Hydrodynamic Aspects of Moored Semi-Submersibles and TLPs

Hydrodynamic Aspects of Moored

Senti-Submersibles and TLP's

J.A. Pinkster, A. Dercksen and

A.K. Dey

Offshore Technology Conference,

Houston, Texas, USA, May 1993

Report No. 966-P, May 1993

Delft University of Technology

Ship Hydromechanics Laborato,y Mekelweg 2

2628 CD Delft The Netherlands Phone 015 - 7868 82

VOLUME I

25th ANNUAL

OFFSHORE TECHNOLOGY

CONFERENCE

LJt1i

ANNIVERSARY

1993

PROCEEDINGS

VOLUME i

ABSTRACT

The mean and low-frequency horizon-tal wave drift forces on 2 types of semi-submersible structures in regu-lar and in irreguregu-lar waves are

de-termined from model tests and

calcu-lations. For the measurement of the low-frequency drift forces in irre-gular waves use is made of a special

dynamic system of restraint.

Comparison of measured and computed drift forces in irregular waves show

increasing divergence between

predictions based on 3-dimensional potential theory and results of

ex-periments with increasing severity of

the irregular sea conditions.

Comparison between computed and

measured mean drift forces in regulat

waves show increasing divergence at

lower wave frequencies. A simple

model for approximating viscous

contributions to the drift forces in

irregular waves is applied to some

test results and it ís shown that the

correlation between measurements and predictions is improved.

In order to gain more detailed insight in the mechanisms of the

viscous contribution to the drift

force tests were carried out with single fixed vertical cylinder in

Hydrodynamic Aspects of Moored Semi-Submersibles and TLP's

J .A. Pinkster

Deift University of Technology

A. Dercksen

Maritime Research Institute Netherlands

A. K. Dey

Deift University of Technology

regular waves. The results of tests confirm that in conditions of waves

without current the major part of the viscous contribution to the drift force is confined to the splash zone of the cylinder.

INTRODUCTION

The motions and mooring forces of

Semi-Submersibles and TLP's moored in exposed locations are often dominated

by wave effects. These may be

sub-divided in first order wave frequency forces with frequencies corresponding

to the individual waves and mean and

low-frequency second order wave drift forces related to wave groups.

From the point of view of the design

of mooring systems both first and

second order wave loads and the mo-tion and mooring load response need

to be taken into account. At the

design stage predictions of these

quantities for a particular design

can be based on computational

methods, model tests or a rational

combination of both.

Computational methods for wave

frequency loads and motion response for semi-submersibles have been in

development since the early 70's.

based on linear hydrodynamic theory

and have proved their worth on many

occasions.

Non-linear, mean wave drift forces on

semi-submersible type structures can be computed based on the application of linear, 3-dimensional diffraction

theory computational methods combined

with either a far-field method or a near-field method for the evaluation

of the second order wave loads on the structure. In case a far-field method

is applied, generally only the mean second order horizontal drift forces

can be calculated. See reference [2] and reference [3] . If a near-field or

pressure integration method is

applied the mean and low-frequency

components of the drift forces can be

computed for 6 degrees of freedom.

See reference [4] . This type of

computational method assumes the flow

to be inviscid thus excluding any

effects which might arise from

separated flow around the structure. In the past efforts have been made to

verify the computational methods for the mean and low-frequency or slowly varying wave drift forces on

semi-submersible type structures. See

reference [5] . It has been surmised

that the drift forces on

semi-submersible type structures, which

consist of relatively slender

surface-piercing columns and

sub-merged floaters, are in some cases

significantly affected by viscous

effects in the flow around the

structural elements. Very little

ex-perimental data is available which

can give insight in such effects. See reference [10]

In order to increase insight in these effects MARIN, in co-operation with a number of offshore operators,

design-ers and manufactures, carried out

extensive model test programs and

computations among others of the mean

and slowly varying wave forces on

slender and full semi-submersibles. In this paper a number of aspects of

this research including the model

test programs and the correlation

between model test results and

results of computations are

discussed.

The findings of these studies have

confirmed that significant viscous

effects can be present in the

low-frequency wave forces on such

structures. As a result, a research

program has been initiated by the

Deift University of Technology into

determining such effects on

structu-ral elements of semi-subrnersibles

such as the columns and the

pon-toons. The purpose of this research

is to determine for which of these

elements the viscous effects play an

important role and, if possible, to

develop a rational computational

procedure for taking such effects

into account when determining the

mean and slowly varying drift forces

on the complete structure. In this

paper some results of recent model tests carried out on a fixed

verti-cal cylinder in waves are presented.

SECOND ORDER WAVE DRIFT FORCES

ON A SEMI-SUBMERIBLE

The subjects of this investigation were a slender 8-column Semi-Submer-sible I with circular columns and a displacement of 23,270 tonnes and a

full 6-column Semi-Submersible II

wíth square columns and a

displace-ment of 56,300 tonnes. The body plans of the semi-submersibles are given in Figure 1 and Figure 2. In the

follow-ing all results of measurements and computations will be given for the

full scale structures.

The aims of the study were as

follows:

-To increase insight in the mean and

low-frequency horizontal wave

excit-ing forces and motion responses of

large semi-submersibles

-To check the validity of

computa-tional methods for the prediction

of first order wave frequency

mo-tions and low-frequency wave drift forces based on 3-dimensional

The total scope of the research does not allow all aspects to be treated

here. In this paper the results of

the following investigations are

presented:

-Results of model tests in regular

waves to determine the mean hori-zontal wave drift force response. -Results of tests in irregular waves

to determine the mean and low-fre-quency wave drift force records.

The model tests were carried out at a scale of 1:40 in the Seakeeping Basin of MARIN. This basin measures 100 m x 24 m x 2.5 m.

MODEL TEST SET-UP IN THE BASIN

Measurements of the mean horizontal wave drift forces on a model in re-gular waves can be carried out using a soft-spring restraining or mooring system which consists of horizontal

wires incorporating soft linear

springs which are connected to force

transducers mounted on the model. The

mooring wires are connected at deck

level. The set-up for tests in regular waves is shown in Figure 3.

In order to measure the mean and

slowly varying horizontal wave drift

forces in irregular waves, ideally

the model should be moored in such a

way that all low frequency motion

response is suppressed while leaving the model completely free to carry

out the motions at wave frequencies.

The first requirement ensures that

the measured force is not affected by

dynamic magnification effects. The

second requirement can be deduced

from theoretical analysis of the

second order wave drift forces which show that part of the total second order excitation forces are directly dependent on the structural motions

at wave frequencies. See reference

[4]

As a consequence, the model

re-straining system must possess the

characteristics of an ideal Dynamic

Positioning system. For the model

tests a system consisting of

hori-zontal restraining wires connected to controllable tension winches was selected. See Figure 4. The winches

were operated based on an active

control system with a feed-back loop

supplemented by a _feed-forward

control loop. See Figure 5.

The feed-back loop acted on the

hori-zontal position error and the time

derivative of the error (Proportio-nal-Differential control) . The

feed-forward control loop was based on the real-time measurement of the

re-lative wave elevation on the up-wave columns of the semi-submersible. It has been shown that a major part of the mean and slowly varying second order wave drift forces as predicted by potential theory, is due to terms

related to the square of the instan-taneous relative wave elevation

around the waterline of a floating

structure. This has been

demon-strated, among others, from model

tests on a tanker. See reference [4]

Application of back and

feed-forward control still does not result in full suppression of low-frequency motions however. This due to the fact that the feed-forward loop is

supply-ing an imperfect estimate for the

instantaneous low frequency

horizon-tal force. As a result, the total

restraining force is not equal and opposite to the low frequency wave

exciting force thus resulting in

residual low frequency motions. See

Figure 6. In order to obtain a best estimate of the total low frequency

wave force on the model, the measured

restraining force is corrected for

the residual horizontal motions of

the vessel. This is carried out

off-line after a test has been carried

out. The basic assumption behind this process is that the instantaneous

discrepancy between the true wave

force and the measured restraining force results in horizontal motion accelerations which are described by

the following relationship:

in which m represents the virtual

mass of the vessel and x(t) the

¡no-tion accellera¡no-tion. Assuming that the virtual mass is constant, the accele-ration force can be determined in the

time domain by passing a

double-differentiating filter over the time record of the low frequency horizon-tal motions. The best estimate of the

time record of the horizontal drift

force then follows from:

Fd(t) = Fm(t)

+ m (t)

An example of time records of

measured restraining force, residual

surge motion, correction force and

total drift force are shown in Figure 7, The results apply to Semi-Submer-sible II.

In order to verify the accuracy of

this procedure model tests were

repeated using different settings of

the dynamic restraining system.

An example of the results found for the spectral density of the slowly

varying wave drift force on the Semi-Submersible II in irregular head seas is shown in Figure 8.

The results apply to the system of

restraint being adjusted to

repre-senting a spring system

(Proportio-nal control), a spring and damper

system (Proportional -Differential

control) and a P-D control including Feed-forward based on the relative

wave elevation measurements. The

results shown in the figure indicate

that the spectral density of the

drift force obtained from tests with

significantly different

characteris-tics of the restraining system are

reasonably consistent.

TESTS IN REGULAR WAVES

Tests in regular waves were carried out for both semi-submersibles for a

range of wave frequencies, wave

amplitudes and wave directions. For

the slender Semi-Submersible I tests

in head seas were carried out

with-out and with bracings. The results

are given in Figure 9 through Figure 13 for both structures in head waves and in beam waves as mean drift force transfer functions.

In the figures the theoretical values

found on the basis of 3-dimensional

potential theory computations

ex-cluding the contribution from the

bracings are also given.

Comparison between the results of mo-del tests and computations show that in the lower wave frequency range the

mean drift forces tend to be

consi-stently underestimated by the

compu-tations. The effect of the bracings

on the mean drift forces on the

slender Semi-Submersible I in head

seas is to increase slightly the mean drift forces as can be seen from the comparison between the measured results shown in Fígure 9 and in

Figure 11.

TESTS IN IRREGULAR WAVES

Tests were again carried out for both semi-submersibles. The slender Semi-Submersible I was tested with-out

bracings.

Results of tests in irregular waves

are given in the form of time traces

of the measured low frequency drift forces compared with time traces of

the corresponding predicted low

fre-quency force based on 3-dimensional potential theory. In some cases the

spectral density of the computed and measured forces are compared. For the time domain predictions, use was made

of second order impuls response functions combined with the measured

time trace of the undisturbed

ir-regular wave record in the basin. See reference [5] . The time domain second order impuls response functions for the drift forces are obtained from the complete second order quadratic transfer functions computed in the

frequency domain. The quadratic transfer functions were computed

based on the pressure integration

Tests were carried out in different irregular sea conditions in order to determine the dependency of the cor-relation between computed and mea-sured forces on the sea condition. Time traces of the measured and com-puted drift force records for irre-gular haed seas are given in Figure

14 through Figure 17. The spectral

densities of the computed and

mea-sured surge drift forces on

Semi-Submersible II are compared in Figure

18 through Figure 20. Comparison

between measured and computed data

show that the correlation is good in low sea conditions with relatively

short mean periods. En higher sea

conditions combined with

corres-pondingly longer mean wave periods

the correlation worsens. Beside

significant differences in the force peak values, the phase shift between

peaks in the measured and computed

records increases in higher sea con-ditions. The trend is more or less

the same for both types of

semi-submersibles.

DISCUSSION

OF

RESULTS

FROM

TESTS IN REGULAR AND IRREGULAR

WAVES

The results found from tests in

regular and irregular waves with

respect to the drift force seem to support each other in that in both

cases the correlation between

measurement and computations worsen with an increase in the wave period.

The reduction in the correlation

seems to be accompanied by an

in-creasing phase shift between measured and computed forces. The peaks in the

computed drift forces tend to shift

relative to the wave groups. See

Figure 17.This is related to the fact

that in longer waves diffraction

effects which make up the major part of the drift forces in shorter waves

are reduced. In longer waves the

low-frequency drift forces are to a

larger extent dominated by the compo-nents related to the second order

set-down waves present in the irregu-lar wave field. The troughs of these

waves, which have periods comparable

to the wave group periods, are in

phase with the peaks of the wave

groups. See Figure 21. The

low-frequency wave force components due

to these waves are in phase with the

horizontal acceleration of the fluid which is largest when the slope of

the set-down waves is greatest. This

occurs after the peak in the wave

group passes the structure. Due to

this effect the peaks in the computed

wave drift forces in irregular waves with longer mean periods tend to lag

behind the peaks in the wave groups.

The measured records, however, still

shown that the peak forces co-incide

with the peaks in the wave groups. A

possible explanation is that in

longer waves, viscous forces, which

are dominated by velocity related

effects, are becoming relatively more important. Since the fluid velocities

are highest near the peaks in the

wave group, peaks in viscous contri-butions to the drift forces will also

tend to co-incide with the peaks in

the wave groups.

In order to investigate this effect,

a simple model for the viscous con-tributions in the horizontal drift

forces has been investigated and some further comparisons between the

measured and computed drift forces, including viscous contributions, on the slender semi-submersible carried

out.

VISCOUS COMPONENTS OF THE DRIFT

FORCES IN IRREGULAR WAVES

The model used to describe the

vis-cous contribution to the drift forces

is based on the assumption that

Morison's equation for the drag force

on a vertical cylinder in waves can

be applied to the surface-piercing parts of the columns of a

semi-sub-mersible. See reference [6] through

reference [10] . For the case of waves

without current it can be shown that, as a first approximation, the viscous

drag force contribution to the drift

forces is confined to the splash zone on a column. The viscous drag terni is determined from the following equation:

ç (t)

Fvdft)

_{p Cd5 v(t),Iv(t)}

_{D dZ}

in which:

v(t) = relative horizontal velocity between the

fluid and the column

D = column diameter

Ç(t) = relative wave elevation

Cd drag coefficient

This contribution to the drift force

could be evaluated for each column in

the time-domain based on the

un-disturbed wave elevation record, the

frequency domain motion

characteris-tics of the semi-submersible and an assumption regarding the drag coef-ficient in the above equation. The summation of the drag force on each

column results in the estimated

vis-cous drag force contribution to the

drift force. The total drift force is found by adding the viscous and potential contributions.

The results of thes computations are

shown in Figure 22 for the slender

Semi-Submersible I in irregular head

seas. In this figure the wave

eleva-tion record, the potential part of

the drift force and the viscous part of the drift force are shown in the

top three traces, The lower trace

shows the sum of the viscous force

and the potential force compared with the total measured force.

It is clear that the result of adding the viscous contribution is a clearly improved correlation with the

measured force. In order to show the overall effect of adding a viscous

contribution to the potential

contri-bution spectra of the low-frequency

surge force in irregular head seas

and sway force in beam seas on Semi-Submersible I are given in Figure 23

for three different sea conditions. Each figure shows the drag

coeffi-cient Cd used for the computations of

the viscous force contribution. The Cd values used for the computations of the viscous contribution in some

cases had to be adjusted in order to achieve a reasonable fit with the

measured data. This clearly is an

unsatisfactory aspect of the

simpli-fied model for the viscous effect

which will need to be refined in the

future. An important effect not

accounted for is for instance, the

shielding effects due to the

proximity of the columns. However,

the above results tend to confirm

that there is a significant viscous effect in the drift forces on

semi-submersible type structures which, in irregular waves without current seems to be concentrated in the splash zone of the columns. The analysis has been

based on a rather simple model for the viscous contribution which has

not been verified to any great

extent. In the next section some

results of ongoing detailed research

carried out at the Deift University of Technology into such effects is

described.

VISCOUS EFFECTS IN DRIFT FORCES

ON A FIXED VERTICAL CYLINDER

In the previous section it was indi-cated that the most significant

vis-cous contribution to the horizontal

drift force on a semi-submersible

seems to originate from the splash

zone of the columns. In order to gain

more insight in such effects, model

tests have been carried out to deter-mine the distribution along the

ver-tical of the mean horizontal drift force on a single vertical cylinder

in regular waves. The work is part of an on-going Ph.D. project.

See reference [11]

The model tests were carried out in

Hydro-mechanics Department. This facility measures 80 m x 2.75 m x 1.25 m and

is equiped with a single flap

hydraulically operated wave-maker

capable of generating regular and

irregular waves. The basin is fitted

out with a towing carriage with a

special low speed carriage control

for the simulation of current effects by towing.

The model cylinder which had a dia-meter of 0.075 m is shown in Figure

24. At scale 1:100 this could be re-presentative of a column with a 7.5 m

diameter. The splash zone and the

sub-surface part are independently

attached to a central core through

force transducers measuring the

horizontal force on each of the two

sections.

Model tests were carried out in regu-lar waves with and without current.

For each test the vertical position of the cylinder was adjusted so that the through of the wave passing the cylinder passed just above the sepa-ration between the splash zone part of the cylinder and the sub-surface part of the cylinder. This ensured

that the sub-surface part of the

cylinder was fully submerged at all

times. Results of measurements in

regular waves without current of the mean horizontal drift force on the splash zone and the sub-surface zone

are compared with results of

calcula-tions of the relevant contribucalcula-tions

to the drift forces based on

3-dimen-sional potential theory and the

ap-plication of the pressure integration or near-field method in Figure 25 and Figure 26 respectively.

According to the near-field theory

for drift forces, the splash zone

contribution is dependent on the

square of the relative wave elevation

around the cylinder while the drift force on the subsurface element is due to the non-linear pressure con-tribution in the Bernoulli pressure

equation. For this reason the results

of mean force measurements have been divided by the square of the

undis-turbed wave amplitude. Results are

given for the model scale.

The model tests were carried out for a range of wave frequencies

corres-ponding to the longer waves for a

semi-submersible. At scale 1:100 the

wave frequencies tested in the model correspond to 0,3 r/s to 0.8 r/s at

full scale. This is a range of

frequencies relevant for extreme sea

conditions.

The results shown in Figure 25 and Figure 26 confirm that the greatest discrepancies between the potential computations and the measurements of

the mean forces are found for the

splash zone of the cylinder. The

measured mean forces are consistently

significantly larger than the

com-puted values. For the sub-surface

part of the cylinder, differences

also occur between measurements and

computations. In a relative sense

they appear to be of the same order

as for the splash zone part. However,

the absolute value of the forces is

considerable lower and the

differen-ces between measurements and

compu-tations are less consistent.

It can be concluded that these model tests poit to the splash zone

contri-bution to the viscous part of the

mean drift force as being the most

important one.

FINAL REMARKS

In thís paper we have shown some

results of an extensive series of

model tests on two semi-submersibles

which confirm differences between

computed and measured mean and

low-frequency horizontal wave drift forces in regular and irregular

waves.

Application of a simple model for the viscous contribution to the drift forces indicated that irregular waves

without current the major source of the viscous contribution was to be

found at the splash zone part of the columns of a semi-submersible.

Model test in regular waves with a

fixed vertical cylinder representing a single column of a semi-submersible

or a TLP confirm that the largest

discrepancies between computed and measured drift forces are indeed to be found in the splash zone.

Further experimental investigations are required in order to be able to formulate a more detailed model for the viscous effects which can also

take into account such aspects as the interaction effects due to the proxi-mity of the columns of a

semi-submer-sible.

REFERENCES

Hooft, J.P. : 'Hydrodynarnic

Aspects of Semi-Submersible Platforms' , Publication No.

400, Netherlands Ship Model Basin, 1972

Newman, J.N. : 'The Drift Force and Moment on Ships in Waves'

Journal of Ship Research, 1966 Faltinsen, 0M, and Michelsen,

F.C. : 'Motions of Large

Struc-tures in Waves at Zero Froude

Number' , Symposium on Marine Vehicles, London, 1974

Pínkster, J.A. : 'Low-Frequency Second Order Wave Exciting Forces on Floating Structu-res' , Publication No. 650,

Netherlands Ship Model Basin, Wageningen, 1980

Pinkster, J.A. and Hutjsmans,

R.H.M.: 'The Low Frequency

Mo-tions of a Semi-Submersible in Waves', Boss'82, Boston, 1982

Pijfers, J.G.L. and Brink, A.W. : 'Calculated Drift Forces

of Two Semi-Submersible Plat-form Types in Regular and

Ir-regular Waves', Paper No. OTC

2977, Offshore Technology

Conference, Houston, 1977

Huse, E. : 'Wave induced Mean

Force on Platforms in

Direc-tion Opposite to Wave

Propa-gation' , Norwegian Maritime Research, Vol.5, No.1, 1977

Standing, R.G. , Brendling, W.J. and Jackson, G.E.:

'Full-scale Measured and Predicted

Low-Frequency Motions of the

Semi-Submersible Support

Ves-sel 'Uncle John' ' , First

In-ternational Offshore and Polar

Engineering

Conference,

Edinburgh, 1991

Ferretti, C. and Berta, M.:

'Viscous Effect Contribution to the Drift Forces on

Float-ing Structures' , International

Symposium on Ocean Engineering Ship Handling, Gothenburg, '80

Chakrabarti, AK.: 'Steady

Drift Force on Vertical

Cylin-der - Viscous vs. Potential',

Applied Ocean Research, Vol.6, No.2, 1984

Dey, A.K. : 'Experimental

In-vestigations of Viscous Mean

Drift Forces on a Fixed Verti-cal Circular Cylinder in Waves

and Currents Part I', Report

No. 928-M, Ship Hydrodynamics

Department, Delft University of Technology, 1992

9.14 20 57 59.44 3 05 Dimensions in m OC 05 9.14 5.49 5 .49 9 14 22.86 i .83 22.36 22.86

Fig. 1 - General arrangement of Semi-Submersible I.

14.0 25.0 73.5

-I I I I I I ---J

L__

--Dimensions are given in metres

25.0

r--i

__L_

L

90.0

15.0

Fig. 2 _{- General arrangement of Semi-Submersible II.}

9.0

28.5

10.5

13.5

/

/

E

Tre ransducer

xi

Fig. 3 - Test set-up for tests in regular waves

45

450

Force crCsducer

Servo unit Feed- foGard control svs tern Feed-back control system Vessel:

Mass and damfling characteristics Waves Drift forces Relative wave elev. Horizontal motion

Fig. 5 _{- Block diagram of control system for}

tests in irregular waves

F _{(Wave drift force)} w

Motions

FB (Force from control system)

Flg.

6- B1ok diagram

of forces actitig on the

tructure.

Horizontal motions

relative wave elevazon Vessel

Force

= 3.1 in

=7.0 s

dir

= 1800 5.00 _Wave o . . . . :.\; 100.00

tf Measured restraining force a

0-",\

i

/

_\ ', \\ ¡ d 'I 100.00 ti 100.00 ti f'

°

'V \fi

III ej 0 50 100 t in S Correction force b m .

I"

_'"J

'i _V Total force a + b

I

,i'\

_\

'j 'J A ;

V \i 'I

V' A z_

f.

\J\ /JJj

Fig. 7---Example of measuredrestraining frce(aorrectjofcfor

motions(b) and total drift force record(a+b) m

5.00

20 o CN C -4 X i., 3J VC C e

Wave dIrection L8O degrees

Wave charactertstics: 5.49 ni

wl/3

= 11.30 s

System of restraint based upon:

15-10 S k Sorings

-

springs motion + + 1,

-

sprinas motion relative wave 0 0.25 0.50 0.75

Wave frequency in rad/s

Fig. 8 _{- Spectra of drift forces obtained for different restraining}

-20 -lo o 20 lo o

Calculation

O

Regular waves. 0.5 1.0 in rad.sec.1 1.5

Fig. 9 - Mean Surge drift force on Semi-Submersible I in regular head

waves

Fig. 10 - Mean sway drift force _{on Semi-Submersible I in regular beam}

wave s

o 0.5 1.0 1.5

-20 -lo o ('1 culat±c'n L O Reu1ar wa',es o D o

L

O 0.5 1.0 1.5 C')

in radsec1

Fig. 11 _{- Mean surge drift force on Semi-Submersible I in regular head}

waves including effect of bracings o

-50 N -25 rd N o

50

N E s-.-A

25

rd N -s-. o I cuìatiou

O Q L Reii1ar waves (ascenrig wave height)

8

o

s

o

0.5

1.0

1.5

w rad / s

Fig. 13 _{- Mean sway drift force on Smi-Submersib1e II in regular beam}

waves

o

_{0_5}

1.0

1.5

w _radis

Fig. 12

- Mean surge drift force on

Semi-Submersible II in regular

head waves

Cajcuiation

O Q Re1ar waves. (ascending wave height)

s

o

i

o '-, o

5.00 - Wave

4JMç5' = 3.09;

= 7.12 s

-

-m0

-5.00 1

- Measured

Calculated 10.00 -Force t f O

-- :

-10.00 J

-. 5.00 - Wave A _j O t

i.

riI'!\\

i\/\

c'.tJ\t\At!\

e\-H.,i',

r,

'n O

v \!

\i

.J\/\_

4J

= 5.85 'n;

= 11.30 s

10.00-tf O .--10.00 5.00

'no

-5.00 10.00-tf O -10.00-Force 0 50 100 t in seconds

/

--Wave 1 A !

í\At

/1A1-AA-Al

A R ;V\ r\ 1\ _I!

'\

/;

\J\\!U\il\l\j/\ij/\J\/

4J'

11.24 'n;

- 1425 s

Force r.

r

N ;

-/

Fig. 14 - Low-Frequency surge drift force _{on Semi-Submersible I in}

5.00 -, 100.00 tf ₀ -100.00

I

0 50 100 t

in s

Fig. 15 _{- Low-Frequency surge drift force on Semi-Submersible II}

in

irregular head seas, 1-Is 3.1 m, Ti = 7.1 s

5.00-nj Wave

¡ H

A ÏAfl

iL,

AIHth

m H

Vj

-S.00 Force 0 50 100 t

in s

Force - -' ____:---

-___-

'N___'

/T

-/

-

---

Measured Calculated

"z-= 5-5

m T]

=ll..3s

dir

= 1800

Fig. 16 _{- Low-Frequency surge drift force on Semi-Submersible II in}

irregular head seas, Hs = 5.5 ru, Ti 11.3 s

O,'

Wave

.

... . -5.00

-

Measured

= 3.1 in

Calculated

=7.ls

dir

= 180°

100.00

_T

tf

L..

100.00

-5.00 -5.00 100.00

T

tf O -100.00 N

Force

I II 0

50 100

t in s

Fig. 17 _{- Low-frequency surge drift force}

on Serai-Submersible II in

irregular head seas, Hs _{10.3 ra, Ti 14.5 s}

Wave J\Ì

/rN-c -

- - -;---

_{- -} -.f j ( f1¡ f _r 4JWj = 10.3 in

- Measured

_{= 14.5 s}

Calculated

_dir

_{= 1800}

rj N A

5000

3 rj

10000

o

Measured

Calculated

Measured

Calculated

O c)

rad/s

Fig. 19 _{- Spectral density ot surge drift force of Figure 16 on}

Semi-Submersible II

10000

N L I' 5000 I' I'l1 I t I J t I V 's I'_I t-O O

0.25

0.50

()

rad/s

Fig. 18 _{- Spectral density of surge drift force of Figure 15 on}

Semi-Submersible II

10000

ii

ij'

j j

il

I I i t ' 50004.- t I I t I

j

0

0.25

rad/s

Measured

Calculated

20000

rl

ji

iii

0.50

Fig. 20 - Spectral density of surge drift force of Figure _{17 on}

Semi-Submersible II i i

15000 -

I i i I I N cf

5.00

in O

-5.00

25.00

Fig. 21 - Wave set-down in irregular waves

T

J

Wave

CD = 0.80

\f

Force Calculated (potential part) a

Fig. 22 _{- Low-frequency surge drift force on Semi-Submersible}

I in irregular head seas

j

_{- - - Measured}

Calculated (viscous part) b

tf O -25.00 25.00 tf -25.00 25.00

-Force Total calculated

a +b

tf O -

-=---25.00

0 50 100

160

0

400

O

)4easured durino riiodel test

Calculated, otential contribution only Calculated, viscous effect included

Head seas beam seas

4000

2000

O

Fig. 23 _{- Spectra of low-frequency drift forces on Semi-Submersible I}

H5 = 3.09; T1 = 7.1 s C =0.60 D \ H5 = 3.09; D T1 = 7.1 s \ \\ Rs = 5.85; T1 = 11,3 C =0.20 s H5 5.85; T1 = 11.3 s CD = 0. 75 H5

11.24; T1 = 14.1

CD = 1.00 s H5 = 11.24; T1 = 14.1 = 0. 1 5 s o 0.2 0.4 o 02 0.4 W in rad/s CL) in cad/s Cn N 200 160 80 o 800 400 o 400 - 200 o

loo

80

60

40

20

2750 75

MEAN DRIFT FORCE [NIM21

UPPER DUMNY CYLISIIER UPPER TEST CYLISOER

FORCE TRANSDUCER

LOWER TEST CYLINDER

FORCE TRANSDUCER

LOWER SliMilY CYLINDER

Fig. 24 _{- Arrangement and model set-up of fixed vertical cylinder}

HIGHEST SET OF WAVE AMP.

NTEMEDIATE SET OF WAVE AMP.

(L) LOWEST SET OF WAVE AMP.

O

A

O

Fig. 25 - Mean drift forces on the splash zone part of the cylinder

0 1 2 3 4 5 6 7 8

OMEGA R/S1

MEASURED(I) MEASURED(H)

POTENTIAL THEORY A MEASURED(L)

O O O

60

40

20

o

-20

MEAN DRIFT FORCE [NIM2]

0

1 2 3 4 5 6 7 ₈

OMEGA ER/SI

0 _MEASURED(I) O _MEASURED(H)

- POTENTIAL THEORY

0 _MEASURED(L)

Fig. 26 - Mean drift forces on the subsurface part of the cylinder

80

-s _{HIGHEST SET OF WAVE AMP.}

NTEMEDIATE SET OF WAVE AMP.

(L) LOWEST SET OF WAVE AMP.

O _O O

O