X
Q
Q
Q
QQQ
QQQ
NNII
Nii
N
I
iNINiN
Ii Eq. 59
in matrix form:
[ ] [ ] [ ] [ ]( ) [ ]0.. =−∆∂∂+∂∂ QXQQQ TV Eq. 60
this can be rearranged to find the recommended value of ∆X:
[ ] [ ][ ]( ) [ ] [ ]( )QQQQQX VT −∂∂∂∂∂∂−=∆
−...
1 Eq. 61
The recommended change in the wavelength, λ, can also be found, and incorporated into the
matrix expression, Eq. 59, and solved in the same way:
���
�
�
�
=����
�
�
����
�
�
−���
�
�
�
∆∆
���
�
�
�
∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂
+���
�
�
�
���
�
�
�
∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂
000
..1
,,1,
,,1,
,1,11,11
,,,1
,,,1
1,1,1,1
Q
k
X
X
QQQ
QQQ
QQQ
Q
Q
Q
QQQ
QQQ
QQQ
N
kINII
kiNii
kN
I
i
kIkik
NINiN
Ii
Eq. 62
Three errors need to be minimised:
(i) wave height
(ii) mean surface level
(iii) Bernoulli pressure
The above algorithm reduces the error in the Bernoulli pressure.
Duwind 2003.025 Page 146 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
The error in the wave height and mean surface level can be reduced in the same manner as
used previously, by:
(i) multiplying the Deans stream function factors by an equal value until the wave height
is correct
(ii) varying the stream function value, ψη, at the surface (and hence the surface elevation
at the crest, ηx0) until the mean level is zero
If these steps are repeated, both errors will converge to zero.
App D.5 For Irregular Wave Profiles
The first stage of extending the stream function theory to irregular wave profiles is described
in this section, however time constraints meant that it could not be completed and
implemented. Starting with the main stream-function equation, additional asymmetrical sine
terms are added to the regular function, Eq. 1:
( ) { }( ) ( )( )�
= +���
����
�
+++=
N
n nN
n
nkxX
nkxXdynkcyyx
1 sin
cossinh,ψ Eq. 63
for regular waves, there are three conditions:
(i) wave height
(ii) mean surface level
(iii) Bernoulli pressure
of which only the last is applicable to irregular wave profiles, since the second condition
should be met by the prescribed profile and the first condition is replaced by the prescribed
profile.
App D.5.1 Wave Kinematics
Using the same conventions as above, the velocities are:
( ) ( )( )�
= +���
����
�
−−
+−=+−==−N
n nN
n
nkxX
nkxXsknnkcucu
dyd
1 sin
cos...cosh'
ψ
( ) ( )( )�
= +���
����
�
+−
==N
n nN
n
nkxX
nkxXnksnkv
dxd
1 cos
sin.sinh
ψ
Eq. 64
Duwind 2003.025 Page 147 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
To find the accelerations, Eq. 11 is applied in the global axes [x',y], as before, the terms are:
( ) ( )( )�
= +���
����
�
+−
=∂∂ N
n nN
n
nkxX
nkxXnksnk
tu
1
2
cos
sin.cosh.
' ω
( ) ( )( )�
= +���
����
�
−=
∂∂ N
n nN
n
nkxX
nkxXnksnk
xu
1
22
sin
cos.cosh
''
( ) ( )( )�
= +���
����
�
−−
=∂∂ N
n nN
n
nkxX
nkxXnksnk
yu
1
22
sin
cos.sinh
'
Eq.s 65
and Eq. 11 becomes:
( ) ( )( )
( ) ( )( ) ( ) ( )
( )
( ) ( )( ) ( ) ( )
( )��
��
�
= += +
= += +
= +
���
����
�
+−
���
����
�
++
���
����
�
−−
���
����
�
−+
���
����
�
+−
=
N
n nN
nN
n nN
n
N
n nN
nN
n nN
n
N
n nN
n
nkxX
nkxXnksnk
nkxX
nkxXnksnk
nkxX
nkxXnksnk
nkxX
nkxXnksnk
nkxX
nkxXnksnk
dtdu
11
22
11
22
1
2
cos
sinsinh.
sin
cossinh
sin
coscosh.
cos
sincosh
cos
sin.cosh.
' ω
Eq. 66
and in the local co-ordinate system, the terms of Eq. 10 become:
0=∂∂
tu
( ) ( )( )�
= +���
����
�
−=
N
n nN
n
nkxX
nkxXnksnk
dxdu
1
22
cos
sincosh
( ) ( )( )�
= +���
����
�
−−
=N
n nN
n
nkxX
nkxXnksnk
dydu
1
22
sin
cossinh
Eq.s 67
( ) ( )( ) ( ) ( )
( )
( ) ( )( ) ( ) ( )
( )��
��
= += +
= += +
���
����
�
−−
���
����
�
−−
+
��
�
����
����
�
−−
+−���
����
�
−+
=
N
n nN
nN
n nN
n
N
n nN
nN
n nN
n
nkxX
nkxXnksnk
nkxX
nkxXnksnk
nkxX
nkxXnksnkc
nkxX
nkxXnksnk
dtdu
11
22
11
22
cos
sinsinh
sin
cos.sinh
sin
coscosh
cos
sin.cosh
0'
Eq. 68
Duwind 2003.025 Page 148 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
The general expression for the vertical acceleration in the global axes, Eq. 16, can be applied
as previously:
( ) ( )( )�
= +���
����
�
−−
=∂∂ N
n nN
n
nkxX
nkxXnksnk
tv
1
2
sin
cossinh.ω
( ) ( )( )�
= +���
����
�
−−
=∂∂ N
n nN
n
nkxX
nkxXnksnk
xv
1
22
sin
cos.sinh
`
( ) ( )( )�
= +���
����
�
−−
−=∂∂ N
n nN
n
nkxX
nkxXsknnk
yv
1
22
sin
cos..cosh.
Eq.s 69
and hence Eq. 16 becomes:
( ) ( )( )
( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )��
��
�
= +=
= +=
= +
���
����
�
−−
+
���
����
�
−−
+
���
����
�
−−
=
N
n nN
nN
n
N
n nN
nN
n
N
n nN
n
nkxX
nkxXnksnknkxnksnk
nkxX
nkxXnksnknkxnksnk
nkxX
nkxXnksnk
dtdv
11
22
11
22
1
2
sin
cossinhsincosh
sin
coscoshcossinh
sin
cossinh.ω
Eq. 70
Likewise, to check this expression, the differentiation can be performed in the local axes
[x ,̀y ]̀, Eq. 10, where:
0=∂∂
tv
( ) ( )( )�
= +���
����
�
−−
=∂∂ N
n nN
n
nkxX
nkxXnksnk
xv
1
22
sin
cossinh
'
( ) ( )( )�
= +���
����
�
−−
=∂∂ N
n nN
n
nkxX
nkxXnksnk
yv
1
22
sin
coscosh
Eq.s 71
hence Eq. 16 becomes:
( ) ( )( ) ( ) (
(
( ) ( )( ) ( ) ( )
( )��
��
= += +
= += +
���
����
�
−−
���
����
�
−−
+
���
����
�
−−
+���
����
�
−−
+=
N
n nN
nN
n nN
n
N
n nN
nN
n nN
n
nkxX
nkxXsknnk
nkxX
nkxXnksnk
nkxX
nkxXnksnkc
nkxX
nkxXnksnk
dtdv
11
22
11
22
sin
cos..sinh
sin
coscosh
sin
coscosh
sin
cossinh0
Eq. 72
Duwind 2003.025 Page 149 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
The pressure can be found as previously:
( ) ( )( )
( ) ( )( ) ( ) ( )
(
( ) ( )( ) ( ) ( )
(nkxX
nkxXnksn
nkxX
nkxXnksnk
nkxX
nkxXnksn
nkxX
nkxXnksnk
nkxX
nkxXnksnk
P
N
n nN
nN
n nN
n
N
n nN
nN
n nN
n
N
n nN
n
sin
cossinh
sin
cossinh
sin
coscosh
sin
coscosh
sin
coscosh.
.
11
23
11
23
1
2
��
��
�
����
�
����
�
�
���
�
−−
���
����
�
−−
+
���
�
−−
���
����
�
−−
+
���
����
�
−−
−=
= += +
= += +
= +ω
ρ
Eq. 73
App D.5.2 Finding the Solution
Substituting into the expression for the Bernoulli pressure Eq. 49:
( )( )( ) ( )
( )
( )( ) ( )( ) �
�����
�
�
������
�
�
���
����
����
����
�
+−
++
���
����
����
����
�
++
++
+=
�
�
= +
= +
2
1
2
1
cos
sin.sinh
sin
coscosh
.21
,N
n nN
n
N
n nN
n
nkxX
nkxXdnknk
nkxX
nkxXdnknkc
gxQ
η
ηηη Eq. 74
Differentiating with respect to the Deans stream function parameter:
( )( ) ( )( )( )( ) ( )( ) (
( )( ) ( )( )( )( ) ( )( ) (
(������
�
�
���
����
�
+−
+���
����
�
≤<−≤
++
���
����
�
+++��
�
����
�
≤<−≤
+
=∂∂
�
�
= +
= +
N
n nN
n
N
n nN
n
X
nkxXdnknk
NNkx
Nkxdk
X
Xdnknkc
NNkxNkx
dk
gk
XQ
1
1
cos
sinsinh.
2sincos
sinh
sin
coscosh.
2sincos
cosh
.
ηη
ηη
nnnn
nn
nnnn
nn
n
Eq. 75
Differentiating with respect to the wave number:
( )( )( ) ( )( )
( )( )
( )( ) ( )( )
( )( ) ( )( )
( )( )( ) ( )( )
( )( )
( )( ) ( )( )
( )( ) ( )( )
������������
�
�
������������
�
�
���
����
����
����
�
+−
+
�����
�
�
�����
�
�
���
����
�++
���
����
�
+−
���
����
�
+++
++
���
����
����
����
�
−−
++
�����
�
�
�����
�
�
���
����
�
+−
++
���
����
����
����
�
+++
+
=∂∂
��
��
= +=
+
+
= +=
+
+
N
n nN
nN
n
nN
n
nN
n
N
n nN
nN
n
nN
n
nN
n
nkxX
nkxXdnknk
nkxX
nkxXdnkkxn
nkxX
nkxX
dnkdnkdnk
n
nkxX
nkxXdnknkc
nkxX
nkxXdnkkxn
nkxX
nkxX
dnkdnk
dnkn
gkQ
11 2
11 2
cos
sinsinh
sin
cossinh.
cos
sin
coshsinh
.sin
cos.cosh
cos
sincosh
sin
cos
sinhcosh
.1
ηη
ηηη
ηη
ηηη
Eq. 76
To minimise the error in the Bernoulli expression, Eq. 51, the differential is set to zero:
( ) ( )( )
( )0...2.
1
1 1
1
=���
�
�
��
���
��
���
−���
����
�′∆
∂∂
+∂∂
=′∆∂
∂� �
= =
+ I
i
N
nn
n
jij
i
ji
j
QXXQ
QXQ
IXE
nn
Eq. 77
Duwind 2003.025 Page 150 Hydrodynamic Loading on Offshore Wind Turbines
OWTES-Blyth Project March 2003 Section Wind Energy, TUDelft
Solve by:
(i) estimate preliminary parameters from shallow water solution
(ii) apply Eq. 77 iteratively to find solution
The error in the value of the stream function at the surface should be solvable in the same
manner:
Differentiating the stream function with respect to the Deans stream function parameter:
( )( ) ( )( )( )( )���
����
�
≤<−≤
+=∂∂
NNkx
Nkxdnk
X 2sincos
sinhnn
nn
nηψ
Eq. 78
Differentiating with respect to the wave number:
( ) ( )( ) ( )( )
( )( ) ( )( )�
�
= +
= +
���
����
�
+−
++
���
����
�++=
∂∂
N
n nN
n
N
n nN
n
nkxX
nkxXdnkn
nkxX
nkxXdnkdn
k
1
1
cos
sin.sinh
sin
cos.cosh
η
ηηψ
Eq. 79