Viscous Mean Drift Forces on Moored Semi-Submersibles

Viscous Mean Drift Forces on

Moored Semi-Submérsibles.

Dey, A.K. and J.A. Pinkster

Report No. 1026-P

ISOPE-1995, Offshore and Polar Eng.

Conference, The Hague, The Netherlands

TU Deift

Faculty of Mechanical Engineering and Marine Technology Ship Hydromechanics Laboratoiy

The PÏoceedings of

the Fifth

(1995)

_{International}

OFFSI. ORE AND POLAR

ENGINEERING

_CONFERENCE

VOLUME III,

1995.

&UMERJCAL WAVES, WAVE MEASUREMENTS,

WAVE BkEAKING'& STATISTICS,,

WAVE-BODYINTERACflONS HYDRODYNAMIC FORCES,

DYNAMiC RESPONSES, HIGHER-ORDER EFFECTS, VORTEX & VIBRATIONS,

_COASTAL

HYDRODYNAMICS, L BORATORy & OCEAN MEASIJREMENTS

edited by:

Jin S.. Chung, Colorado School of Mines, Golden, Colorado, USA

Hisaaki Maeda, University of

_{Tokyo, Tokyo, Japan}

C.H. Kim, Texas A & M University,

_{College Station, Texas, USA}

presented at:

The Fifth(1995) International Offshoreand PolarEngineerrng Cônference

held in The Hague, The Netherlands, June 11-16, 1995

organized by:

International Society of Offshore and Polar Engineers.

sponsored by:

International Society of Offshore and Polar Engineers (ISOPE)

Offshore Mechanics aìid Polar Engineering Council (OMPEC)

with cooperating societies andassociations

International Society of Offshore and 'Polar Engineers (ISOPE).

Cópyright© 1995 by 'International Society of Offshore and Polar Engineers, Golden, Coloradò, USA. All Rights Reserved.

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Printed and bound in USA

Abstract;

Theoretical predictions using 3-D potential theory for the

menu drift forces on moored floating structures like

semi-submersibles and tension leg platforms show discrepancies

in both regular and irregular waves when compared with

re-sults of model tests. Such divergence is fürther pronounced

in the low frequency range (storm condition) where

diffrac-tion effects are less for such structures and is thus believed

to be induced by viscous effects. The viscous drag term of

the Morison equation in combination with the linear (Airy)

theory up to the instantaneous wave elevation has been

considered as the basis in order to develop the theory in

a waves-only field as well as in a wave-current coexisting

flow field.

Experimental investigations have been carried

out with fixed vertical surface piércing model cylinders and

n submerged pontoon in regular waves. Test results

val-idate the theory with a further indication that the mean

forces on completely submerged bodies are not influenced

significantly by viscous effects in a waves-only field.

Fi-nally, experiments have been carried on for a

completesemi-submersible model (ITTC Model) in regular waves (both

head and benin seas) in fixed amid free floating conditions

at zero speed and with forward velocity. Forces have been

mensured for the complete model as wellas for someindivid

ual columns in order to gain an insight into any interference

effect on the viscous mean drift forces. The theory,

incor-porating a relative horizontal velocity and relative surface

elevation coñcept, enhances the theoretical predictions for

the horizontal mean drift forces when experimentally

ob-tained values of the mean drag coefficients for the cylinders

and the submerged pontoon are properly applied for the

complete semi-submersible.

Key Words:. drift forces; potential; viscous; vertical cylinders;

sii biumerged pontoon; model testi ng; meas drag coefficients;

semi-f41! l,uuiersjl,les

Viscous Mean Drift Forces on

Moored Sem'i-Submersibles

A. K. Dcv and J. A. Pinkstcr

Ship }lydrornechanics Laboratory

Dehit University of Technology

Mekelweg 2, 2628 CD Delft

The Netherlands

i

Introduction

Fixed or floating bodies in regular waves are, besides the large linear forces, subject to small second order mean wave forces and in

irreg-ular waves to additional löw frequency slowly varying wave forces due to the continuously varying wave height and frequency. While time first Order force with the wave frequency is linear with the wave height, time mean force being non-linear is quadratic with the wave height. The slowly varying component of the drift forceis of

partic-ular importance for compliant structures like serni-submersibles and tension leg platforms both of which possess low restoring forces in horizontal modes.

Theoretical predictions for these second order forces on a float-ing structûrearedisplayodto be quite adequate when such quadratic forces are dominated by potential effects and as such treated by time linear potential theory using time conservation of momentum princi-pIé (far field approach) (Maruo 1960; Newman l967)or the pressure

integration method (near field approach) (Pinkster 1980). But

ac-cording tosonieliterature discrepancies are noted between the com-puted and the measured mean drift forces in regular waves at lower wave frequencies where diffraction effects are less for slender body

structures like serni-submersibles and tension leg platforms. Such

observations expose the fact that the computational technique with 3-D potential theory underestimates the mean drift forces. Such di-vergence is believed to be caused by viscous effects originating from

thesurface piercing structures.

A semi-submersible consists of two major structures - the

sub-merged underwater hull hereinafter referred to as the subsub-merged

pontoon and the surface piercing vertical columns. Before dealing

with the viscous effects for time complete semi-su binersible, these two

structures - the submerged pontoon and the vertical columns need

to be treated separately.

Most of the works regarding submerged structures reported iii

the available literature deal with circular cylinders except a few.

Koterayama (1979) investigated wave forces oui horizontal circuilár cylinders of different diameters in waves. Values of drag coellicients were found to be satisfactory for large period parameter when conm-pared with those in oscillatory flow ltamberg and Niedzwecku ('1982) presented an investigation on a horizontal cylinder in waves where it

t.i'ady flow a.t Iiiglmr vaIncs of Ketilegan-Carpetiter Nutiibor (Nk_c.). (1lìapliii (1981) reporte(l 1)0th horizontal aiid vertical iiiean forces for a sIuI)Illerge(I circular cylinder beneath waves. Effects of circuJaion

were a!so discussed. Probleiiis of noise in the force measurements wnre itidicated as well. Forces on horizontal circular cylinders in

waves and also in waves and currents, were reported by (Teng and Nati! 1985; Twig and Nath 1988). Values of drag coefficients in the

said flow field were compared with those in planar oscillatory flow

showing the efFects of the vertical velocity component. CD values were also shown as a function of Moe-Verley Niilnber(NAI_y).

A kw investigations were carried out with fixed liorzontalirectan-guiar cylinders in waves. Ikeda et al. (1988) and Otsuka et al. (1993) reported the values of drag coefficients as a function of NK_C and

they were found to be higher than those in planar oscillatory flow.

l'resence of circulation affecting the inertia forces Was also indicated. Otsllk;L et al. ( 1990) Further (liscussed the preseiice.oI viscous effects

when the cylin(ler (rectangular and circular section) aiid a coniplete

seIiii-siihnlersjl)le iiiodel aro subject to a low frequency motion in

waves. Viscous drag forces originate froiii both tinderwater hull and verlicalcoliiiii lis. Wave forces on rectangular horizoiitalcylinder were IiieasIire(l by Arai ( 1993)sliowiiìg reduction in the viscous drag forces vitli decreasein submergence depth.. Circulation phenomenon was hinted at as the cause of îeduction in the inertia forces. A detailed experimental study was conducted by llainel-Derouich (1993) for ver-tical and horizontal rectaiigular cylinders of different aspect ratiosin steady flow, in waves and in waves and currents. Presence of

cur-rents was shown to affect the values of the drag coefficients. The

force coefficients were also shown as a function of NK_C.

In all the above works, viscous forces discussed are of first ordêr. No jiidication was giveli for viscous effects on the mean drift force.

Several authors have treated viscous effects in the iiieau drift

force oli floating structures stich as senii-subinersibles and tension leg platfornis. Pijfers aiid Brink ( [9.77) treated the viscous drift force (lite to waves and currents in their analysis of two senii-subinersibles'

(Irift forces. t)eiiise and Ileaf (1979) took account of the drag force

using eIiiI)irical drag and friction coefiìcieiits for the analysis of the

response of a tension leg l)latlorin. I?erretti and [lerta (1980)

ap-plied the Morison equation (Morison el al. 1950) to calculate tite titeatt drift force ott a vertical cylittder (lite to viscous effects to show the influence of wave height on tite splash zone. Luttdgren et al. (1982) (lisCiISse(l the different contributions for tite potential and vis-couts drift force on a fixed cylinder providing approximate analytical exprcssiotts. The Itorizontal relative velocity 1110(1cl fl the Morisoit e(1uation was applied by Burns (1983) while comparing the extreme Itorizoittal excursion of a teiisioii leg platform in botit reguilar and irregutlar waves in frequency and time domain. Chakrabarti (1984) gave closed fortii analytical solutions for both potential atid viscous

drift forces on a fixed vertkal cylinder to find their relative iunpor-talute. Kobayashi et al. (1985) iitvestigated the response of a.tension leg platform iti regular ari(l irregular waves considering tite viscous

couttributiotis to tite wave drift forces based ott tile ltorizotttai rela-tive velocity in the drag term of tite Morison equation Standing

et al. (1991) 1)rovided an expressiott for tite tnean drag force on a situgle column of a semi-submersible usittg both relative

horizon-tal velocity attd relative surface elevation. Comparison (Pinkster

et ai. 1993) between measured and computed miteatt drift [orces on two types of semisttbtnersibles itt both regutlar aitd irregular waves ittdicates consistent divergence hetweett 3-D predictions and' the re-sutits of experitttents. Cititrapui et ai. (1993) presented a method to totnpnte tite wave atid current induced viscous drift forces and

tuuoiitettts oti a tension leg platforiti in regular atid irregular waves.

1'or a sum bmtmergedstructurehike tltepontoon oía settli-sul btitersible, tite viscous ehfects oui tite mtmeaum drift force are not considered duc to tite fact that tite tittie averaged valute of the drag force thrill of tile

Morisott equtatiomi is zero. In case of a vertical cylinder stich as the

-2

column ofaseini-subniersible, wave elevation up to time instantaiteouts

sea Ievel'jsconsidered as the sourceof the'viscous mucan drift force dite

to waves'only. Such forces are calculated by exploiting the drag force term of the Morison equation. In a waves-only fielil and in a'

wave-current coexisting flow field, attention is paid to tite experimental

assessment of the values of the mean drag coefficients.

in this paper, a theoretical evaluation has been carried out for thedetermination of the viscous contributions to the horizontal meaui drift force for a vertical cylinder and for asubmerged pontoon. Tite cylinder is considered divided into two parts namely the splash zomie (from the mw! up to the actual sea level) amid tite submerged zone

('froni the mwl down to the bottom of the cylinder). In theory, the

value of CDO is suppressed by taking its value as unity. Such ba-sic theory can further be extended for a complete semi-submuiersibles

Model tests huaveibeen carried out to deal with such evaluatiomi

exper-iinentaUy for fixed cyliuiders and for a sumbunerged pontoon; Finally, model tests have been carried out for a complete semi-submersible in fixed as well as in free floating (soft moored) condition in regular waves at zero speed and with forward velocity to simWate cutrrents.

2

Theoretical Evaluation

2.1

Viscous Mean 'Drift Forces on a Fixed Cylinder

Reguilar wave kinetuiatics for deep water condition are incorporated using the following expressions. Constant velocity in tite wave crest

of the linear (Airy) theory is applied.

= (gcos(kxwt)

(1)

U = (awetos(kx-'.t)

= u,, cos(kx --wt) (2)

2.1.1

In Waves Only

The oscillatory viscous drag force terni of tite Morison eqitation for a unit length cylindrical section is as 'follbws:

I'D = p C, D (u,,1 cost) Urn coswl

(3)'

=

!pCDD(2tCos(,t

+ higher harniouiic ternis (4) Splash Zone: The nican drift force due to viscous effects in the splash zone is found by integrating the unit length force on thtecylin-der over tite splash zone which is as follows:

T(

ED

--pCDOD(W2JJco5Wtdzdt

(5)

3ir

00

= -PUkCDODC (6)

The mucan drift force on the splaslu zone is tints found to vary with

cube of the wave height and for a particular wave height, it would increase linearly with wave frequtency squared.

2.1.2

In Waves and Currents

Splash Zone: The viscous dragforce per unit length of the wetted

cylinder in presence of currents, U, is as follows:

FD=PCDD(u+U)I1L±UI

(7)

The application of eq. (7) depetids ou the magnitude of U with

PCDOD

JTJ((U2+2Uu

coswt + u, cos2ct)dz dt

For U

I rn

:

FD = jPCDo D(a (J u

For

I U I < tIm

i, CDOD ( u {72 sin 0

+

jin30

+9siii0)

+ (2Ø ir

+ sin20))

(10)

Submerged Zoiie:

The liteau

drift force due to

wavêcurreuit

co-existing flow field at unwl (z=0) is given by the following equatiOns. l'or the couiuplete submerged zone, the. wave particle velocity ù,,1 is Lobe replaced by

(,wc

and as such calculations are to be repeated

for a numberof vertical segments down to the cylinder draft.

pCDOb

10T(U2+2UUm cos wt + ui., cos2 wt) dt

For Uj

Um

1

221

PCDODU,,I(7 ±)

Fori U

i< Urn

pCbo Du,, { 72(20 ir) +

(.2(3 ir + sin 20) + 47 sil! O)

PD =

FD =

=

The value of O is cos' (U/u,,1

=

--7)and for U, the value of

O is cos (U/Urn = ).

In lact

eq.

(13) an(l eq. (10) turns to eq.

(12) and eq. (.9) respectively when U equals u,,,.

2.2

Viscous Mean Drift Forces on a Floating Cylinder

In case of a floating cylinder, only translatory motions, i.e.

only

horizontaian(l vertical modes ofuiuotioiisareconsidered. Accordingly the cylinder is sill) ject toa relativo horizontal velocity anti a relative wave elevation.

Por a floating cylinder, the relative wave elevation ( is replaced by (r,,, cos(ci.'i

+).

(r,,i

=

(a 'J { 1 +

(IlAO)

2(IlAO) cog J

arctan { z, sin

_,/((,,

z0 cost _J (14)

lii a similar way, tite horizontal relative velocity u,. is replaced by

ri

cos(wt + t,,).

IL,.,,,

₌

(,,w 1 + (RAO)

2(RAO)rsin (z)

(u

=

arctaui { z,,, cos

/(u,,,

rn sill e,,. J (15)

2.2.1

In Waves Only

splash Zone:

Similar to a fixed cylinder except replacing u by u,., tite iuieaEt drift force due to viscous effects on a floating cylinder is as

follows:

pCDO D I

1T

f 2

Icos(wt + e,,)

{cos(wt + t,,))dzdt

(16)

=

p

CoD

,,, (rrn

cos(,,

e)

(17)

2.2.2

In Waves and Currents

Pf)=

,pC1)D(u,.

+ U) i

'

+ U I (18)

-, 3,

-Similar to the fixed cylinder, the application of eq.

(18) is to

be perforined depending on the magnitude of tite current velocity If with respect to that of the relative velocity u,.,,,.

Splash Zone:

= _fiCJ)OD

1 1T

U2 + Wurm

cos(wt + e,,) + n,,, cos2(tiut + ,,) )dzdl (19)

Fori U

I Urm

FD

pCboDUurm(r,ncos((u

(.20)

Fori U

<Urm

= 'PCDO(r,,,

1m

L125h1Ocose + {(2O

ii)

cos(,,

ec)+sin2øcos(u +)) + (sin0.)

{cos.( cos2ø(cos2e,, + 1)) + cost,,

(2cos(e,,

e) + (cos20 + 1)sin,, sin co J]

(21)

Submerged Zone:

PCDOD

10T

cos(it + e) + u,,, cos2(tt + ,,) J dt

Fori U

,

Urm

1 ₂

2!

pCoDti,.m(7+)

For

I U < Urn1

¡"D =

p Cno Du,,,

(20

ir) + (20

-ir +Sill 20 cos 2e,,)± 47 5ll Ocos t,,)

2.2.3

Computations

Coniputations have been done for a vertical cylinder, fixed.auid float-ing, of lOin diameter and 20m draft anti results have been.Iiroduced.

in (Dey and Pinkster 1994). The mncall drift force due to the

vis-cous drag force term in tite Morison equation shows that with tite

increase of the wave amplitude, the nican driFt force increases with tite cubic power ofthe wave amplitude whereas potential effects are proportional to the square of time wave amplitude. The. nican drift force due to potential effects: has been calculated by DELFRAC, :a 3-D diffraction progrant developed by Delft University of Technology. Time wave-cürrent interaction effects are much more pronounced at or inimnediateiy below the mean water level. lt increases with the increase of wave frequencies for a particular wave amplitude. As the draft increases, tite interaction becomes weaker.

The basic pattern of the mean drift force (hue to viscous effects for a floating cylinder is similar to that of a fixed cylinder i.e. with the increase of wave height, tite viscous effects would increase in a similar

manner as for a fixed cylinder. For a fixed wave amplitude, the

interaction effects increase with the increase of tite current velocity. This is the saine for a fixed as well, as for a floating cylinder.

2.3

Viscous Mean Drift Forces on a Fixed Submerged

Pontoon'

For the submerged body, the mean drift forces 'are expected to be

mainly of potential origin and as such formulatiOns are given for the nican drift forces due to viscous effects' only in a wave-current coex-isting flow field only in beato and head seas. A submerged pontoon of rectangular section with triangular ends is coitsidered hiere. The overall length is i with. that of tite triangular ends as i and i)readl.h

and depth are b and h respectively.

FD =

FD =

2.3.1

Iii Waves and Currents

Benin Seas:

'DY

"DY

"D)'

Head Seas:

FDX I'DX "DV

Head Seas:

"DX "DX FDX

ph1Cnyo4 J(u+ U)

I

u+

UI dt

For

(J I, Up11 2

21

phi

CDYO.U,,,(-y + )

For U

I< it,,,,

p hiC l3YO.0

{ 'y(20 - ir) +

(20 - ir + sin 20) + 47,-sin 0))

PhCDX4

jT

Iu+UI

-(I-21)/2 '+(l-2I)/2

J

dx-

J

dx)dt (28) -(/2 +1/2

For

I U

I u,,,,

p h CDXO

(2i + '12b)

For

lU 1< u,,,,:

p hCDXo.u,,,[72(20 - ir)b + 16 sinO

(sin

kl/2

sin.k(i -

2i)/2} + 2i(20 - ir)

+ - sin 20 (sin

ki -

sin k(1 - 21e) }1 (30)

2.4

Viscous Mean Drift Forces on a Floating Submerged

Pontoon

2.4.1

In Waves and Currents

Beam Sens:

'DY

phiCDyo

J(itr + U) I

Ur + U I dt

For

I U I Ur,,,

phiCj,yo U,,,

(72+ )

For

I U1-< iL,,,1

---

phlCyiL,,,

172(20 - ir) + 4sin0

2ir

cosc,, + (20 - ir ± sin 2Ocos2,,)J

phCBX4 J0T2(Ur+U)

_Iu,+UI

-(1-21,)/2 +(1-2I)/2 (

J

dx-

J

dx)dt -1/2 +1/2

For

I -U I u,,,, ph CDXO U,,, (21, + 2b)

Fòr

I U I< u,,,,

=

!_PhCDXO,UI 112(20

- ir)b+ I6

sin O sin ,,(siii.ki/2 - sin k(i - 21,)/2)

+ [21,(20 - ir) ± sin 20'sin 2c,,

sin 20 (sin kf - sin.k(i - 21,) ) ]J

2.5

Viscous Mean Drift Forces on a Semi-submersible

Based on the theory outlined through Sectioii 2.1 to 2.4 for the vis-cous mean' drift forces on a fixed -and, floating cylinder in waves only

andin waves and currents and also for a subnierged pontoon 'in a

wave-cürrent coexisting flow field only, the viscous mean drift forces on a-seau-submersible, fixed' or free floating, can be modeled in a sim-ilar way wheD the horizontal velocity and wave elevation are treated

at respective, columns. Furthermore, the relative velocity and

rel-ative surface elevation including their phases at respective columns fora free floating semisubmersible are takeii into account.

(27)

26' 'Modification of Hydtodynamic Parameters

The newly proposed -Keulegan-Carpenter number, (Iwagaki

et al. 1983)-has an advantage of defining a flow field under a single

parameter rather -than introducing additional parameters. In

addi-tion, flow fields-defined by Nkc for a-coexisting flow 'field 'can easily be compared -to 'the -similar one under a waves-only field (Sarpkaya and lsaacson 198-1)

NKÇ for, a waves-only field is different from that in a

wave-current coexisting field not only by the additional magnitude of

the combined velocity but also by the influence of the- crest and

(29) trough phases of the interacting waves.' Formulations-have been given

in (Dey and Pinkater 1994). Besides, N,-_

in the coexisting flow

field is governed not only by the magnitude but by the direction of

U compared toUrn.

3

Model 'Testing

A detailed experimental study was carried out in order to

evalu-ate the presence and extent of the viscous mean drift forces omm fixed cylinders of-different diameters and on a submerged pontoon in waves

only as well as in waves and currents. Both positive' and negative

currents were 'used, i.e by towing the carriage into the waves and

out of time waves.during tIme tests. The main objectives of tIme exper-itnental investigations for the cylinders amid time pontoon were (a) to--'--'

-assess tIme magnitude of the viscous effects for tIme immean force iO a' -

-waves-only flow field as well as in a wave-curreüt coexisting flow field

-including their elfects on- the two separate immmportant hydrodymmamic

zones of timecylinder, (b) to obtain the time averaged values of the

viscous mucan drag coeFficients over a domnimiaiit ramige of the wave

frequencies in the said flow fields and finally (c) to find'

the-suitabil-ity of time single controlling hmydrodymmamic parameter to express the

viscous mean drag coefficients due to wave-current interactions so that time coexisting flow field can be niade analogous to the flow field de to a waves only flow field.

Finally, model testing has been carried for a semi-submuersible

(ITTC Model), fixed as weil as free-floating (soft mmìoored), in regular

waves-with positive and negative currents. The purpose was to assess and compare the viscous macan drift force for the semni-submuersible

'by applying tIme fimidings of the individtmal model tests' results for the

cylinders -and the pontoon. Besides, motions have been measured

to check the validity of the computation method (DELFIIAC) for

predicting the first order motions.

3.1

Expèrimental Set-up

3.1.1

Fixed Vertical Cylinders

Two cylinders of. different diameters, 75 mm and 3-15 mum respectively

wereused in the model tests. The basic construction- is srne,for both

of tiucimi. The model cylimmder was mimado up of Four segmnemits - tIme

lowest and the topmost being the dummy ones and the intermediate ones as the test sections. Both the test sectiomms contained their

utitlivitl,ua.l load cell to ineasit re the forces. 'i'ó eliminate the end elfecis of tIte circular cylinder,, tite lower (i uiniiiy cyli 0(1er *115 used.

i'lie slits between tite cyliiidricai sections of the whole construction Were covered willi thin rubber.

l)etalls of tite, test set up and others have been discussed in details iii ( l)ev aitd l'inkster 1994).

3.1.2

Fixed Submerged Pontoon

Figure i shows tite set up of tite model. The p0111000 was2.3 tu long, etti wide and 18 cm high. [loth ends of the model were made triati-guIar shaped over a length of 14.6 cm aiid all tite eulges liad a radius nl II) ititti attui tite (Irait amoutitted to 40 ciii. The pontoon was fixed Oit two vertical .cylitiders by a set of two force transducers (temple

shape type) catit. Otte transducer for mea.suring tite longitudinal luiiue and the other for the transverse force. TIte traitsdncers were itioti tiled opt top of aitotiter inside tite cylinder and one of them was titoli it ted oit a lotigittidinal sleigh to prevent distortion tension. Two forte tratts(htpcers for tite vertical forces were added.

lut fact tIte cylin(lers were not in any way connected to tite pon-tonti bo(ly so as not to attract any load and the gaps between time potttoott to1) and tite cylinders were mimade water tight with thin

rub-ber seals atid also applyitig rubrub-ber catings. The cyliitders were of

57.5 ciii bug aitd 10.8 citi diaputeler. They were directly mounted to lite carriage.

'i'csts1ltave been conducted in tIte Towiiig Tank No.! of Ship llydtoittecltaitics Laboratory of E)clft University of Technology for a freqtiettcy ratigeof 3.536 - 7.071 r/s (Scale:!/50). For each wave

fret1ttritcy, at least three wave atitplitu(les vere used. For each wave frcquuettc.y with two wave aiitl)ljttt(ies, tests have beti cotiducted for litait positive and negative cit rreitts iii presence of waves. Three car-riage speeds were used and they are 0.146, 0.2!8 and 0.291 tn/s.

3.1.3

Semi-submersible

'l'lie itt lii it' di itteti sinns ait d tite stability data of tite sein i-sii bni ersi hIe

(li

I ( ) lits l)ePti giveil in labie i '1 lie model scale was I to 75

flic

iticaiel was cotistriicted of steel for tite columns and wood for the

ptitntui

Foui r out of eight colutitiits were connected to tite deck and thus were uttade free frotit tite poittoott top. These commits carrie(i

tIn t t i id ivtd ti -ii ior e trutsd ticers for tiieisu ring the hori7ontai force

oit thecohti tu its iuidepeitdeittiy. '['ests' were condticted in regular waves

willi positive amid negative cit rrents in the Towing Tatuk No.1 of Ship I iydroiiiechianics Laboratory.

Fixed Semi-submersible:

For hiC deck structutre, a steel frame

wa.s used to give eitoutgh strttctuiral rigidity and to lux the utiodel

to the carriage via steel bea.itts. The model was fitted with six force tratisthttuers - three to itleastire tite itorizoittal forces and three for tite vertical forces. liesides, two wave probes werecoititecte(i to tue model to itteasitre the relative wave elevation. Tests iiavo been conducted in reguilar waves for a freqtiettcy range of 3:464 - 6.928r/s. At least three wave attiphittides were itsed for cacti wave freqiteitcy and twocarriage

slu('eds, 0.118 aitd 0.178 ill/s were used. Figture 3 amid Figure 4 shows

tIte ttiodel 'iii 1)1ml view a,ttd coluimtis mitt ittbered as 2, 4, 6 and 8 liad

their owut force tratts(luicers. Similar was tite case for the model in l!ee bloating cojiditiomi. 'l'he tito(lel (hraft was 28:0 ciii.

Free Floating Semi-submersible:

Tite deck was changed to a

lighter tutaterial called A ri!titOIe F- Board (altuitmittututi lioneycoitub core

willi woveiu glassy epoxy skium). Tite 1110(1cl was weighed and

(ly-ti;i.ttiica.IIy balaitced to tileastire tite radii ofg,yratiout in air abotit tite t'euler of gravity. lut tite taiik, iutcliutiuig experituietits were carried out

lo check tite vertical position of tIle center of gravity. Natural

5-Table l Main Particulars of Semi-stibmersibie

periods of heave, pitch and roll were also mneasmured. Frequency range

and carriage speeds were same as for tite fixed model except a few more frequencies were used for the waves only field. The model was connected to two force transducers - fore and aft - by meamis of two

horizontally deployed springs between tite model amid the Lratmsdticer.

A perspective view of the set-up 'is shown in Figure 2. Tite draft of the model was 28.25 cm.

3.2

Data Analysis of Measurememits

The measured forces in waves only or in waves and curreuitscan be-expanded in a Fourier Series up to third order as follows:

F = io + I(F, sui TIWILI + ¡'b,.

tos tùt') (37)

One of the ham objectives is to find time viscous contribiutiomi

towardsthe measured mucan force, i.e. to express tite utteasumred mean

force as tile following

Po = p +

FD

The potential niean drift force oui the splasit zoiie is due to time contribution ofthe relative waveelevation and the secondorder pres-sure (velocity squared terni of Bernoulli's equation) amid is calculated by the Program DELFRAC. So, the viscous mean drift force is

cal-clated out as follows ¡"D =

F0 - Pp

From the above, the' timneaveraged mean drag coelTicients can be

òbtained by using the expressions as outlined earlier for the thieoret-icai viscous mean drift 'forces on a fixed cylinder and a submimerged pontoon. however for the splash zone of a cylinder, corrections are to be unade for tite tacan drift force dite to the second order

pres-sure. For lije submerged zone of the cyliumder and for thesubummerged

pontoon in a waves-only field, tue cotitribiution to the uneati force is purely of potential origin and can thus be calculated using the sec-oitd order pressureasiuentioned before. For the models in waves and currents, similar treatment can be applied except that the theoreti-cal siiscous meañ drift fórces need to be considered' for tite couiditions wheti U is greater or equal to IL,,1 or less than u,,,. Poteuitial con-tributiouts are to be coutsidered properly with forward speed effects using the equations proposed by Clark 'et al. (1993). The values of tite iuiean drag coefficients can then be obtaiuted from tite nueasured mucan forces and the theoretical ones.

Designation Model Prototype

Loa

Boss. 1.533 in 1.000 in 115.000mo 75.000' iii D 0:600 m 45.000 m Pontoon: 1.533 ni 115;000m B5 0200 m 15.000 in D 0:120m 9.000 in Columns:

D,

0150in [1.000 m

D3

O.125ni 9500m T 02825m 2!.1875m

089 t

37546875 t KG 0277 m '20.775' iñ GM1 0:035 mmm 2.625 in GM1 0029 ni 2.175 in I(,.-. 0.485 ni 36.394 un ¡Ç,,, 0;451 mn 0.536 m 33.816m 40.228 in

T

2.557 s 22.144 s

Ti

7.120 s 61.661s T9 ' 6:935 s 60.059 s

PONTOON crorvIEw).

PONTOON (ELEVATION)

Figure 1: Experimental Setup of the Submerged Pontoon Model

IIOItIZONTAI. CONNECrIIW BEAM

Figure 2: Experimental Setup of the Free Floating Semi-sub Model

-6-For the seinisiibiiiorsible, an identical approach is used except

that the theoretical mean force on the individual coluniiis due to

potential effects have beeii obtàiued for those columns fitted with a

force transducer by using the relative wave elevation around theni

and the second order pressure.

For the floating seiiuisiibinersiblè, imiotions have beeii measured directly. The pitch motion has been derived based on the two verticaidisplacenients iuieasured fore and aft and the distance between them. horizontal motions were measured at deck lèvel and accordingly thetheoreticalcalculations at the center of gravity have beemi corrected for the deck reference point. 4,

Experimental Results

4.1

Fixed Vertical Cylinders

The results for time mucan drift forces for time three sets of tests car-ried out with and without forward speed have beemi shown iii details in (Dey and Pinkster 1994). The results clearly iiidicate that the di-vergence between the theory and tIme experiment occur in tIme splash

zone while for the submerged zone, time resultsshmow reasonable

agree-ment with the results of 3D potential calculations. So, the splash

zone of vertical columns is thus found to be the principal comitributor for the viscous effects in waves even without currents.

It has also'been shown that the prediction of time mean drift forces for a SEDCO-700 type semni-subnmersible indicate better agreement with the experiiumental results when experimentally obtained values of the mean drag coefficients are used for the splash zone in waves only. Similarly, prediction for the mean drift force on a fixed vertical cylinder (splash zone and submerged zoime) in waves and currents

show better colmmparison when experimemitahly obtained values of the meamm drag coefficient iii tImo sai4 flow field are suitably applied.

4.2

Fixed Submerged Pontoon

ilorizontal mean drift forces for the submnemged pontoon in a

waves-only field in beaiii seas amid head seas arc shown iii Figure 5 and

Figire 6. Though data show scatter, there is no iimdication- that

measured forces are higimer than time theoretical calculations either

by "far field umetimod" or by "near Field umetimòd". Iii addition, time

immeami drift force for stich submimerged body (only second order

pies-surecomitribution) is of conmparatively lower mimagmmitude wiuidi is nmoÈe affected by tiiemneaim offset (nOise) omm time nieasuremumeumt systenu.

Ver-tical nican drift forces also have been simowmm iii Figure 7 amid Figure 8.

Comparison seem quite satisfactory as far as potential contributions

are concermmed. Though circulation pliemmomimemia do exist for sucim a

single body cylindrical structure, its effects are still within the

po-tential theory affecting omuly the inertia force caumsimmg mmomm-limiearity imi

both first order liorizommtal and vertical forces.

Figure 9 shows the values of the nmeamm drag coefficients in beam

seas in a wave-current coexisting flow field for both positive and

negativecurremuts. Values havebeen produmced as a function of N7-_c.

The mulcan values cami be takeuu as equal to unity except for somne higher waveaunplitudes in positive currents iii which case, the values areslighitly higIier Comparison has been shown in Figure 10 between the present theory using the experimentally obtained CDYO values.

Prediction appears reasonably good.

4.3

Semi-submersible

4.3.1

Fixed Semi-submersible

Through Figure 11 througim Figure 14, relative (absolute) wave el evation immeasured at positiomi Pl and P2 Imave been produced for a

waves-only field in beani seas and head seas which reveals LIje fact

that the theoretical calculations of the mucan drift force due to

o 20 -u -20 o ru

Figure 3: PIan View

3-D(nf)

-- 3-D(tf)

o exp(Ç1) V exp(Ç2) A _exp(Ç,) ° exp(Ç4) . exp(Ç5)

-.0 .4 .8 1.2

-7--2.5 C w 0 - -5.0 w E C

j

-10.0 200 o o w E ioo s, > -12.5 0 .25 50 .75 ü ¡nrad/sec

Figure 6: Comparison

fixed submerged pontoon In beam seas

300 o

fixed submerged pontoon in head seas

3-D(nI) 3-D(tt)

\

D exp(Ç1) V exp(Ç) A _exp(Ç3) H O exp(Ç14) 4 V 1.00 3-D(nf) O _exp(Ç,) V exp(Ç2) exp(Ç3) ° exp(Ç4) 9 exp(Ç5)

- 3-D(nI)

o exp(Ç,) V _exp(Ç) A _exp(Ç3) O _exp(Ç4) exp(Ç5) 1.25 w E o N o -c o - _{4_}

W9

A 0 _y O a oinirad/sec

Figure 4: Plan View

Figure 7: Comparison

fixed submerged pontoon In beam seas fixed submerged pontoon In head seas

oin rad/sec o,in!rad/sec

Figure 5: Comparison

Figure 8: Comparison

8

7 6

e

J

.4

o

00 2.0 2.0 E E 15 C C o (s a - I.0 51 > (u SI > . .5 SI

fixed submerged pontoonin beam seas

Figure 9: Experini'iital Results

fixed iflc s mi-sub in beam seas (positIon 1)

o

- 3-D(nI)

D exp(Ç1) V exp(Ç2) A exp(Ç1) ° exp(Ç) a exp(Ç) .2 .4

i

û) in rad/sec

Figure 11: Comparison

8

- 3-D(nf)

exp(Ç1) V _exp(Ç2) A _exp(Ç) ° exp(Ç4) exp(Ç5) .2 .4 .6 .8 10 0) In rad/sec

Figure 10: Comparison

fixed ittc semi-sub ¡n beam seas (positIon 2)

.6 .8 10

-8-a e 5000 2500 2500 -5000 -a a

fixed submerged pontoon ¡n beam seas

U-EJ

theory(+U2) exp(+U2) 13---EJ theory(-U2) exp(-U2) D ti theory(+U3) exp(+U3) a- ---a theory(U3) exp(U3) o-o theory(+U1) exp(+U1)

o.- --O theoty(U1}

exp(-U1) U1IO3mis U2.IS4mfs U3-2.52m1s .2 .4 .6 .8 lO Olin rad/sec

Figure 13: Comparison

fixed ¡ttc semi-sub ¡nhead seas (positIon 2)

2,0 E E 1:6 3-D(nf) exp(Ç1) exp(Ç2)

-

_°

V C A _exp(Ç3) C o a ° a exp(Ç,) exp(Ç) - 1.0 ID a > ..i .5 2.0 Olin rad/sec

Figure 12:

Comparison

fixed ftc semi-sub inhead sens (position 1)

E 1.5 3-D(nI) exp(Ç1) exp(Ç2)

-

O

'

C A _exp(Ç) C o e > w 0

°

exp(Ç4) exp(Ç5) A 1.0 2.0 O _{waves + U} D _o waves-U 1.5 n D a O O o o 1.0 OO Oe o .5 6 4 o o 0 .2 .4 .6 .8 Io .8 IC o 0 .2 .4 .6 w inrad/sec

Figure 14: Comparison

75

25

fixed ittc semi-sub (calif 2)'in beam seas

3-D(nI)

°

exp(Ç1) V _exp(Ç2) exp(Ç3)

°

exp(Ç B _exp(Ç5) R

u

-25 -0 .2 .4 .6 .8 oi ¡n rad/sec

Figure 15: Comparison

fixed ittc sem-sub(col# 6) in beam seas

e A n w in rad/sec

Figure 17: Comparison

8 a V A .4 .6 .8 10

-9-10 2 .4 .6 .8 10 w In rad/sec

Figure 18: Comparisón

flxedittc semi-sub in beam seas

3-D(nI) 3-D(t1) exp(Ç1) exp(Ç2) exp(Ç) exp(Ç) exp(Ç5)

fixed ftc seml-subi(colif 6) Inlhead seas

3-D(nt) o exp(Ç1) V exp(Ç2) exp(Ç) 0 exp(Ç1) O _exp(Ç5)

9v

A

000

.9 9 9

o'

o o e E c -2

't

-10 250 200 Iso 10 0

-°

'

A o B 50

tixed ittc semi-sub in head seas

V S 5 .2 -lOO 3-D(ff), -o 3-O(nI)

--a -I60 -200 O V o exp(Ç1) exp(Ç2) exp(Ç) exp(Ç4) exp(Ç5) -250 -75 3-D(nt)

-E C o

't

50 o V ° exp(Ç1) exp(t2) exp(Ç3) exp(Ç4)

exp()

25 -5 3-D(nl)

--10

_°

exp(Ç1) V _exp(Ç) A exp(Ç3) -15

°

exp(Ç4) exp(Ç5) -20 C Is e E -a C o o o -C -25 o .2 .4 .6 .8 10 w in rad/sec

Figure 16: Comparison

fixed Hic semi-sub (col#4) In head seas

o 0 .2 .4 .6 1.0 winrad/sec

Figure 20: Comparison

o .2 .4 .6 .8 10 w in rad/sc

Figure 19: Comparison

o .2 100 -a C o 50 o

E E C.

o

w O) In 2.0 2.0

free floating ¡tic semi-sub in head seas

3-D(nl)(drp) 3-D(nl)(cog) exp(Ç1) exp(Ç2) exp(Ç) w in rad/sec

Figure 21:

Comparison

free floating lttc semi-sub in head seas

- 3-D(nl)

V exp(Ç1) ° exp(Ç2) exp(Ç3) e .5 1.0 c» in rad/sec

Figi.ire. 22: Comparison

free floating lttc semi-sub Inhead seas

w in rad/sec

Figure 23: Comparison

-

3-D(nl) exp(Ç1) ° exp(Ç2) A _exp(Ç) 15

lo

400 rl

::

I

400 o o 200 w E 100 N -c O w In rad/sec

Figure 24: Comparison

free floating lUc semi-sub in head seas

exp(U=i-1 02m/s) oO theory(U=+ i 02m/s) exp(U=+1.55m/s) D-5I theory(U= I .55m/s)

U.

.

w in rad/sec

Figure 25: Comparison

free floating Ittc semi-sub In head seas

Io

f-' 3-D(nl)

o-Q 3-D(nl)U

I-O 3-D(ni)+U

exp(U=+l .O2mIs) exp(U=+l .SSmIs) U e o 0 .2 .4 .6 w In rad/sec

Figure 26: Comparison

freetloating ittcserni-sub In head seas

100 Q ?----V exp (Ç1)

o---. exp(Ç)

80

_{A---a exp(f)}

thoery(Ç1)

A

theory(Ç2) 60 _e lheory(Ç3)

I'

I

¡U

V I t 3D(nQ 40

_/iII."t

20 e e _s e 71 u o o .6 .8 I .5 1.0 1,5 .2 .4 .6 .8 IO .8 l'0 1.5 E E C

o

1M n -c

I)uriiig iiiodel testing, horizontal forces on some columns were

also ninasured. Figure 15 and Figure 16 show the iiieaii drift forces

iii l,eaiii seas atid Figure 17 and Figure 18 show the saine iù head sea.s. Measured forces for all the coluntiis show consistently higher

forces than the potential contribution except minor scatter which

uiiay he (lue to the interference effects.

In Figure 19 and Figure 20, horizontal mean drift forces:in waves only have beeii shown for beam seas and head seas. Measured mean forces are higlier than the potential one showing the marked viscous ell'ects on the horizontal mean drift forces.

4.3.2

Free Floating Semi-submersible

One of the piirjoses of the experiments was to measure the motions

to validate 3-D predictions. In computer coding for calculating the

viscous mucan drift force, theoretical motions and phases are used. Figure 21 through Figure 23 show excellent comparison between the-ory and iiieasnreinent for surgej heave and pitch response amplitude operators in head seas. Surge ¡1AO lias been corrected for the deck reference point.

In Figure 2, the horizontal mean drift forces have been prodùced showing time calculations based oit 3-D potential theory, the measure-nients and time results of calculations after viscouscoutributions have beemi added to the potential oiie. Comparatively, prediction is much iiiiproved when such viscous contributions are taken into account.

Finally iii Figure 25, comparison lias been shown for the mean

drift force iii waves and currents. Using experimentally obtained

values of time mucan drag coelhicients for the vertical cylinders and the submerged pontoon, theoretical prediction can be tuch improved. Omm tIme other hand, a generally accepted approxinuateapproach based oui summmmmiatioim of time potential mucan drift force in waves and the

cii rremut force iii still water (loes lint aPpear to be applicable.

5

Concluding Remarks

Model tests with dilFeremit (lialuleter cyliuuders clearly signify that the

51)laSim zone is the iiiaimu source of time viscotus contributiomus iii the luorisniutal nican drift force. lhowever, tuse of time values of time imuean drag coelilcients should be carefully juu(lged as it is a futictiomi of

Keumlegamm-Carpeimter Number (N,_) at very low diffraction param-eter whereas with time ilucrease of time (liliraCtiolu paraimiparam-eter, they should be evaluated as a coiumbimuation of both viscous paraimieter

(lI/D) amid the diffractiomu paraummeter (k * D) i.e. wave steepness

(k* Il).

lu case of suubmuierged structures like tite submerged zone of a cylimuder or a submerged pontoon in waves, the experitmuental inves-tigatiomus show similar trends in the nican drift force as predicted

by presemut 3-l) cotumpumtatiomual techuiuique. lui waves amid currents,

time preselut theory can predict the horizontal mean drift force with

coutuparatively better accuracy whuemm experimuientaihy obtained values of time uueaiu drag coefliciemits are enuployed. Iustea(l of using sep-arate hydrodynamnic paraimieters like Ketulegan Carpenter Number

aml(l Moe-Verley miummiber, a mnodil'med Keuhegan-Carpenter Number

camu take care of both with additional effects of crest and trough

l)huases of time interacting waves with currents.

lt is expected that time above analogy to behave in asimilar

fash-iou for a floating structure and accordingly they can be applied for

a floatiuug seini-suubmiuersible. 'lime differences in the nueami drift force

calculation amid time experimental resumlts are due to interference

ef-fects iii wimichi tlme'diffracted waves play anuajor role. lui longer waves

(hilfractiouu effects are reduced and viscous effects beconme relatively

i mii por t alu t.

l'ue actual flow around vertical cohuinuns amid subumerged pontooli iii waves an(l aleo in waves an(l currents imuchuding viscous effects is in fact aim extreummely complex PheI!omfleuuon. Usimug time drag force terni

of the Morisomi equation and overcoming tite mitain uncertainties in

the choice of the values of nican drag coeflicients from experimemutal

iimvestigations 'is not a robust technique because of its fragile uiatumre.

But at this time, aim improved indication can be giveut with regard to such viscous effects for the horizontal nican drift force for moored

structures like semumi-submnersibles.

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