Journal of Physics: Conference Series

PAPER • OPEN ACCESS

Numerical prediction of hydrodynamic coefficients for a semi-sub
platform by using large eddy simulation with volume of fluid method andRichardson extrapolation
To cite this article: Jia Pan and Takeshi Ishihara 2019 J. Phys.: Conf. Ser. 1356 012034

View the article online for updates and enhancements.

This content was downloaded from IP address 158.46.222.72 on 25/10/2019 at 02:19

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

Numerical prediction of hydrodynamic coefficients for asemi-sub platform by using large eddy simulation with volumeof fluid method and Richardson extrapolation
Jia Pan1, Takeshi Ishihara1
1

Department of Civil Engineering, School of Engineering, The University of Tokyo, Japan,7-3-1
Hongo, Bunkyo, 113-8656 Tokyo, Japan
E-mail: ishihara@bridge.t.u-tokyo.ac.jp
Abstract. Hydrodynamic coefficients for predicting hydrodynamic loading is significant forthe design of offshore wind turbine floaters. In this paper, hydrodynamic coefficients for asemi-submersible floating offshore wind turbine are investigated experimentally andnumerically. In the water tank test, added mass and drag coefficients of the semi-submersiblemodel are identified by the forced oscillation tests. In numerical simulation, large eddy
simulation (LES) with volume of fluid method (VOF) is adopted to predict added mass anddrag coefficients. Firstly, numerical errors in the predicted hydrodynamic coefficients are
systematically studied, and Richardson extrapolation is employed to obtain the gridindependent solution. The predicted added mass and drag coefficients varying with KC number
are then validated by the water tank tests and the mechanism of the hydrodynamic force areclarified by vortex shedding patterns. The hydrodynamic coefficients for each element are also
investigated. Finally, effect of free surface on the hydrodynamic forces is discussed and thepredicted added mass and drag coefficients are compared with those obtained from the watertank tests.

1. Introduction
Floating offshore wind turbine (FOWT) is a promising innovation for generating electricity fromrenewable energy. Comparing with other types of floaters, the semi-submersible type of FOWT
containing columns, heave plates, pontoons and braces, is one of the most complex types and received
main attention, as there are some advantages such as the high adaptedness for both shallow and deep
water, and the competitive cost. In WindFloat project [1], a 2MW FOWT with a semi-submersible
foundation was installed in 2011. In Japan, a 2MW compact semi-submersible FOWT and a 7MW Vshape semi-submersible FOWT have been constructed and operated in Fukushima Project [2]. The
platforms are set in the sea states where the wave period is near to the natural period of floater motion,
so hydrodynamic coefficients for predicting nonlinear hydrodynamic loading by Morison’s equation[3] and potential theory [4] is significant for the design of FOWTs.
A number of numerical studies on the hydrodynamic coefficients has been conducted for simple
heave plates or cylinders, and compared with the water tank tests by Tao et al.[5], [6], [7]. LopezPavon and Souto-Iglesias [8] predicted the hydrodynamic coefficients for a heave plate by RANS
model, and provided reasonable results for plain plate, but there were some errors for reinforced plate.

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

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

Benitz et al. [9] predicted the hydrodynamic coefficients for the simplified OC4-DeepWind semisubmersible, and highlighted the role of free surface in the fluid flow near the structure. Zhang andIshihara [10] investigated the hydrodynamic coefficients of multiple heave plates by large eddy
simulation (LES) turbulent model with volume of fluid (VOF) method and the numerical results
matched well with the experimental data. Unlike a simple plate, numerical prediction of hydrodynamic
coefficients for a semi-submersible platform by CFD is a challenge due to the complexity of platform
and interaction between the elements. It is difficult to obtain the cost-effective and reliable solution byrefining grid only, the methodology to provide an accurate solution is necessary. Furthermore,
systemic research of free surface on the hydrodynamic coefficients in the horizontal and verticaldirections has not been conducted.
Klaka et al. [11] indicated that the hydrodynamic coefficients strongly depend on Keulegan
Carpenter (KC) number [12], and the added mass coefficients predicted by the potential theory were
not accurate comparing with the experimental data. However, dependency of hydrodynamic
coefficients on KC number for a semi-submersible platform has not been discussed, and the
distribution of hydrodynamic coefficients has not been investigated yet.
Section 2 describes the numerical model used in this study, including the model description,
governing equations, numerical setup and water tank tests. In Section 3, numerical errors in thepredicted hydrodynamic coefficients based on three levels of grid are systematically studied. The gridindependent solution by Richardson extrapolation is investigated and validated by the experiment. The
dependency of added mass and drag coefficients on KC number are then numerically examined andcompared with the water tank test. The distribution of hydrodynamic coefficients for each element isalso investigated. Finally, effects of free surface on the added mass and drag coefficients are discussed
and compared with those from the water tank tests. Conclusions are summarized in Section 4.

(a)

(b)

Fig. 1. (a) Top view and (b) side view of the model and their dimensions, units (mm).

2. Numerical model
Numerical model of a down-scaled semi-submersible floating platform is described in section 2.1.
The governing equation by using LES with VOF method are introduced in section 2.2 andcomputational domain including the grid system used in the numerical simulation is described in
section 2.3. Numerical schemes and boundary conditions are presented in section 2.4. The definitionof the hydrodynamic coefficients is provided in section 2.5 and the water tank test is described in
section 2.6.

2

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

2.1 Model description
A 1/50 down-scaled semi-submersible type of platform based on the 2MW floating offshore windturbine system in Fukushima Forward Project is used in the water tank tests and numerical simulations.
The floater contains four columns connected by three pontoons, three braces and three decks. The
overview of the floater and its dimensions are shown in Table 1. The total weight of the platform
model is 42.2 kg and the model draft is 0.38m as shown in the reference [13]. A summary of thegeometry, including diameter and height for each element is illustrated in the Table 1, and thecharacteristic volume and area of elements in the vertical and horizontal directions for normalizinghydrodynamic coefficients in this study are listed in Table 2.
Table 1. Specifications of the 1/50 scaled model geometry of the FOWT model.
Elements
Total draft of the platform
Elevation of center column(tower base) and side columns(SC) above SWL
Spacing between side columns
Height of center columns (CC) and side columns
Diameter of center column
Diameter of side columns
Depth to top of heave plates(Hp) below SWL
Height of heave plates and pontoon (Pntn)
Diameter of heave plate
Width of pontoon
Length of pontoon
Diameter of brace (Brace)
Height of deck
Width of deck
Center of gravity below SWL
meta-centric height above SWL
Radius of gyration Kxx
Radius of gyration Kyy

Dimension (m)
0.38
0.2
1.005
0.5
0.1
0.136
0.3
0.08
0.28
0.06~0.12
0.39
0.045
0.045
0.045
-0.166
0.086
0.52
0.51

Table 2. Characteristic volume and area of each element in the vertical and horizontal directions fornormalizing hydrodynamic coefficients of 1/50 scaled model.
VolumeArea in
Area in
Area of water
3
2
2
(m )
horizontal ( m )vertical ( m )plane ( m 2 )Side column SC-1
0.004358
0.040800.014527
Side column SC-2
0.004358
0.040800.014527
Side column SC-3
0.004358
0.040800.014527
Center column Cc
0.0023560.03
00.007854
Heave plate Hp-1
0.004926
0.02240.061575
0
Heave plate Hp-2
0.004926
0.02240.061575
0
Heave plate Hp-3
0.004926
0.02240.0615750Center heave plate Hp-C 0.002993
0.01920.037412
0
Pontoon Pntn-1
0.002433
0.030409
0.02690
Pontoon Pntn-2
0.002433
0.030409
0.02690
Pontoon Pntn-3
0.002433
0.030409
0.02690Brace Br-1
0.000524
0.014764
00.002249
Brace Br-2
0.000524
0.014764
00.002249
Brace Br-3
0.000524
0.014764
00.002249
Total value [13]0.042190.32160.185
0.0581823

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

2.2 Governing equation
Large-eddy simulation (LES) is adopted in this study The Boussinesq hypothesis is employed, andthe standard Smagorinsky-Lilly model is used to calculate the subgrid-scale stresses. The governing
equations in Cartesian coordinates are expressed in the form of tensor as

ui
xi
r

ui
t

r

ui u j
xj

-

(1)

0

p
xi

m

xj

ui
xj

tij
uj
xi
xj

(2)

where u i and p are the filtered velocity and pressure, respectively.  is the molecular viscosity, and is the density of fluid. tij

r ui u j - uiu j is the subgrid-scale stress resulting from the filtering

operations, and is expressed as

1
3

 ij  -2t Sij   ii ij(3)

In which, t is the subgrid-scale turbulent viscosity, and Sij is the rate-of –strain tensor for the
resolved scale defined as

Sij

ui
xj

1
2

uj
xi

(4)

Smagorinsky-Lilly model is used to calculate the subgrid-scale turbulent viscosity, t defined as

mtr L2S Sr LS 2Sij Sij(5)where, LS is the mixing length for subgrid-scales, defined asLS  min  , CsV 1/3 (6)In which,  is the von Karman constant, 0.42, Cs is Smagorinsky constant and is set as 0.032

following the suggestion in the reference [14],  is the distance to the closest wall and V is thevolume of a computational cell.
Volume of Fluid (VOF) model is used to model air and water and to capture interface between air
and water. Volume fraction of water will be solved to capture the interface between water and air.
Continuity equation for the volume fraction of water  w is expressed as

1  w w     w wvw   0
w  t(7)where  w is water density.
2.3 Computational domain and grid arrangement
The numerical setup of simulation follows the water tank test. The whole computational domain andgrid around the model are displayed in Fig. 2. The computational domain is divided into two
subdomains with respect to the simulated phases. The lower subdomain is used to simulate the phaseof water and bottom of the subdomain is 4.1h away from the still water level (SWL), where h is themodel height of 0.5m. The upper subdomain is utilized to consider the phase of air and top of the
subdomain is 1.3h above the SWL. In order to mitigate the reflecting flow from the boundary, sidewalls are located sufficiently far away from the model. The distance between the model and inlet andoutlet is the same value of 20.7h. Hexahedral meshes are used for the whole computational domain
except for the region near the center column, where hybrid meshes are employed.
416th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

Quality of grid plays a crucial role in the accuracy of numerical results. Grid independences arestudied to choose a reasonable grid system. In this study, three different levels of grid system are
employed as shown in Table 3. The grids are refined at the locations where the substantial flow
separations are expected. As discussed in section 3, the grids are refined near the heave plates andcenter heave plate since they mostly contribute to the hydrodynamic forces in the vertical direction.
The area of 5cm around the heave plates and center heave plate is chosen to be refined considering thevortex shedding. The grids are also refined in the area of 5cm close to the side columns and center
column because they play an important role in the hydrodynamic forces in the horizontal direction.
For LES turbulent model, the first layer of mesh near the model is significant, thus more grid points
are designed in the boundary layer. In order to refine cells in the selected zones, the heave plates andcenter heave plate are selected for the boundary adaption during the vertically forced vibration, while
the side columns and center column are marked during the horizontally forced vibration. An example
of grid refinement with the boundary adaption near the heave plate HP-1 in the vertical direction andthe side column SC-1 in the horizontal direction are shown in Fig. 3 and Fig. 4, respectively, in which
the number of grid is increased from 13.7 million to 18.8 million. The grids are refined twice by thesame regulation, and the grid number of level 3 reaches 63.8 million as shown in Table 3. It is noticed
that the number of grid increases rapidly as the grid size decreases. The gird information aresummarized in Table. 4. The grid size of first layer in the vertical and radial directions are 1.3 mm and
2 mm, respectively. The grid size is 8 mm for the grid level 1 and 2 mm for the grid level 3 with the
expanding factor from 1.0 to 1.2.
(a)(b)
Symmetry

Outflow

Wall

Fig. 2. (a) Computational domain and (b) grid around the model.
Table 3. Description of grid size and grid number for three different grid levels.
Grid level1
2
3
Grid size
h1  8mm
h2  4mm
h3  2mm
Grid number (million)

13.7

18.8

Table 4. Summary of grid information.
Parameters
First layer in the radial direction (mm)First layer in the vertical direction (mm)Grid size (mm)Expanding factor
Grid number (million)

Value
2
1.3
2~8
1.0~1.2
13.7~63.8

5

63.8

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034(a)

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

(b)
z

z
y

x

y

x

Fig. 3. Local view of mesh around the edge of heave plate HP-1 with (a) grid level 1 and (b) gridlevel 2 in x-z plane.
(a)

(b)
y

y
z

x

z

Fig. 4. Local view of mesh for the side column SC-1 with (a) grid level 1 and (b) grid level 2 in x-yplane.
2.4 Numerical schemes and boundary conditions
A finite volume method is employed for present time-dependent numerical simulations shown in the
Table 3. A second order central difference scheme is used for the convective and viscous term. A firstorder implicit scheme is employed for the unsteady term in the momentum equation as shown inEq.(2), while an explicit approach is adopted for time discretization in the volume fraction equation asshown in Eq.(7). The courant number is 0.25, which means the time step for VOF simulation will bechosen to be one-fourth of the minimum transit time for any cell near the interface. A pressure-based
segregated algorithm is used to solve the non-linear and coupled governing equations. A PressureImplicit with SIMPLE algorithm is chosen to decrease the iterations for the pressure-velocity coupling
solutions as shown the reference [15]. The Table 5 summarizes the numerical schemes used in thisstudy.
The forced vibration tests are carried out in the numerical simulations to evaluate the hydrodynamic
coefficients, and dynamic mesh with layering mesh update method is utilized to simulate themovement of the model. The top of air phase and the bottom of water phase are treated as stationary
boundaries. Both split and collapse factors for the cells near the boundaries are 0.4 with respect to thefirst cell height at the boundaries as shown in the reference [10].
Boundary conditions are summarized in Table 6. No-slip wall condition is adopted for the surface ofmodel and bottom of the domain. Symmetry conditions are applied for top and side wells, while the6x16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

outflow boundary condition is applied to the inlet and outlet. The static pressure profile in the verticaldirection is used to avoid the reflection of water in the horizontal direction.
The governing equations are solved by a software ANSYS Fluent [16]. Near-Wall treatment is used
for the wall-adjacent cells. When the mesh is fine enough to resolve the laminar sublayer, the wall
shear stress is obtained from the laminar stress-strain relationship. If the mesh is too coarse to resolve
the laminar sublayer, it is assumed that the centroid of the wall adjacent cell falls within thelogarithmic region of the boundary layer and the law-of-the-wall is employed.
According to the reference [14], five periods of time histories of predicted hydrodynamic force areconsidered. The predicted hydrodynamic forces are stable enough after the first period as shown in
Table 7, The differences between the predicted hydrodynamic coefficients by using the time histories
in the last 4, 3 and 2 periods are less than 1%. It means the simulation converges to a stable state after
the first period. In order to avoid the effect from the initial unstable solution on the accuracy ofhydrodynamic coefficients, the simulated data from the second periods are chosen in the analysis, andthe averaged coefficients are used in this study.
Table 5. Summary of numerical schemes.
Items

SchemeSmagorinsky-Lilly ( Cs =0.032)
Second order central difference scheme
First order implicit scheme

Turbulence model
Spatial discretization method
Time discretization for momentum equation
Time discretization for volume fraction
equation
Pressure-velocity coupling
Courant number
Dynamic mesh

Explicit scheme
SIMPLE
0.25
Layering

Table 6. Summary of boundary conditions.
Items
Model walls
Top of air phase
Bottom of water phase
Side walls
Inlet and Outlet

Boundary conditions
No-slip wall
Symmetry
Symmetry
Symmetry
Outflow

Table 7. Error estimation for the predicted hydrodynamic coefficients in the horizontally forced
vibration with a=0.1m and T=3.0s.
Periods
Mean force
Predicted Ca
Error of Ca
Predicted Cd
Error of Cd
2~5-0.01160.779401.110603~5-0.00830.77950.01%
1.11230.15%
4~5-0.0092
0.77990.06%
1.11410.32%
2.5 Definition of hydrodynamic coefficients
The added mass and added inertia moment coefficients matrix, Ca , consists of 36 components forthe six degrees of freedom and Caij represents the component in the i th direction due to an unit
acceleration in the j th direction. The drag force and drag moment coefficients matrix, Cd , can besimilarly defined as Ca .
Due to the symmetry of coefficients matrices, it can be concluded that Caij  Caji and Cdij  Cdji .
The surge motion caused by sway, heave and yaw motions induces no transversal force due to7

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

symmetric geometries of floater. and the same consideration can be applied for the heave motion, theadded mass and drag force s matrices can be simplified as.
 Ca11
 0.0

 0.0
[Ca ]  
 0.0
Ca 51 0.0

0.0

0.0

0.0

Ca15

Ca 22

0.0

Ca 24

0.0

0.0

Ca 33

0.0

0.0

Ca 42

0.0

Ca 44

0.0

0.0

0.0

0.0

Ca 55

0.0

0.0

0.0

0.0

0.0 
 Cd 11

 0.0
0.0 

 0.0
0.0 
 ,[Cd ]  
0.0 
 0.0

Cd 510.0
Ca 66  0.0

0.0

0.0

0.0

Cd 15

Cd 22

0.0

Cd 24

0.0

0.0

Cd 33

0.0

0.0

Cd 42

0.0

Cd 44

0.0

0.0

0.0

0.0

Cd 55

0.0

0.0

0.0

0.0

0.0 
0.0 
0.0 

0.0 
0.0 

Cd 66 (8)

In order to obtain the added mass and drag coefficients of the model. The horizontally and verticallyforced vibrations in the numerical simulations are carried out as conducted in the water tank test. The
model is forced to oscillate in the horizontal and vertical directions as(9)x(t )  a sin(t )where   2 /T is the frequency of oscillation and a is the amplitude of oscillation as shown in
Table 8.
The time series of predicted hydrodynamic force on the wholemodel, FH (t ) by CFD is obtained bysubtracting the buoyancy force, Fb , and hydrostatic force, FK (t ) , from the total predicted force, F (t ) ,which is obtained from the surface pressure and shear on the model.

FH (t )  F (t ) - Fb - FK (t )

(10)

(11)
Fb  w g
where  is the displaced volume of water and g is the gravitational acceleration.
The hydrodynamic force on the whole model, FH (t ) , can be expressed in the form of Morison’sequation as
FH (t ) -Ca Mx(t ) - 0.5 Cd rw A x(t ) x(t )(12)1
Ca rw a w 2 sin(wt ) - Cd rw A (aw ) 2 cos(wt ) cos(wt )2where Ca is the added mass coefficient, Cd is the drag coefficient,  and A represent characteristic
volume and area according to the definition of Ca and Cd . x(t) and x(t) are the velocity andacceleration of the model motion, respectively
As shown in the reference [10], Fourier averages of hydrodynamic coefficients of Ca and Cd are
obtained asTCa  F (t )sin(t )dt  1
F (t )sin(t )dt




a
  a   sin (t )dt
0

T

H

2

w

T

2

0

w

H

(13)

0

T

Cd  -

 F (t ) cos(t )dt0

H1
 w A( a)2  cos(t )  cos(t )  cos 2 (t )dt
0
2
T

-

T
3
FH (t ) cos(t )dt
2 0
4  w A a

(14)

2.6 Water tank tests
A water tank test is carried out in this study to validate the predicted Ca and Cd by the numericalsimulations. Dimensions of the water tank are 100 m length, 5 m width, and 2.65 m depth. The water
8

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

depth in this experiment is set as 2.0 m. Overview of the model is shown in Fig. 5. The detailed
dimensions of the semi-submersible model are described in Table 1. The parameters of water tank test
are summarized in Table 8.
The time series of measured hydrodynamic force, FH (t ) , is obtained by subtracting the buoyancy
force, Fb , inertia force, FI (t ) , and hydrostatic force, FK (t ) , from the measured total force, F (t ) as

FH (t )  F (t ) - Fb - FI (t ) - FK (t )

(15)

(16)
FI (t ) -M x(t )where M is the model mass including the mass of attachment used to connect the force balance andthe model, Fb and FK (t ) are evaluated with the same method used in the numerical simulation.
(a)

(b)

Fig. 5. Overview of the water tank test in (a) horizontally and (b) vertically forced vibrations
Table 8. Parameters and cases used in the forced vibration tests.
Parameters
SymbolHorizontal
Mass of platform (kg)
42.2
M
KR
Hydrostatic stiffness (N/m)
aOscillating amplitude (m)0.1, 0.2
KC numberKC
4.62, 9.24
Oscillating period (s)1.6~3.0
T
Characteristic area (m2)0.3216
A
Displaced volume of water (m3)0.0422

Vertical
42.2
570.6
0.04, 0.08
0.9,1.8
1.6~3.0
0.185
0.042280F

C (N/N)

4-4

Measured hydrodynamic force
Reproduced hydrodynamic force

-8
0

T/2

T/4

3T/4

T

Time (s)Fig.6. Comparison of measured and reproduced hydrodynamic forces obtained from the water tanktest with a  0.04m and T  1.8s in the vertical direction
The time history of measured and reproduced hydrodynamic force is presented in a nondimensional form as

99

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

CF (t * ) 

FH (t )
1
2
 w A  a 
2

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034; t* 

tT(17)

where, FH (t ) is the measured or reproduced hydrodynamic force; A is the characteristic area in theoscillating direction; and t * is the non-dimensional time. Fig. 6 shows one example, in which thereproduced hydrodynamic force matches well with the measured force in the water tank test with theoscillating amplitude of 0.04m and the period of 1.8s in the vertical direction.

3. Results and discussions
The hydrodynamic coefficients of a semi-sub floater predicted by grid refinement and Richardson
extrapolation are discussed in section 3.1. KC number dependent hydrodynamic coefficients are
examined and compared with experiment in section 3.2. The distributed hydrodynamic coefficients
considering effect of interaction among each component are presented in section 3.3. The effect of freesurface on the hydrodynamic coefficients is investigated in section 3.4.
3.1 Effect of grid refinement
Systematic error estimation for the predicted hydrodynamic coefficients based on the three different
grid levels is conducted to obtain a grid independent solution as shown in reference [15]. In thisanalysis, an approximate solution is obtained by using the fine grid and the grid independent solution
is then estimated by Richardson extrapolation. The exact solution  and the discretization error  hcan be expressed as(18)  h   hh  h p  H(19)

where h stands for the approximate solution with a grid size of h , H means the higher order terms,
 represents the derivative and is independent of h , The exponent p indicates the order of thenumerical scheme. The exact solution  based on the solution hi with the grid size of h i is thenobtained as  hi   hi  H
p(20)

The exponent p and  are derived by the solutions on the three grid levels as h  h1 
log  2 h  h 
2 
 3
p
log 
h  h p3 p 2
h3   1





(21)
(22)

where  means the ratio of grid size, and h1   h2   2 h3 . The discretization error on the third gridcan be derived ash 
3h  h p 1
3

2

Finally, the exact solution based on the third grid level is obtained ash  h  h3  3 p 2
 1

10

(23)

(24)

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

 can be obtained from the solutions on the two grid levels once the exponent p is obtained .
The predicted Ca and Cd in the case of vertical vibration with the amplitude of 0.04m and theperiod of 1.8s are presented as one example. The grid is refined as shown in Table 3 andh1  8mm, h2  4mm, h3  2mm represent three different grid sizes respectively. In general, p will be
1 according to the first order scheme of time discretization if the grid velocity is constant. However,p is 0.5 less than 1 since the dynamic mesh with a variable grid velocity is used in this study.
Fig. 7 shows variation of the predicted hydrodynamic coefficients with the grid size. It is found thatthe predicted drag coefficient based on the grid level 1 is significantly overestimated and the predictedadded mass coefficients show relatively weak dependence on grid size. As expected, the error is
reduced when the grid is refined, however. the predicted drag coefficient by the finest grid stilloverestimates that obtained from the water tank test. On the other hand, the predicted drag coefficientby Richardson extrapolation show good agreement with the measurement.
In order to clarify the effect of grid refinement on the hydrodynamic coefficients, instantaneous
vorticity is investigated and obtained as

wy 

u v

z x

(25)

Fig. 8 illustrates the normalized dynamic pressure obtained from the numerical simulation as shownin the reference [10]. There is no large difference between the dynamic pressures on the two gird
levels because the added mass is dominant in hydrodynamic force and is almost independent on thegrid level.
Fig. 9 presents a snapshot of the vorticity at t=0 in x-z plane to explain why the prediction error
decreases as the grid number increases. The two cases are obtained at t/T=1 when the velocity of platemotion reaches maximum. It can be seen that the vorticity is well captured by the fine grid. The
distribution of vorticity is useful to determine the area for the grid refinement near the edge of plate.
The predicted added mass and drag coefficients in the vertically and horizontally forced vibrations
are presented in Fig. 10 and Fig. 11. For all the cases, Richardson extrapolation is applied. “Coarse
grid” and “Extrapolation” indicate the results obtained by the grid level 1 and Richardson
extrapolation. It is found that the drag coefficients in the vertically forced vibration is sensitive to thegrid size because the vortex shedding around the edge of plate strongly depends the grid size. The
hydrodynamic coefficients predicted by Richardson extrapolation show good agreement with themeasurement, however those by the grid level 1 is significantly overestimated in the vertically forced
vibration as shown in Fig. 10 (b).
(a)

(b)
1.5

8
7
6

Extrapolated value: 1.16

1

Cd

Ca

5

Extrapolated value: 5.36

4
3

0.5

2
Cal.

Cal.
1

Exp.

Exp.

0

0
0

2

4

6

8

0

Grid level

2

4

6

8

Grid level

Fig. 7. Variation of the predicted hydrodynamic coefficients of (a) Ca and (b) Cd with the different
grid size in the vertically forced vibration with KC=0.9 and T=1.8s

1116th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

(a)
z

(b)
z

x

x

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

yyFig. 8.The normalized dynamic pressure on the surfaces of HP1 and SC1 at t/T=1 calculated with(a) grid level 1 and (b) grid level 2 in the vertically forced vibration with KC=0.9 and T=1.8s
(a)

(b)
z

z

y

x

y

x

Fig. 9. Instantaneous vortisity around the edge of HP1 at t/T=1 calculated by (a) grid level 1 and (b)grid level 2 for the vertically forced vibration with KC=0.9 and T=1.8s
(a)

(b)
2

8
Exp.
Cal.-Coarse gridCal.-Extrapolation

7

1.5

6

d33

4

C

1

C

a33

5

3
0.5

2

Exp.
Cal.-Coarse grid

1
0
1.5

2

2.5

3

0
1.5

3.5

Period (s)Cal.-Extrapolation
2

2.5

3

3.5

Period (s)Fig. 10. The hydrodynamic coefficients of (a) Ca and (b) Cd predicted by the grid level 1 andRichardson extrapolation for the vertically forced vibrations with KC=0.9 for the four oscillating
periods

12

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034(a)

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

(b)
1

2

0.8
1.5

C

C

a11

d11

0.6
1

0.4
Exp.
0.2

0.5

Cal.-Coarse gridExp.
Cal.-Coarse gridCal.-Extrapolation

Cal.-Extrapolation
0
1.5

2

2.5

3

0
1.5

3.5

2

2.5

Period (s)33.5

Period (s)Fig. 11. The hydrodynamic coefficients of (a) Ca and (b) Cd predicted by the grid level 1 andRichardson extrapolation in the horizontally forced vibration with KC=9.24 for the four oscillating
periods
3.2 Effect of KC number
The effects of KC number on the hydrodynamic coefficients are discussed for the side column andthe whole semi-sub floater. Firstly, the normalized dynamic pressure under the free surface on the sidecolumn 2 are shown in Figs. 12 and 13 for the vertical and horizontal vibrations. It is noticed that there
is no large difference between the dynamic pressures around the heave plate with different KC
numbers for the vertically forced vibration, however, large radiated waves as shown in Fig. 12 (a) are
generated near the free surface and result in the change of dynamic pressure on the surface of sidecolumn in the horizontally forced vibration.
(a)
z

(b)
z

x

x
y

y

Fig. 12. The normalized dynamic pressure on the surfaces of HP1 and SC1 at t/T=1 under the freesurface in the vertically forced vibrations with (a) KC=0.9 and (b) KC=1.8
Figs. 14 and 15 show the predicted Ca and Cd in the vertically and horizontally forced vibrations
with different periods and amplitudes, respectively. The predicted added mass coefficients bypotential theory [16] and the drag coefficients used in the reference [17] are also plotted forcomparison. As shown in Fig. 14, Ca and Cd are almost independent on the period, but a function with
KC number. It is clear that the added mass coefficients predicted by potential theory is underestimatedbecause the potential theory is based on linear assumption and is only applicable for the small
amplitude. The drag coefficients decrease as KC numbers increase. From Fig. 15, it is concluded thatthe predicted added mass and drag coefficients for the larger oscillating amplitude weakly depend on
13

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

the oscillating periods as mentioned in the previous study [8]. In addition, the added mass coefficientsdecrease as the KC number increases, while the drag coefficients increase as the KC number increases.
The predicted hydrodynamic coefficients by CFD well demonstrate KC number dependent
characteristics and show favourable agreement with those from the water tank tests.
(a)

(b)

z

z
y

x

x

y

Fig. 13. The nomalized dynamic pressure on the surfaces of HP1 and SC1 at t/T=1 under the freesurface in the horizontally forced vibrations with (a) KC=4.62 and (b) KC=9.24
(a)

(b)
2

8
7

1.5

6

d33

4

C

1

C

a33

5

0.5

0
1.5

3

Exp.-KC=0.9
Exp.-KC=1.8
Cal.-KC=0.9
Cal.-KC=1.8
Potential theory
2

2
1

2.5

3

0
1.5

3.5

Exp.-KC=0.9
Exp.-KC=1.8
Cal.-KC=0.9
Cal.-KC=1.8
Robertson et al. (2014)
2

Period (s)

2.5

3

3.5

Period (s)Fig. 14. The predicted hydrodynamic coefficients of (a) Ca and (b) Cd in the verticallyforced vibration for two KC numbers and the four oscillating periods.
(a)

(b)
1

2

0.8
1.5

C

C

a11

d11

0.6

0.4

0.2

0
1.5

Exp.-KC=4.62
Exp.-KC=9.24
Cal.-KC=4.62
Cal.-KC=9.24
Potential theory
2

1Exp.-KC=4.62
Exp.-KC=9.24
Cal.-KC=4.62
Cal.-KC=9.24
Robertson et al. (2014)

0.5

2.5

3

0
1.5

3.5

Period (s)

2

2.5

3

3.5

Period (s)Fig. 15. The predicted of hydrodynamic coefficients of (a) Ca and (b) Cd
horizontally vibrations for two KC numbers and the four oscillating periods.
14

in the16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

The potential theory with the linear assumption accurately predicts Ca with a small oscillating
amplitude, but not for the case with a large KC number due to the strong nonlinearity. Besides, thedatabase of Cd provided in the literature [17] did not considered the KC dependency and generally
give underestimated predictions. The CFD simulation with VOF not only confirmed the influence ofoscillating amplitude (KC number) but also reproduced the oscillating period dependent Ca with agood accuracy. It can be found that the prediction error in terms of Ca and Cd is less than 5% for allcases.
3.3 Effect of interaction between elements
The distribution of hydrodynamic coefficients considering interaction between elements areinvestigated here. In order to evaluate the distribute hydrodynamic coefficients for each element, thechange of the force due to hydrostatic force should be excluded. Special attention should be paid thatthe area of the lower and upper surface contacted with water is different since the heave plate is
attached by one column above. The hydrodynamic force, FH (t) , associating with the Ca and Cd isobtained as

FHi (t) = Flower (t )  w ghlower A lower  w gz (t ) A ower Fupper (t )  w ghupper A upper w gz (t ) A upper(26)

where Flower (t ) and Flupper (t ) are calculated from the instantaneous forces on the lower and uppersurfaces of each element. hlower and hupper are the lower and upper surface of each element in thevertical direction when the floater is located at the still water. A lower and A upper are the area of thelower and upper surfaces of each element. z(t) is the instantaneous displacement of the floater .
The distribution of hydrodynamic coefficients for each element are defined asTo1kFki (t ) sin(t )dt
i Ca 
0
 wi ak
i

Cdk  -To
3
Fki (t ) cos(t )dt
2 0
4 w Ai ak

(27)
(28)

where i expresses the element number, k presents the direction, that is, k  n is in the normal
direction and k  t is in the axial direction of element, the characteristic area and volume of eachelement for normalizing the hydrodynamic coefficients are shown in Table 2. i Can and i Cat are theadded mass coefficients of each element in the normal and axial directions, and i Cdn and i Cdt refer todrag coefficients of each element in the normal and axial directions, respectively. The hydrodynamic
coefficients for each element in the global coordinate can be translated into the local one, thenhydrodynamic coefficients predicted by CFD can be directly used for prediction of dynamic responseof FOWTs.
The hydrodynamic coefficients for each element in the normal and axial directions are summarizedin Table 9. Since the pontoons are rectangular cylinders with variable cross section as shown in Fig. 1,
its hydrodynamic coefficients are presented in the horizontal and vertical directions. For instance, thePntn-1-h and Pntn-1-v represent the values in the horizontal and vertical directions for the Pontoon 1.
The interaction between elements for the predicted hydrodynamic coefficients are also shown in
Table 9, where the interaction factor is defined as the ratio of each element to reference element
marked by a symbol of *. The hydrodynamic coefficients of SC2 and SC3 are same because there is
not any interaction effect and SC2 is chosen as the reference element with the interaction factor of 1.
Similarly, SC-2, Hp-2 and Pntn-2 are selected as the reference elements. The hydrodynamic
coefficients of SC-1 are smaller than those of SC-2 and SC-3 due to the interaction with the center
column. The hydrodynamic coefficients of heave plate Hp-1 and brace Br-1 in the normal direction
decrease due the similar reason. It is noticed that though the hydrodynamic coefficient of brace in15

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

normal direction by the horizontally and vertically vibration tests are slight different, the value
obtained from the horizontally vibration is adopted in this study.
Table 9. Summary of hydrodynamic coefficients for each element
Element
Side column SC-1
Side column SC-2 *Side column SC-3
Center column CC
Side heave plate Hp-1
Side heave plate Hp-2 *Side heave plate Hp-3
Center heave plate Hp-C
Pontoon Pntn-1-hPontoon Pntn-2-h *Pontoon Pntn-3-hPontoon Pntn-1-v
Pontoon Pntn-2-v *Pontoon Pntn-3-v
Brace Br-1
Brace Br-2 *Brace Br-3
Global value

i

Can
0.88
1.01
1.01
0.98
0.33
0.44
0.44
0.36
1.66
1.66
1.66
1.82
1.82
1.82
0.75
1.25
1.25
0.70

i

 an
0.87
1.00
1.00
0.75
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.60
1.00
1.00
-

i

Cdn
0.49
0.93
0.93
0.54
0.93
1.50
1.50
1.10
3.28
3.28
3.28
3.22
3.22
3.22
0.73
1.73
1.73
1.19

i

 dn
0.53
1.00
1.00
0.62
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.42
1.00
1.00
-

i

Cat
0
0
0
0
1.79
1.79
1.79
2.10
0
0
0
0
0
0
0
0
0
1.11

i

 at
0
0
0
0
1.00
1.00
1.00
0
0
0
0
0
0
0
0
0
--

i

Cdt
0
0
0
0
3.09
3.09
3.09
3.69
0
0
0
0
0
0
0
0
0
5.55

i

 dt
0
0
0
0
1.00
1.00
1.00
0
0
0
0
0
0
0
0
0
-

3.4 Effect of free surface
LES with VOF is employed to consider the effect of free surface on the hydrodynamic coefficients.
Multiple phase and the interface between water and air can be described by VOF model. LES model isalso used as single phase model, in which only phase of water is considered and the boundary
condition of surface is set as symmetry as used in the literature [18].
The normalized dynamic pressure on SC1 and Hp1 under water with and without the effect of freesurface in the vertically forced vibration at t/T=1 are revealed in Fig.16. There is no large radiatedwave observed during the vertically forced oscillation since the shape of free surface is not disturbed
by the vertical columns of the semi-sub platform model and the generated waves are negligible asshown in Fig. 5 (b). It implies that the effect of free surface on the hydrodynamic coefficient invertical direction can be neglected in the numerical simulation.
However, the shape of free surface around the columns during the horizontally forced vibration
changes significantly due to the radiated waves as shown in Fig. 5 (a). The change of dynamic
pressure near the free surface is observed in Fig17 (a). It can be seen that some vortices appear near
the free surface and lead to the change of dynamic pressure. On the other hand, the change of dynamic
pressure is not be observed in Fig17 (b). It implies that accurate prediction of the shape of free surface
has great importance for evaluation of the hydrodynamic force on the column since the free surface
determines the distribution of the dynamic pressure around the column in the horizontal direction.
Fig. 18 illustrates the predicted Ca and Cd with and without the effect of free surface for several
oscillating periods in the vertically forced vibration. It can be seen that the predicted hydrodynamic
coefficients with and without the effect of free surface show favourable agreement with those obtained
from the water tank tests. This implies that in this condition, CFD without VOF can also be suitable
for the hydrodynamic coefficients prediction because the predicted hydrodynamic coefficients keep nochange with and without the effect of free surface for a fully submerged heave plate in deep draft.
However, the impact of free surface increases as the heave plate is closer to the free surface.

16

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

(a)
z

(b)

x

x

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034zyyFig. 16. The normalized dynamic pressure on SC1 and Hp1 under water at t/T=1 (a) with and (b)without the effect of free surface in the vertically forced vibration for KC=1.8 and T=1.8s.
(a)

(b)

z

z

x

y

x

y

Fig. 17. The normalized dynamic pressure on SC1 and Hp1 under water at t/T=1 (a) with and (b)without the effect of free surface in the horizontally forced vibration for KC=9.24 and T=1.8s
(a)

(b)
2

8
Exp.
Cal.-W/O F.S.
Cal.-With F.S.

7

1.5

6

Exp.
Cal.-W/O F.S.
Cal.-With F.S.

d33

4

C

1

C

a33

5

3
0.5

2
1

0
1.5

2

2.5

3

0
1.5

3.5

Period (s)

2

2.5

3

3.5

Period (s)Fig. 18. The predicted hydrodynamic coefficients of (a) Ca and (b) Cd for KC=1.8 in the verticallyforced vibration with and without the effect of free surface.

17

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034(a)

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

(b)
1

2
Exp.
Cal.-W/O F.S.

0.8

Cal.-With F.S.

1.5

1

C

C

a11

d11

0.6

0.4
0.5
0.2

Exp.
Cal.-W/O F.S.
Cal.-With F.S.

0
1.5

2

2.5

3

0
1.5

3.5

Period (s)

2

2.5

3

3.5

Period (s)Fig. 19. The predicted hydrodynamic coefficients of (a) Ca and (b) Cd for KC=9.24 in thehorizontally forced vibration with and without the effect of free surface.
Fig. 19 shows the predicted Ca and Cd with and without effect of free surface in the horizontally
forced vibration. It is obvious that in the simulations without the effect of free surface, the added masscoefficients are underestimated and the drag coefficients are overestimated due to the defect of thepredicted hydrodynamic forces on the columns. By contrast, the numerical simulation withconsideration of two phase matches well with the measurement. The change of free surface affect both
Ca and Cd. This implies that the generated waves induce inertia and radiation damping forces andconsideration of the free surface in the numerical simulations is necessary.

4. Conclusions
Numerical studies of hydrodynamic coefficients for a semi-submersible floating offshore windturbine platform are systematically conducted by using LES with VOF model, and following
conclusions are obtained:
1. The grid dependency of predicted hydrodynamic coefficients is observed and the fine grid
improves accuracy of prediction because the vortex shedding near the model is captured. The
grid independent hydrodynamic coefficients are obtained by Richardson extrapolation and are
validated by the water tank tests.
2. The effects of KC number on the added mass and drag coefficients are systematicallyinvestigated and the predicted hydrodynamic coefficients by CFD match well with theexperimental data, while potential theory underestimates the added mass coefficients in thevertical direction and overestimates that in the horizontal direction for the large KC number.
3. The hydrodynamic coefficients in the normal and axial directions for all the elements of platform
are investigated by the vertically and horizontally forced vibrations. The interaction betweenside and centre columns reduces the added mass and drag coefficients.
4. The effects of free surface on the predicted added mass and drag coefficients are clarified by CFD.
In the vertical direction, the hydrodynamic coefficients predicted by LES with and without freesurface coincide well with those form the water tank tests since free surface has limited effect inthe vertical direction for the deep draft model. However, the hydrodynamic coefficients in thehorizontal direction by LES with free surface show a good agreement with the experimental
data, while those predicted by LES without free surface show significant discrepancy.

Acknowledgement18

16th Deep Sea Offshore Wind R&D conference
IOP Conf. Series: Journal of Physics: Conf. Series 1356 (2019) 012034

IOP Publishingdoi:10.1088/1742-6596/1356/1/012034

This research is carried out as a part of the Fukushima floating offshore wind farm demonstration
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.


References[1] WindFloat:
http://www.principlepowerinc.com/en/news-press/press-archive/2016/06/02/principlepowers-windfloat-prototype-celebrates-its-5-year-anniversary-and-the-final-stage-of-technologydemonstration[2] Fukushima FORWARD: http://www.fukushima-forward.jp/english/index.html[3] Waris, M.B. and Ishihara, T. "Dynamic response analysis of floating offshore wind turbine withdifferent types of heave plates and mooring systems by using a fully nonlinear model." Coupled
Systems Mechanics. Vol 1(3): pp. 247-268, 2012[4] Jonkman, J.M. Dynamics modeling and loads analysis of an offshore floating wind turbine.
Technical Report, NREL/TP-500-41958, 2007
[5] Tao, L. and Cai, S. "Heave motion suppression of a Spar with a heave plate." Ocean Engineering
Vol. 31, No.5, pp: 669-692, 2004
[6] Tao, L., Lim, K. Y. and Thiagarajan, K. "Heave response of classic spar with variable geometry."
Journal of Offshore Mechanics and Arctic Engineering Vol. 126, No.1, pp: 90-95, 2004
[7] Tao, L., Molin, B., Scolan, Y.-M., and Thiagarajan, K. "Spacing effects on hydrodynamics of heave
plates on offshore structures." Journal of Fluids and Structures, Vol. 23, No.8, pp: 1119-1136,
2007[8] Lopez-Pavon, C. and Souto-Iglesias, A.. "Hydrodynamic coefficients and pressure loads on heave
plates for semi-submersible floating offshore wind turbines: A comparative analysis using large
scale models." Renewable Energy, Vol.81, pp: 864-881, 2015[9] Benitz, M. A., Schmidt, D.P. Lackner, M.A., Stewart, G.M., Jonkman, J. and Robertson, A.
Validation of hydrodynamic load models using CFD for the OC4-DeepCwind semisubmersible.
In ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering.
American Society of Mechanical Engineers, 2015[10] Zhang, S., and Ishihara, T. "Numerical study of hydrodynamic coefficients of multiple heave plates bylarge eddy simulations with volume of fluid method." Ocean Engineering, Vol. 163, pp: 583-598,
2018[11] Klaka, K., Penrose, J.D., Horsley, R.R. and Renilson, M.R. "Hydrodynamic tests on a plate inforced oscillation." Ocean Engineering, Vol. 34, No. 8, pp: 1225-1234, 2007[12] Sarpkaya, T. In-Line And Transverse Forces, On Cylinders In Oscillatory Flow At High Reynolds
Numbers. In Offshore Technology Conference. Offshore Technology Conference, 1976
[13] Ishihara, T. and Zhang, S. "Prediction of dynamic response of semi-submersible floating offshore
wind turbine using augmented Morison’s equation with frequency dependent hydrodynamic
coefficients." Renewable Energy, Vol. 131, pp: 1186-1207, 2019.
[14] Oka, S. and T. Ishihara. "Numerical study of aerodynamic characteristics of a square prism in a
uniform flow." Journal of Wind Engineering and Industrial Aerodynamics, Vol. 97, pp: 548-559,
2009[15] Ferziger, J. H. and Peric M. Computational methods for fluid dynamics. Springer Science &Business Media, 2012.
[16] ANSYS, AQWA. "AQWA theory manual." ed. Canonsburg, PA15317, 2013[17] Robertson, A., Jonkman, J., Musial, W., Vorpahl, F. Popko, W. Offshore code comparison
collaboration, continuation: Phase II results of a floating semisubmersible wind system. EWEA
Offshore. 2013.
[18] Chen, C.R. and Chen, H.C. "Simulation of vortex-induced motions of a deep draft semisubmersible in current" , Ocean Engineering 118, 107-116, 2016.

19