VIM Time-domain Simulation on a Semi-submersible Floater

率の評価を実施した．風見鶏の効果を考慮した場合

と全方向から一様に波が入射する仮定の下での稼働

率は，風見鶏の効果を考慮した場合において若干の

向上は確認されるものの，両者に大差無い結果が得

られた．また，FLNG を波上側に配置し，斜め向波

を受けるようにスラスターによる操船を行うことで，

稼働率が向上することが確認された．

謝辞

本論文の模型実験に際して，横浜国立大学大学院工学研究

院の高山武彦特別研究教員，ならびに横浜国立大学海空制御

システム研究室の諸氏に協力頂きました．ここに謝意を表し

ます．

参考文献

1) R.H.M. Hujismans, J.A. Pinkster & J.J. Wilde：Diffraction

and Radiation of Waves Around Side-by-Side Moored Vessels, Proceedings of the 11th International Offshore and Polar Engineering Conference, 2001.

2) J.N. Newman：Application of generalized modes for the simulation of free surface patches in multiple body interactions, Proceedings of the 4th Annual WAMIT Consortium Report, 2003.

3) X.B. Chen ： HYDRODYNAMIC ANALYSIS FOR OFFSHORE LNG TERMINALS, 2nd International Workshop on Applied Offshore Hydrodynamics, 2005.

4) W.H. Pauw, R.H.M. Hujismans & A. Voogt：ADVANCES IN THE HYDRODYNAMICS OF SIDE-BY-SIDE MOORED VESSELS, Proceedings of the 26nd International Conference on Offshore Mechanics and Arctic Engineering, 2007

5) Ŝ. Malenica, M. Zalar & X.B. Chen：Dynamic coupling of seakeeping and sloshing, Proceedings of the 13th International Offshore and Polar Engineering Conference, 2003.

6) Bureau Veritas：HYDROSTAR FOR EXPERTS USER MANUAL, 2014

7) B. Molin：Experimental study of the wave propagation and decay in a channel through a rigid ice-sheet, Applied Ocean Research, Vol. 24, pp. 247-260, 2002.

8) PETROBRAS ： METOCEAN DATA, I-ET-3000.00-1000-941-PPC-001, 2005.

VIM Time-domain Simulation on a Semi-submersible Floater Using Wake Oscillator Model

by Toshifumi Fujiwara*, Member

Summary

This paper presents a time-domain simulation method using the wake oscillator model for prediction of Vortex-Induced Motion

(VIM) of a four columns semi-submersible. VIM effect should be assessed in an appropriate manner since the VIM causes fatigue damage of the floating structure’s mooring lines. There is no way, however, to assess VIM phenomenon on semi-submersible type floating structures in time-domain. Then this paper represents the method to simulate the VIM motion to be able to use in the time-domain simulation of a moored floater in ocean engineering topics. This method for a four columns floater is proposed by modifying the VIM simulation method on a cylindrical column type floater. Some empirical parameters in the method are obtained from the systematic model tests used many type semi-submersible floaters. This simulation method is easy to use and to understand since the method is based on a rational physical modeling. The validity of this method is shown comparing the simulated results of in-line and transverse VIM amplitudes with experimental ones.

1. Introduction

Semi-submersible type offshore floating structures, i.e. semi-

subs, are often used for drilling and production supports in the development of oil and gas fields all over the world. In Japan, the semi-subs are also expected to be used in the coastal area and at sea off Japan for promoting resource exploitation and development, and for renewable energy facilities, e.g. as a floating body for a wind power generation plant etc. Considering the operation of the semi-subs for a long-term, VIM evaluation is the necessary requirement for the safety assessment of mooring lines.

In the ISO and API standards1,2), the VIM assessment of the floaters is treated as an important element for the safety assessment of mooring lines. VIM investigations of individual semi-subs have been usually conducted in the experimental tanks3~6). Although VIM simulations of a semi-sub have been tried using the potential theory and the CFD calculations7~9) in some cases, it seems to be difficult to calculate in a short-term under reliable conviction at the initial design stage of it. Then a reasonable and simple VIM simulation method of semi-subs, which becomes effective in the design sea condition, is needed for the safety assessment of semi-sub operation and mooring lines.

To answer this request, the author and the others had conducted the VIM model tests on the various types of semi-sub models with four circular form columns with/without lower hulls to understand basic VIM specification of semi-subs10). As lower hull volume had influence on the VIM amplitude largely, the estimation equations on maximum in-line and transverse VIM amplitudes to be able to obtain the VIM amplitude in the directed current velocity deterministically were proposed using the column ratio parameter of the semi-sub.

Using the specification of the model test results of VIM

phenomenon on the semi-subs, the time-domain simulation method with wake oscillator model of a semi-sub is proposed in this paper. The method is based on the one cylindrical floater VIM simulation model proposed by the author11), and VIM amplification qualities of the semi-subs obtained from model tests are treated in the simulation method. This method is effective to do the time-domain simulation of a semi-sub floater with a mooring system in wind, waves and current. That is to say, the method is able to contribute the strength assessment of mooring lines in the heavy design weather conditions and the fatigue assessment of them from the safety aspect of semi-subs’ long-term operation.

At first in this paper, the VIM time-domain simulation method of a semi-sub using the wake oscillator model is introduced. Here, the semi-sub simulation targets four columns type of a floater with/without lower hulls since the semi-sub is major in the worldwide for offshore engineering12,13). Some empirical parameters in the motion equations of the simulation are obtained from the systematic model tests used eight semi-subs. In the final stage, comparing the simulated results of VIM amplitudes with experimental ones, the validities of this method are shown clearly.

2. VIM time-domain simulation method on a semi-sub

using wake oscillator model

2. 1 Basic concept A four columns semi-sub was selected as a subject from the

resent semi-sub rig built trend12,13). The semi-sub equips round form columns as safe side consideration for the reason that VIM seemed to be caused easily by unstable vortex shedding at each column.

On the other hand, it is well known that the maximum VIM amplitude level seems to be about representative length3~6,10,12etc.), here it is the diameter of each column, not four times column diameter. In this paper, the VIM amplitude is treated as averaged semi-sub motion against one column diameter to get a simple solution. At this time, the motion equations of a semi-sub are

* National Maritime Research Institute, National Institute of Maritime, Port and Aviation Technology, Japan

Received DD/MM/YYYY Received 11 December 2017

49

obtained from the same scheme of the single circle floater VIM simulation already presented in the previous paper11), and includes column and lower hull interaction effect.

As shown in the later section, column and column interaction, that is columns interval effect, is not clear from the model tests under the condition that the sample semi-sub range comprehends most semi-sub forms already built in recent. Then the columns interval effect is not considered in the simulation.

2. 2 Mathematical modeling of motion equations of a semi-sub and wake oscillation 2. 2. 1 Motion equation of wake oscillation11)

The coordinate systems of the simulation are shown as Fig. 1, where a semi-sub treated mainly at later section is set at the origin, and a 2-dimensional circle with diameter D , that represents one column cross section of the semi-sub, is placed at the second origin of the coordinate systems. The explanation of wake motion equation on the column is described based on the second origin.

A group of vortex shedding regarded as a wake region, which is able to swing in the transverse direction with expansion and contraction in the longitudinal direction of the wake region, is occurred behind the circle column with the rotation angle and the non-dimensional average length of wake oscillator *l . The motion equation of the wake angle working at the column is represented as follows:

YYmP **22 )'1(2 , (1)

where

.10.1,16.1

,2,)5.0/(1

,)/2('

,/2,/,)22/(

*

*

**

20

00

*2

lfS

lmCfP

Tlf

v

L

Nv

(2)

The Y is non-dimensional transverse position (= y / D ) of the column. The non-dimensional circular frequency (= v / 0 ), that means non-dimensional velocity, where the 0 is circular frequency of mooring lines keeping floater at prescribed area in current, and the natural period of the mooring floater NT , the lift slope f , equal to 1.16, the lift coefficient on a fixed floater 0LC ,

the 1/2 non-dimensional average length of wake oscillator *l , that is 1.10, are used in the equations. These parameters, f and *l , may be changed by the effect of the column-column and column-lower hull interactions. As the previous paper by the author11), however, showed that the constant f and *l values were effective for some types of floaters and current velocity range, these parameters were used as same and fixed values continuously.

The v was decided from the trend of the model test data like as follows:

,/1,0.9,/

,/

,)(2.02

,2

rinv

rinNrincrin

Nr

rinrcrincrinv

v

rinrv

v

VSVTDVV

DVTV

VVinVVVDS

VVinVDS

(3)

where the rV is reduced velocity, the V is current velocity and the vS is the Strouhal number on a stationary cylinder relating to the reduced velocity in the vortex shedding lock-in situation, rinV , which is rinV =9.0. 2. 2. 2 Motion equations of a semi-sub

The motion equations of a semi-sub are represented using VIM external force F with subscriptions of each direction as follows:

,)(,)(

yccya

xccxa

FyKyCyMMFxKxCxMM

(4)

where the M , aM are mass and added mass of an objected floater, the cC s, namely cxC , cyC , are damping coefficient, that is decided from free surge and sway tests etc. for instance, and the

cK is restoring coefficient of mooring stiffness. The external force F consists of the subscript characters

caused by lift L and drag D , that is:

.,

DyLyy

DxLxx

FFFFFF

(5)

Considering x, y-directions separately, the x-directional induced drag LxF , caused by lift relating to the wake, is also assumed to be proportional to ( ), where the is the relative angle for current velocity.

.)()(21 222

fxVDFLx (6)

Here, the means the water density. The LxF is included the effect of four times column number and half volume of a mirror image on wing lifting theory.

The x-directional drag DxF acting on a floater is represented as follows:

.)(21 2xVCAF DxWFDx (7)

The WFA is frontal projected area in the water and the DxC is

Fig. 1 Coordinate systems of the motion simulation of a semi-

sub with wake oscillator model at one column.

50 日本船舶海洋工学会論文集　第 27 号 2018 年 6 月

obtained from the same scheme of the single circle floater VIM simulation already presented in the previous paper11), and includes column and lower hull interaction effect.

As shown in the later section, column and column interaction, that is columns interval effect, is not clear from the model tests under the condition that the sample semi-sub range comprehends most semi-sub forms already built in recent. Then the columns interval effect is not considered in the simulation.

2. 2 Mathematical modeling of motion equations of a semi-sub and wake oscillation 2. 2. 1 Motion equation of wake oscillation11)

The coordinate systems of the simulation are shown as Fig. 1, where a semi-sub treated mainly at later section is set at the origin, and a 2-dimensional circle with diameter D , that represents one column cross section of the semi-sub, is placed at the second origin of the coordinate systems. The explanation of wake motion equation on the column is described based on the second origin.

A group of vortex shedding regarded as a wake region, which is able to swing in the transverse direction with expansion and contraction in the longitudinal direction of the wake region, is occurred behind the circle column with the rotation angle and the non-dimensional average length of wake oscillator *l . The motion equation of the wake angle working at the column is represented as follows:

YYmP **22 )'1(2 , (1)

where

.10.1,16.1

,2,)5.0/(1

,)/2('

,/2,/,)22/(

*

*

**

20

00

*2

lfS

lmCfP

Tlf

v

L

Nv

(2)

The Y is non-dimensional transverse position (= y / D ) of the column. The non-dimensional circular frequency (= v / 0 ), that means non-dimensional velocity, where the 0 is circular frequency of mooring lines keeping floater at prescribed area in current, and the natural period of the mooring floater NT , the lift slope f , equal to 1.16, the lift coefficient on a fixed floater 0LC ,

the 1/2 non-dimensional average length of wake oscillator *l , that is 1.10, are used in the equations. These parameters, f and *l , may be changed by the effect of the column-column and column-lower hull interactions. As the previous paper by the author11), however, showed that the constant f and *l values were effective for some types of floaters and current velocity range, these parameters were used as same and fixed values continuously.

The v was decided from the trend of the model test data like as follows:

,/1,0.9,/

,/

,)(2.02

,2

rinv

rinNrincrin

Nr

rinrcrincrinv

v

rinrv

v

VSVTDVV

DVTV

VVinVVVDS

VVinVDS

(3)

where the rV is reduced velocity, the V is current velocity and the vS is the Strouhal number on a stationary cylinder relating to the reduced velocity in the vortex shedding lock-in situation, rinV , which is rinV =9.0. 2. 2. 2 Motion equations of a semi-sub

The motion equations of a semi-sub are represented using VIM external force F with subscriptions of each direction as follows:

,)(,)(

yccya

xccxa

FyKyCyMMFxKxCxMM

(4)

where the M , aM are mass and added mass of an objected floater, the cC s, namely cxC , cyC , are damping coefficient, that is decided from free surge and sway tests etc. for instance, and the

cK is restoring coefficient of mooring stiffness. The external force F consists of the subscript characters

caused by lift L and drag D , that is:

.,

DyLyy

DxLxx

FFFFFF

(5)

Considering x, y-directions separately, the x-directional induced drag LxF , caused by lift relating to the wake, is also assumed to be proportional to ( ), where the is the relative angle for current velocity.

.)()(21 222

fxVDFLx (6)

Here, the means the water density. The LxF is included the effect of four times column number and half volume of a mirror image on wing lifting theory.

The x-directional drag DxF acting on a floater is represented as follows:

.)(21 2xVCAF DxWFDx (7)

The WFA is frontal projected area in the water and the DxC is

Fig. 1 Coordinate systems of the motion simulation of a semi-

sub with wake oscillator model at one column.

drag coefficient for in-line direction. The y-directional lift LyF based on the wake is also assumed

to be proportional to ( ), then,

.)()(2 2 fxVDdF cLy (8)

Here, the cd means draught of the column. This LyF is also considered four times column effect.

The drag DyF in y-direction in the VIM is represented as follows:

,)(21

)()(

21

2222

yxVCA

yxV

yyxVCAF

DyWL

DyWLDy

(9)

where the WLA is lateral projected area in the water, and the DyC is drag coefficient for transverse direction, but it is difficult to decide the DyC in the VIM oscillation. Then in this paper, the

DyC is set as the constant representative value, 1.0. As a result, the motion equations caused by VIM on a semi-

sub are represented as:

,2

22

*2

2*

2

*2

*

YfCDdAn

XXYfCDdAnX

Dxc

WF

Dxc

WF

(10)

,24

242

**

**

Xfn

YYXnnDdAXnnfY

c

WL

(11)

where the X is non-dimensional in-line position (= x / D ) of the column as same as the Y , and the (= ))(2/( 0 ac MMC ) is non-dimensional damping coefficient of a floater, the n (=

))(2/(2ac MMdD ) is mass ratio, the (= Ddc / ) is

constant parameter that means the aspect ratio of a column. Here, the damping coefficients cC were set as the same value in the x, y-directions since the measurement of the cxC had not been conducted in the model test and the two ‘P’ models shown as later only comes under the influence.

Moreover, this study made clear that the effects on the interaction of X , Y terms and high order term of the wake oscillator angle, square term 2 , are small for contribution against VIM amplitudes (See Appendix). Then omitting the interaction of X , Y terms and 2 term of the wake oscillator angle, the

motion equations are rewritten as follows:

,2 2*

2

*

n

DdACXXn

DdACX

c

WFDx

c

WFDx

(12)

.442 2*

2

*

fnYYn

DdAfY

c

WL

(13)

2. 3 Parameter settings based on the model test of semi-subs

In the previous study10), it was clear that the semi-sub VIM

amplitudes are affected by the column volume ratios in the whole displacement. The summary of the model test to be conducted to obtain the relationship between the VIM amplitudes and the column volume ratio is presented in this section. 2. 3. 1 Sample models

Semi-sub model test on VIM investigation was conducted in our research institute, NMRI. Plural semi-sub models were prepared for the model test.

Table 1 shows the model specifications, and Fig. 2 shows the appearances of the models. Each column has 0.2m diameter . The columns were connected by oval sectioned thin braces to avoid making large drag and flow turbulence. Two types of column intervals were set using different length braces. Short brace semi-subs were named as ‘C05’ series, which had 0.5m column interval, and relatively long brace ones were ‘C08’ series, which had 0.8m one. Moreover, two types of lower hulls were prepared in the tests, that one was a parallel lower hull, added the character ‘P’ in the model name, and the other was a square shaped lower hull ‘S’.

D

Table 1 Model specifications on sample semi-subs.

Fig. 2 Model forms on the C05 (left) and C08 (right)

series (The plans represent S14 conditions).

Item UnitLowerHull － ColumnOnly Parallel Square Square

Diameter (D) mDraft (d) m 0.15 0.22 0.22 0.29

Column draft (dC) mLower hull height (HLH) m 0 0.07 0.07 0.14

Model name － C05d15 C05d15P07 C05d15S07 C05d15S14Length overall (L) m

Column interval (LC) mMass (M) kg 19.3 37.7 46.7 74.1

Column mass (MC) kg 18.8 18.8 18.8 18.8Lower hull mass (MLH) kg 0.5 18.9 27.9 55.3Column ratio (RCLM) - 0.98 0.50 0.40 0.25

Frontal projected area (AWF) m2 0.063 0.091 0.112 0.161Lateral projected area (AWL) m2 0.063 0.112 0.112 0.161

Sway damping ratio (γ) % 12.7 11.8 13.2 14.4Sway natural period (TN) s 7.1 9.6 10.5 13.7

Image(Top and Side view) －

Model name － C08d15 C08d15P07 C08d15S07 C08d15S14Length overall (L) m

Column interval (LC) mMass (M) kg 19.6 46.4 63.8 108.0

Column mass (MC) kg 18.8 18.8 18.8 18.8Lower hull mass (MLH) kg 0.8 27.5 45.0 89.1Column ratio (RCLM) - 0.96 0.41 0.30 0.17

Frontal projected area (AWF) m2 0.066 0.094 0.136 0.206Lateral projected area (AWL) m2 0.066 0.136 0.136 0.206

Sway damping ratio (γ) % 11.1 13.2 12.6 12.5Sway natural period (TN) s 6.8 10.6 11.2 15.6

Image(Top and Side view) －

1.00.8

Semi-sub models

0.2

0.15

0.70.5

51VIM Time-domain Simulation on a Semi-submersible Floater Using Wake Oscillator Model

Each lower hull can be changed its thickness, 0.07m and 0.14m. The column ratio CLMR in the table, that has an important role to assess the VIM characteristics, means the ratio that total columns displacement CV divided by the whole displacement V . The NT is the natural period of mooring condition without current. As a basic characteristic of the models, damping ratios of each model against critical damping are also shown in the table. Here, the damping ratio was fitted to the function NTte /2 using duration time t . The test set-up of model mooring condition in the tank is shown in Fig. 3. It is expected to refer to the Reference 10 for the details about the information of the semi-sub model test. 2. 3. 2 Model test results and decided parameters

The maximum values of the transverse VIM ratio of the C05 series, divided by the model column diameter, DAT / , are shown in Fig. 4 as one example. Here, the ‘maximum’ plotted data mean the maximum values in peak VIM amplitudes measured in one-time history test data, where peak VIM amplitudes were counted about 20~30 times averagely, that is about 200 s duration time in one model test.

From Fig. 4, it became clear that lower hull volume in the displacement of semi-subs had an important role in the development of VIM amplitude in the current. Using the column volume ratio in the whole displacement, CLMR , maximum amplitude ratios of the transverse VIM in rV 16 are summarized as shown in Fig. 5.

The transverse VIMs have a linear relationship with the CLMR generally. In the same way, VIM amplitude is closely

connected with the lift force parameter 0LC . The 0LC is, therefore, newly set as the following form using the experimental parameter.

,/,030.0045.00

ALLCLMCLM

CLML

VVRRC

(14)

where the CLMV is total column displacement, and the ALLV is the whole one of a semi-sub. The 0LC is used in Eq. (1), and has an influence on wake angle. As a result, VIM amplitude is affected from the interaction effect of columns and lower hulls through the wake angle. This 0LC is simple formulation using the CLMR . It is originally assumed that the tendencies of the 0LC are different by lower hull shapes, however, the influence of the shapes did not

Fig. 4 Maximum amplitude ratios of the transverse VIMs

of the C05 series model tests.

Fig. 5 Maximum amplitude ratios of the transverse VIM on

column ratio ( rV 16). Table 2 Damping coefficients of the sample semi-sub models

obtained from the model tests.

Model a b C05d15 0.779 -0.671 C05d15P07 0.886 0.0 C05d15S07 0.991 0.0 C05d15S14 0.908 0.0 C08d15 1.294 -0.139 C08d15P07 0.569 -0.537 C08d15S07 0.547 -0.573 C08d15S14 0.693 -0.377

Fig. 3 Mooring condition of the model in Ocean engineering

tank in NMRI (upper) and side view of mooring condition of the model (lower).


Each lower hull can be changed its thickness, 0.07m and 0.14m. The column ratio CLMR in the table, that has an important role to assess the VIM characteristics, means the ratio that total columns displacement CV divided by the whole displacement V . The NT is the natural period of mooring condition without current. As a basic characteristic of the models, damping ratios of each model against critical damping are also shown in the table. Here, the damping ratio was fitted to the function NTte /2 using duration time t . The test set-up of model mooring condition in the tank is shown in Fig. 3. It is expected to refer to the Reference 10 for the details about the information of the semi-sub model test. 2. 3. 2 Model test results and decided parameters

The maximum values of the transverse VIM ratio of the C05 series, divided by the model column diameter, DAT / , are shown in Fig. 4 as one example. Here, the ‘maximum’ plotted data mean the maximum values in peak VIM amplitudes measured in one-time history test data, where peak VIM amplitudes were counted about 20~30 times averagely, that is about 200 s duration time in one model test.

From Fig. 4, it became clear that lower hull volume in the displacement of semi-subs had an important role in the development of VIM amplitude in the current. Using the column volume ratio in the whole displacement, CLMR , maximum amplitude ratios of the transverse VIM in rV 16 are summarized as shown in Fig. 5.

The transverse VIMs have a linear relationship with the CLMR generally. In the same way, VIM amplitude is closely

connected with the lift force parameter 0LC . The 0LC is, therefore, newly set as the following form using the experimental parameter.

,/,030.0045.00

ALLCLMCLM

CLML

VVRRC

(14)

where the CLMV is total column displacement, and the ALLV is the whole one of a semi-sub. The 0LC is used in Eq. (1), and has an influence on wake angle. As a result, VIM amplitude is affected from the interaction effect of columns and lower hulls through the wake angle. This 0LC is simple formulation using the CLMR . It is originally assumed that the tendencies of the 0LC are different by lower hull shapes, however, the influence of the shapes did not

Fig. 4 Maximum amplitude ratios of the transverse VIMs

of the C05 series model tests.

Fig. 5 Maximum amplitude ratios of the transverse VIM on

column ratio ( rV 16). Table 2 Damping coefficients of the sample semi-sub models

obtained from the model tests.

Model a b C05d15 0.779 -0.671 C05d15P07 0.886 0.0 C05d15S07 0.991 0.0 C05d15S14 0.908 0.0 C08d15 1.294 -0.139 C08d15P07 0.569 -0.537 C08d15S07 0.547 -0.573 C08d15S14 0.693 -0.377

Fig. 3 Mooring condition of the model in Ocean engineering

tank in NMRI (upper) and side view of mooring condition of the model (lower).

become clear from the model test results as shown in Fig. 5. Under this circumstance, the author thinks it to be enough practically as initial examination to estimate the semi-sub VIM, though it is a rough approximation.

It is known that the drag coefficient DC is largely effected from the transverse motion of a floater1,2,10,12 etc.). Then, the drag coefficient DxC is represented using the with an empirical parameter DirC depending on the Reynolds number eR on the basis of the model test results10,11).

.13.0in 6.338 log-1.103,||0.5

10

0

DireDir

DDirDx

CRCCCC

(15)

The drag coefficient 0DC without VIM has to be decided

according to the subjected floater condition. Here, the 0DC was set using the model test results10), and fitted by the following definition in the V 0.5 m/s:

.0.400 Db

D CinaVC (16) The coefficients, a and b , are shown in Table 2.

As it was previously mentioned, strictly speaking, some treatments lack and are not enough in the simulation, which is the interaction effect of the wake for columns interval with free surface disturbance, column and lower hull form specifications, draft etc. For example, the model wake of fore side columns shown in Section 2.2.1 contacts with the aft columns. From the results of Fig. 5, however, it seems that the VIM amplitudes have fortunately dull trend except for the effect of the CLMR that has an important role for the VIM amplitude rather than the column interval effect. Some problems in the formulations are placed as future works recognizing this simulation method represented in this paper is useful for the typical semi-sub forms in the present situation.

3. Confirmation of the accuracy of the simulation method

At first, the time history of calculated results of the in-line,

transverse displacements and the wake oscillator angle compared with the model test ones on the C05d15 in rV =9.1 are

shown in Fig. 6. The simulation, starting from no current and no VIM, was conducted by the 4th order Runge-Kutta method with 0.05 s time step, 1500 s in model scale, which was able to get the steady floater motion. The = 0.001 rad in t =0.0 s as the initial disturbance was set in all calculation. As a careful point, indicated initial time, 0 s, in the horizontal axis is a reference point, not the start time for measurement of the model test and simulation.

On the time history of in-line motion in the model test, the amplitude greatly varies to the difficulty level to find the maximum and minimum values of the amplitude as shown in the figure unlike it of the calculated result in no current disturbance. In the model test, the current disturbance, that is about 20% of the mean current velocity, seems to affect the floater’s motion. Free surface disturbance effect caused by multi-columns may work for the VIM motion. The variance of displacement in the model tests is also caused from the Reynolds number effect. This effect is averagely included in the drag inclination term in Eq. (15), but it is shown that to catch condition that changes moment by moment is difficult.

In the steady situation of the transverse amplitude in the model test, although the transverse amplitude also fluctuates to some extent, the calculated result amplitudes generally coincides with the model test ones of the maximum level. The profile in the wake angle is asymmetry slightly. This seems to be caused by the relative velocity of the transverse motion between the floater and wake.

Figs. 7 and 8 show the calculated results of maximum amplitude ratios of the in-line, transverse VIMs of the C05 and C08 series models compared with the experimental ones. The test

Fig. 6 Time histories of the calculated results (Cal.) of the

C08d15 comparing with the model test ones (Exp.).

Fig. 7 Maximum amplitude ratios of the in-line, transverse

VIMs of the C05 series models comparing with the calculated ones.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18

AI/D

Vr

In-line VIM (Max) C05d15 (Exp.) C05d15P07 (Exp.) C05d15S07 (Exp.) C05d15S14 (Exp.) C05d15 (Cal.) C05d15P07 (Cal.) C05d15S07 (Cal.) C05d15S14 (Cal.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18

A T/D

Vr

Transverse VIM (Max) C05d15 (Exp.) C05d15P07 (Exp.) C05d15S07 (Exp.) C05d15S14 (Exp.) C05d15 (Cal.) C05d15P07 (Cal.) C05d15S07 (Cal.) C05d15S14 (Cal.)


results of 4 kinds of models, ‘Exp.’, are shown in the figures, where the circle and triangle marks etc. stand for non-dimensional in-line and transverse VIM amplitudes, respectively. The calculated results are shown in the figure with lines. Current velocities were set in 0.05~0.37m/s.

The calculated results of the C05d15 and C08d15 coincide with the experimental ones enough for trend and value level. For the others of transverse cases, the calculated results slightly overestimate for the experimental ones in some rV areas. That shows the modeling of the 0LC etc. is not enough to represent the VIM phenomenon of semi-subs at present.

The calculated results of the semi-subs with large lower hulls like the C08d15S14 have the trend of dropping down in the larger range of rV =12, though the VIM amplitudes of the model tests continue to increase. In the calculation, the VIM amplitude is depended on the column vortex shedding only. Lower hull VIM seems to affect the VIM increasing in the larger rV range in the model test. It may be necessary to include the lower hull VIM effect in the simulation modeling in the future.

As a generalization, it is thought that the simulation method is useful for estimating in-line and transverse VIMs from the viewpoints of practical use in the range of about rV 14, where such range is frequently used in the ocean engineering.

4. Conclusions

The time-domain VIM simulation method on a wide variety form of four columns semi-subs in the current has been proposed on the basis of the single cylindrical floater’s wake oscillator model. This method is the first trial to easily evaluate in-line and transverse VIMs on semi-subs. The lift force coefficient with the column ratio parameter, used in the simulation and has an important role, is presented originally on the basis of the model test results. This method is remarkable one that can calculate VIM motion from the outline external form of a semi-sub since there was not a method to evaluate VIM of the semi-subs so far.

As a result, calculated results of the in-line and transverse VIM amplitudes have generally shown good agreement for the model test results conducted in the experimental basin in the range of about rV 14. It is demonstrated that this simulation method is an effective technique for the VIM simulation of semi-subs, and the method can be applied to the time-domain analysis of a semi-sub in the fluctuating current and quasi-static one recommended in the ISO19901-7 etc. in the design stage of the semi-subs.

Acknowledgments

This research was supported by the Japan Society for the

Promotion of Science (JSPS KAKENHI Grant Number 26289344). The model test was conducted with corporation by Mr. Tadashi Nimura, Mr. Masakatsu Saito, Researchers in NMRI, Mr. Fumitoshi Kitamura, Researcher in NMRI at the time of this research and Mr. Kohei Shimozato, Mitsui Engineering & Shipbuilding Co., Ltd., Researcher in NMRI at the time of this research, et al. The author expresses the deepest appreciation for them.

References

1) ISO: International Standard 19901-7 (Petroleum and natural gas

industries -Specific requirements for offshore structures -, Part 7: Stationkeeping systems for floating offshore structures and mobile offshore units), 2013.

2) API: Recommended Practice 2SK - Design and Analysis of Stationkeeping Systems for Floating Structures, Third Edition, 2005.

3) Waals, O. J., Phadke, A. C. and Bultema, S.: FLOW INDUCED MOTIONS OF MULTI COLUMN FLOATERS, OMAE2007-29539, 2007.

4) Gocalves, R. T., Rosetti, G. F. and Fujarra, A. L. C. et al.: EXPERIMENTAL STUDY ON VORTEX-INDUCED MOTION (VIM) OF A LARGE VOLUME SEMI-SUBMERSIBLE PLATFORM, OMAE2011-49010, 2011.

5) Xu, Q.: A NEW SEMISUBMERSIBLE DESIGN FOR IMPROVED HEAVE MOTION, VORTEX-INDUCED MOTION AND QUAYSIDE STABILITY, OMAE2011-49118, 2011.

6) Magee, A. et al.: MODEL TESTS FOR VIM OF MULTI-COLUMN FLOATING PLATFORMS, OMAE2011-49151, 2011.

7) Bratland, A. K., Haver, S. and Stansberg, C. T. et al.: A SEMI



0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18

AI/D

Vr


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18

A T/D

Vr



results of 4 kinds of models, ‘Exp.’, are shown in the figures, where the circle and triangle marks etc. stand for non-dimensional in-line and transverse VIM amplitudes, respectively. The calculated results are shown in the figure with lines. Current velocities were set in 0.05~0.37m/s.

The calculated results of the C05d15 and C08d15 coincide with the experimental ones enough for trend and value level. For the others of transverse cases, the calculated results slightly overestimate for the experimental ones in some rV areas. That shows the modeling of the 0LC etc. is not enough to represent the VIM phenomenon of semi-subs at present.

The calculated results of the semi-subs with large lower hulls like the C08d15S14 have the trend of dropping down in the larger range of rV =12, though the VIM amplitudes of the model tests continue to increase. In the calculation, the VIM amplitude is depended on the column vortex shedding only. Lower hull VIM seems to affect the VIM increasing in the larger rV range in the model test. It may be necessary to include the lower hull VIM effect in the simulation modeling in the future.

As a generalization, it is thought that the simulation method is useful for estimating in-line and transverse VIMs from the viewpoints of practical use in the range of about rV 14, where such range is frequently used in the ocean engineering.

4. Conclusions

The time-domain VIM simulation method on a wide variety form of four columns semi-subs in the current has been proposed on the basis of the single cylindrical floater’s wake oscillator model. This method is the first trial to easily evaluate in-line and transverse VIMs on semi-subs. The lift force coefficient with the column ratio parameter, used in the simulation and has an important role, is presented originally on the basis of the model test results. This method is remarkable one that can calculate VIM motion from the outline external form of a semi-sub since there was not a method to evaluate VIM of the semi-subs so far.

As a result, calculated results of the in-line and transverse VIM amplitudes have generally shown good agreement for the model test results conducted in the experimental basin in the range of about rV 14. It is demonstrated that this simulation method is an effective technique for the VIM simulation of semi-subs, and the method can be applied to the time-domain analysis of a semi-sub in the fluctuating current and quasi-static one recommended in the ISO19901-7 etc. in the design stage of the semi-subs.

Acknowledgments

This research was supported by the Japan Society for the

Promotion of Science (JSPS KAKENHI Grant Number 26289344). The model test was conducted with corporation by Mr. Tadashi Nimura, Mr. Masakatsu Saito, Researchers in NMRI, Mr. Fumitoshi Kitamura, Researcher in NMRI at the time of this research and Mr. Kohei Shimozato, Mitsui Engineering & Shipbuilding Co., Ltd., Researcher in NMRI at the time of this research, et al. The author expresses the deepest appreciation for them.

References

1) ISO: International Standard 19901-7 (Petroleum and natural gas

industries -Specific requirements for offshore structures -, Part 7: Stationkeeping systems for floating offshore structures and mobile offshore units), 2013.

2) API: Recommended Practice 2SK - Design and Analysis of Stationkeeping Systems for Floating Structures, Third Edition, 2005.

3) Waals, O. J., Phadke, A. C. and Bultema, S.: FLOW INDUCED MOTIONS OF MULTI COLUMN FLOATERS, OMAE2007-29539, 2007.

4) Gocalves, R. T., Rosetti, G. F. and Fujarra, A. L. C. et al.: EXPERIMENTAL STUDY ON VORTEX-INDUCED MOTION (VIM) OF A LARGE VOLUME SEMI-SUBMERSIBLE PLATFORM, OMAE2011-49010, 2011.

5) Xu, Q.: A NEW SEMISUBMERSIBLE DESIGN FOR IMPROVED HEAVE MOTION, VORTEX-INDUCED MOTION AND QUAYSIDE STABILITY, OMAE2011-49118, 2011.

6) Magee, A. et al.: MODEL TESTS FOR VIM OF MULTI-COLUMN FLOATING PLATFORMS, OMAE2011-49151, 2011.

7) Bratland, A. K., Haver, S. and Stansberg, C. T. et al.: A SEMI



0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18

AI/D

Vr


0.0

0.2

0.4

0.6

0.8

1.0

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1.6

1.8

0 2 4 6 8 10 12 14 16 18

A T/D

Vr


SUBMERSIBLE IN COMBINED EXTREME WAVES AND CURRENT – COMPARISON OF MODEL TESTS AND EXISTING SOFTWARE, OMAE2007-29652, 2007.

8) Kyoung, J., Kim, J. W. and Jang, H. et al.: INVESTIGATION ON THE VIM MITIGATION OF THE HVS SEMISUBMERSIBLE, OMAE2015-41188, 2015.

9) Liu, M., Xiao, L., Lyu, H. and Tao, L.: NUMERICAL ANALYSIS OF PONTOON EFFECT ON FLOW-INDUCED FORCES OF THE DEEP DRAFT SEMISUBMERSIBLE IN A CROSS-FLOW, OMAE2015-41764, 2015.

10) Fujiwara, T., Nimura, T. and Saito, M.: VIM Model Test and Mooring Line Fatigue Assessment on Semi-submersible Floaters, Journal of the Japan Society of Naval Architects and Ocean Engineers Vol.26, 2018.

11) Fujiwara, T.: VIM simulation on a cylindrical floating structure, Journal of the Marine Science and Technology Vol.22, DOI: 10.1007/s00773-017-0470-x, 2017.

12) Fujiwara, T., Nimura, T., Shimozato, K. and Matsui, R.: VIM MODEL TEST AND ASSESSMENT ON A SEMI-SUBMERSIBLE TYPE FLOATER, OMAE2016-54308, 2016.

13) The Mobile Offshore Drilling Units Register 2014 (15th Edition), Clarkson Research Services, 2014.

Appendix

Contribution of the X, Y interaction terms and high order terms of wake oscillator angle α

Effects on the interaction of X , Y terms and high order

terms of wake oscillator angle were investigated in this section. The original motion equations of a floater are represented as follows:

,2

22

*2

2*

2

*2

*

YfCDdAn

XXYfCDdAnX

Dxc

WF

Dxc

WF

(10)

.24

242

**

**

Xfn

YYXnnDdA

XnnfYc

WL

(11)

Omitting the interaction of X , Y terms and 2 term of wake oscillator angle, the motion equations are rewritten as follows:

,2 2*

2

*

n

DdACXXn

DdACX

c

WFDx

c

WFDx

(12)

.442 2*

2

*

fnYYn

DdAfY

c

WL

(13)

VIM amplitudes of in-line and transverse were shown in Fig.

A1 respectively comparing with the effect of the terms of the interaction of X , Y and wake oscillator angle 2 . The results of ‘with inter.’ mean the calculated ones from Eqs. 10, 11, and those of ‘w/o inter.’ were from Eqs. 12, 13. It can be said that the differences of the results between the two set equations are very small. Then in this paper the motion equations, 12 and 13, omitting the interaction terms of the X , Y and high order wake oscillator angles, were used as simple representations.

Fig. A1 Maximum amplitude ratios of the in-line, transverse VIMs of the C05 series models on effects of interaction terms X , Y and high order terms of wake oscillator angle.


Documents

VIM Time-domain Simulation on a Semi-submersible Floater