Calculated drift forces of two semi-submersible platform types in regular and irregular waves

OTC

-.WV 1978

HIEF

CALCULATEGL,OR I FT FORCES: .OF NO 'SEMI SUBMERS) BLE

PLATFORM TYPES IN REGULAR AND IRREGULAR WAVES _

by J: G. L. ;Pijfers-and A. W. Brink,-

-" Institute TNO:-.for-Mechanical Constructions

Cdpiiight 1977, :Offshore Technology Conference

eepsbouykuptle

!1o9 eschool

Delft

2

ic-

er-eta -t7"

Ti's paper was presented at the 9th Annual OTC in Houston, Tex., May 2-5, 1977. The material is subject to correction by the author. Permission to copy is restricted to an abstract Of not more than 300 words

ABSTRACT

In. this paper'... a method is given tO calculate. the .mean and slowly-yarying drift. forces. on ,sami7

Ubieeob to; struturs. ,clUe to the environmental

hydrodynamic loads: i c waves and current. The semi-submersible structure can be-efther free .floating or -captive or j.ixed.

.."

Calculated drift forces of two different

semi-submersible fJ ree_fioati ng..platforms are presented:

The uence of ,pirrept, velocity on the drift

forces:::i is shown, and 'a, comparison is made between

the drift fOrceS.and the forces due to current only, - and due to Wind._

,

Al so ,a me thatl( s..giyen to cal cu I ate

time ddmalq' the slowly _varying ;.:0,0 ft fo.rcet,ip,-.

irregular waves Two time histories are shown, which

have been genera ted:JOKti mule t ion purposes.:

'

. ItimpueliON

-If semisuBmerSible structures are assimiedto be composed of rectangular or circular-cylinderS,-,

the hydiodynamiclO"rces. generated by waves'anA'

-current, on these structures can be calculated Making use of the ,MORFBOh formula. In this form2pla,the,..:4

hydrOdYnamiC.fo'rce on a cylinder can be expressed =

as the sum of al drag and inertia. force. The drag

,force is proportional with the yelbcity squaridtof

the .1.10id relative to the tylinder. The-f lUid

is the:.Sum of-a Con-stanp part, 1;:t.-..-.turrefil..and

mass transOort.VelOcity and a harmonic part Caused

by waVe;motiblis and structure motions.

A.;'mean,h'ydrp-dynamic. force can-be calculated, which is the time

average of all forceson the structuce,....:The

wave ' drift force is the mean force in

Of the waves and can be calculated by subtracting the true current force' from the hydrodynamic' ftirte.

To calculate the. hydrodynamic forces..c11:-,a,

FY.-I inder.; it is split up into a number-of,,,c0-inde elements, ,Which are assumed to ibesmal.1,cd0afed,to.,,

the :wave length Any cylinder of ULe Structure,can

' 1"7,

References and illustrations at end of

" be randomly oriented. However, interactionSbetWeeh,

two or moreclinders' are not

:

InertTa' a k.drag coefficients 'U- 8 1 --mciji) ON for-.

mule Will i',136':determined from

data

1, '

Calculated drift forces are presented for .t:%;;ii, types of semisubmersible structures,'. one7o,them.

having two large' pontoons With -S columns, )whi le the second type, uses 5 floaters with t5 ,cc) 1 uMn

The drift forces in irregular waves" are - calculated

for two -sea spectra, according PiersonMoskowitz

-and JONSWAR. These' drift fortesa're -,illipoet4ht7- to -determine the required power tedYnaffitally positioned

platfOrmS.

Also a Method is desceibed:to_geherate time histories of the slowly Varying. 7,time=dependerit

-drift forces- in irregular waves Results ,for both

. platform types are pres'ented': Such time histories ' can be used in simulation 4Udfds.-,,jh which the low

frequency mat ions of modred"-Pe-AVhaMiCally- poS it ioned platforms are Ahyestigated.

-SYSTEM OE AXES AND DEFINITIONS

Figure) shows the defjnit(Onof.

thel:right_hand--iprthogonal :system of axes Ox'1,C2x,, is -f ikedito ' the floating, structure. The 0)(1-6)(iS,:_!it. Chtise-k in

forward..,direction, the Ox2 axis tb;.Starboard'arld-the

Ox axis is -vertical dowHwards thrOUgh the Center of

gravity G of the floating structure': In the 00 ,

brium-condition' the Oxpq Plane cointideS.WitF6the undisturbed Water level.

-

-;

. cohs'ider' an, element of a :r=ahclOiril`y Orlenied. -cyl Hi-der (either circular Or rectangular) with its Own 'orthOganal system of aims O'xpcixl. The dr..!g n .0' ass:ume&to be the -rioint of appl itg._Von Of the hydro:,,

he'e.1 enien . fhe' IS the 7

-centerline The',0-1:-ii":"...and'Zi 1'x

are normal to the sides in case of , a -rectangular

'rcy-1i nder i t ra -8-aVe of a.:

circular cylinder.

.

: :"," '

The drStanCe yector from- the -origin 0..c.cf:t he

floating structure to,theo.rigin' 0' of the element o-f

';z1

.

,

ifr,

eirvk

..

..44101000

a randomly -oriented cylinder i-S defined-by

13).

Fig. 2 Shows the definitions of the absolute and relative advance angles of waves and current. In

the calculations only relative advance angles are used, which will be:

-for waves : p =

-for current: pc = tpc - b

11,we and tpc are the absolute wave- and current advance angles respectively and 11) is the heading angle, of the

structure with respect to the earth fixed system of axes 00x0y0z0.

Current velocity isdefined as a zontal velocity of the water, uniform depth. The current velocity vector Vc to the body fixed axes 0x1x2x3 is:

[

Vc cos. pc

= Vc sin pc

0

constant hori-over the water-with respect

(1)

Mass transport .velocity can be defined as the non periodic velocity of waves, derived from the second order wave theory. According to Stokes, [1] the mass transport velocity Vm in deep water in the direction of the waves as a function of waterdepth x3 is:

2

Vm = ce k w exp (-2kx3) )

in which:.

.ce = wave amplitude

2. k = wave number (k = w /g)

wave frequency

The mass transport velocity vector will be:

[

Vm cos P = Vm sin u

0 '

HYDRODYNAMIC FORCES ON A. CYLINDER

In general the hydrodynamic-force on a cylinder can be calculated using the MORISON equation L2]. The hydrodynamic force vector' is:

.F(t) =

-ipCd Ap

-IVIV pC VS (4)

- m

in

which:-p = mass density of the fluid Cd = drag coefficient

Cm = inertia coefficient An = projected area

V,S = velocity vector and acceleration vector . of the fluid relative to the cylinder

Iv' = the magnitdde of the -velocity vector V = volume of the cylinder.

Due to current velockty,mass transport velocity of, waves, harmonic wave- and structure motions the velocity vector of the fluid V relative to the,,, cylinder element as defined in fig. 1 will be.:

=V+V+x- Bx

in which: 46- 1 t (3) (5)

x = velocity vector of the water particles in w

the origin 0' of the cylinder element, due to the harmonic wave motion

x = velocity vector derived from the absolute motion vector x (with six components) of the floating structure

B -= translation matrix to transpose the rota-tions of the structure about the Ox. axes into translations of the cylinder dement. The translation matrix can be defined by:

[

1 0 0 0 +13 -12

B = 0 1 0 -13 0

+if

(6)

0 0 1 +12 -11 0

:Due to harmonic wave and structure motions, the

acceleration vector S of the fluid relative to the cy-linder element will Se:

= Rw - ₍₇₎

in

which:-R = d(%)/dt (8)

w

.R = d2(x)/dt2 (9)

The velocity vector xi of the water Particles and the absolute motion vector x of the floating structure are defined in the Appendix..

DRIFT FORCES AND -MOTIONS IN REGULAR WAVES

The 'total drift force or -moment on the structure is the sum of the drift forces or -moments on

cylinder elements. If the number of elements is N, the' total drift force vector is:

N

D = E D (10).rt

-n

n=1

In the.following, the drift force pet cylinder-element is looked at, in which the index n 'is omitted.

The drift force can be defined as the time

average of the hydrodynamic forces acting on an element in consequence.of the waves. Since there are, in . regular waves, only constant and harmonic forces, the drift force can be obtained by determining the mean

hydrodynamic forces over one period of encounter Te.

From eq. (14) _{the mean hydrodynamic force vector.}

F can be described- by:

-H

F

H Teo-

= j_ I e F(t) dt (11)

According to the definition', the mean drift force is found by subtracting the true current force from the mean hydrodynamic force.

An vector notation:

D = F -F (12)

- -c

in which':

F = current force vector.

-c

Since 'the relative acceleration of any cylinder element in eq.. (4) includes only simple harmonic components, the inertia forces' will not contribute to the mean

hydrodynamic forces defined in eq. (11). Because drag forces are non linear and the relative velocity con-tains both a constant and a harmonic part, only the

drag forces contribure to the mean hydrodynamic. for-ces in regular waves:

The constant and harmonic components of the relative velocity vector .V are defined with respect to the body fixed system of axes.0'x1x2x3. A cylinder element, however, on which the relative

velocity.veC-tor is working, can be randomly oriented with respect to the body fixed system of axes. Since the drag_ coefficients of any cylinder can be determined only'

in the directions of the three dirAlerfiesLazces'

O'xf (i = 1,3), the components of the relative velocity vectors V and. Vc have to be expressed inl

-components with respect to-the.cylinder fixed systems of axes.

If one defines a transformation matrix A, in which the three column vectors are the.unit vectors along the cylinder fixed O'xi. axes, expressed in components with respect to the body fixed 0x1-axes,

the relative velocity vectors V' and VC, both with respect to the cylinder fixed O'xi system of axes can be obtained from:

V = AV'

.r.

. .

...

-=AV'

c -c

Because of the orthogonality of both systems of axes, the inverse matrix A-1 is equal to the transpose matrix AT of the transformation matrix so that from

eq. (13) and (14) one obtains:

-V' = ATV-

...

(15)

V' = ATV (16)

c -c

The drift force vector D', with respect to the cy-linder fixed system of axes will become, with use of eq. (4) and eqs. (11) through (16):

1 Te

D' = f -iPc(IvIlvi_{- - -c -c}

e o .

dt . . . (17) in which C is a diagonal matrix with components cjk (j,k = 1,3):

-cjk =0 for j k

c.1( = Cd Ap.

J.

for j = k (18)

J J

Cd- = drag coefficient of the cylinder element J in the direction of the 01x1-axis A = projected area.of the cyllndef element

P.

perpendicular on the O'xi-axis.

According to HOERNER [3] drag forces on cylin-ders are only a function of the velocity component . id'the direction normal to any axis of slenderness.

Deth magnitudes 11/11 and111,11in eq. (17) therefore have to be calculated prom Projections of the velocity vectors V' and V' normal to any axis of slenderness, which can be acil%ved by making the 3 components al', of the matrix AT equal to zero in eqs. (15) and

46),

if the O'x!-axis As an axis of slenderness of the cylinder element.

The drift force vector D' should be transformed to the structure fixed coordinate system 0x1x2x3. The contribution of the drift forces and -moments of the n-th cylinder element to.the structure is

defined by vector Dn:

D = BTAD' (19)

-n

in which:

BT = transpose matrix of translation matrix B.

-From the total drift-force vector of the

structure-2 according eq.. (10) the drift force coefficients are defined by vector C:

2

= g/(ipga)

DRAG COEFFICIENTS.

Since the relative velocity components in the drift force calculations contains both a constant part and a harmonic part, drag coefficients have to be known as a function of the Reynolds number Re and the -Keylegan-Carpenter number K [8]. Sarpkaya _[7] presents drag coefficients as a function of both Re and K, however,-the results reported herein are applicable only to cylinders in harmonic flow with zero mean velocity. No drag coefficients were found in the literature In which drag forces on cylinders were measured In a harmonic flow with a mean velocity

superimposed on it.

Drag coefficients used in the drift force calcu-lations are obtained in the following way:

The drag coefficient due to the constant velocity Cd (Re) is calculated according to Hoerner [3] as-a function of the Reynolds number Re:

Re = IV' + 1/11.d/v (21)

-c -m

in which:

IV' + VI

the magnitude of the vector sum of

-c -m

.both constant-velocities, i.c. current and mass transport of

. waves, normal to the cylinder axis

d = diameter of the cylinder v = viscosity of the fluid.

The drag coefficient due to the harmonkc velocity

d (K) is taken according to Keulegan and Carpenter

[8], in which the. Keplegan-Carpenter number K is calculated from:

K = IV' leTe /d (22)

-H

in which !WI is the amplitude of the relative harmo-nic velocity vector, of the water normal to the-cylin-der axis.and can be calculated with-use. of eqs. (5) and

(15).-C(Re'K) - . (23)

d

- IV' + V'l +

IV*

.

-c -m -H

Drag coefficients for rectangular cylinders can be calculated in a similar way. Keulegan and Carpenter measured the forces on flat plates in a harmonic fluid However, the flow-pattern about (one dimensional) flat plates can not be compared with the flow pattern about (two dimensional) rectangular cylinders, so that it is not justified to use these data:Therefore, drag coefficients for rectangular cylindirrare taken only

'(20)

The drag coefficients Cd (Re, 10 used in the drift force calculation's is an average of- both Cd (Re) '

and Cd (K), calculated as follows:

a function, of Riy-nol di number, 'accord i n to oerner.

- _

DRI FT FORCEAND -MOMENTS IN; IRREGULAR WAVES

Fr`oin fi'Lforces _arid -moments

in regular waves, ineeh: drift - forces and -' moments in regiji a rY waves can -'becleterminedW1 t h. -Use' of...the'

sfadetra 1 analysis, 'in WhicKfthe; dr if.f forces

in regular waves have to be proportional with :t he wave

arriOli tilde% s-quared However ,, the dFi rces ih

regu-lar waves depend on current velocity squared, mass

traiiport.'!Vel.clocity squaredand Wave=.' endi.-structu re motions squared

The drift forces calculated in this

way are n424proportional with the- WaVjampl:i tilde

squared', .

-To- useTleV the.linear spec t ra1.7'.ena I yslTs the drift forces . are calculated with a: mean wa.4;amplitude

wh representative for the wave spectrum in

which the .7Mean.-d,r(rfi...f.o rces in irregular waves will

be ca I cid atedylh-,order-'to minimize err-Or':

For ad-spectra:1; den§i ty St-(w) of the waves the . mean' square amplitude can 6e defined equal to the

total, energy: per unit of!-carea!Jof.::Ithd,;Weter::surface:

. --2.,

E= f

S -(w) dur = mo a o

t:

- -. (25) In a now-narrow band, spectrum this should be

corrected accord irig to. Ochi n -the fol lowing way:

=1(1

- iE2) m . .

.. ...

(26)

a , o

in which e the spectral.,,band; width defined by:

, 11

M /m m

with,..in general: for' it -= 0, 1,

S (w ) . . . (28)

The ,s gn icant .waveheight /3 of, the spectrum can

be.'clef i fled by:

- lc2)

mo . . . (29)

1/3

A_ method. to ca 1 cul ate, the drift forces and

nmomeht,il,in rregularwaNies-is -suggested by- Hsu and

Blenkath--,[5] . A more!_genera 1 ;approach is .described

by F'inkgter [6] and lead to the following

expres-sions for. the specere 1, -dens ity SD (w .F) of the low

: frequen-4,!-dri-ft force 'and --Ehe, Mean dr i ft _force D in

' i reg ul a r waves.

-The!spectral -density of the drift force or

-momen-t. in the t i on is

(24)

j

lf

) 8-I (w) ,S

S(+ wlfj

)ID. (w +

o.-; } do> .

The mean drift. force or -moment::i n the: j-difeCtion of the. body fixed system .of: axes

:,' D. = 2 S (w){D.N)/ca}dw J

'

0 J 158 )/ (30) (31)

METHOD TO: CALCULATE THE OF S1.01a11:Y:VARY.

I NG -DRI ET FORCES IN IRREGULAR:WAVES

. . .

of-plat-fort:11s (or any Other

floating objects) in '-the.:"-t 4-m4-domain, computer

simula-tion techniques re used A mathematical model of the

,platforms - add:. of; theen?Pronmeht- must be developed

for this

rpdse .4.

-- 4z, - ---.

especially When the low' f

requen-_ .

ty behay, lour' is -stUdieck,,--the model should include al so

a_ mathematical representation of the low frequency drifti-farce.-p and momentas a function of time On this

- _

subject ,s- papers ..hdvec been published by geyera I authors:

Verhegen L9.1, Piiikster'"[6] , Ndienah' [10]

The mettiOd',described :.here.-',1t, basedl:'; on the assump-tion that the wavespeètrum is sufficiently narrow.

An

far:,Maje, then be wri tieh as

_{, .}

an ampp.,ts.icle:onodi., 1 a tecf

c(t)=-'Reg.(t)'..=.Rd[12.(1)7.ie;cp{ (32)

(33)

The slowly varying drift 'force . Or 4r:orient kn the .. .

horizontal -:p,larie = 1, 6) s' calculated,

- .2 , . '

-the drift .forC6--;, br noméhi prcipoTt td -the square of the -w-aiieenvel 01: R(t)( -:47-<

the,'-cirj force'f-or'=inonidnt. coefficient Cj 1,- 2,

.61) is equi valen-t.to the _drift force or-:_=moment

coefficient in 'reguldr, waveat

the,moMentari?P--f requeney- (t) . ' t=

Thus

the drift force lf 6-- longitudinal

rectIR.6,".

i$2 in traniversey'direction and the drift moment06

a bdu t. the vertical 'ex s: are, reskclt Lvdi

= ipgC.-(;), * with 2 6) :

(34)J

Evidently, i a timet hi story Ofi;A: ea' must be

calcu-lated This is done in the,,Fdl roWing-way:1 ,

The wave envelope R is the modulus *of he' c'ompleic

variable _ Thus 2 2 R

= (Re)

+ ( ImE) Al soi : n 1 7 r -L(ReE) FrnE) (Re) 4-:.( 171E) -= *PReEI..(imE)

-(1m)-(6c)]*.

R -tan .,[1mE/Re]7:1 (36) ImE) (ROE)]

e:-(t) -= wave amp tl'tuder. ,

R(t) =,sloWly:varyi ng, ;wave s:envelope

n (t) =16 cat + e (t),

where', i s. f'eeeluency of

'the narrow bend -sj:;iect rum ,arid

(t)

-I a,-;-s1 owl j;:': veil; Ing phis'

e angle

I t fol lows that-, the -so-clallecPhicliinen'tary Frequency of the spectrum.i s:.

= w ,+ cAt) . . .

. c

It was al ready stated_ that the irregular wave ampl

tude (t). was identical with. ReC(t).

Now c(t) can also be represented4by the sum Of a (large) number of regular waves:

C(t) =.-Ite(t) =

E

n1

=n

A cos(wnt cp-) (38)

in which A calculated as a function of the wave

spectral density -S4' (w according to the well-known

- '

formula

An = /2y6n) ,!,r4wn =.; /2ycon) * (39)

because Aw is taken constant.

is a random phase angle between 0 and 27r radians.

Along the same line, the ima6inary part of is found

to be_ _-

'-N

IM = E

A sin(wt + o)

(40

n=1 n n n

Only the angle w t is time-dependent in these _ex-pression's. The time derivatives of Re t and IK are 'therefore . e -. A w

+)

.r1=1.2:n n n Ini = `E A w'Cos(6 t + .n n n n

In this Way expressions are now available for

R (t) and n(0. At each moment in time four sums must

.be evaluafed for Re, lm, Re and !MC. However,. the

components of iteE aid .1mCare fouricl-directlyLfram

the components of:1K, resp. through.,,mult

ip1y-;rig by -wn, respi. wn. The Overall' computing time Is

reasonable, even case the riUmber"of-CoMPonents.N is large (e .g. around 200)-.

Usually, sUch. a_tirne_historY computedoff-line

and filed on disk: During the simulation, the file is

read back from disk, so that the drift forces and/or.

-moment can be added to the other low frequency

-disturbances (such as wind forces and -moment, etc.):

-CALCULATED 'RESULTS

_

Drift forces, 'according to the method described ' before, are calculated for two types of semisubmersi,-. ble structures. Type 1 is a structure with-two- large-pontoons with 8 ..vertical columns piercing the.water-level while tyPe_2 is a structure with 5_separate big floaters with .5 vertical columns. Figures 3 and 4 show a general view of both tYpes. The Most important data of the platform structures are:

- Type 1

-Type 2

Length m 108.2 _117.

Breadth in 67.4 161.5

-Draught 21:3 2510

Moulded volume m ]"" 19700 - 27400.

Type 2 is according to a ,draft of MARCON,Jhe Hague.

Drift forces and "-moment's- in 'regular Waves are calcu-.

lated for both platform types for the following com-,

,..-binations of wave-ampritudes, wave directions, .

current 'velocit ies 'and -di rect ions:

wave amplitudes

= 1, 2, 3m

wave directions i = 0 and 9.60 degrees

current velocities

Vc = 0.5 (0.5) 2.0, m./S

current 'direction's uc = 0, 90,- 180, and 27.p_.

. . (41)

The, required wave_ induced structure motions in regular' waves 'are calculated- for the free .floating platforms. Some significant.res-ults-ofthe drift. force calcula-tions of both platform. types.are, presented in figures 5 through _17.

_ -

;-Drift moments.have been calculated_too.. However, because of symmetric or nearly symmetric structures with respect to the body fixed axes the drift moment about the Ox3_axls, which is th.e-only important drift moment, was,very small-,for-,the wave- and current

directions conssi here:0 Therefor-e no f igures are

presented for the drift moments.

Fig. 5 shows the transverse drift force for

,,,

various wave ampl itudes.and-two, current velocities Vc = 0 and 1 m/s. Note that-, where the drift forces--at a current velocity Vc = 0 are not proportional with the-wave amplitude squared, the_drift_forces at

Vc = 1-rn/s are,nearly so.' ,

Fig'. ,6 show, the. lon§itudinal ancktransverse

drift force coefficients respectively for_ type- 1-platform at various, current;velocities as_aJunction of frequency of encounter. A positive current velocity means that current and waves traYel with the same advance angles, while a negative current velocity corresponds with,wave and current direction being 1809 ,apart. kpositi,ve drift force- means,, that the force i.s acting' in. the same direction as- the- waves.

Mean drift forces and -moments ih irregular waves are.calculated for two spectra, both using the '..IONSWAP

[11]_forMula: tw_w 12 m/ exp{- } -w ,} y 2a w

_{2ri; -(43)}

with: a = parameter (see [11])

wm = frequency at the maximum of the spectrum y , =ratio...of-the.rnaximum spectral energy to

thiesmaximumof,the corresponding P,ierson .MosIsowitz spectrum

ce' =10:676.(gx/Uj0)70.22 .(44)

-in which:

x fetch

Ulo = wind,velocity at 10-m height...above water surface.

The two spectra .u%ed in ,the,,calculat ions, and

in the figures,,are:

- Pierson-Mosicipwitz : 1:

.,JONSWAP:, = 3,3, which,is, a mean value,'

.recommended to describe the North Sea

spectrum.

.For,both spectra.a fetch hasbeen-chosen of ,x = 480 Ipri,,The.-only parameter'left,,nowjs the wind

velocity U1.

Fig. 8shows-the signifidantwave--heights of both s_pectra.,, asl, a tun_ct i on, of ,wind.,velocity,,.using formulae- (43) and (29).

' ..;

The mean period 71' of both .spectra is calculated from

. .

r/11:37r7.;

. ,.),

. . (45)

For m_.o and m2: see eq. (28).

5'ericl-10 show the longitudinal and trans-verse'drIff- force coefficients in irregular waves as

a. funct i CU &ant. velocity actOrd ng to fb rm. ' (31).

Al so in irregular waves the current is the most i

m-por tent:. parameter, i n: the drift:force calculations Note that: drift, forces can "eVerOpe in the:oppos te

dir'ecti'on as the wive:eUvance4t negat ve Current

velocities::

Fig.%11,and 12 show the spectral- densities of the low frequent loh4itudinal-',aneteansverse drift forces respectively according

:Fronithese.s.pectra of the row frequenCy dens

i-ties,

the=low- freqUencyntotionS can%ble,dalcUlated,

if virtual MaSt,',dentpihg%and .restohingoefficients of

the_sttUctUrelare-

known..

-Figures- 13 through 16 show the' dni ft:forces at

a cur_reht,velocity Vc = I m/s,- coMpariSoh \With

the other environmental. forces . c . ,wind and current

-fortes; The' forces are ,given fUhCtion ofwind

velocity and Beaufort ' scale.

-The wind forces. F,1 are eitimated=witnItYie Of

-F =-46 CA,VL 'A?

wwww

in whith

A%;Li'S'''the- projected WinareaYand--i.S..taken from

r

the

geneeiarrangements_'C

'of"both.r-platformi:

c has been taken" according the regulations of DET

N0RSKt4EillIAS'112]..'

ThecUerent forces have been calculated in the_ . _- .

Same waY:.as theAriftjortes according form. (10) throuqk (0)-. Only forMt(17) 'has' to be 'changed into

the cUrreht.jorce vector-.FL:

.I.

e

77. 6. c _-c Fv,_-c dt .

. (46)

:The tirag.:coefficientS the-dagonal matrix

C-:arethe same-as calculated 'in foem 117), so as a

fUriction.,of .Reynolds 'number and

the=-KeUlegan-Carpen-- tee-n0Mbee..:'The Current forces, .' calculated in this

way' are.:then a function of WavejleIght-and -frequen-Cy-antk-structure motions. However, the current '.forces:given in the figures are theaverage forces

over :the-frequency range and the wind Velocity range,

:anclare.therefciee given a'sconStarit-lifiesin' the

figures (l.3) through (16)

All :figures show that the difference in the

drift forces of both

platforms: is not °Signi f icant.

However, the drift forces calculated with the JONSWAP spectrum are about 3p-kor h46ber than'fileCtrift

' forces taicti lifed 'With the' P ithson-Moskow i 'spectrum.

- at the-seine:wind' Velotity.]

Figure17-shows the results of the time-history calculation of -thele-JOWly Vdryihg 4:11.1ft forces in

irregular waves . for 'botK=IplalfbeW-types

Run dUratiOnwas 'about 20 mm The . first

.three./charil:-represen-v.theXeregUlar wave amplitude 4, the ' Wave: envelOpei squared R and thetildmenfa:1'Y frequency

n. (rad/s)A7:.Bieson=MoskOW44,dpectrunt was :used,

s[ghif

leant wave heighi).6.5'M..:

Channel A-irid 5 represent the longftridinal drift

force-hi end t raniverse -dr ift ftorde-.h2';',TespectiVely,

. for ,Type 1 platform.

Channe1'76 and 7 represent the same for the Type 2 -platfOrM.

160

'was cmputed with iwaves and 4-"il'-rn/S-Cdrret'-from astern ;. 2with .Waves and' -a-1:M/s-tUe'rent- f etim- .

The' computed- time-averaged drift forces ,_{_} 'Di and

are'

2

These' values torrespOndWith the mean :valbes,. .

were found- v ia the 1 ihea tra4. anal yals.

-The recorded time histories lead to the follow-ing comment:

= the wave amPl itude is between 5-6 meters at the, maximum, which corresponds with a,significant

. wave height of '6.5::.meters'; _.

,

the enverope squared' and also the_dri ft forces -show, apart from the large arit-014tude low frequency variations, also relatively high frequency vaeia-.. t ions-_w ih small amplitudes. This is due to the

fact that . the 'real,Wave.;-peC t run,' i nOt. as narrow

as theory requires.

' '

However, those variation's of relatively high

-f requenCy are '-filtered :OUt by the control led plat-form itself,- because of-its large Mass and cor-responding low natural frequency;

thef'momentarY,frequeriCy:ShOws''''.csiites9, whenever R2

approaches zero Th(S.'1,1-however; does not affect 2

_

the drift forces, becaUSe.ortheaMall value C4 '-R .

The "aveeage:momentarry.4eequeriCy.:-.6r. center

fre-quency is. of the order:' of 1.61,7rrad/s,-which cor-responds to a-pieiod of 9-10-seCorldS'.-±,

The records indicate that the time varying- drift forte can be as high as 5 to 6 times the averaged

values. -It-Means -that theie 'fortes are of Such',order,

that they should be taken Into decount.When, the low,

frequency 'behaviour =of -moored or dynamicdlly

positioned platforms is investigated.

CONCLUSIONS

A method to ca lculate_driff forces on seMi7 submerSible structures in waves and 'Clireeht has been

-presented, -,The. result lead 7to,.'.the:fol.lowin.§ .most,

.significant

COhtlasithe:

- Besides the:mass:,transport- velocty of waves,

wave and struCEUTe Mot iorci,-,-.Whith al I are a function of wave height and, -frequency,' the:current velocity

is the most important parameter.jthe.,mean and

slow-ly Varying drift fOrce'cilCUlatiO4:,

- The mean drift, fories=at,-the Weather conditions

con's idergd .heeer.(cureerit velcitity 1 m/s;, Beaufort

8L9) are of the sethe -order as thd.turrent forces and

wind fortes and can nô be neglected when determining

the total envi ronmerital' 'forces.

- ,,- '= ..._

.- Though the 0:1 ag',' coefficients are not cohatani:.

, throughout the WaVe;Oirrod,, the time ii.',e1 eliminated as

. ._ , ,

an: independent= Vartablein the deag Coefficients _

-,

priientea in f7ilarid181.'

--. . ,.... , . : ... Platforri-v_ '-- D'-(kN) . - D2'.(kN) . Type-1---1 - -

437.;,

, :'.--702 -Type 2 ' ---,-,Lt54'. - -.'641 2-

--Because drag coefficients are not available for cy-linders in a constant and a harmonic flow superimposed on it, they are composed from those data which include either the effect of the Reynolds number or the

effect of the Keulegan and Carpenter number. The drag coefficients, therefore, may introduce a certain error in the drift force calculations.

- Model experiments are necessary to determine drag

coefficients of circular and rectangular cylinders in a constant and a harmonic flow superimposed on it. Moreover, model and full scale experiments, in which the influence of the current on the drift forces of semisubmersible structures is investigated, are needed to verify the calculations.

NOMENCLATURE A = projected area Cd = drag coefficient Cm = inertia coefficient D = drift force g = acceleration of gravity W1/1 = significant wave height

k = wave number

R = slowly varying wave envelope = spectral density of waves

SD = spectral density of the low frequent drift force

T = mean wave period Te = period of encounter

t = time

U10 = wind velocity at 10 m height above the water surface

Vc = current velocity

Vm = mass transport velocity in regular waves WA = complex wave amplitude

Wx = complex motion amplitude x = fetch

e = width of the wave spectrum c = wave amplitude

n = momentary frequency

p = wave direction relative to the structure pc = current direction relative to the structure

p = mass density of fluid w = circular wave frequency we = frequency of encounter ACKNOWLEDGEMENT

The authors wish to thank J. Bos for his con-tribution to the development of the time-history calculation method.

REFERENCES

1. Stokes, G.G.: "On the theory of oscillatory

waves", Mathematical and Physical Papers I, Cambridge University Press.

2, Morison, J.R. et al.: "The forces exerted by surface waves on piles", Journ. Petrol. Technol. Am. Inst. Mining 189 (1950).

Hoerner, S.F.: "Fluid dynamic drag", 1965.

Ochi, M.K. and Bolton, W.E.: "Statistics for predicting of ship performance in a seaway",

International Shipbuilding Progress.

Hsu, F.H. and Blankarn, K.A.: "Analysis of peak mooring forces caused by slow vessel drift oscillations in random seas", OTC paper 1159,

1970.

Pinkster, J.A.: "Low frequency phenomena associ-ated with vessels moored at sea", Paper SPE 4837,

Society of Petroleum Engineers of AIME, Spring meeting Amsterdam 1974.

Sarpkaya, T.: "In-line and transverse forces on cylinders in oscillatory flow at high Reynolds numbers", OTC paper 2533, 1976.

Keulegan, G.H. and Carpenter, L.H.: "Forces on cylinders and plates in an oscillating

Journal of the National Bureau of Standards, Vol. 60, No. 5, 1958.

Verhagen, J.G.H. and Van Sluis, M.F.: "The low frequency drifting force on a floating body in waves", Int. Shipbldg. Progress, No. 188, 1970.

Newman, J.N.: "Second order, slowly varying forces on vessels in irregular waves", University College London, April 1974.

Hasselman, K. et al.: "Measurements of wind-wave' Growth and Swell decay during the Joint North Sea Wave Project (JONSWAP)", Deutsches Hydro-graphisches Institut - Hamburg (1973).

Det Norske Veritas: "Rules for the construction and classification of mobile offshore units",

1975.

APPENDIX

Wave motions

To calculate the hydrodynamic forces on a random-ly oriented cylinder element, the wave motions have to be known. If the water is assumed to be an incom-pressible, irrotational and inviscid fluid, the velocity vector xw of the water particles in deep water waves, derived from the velocity potential is:

i cos p ;(' = Re fwe eW exp(iw -i sin p } (48) -w 1 in which:

WA = complex wave amplitude.

At a point (x1, x2, x3) in the fluid with respect to the Ox1x2x3 coordinate system, it can be described

by:

WA = ce exp{i(-kxicos p - kx2sin p) - kx31 . (49) The frequency of encounter, we can be defined by:

We = w{1 + wVccos(p - pc)/g} (50)

in which Vccos(p - pc) is the component of the current velocity in the direction of the waves.

Structure motions

To calculate the hydrodynamic forces on a cylin-der element the wave induced structure motions in six degrees of freedom have to be known.

These six harmonic motions can be calculated from linear (or linearized) motion theory. The six components of the motion vector x will be described

by:

x. = Re{W

exp(iwe . .

.(J = 1.6)

. . . (51)

x.

,

,-.. BODY F I iED AN CYLINDER FIXED. SYSTEMS. OF. AXES'..

_

-eLement

162 _....

- .

4.4 - .

rt-the complex amP I tude

-of-the smotfonH n4 -of-the j=diTeCt ion

4., = real amp tu e- o e, mot ion in the.

j-j, direction

7 Phase 'd i'fference'- of:--t he Mot i n the j

-d rect.; on 'wilh respect- to a harmonic

reference - s:igna 1.

:

' , -

..-If j =

'1'47-2 '',?3;4t Fie7Compoilen t x?= can '-be def i ne'd

i.splacemenr)c;f:,7the4st rutur-e %in the

(ad

ret t

If

are he rotations

of the>,;striictyrelf,a_bouti'the,Q1, Oxj and On axes .

respect:i.141.y1- 6 TO1-4:-.?;- .! t 7.5 ti 'Yr!' 6 . X _f ANGLES EARTH,:,

FIXED 'iY'S.TEM*..axEs:..;O0XY0Z0 AND BODY.F1XED '

STEM OF Aiiii-O'Xi5(2i(j;-5?1;:;.;!:, .=,. P.', Cr5 Sr.:74 15 Eel'

i G'17 4:3TftLAN SVER SE FY-=FOitei;t01-,EFF I C I ENT.rASR-A ifUNCT, I ON OF, FREQUENCY

-coutijp, ,LiidflFoR DIFFER EN CT WAV,E, ArIPL-I tUDES ;. -,-.7.,- 7 :'Ti-"...-3;p,,e,:. '0`ft .1.5---.1"ni-F; m-d 0E-f- .,.,"; - " er87-71.-.; ''. --T , -1,,,,,Tt -:-.r.;::-.f-r. eq ' ..t -i....,,,,.. ..-, . 1;'==

-,...--40 5 5 0 3 3 20 20 7 0 05 10 15 We [Si]

FIG. 9 - LONGITUDINAL DRIFT FORCE IN IRREGULAR WAVES AS A FUNCTION OF

CUR-RENT VELOCITY Vc. 40 Di o. o C', 120 20 -10 _-10, -2 -1 0 1 ' - 2 -2 -1 0 ' 1 2

Current velocity Vw [rnis] _{Current velocity Vc [m/s]}

FIG. 10 - TRANSVERSE DRIFT FORCE IN

IRREGULAR WAVES AS A FUNCTION OF

CUR-RENT VELOCITY VC. SDI Nre.s] 3 1 15 Pierson - Moskowi tz Jonswap 20 30 Winclvelocity Vw [m/s] 11,23.4 5 6 7 8 9 10 .11 Beaufort 11 . 02 03 Wif [11

FIG. 11 - SPECTRAL DENSITY OF LOW FRE-QUENT LONGITUDINAL DRIFT FORCE SD FOR TWO WAVE SPECTRA.

Type 1 tci=1m p. =0" 1.tc =0(4c) p., =1801-Vol Vc .2 m/s Vc=1m/s

F

A

cr -2 m/s v , . . AIIII

lwillr,

Vc= -1 m/s Type 1 Ea.1m pc .50.(40 lic=-90%-VO Vc..2rras Vc..-1m/s Vc= - 2 m/s Vc= 0

II'V

_c...-1m/s - - Pierson-Moskowitz --- Type1 --- Type2 Vw..21m/s .11 A 0. 0. cVc ) .p.c =180' (-Vc ) 1 .../11 Jonswap

rill

4S1!!!!!!!!!

_ Pierson-Moskowitz

-- Type 1

---- Type 2 V w . 21 m/s JJ, =BO. pc='90"(Ve) ki.c.--90*(-Va.) -../

/

./

Z

,

Jonswa Jonswa Pierson-Moskowitz Type'''. ' 2 VC .. 1 m/s Vw= 21 m/s J.1 =Pc=O. ---- Type ----... .---Jonswop ... -: Pierson-Moskowitz

FIG. 6 - LONGITUDINAL DRIFT FORCE FIG. 7"- TRANVERSE DRIFT- FORCE tOEF:-. _FIG. 8=

SIGNiFICANT WAVE HEIGHT AND COEFFICIENT AS A FUNCTION OF FRE- FICIENT AS A FUNCTION OF FREQUENCY _{MEAN PERIOD AS A FUNCTION OF WIND} QUENCY OF ENCOUNTERLJeFOR DIFFER- OF ENCOUNTER WeFOR DIFFERENT CURRENT _{VELOCITY AND BEAUFORT SCALE.}

ENT CURRENT VELOCITIES Vc. VELOCITIES VC.

0 05 10 15

-1,3i[e7.7"..;

-F S_PECTRAL.DENS I TY; OF;LOV-1

.1FRE-'GWENT -,TRANSVERSE DRIFT FORCE S2 FOR TWO ,W'AVE-,,SEECTNN. _ -,,

=

.,.

44,10,, ,'"-->

20--/3D

v-30 T

...;',Wrbi-c,ity.,N,,,[rn./cs].,..71,,,

...,S ITZS ' ,11;-"F-417.V4-'-9ti iorr, II si.14-;16117 ''"'' -- . '

'TT" -;7.T'._,,14.-.4,

,

FIG., ,_-,--pt.:T.73,4'1:f_44--'_--,-=,'-.4.-.;',:._:.:1;<'4; -'" 17 'LONG I;TUD I NAC,ENV I RONMENTA,L' ..

FGROES:,'AS". ...th.lijOiION. OF WIND VELOCITY. ... ,.., . '-'-(TYPE_.2...p.i...AT.O.oRr4).. . _ _{...,._.,} . [ 2,34

557-

§' 1 10 Becki-TO4 . . FIG. 13 liNarrimItiAi,ENVIIONMENTAL FORCES AS ,A,,F UNCTIONA.Fil-wi ND. ,VELOC I Y.

(TYPg.,

3..10

- 20 4,_'... .- . 732E ---. 'Viindyelociti YEil m/5.1. , ...,',.x.,.-: ,' 2 , 3 ,.1. , 5 , 6 ,, 7 ,,,.1 -,. -19 ' ,4.-10_.,..14, -. ---... Bealifert ',: -',.:. - -":.`_,...: 16 TRANSVERSE ENVIRONMENTAL 7. tr:Nr.-').. 72 -Afree/s1' ' - !,Langitydinal drifthnte 5,400,3 lop) ; , , ',-bi,"1-4444ty i-5 1. 81.7 L8 i9 I-10:z Beoufort , . :, -;.2!:-5,',"; ;'-) - -1 ' ' " ..iumkLALLAAJwadAWARIL 6,11441.0NAL _ IS T le in riinUteS .14 ..'"TRAN.SVERSVIENVIFIONMENTAL FORCE'S ,F UN CTION":6F;,iii ND "'VELOC I TY5TY-Fi'E ,,+1

'PLATFORM)-!.4-"

FIG. . .

FORCES AS A FUNCTION, OF WIND VELOCITY Fi6. .17 --TIME HISTORY OF WAVE 13i1TkAND',13NI FT FORCES" .klIR TYPE

'(TYPE 2 PLATFORM) ... . .

'

1 AND TYPE 12 ;r5PCATFORM:sg...::;zt

-, .. -41 " " i::: '.-=,',-.-: -''!Y<.. - .. -Lo :Xc...Current/Ora, icw;Wineioroe

Dr,iftio..-ce - - ; , ,-,-, pjerson-vosxowitz '' -'Type-I" ,,,,,v,` i tym is ''..p ..: pc= 'O'' ; ---=-Jerit',:iir. ,-.1.''.-- . 4 .,: .. - , ' .. ' . --- "

xDJ

. ...?r, .,_:._,,".. ,44- Cu'ilintfOrr.-1",:.,;,-,,,,,-WinEfo ,.-,-_..c.. 4,-4._ Drif ttarp -'''''-4-1`''_ i.731 ,Mosbowitz ...son Type 1_. Vc= 1 m/s, -I" ' J"b '446 4 `Fie _ .. Jobswitp-,-,t. '''' __4-.a.t.-_-.A - .r..7 ',it,

,

c -..i. ./ -1 ,---,: -.---...,.... i -' , -5. 1 mis'. :V..121 m/s '. ',F ..11c ' . Type

--'

TYPE.2 . ...k., r -,..; ,7-op , --..-.7 .- I ,r I.A."-, -. ;.'....-= :',1 .., . , ..: ' ."-,,...-4,; , - -...._. -PiersonF., MPskoWiti''' , , '-- .

....

. .... - -._.-_---Xc.i C _nr X. if gforce - ..Xo=.5;'-'(,4ttsirCe...--.,..'-3---'''''''''ile;sOn'LMbskowi 71;' --"---_ . -V 1 . 'pe2 s,:.. ..-,. . -''-=idnswilp . - . -r'7--' ...-: . .."7;"':- .'..,... .,... _ , 2-r-f-rl'',:'''' -I:::

Fr'

-,,..., _,:i:

Yc._=+Cur rent 1orce

Y,,,=Windloyee

:sec,.:Drit irol-ce

Pierson-Moskow.2 . . Type 2, I , p !!-i-!C ; , 4--1 -, Joniwapi-, 4 r

A

....

_._idiri