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Nonlinear wave effects on dynamic responses of asemisubmersible floating offshore wind turbine in
the intermediate water
To cite this article: Jia Pan and Takeshi Ishihara 2018 J. Phys.: Conf. Ser. 1037 022037

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The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

Nonlinear wave effects on dynamic responses of asemisubmersible floating offshore wind turbine in theintermediate water
Jia Pan1, Takeshi Ishihara1
1

Department of Civil Engineering, School of Engineering, The University of Tokyo,
Japan, 7-3-1 Hongo, Bunkyo, 113-8656
E-mail: pan@bridge.t.u-tokyo.ac.jp
Abstract. The effects of nonlinear wave on the dynamic responses of Floating Offshore Wind
Turbines (FOWTs) in the intermediate water are investigated. Firstly, the predicted dynamic
responses of FOWT by the stream function in various wave conditions are investigated andvalidated by the water tank tests. The effects of nonlinear wave on dynamic responses of
FOWT are then examined by comparing response amplitude operators (RAO) and time
histories of wave elevation, dynamic responses and mooring tensions predicted by linear andnonlinear wave theories. Finally, the power spectrum densities (PSD) of dynamic responses of
FOWT are analysed to explain the nonlinear wave effects in the frequency domain.

1. Introduction
Floating Offshore Wind Turbine is a promising innovation. Some researches show that FOWTs arecost effective and are expected to produce more energy comparing with the fixed foundation at thewater depth greater than 50m [1]. Some FOWTs are built and planed in the intermediate water, where
the water depth is ranged between the deep and shallow waters. In WindFloat project, a 2MW FOWT
with a semisubmersible foundation was installed in 2011 at 5km offshore of Portugal, where the water
depth is about 40m and the waves exceed 17 meters in height [2]. In Japan, NEDO plans to construct a
3MW FOWT with a pontoon foundation at City of Kitakyushu in a site with about 50m water depth.
The wave theory is important for the design of offshore wind turbines and the nonlinearity of waves
are considered for the fixed offshore wind turbines in the shallow water. Aharwal and Manuel (2011)[3] and Van Der Meulen et al. (2012) [4] applied the second order wave theory to predict the extreme
and fatigue loads for the fixed foundations and compared with those by the traditional linear wave
theory. Ishihara and Phuc (2007) [5] investigated the effect of nonlinear wave on the dynamic
responses of a floating offshore wind turbine system and showed the dynamic responses of bending
moment at 50m water depth is 1.8 times larger than these at 200m water depth. It indicated that thenonlinear waves may cause larger extreme loads in the intermediate water. However, the systematicstudies of dynamic responses of FOWTs in the intermediate water are limited.
The several wave theories have been proposed and used in offshore engineering [6][7]. Airy wave
theory is a common approach to produce the linear regular waves. The stream function to generate thenonlinear regular waves was first developed by Dean (1965) [8] and the numerical solution was
obtained from the governing equation of velocity potential to satisfy the Laplace equation andnonlinear boundary conditions.
In this study, the applicability of linear and nonlinear wave theories are investigated to predict the
dynamic responses of a semisubmersible FOWT in the intermediate water. Firstly, numerical models
for prediction of dynamic responses and a water tank test are presented in section 2. The predicted
dynamic responses by the nonlinear wave theory are then validated by the water tank test and are
discussed in the linear, weak nonlinear and strong nonlinear regions in section 3. The conclusions
obtained from this study are summarised in section 4.
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
1

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

2. Numerical models and water tank tests
Hydrodynamic models and wave theories used in this study are described in the section. The dynamic
responses of 1/50 scaled FOWT employed in Fukushima FOREARD project [9] is investigated undervarious sea states conditions in water depths of 1.0m and used to validate the numerical models .
2.1. Hydrodynamic models
The equation of motion for the coupled simulating tower and support platform system can be written
as follows:
M ij + Aij q&&j + Cij q& j + Kij q j = FGi + FBi + FHi + FMi + FRi
(1)

(

)

where M ij , Cij and K ij are the mass matrix, damping matrix and stiffness matrix respectively, Aij isthe full added mass matrix, q j , q& j , q&& j are the unknown displacements relative to the earth at the
specified reference origin and their time derivatives, FGi is the gravitational force, FBi is the buoyancy
force, FHi is the hydrodynamic force, FMi is the mooring tension, FRi is the restoring force, and thewind load is ignored in this study. The subscripts i and j express the degrees of freedom of the modeland 6 degrees of freedom are considered..
The hydrodynamic forces contain Froude-Krylov force, diffraction force and viscous drag force asfollows:
FHi = (1 + Cai ) rVu&i + å1/ 2 rCdik Ak (ui - q&i ) | (ui - q&i ) |(2)
where V is the volume displaced by the model, the added mass coefficient Ca is specified by a fullmatrix obtained by the potential theory and acting on the reference centre of model. The dragcoefficients Cdik are assumed acting on the component k of model in the i degree of freedom inTable1 according to the references[10][11]. u , u& are the local water particle velocity and acceleration
relative to the earth obtained by the wave kinematics models and will be describe in the section 2.2.

Ca11
Ca 33
Cd 11
Cd 33

SF1
0.3
1.6
0.9
4.8

Table 1. Hydrodynamic parameters for each component of the platform.
SF2/3
CF
Pn1
Pn2/3
CC
SC1
SC2/3
Br1
Br2/3
0.4
0.3
0.01
1.0
1.1
1.0
1.1
0.4
1.4
1.6
1.9
1.6
1.6
0
0
0
0.4
0.4
1.5
1.1
0.07
2.1
0.9
0.6
1.3
0.2
1.3
4.8
4.4
0
0
0
0
0
0
0

Moor
1.0
0.5
2.6
1.2

The integrated form for Ca is derived as Eq. (3) and compared with that by the potential theory as
shown in Eq. (4):
1 N i
1 N i
Ca11 =
C
r
V
,
C
=
å
å C rVa 33
rV i =1 a11 i
rV i =1 a33 i
(3)
N
N
N
1
i
2
i
2
i
2
Ca55 =(å C rV z + å C rV x + å C rV R )rVR2 i =1 a11 i i i =1 a33 i i i =1 a 55 i i
0.0 -0.13 0.0 ùé 0.71 0.0 0.0 0.0 -0.13 0.0 ùé 0.72 0.0 0.0
ê 0.0 0.71 0.0 0.13 0.0
ú
ê
0.0 ú
0.0
0.0 úú
ê
ê 0.0 0.72 0.0 0.01
ê 0.0
0.0 0.99 0.0
0.0
0.0 ú integrated ê 0.0
0.0 0.99 0.0
0.0
0.0 ú
Caaqwa = ê
=ê
ú Ca
ú (4)
0.0 ú
0.0 ú
ê 0.0 0.13 0.0 0.09 0.0
ê 0.0 0.01 0.0 0.092 0.0
ê-0.13 0.0 0.0 0.0 0.09 0.0 úê -0.13 0.0 0.0
0.0 0.092 0.0 úêúêú
0.0 0.0 0.0
0.0 0.09 úû
0.0 0.0
0.0
0.0 0.092úûêë 0.0êë 0.02

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

The difference between the potential theory and the integrated force model is negligible if theelements in the distributed force model are small enough, and the detail was shown in the reference[12]. Cd can be obtained by the same integrated form. The amplitude dependency of hydrodynamic
coefficients is neglected since the effect of the nonlinear waves on dynamic responses is the main
topic in this study. The software Orcaflex[13] is used for prediction of dynamic responses.
2.2. Wave theories
Two typical regular wave models representing linear and fully nonlinear waves are used in this study.
Airy wave theory is the simplest wave theory for simulating linear regular wave, and the stream
function represents a fully nonlinear regular wave. The elevation of linear regular wave surface can beshown as follows:Hh LR (t ) = cos(kx - wt )(5)2
where H is the wave height, k is the wave number, and w is the frequency of wave. In the linear
wave, the wave crest height is the same as the wave through depth. And linear wave kinematics is used.
The elevation of nonlinear regular wave surface generated by stream function can be expressed as :N
2p n
h NR (t ) = a n cos( x - ct )
(6)
L
n =1

å

where N is the order of stream function determined by the wave parameters and water depth, a n is
the unknown coefficients that can be calculated according to the boundary conditions, c is the
propagation velocity of wave with respect to the water. Stream function can simulate waves from thelinear wave (with N = 1 ) to the wave close to breaking wave height (with N > 11 ).
φ280

SF2

φ136

Pn2

SF1
Pn1

CF

Br3

Br1
1005

580

500

SC1
SC1

CC

SC3
SC3

φ45
φ100

Pn3
80

SF3
870

(a) Overview(b) Top view(c) Side view(d) Configuration
Figure 1. Configuration and dimension of the model used in the water tank test.
2.3. Water tank test
A 1/50 scaled FOWT employed in Fukushima FOREARD project is used in the water tank test as
shown in Figure 2. The tower and nacelle is simulated by a pole with a weight of 1.42kg and set on the3The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

platform, while the detail of rotor is not considered in this model since the wind loads are not includedin the water tank test. The total weight of the platform is 42.2kg and the model draft is 0.38m. In thissemi-submersible FOWT, four mooring lines are connected with the three heave plates. The length ofmooring line is 9.86m and the water depth is 1.0m. The fairlead is set on the heave plate and is locatedat 0.34m under the water surface. The wave height is measured at the front of model in the water tanktest. The detail dimension of FOWT is shown in the top and side view of model in Fig. 1 (b) and (c).
The mooring lines 1 and 2 are located in the upstream and the mooring lines 3 and 4 are in thedownstream.
The static equilibrium test in still water is conducted to confirm the initial position of the platform andthe initial fairlead tension in the mooring lines. Free decay tests in surge, sway, heave, roll, pitch andyaw directions in the still water are carried out to find the natural periods of platform in each degree of
freedom (DOF). Time histories of platform motions in regular waves are measured. Unidirectional
waves are used in the experiment.
2.4. Cases in water tank test and simulations
The cases of the water tank tests and simulations and various wave theories [7] are shown in Figure 2.
In this study, three regions, namely, linear, week nonlinear and strong nonlinear regions are definedand dynamic responses of FOWT are considered in these regions. In the linear region, Airy wave
theories and 1st order stream function are generally used. In the weak nonlinear region, Stokes’s 5th
and 3rd order stream function are selected in the diagram, where the nonlinearity of waves is not sostrong. The strong nonlinear region is defined near the breaking waves. As the limitation of water tanktest, it is difficult to generate very large wave, even FOWTs may endure such wave height in the realocean environment. In this study, strong nonlinear waves are considered only in the simulations.

H/gT^2T
LinearWeak nonlinear
Strong nonlinear
Deep water breaking limit
H/L=0.14
Hb

0.01

0.9Hb
Shallow water
breaking limit
H/d=0.78

Stokes' 5th or
stream function 3

LinearregionHeight
(m)
0.02

0.001
>11 11 9 7 5

WeaknonlinearregionStrongnonlinearregion3

Stream function
Linear/Airy or
stream function

0.0001

0.001

0.01

Shallow waterIntermediate depth

0.1

d/gT^20.20

0.50

Period
(s)
1.5,
2.0
2.8
1.5
2.0
2.8
1.5
2.0
2.8

Ursellnumber
0.2
0.5
1.3
2.2
5.4
12.9
5.5
13.5
36.1

Deep water(a) Regular wave theory selection diagram in (b) Typical regular wave conditions in the water tankdifferent water depths and wave conditions
tests and simulations.
Figure 2. Typical regular wave conditions and that in regular wave theory selection diagram.
The Ursell number [14], which relates with the wave steepness parameter and shallow water parameter
representing the nonlinearity of regular waves, is used to identify the nonlinearity of waves and isdefined as follows:
U R = S / m 3 = HL2 / d 3(7)where S is the wave steepness parameter and m is the shallow water parameter, and H , L and d
are the wave height, wave length and water depth respectively. It is clear that Ursell number increases

4The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

if the wave height and the wave length increase and the water depth becomes shallower. Some strong
nonlinear waves cannot be generated in experiment due to the limitation of the water tank test in water
depth of 1.0m.
Response amplitude operators (RAO) in the regular waves are used in this study. Fast Fourier
Transform is employed to transform the time series to the amplitude of response in the frequency
domain, and the dimensionless amplitude normalized by the half of wave height is shown as:(8)
RAOx = Ax / ( H / 2) , RAOz = Az / ( H / 2) , RAOq y = Aq y / ( H / 2) , RAOTi = ATi / p0 BS
where Ax , Az and Aq y are the amplitudes corresponding to the wave frequency component in surge,
heave and pitch directions, and H refers to the wave height, ATi refers to the amplitude of fairlead
tension in each mooring line, p0 is the dynamic pressure at the water surface and is estimated as
p0 = r gH / 2 , B and S are the floater span and the submerged depth.
The non-dimensional time histories of dynamic motion are defined as follows:x* (t ) = x(t ) / ( H / 2) , z* (t ) = z (t ) / ( H / 2) , q y * (t ) = q y (t ) / ( H / 2) , Ti* (t ) = Ti (t ) / p0 BS(9)where x(t ) , z (t ) , q y (t ) , Ti (t ) correspond to the time histories of dynamic motions in surge, heave andpitch directions, and mooring tension respectively.
3. Results and discussions
The predicted dynamic responses by the nonlinear wave theory are validated by the water tank test.
The dynamic motions predicted by the linear and nonlinear wave theories are then compared in thelinear, weak nonlinear and strong nonlinear regions in order to investigate the effects of nonlinear
waves.
3.1. Validation
The static equilibrium test by the numerical model is performed and the initial displacements of thefloater in the water tank with a depth of 1.0m are shown in the Table 2. It is can been seen that theinitial displacements in six degrees of freedom are small and the model is balanced at the initialsituation.
Table 2. Comparison of predicted and measured initial displacements of the floater model.
DOF
Surge(m) Sway(m) Heave(m) Roll(rad.) Pitch(rad.) Yaw(rad.)Experiment
0.005
-0.02
-0.00070.00010.0009-0.003
Simulation-0.0020.004
0.00050.0006
0.002
0.001
The free decay tests in six degrees of freedom are performed for comparing the predicted natural
periods of floater with the experimental data as shown in Table 3. It is clear that differences of natural
periods in the surge, heave and yaw directions are very small, while the natural periods show somedifferences in the sway, roll and pitch directions. These differences are due to the imperfection ofexperiment, especially in the sway and roll directions.
Table 3. Comparison of predicted and measured natural periods of the FOWT.
Natural periods(s)
Surge
Sway
Heave
Roll
Pitch
Yaw
Experiment
5.85
18.51
2.38
3.32
3.23
5.53
Simulation
5.84
19.80
2.38
3.59
3.40
5.54
Figure 3 shows the measured and predicted RAOs of dynamic motions and tensions of mooring lines.
RAOs predicted by the numerical model show good agreement with those from the water tank test .
The difference between experiment and simulation in pitch for the case with a wave height of 0.02m

5The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

may be caused by the measurement errors in the water tank test because the wave height is very low inthis case. It is found that the normalized surges increase linearly as the wave period increases because
the wave periods are far away from the natural period of floater. A peak appears in the heave motion
as a result of resonance since the wave period is close to the heave natural period of floater. It is clearthat RAOs of heave motions significantly decrease with the increase of wave height. The reason is thatthe nonlinear hydrodynamic force is dominant in the natural period.
3

3

3

2.5

2.5

0.04

Stream function H=0.02m
0.035

Stream function H=0.14m

2.5

0.03

Exp H=0.20m
1.5

1

2

1.5

1

2

RAO-Tension (N/N)

Exp H=0.14m

RAO-Pitch (radm/m)

Exp H=0.02m

RAO-Heave (m/m)

RAO-Surge (m/m)

Stream function H=0.20m
2

1.5

1

0.025

0.02

0.015

0.01

0.5

0.5

0.5
0.005

0

0
1

1.5

2

2.5

3

0

0
1

1.5

2

2.5

3

1

Period (s)Period (s)

1.5

2

2.5

3

Period (s)

1

1.5

2

2.5

3

Period (s)(a) Surge motion(b) Heave motion(c) Pitch motion(d) Mooring tension
Figure 3. Comparison of RAOs of surge, heave and pitch motions, and mooring tensions between
experiments and simulations in the intermediate water.
3.2. Effects of nonlinear waves on the dynamic responses of FOWTThe comparison of RAOs of dynamic motions predicted by Airy wave theory and stream function are
shown in Figure 4 for various wave heights at the wave period of 2.0s. It is clear that there are no large
difference between dynamic motions by the linear and nonlinear wave theories for the waves heightsfrom 0.02m in the linear region to 0.20m in the weak nonlinear region. In the strong nonlinear region
defined near the breaking wave as shown in Figure 2, the RAOs of heave and pitch motions of the
FOWT predicted by the linear wave theory are overestimated comparing with those by the nonlinear
wave theory, and the overestimations become large with the increase of wave height. Furthermore,
RAOs of surge motions are almost constant, while RAOs of heave and pitch motions increase with theincrease of wave height at T=2.0s.
Airy wave
Stream function

Airy wave
Stream function
1

0.8

0.8

0.8

0.6

Pitch(rad/m)

1

Heave(m/m)

Surge(m/m)

Airy wave
Stream function
1

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

H=0.02m H=0.14m H=0.20m H=0.30m H=0.40m H=0.50m

0

H=0.02m H=0.14m H=0.20m H=0.30m H=0.40m H=0.50m

0

H=0.02m H=0.14m H=0.20m H=0.30m H=0.40m H=0.50m

Figure 4. Comparison of RAOs of dynamic motions in surge, heave and pitch directions by Airy wave
theory and stream function for different wave heights at the wave period of 2.0s.
The predicted RAOs of mooring tensions in mooring lines 1 and 3 by Airy wave theory and stream
function are shown in Figure 5 for various wave heights at the wave period of 2.0s. The mooring
tensions by linear wave theory are overestimated in the upstream side and underestimated in thedownstream side, comparing with those by the nonlinear wave theory.
Table 4 shows the differences between RAOs of surge, heave and pitch motions and mooring tensions
predicted by the linear and nonlinear wave theories. Difference between RAOs of surge by the linear
and nonlinear waves is less than 4% since the surge motions are not sensitive to the nonlinear wave
effects. It is obvious that there are no differences between the dynamic motions in the linear region.
However, RAOs of heave and pitch motions are overestimated by 17.6% and 24.6% if the nonlinearity

6

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

of waves are not taken into account, which lead to a conservative design in the strong nonlinear region.
RAOs of mooring tensions by the linear and nonlinear wave theories show no differences in the linear
region, and little differences in the weak nonlinear region. In the strong nonlinear region, the linear
wave theory overestimates the tension of mooring line 1 by 7.1% and underestimates that of mooringline 3 by 8.2%.
250

140

Airy wave
Stream function

120

Airy wave
Stream function

200

Tension(N/m)

Tension(N/m)

100

150

100

80

60

40

50
20

0

0

H=0.02m H=0.14m H=0.20m H=0.30m H=0.40m H=0.50m

H=0.02m H=0.14m H=0.20m H=0.30m H=0.40m H=0.50m

Figure 5. Comparison of RAOs of mooring tensions of line 1 (left) and line 3(right) by airy wave and stream function for different wave heights.
Table 4. Differences of RAOs predicted by the linear and nonlinear wave theories .
Ratio
H=0.02m H=0.14m H=0.20m H=0.30m H=0.40m
H=0.50m
Surge
0
0
-1%
-1%
-2%
-4%
Heave
0
0
1.8%
4%
8%
17.6%
Pitch
0
0
3%
5%
13.6%
24.6%
Line 1
0
0.8%
3.6%
6%
3.4%
7.1%
Line 3
0
-0.7%
-2.4%
-5.2%
-7.6%
-8.2%
To explain the effects of nonlinear wave on the dynamic responses of floater, time histories and PSD
of wave elevations, dynamic responses and mooring tensions in various wave heights are shown in
Figure 6 and Figure 7 in the linear, weak nonlinear and strong nonlinear regions. Ten periods of data
are used for the spectrum analysis. The linear and nonlinear wave theories show the same wave
elevation profile and spectrum in the linear region and result in the same dynamic responses andmooring tensions. No second order motion is excited in this region because both waves and mooringlines are linear. The dynamic motions in the surge and pitch directions in the low frequency region iscaused by the motions with the natural periods of floater.
In the weak nonlinear region with a wave height of 0.20m, differences of dynamic motions are lessthan 3% in the surge, heave and pitch directions. The nonlinearity of wave has no much effect on
dynamic responses in the region, even the crest predicted by the stream function is higher than that by
Airy waves and some high order components appear in the wave elevation and mooring tension. PSD
of dyanmic responses shows that the pitch motion is slightly sensetive to the nonlinear wave effects,
since the second order pitch motion is observed in the weak nonlinear region.
In the strong nonlinear region, the strong nonlinearity of wave is observed by comparing with wave
elevations predicted by the linear and nonlinear wave theories. The nonlinear regular wave show
obvious asymmety for the crest and trough and the high order components are significant. The highorder components in the dynamic motions are also excited by the strong nonlinearity of mooring
tensions.
The reason of overestimations of RAOs by the linear wave theory in the heave and pitch directions isthat the first order component of wave elevation in the linear wave is larger than that in the linear
wave. The higher order components may be important for some structures with a natural frequency inthese frequency regions. The mooring tensions by Airy wave theory and the stream function havesome phase differences due to the asymmetries in the wave velocity and acceleration by the stream

7

The Science of Making Torque from Wind (TORQUE 2018)
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IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

function.
1.5

1.5
Airy w ave
Stream function
Exp.

1

-0.5

-1

-1

-1

-1.5
T/4

T/2
Time (s)

3T/4

T

T

0

0

-1

-1

-1

-1.5
T

-1.5
0

1.5

T/2
Time (s)

T/4

3T/4

T

0

1.5
Airy wave
Stream function
Exp.

1

Heave(m/m)

Heave(m/m)

0

0

-0.5

-1

-1

-1

-1.5
T/4

T/2
Time (s)

3T/4

T

-1.5
0

1.5

T/4

T/2
Time (s)

3T/4

T

0

1.5
Airy wave
Stream function
Exp.

1

Pitch(rad/m)

Pitch(rad/m)

0

0

-0.5

-1

-1

-1

-1.5
T/4

T/2
Time (s)

3T/4

T

-1.5
0

0.006

0.006

0.004

0.004

0.002

0.002

T

0.5

-0.5

0

3T/4

Airy w ave
Stream function

-0.5

-1.5

T/2
Time (s)

1

0.5

0

T/4

1.5
Airy wave
Stream function
Exp.

1

0.5

T

0.5

-0.5

0

3T/4

Airy w ave
Stream function

-0.5

-1.5

T/2
Time (s)

1

0.5

0

T/4

1.5
Airy wave
Stream function
Exp.

1

0.5

T

0.5

0

-0.5

3T/4

3T/4

Airy w ave
Stream function

-0.5

T/2
Time (s)T/2Time (s)

1

-0.5

T/4

T/4

1.5
Airy wave
Stream function
Exp.

Surge(m/m)

Surge(m/m)

Surge(m/m)

3T/4

0.5

0

0

Heave(m/m)

T/2Time (s)1-1.5

Pitch(rad/m)

T/4

1.5

0.5

T/4

T/2
Time (s)

3T/4

T

0

T/4

T/2
Time (s)

3T/4

T

0.3
Airy wave
Stream function

0.25

Tension (N/N)

Tension (N/N)

0.2

0-0.002Tension (N/N)

Surge

-1.5
0

Airy wave
Stream function
Exp.

1

Heave

0

-0.5

1.5

Pitch

0

-0.5

0

Stream function

0.5
Elevation (m/m)

0

-1.5

Mooring tension

Airy w ave
1

0.5
Elevation (m/m)

0.5
Elevation (m/m)

Wave elevation

1

1.5
Airy w ave
Stream function
Exp.

0

0.15

0.1

-0.0020.05-0.004

Airy wave
Stream function
Exp.

-0.004-0.006Airy wave
Stream functionExp.

0-0.0060

T/4

T/2
Time (s)

3T/4

T

-0.05
0

T/4

T/2
Time (s)

3T/4

T

0

T/4

T/2
Time (s)3T/4T(a) H=0.02m(b) H=0.20m(c) H=0.50m
Figure 6. Comparison of time histories of normalized elevations, dynamic responses andmooring tensions, by linear and nonlinear wave theories for different wave heights.

8

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

100
Airy wave
Stream function
Exp.

-6

1.5

2

0

2

0.0001

-6

-8

10
0

0.5

1
Frequency (Hz)

1.5

2

0

2

0.0001

-6

-8

-8

10
0

0.5

1
Frequency (Hz)

1.5

2

0

100

2

0.0001

-6

-6

-8

10
1
Frequency (Hz)

1.5

10
0

2

0.01

0.5

1
Frequency (Hz)

1.5

2

0

0.01
Airy wave
Stream function
Exp.

-4

2

-10

-6

10

-8

10

-10

10

2

-4

2

2

-8

10

1.5

10

2

-6

1
Frequency (Hz)

-6

10

2

S-tension(N /N /Hz)

10

10

0.5

0.01
Airy wave
Stream functionExp.

S-tension(N /N /Hz)

-4100.0001

10

-8

-8

0.01

2

2

0.01

10

10

2

Stream function

2
-6

10

1.5

1

S-pitch(m /m /Hz)

S-pitch(m /m /Hz)

20.00010.5

1Frequency (Hz)Airy w ave

1

0.01

0

0.5

100
Airy wave
Stream function
Exp.

Airy w ave
Stream functionExp.

1

0.0001

10

10
2

100

0.01

2

2

2

0.01

-6

-8

2

Stream function

10

10

1.5

1

S-heave(m /m /Hz)S-heave(m /m /Hz)

2

-6

10

1.5

1
Frequency (Hz)Airy w ave

1

0.00011Frequency (Hz)

0.5

100
Airy w ave
Stream function
Exp.

Airy wave
Stream function
Exp.

0.5

0.0001

10

100

0

0.01

2

2

0.01

2

0.01

2

Stream function

-8

1

1.5

Airy w ave

10
1.5

1
Frequency (Hz)

1

-6

1
Frequency (Hz)

0.5

100

10

100

2

2

2

0.0001-8

S-heave(m /m /Hz)

1.5

S-surge(m /m /Hz)

S-surge(m /m /Hz)

2

2

S-surge(m /m /Hz)

0.01

10

2

1
Frequency (Hz)1

-6

S-pitch(m /m /Hz)

-8

0.5

Airy w ave
Stream function
Exp.

Airy wave
Stream function
Exp.

0.5

0.0001

10
0

10

2

2

2
100

0

0.01

-6

-8

1
Frequency (Hz)Stream function

10

10
0.5

1

S-tension(N /N /Hz)

0.0001

10

100

Surge

0.01

-6

10

0

Heave

S-elevation(m /m /Hz)

20.0001-8

Pitch

Airy wave
1

2

2
2

0.01

10

Mooring tension

100
Airy wave
Stream functionExp.

1
S-elevation(m /m /Hz)

1
S-elevation(m /m /Hz)

Wave elevation

100

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

-8

10

-10

10

10

Airy w ave
Stream function
-12

-12

10

-12

10
0

0.5

1
Frequency (Hz)

1.5

2

10
0

0.5

1
Frequency (Hz)

1.5

2

0

0.5

1
Frequency (Hz)1.5(a) H=0.02m(b) H=0.20m(c) H=0.50m
Figure 7. Comparison of PSD of normalized elevations, dynamic responses and mooring
tensions, by linear and nonlinear wave theories for different wave heights.

92

The Science of Making Torque from Wind (TORQUE 2018)
IOP Conf. Series: Journal of Physics: Conf. Series
1037 (2018)1234567890‘’“” 022037

IOP Publishingdoi:10.1088/1742-6596/1037/2/022037

4. Conclusions
A numerical model to predict the dynamic responses based on the potential theory and nonlinear dragforce is validated by the water tank tests. Waves generated by Airy wave theory and the stream
function are applied for prediction of dynamic responses of a semisubmersible floating offshore windturbines in the intermediate water. Conclusions are obtained as follows:
1) The dynamic responses of a semisubmersible FOWT predicted by the nonlinear wave theory show
good agreement with those from the water tank tests.
2) The dynamic responses and mooring tensions in the linear and weak nonlinear regions areaccurately predicted by both linear and nonlinear wave theories. However, in the strong nonlinear
region, the linear wave theory overestimates the heave and pitch motions by 17.6% and 24.6% andunderestimates the surge motion by 4%. The mooring tensions in the upstream are underestimated,while those in the downstream are overestimated.
3) The PSDs of dynamic responses of the floater motions and mooring tensions only show theharmonic component in the linear region. The second and third order components appear in the
PSDs of wave elevation, pitch motions and mooring tensions in the weak nonlinear regions, while
all PSDs reveal the higher order components in the strong nonlinear wave region.
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