energies
ArticleDynamic Response Analysis of a Semi-Submersible
Floating Wind Turbine in Combined Wave andCurrent Conditions Using Advanced
Hydrodynamic Models
Takeshi Ishihara

and Yuliang Liu *Department of Civil Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo,
Tokyo 113-8656, Japan; ishihara@bridge.t.u-tokyo.ac.jp* Correspondence: yuliang.liu@bridge.t.u-tokyo.ac.jp
Received: 12 October 2020; Accepted: 4 November 2020; Published: 6 November 2020


Abstract: In this study, advanced hydrodynamic models are proposed to predict dynamic response of
a floating offshore wind turbine (FOWT) in combined wave and current conditions and validated bylaboratory and full-scale semi-submersible platforms. Firstly, hydrodynamic coefficient models are
introduced to evaluate the added mass and drag coefficients in a wide range of Reynolds numbers.
An advanced hydrodynamic model is then proposed to calculate the drag force of cylinder in
combined wave and current conditions. The proposed model is validated by the water tank tests inthe current-only, wave-only and current-wave conditions and is used to investigate the effect of current
on the dynamic response of FOWT. Finally, the full-scale semi-submersible platform used in the
Fukushima demonstration project is investigated. It is found that the predicted dynamic responses ofplatform by the proposed hydrodynamic models are improved by the directional spreading functionof the sea wave spectrum and show favorable agreement with the field measurement.
Keywords: semi-submersible floating offshore wind turbine; hydrodynamic models; dynamic
response; mooring tension; combined wave and current conditions; directional spreading function

1. Introduction
Hydrodynamic forces due to wave and current, such as inertia force and viscous force,play an important role in the design of floating offshore wind turbines (FWOTs), and the dynamic
response analyses are required for the specific design load cases according to design standards andguidelines [1–4] to ensure FOWTs have adequate stability and structural strength.
The added mass and drag coefficients significantly affect the motion of FOWT and associatewith the Keulegan-Carpenter number, KC, Reynolds number, Re, and surface roughness [5].
Sarpkaya [6,7] conducted a series of experiments and investigated the hydrodynamic coefficients ofthe two-dimensional cylinders in the oscillatory and steady flows with a wide range of Reynolds
numbers. It was found that added mass and drag coefficients of the cylinders approached a constantvalue as Reynolds number increased. Zhang and Ishihara [8] examined the effects of KC number,thickness and diameter of heave plate on the hydrodynamic coefficients by the numerically forced
oscillation tests. Tanaka et al. [9] displayed the variation of the coefficients of rectangular cylinders
with ratios of width to height and circular corners at low KC numbers. Liu and Ishihara [10] proposed
a hydrodynamic coefficient model to estimate the hydrodynamic coefficients for different components
in a semi-submersible FOWT to take the influences of Reynolds and KC numbers into account.
The proposed hydrodynamic coefficients were based on the relevant experiments and simulations at
low Reynolds numbers for laboratory-scale models and presented as functions of Re and KC numbers.
Energies 2020, 13, 5820; doi:10.3390/en13215820

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Energies 2020, 13, 5820

2 of 22The laboratory-scale model for a floating structure is generally based on Froude number to ensure
similarity between the laboratory-scale model and prototype. However, the equality in Reynolds
number cannot be achieved by Froude scaling law. Thus, new hydrodynamic coefficient models arenecessary to cover a wide range of Reynolds numbers and consider the effects of KC number for the
dynamic response analyses of the laboratory- and full-scale platforms.
Accurate prediction of the current load on the platform is critical for the safety of mooring lines.
Chen and Basu [11] investigated the effects of current load and wave–current interactions on thefatigue load of a spar-type FOWT. It was found that the current had a notable influence on tower
responses and mean tensions of mooring lines due to the static offset. The fatigue life of mooringline was overestimated if the wave–current interaction was neglected. Soeb et al. [12] observedthat the current loads compressed the oscillation of surge, heave and pitch for a spar-type platform.
However, the predicted dynamic responses of the platform and mooring line in these researches
were not validated by the experiments and the same drag coefficient was used to predicted currentand wave-induced drag forces. However, Sarpkaya [13] indicated that the drag coefficient of acircular cylinder in a oscillatory flow is different from that in a steady flow. It implies that the dragforces from the steady and oscillating flows should be considered at the same time in combined
wave-current conditions.
The predicted dynamic responses have been compared with numerous laboratory experiments.
However, validations by the full-scale FOWTs are limited. The predicted dynamic response wascompared partly with the measured data from the spar-type floating wind turbine in the Hywind demo
project [14,15]. Some disagreements in the prediction of dynamic response occurred due to the strong
nonlinear phenomenon, which was not included in the hydrodynamic model. The low-frequency
motion of a spar-type substation was analyzed recently based on the measurement data [16] inthe Fukushima demonstration project. However, the semi-submersible platform in the Fukushima
demonstration project, which is much more complex than the spar-type platform, has not been studied
yet. In addition, ocean waves cannot be adequately described by the frequency spectrum alone due tothe existence of wave propagating in various directions. Driscoll et al. [14] examined the numericalmodel through the field measurement and observed that the roll motion in the low-frequency range
was underestimated. The possible reason was supposed that the directional spreading of the wave
spectrum was neglected in the numerical model, while it exists in the ocean wave. Zhang andIshihara [17] pointed out that the spreading function with the exponent of 2 in the wind wave condition
has a significant effect on the dynamic response of motion on a 2 MW FOWT used in the Fukushima
demonstration project. However, the exponent in the spreading function should be identified based on
the measurement data since the wind wave and swell exist at the Fukushima site facing the Pacific
Ocean [18]. The effect of the spreading function on the motion of semi-submersible platform in different
wave conditions has not been validated by the field measurement.
The aim of this study is to propose advanced hydrodynamic models and predict dynamic responseof a floating wind turbine in the combined wave-current conditions for laboratory and full-scalesemi-submersible platforms. In Section 2, global hydrodynamic coefficient models considering theeffect of Reynolds and KC numbers are firstly proposed for laboratory- and full-scale semi-submersible
platforms. Advanced hydrodynamic models are then proposed to evaluate the drag force in the
combined wave-current condition. Finally, the wave model considering the directional spreading
function is introduced. In Section 3, the proposed global hydrodynamic coefficient models are validated
firstly by the added mass and drag coefficients from the forced oscillation tests. The effects of the
combined wave and current on the platform motion and mooring tension are then examined by thenumerical model and validated by the water tank tests. Finally, the effect of the directional spreading ofwave spectrum on the dynamic responses of the 2 MW semi-submersible FOWT used in the Fukushima
demonstration project is investigated and compared with those obtained from the field measurementsin parked conditions. Conclusions from this study are summarized in Section 4.

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2. Numerical Models
2. Numerical Models
Two laboratory-scale semi-submersible platforms are presented in Section 2.1. One is a 1:50 scaleTwosemi-submersible
platforms
arecurrent
presented
in SectionThe2.1.other
One isis aa 1:60
1:50 scalemodel
andlaboratory-scale
is used in the experimentswith thewave andconditions.
is usedused inin thetheforced
experiments
withtests
the waveand current conditions.
The other
is aSection
1:60 scalemodel and isoscillation
by Computational
Fluid Dynamics(CFD).
2.2
model theandmodelsis used toin evaluate
the forcedthe
oscillation
tests bycoefficients
Computational
Fluid
Dynamicsof(CFD).
Section 2.2
showshydrodynamic
for eachcomponent
the platform,inshows theevaluate
the hydrodynamic
coefficients
eachKC
component
the platform,
which
and Cdto aremodelled
as functions ofReynoldsfor
andnumbersofbasedon theCa modelsin which Ca andCdobtained
are modelledof Reynolds
and KC
numbers
on the experimental
experimental
datafrom asthefunctions
laboratoryand full-scale
platforms
in based
the oscillating
and steadydata obtained
from thelaboratoryfull-scale
platforms
in the oscillating
and steadyflows.
flows.
The formulas
to calculate
the andglobalhydrodynamic
coefficientsare described.
A modeltoTheformulastocalculatetheglobalhydrodynamic
coefficientsaredescribed.
A
model
to
calculatecalculate the drag force in the combined wave-current condition is then proposed for the dynamic
the drag force
in thewave-current
conditionis thenproposed
forfromthe dynamic
responses
responses
analysisincombined
Section 2.3,
which combines
the drag
forcesobtainedthe oscillatory
andanalysisflows.
in Section
2.3,thewhich
combines
drag forcesin
obtainedsteadyFinally,wave
model isthesummarized
Section from
2.4. the oscillatory and steady flows.
Finally, the wave model is summarized in Section 2.4.
2.1. Description of Platform Models
2.1. Description of Platform Models
The two scale models are developed according to the 2 MW semi-submersible FOWT used in
The two scalemodels are developed
accordingto the2 MW ofsemi-submersible
FOWT
thethe Fukushima
demonstration
project. The
platform
consistsa center column
and used
threeinside
Fukushima
demonstration
project.
The
platform
consistsofacentercolumn
and
three
side
columns,
columns, which are connected to the center column by pontoons, braces and decks. The pontoons arewhich areas
connected
to thecenter column
by pontoons,and decks.
The pontoons
are designed
designed
rectangularcylinderswith tapered
sections.braces
The skirts
are attachedto the bottomof theasrectangularcylinderswithtaperedsections.
The
skirtsareattachedtothebottomof theheave
heave plates and pontoons for suppression of the platform motion and shift of the natural
periodinplatesanddirection,
pontoonswhich
for suppression
the2platform
motionand scaleshift ofthe naturalin thetheheave
are used forofthe
MW FOWTand 1:60
model,
but notperiod
attached
for
heave
which are used for the 2 MW FOWT and 1:60 scale model, but not attached for thethe
1:50direction,
scale model.
1:50 scalemodel.
Figure 1 shows the schematic of the 1:60 scale platform and the right-handed coordinate, where
Figurelocates
1 showsatthe
schematicof thecenter
1:60 scale
the right-handed
coordinate,
where
the
the origin
thegeometricalon platformthe waterandplanearea, with positive
x-axisin theoriginlocatesatthegeometricalcenter
on
the
water
plane
area,
with
positive
x-axisinthedirectionof
direction of wave propagation and z-axis upward. The geometry of the 1:50 scale platform is similar
wave
propagation
andz-axisupward.
The
geometry
of
the
1:50
scale
platformissimilartothatofto that of the 1:60 scale model except that the skirts are not attached. The detailed geometry andthe 1:60 scalemodel except
skirtsarearenotdescribedattached. inThegeometryhydrostatichydrostatic
propertiesof thethat
1:50thescale
model
Liudetailed
and Ishihara[10]. andTable
1 lists theproperties
of
the
1:50
scale
model
are
described
in
Liu
and
Ishihara
[10].
Table
1
lists
the
summaryofsummary of geometry and hydrostatic properties of the 1:60 scale platform.
geometry and hydrostatic properties of the 1:60 scale platform.

(a)

(b)

(c)

Figure 1.
1. Schematic
Schematicofofthe
the semi-submersible
semi-submersible platform.
platform. (a)(a)BirdBirdview,view, (b)(b) top
top viewview andand (c)(c) side
side view.
Figure2.2. Global Hydrodynamic Coefficient Models Considering the Effects of Reynolds and KC Numbers
Table 1. Dimension and hydrostatic properties of the 1:60 scale semi-submersible platform.

Numerical simulations using the large eddy simulation as described by Pan and Ishihara [19] are
Elements
Dimension (m)conducted to calculate the hydrodynamic coefficients of the 1:60 scale platform at various Reynolds
Draft of the platform
0.2667and KC numbers. The numerically forced oscillation tests are carried out and the movement of the
Height of center column (Hcc) and side columns (Hsc)
0.4667model is simulated by dynamic mesh with the layering mesh update method [19].
Diameter of center column (Dcc)
0.0823
Diameter of side columns (Dsc)
0.1250Height of heave plate (HHp) and pontoon (HPn)0.0667
Diameter of heave plate without skirt (DHp)
0.2333
Width of pontoon
0.05~0.10
Length of pontoon (LPn)
0.2827
Diameter of brace (Dbrace)
0.0375

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Table 1. Dimension and hydrostatic properties of the 1:60 scale semi-submersible platform.
Elements

Dimension (m)Draft of the platform
Height of center column (Hcc ) and side columns (Hsc )Diameter of center column (Dcc )Diameter of side columns (Dsc )Height of heave plate (HHp ) and pontoon (HPn )Diameter of heave plate without skirt (DHp )Width of pontoon
Length of pontoon (LPn )Diameter of brace (Dbrace )Length of brace (Lbrace )Center of gravity below still water levelRadius of gyration Kxx
Radius of gyration Kyy

0.2667
0.46670.08230.12500.06670.23330.05~0.10
0.28270.03750.3887−0.0687
0.370
0.374

The KC number, Reynolds number and frequency parameter, β, for the oscillating platform modelare defined as:
uT
2πa
KC =
=
(1)
L
L
ρuL
uL
2πaL
Re =
=
=
(2)
µ
υ
Tυ
β=

Re
L2
=
KC
υT

(3)

where L is the characteristic length of the model, and u and T indicate the amplitude of velocity andperiod of oscillation, respectively. The product of u and T can be replaced by 2πa in the sinusoidal
oscillation, where a is the amplitude of the movement. µ and υ denote the dynamic and kinematic
viscosities of fluid. Diameters of side column and heave plate are used as the characteristic length, L,
for the platform in the horizontal and vertical directions, respectively.
In the forced oscillation tests in still water, the displacement of the model is described as:x(t) = a sin(ωt)(4)

where ω = 2π/T is the frequency of oscillation and a is the amplitude of oscillation. The hydrodynamic
force acted on the platform can be expressed as:
FH (t) = F(t) − Fb − FK (t)(5)

where total force, F(t), is integrated from pressure on the surface of the model, Fb is constant buoyancy
force of the platform at the equilibrium position in still water and FK (t) indicates variation of buoyancy
force due to movement of the platform.
According to Morison’s equation, the hydrodynamic force FH (t) can be expressed as:..
.
.
1
FH (t) = −Ca Mx(t) − Cd ρw A x(t) x(t)2(6)

where M and A represent mass of displaced water and characteristic area of the model in corresponding
movement direction respectively, and ρw indicates the water density. In this study, the characteristicarea is defined as projected area in the YZ plane in the horizontal direction and sum area of heave
plates without skirts in the vertical direction. They are expressed as:
HHp + NPn LPn HPn sin(θX ) + Nsc Dsc Hsc + Dcc Hcc + Nbrace Dbrace Lbrace cos(θYZ ); NHp DHp
A=
πD2Hp

 N
;
Hp

4

in horizontal direction
in vertical direction(7)

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where NHp , NPn , Nsc , Ncc and Nbrace are the number of heave plates, pontoons, center columns,
side columns and braces, DHp and HHp indicate the diameter and height of heave plates without theskirts, LPntn and HPntn are the length and height of pontoons without the skirts and Dcc , Dsc and Dbrace
are the diameters of center column, side column and brace. Hcc and Hsc are the heights of center
column and side column, Lbrace is the length of brace and θX represents the angle between the axial
direction of element and the global X coordinate, as described in Ishihara and Zhang [20].
Fourier averaged added mass and drag coefficients can be calculated as:
Ca =

1
πMωa

Z T

3
Cd = −
4ρw Aωa2

FH (t)sin(ωt)dt

(8)

0

Z T

FH (t)cos(ωt)dt(9)0

The hydrodynamic coefficients of the cylinders are influenced by several factors, including geometry,interaction between cylinders, KC and Reynolds numbers. The hydrodynamic coefficient models
proposed by Liu and Ishihara [10] focus on the added mass and drag coefficients for each componentof the FOWT, with consideration of these factors as:
k k
k k
k k
k
k
k
i Ca (β0 , KC0 , η ) = r Ca (β0 , KC0 ) × r γa × i ηa

(10)

k k
k k
k k
k
k
k
i Cd (β0 , KC0 , η ) = r Cd (β0 , KC0 ) × r γd × i ηd(11)where subscripts i and r denote the component i and the referenced component r, and superscript k
indicates the direction of hydrodynamic coefficients, which can be expressed as the normal direction byn and the axial direction by t. i Cka (βk0 , KCk0 , ηk ) and i Ckd (βk0 , KCk0 , ηk ) represent the added mass and dragcoefficients of the component i at the frequency parameter β and KC number in the k direction, including
the contribution of the interaction effect η. r Cka (βk0 , KCk0 ) and r Ckd (βk0 , KCk0 ) mean the representative Ca
and Cd for the referenced component r at the typical frequency parameter β0 and KC0 number in the kdirection. Correction factors i ηka and i ηkd are introduced to account for the interaction phenomenon
between components. Correction factors r γka and r γkd represent the hydrodynamic coefficients of thecylinder as functions of KC and Reynolds numbers.
The original correction factors r γka and r γkd were proposed based on the experimental data at the
low Reynolds numbers in the oscillating flow in Liu and Ishihara [10] and were presented as thefunctions of KC and Reynolds numbers. Since the correction factors are not available for the full-scalemodel at the high Reynolds numbers, the model is extended to a wide range of Reynolds numbers
based on both laboratory- and full-scale models in this study.
For circular cylinders, the hydrodynamic coefficients are mainly dependent on Reynolds numbers.
The correction factors r γka and r γkd for circular cylinders are extended to the high Reynolds number toevaluate the Ca and Cd for full-scale models. The formulas for r γka and r γkd are obtained by the curve
fitting based on the coefficients from the numerical simulations of cylinders [19] at the low Re andforced oscillation tests of cylinders at the high Re. Figures 2 and 3 illustrate Ca and Cd for SC-2 as shownin Figure 1 as well as data obtained from the numerical simulations and experiments. The predicted
coefficients are consistent with the experimental data from Re = 105 to Re = 106.17 at KC = 40 and
KC = 60. The added mass and drag coefficients approach constant values as the Reynolds number
increases. The added mass and drag coefficients are also assumed to approach constant values in therange of Re < 104 .

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66 of
of 22
22

SC-2
CCa a ofof SC-2

33

Exp.
Exp.
CFD
CFD
Cal.
Cal.

22

11

00

3
10
103

4
10
104

5
10
105
Re
Re ==  *KC
*KC

6
10
106

7
10
107

Figure
for
SC-2
with
Reynolds(Re)
numberintheoscillatory
flow.
Figure2.2.Variation
Variation
ofof C
forSC-2SC-2with
withReynolds
Reynolds(Re)(Re)number
numberin
inthe
theoscillatory
oscillatoryflow.
flow.
Figure2.
Variationof
Caaa forC

SC-2
CCd d ofof SC-2

33

Exp.
Exp.
CFD
CFD
Cal.
Cal.

22

11

00

3
10
103

4
10
104

5
10
105
Re
Re ==  *KC
*KC

6
10
106

7
10
107

Figureof
SC-2
with
Reynolds
numberintheoscillatory
flow.
Figure3.
Variationof
Cdd ofFigure3.3.Variation
Variation
ofof C
ofSC-2SC-2with
withReynolds
Reynoldsnumber
numberin
inthe
theoscillatory
oscillatoryflow.
flow.
C
d

Table 2 summarizes Table
the correction
factors of circularcylinders, which change with Reynolds
Table 2.
2. List
List ofof correction
correction factors
factors forfor eacheach component.
component.
number. Since pontoons, heave plates and skirts have the sharp edges, the hydrodynamic coefficients
k
k
k and
k
Component
r
aThe
rd
of them do not dependon the oscillation frequency.
hydrodynamic
coefficients are dependent
Componentr  a and r  d
4
on KC number, which have the same values
for the laboratory- and full-scalemodels.
It should be
1.12
Re
1.12
Re  10104n
4
5 by the oscillation
noted that the hydrodynamic coefficients
of
the
heave
plateswillbelargelyaffectedr an  0.85sin(20.85sin(2 log
log ReRe 0.7)0.7)  0.40
0.40 10104  Re
Re  10105
frequency if the free surface
effectrisasignificant,as mentioned by Wadhwa and Thiagarajan[21], such as
SC-25
0.28 tanh(log Re 4.9)  0.47
SC-2
5
Re
 10
0.28
tanh(log
Re

4.9)

0.47
Re
10

the plates close to thewater surface. This phenomenon does not occur in this study since the draft of
Hp-2
Hp-2
4.8

Re
the platform is large enough.

n
1.05
1.05 tanh(2.5log
tanh(2.5log ReRe11.4)11.4) 
1.68
1.68
Re
 10
104.8

n  
r
d
r  d  0.87 tanh(6.0 log Re 30)  1.52
4.8
Re>104.8
 0.87 tanh(6.0 log Re 30)  1.52of correction factors for each component. Re>10
Table 2. List 
3.5
1.10Re
1.10
Re  10103.5
k and γkn3.5γComponentn0.44sin(1.4 log Rer a3.2) r 0.66
10 3.5  Re  1055
d
Br-2
r
a  
0.44sin(1.4 log Re  3.2)  0.66 10  Re 4 10
Br-2
ra 
 1.12
5

Re Re
≤ 10

5
0.22
tanh(logRe 4.9) 0.37
 510
0.22
tanh(log
Re
4.9)
0.37
Re
10

n
4

 0.85 sin(2 log Re − 0.7) + 0.40 10 < Re < 10r γa = n
SC-2 0.28tanh(log Re Re−)5.28)+ 0.47
Re ≥ 105r dn  0.40
0.40 tanh(1.2log
tanh(1.2log
Re4.9
 5.28)
 0.73
0.73
Hp-2

r

d

1.05tanh(2.5 log Re − 11.4KC) + 1.68
n0.77
KC  1.0
1.0
r γdn = 
0.77(Re ≤ 104.8

n−0.87tanh(6.0 log Re − 30) + 1.52 Re > 104.8ra  
0.34 KC  1 0.3
0.3  0.35ra 
3.5
KC  1.0
 0.34
 1.10 KC  1  0.35 KC  1.0 3.5 Re ≥ 10 50.44 sin(1.4 log
10 < Re < 10
0.33Re − 3.2) + 0.66
0.45 KC  1 0.33  1.93
n
Br-2

r
d 



0.45
KC

1

1.93
0.22tanh
(
log
Re
−
4.9
)
+
0.37
Re ≥ 105r d
nr γd = −0.40tanh(1.2 log Re − 5.28) + 0.73(
0.77
KC <a 1.0
Figure4
illustrates
variationofhydrodynamic
coefficientsnFigure 4 illustrates variationofhydrodynamic
coefficients ofof a squaresquare cylinder
cylinder with
with KC
number.
KC number.
γ
=
r a
0.3
(
)
0.34
KC
+
1
+
0.35
KC
≥
1.0
Pn-2
The
coefficients
predicted
bytheproposedmodel
areconsistent
with
those
obtainedfromtheThe coefficients predicted by the nproposed model 0.33
are consistent with those obtained from the forced
forced
+ 1.93r γd = −0.45(KC + 1)
oscillation tests [9].
Pn-2
Pn-2

oscillation tests [9].

nr γan = 






Energies 2020, 13, 5820

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Figure 4 illustrates variation of hydrodynamic coefficients of a square cylinder with KC number.
Thecoefficients
predicted
by the proposed model are consistent with those obtained from the forced
Energies
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22
oscillation tests [9].
33

33

Exp.
Exp.
Cal.
Cal.

Exp.
Exp.
Cal.
Cal.

22
d

a

C
C
d

C
C
a

22

11

11

00

00
00

55

10
10

15
15

00

55

KC
KC

10
10

15
15

KC
KC

(a)
(a) added
added mass
mass coefficient
coefficient(b)(b) drag
drag coefficient
coefficient

FigurenumberKC
Figure 4.
4.
Variation
of
(a)
added
mass
and
(b)
drag
coefficientsofsquarecylinderwith
numberKCnumber
4.Variation
Variationofof(a)(a)added
addedmassmassandand(b)(b)dragdragcoefficientscoefficientsof
ofaaasquaresquarecylinder
cylinderwith
with KCinin
oscillatory
flow.
in
the
oscillatoryflow.
thetheoscillatoryflow.

<<3.08
2.4
2.4
3.08

interpolation
3.08



<

interpolation
3.08



<
4.87
4.87
2.4
Re

190
2.4
Re  190

β < 3.08
 2.4




11



,
C
(

)

8.10

3.7
tan
(0.06
Re

12.6)


=

4.87
C

C
(

)
190

<

Re

<

708
,
C
(

)

8.10

3.7
tan
(0.06
Re

12.6)


=

4.87
C

C
(

)
190

<

Re

<

708

 dd

dd
(12)
dd 2.4Re ≤ 190
3.08 ≤ β < 4.87 (12) interpolation 2.4
−1interpolation
4.87

<



7.30
Re

708
interpolation
4.87

<



7.30
(
Cd = 
,
C
(
β
)
=
(12)
C
β
)
190
<
Re
<
8.10
−
3.7
tan
(
0.06Re
−
12.6
)
β
=
4.87
2.4
Re

708


d
d





 2.4

11

interpolation
4.87
<
β
≤
7.30
Re ≥ 708
>>7.30
8.068.06  4.0
4.0tantan (0.015(0.015Re
Re 3.9)
3.9)
7.30




8.06 − 4.0 tan−1 (0.015Re − 3.9)
β > 7.30
The
The C
of aa studlessstudless chain
chain at
at various
various amplitudes
amplitudes andand frequencies
frequencies is
is investigatedinvestigated inin thethe
Cdd ofTheCofastudlesschain
atvariousamplitudes
andfrequencies
isinvestigatedintheexperimentdexperiment
by
Yang
[22]
and
is
shown
in
Figure5.
Theisdependent
on
not
only
Reynolds
Cexperiment by Yang [22] and is shown in Figure 5. The Cdd is dependent on not only Reynolds
by Yang [22]also
and is shown parameter,
in Figure 5. The Cd isthedependentInon
not onlyofReynolds
numberbut708also
numbernumber but
but also frequency
frequency parameter,  ,, fromfrom the figure.
figure. In thethe range
range of Re
and ReRe 190
190 andRe  708 ,,
frequency parameter, β, from the figure. In the range of Re ≤ 190 and Re ≥ 708, the Cd is a nearly
the
the C
is aa nearly
nearly constantconstant value
value ofof 2.4.
2.4. InIn thethe range
range ofof 190the frequency
frequency parameter,parameter, 
Cdd is190  Re
Re  708
708 ,, theconstant
value of 2.4. In the range of 190 ≤ Re ≤ 708, the frequency parameter, β, significantly influences,, significantly
influences
the drag
coefficient,
for
the
chain
differentCsignificantly
influences
coefficient,
The formula
formulafor different
the studlessstudless
chain with
withdifferent
Cdd .. Thethedrag coefficient,
Cd . theThedrag
formulafor the studlesschain withfrequency
parameter,β,frequency
parameter,
,
in
the
range
of
is
expressed
as
shown
in
Equation
(12)
and
190

Re

708

frequency
parameter,
,
in
the
range
of
is
expressed
as
shown
in
Equation
(12)
and
190

Re

708

in the range of 190 ≤ Re ≤ 708 is expressed as shown in Equation (12) and linear interpolation is used
linearinterpolationispredict
the
range
of
and
.. The4.87
   7.30
to predict
Cd in the range
3.08≤ β <C<β7.30. The
added
mass
the
linearinterpolation
is usedusedoftotopredict
inandthe 4.87
rangeof≤3.08
and
The
C4.87
3.08
  4.87
4.87
4.87coefficient
7.30of
dd in
chain
is
consideredtobeconsistentwiththetheoretical
value
of
1.0
for
cylinders,asprovidedby
added
added mass
mass coefficient
coefficient ofof thethe chainchain isis considered
considered to
to be
be consistentconsistent with
with thethe theoreticaltheoretical value
value ofof 1.01.0 forfor
DNV-PR-C205[5] at any
number.
cylinders,asby
DNV-PR-C205[5]cylinders,
as provided
provided
byReynolds
DNV-PR-C205[5] atat anyany Reynolds
Reynolds number.
number.
15
15

Cd of
of chain
chain
C
d

Exp.
Exp. (
(
Exp.
Exp. (
(
Exp.
Exp. (
(

Cal.
Cal. (
(
Cal.
Cal. (
(
Cal.
Cal. (
(

10
10

55

00
00

200
200

400
600
400
600
Re
Re ==  ** KC
KC

800
800

1000
1000

Figurechain
withReynolds
numberintheoscillatory
flow.
Figure5.
Variationof
Figure5.5.Variation
Variation
ofofC
of studlessstudlesschain
chainwith
withReynolds
Reynoldsnumber
numberin
inthe
theoscillatory
oscillatoryflow.
flow.
CCddd of

Regarding
Regarding thethe Cof thethe stud
stud chain,chain, itit isis estimatedestimated to
to be
be 1.08
1.08 times
times thatthat ofof C
for thethe studlessstudless
Cdd of
Cdd forchain
chain based
based onon DNVGL-OS-E301
DNVGL-OS-E301 [23][23] forfor thethe mooring
mooring chain,chain, which
which recommends
recommends 2.4
2.4 forfor thethe studlessstudless
chain
and
2.6
for
the
stud
chain
at
high
Reynolds
numbers.
chain and 2.6 for the stud chain at high Reynolds numbers.

Energies 2020, 13, 5820

8 of 22

Regarding the Cd of the stud chain, it is estimated to be 1.08 times that of Cd for the studless chain
based
on DNVGL-OS-E301[23] for the mooring chain, which recommends 2.4 for the studless 8chain
Energies 2020,
13, x FOR PEER REVIEW
of 22
and 2.6 for the stud chain at high Reynolds numbers.
Figure 66 shows
shows thethe variation
variation ofof dragdrag coefficientscoefficients ofof circular
circular andand squaresquare cylinders
cylinders with
with Reynolds
Reynolds
Figurenumbers
insteady
flow,
which
are
obtainedfromtheexperiments[9,13].
Thedragcoefficientof thethenumbers in steady flow, which are obtained from the experiments [9,13]. The drag coefficient ofcircular cylinder
cylinder increases
increases as
as Reynolds
Reynolds numbers
numbers increases,
increases, reaches
reaches aa maximum,
maximum, andand thenthen gradually
gradually
circulardecreases andand approaches
approaches aa constantconstant value.
value. The
Thedragdragcoefficient
coefficient ofof thethe squaresquare cylinder
cylinder has
has aa constantconstantdecreases
value ofof 2.0.
2.0. TheThe formulas
formulas for
for C
value
thesteadysteadyflow
flowarearesummarizedsummarizedininTable
Table3.3.
Cdd ininthe
3
Square cylinder
Circular cylinder

Cd

2

1

0
103

104

105
Re

106

107

Figure 6. Variation of Cd of square and circular cylinders with Reynolds number in the steady flow.
Figure 6. Variation of Cd of square and circular cylinders with Reynolds number in the steady flow.
Table 3. Drag coefficients of circular and square cylinders as functions of Re in steady flow.

Table 3. Drag coefficients of circular and square cylinders as functions of Re in steady flow.
Drag Coefficients
DragCoefficients
tanh(log Re − 5) + 1.85
Re ≤ 104



4 4

+ 0.76
 0.08
Ctanh(logRe log5)Re 1.85Re10
 10< Re ≤ 105
Circular cylinder5 −tanh(3 log Re − 16.8) + 0.2
 d=
10 < Re ≤ 105.57

Re  0.76
104  Re  1055.57
 0.25tanh
0.08log(2.5 log Re − 15.75) + 0.53
10
< Re

Circular cylinder
Square cylinder

Cd  
5
5.57
 tanh(3log Re  16.8)  0.2 Cd = 2.0 10  Re  10
0.25 tanh(2.5log Re  15.75)  0.53
105.57  Re

The distributedhydrodynamic coefficients need be C
integrated
to obtain the global coefficients for
Square cylinderd  2.0
the dynamic response analysis of the whole platform in the global coordinate (X, Y, Z). The platform inthis study
is divided into
79 elements,coefficientsand Ca and need
Cd of beeachelement inlocalthecoordinate(x, y, z) can
The distributedhydrodynamic
integrated
tothe
obtainglobal coefficients
be computedby Equations(10) and of(11).
and Cdplatformof the platformin the surgeand heaveforthe dynamic
response analysis
theCawholein the global
coordinate(X, Y,directions
Z). Theareintegratedfromthehydrodynamic
coefficientsofeachelement
based
on
the
formulasproposedbyplatform in this study is divided into 79 elements, and Ca and Cd of each element in the local[Cd ] forIshihara and[20].
integrated
added masscoefficient
coefficientthecoordinate(x,Zhangy, z) canbeThecomputedby Equations(10)
and (11).[CCaa] andanddragthe platformin theCd of[][platform
canbeexpressedby6
×6
matrices.
The
components
intheglobalmatrices
of
C
and
C
a
d]
surge and heave directions are integrated from the hydrodynamic coefficients of each element
based
are presented as:on the formulas proposed by Ishihara and Zhang [20]. The integrated added mass coefficient Ca 
0.06  0.0
0.0 Cd15The0.0
0.0
0.0
0.0 Ca15 can
0.0 be Cd11 by expressed Ca11and drag coefficientcomponentsin
6 matrices.
 Cd  for the platform









 0.0 C
 0.0 C
0.0 Cd24 0.0
0.0 
0.0 Ca24 0.0
0.0 


a22
d22



C
C
the global matrices ofandarepresented
as:

0.0 Cd33 0.0
0.0
0.0 
0.0 a0.0 Ca33 0.0
0.0
0.0 d 0.0[Ca ] =  [Cd ] =  0.0 Cd42 0.0 Cd44 0.0
 0.0 Ca42 0.0 Ca44 0.0
0.0 
0.0 







Cd 11 0.00.0 0.00.0 0.00.0Cd55Cd 15 0.0 0.0
 Cd51  Ca11Ca510.0 0.00.0 0.0 0.00.0Ca15Ca55 0.00.0
 




 0.0


0.00.0 0.0
0.0Ca 22 0.00.0 0.0Ca 240.0 0.00.0 0.0
Ca66
Cd 22 0.00.0 0.0Cd 24 0.00.0Cd660.0   0.0 0.0
[Ca ]  
 0.0 Ca 42
C0.0 a 51
 0.0 0.0 

Ca 33 0.0
0.0 Ca 44
0.0 0.0
0.0 0.0

0.0
0.0
Ca 55
0.0


0.0 

0.0 
0.0 

Ca 66 
 0.0 0.0
[Cd ]  
 0.0 Cd 42
C0.0 d 51
 0.0 0.0

Cd 33 0.0
0.0 Cd 44
0.0 0.0
0.0 0.0

0.0
0.0
Cd 55
0.0

The non-zero components of global added mass coefficients are calculated by:


0.0 

0.0 
0.0 

Cd 66 

(13)

(13)

Energies 2020, 13, 5820

9 of 22

The non-zero components of global added mass coefficients are calculated by:
Ca11 = ρw1 ∀

N
PW

NW

w

1
Ca66 = ρ ∀R
2

P

i=1
N
PW
i=1

t
n
i Ca ρ∀i ;

P

i=1

i=1
plate
NW h

1
Ca44 = ρ ∀R
2

w

plate

n
1
n
i Ca ρ∀i ; Ca22 = Ca11 ; Ca33 = ρw ∀

plate

NW h
i
i
P
2
2
t
n
t
n
1
t 2
n
t 2
n
i Ca ρ∀i yi + i Ca ρ∀i (zi − zG ) ; Ca55 = ρ ∀R2
i Ca ρ∀i xi + i Ca ρ∀i (zi − zG ) ;
w

i=1

(14)

NW
h
i
n
n
1 P
n 2
2
n
i Ca ρ∀i xi + yi ; Ca15 = ρw ∀R
i Ca ρ∀i (zi − zG ); Ca51 = Ca15 ;i=1

Ca24 = −Ca15 ; Ca42 = Ca24 .
platewhere NW is the number of element in water, NW is the number of plates including heave plates andpontoons, ∀nι and ∀ti represent the displaced volume of element i in the surge and heave directions, xi ,yi and zi are the local coordinates for the element i and zG is the z-coordinate of gravity center for thewhole platform.
The non-zero components of global drag coefficients are calculated by:
plate

NW
N
PW n nt n
1 P
Cd11 = A1x
i Cd Ai ;i Cd Ai ; Cd22 = Cd11 ; Cd33 = Az
i=1
i=1
plate
plate

NW 
NW h
i
3
3
t t 3
t t 3
n n
n n
1 P
1 P
Cd44 = Ax
i Cd Ai |x| + i Cd Ai |zk − zG | ;i Cd Ai y + i Cd Ai |zk − zG | ; Cd55 = Ax
i=1
i=1


NW
NW
3
2
3
n n
n n
1 P
1 P
Cd66 = Ax
i Cd Ai |x| + y ; Cd15 = − Ax
i Cd Ai (zk − zG ) ; Cd51 = Cd15 ;
i=1
i=1

(15)

Cd24 = −Cd15 ; Cd42 = Cd24 .
where Ax and Az are the characteristic area of the floater in the surge and heave directions, and Ani
and Ati represent the characteristic area of element i in the surge and heave directions, respectively.
The detailed information about the derivation of formulas was shown in Ishihara and Zhang [20].
2.3. Advanced Hydrodynamic Force Models Considering the Combined Wave-Current Condition
The equation of motion for the floating wind turbine system including the wind turbine platform
and mooring lines can be expressed as:n .. on.o[M] x + [C] x + [K]{x} = {FG } + {FB } + {FH } + {FM } + {FR }

(16)

n.o
n .. owhere M, C and K are mass, damping and stiffness matrices, respectively. {x}, x and x standfor displacement, velocity and acceleration vector of the platform. {FG }, {FB }, {FH }, {FM } and {FR }indicate the gravitational, buoyancy, hydrodynamic, mooring line and restoring forces, respectively.
The restoring force is calculated as the product of hydrostatic stiffness and displacement vector.
The hydrodynamic force including the radiation, diffraction loads and drag forces are expressed as:
FH = Fm,a + Fm,d + Fw,F−K + Fw,d + Fd

(17)where Fm,a , Fm,d and Fd are the hydrodynamic inertia force, linear radiation damping force and viscosedrag force due to movement of the platform, and Fw,F−K and Fw,d indicate the Froude-Krylov (F-K)
force and diffraction force due to wave excitation. Fm,a is calculated by:n .. oFm,a = ρw ∀[Ca ] x(18)where ρw is density of water and ∀ is volume of displaced water. The sum of Fw,F−K and Fw,d , named aswave load, is commonly predicted by the potential theory. The wave load is given by:
Fw = Fw,F−K + Fw,d =

h

F1w

F2w

F3w

F4w

F5w

F6w

i

(19)

Energies 2020, 13, 5820

10 of 22

Fiw = Ri a cos(ωt − φi ) i = 1, 2, · · · 6(20)where superscript i denotes the six degrees of freedom in the global coordinate, Fiw indicates the force
and moment in each direction, Ri and φi are the amplitude and phase of wave loads, a and ω represent
the wave amplitude and frequency and t is time. All terms in Equation (17) are computed as theintegrated force and moment based on the global coordinate, except drag force, Fd , which is calculated
as the distributed force based on the local coordinate of each element by:
.n
.n
1
n n n
n
n
i Fd = ρi Cd Ai (ui − xi ) ui − xi

(21)

2

.n

where i Cnd is the drag coefficients of the structure in the normal direction, uni − xi represents the relative
velocity of the structure to water particle and Ani denotes the characteristic area of the element in thenormal direction. The drag force in the axial direction for the heave plates is calculated by the equation
similar to Equation (21).
The drag force acted on the platform is widely calculated by the Equation (21) in the combined
wave-current condition. However, Cd obtained from the oscillating and steady flow is inapplicable forthe combined wave-current condition since they have different values, as mentioned by Sarpkaya [13].
In addition, Cd for the combined wave-current condition is difficult to be measured by the experiments.
In this study, an advanced drag force model is proposed to calculate the drag force contributed
from the wave and current as:
Fd =

1
1
ρACd (v + vc )|v + vc | + ρA(Cdc − Cd )vc |vc |
2
2

(22)

where Cd is the drag coefficient of cylinder obtained from the forced oscillation test, Cdc means the dragcoefficient of cylinder measured in the steady flow and both Cd and Cdc are the functions of Reynolds
number. ν and νc denote the relative velocity and current velocity, respectively. In the wave-only andcurrent-only conditions, the drag forces by the proposed formula are the same as those calculated by
Equation (21). In the combined wave-current condition, the current-induced drag forces are calculatedusing Cd from the steady flow, while the wave- and motion-induced drag forces are predicted using Cdfrom the forced oscillation test.
2.4. Wave Model Considering the Directional Spreading Function
Ocean waves have multidirectional components. The directional spreading function can provide
the directional distribution of wave energy. The directional spectrum of the sea surface is commonly
presented as the frequency spectrum times a function of angular spreading, expressed as:S( f , θ) = S( f )D(θ)

(23)

"
#

! 
f − fp 2
fp4
 5 f −4  exp −0.5( σ fp )5
γ
S( f ) = Aγ Hs 5 exp−

16
4 fp
f
(
0.07 f ≤ fp
Aγ = 1 − 0.287In(γ) σ =0.09 f > fp
Γ(1+n/2)

D(θ) = √πΓ(1/2+n/2) cosn (θ − θp )− π2 ≤ θ − θp ≤ π2

(24)

(25)

where S( f ) is the frequency spectrum of the wave and is described by the JONSWAP spectrum. D(θ) is
the directional spreading function. Hs , fp and f represent the significant wave height, peak frequencyof wave and wave frequency. γ implies peak enhancement factor, Γ is the Gamma function and θ andθp are the direction of elementary wave trains and the main wave direction, respectively.

Energies 2020, 13, 5820

11 of 22

The wave frequency spectrum is discretized with the number of frequencies, N, and the transient
wave elevation, η, on a location can be calculated by a sum of sinusoidal components, presented as:η=NX

an cos(kn x − ωn t + ϕn )

(26)

n=1

where an is the amplitude of wave component, kn , and ωn and ϕn indicate wave number, wave frequency
and random phase, respectively. The detailed information can be found in Ishihara and Zhang [20].
3. Results and Discussion
The proposed hydrodynamic coefficient models are validated by the integrated added mass anddrag coefficients obtained from the forced oscillation test of a 1:60 scale model in Section 3.1. The effectof current on the dynamic response of the platform is then investigated by the proposed hydrodynamic
model and validated by the water tank tests of a 1:50 scale model in Section 3.2. Finally, the effectof directional spreading of wave on the dynamic responses of a full-scale platform is examined andvalidated by the field measurements in Section 3.3.
3.1. Validation of the Proposed Global Hydrodynamic Coefficient Models Considering the Effects of Reynolds
and KC Numbers
The global added mass and drag coefficients predicted by the proposed model is compared tothose obtained from the numerical forced oscillation test by Pan and Ishihara [19], which showed
good agreement with the experimental data from the water tank tests. The numerical simulations
were conducted at four Reynolds numbers in the surge direction and three KC numbers in the heave
direction. The global added mass and drag coefficients in the surge and heave directions from CFD are
calculated according to Equations (8) and (9). The global hydrodynamic coefficients from the proposed
formulas are calculated by Equations (14) and (15).
Figure 7a,b display variation of global added mass and drag coefficients with Re in the surge
direction, and Figure 7c,d show variation of those with KC in the heave direction. It is found thatthe global drag coefficient in Figure 7b shows a peak near Re = 1 × 105 , since the drag coefficient ofcircular cylinder has maximum value near this Reynolds number. In Figure 7c,d, the variation of global
added mass and drag coefficients in the heave direction shows different trends with KC. The global
added mass coefficient, Ca , increases as KC increases, while the global drag coefficient, Cd , decreases as
KC increases.
3.2. Validation of the Advanved Hydrodynamic Models by the Water Tank Tests in the Combined Wave andCurrent Conditions
The proposed hydrodynamic model with Cd in steady and oscillatory flow is validated by thewater tank tests in the current-only, wave-only and combined wave-current conditions. The influenceof the current on the dynamic responses of platform motion and mooring tension is also investigated.
The dynamic responses of the platform for the 1:50 scale model [10] in the wave and current
conditions are studied. Figure 8 shows an overview of the water tank and platform. Figure 9 presents
the arrangement of the experiment and layout of the mooring lines. The platform motion and themooring tension are measured in the experiments. The wave elevation is measured by two wave
height meters, which are installed upstream and beside the platform. A current meter, which locatesdownstream, records the current velocity. Table 4 provides the environmental states in the experiments.
In the free decay tests (Case 1), the natural periods of six degrees of the motion are measured first.
The responses of the platform in the current-only (Case 2) and wave-only conditions (Case 3 withregular waves and Case 4 with irregular waves) are then investigated. Finally, the dynamic responsesof the platform in the combined wave-current conditions (Case 5 with regular waves and Case 6 withirregular waves) are examined.

Energies
FOR
Energies2020,
2020,13,
13,xx 5820
Energies
2020,
13,
FOR PEER
PEER REVIEWREVIEW

12
of
22
12 12
of of
2222

33

33
CFD
CFD
Cal.
Cal.

CFD
CFD
Cal.
Cal.

d11

a11

CCd11

22

CCa11

22

11

11

00
3
10
103

4

4
10
10

5

6

5
10
10

6
10
10

Re
Re ==ββ ** KC
KC

00
3
10
103

7

7
10
10

4

4
10
10

5

40
40
CFD
CFD
Cal.
Cal.

CFD
CFD
Cal.
Cal.

30
30
d33

CCd33

33
a33

7

7
10
10

(b)
(b)

44

CCa33

6
10
10

Re
Re == ββ ** KC
KC

(a)
(a)

22
11
00

6

5
10
10

20
20
10
10

00

11

KC
KC

(c)
(c)

22

33

00

00

11

KC
KC

22

33

(d)
(d)

FigureGlobalhydrodynamic
coefficients
inanddirections.
(a)mass
Figure7.
Globalhydrodynamic
coefficients
insurgethe surge
and heave
directions.
(a) Added
Figure
7.7.
Globalhydrodynamic
coefficientsin thethesurgeand heaveheave
directions.
(a) Added
Added
mass masscoefficient
in
the
surge
direction,(b)
dragcoefficient
in
the
surge
direction,
(c)
added
mass
coefficient
in
the
surge
direction,(b)
dragcoefficient
in
the
surge
direction,
(c)
added
mass
coefficient in the surge direction, (b) drag coefficient in the surge direction, (c) added mass coefficientin the heave
directionand (d) dragin the heave
direction.
coefficient
in
heave
and
(d)
in
heave
coefficientin thetheheave direction
direction
andcoefficient(d) drag
drag coefficient
coefficientin thetheheave direction.
direction.
Energies 2020, 13, x FOR PEER REVIEW

13 of 223.2.
3.2. Validation
Validation ofof thethe Advanved
Advanved Hydrodynamic
Hydrodynamic Models
Models byby thethe Water
Water Tank
Tank Tests
Tests inin thethe Combined
Combined
Wave
andCurrent
Conditions
Wave and Current Conditions
The
The proposed
proposed hydrodynamichydrodynamic model
model with
with C
Cdd inin steadysteady andand oscillatory
oscillatory flow
flow isis validated
validated byby thethewater
water tanktank tests
tests inin thethe current-only,
current-only, wave-onlywave-only andand combined
combined wave-current
wave-current conditions.
conditions. The
The influence
influenceofof thethe current
current on
on thethe dynamic
dynamic responses
responses ofof platform
platform motion
motion andand mooring
mooring tension
tension isis also
also investigated.
investigated.
The
dynamic
responsesoftheplatform
for
the
1:50
scale
model
[10]
in
the
wave
The dynamic responses of the platform for the 1:50 scale model [10] in the wave andand current
current
conditions
arestudied.
Figure8shows
anoverview
of
the
water
tank
and
platform.
Figure9presents
conditions are studied. Figure 8 shows an overview of the water tank and platform. Figure 9 presents
thethe arrangement
arrangement ofof thethe experimentexperiment andand layout
layout ofof thethe mooring
mooring lines.
lines. The
The platform
platform motion
motion andand thethemooringmooring tension
tension areare measured
measured inin thethe experiments.
experiments. The
The wave
wave elevation
elevation isis measured
measured byby twotwo wavewave
heightheight meters,
meters, which
which areare installed
installed upstream
upstream andand besidebeside thethe platform.
platform. A
A current
current meter,meter, which
which locateslocatesdownstream,
records
thecurrent
velocity.
Table
4providestheenvironmental
statesdownstream, records the current velocity. Table 4 provides the environmental states inin thetheexperiments.
experiments. In
In thethe freefree decay
decay tests
tests (Case(Case 1),1), thethe natural
natural periods
periods ofof sixsix degrees
degrees ofof thethe motion
motion arearemeasuredfirst.
The
responsesoftheplatforminthecurrent-only
(Case
2)
and
wave-only
conditions
measured first. The responses of the platform in the current-only (Case 2) and wave-only conditions(Case(Case 33 with
with regular
regular waves
waves andand Case
Case 44 with
with irregular
irregular waves)waves) areare thenthen investigated.
investigated. Finally,
Finally, thetheFigure 8. Overview of the water tank and platform.
Figure8.
Overview
of
the
water
tank
and
platform.
dynamic
responsesoftheplatforminthecombinedwave-current
conditions
(Case
5
with
regulardynamic responses of the platform in the combined wave-current conditions (Case 5 with regular
waves
waves andand Case
Case 66 with
with irregular
irregular waves)waves) areare examined.
examined.

Figure 9. Arrangement of the experiment and layout of the mooring lines in the wave and current
conditions.

Energies 2020, 13, 5820

13 of 22

Figure 8. Overview of the water tank and platform.

Figure 9.9. Arrangement
Arrangementof theexperimentlayoutthe mooringin theandwave
andFigureof theexperimentand andlayoutof theofmooringlines inlines
the wavecurrent
current
conditions.
conditions.
Table 4. Wave and current conditions in the experiments.
Table 4. Wave and current conditions in the experiments.
Case No.

Case
No.
1
2
3
1
4
25
6

Current Velocity
Current(m/s)

Conditions

Conditions

Still water
Current-only
Wave-only
Still water
Wave-only
Wave
and current
Current-only
Wave and current

Velocity(m/s)
0.05~0.20

0.20
0.05~0.20
0.20

Wave Height
Wave
(m)

Height
(m)
0.06
0.15
0.06
0.15

Wave Period
Wave
(s)

Period(s)-

1.4~2.8
1.82
2.0~2.8
1.82

Description

Description

Free decay tests
Free
decay
Regularwave
Irregular
tests wave
Regular- waveIrregular wave

3
Wave-only
0.06
1.4~2.8
Regular wave
4
Wave-only
0.15
1.82
Irregular wave
The calculated
dynamic
Wave
and responses of the platform in different wave and current conditions are
5
0.20
0.06
2.0~2.8
Regular wave
compared with thecurrent
experimental data to validate the proposed model. Two numerical models are built.
The first model isdeveloped
Wave
and according to the proposed model, which utilizes the drag coefficients of
6
0.20
0.15
1.82
Irregular wave
the cylinder in thecurrent
oscillatory and steady flows, as shown in Equation (22), and applied to 112 positions
on the platform. Another model only uses the drag coefficients in the oscillatory flow. The added mass
The calculated
dynamic
responses
the platform
different wavedragand coefficientscurrent conditionsarecoefficientsobtainedfrom thepotentialoftheory
and theinrecommended
from DNV
comparedthe experimental
data to The
validate
theReynolds
proposednumbers
model. Twonumerical
models are
standards withare usedas the initial values.
KC anddependenthydrodynamic
built.
The
first
model
is
developed
to theproposedmodel,
which
utilizes thedragcoefficientsarethenacquired
based onaccording
the predicted
responses.
These
updatedcoefficientsareEnergies
2020, 13,
x FOR
PEER
REVIEW
14used,of 22
coefficientsof the cylinder
in thesteady flows,
as shown inEquation(22),
and applied
and the simulation
is carriedoutoscillatory
again. Theandsimulations
are performed
using
Orcaflex[24].
Figure10comparisonofpredicted
andmeasured
mean
of
the
to 112
positionson the platform.
Anothermodel
only uses
drag coefficientsin theoscillatory
flow.
Figure10presents
presents
comparison
ofthe
the
predicted
andthemeasuredmeanvalue
value
ofthe
thesurge
surgeininthecurrent-only
condition.
found
the conventionalusingobtained
fromdragthe
Cd recommended
The
added masscoefficientsobtained
the potential
theoryandthecurrent-only
condition.
ItItisisfoundthatthatthefromconventionalmodelmodelusing C
from the oscillatory
d obtained
KCthe
coefficientsfrom overestimates
DNV
standards
used
as
the
initialvalues.wellThe
andReynolds
numbers
flow overestimates
the surge,
while
the
proposedmodel
withexperimental
data.
It is
oscillatoryflow
thearesurge,
while
the matches
proposedmodel
matcheswell withthedependenthydrodynamic
coefficients
areproposed
then
based
on
the
responses.
These
attributed that
the Itproposedmodel
calculates
theacquired
current
according
to Cdloading
obtainedfromtheexperimental
data.
is attributedthatthemodelloading
calculates
thepredictedcurrent
according
updatedcoefficients
used,
and
the
is carried
outCagain.
Thesimulations
areperformed
steadyflow,
whichsmallerthan
Cd simulation
obtainedfromthethanoscillatory
flow
at
the
Reynolds
numberintofromisarethesteadyflow,
which is smaller
obtainedfromtheoscillatoryflow
Cd obtaineddusing
Orcaflex[24].
theexperiment.
at the Reynolds number in the experiment.
0.2

Surge (m)

0.15

Exp.
Cal. (proposed model)
Cal. (conventional model)

0.1
0.05
0
0.05

0.1

0.15
Current velocity (m/s)

0.2

Figure 10. Comparison of predicted and measured surge displacement in the current-only condition.
Figure 10. Comparison of predicted and measured surge displacement in the current-only condition.

Figure 11 shows a comparison of the predicted and measured mean values of surge, pitch andtensions of mooring lines ML2 and ML3 in the regular wave with the current. The mean values arenormalized by the experimental data in Figure 11. The value of surge predicted by Cd from theoscillatory flow is overestimated, which is consistent with the conclusion from Figure 10. The mean

0.05
0
0.05

0.1

0.15
Current velocity (m/s)

0.2Energies 2020, 13, 5820

14 of 22

Figure 10. Comparison of predicted and measured surge displacement in the current-only condition.

Figure 11
11 shows
shows aa comparison
comparison ofof thethe predictedpredicted andand measured
measured mean
mean values
values ofof surge,
surge, pitch
pitch and
and
Figuretensions ofofmooring
mooringlines
linesML2
ML2andandML3ML3inintheregularwave
withcurrent.
meanvaluestensions
theregularwave
withthethecurrent.
TheThemean
valuesareare normalized
by theexperimental
in Figure
value
surge
predicted
from thenormalized
by theexperimental
datadata
in Figure11. 11.
TheThe
value
of of
surge
predicted
bybyCCd d from theoscillatoryflow
isoverestimated,which
isconsistentwith
theconclusion
from
Figure10.
Themean
oscillatory flow is overestimated, which is consistent with the conclusion from Figure 10. The mean
value
of
pitch
is
also
overestimated
due
to
the
overestimation
of
the
mean
value
of
surge.
value of pitch is also overestimated due to the overestimation of the mean value of surge.

Mean value of response

2
Exp.
Cal. (proposed model)
Cal. (conventional model)

1.5
1
0.5
0
Surge

Pitch

T1

T3

FigureFigure 11.
11. Comparison
Comparisonofof predictedpredicted andand measured
measured mean
mean values
values ofof thethe displacement,
displacement, rotation
rotation andandmooringmooring tension
tension inin thethe combined
combined wave-current
wave-current condition (regular wave with T == 2.5
2.5 s).
s).

Figure 12
12 shows
shows aa comparison
comparison ofof thethe predictedpredicted andand measured
measured response
response amplitude
amplitude operators
operators
Figure(RAOs).
The
RAOs
of
the
surge,
heave
and
pitch
motions
as
well
as
the
mooring
tensionin the(RAOs). The RAOs of the surge, heave and pitch motions as well as the mooring tension in the wavewave-onlyandcombinedwave-current
conditionsarecalculated.
In
the
water
tank
tests,
theresponses
only and combined wave-current conditions are calculated. In the water tank tests, the responses at
9 waveperiods
wave-only
conditionand
andatat3 3wave
waveperiods
periodsinin thethe combined
combined wave-current
wave-current
9atwave
periods
in inthethewave-only
condition
conditionareexamined
and
used
for
validationoftheproposedmodel.
Sincetheproposedmodel
is thecondition
examined and used for validation of the proposed model. Since the proposed
model
is
same
as theasconventionalmodel inthe wave-only
condition,condition,
only the predicted
responses
by
the
same
the conventionalmodel
in the wave-only
only thedynamic
predicted
dynamic
the proposedmodel
are shownin arethe shownfigures.inInthethefigures.
combinedwave-current
condition, thecondition,
predicted
responses
by theproposedmodel
In thecombined wave-current
dynamic
responsesbybothmodelsareplottedinthefiguresforcomparison.
the predicted dynamic responses by both models are plotted in the figures for comparison.
The dynamic response of surge motion illustrated in Figure 12a shows similar values for the caseswith and without the current since the experiments near the natural period of the surge motion are
not conducted due to the limitation of the water tank. The dynamic responses of heave and pitch
motions in the case with the current as shown in Figure 12b,c are smaller than those in the case without
the current, while the dynamic responses of mooring tension as illustrated in Figure 12d increases in
the case with the current. The current increases the velocity of water particles passing through the
pontoon, which leads to the increment of damping in the vertical direction. The current also moves
the equilibrium position of the platform to the downstream and results in the increase of the tension
of upstream mooring line and affects the pitch response. It is suspected that this is caused by thechange of mean tension of the mooring line and the equilibrium position of pitch. It is noticed that theconventional model using Cd obtained from the oscillatory flow slightly overestimates the response of
the mooring tension due to the overestimation of mean values of surge and pitch motions, as shown in
Figure 11, while the proposed model shows favorable agreement with the experimental data.
In the irregular wave condition, the power spectral density (PSD) of wave elevation without andwith the current is shown in Figure 13 and the dynamic responses of platform and mooring line are
given in Figure 14. It can be found that the wave height of the combined wave-current condition isslightly lower than that of the wave-only condition. To minimize the effect of the wave height on
the motions of platform and tension of mooring, the PSD of the responses are normalized by thesquared significant wave heights, as shown in Figure 14. Similarity to dynamic responses in theregular wave conditions, the current has a limited effect on the dynamic response of the motions ofplatform, as illustrated in Figure 14a–c, while it significantly increases the dynamic response of thetension of the upstream mooring line due to the movement of the mean position of surge, as shown in
Figure 14d. At the natural period of surge motion, the predicted dynamic responses with the current

Energies 2020, 13, 5820

15 of 22decrease due to the increase of damping in the surge direction. It is the same as what was mentionedby Soeb et al. [12]. On the other hand, the dynamic response of the tension of the upstream mooringline significantly increases due to the existence of the current. It implies that the fatigue life of themooring line is overestimated if the effect of current is neglected for the semi-submersible FOWT,
Energies 2020, 13, x FOR PEER REVIEW
15 of 22
which is consistent with the conclusion by Chen and Basu [11] for the spar-type FOWT.
3Exp. w/o current
Exp. with current
Cal. w/o current (proposed model)
Cal. with current (proposed model)
Cal. with current (conventional model)

2

Heave (m/m)

Surge (m/m)

3

2

1

1

0

0
1
Full scale
7.07

1.5
10.61

2

2.5

14.14

17.68

1

1.5

Full scale
7.07

10.61

3
21.21

2.5

3

14.14

17.68

21.21

Period (s)

(a)

(b)
50

Exp. w/o current
Exp. with current
Cal. w/o current (proposed model)
Cal. with current (proposed model)
Cal. with current (convetional model)

40
T2 (N/m)22

Period (s)3Pitch (rad/m)Exp. w/o current
Exp. with current
Cal. w/o current (proposed model)
Cal. with current (proposed model)
Cal. with current (conventional model)

Exp. w/o current
Exp. with current
Cal. w/o current (proposed model)
Cal. with current (proposed model)
Cal. with current (conventional model)

30

20

1

Energies 2020, 13, x FOR PEER REVIEW

10

16 of 220 current has a limited effect on the dynamic response0
conditions, theof the motions of platform, as
1
1.5
2
2.5
3
1
1.5
2
2.5
3
Full scale
illustrated in Figure 14a–c, while it significantly increases Fullthescaledynamic response of the tension of the
7.07
10.61
14.14
17.68
21.21
7.07
10.61
14.14
17.68
21.21
upstream mooring line due
as (s)
shown in Figure 14d.
Periodto(s) the movement of the mean position of surge,
PeriodAt the natural period of surge motion, the predicted dynamic responses with the current decrease(c)(d)was mentioned by Soeb
due to the increase of damping
in the surge direction. It is the same as what
et al. [12]. On the other hand, the dynamic response of the tension of the upstream mooring line
FigureComparison ofof predictedpredicted andand measured response
amplitudeoperators (RAOs)of theFigure 12.
12.
Comparison
response
amplitude(RAOs)
thesignificantly
increases
due to theexistence of measured
the current.
It implies
that theoperators
fatigue lifeof theofmooring
platforminthewave-only
andcombinedwave-current
conditions.
(a)
RAO
of
surge
motion,(b)platform in the wave-only and combined wave-current conditions. (a) RAO of surge motion, (b) RAO
RAO
line isofoverestimated
if the effectof currentand
is neglected
for the tension
semi-submersible
FOWT, which isof heaveheave motion,
motion, (c)(c) RAO
RAO ofof pitch
pitch motion
motion and (d)(d) RAO
RAO ofof mooingmooing tension forfor ML2.
ML2.
consistent with the conclusion by Chen and Basu [11] for the spar-type FOWT.

2

Wave elevation (m /Hz)The dynamic response of surge
motion illustrated in Figure 12a shows similar values for the
10-1
w/o current
cases with and without the current since the experimentswave
near
the natural period of the surge motion
wave with current-2
10
are not conducted due to the limitationof the water tank. The dynamic responses of heave and pitch
motions in the case with the current-3 as shown in Figure 12b,c are smaller than those in the case10
without the current, while the dynamic responses of mooring tension as illustrated in Figure 12d-4 The current increases the velocity of water particles passing
increases in the case with the current.
10
through the pontoon, which leads to the increment of damping in the vertical direction. The current-5also moves the equilibrium position10of
the platform to the downstream and results in the increase of
the tension of upstream mooring line-6and affects the pitch response. It is suspected that this is caused
1000.2 line
0.4and0.6
0.8
1
by the change of mean tension of the mooring
the equilibrium
position of pitch. It is noticed
that the conventional model using Cd obtainedfromtheoscillatoryflow
slightly overestimates theFrequency (Hz)

Hz)

Hz)

response of the mooring tension due to the overestimation of mean values of surge and pitch motions,FigureFigure13.
13.PSD
PSDof
ofmeasured
measuredwave
waveelevation.
elevation.
as shown in Figure 11, while the proposed model shows favorable agreement with the experimental
data.
0
-1
In the irregularwave condition, the power spectral density10
10 (PSD) of wave elevation without andwith the current is shown in Figure 13 and the dynamic responses of platform and mooring line are
given in Figure10-1 14. It can be found that the wave height of the
10-2 combined wave-current condition is

10

-6

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)Figure 13. PSD of measured wave elevation.

Energies 2020, 13, 5820

10

10-1

10-2

2

Heave (m /m /Hz)

2

2

-1

10

2

Surge (m /m /Hz)

0

10-2

10

-3

10

-4

Exp. w/o current
Exp. with current
Cal. w/o current
Cal. with current
0

0.2

0.4

0.6

0.8

10-3

10

-4

10

-5

1

Exp. w/o current
Exp. with current
Cal. w/o current
Cal. with current
00.2

Frequency (Hz)

0.6

0.8

1

0.8

1

(b)

3

10

2

102

101

T2 (N /m /Hz)

10

2

2

101

2

2

0.4

Frequency (Hz)(a)

Pitch (deg /m /Hz)

16 of 22

Exp. w/o current
Exp. with current
Cal. w/o current
Cal. with current

0

10

10

-1

0

0.2

0.4

0.6

Frequency (Hz)

(c)

0.8

1

100

10

-1

10

-2

Exp. w/o current
Exp. with current
Cal. w/o current
Cal. with current
0

0.2

0.4

0.6

Frequency (Hz)(d)

Figure14.14.
Comparison
of ofpredicted
andmeasuredPSD
of ofmotionsandmooring
tensionin inwave-only
FigureComparison
predicted
andmeasuredPSD
motionsandmooring
tensionwave-only
andcombinedwave-current
conditions.
PSD
ofof
surge
motion,(b)(b)
PSD
ofof
heave
motion,(c)(c)
PSD
ofof
and
combinedwave-current
conditions.(a)(a)
PSD
surge
motion,PSD
heave
motion,PSD
pitchmotion
and
(d)
PSD
ofof
mooring
tension.
pitch
motionand(d)PSDmooring
tension.

3.3. Validation of the Proposed Hydrodynamic Models and the Directional Spreading Function by the
Full-Scale Platform
To validate the proposed hydrodynamic models for the full-scale platform, the 2 MW semi-submersible
FOWT used in the Fukushima demonstration project is selected. The effect of directional spreading
function on the dynamic responses of FOWT in parked condition is also investigated.
The Fukushima Floating Offshore Wind Farm Demonstration Project (Fukushima FORWARD) has
been conducted since 2011. The 2 MW semi-submersible FOWT was installed in 2013 in 120 m water
depth. The platform is moored by six catenary mooring lines using stud chains, which are different
from the laboratory-scale model using studless chains. The overview of the 2 MW semi-submersible
FOWT and configuration of the mooring lines are shown in Figure 15.
The simulations for the 2 MW semi-submersible FOWT are performed using the coupled
Orcaflex [24] and FAST [25]. The aerodynamic loads on the wind turbine and tower in air are simulated
by FAST and the hydrodynamic loads on the platform and mooring lines in water are computedby Orcaflex. The added mass and drag coefficients in the numerical model are evaluated using the
proposed hydrodynamic coefficient models.
The numerical model involves the wind turbine, including three blades, hub, nacelle, generator,
tower, platform and mooring lines. The variation of current velocity with depth is expressed by apower law with an exponent of 1/7. Three blades are fixed at the pitch angle of 90 degrees in theparked condition.

The Fukushima Floating Offshore Wind Farm Demonstration Project (Fukushima FORWARD)
has been conducted since 2011. The 2 MW semi-submersible FOWT was installed in 2013 in 120 m
water depth. The platform is moored by six catenary mooring lines using stud chains, which are
different from the laboratory-scale model using studless chains. The overview of the 2 MW semiEnergies
2020, 13, 5820
17 of 22submersible
FOWT and configuration of the mooring lines are shown in Figure 15.

(a)

(b)

Figure2 MWsemi-submersible
FOWT
inFukushima
the Fukushima
demonstration
project.
(a)Figure15. 15.
2 MW
semi-submersible
FOWT
usedusedin thedemonstration
project.
(a) Overview
Overviewoffull-scale
platformand(b)configurationofmooringlines.
of full-scale platform and (b) configuration of mooring lines.

In thethe platform
motions insix degrees
of freedom
at the
TheFukushima
simulationsFORWORD,
for the 2 MW
semi-submersible
FOWT
are performed
using
thegravity
coupledcenterare recorded
byFASTa GPS,[25].
a 3-axis
and a 3-axis
accelerometer.
Tower
moments
are measured
Orcaflex[24] and
Thegyroaerodynamic
loads
on the windturbineand tower
in air are
by simulatedusing the strain
gaugesdifferent elevations.
floating
substationnear the2 MWwindareby FASTand atthetwohydrodynamic
loads onAthe
platformand mooringlines
in waterturbineis equippedwith anemometers,wave
andcurrent
measurement
devices. The
waveareelevation
computedby Orcaflex.
The added massanddragcoefficientsin the numericalmodel
evaluatedis measured
the wavemeter. The unexpectedcomponent in the measured spectrum
using the byproposedhydrodynamic
coefficientlow-frequency
models.
of waves,which
is generated
by the motionof thesubstation[26]threeand isnot observedby thebuoy
Thenumerical
model involves
the windturbine,includingblades,hub, nacelle,
generator,
measurement,is removed
in this lines.
study.The
Thevariation
motion of
substation
haswitha significant
on thetower, platform
and mooring
ofthe
current
velocity
depth is effectexpressed
by aturbulence
intensity
in exponent
the verticalthe effecton thevelocity
component
power lawwith anof direction,
1/7. Threebut
bladesare fixed
at horizontal
the pitch angle
of 90
degrees inisthe
limited,as condition.
mentioned by Yamaguchi and Ishihara [27,28]. The steady wind and current models areparkedused in thesimulations
andFORWORD,
the validationtheof theproposedhydrodynamic
modelofinfreedom
parked conditionisIn theFukushimaplatform
motionsin six degrees
at the gravity
focusedin recorded
this study.byGPS
occasionally
fails andto record
data
and leads to missing
data. To compensate
centeronarea GPS,
a 3-axis gyroa 3-axis
accelerometer.
Tower moments
are measured
forby
theusing
missingthedata byaccelerometer
areAcombined
to obtain thecontinuoustime
thedata,straingaugesatGPS
two anddifferentelevations.
floating substationnear
the 2 MWwindhistoryof platform
motionGPS data wave
is missing,as describedby Yamaguchi
etTheal. [29].
turbineis equippedwithwhenanemometers,
and current
measurement
devices.
wave elevation
20-minbydatasetsin meter.
parkedThe
condition
are selected
for the low
and high wave
conditions.
is Twomeasured
the wave
unexpected
low-frequency
componentin theheightmeasured
spectrum
Theofinformation
on is
wind,
wave byandthecurrent
conditions
are shown
Table
5, where
the bydirection
waves, which
generatedmotionof the substation[26]inandis notobserved
the buoyin themeasurement
is defined
the norththe clockwise
direction.
γ in the JONSWAP
measurement,is removed
in from
this study.
The inmotionof the substation
hasThea significant
effect on the
spectrumand intensity
exponentin
n inspreading
functionby theleast square
methodbased isturbulence
thetheverticaldirection,
but arethe identified
effect on thehorizontal
velocity
componenton limited,
the measured
wave spectrum.
The exponent
n at[27,28].
the Fukushimasite
is slightly
largermodels
than 2,areas mentioned
by Yamaguchi
and Ishihara
The steadywind
and current
as shown
Table
5, whichandis consistent
with ofthethetypicalvaluehydrodynamicof the exponentn as recommended
inused inin
thesimulations
the validation
proposedmodel
in parked condition
DNVGL-CG-0130[30].
is focused on inthis study. GPS occasionally fails to record data and leads to missing data. To
compensate for the missing data, the data by GPS and accelerometer are combined to obtain the
Table 5. Description
on theofwind,
wave motion
and current
conditions
at thesitedescribed
used in thiscontinuous
time history
platformwhen
GPS data
isFukushima
missing, as
bystudy.
Yamaguchi
etEnvironmental
al. [29].
HeightHighandWave
HeightCase
Two 20-min datasets Description
in parked conditionLow
areWaveselected
forCasethe lowhigh
waveheightConditions
conditions. The information
onspeedwind,
wave and current conditions
are shown in 27.15
Table 5, where the
Wind
(m/s)
6.44
Wind

Wind direction (deg.)

234.2

158.9

Wave

Wave height (m)Wave peak period(s)Wave direction (deg.)
Peak enhancement factor γ
Exponent n

1.37
11.6
84.0
1.30
2.33

5.31
12.8
155.3
0.77
2.91

Current

Current velocity (m/s)
Current direction (deg.)

0.10
136.5

0.38
310.7

Wave

Wave peak period(s)Wave direction (deg.)
Peak enhancement factor 

Energies 2020, 13, 5820

Exponent n
Current velocity (m/s)
Current direction (deg.)

Current

11.6
84.0
1.30
2.33
0.10
136.5

12.8
155.3
0.77
2.91
0.38
310.7

18 of 22

Figure 16 presents the comparison between the simulated JONSWAP wave spectrum and spreading
Figure 16 presents the comparison between the simulated JONSWAP wave spectrum andfunction and the measurement data for low and high wave height cases. The JONSWAP spectrum andspreading function and the measurement data for low and high wave height cases. The JONSWAP
the spreading function match well with the measurement.
spectrum and the spreading function match well with the measurement.
1.2

Obs. (Low wave height case)Obs. (High wave height case)
JONSWAP (Low wave height case)
JONSWAP (High wave height case)Obs. (Low wave height case)Obs. (High wave height case)Spreading fun. (Low wave height case)Spreading fun. (High wave height case)

30

0.9

Wave spreading

2

Wave spectrum (m /Hz)

40

20

10

0.6

0.3

0
0

0.1

0.2

0
-90

0.3

-60

Frequency (Hz)

-30

0

30

60

90

Direction (deg)

(a)

(b)

FigureFigure 16.
16. Comparison
Comparisonofofthe
the(a)(a)frequency
frequencyspectrum
spectrumandand(b)(b)directional
directionalspreading
spreadingbetween
betweenthe
theobserved
observedandandsimulated
simulatedwave
waveatatlow
lowandandhighhighwave
waveheightheightcases.
cases.

Thefreefreedecay
decaytest
testisis performed
performed forfor thethe full-scale
full-scale model.
model. The
Thesimulated
simulatednatural
naturalperiod
periodof
ofrollroll
Themotionisis22.4
22.4s,s,which
whichisisconsistentconsistentwith
withthethemeasured
measurednatural
naturalperiod
periodofofrollrollmotion
motionfor
forthethe22MWMW
motionFOWTthroughthroughananemergency
emergencystop
stopininthe
theFukushima
Fukushima demonstration
demonstration projectproject [31].
[31].
FOWT
Two numerical simulations
thethespreading
function
are thenconducted.
The PSDs
Two
simulationswithout
withoutand
andwithwithspreading
function
are thenconducted.
Theof theof
platform
motions
for the lowheightcase
are given
Figuredynamic
responses
are
PSDs
the platform
motions
for wavethe low
wave
heightcase inaregiven17.inThe
Figure17. Thedynamic
normalized
the squaredby
significant
wave
height. Inwave
this case,
the Inwave
closedirection
to the surge
responses
arebynormalized
the squared
significant
height.
thisdirection
case, theiswave
isdirectionandthespreading
functionstrongly
affectsthemotions
in
the
sway,
roll
and
yaw
directions.
close to the surge direction and the spreading function strongly affects the motions in the sway, rollWithout
considerationof directional
spreadingof the wave
spectrum,and yaw
andyaw directions.
Without
considerationof directional
spreadingof thethe sway,
wave roll
spectrum,themotions
sway,
are andsignificantly
underestimated.
The underestimated.
PSDs of platformThemotions
the high motions
wave height
case
are
roll
yaw motions
are significantly
PSDs forof platform
for thehigh
shownheightin FigureTheeffect of
spreading
function
responses
of on
platform
motionswave
case18.areshown
in theFigure18. Theeffect onof thethedynamic
spreading
function
the dynamic
is relatively
in thiscase, issince
the primarywave
is locatedbetweenthesurge and
responses
of smallplatform
motions
relatively
small inthisdirection
case, since
the primarywave
directionisswaydirections,
while
the
improvement
for
the
yaw
motionbythespreading
functionissignificant,located between the surge and sway directions, while the improvement for the yaw motion by theas shown
inFigureThe
low-frequency
observed
the PSDs of platform
motionsinthe
thespreadingis18f.
significant,as shown inpeaks
Figure18f. Theinlow-frequency
peaks observedinEnergiesfunction
2020,
13, x FORPEER
REVIEW
19 of 22surgeofandsway directions,
illustrated
in sway
Figuredirections,
18a,b, correspondto theinnaturalin these
PSDs
platform
motions inasthe
surge andas illustrated
Figurefrequencies
18a,b, correspondto the natural
frequencies
in these
two aredirections.
The low-frequency
peaks
are not
obviousininthe
theheave,
two directions.
The
low-frequency
peaks
not obvious
in the PSDs ofplatform
motionsPSDsofplatform
motionsintheheave,
roll
and
pitch
directions,
as
shown
in
Figure18c–e,
since
the areroll and pitch directions, as shown in Figure 18c–e, since the natural frequencies in these directions
natural frequencies in these directions are close to the peak frequency of the wave. The dynamic
close to the peak frequency of the wave. The dynamic responses of 2 MW FOWT in the six degrees of
responses of 2 MW FOWT in the six degrees of freedom are well predicted by the proposed
freedom are well predicted by the proposed hydrodynamic models.
hydrodynamic models.
2

2

10

10
Obs.
Cal. with spreading
Cal. w/o spreading

100

Sway (m /m /Hz)

2

10-2

10-2

2

2

2

Surge (m /m /Hz)

100

10

-4

10

-6

10

-8

0

0.05

0.1

0.15

0.2

10

-4

10

-6

10

-8

0.25

0

Frequency (Hz)

2

(b)
2
Figure 17. Cont.
10

100
10-2

2

2

Roll (deg /m /Hz)

2

2

Heave (m /m /Hz)

100
10-2

10

-4

10

-6

0.1

0.15

Frequency (Hz)

(a)
10

0.05

10

-4

10

-6

0.2

0.25

-4

10

-6

10

-8

Swa

Surg

10

0

0.05

0.1

0.15

0.2

10

-4

10

-6

10

-8

0.25

0

0.05

Frequency (Hz)Energies 2020, 13, 5820

2

10

0.2

0.25

19 of 22

2

100

100

Roll (deg /m /Hz)

10

2

10-2

10-2

2

10

-4

10

-6

10

-8

0

0.05

0.1

0.15

0.2

10

-4

10

-6

10

-8

0.25

0

0.05

Frequency (Hz)

0.1

0.15

0.2

0.25

0.2

0.25

Frequency (Hz)

(c)

(d)

2

10

2

100

100
2

Yaw (deg /m /Hz)

10

10-2

10-2

2

2

2

0.15

(b)

2

2

Heave (m /m /Hz)(a)

Pitch (deg /m /Hz)

0.1

Frequency (Hz)

10

-4

10

-6

10

-8

0

0.05

0.1

0.15

0.2

10

-4

10

-6

10

-8

0.25

0

0.05

Frequency (Hz)

0.1

0.15

Frequency (Hz)

(e)

(f)

Figure 17. Normalized PSD of observed and simulated motions in the six degrees of freedom at the

Figure 17. Normalized PSD of observed and simulated motions in the six degrees of freedom at the
low wave height case. (a) PSD of surge motion, (b) PSD of sway motion, (c) PSD of heave motion, (d)low wave height case. (a) PSD of surge motion, (b) PSD of sway motion, (c) PSD of heave motion,
PSD of roll motion, (e) PSD of pitch motion and (f) PSD of yaw motion.
Energies
13, xmotion,
FOR PEER
REVIEW
20 of 22(d) PSD2020,of roll(e)
PSD of pitch motion and (f) PSD of yaw motion.

2

2

10

10
Obs.
Cal. with spreading
Cal. w/o spreading

100

Sway (m /m /Hz)

2

10-2

10-2

2

2

2

Surge (m /m /Hz)

100

10

-4

10

-6

10

-8

0

0.05

0.1

0.15

0.2

10

-4

10

-6

10

-8

0.25

0

0.05

Frequency (Hz)

2

10

0.2

0.25

0.2

0.25

2

100

100

Roll (deg /m /Hz)

10

2

10-2

10-2

2

10

-4

10

-6

10

-8

0

0.05

0.1

0.15

0.2

10

-4

10

-6

10

-8

0.25

0

Frequency (Hz)

100

100
2

10-2

2

2

Yaw (deg /m /Hz)

Figure 18. Cont.
2
10

10-2

10

-4

10

-6

0.1

(d)

10

2

0.05

0.15

Frequency (Hz)

(c)

2

0.15

(b)

2

2

Heave (m /m /Hz)(a)

Pitch (deg /m /Hz)

0.1

Frequency (Hz)

10

-4

10

-6

-4

10

-6

10

-8

Roll

Heav

10

0

0.05

0.1

0.15

0.2

10

-4

10

-6

10

-8

0.25

0

0.05

Frequency (Hz)Energies 2020, 13, 5820

(c)

0.15

0.2

0.25

20 of 22

(d)

2

10

2

100

100
2

Yaw (deg /m /Hz)

10

10-2

10-2

2

2

2

Pitch (deg /m /Hz)

0.1

Frequency (Hz)

10

-4

10

-6

10

-8

0

0.05

0.1

0.15

0.2

0.25

10

-4

10

-6

10

-8

0

Frequency (Hz)

(e)

0.05

0.1

0.15

0.2

0.25

Frequency (Hz)(f)

Figure
18. PSDof observedandandsimulated
simulated motions
motions ininthe
degreesof freedom
at thewavewave
Figure
18. PSDof observed
thesixsix
degreesof freedom
at high
the highheight
case.
(a)
PSD
of
surge,
(b)
PSD
of
sway
motion,
(c)
PSD
of
heave
motion,
(d)
PSD
of
roll
motion,height case. (a) PSD of surge, (b) PSD of sway motion, (c) PSD of heave motion, (d) PSD of roll motion,(e) PSDof pitchmotionand
PSDofofyaw
yawmotion.
motion.
(e) PSDof pitchmotionand(f)(f)PSD

4. Conclusions
4. Conclusions
advancedhydrodynamicmodels
models were
were proposed
dynamic
responsesof floating
The Theadvancedhydrodynamic
proposedtotopredict
predict
dynamic
responsesof floating
wind turbines in combined wave-current conditions and validated by laboratory- and full-scale semiwind turbines in combined wave-current conditions and validated by laboratory- and full-scale
submersible platforms. The conclusions are obtained as follows:

semi-submersible platforms. The conclusions are obtained as follows:

The global hydrodynamic coefficient models considering the effects of Reynolds and KC
Thenumbers
global hydrodynamic
the effects
Reynolds
and KC numbers
were proposedcoefficientfor the firstmodels
time to considering
evaluate the added
massofanddrag coefficientsforweretheproposed
for andthe first
time platforms.
to evaluateThethepredictedadded mass
anddrag coefficientsfor theby
laboratorylaboratoryfull-scale
globalhydrodynamic
coefficients
theshowThegood
agreementwith hydrodynamic
those obtained fromthe forced
tests. models
andproposed
full-scalemodels
platforms.
predicted
globalcoefficients
byoscillation
the proposed
2.show
Angoodadvancedhydrodynamic
model
was
proposedtocalculate
the
drag
force
in
the
combined
agreement with those obtained from the forced oscillation tests.
wave-current condition. The predicted platform motions and mooring tensions by the proposed
An advancedhydrodynamic model was proposed to calculate the drag force in the combined
wave-current condition. The predicted platform motions and mooring tensions by the proposedmodel showed favorable agreement with the experimental data obtained from the water tanktests, while the conventional Morison’s equation overestimated them.
The predicted dynamic responses of platform were improved with consideration of the directional
spreading of the sea wave spectrum, while those by the wave model without the directional
spreading function were significantly underestimated in the sway, roll and yaw directions.
The predicted dynamic responses by the proposed model showed good agreement with the fieldmeasurements on the 2 MW FOWT used in the Fukushima demonstration project.
1.

1.

2.

3.

Author Contributions: Conceptualization, T.I.; formal analysis, Y.L.; investigation, Y.L.; visualization, Y.L;
writing—original draft preparation, T.I.; writing—review and editing, T.I.; project administration, T.I. All authors
have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Acknowledgments: This research was carried out as a part of the Fukushima floating offshore wind farmdemonstration project funded by the Ministry of Economy, Trade and Industry. The authors wish to express their
deepest gratitude to the concerned parties for their assistance during this study.
Conflicts of Interest: The authors declare no conflict of interest.

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