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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


Duwind 2003.025 Page i Hydrodynamic Loading on Offshore Wind Turbines


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


• 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


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


Duwind 2003.025 Page v Hydrodynamic Loading on Offshore Wind Turbines


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


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


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


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


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



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



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.



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?



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



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



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



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

]



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



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



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.



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



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:



• 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



(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



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



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.



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



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.



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.



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



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



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]



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).



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


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


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


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]



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.



( )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)



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%



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 √



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.



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



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]).



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



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



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


Time [s]

Shea

r For

ce [k

N]

0 5 10 1520

10

0

10

20

30


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%.



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



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.



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.



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.



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



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.



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.



-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



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]



(ii) IEC [43].



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



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.



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.



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

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10 10 10

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11

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19

19

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20

20

Figure 30: Bathymetry at Blyth

East - West [x10m]

North - South [x10m]

Southern Windturbine

Southern Windturbine



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



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.



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.



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



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]



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].



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



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.





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]

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eigh

t [m

]

Breaking

Waves

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Wav

e H

eigh

t [m

]

Breaking

Waves



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GrDataT

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2 1071.5 107

1.5 107

1 107

1 107

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1 107

5 106

5 106

5 106

Figure 39: Overturning Moment Extrapolated (left) and Constant (right) Crest

GrDataT

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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



GrDataT

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2 1062 106

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1 1061 106

1 106

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5 105

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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



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

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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



Gr ZeroT

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Gr ZeroT

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35

35

35

40

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40

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45

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55 55

60

60

Figure 44: Error in Overturning Moment: Wheeler (left) & Chakrabarti (right)

Gr ZeroT

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Gr ZeroT

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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



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


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


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







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

]



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

]



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

]



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



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



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].



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.



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



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



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



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.



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



• 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.



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.



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

]



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]



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



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

]



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]



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.



0

100

200

300

400

500

600

5 7 9 11 13 15Wave Period [s]

Sur

ge F

orce

[kN

]



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

]



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



Figure 75: Inertia Coefficients (2m)

0

2

4

6

8

10

12

5 7 9 11 13 15Wave Period [s]

OTM

[MN

m]



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.



0

200

400

600

800

1000

1200

1400

5 7 9 11 13 15Wave Period [s]

Sur

ge F

orce

[kN

]



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

]




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



Figure 79: Inertia Coefficients

0

2

4

6

8

10

12

5 7 9 11 13 15Wave Period [s]

OTM

[MN

m]



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



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

]



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

]



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%.



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



Figure 85: Error in Froude-Krylov

01

23

456

78

910

5 7 9 11 13 15Wave Period [s]

OTM

[MN

m]




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

]



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

]




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.



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



Figure 89: Error in Froude-Krylov

01

23

45

67

89

10

5 7 9 11 13 15Wave Period [s]

OTM

[MN

m]




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

]


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

]



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



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



Figure 93: Inertia Coefficients (Morison)

0

2

4

6

8

10

12

5 7 9 11 13 15Wave Period [s]

OTM

[MN

m]



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

]




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



Figure 96: Inertia Coefficients (Morison)

The recommended force coefficients are collated in to the following Table VIII.



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

]



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

]



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



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



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

]



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

]


18m Diffraction 12m Froude-Krylov 15m Morison


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



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:



- 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


Hea

ve F

orce

[kN

]


Figure 103: Surge (left) and Heave (right) Forces on Deepwater GBS (at 15m depth)

0

1

2

3

4

5

6

7


OTM

[M

Nm

] DiffractionAiry 2mWheeler 2mDeans 2m

Figure 104: OTM

0

1000

2000

3000

4000

5000

6000


Sur

ge F

orce

[kN

]


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



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


Hea

ve F

orce

[kN

]


0

5

10

15

20

25

30


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


Sur

ge F

orce

[kN

]


0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000


Hea

ve F

orce

[kN

]


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



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



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



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



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.



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)



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)



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



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



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



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


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


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



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



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.



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



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



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.



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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90 Spell, C.A., Zhang, J. & Randall, R.E., Hybrid wave model for unidirectional irregular

waves - Part II. Comparison with laboratory measurements, Applied Ocean Res, Vol 18,

pp 93-110, 1996

91 Stansberg, C. T., Nonlinear extreme wave evolution in random wave groups, ISOPE,

2000

92 Stokes, G. G., On the theory of oscillatory waves, Mathematical and Physical Papers, 1,

Cambridge University press, 1880

93 SWAN, Simulated Waves Nearshore, http://swan.ct.tudelft.nl/

94 Tanimoto, K., Takahashi, S., Kaneko, T. and Shiota, K., Impulsive breaking wav forces

on an inclined pile exerted by random waves, Proc. 20th Intern. Conf. On Coastal

Engineering, Vol. 3, pp 2288-2302, 1986.



95 Tayfun, M.A. Joint distribution of large wave heights and associated periods, Journ.

Waterw. Port, Coast. & Ocean Eng., Vol. 119, No. 3, p 261, 1993.

96 Tromens, P.S., Anaturk, A.R., Hagemijer, P.M., A new model for the kinematics of large

ocean wave-application as a design wave. ISOPE, 1991.

97 Tucker, M.J., Waves in Ocean Engineering; measurement, analysis, interpretation; Ellis

Horwood, ISBN 0-13-932955-2, 1991

98 Tucker, M.J., Nearshore wave heights during storms, Coastal Engineering 24, pp 111-

136, 1994.

99 Tucker, M.J., Challenor, P.G. and Carter D.J.T., Numerical simulation of a random sea: a

common error and its effect upon wave group statistics, Applied Ocean Research, Vol. 6,

No. 2, pp 118-122, 1984

100 Verruijt, A., Offshore Soil Mechanics, Delft University of Technology, Delft, August

1998.

101 Vrijking, J.K., Probabilistic design in Hydrodynamic Engineering (in Dutch), Delft

University of Technology, 1996

102 von Kármán, Th., The impact forces on a seaplane float during landing, National

Advisory Committee for Aeronautics, NACA TN 3221, 1929.

103 Wagner, H., Über Stoß- und Gleitvorgänge an der Oberfläche von flüssigkeiten,

Zeitschrift für angewandte Mathematik und Mechanik, Vol. 12, Nr. 4, pp 193-215 (ihnj

German), 1932.

104 Wienke, J. et al, Breaking Wave Impact on a Slender Cylinder, Coastal Engineering

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105 Wienke, J. Sparboom, U. and Oumeraci, H., Large-Scale Experiments with Slender

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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.



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.



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



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



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.



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



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



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.



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.



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]



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


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).



0.01 0.1 1 100.01

0.1

1

10

pi.D / wave-length

Keu

lega

n-C

arpe

nter

Num

ber


Figure 147: Recommended Waveload Model



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,



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



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



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



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



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.



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



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)



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



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



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



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



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



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..


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..


11

1

2

1

��

��

==

==

=

N

nn

N

nn

N

nn

N

nn

xknsknXnk

xknsknXn

dxxknsknXnxknsknXn

Eq. 36



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:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )


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




( ) ( ) ( ) ( )

( ) ( ) ( ) ( )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:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )


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


( ) ( ) ( ) ( )

( ) ( ) ( ) ( )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



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.



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



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

QQ

QQ

QQ

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

QQ

QQ

QQ

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.



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



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



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



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


( )( )( ) ( )( )

( )( )

( )( ) ( )( )

( )( ) ( )( )

( )( )( ) ( )( )

( )( )

( )( ) ( )( )

( )( ) ( )( )

��

�

�

��

�

�

��

��

��

��

�

+−

+

��

�

�

��

�

�

��

��

�++

��

��

�

+−

��

��

�

+++

++

��

��

��

��

�

−−

++

��

�

�

��

�

�

��

��

�

+−

++

��

��

��

��

�

+++

+

=∂∂

��

��

= +=

+

+

= +=

+

+

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



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


( ) ( )( ) ( )( )

( )( ) ( )( )�

�

= +

= +

��

��

�

+−

++

��

��

�++=

∂∂

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

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