OTC 3951

MEAN AND LOW FREQUENCY WAVE FORCES ON SEMI -SUBMERS I BLES

by J.A. Pinkster, Netherlands Ship Model Basin

Ocopyrlght 1981 Offshore Technology Conference This paper was presented at the 13th Annual OTC in Houston, TX, mission to copy 1s restricted to an abstract of not more than 300 words.

ABSTRACT - -

Mean and low frequency wave d r i f t forces on moored s t ructures have been shown t o be of importance with respect t o low frequency motions and peak mooring loads. This paper i s concerned with lxedict ion of these forces on semi-submersible rtype s t ructures by computa- t ions based on three-dimensional potent ia l theory. A discussion i s given of a method t o compute such forces based on d i rec t in tegrat ion of pressure on the wetted par t of the h u l l of a r b i t r a r i l y shaped struc- tu res . Results o l computations of horizontal d r i f t forces on a six-column semi-submersible are compared with model t e s t s i n regular and i r regu la r waves. The mean v e r t i c a l drif t forces on a submerged horizontal. cylinder obtained from model t e s t s are a l so compared t o resu l t s of computations. On the basis of these comparisons it i s concluded t h a t wave d r i f t forces on semi-submersible type s t ructures i n conditions of waves without current can be predicted t o a reasonable degree of accuracy by means of computations based on po ten t ia l theory.

INTRODUCTIOrJ

Stationary vessels f loa t ing or submerged i n ir- regular waves are subjected t o l a rge , f i r s t order, wave forces and moments which are l inear ly proportional t o the wave height and contain the same frequencies as the waves. They are a l so subjected t o small, second order, mean and low frequency wave forces and moments which are proportional t o the square of the wave height. The frequencies of the second order low fre- quency components a re associated with the frequencies of wave groups occurring i n i r regu la r waves. -

The f i r s t order wave forces and moments are the cause of the well known f i r s t order motions with wave frequencies. Due t o the importance of the f i r s t order wave forces and motions they have been subject t o investigation f o r several decades. As a r e s u l t of these investigations, methods have evolved by means of which these may be predicted with a reasonable degree of accuracy f o r many different vessel shapes.

- - References and i l l u s t r a t i o n s a t end of paper.

May 4.7, 1981. The material Is subject to correction by the author. Per-

U

For semi-submersibles which consist of a number of re la t ive ly slender elements such as columns, f l o a t e r s and bracings, computation methods have been developed which determine the hydrodynamic loads on such elements without taking i n t o account interact ion e f fec t s between the elements. For the f i r s t order wave loads and motion problem t h i s has been shown t o give accurate resu l t s . See Hooft [ l ] .

This paper deals with the mean and low frequency second order wave forces act ing on s ta t ionary vessels i n regular and i r regu la r waves i n general and, i n par t i cu la r , with a method t o predict these forces on bas i s of computations.

The importance of the mean and low frequency wave d r i f t forces from the point of view of motion behaviour and mooring loads on vessels moored a t sea has been recognized only within the l a s t few yeass. Verhagen and Van S l u i j s [ 2 ] , Bsu and Blenkasn E33 and Remery and Hennans [ h ] showed t h a t the low frequency components of the wave d r i f t forces i n i r regu la r waves could, even though re la t ive ly small i n magnitude, exci te l a rge amplitude low frequency horizontal mo- t ions i n moored s t ructures . It was shown t h a t i n i r regu la r waves the d r i f t forces contain components with frequencies coinciding with the natural frequen- c ies of the horizontal motions of moored vessels. Combined with the fac t t h a t the damping of low fre- quency horizontal motions of moored s t ructures i s generally very low, t h i s leads t o large amplitude resonant behaviour of the motions. See Fig. 1. Remery and Hermans [4 ] established t h a t the low frequency components i n the d r i f t forces are associated with the frequencies of groups of waves present i n an ir- regular wave t r a i n .

The v e r t i c a l components of the second order forces a re sometimes known as suction forces. This term is generally applied i n connection with t h e mean wave induced v e r t i c a l force and pitching moment act ing on submarine vehicles when hovering or t r ave l l ing near the f ree surface. It i s shown by Bhattaeharyya [ 5 ] tha t i n extreme cases the upward act ing suction force due t o waves can cause a submarine vehicle t o r i s e and broach the surface, thus posing a problem concerning the control of t h e vehicle i n t h e v e r t i c a l plane.

The v e r t i c a l components of the second order wave forces have a l so been connected with the phenomena of t h e steady tilt of semi-submersibles with low i n i t i a l s t a t i c s t a b i l i t y as indicated by Kuo e t a 1 [61. Depen- ding on the frequency of the waves it has been found t h a t the difference i n the suction forces on t h e f l o a t e r s of a semi-submersible can resu l t i n a t i l t i n g moment, which can cause the platform t o tilt towards o r away f romthe oncoming waves. Such e f fec t s a re of importance i n judging the minimum s t a t i c s t a b i l i t y requirements f o r such platforms.

I n order t o determine the behaviour of moored s t ructures i n waves, model t e s t s a re of ten carr ied out. Simulation techniques based on numerical compu- t a t ions are becoming of increasing importance i n the design phase of many f loat ing s t ructures however. For such simulation s tudies accurate numerical data on the behaviour of t h e mean and low frequency wave forces are desirable, so t h a t meanin~ful r e s u l t s can be given regarding t h e systems under-investigation. See f o r instance Van Oortmerssen L71 and Arai e t a1 [81 and Wichers [g] . In order t o produce numerical resu l t s , however, a r e l i a b l e theory must be available on which calculations can be based.

Wahab [ 101 and P i j f e r s and Brink [ 11 1 formulated expressions f o r the d r i f t forces on semi-submersibles based on Morisonts equation. The d s i f t force on the t o t a l s t ructure i s assumed t o be the sum of the forces act ing on elements such as the columns and f l o a t e r s which are determined without taking i n t o account interact ion e f fec t s . Such formulations indicate t h a t

submerged cylinder. For t h e semi-submersible r e s u l t s of model t e s t s i n i r regu la r head waves w i l l a l so be compared with computations. Before t r e a t i n g r e s u l t s of model t e s t s and computations a b r i e f review of the computation method i s given here. A complete review i s given i n r e f . [ 131.

THEORY

The computations are based on po ten t ia l theory. It i s assumed t h a t the f l u i d i s inviscid , incompress- i b l e and i r ro ta t iona l . The f l u i d motions a r e described by a veloci ty po ten t ia l Q:

where :

Q" ) = f i r s t order velocity po ten t ia l from which f i r s t order pressures and forces a re de- rived

Q(2) = second order veloci ty po ten t ia l

E = a small parameter << 1

Both of these potent ia ls may be wri t ten i n the follow- ing form:

Q Q Q @ . . . . . . . . . . . . . ( 2 )

i n which:

Q-- = velocity po ten t ia l of incoming waves

the wave amplitude ra ther than a quadratic function l Qb = veloci ty po ten t ia l due t o the motion of t h e as is predicted by po ten t ia l theory. body

the viscous drag term i n Morisonls equation can r e s u l t i n a mean d r i f t force i n regular waves. Furthermore, the d r i f t force i s shown t o be a cubic function of

Pinkster [ 121 computed the mean and low frequency second order wave d r i f t forces acting on a smi-sub- mersible i n head waves based on po ten t ia l theory

W

ad = veloci ty po ten t ia l due t o d i f f rac t ion of Q on the s ta t ionary body

W

methods which neglect the e f fec t s of viscosity.-Com- parison of r e s u l t s of computations with r e s u l t s of model t e s t s indicated t h a t such methods are capable of giving good quantitative predictions of d r i f t forces on such s t ructures . Since viscous e f fec t s are not included i n po ten t ia l theor$ methods t h i s resu l t indicates t h a t wave d r i f t forces cannot be predicted correct ly by using methods based on Morisonfs equation

In t h i s paper computed and experimental r e s u l t s on the mean d r i f t forces i n regular waves and irreg- u l a r waves on a semi-submersible w i l l be presented. The computations are based on d i rec t in tegrat ion of second order pressure. and force contributions acting on the instantaneous wetted par t of the h u l l of a semi-submersible. The f l u i d motions and pressures are obtained using a three-dimensional di f f ract ion program based on l i n e a r po ten t ia l theory. Results of computations and model t e s t s on mean horizontal drift forces i n regular waves are compared f o r th ree wave direct ions , v iz . : head waves, quartering waves and beam waves.

I n order t o show t h a t drift forces a l so ac t i n the v e r t i c a l di rect ion, r e s u l t s of computations and experiments f o r the same wave directions w i l l a l s o be compared with respect t o the mean v e r t i c a l d r i f t force and t r i m moment i n re@;ulax waves on a horizontal

The second order potent ia ls Q") and @F) are quadrat- W

(If ( 1 ) i c functions of t h e f i r s t order potent ia ls % , 1.11 1 1 1 , and Q,,

In developing-the expressions f o r the wave drift forces and moments use i s made of th ree systems of co-ordinate axes; see Fig. 2. The f i r s t i s a fixed 0-X,-X2-X system with or igin i n the mean f r e e surface, The poten$ials and the l i n e a r motions of the s t ructure are defined re la t ive t o t h i s system. The second i s the body axes G-X,-x2-x3 with or igin i n t h e centre of gravity ( G ) of the s t ructure . The angular motions are defined r e l a t i v e t o t h i s system. The t h i r d i s the G-Xi-XI X' system with or igin i n G but with axes par- a l l e l 20 t h e 0-XI-X2-X3 system. Forces a d moments w i l l be defined re la t ive t o t h e G-X{-%-X> system of axes. This system i s chosen since we are generally interes ted i n , f o r instance, horizontal forces ra ther than i n forces along the body axes which a re contin- ual ly carrying out angular motions.

The hydrodynamic forces are obtained by integra- t i o n of the pressure over the wetted par t of the hull:

i n which:

p = pressure obtainable from Bernoul l i ts equa- t ion

- N = outward pointing normal nit vector of a

surface element dS re la t ive t o the G-Xi-Xi-X; system of axes

- F = force vector

According t o Bernoulli 's equation:

i n which.:

p. = atmospheric pressure

p = mass density of the f l u i d

@ = veloci ty po ten t ia l describing the f l u i d flow; see equation (1)

C ( t ) = a constant depending only on time t

X3 = v e r t i c a l position of point under consid-

erat ion i n 0-XI-X2-X3 system of co-ordi- nates

In t h i s equation p. and ~ ( t ) may be taken equal t o zero without loss of generality.

In order t o determine t h e second order wave d r i f t forces we assume t h a t the s t ructure is carrying out small f i r s t order osci l la tory motions about a mean position. The first order l i n e a r motions of a point on the h u l l of the s t ructure re la t ive t o the 0-X,-X2- X, system i s :

where :

X(') = motion vector of G Q

< ( l ) = angular motion vector with components ( 1 xi ' ) , X(') and x6 5

X = posi t ion vector of the point on the hu l l r e l a t i v e t o t h e body axes G-xl-x2-x3

Since the s t ructure i s a l so carrying out f i r s t order angular motions, the normal vector N of surface ele- ments i s varying re la t ive t o the G-Xi-Xi-Xi system of axes:

where : - n = normal vector re la t ive t o the G-xl-x2-x3

system of axes

Taking i n t o account t h a t a point on the wetted sur- face i s carrying out f i r s t order motions according t o equation ( 5 ) , the pressure i n the point as given by BernoulLils equation ( 4 ) can be expressed i n terms of the pressure pm i n the mean position and the pres- sure gradient :

Substitution of Bernoulli 's equation leads t o the following pressure:

where :

i n which:

pc = constant pa r t of the hydrostatic pressure

Substitution of equations ( g ) , ( I O ) , (1 1 ) and (6) i n equation (3 ) and taking i n t o account t h a t t h e wetted surface S can be subdivided i n a constant part SO and an osc i l l a t ing par t s near the water l i n e (see Fig. 2) f i n a l l y leads t o t h e following expression f o r the second order force vector:

Following a s imilar development the d r i f t moment vec- t o r becomes :

i n which:

<L1) = f i r s t order osc i l l a to ry re la t ive wave elevation a t the water l i n e WL

dR = length element measured along the mean water l i n e

M = mass matrix

I = mass moment of i n e r t i a matrix

Equations (12) and (13) express the forces and moments i n the time domain.

It can be shown t h a t these expressions can be used t o formulate the d r i f t forces and moments i n t h e frequency domain i n terms of quadratic t r ans fe r func- t ions . These functions depend on two frequencies ra ther than on one frequency as i s the case with f i r s t order t rans fe r functions. These t rans fe r functions give the re la t ionship between t h e d r i f t forces and regular wave groups consisting of two reguLas waves with dif ferent frequencies. The quadratic t r ans fe r functions may be used t o determine the mean d r i f t forces and spec t ra l density of the d r i f t forces i n regular waves or t o predict the wave *if% forces i n time domain fo r a rb i t ra ry i r regular waves. Given an i r regu la r wave t r a i n of the following type:

the low frequency par t of the square of the wave ele- vation i s :

2 N N q t ) = C C 25.5. cos{(wi - w . ) t + (g i -E.))

i = l j=l " J J -J (15) . . . . . . . . . .

In t h i s wave t r a i n the wave drift force i s found from:

=

N N = c c 5 . 5 Pij c0s{(ui - W . ) t + (E+ - E.)] +

<=l j=l J J 3 N N

+ C C L - L - Pij sin - ) t + - 1) i = l j=l J 1 3 3 . . . . . . . . . . (16)

i n which:

= in-phase par t of the quadratic = 'ji t rans fe r function f o r the d r i f t

force dependent on the frequencies Wi and w

j - - quadrature par t of the quadratic Qij - -Qji - t r ans fe r function

~ ~ ~ ~ t i ~ ~ (16) shows tha t the second order forces are closely re la ted t o t h e square of the wave elevation given i n equation ( 15 ) . The mean or constant p a r t of the d r i f t forces are found by put t ing wi = w. i n equation ( 16). This becomes : J

2 ( 1 = C P . . . . . . . . . . . (17)

In i r regu la r waves with the spec t ra l density defined as:

2 S5(u)Au C k . . . . . . . . . . . . . . . ( 18)

equation ( I T ) , when wri t ten i n the continuous form, becomes:

m ( 2 ) = 2 1 S5(w) P(u,u) do . (19)

0

where ~ ( w , w ) is the equivalent of Pii.

The low frequency par t of equation (16) may be expressed i n terms of a force spectrum as follows:

m

S ( , ) (U) = 8 $ Sg(u) Si(w+v) T ~ ( u + v , u ) dw F 0 . . . . . . . . . . (20)

where :

2 2 2 T (W+U,U) = P (w+l~,u) -t. Q (w+~,w) . . . . . (21)

i n which ~(w+p,w) i s t h e equivalent of P i j f o r u i - W - = 1-1. The quadratic t r ans fe r f u n c t ~ o n s may a l so be usei to the wave drift forces in the time domain; see r e f . [ Ib] .

I n t h i s paper a t tent ion w i l l be focused on the frequency domain quadratic t r ans fe r functions only. Final evaluation of the quadratic transfer wctions f o r the d r i f t forces i s possible through use of a three-dimensional d i f f rac t ion theory computer program. By means of such computer programs it i s possible t o determine the f i r s t order veloci ty potent ia ls and the

f i r s t order vessel motions from which t h e drift forces a r e then derivable. A description of such computation methods is given i n r e f . [7] and i n r e f . [13].

MODEL TESTS AND COMPUTATIONS

In Table I the main par t i cu la r s of t h e vessels a re given, while t h e body plans are given i n Fig. 3. The semi-submersible i s a conventional six-column, two f l o a t e r design, with c i rcu la r columns and r e c t a - gular f loa te r s . The submerged cylinder i s c i rcu la r i n cross-section with hemispherical ends.

The model t e s t s on t h e semi-submersible were carr ied out i n the Wave and Current Laboratory of the Netherlands Ship Model Basin, Wageningen. This basin measures 60 X 40 X 1 metres. The model t e s t s on t h e submerged cylinder were carr ied out i n the Seakeeping Laboratory of the same i n s t i t u t e . This basin measures 100 X 24 X 2.5 metres.

TESTS I N REGULAR WAVES

For the t e s t s i n regular waves,-the models were res t ra ined by s o f t l i n e a r spring mooring l i n e s which incorporated force transducers t o measure t h e mean d r i f t forces. The spring constants of t h e s o f t mooring system were chosen so t h a t no d3mamic magnification of motions due t o t h e res t ra ining system occurred i n t h e range of wave frequencies tes ted. Tests i n regular waves were carr ied out f o r three wave direct ions , v iz . : head waves ( 1 80 deg . ) , starboard bow quartering waves ( 135 deg . ) and beam waves (90 deg. ) . COMPUTATIONS

In order t o compute the quadratic t r a n s f e r func- t ions f o r mean and low frequency wave drift forces using three-dimensional di f f ract ion theory, the actual shape of the wetted par t of the h u l l i s approximated by a number of plane face t s o r panels. Each panel represents a pulsating source which contributes t o t h e velocity po ten t ia l describing the flow. The water l i n e of the vessel is approximated by a nurnber of l i n e elements. This i s necessary i n order t o e v d u a t e the f i r s t pa r t of the right-hand side of equation (12) and equation ( 13). The facet d i s t r ibu t ion of one s ide of the semi-submersible and the water l i n e d i sc re t i - zation of one column i s shown i n Fig. 4. The facet d i s t r ibu t ion of the cylinder i s shown i n Fig. 5 . Com- putations of the mean d r i f t forces were carr ied out f o r the same wave directions as applied i n the model t e s t s .

RESULTS OF COMPUTATIONS AND MEASUREMENTS I N REGULAR WAVES

The mean longitudinal and transverse d r i f t forces and the mean yaw moment on the semi-submersible are given i n Fig. 6 , 1 and 8. The mean v e r t i c a l d r i f t force and mean trim moment on the cylinder are given i n Fig. 9. A l l r e s u l t s a re given i n non-dimensional. form. The non-dimensional d r i f t forces and moments correspond with t h e non-dimensional form of the qua- d r a t i c t r ans fe r function Pi$. Since both frequencies on which the t rans fe r functions depend a re equal in regular waves, the t rans fe r functions a re f o r t h i s case dependent on one frequency, being i n t h i s case the frequency of the rewlar wave.

9

COMPARISON OF RESULTS OF COMPUTATIONS AND MEASUREMENTS I N REGULAR WAVES P

From the comparison of the resu l t s on the mean d r i f t forces and yaw moment on the semi-submersible, shown i n Figs. 6 through 8, it can be concluded t h a t the d r i f t forces are a lso predicted correctly. With respect t o the r e s u l t s obtained fo r the semi-submers- i b l e it i s noted t h a t the mean d r i f t forces given i n Figs, 6 through 8 show rapid f luctuat ions with the wave frequency fo r a l l wave direct ions . These fluctua- t ions i n the mean d r i f t forces a re re la ted t o in te r - action e f fec t s between the columns. In head waves the r e s u l t s given i n Fig. 6 show a marked seduction i n the mean d r i f t force a t a non-dimensionalwave frequency of 2.2. In beam waves the r e s u l t s given i n Fig. 8 show a similar reduction a t a non-dimensional wave frequency of about 1.8. The wave lengths corre- sponding t o these frequencies fo r the head waves and the beam wave case amount t o 43 m and 62 m respec- t i v e l y f o r the vessel s i ze as given i n Fig, 3. These values are qui te close t o the distance between the columns as measured i n the direct ion of the wave propagation which amount t o 38 m and 60 m respective- ly . I n such cases standing wave e d e c t s occur between the columns.

I n order t o check the quadratic re la t ionship between the mean second order forces and the wave amplitude, experiments with the semi-submersible were carr ied out i n waves with dif ferent amplitudes. In general the quadratic r e l a s o n s h i p between 'the mean forces and the wave amplitude i s conformed with t o a reasonable degree.

The v e r t i c a l d r i f t force and t r i m moment on the submerged horizontal cylinder given i n Fig, 9 show tha t these quant i t ies are a l so predicted correct ly by po ten t ia l theory methods.

In order t o i l l u s t r a t e the importance of t h e several components of the d r i f t forces given i n equa- t ion (12), a breakdown of the mean longitudinal d r i f t force i n head waves on the semi-submersible and the mean v e r t i c a l force on the cylinder i n head and i n beam waves i s given i n Fig. 10 and Fig. 1 1 respec- t ive ly . The numerals i n these figures r e f e r t o the following componen-bs of equation ( 12 ) :

I : F i r s t order r e l a t i v e wave elevation 2

- :pg J C!') .;.a . . . . . . . . . (22) WL

I1 : Pressure drop due t o first order veloci ty

- J - S . . . . . . . . (23)

so 111: Pressure due t o product of gradient of

f i r s t order pressure and f i r s t Order motion

- - p . . . . . . . . (24)

So I V : Contribution due t o products of f i r s t order

angular motions and i n e r t i a forces

= ( l ) ) . . . . . . . . . . . (25) a ( ' ) X ( M . x ~

V : Contribution due t o second order potent ia l$

- 11 t m(2)) .G.& . . . . . . . (26)

So Wt

From the r e s u l t s on the semi-submersible it can be seen t h a t the re la t ive wave elevation contribution I i s dominant. Contribution I1 is directed opposite t o the direct ion of the t o t a l force. Contributions I11 and I V are l e s s important than I and 11. In regular waves contribution V i s always equal t o zero.

Since the cylinder i s f u l l y submerged no conkri- bution a r i s e s from re la t ive wave elevations around a water l i n e . Hence contribution I is zero. Since it is c i rcu la r with the centre of gravity i n the centre of the cylinder no r o l l motions occur. Due t o t h i s e f fec t contribution I V i s zero i n beam waves. The r e s u l t s shown i n Fig. 11 show t h a t i n t h i s case t h e re la t ive importance of the contributions t o t h e t o t a l force can vary qui te s ignif icant ly . In head waves contribu- t i o n I1 is dominant and contributions I11 and IV haye only minor e f fec t on t h e t o t a l . In beam waves, however, contributions I1 and 111 axe of the same order but. opposite i n sign. I n both cases the t o t a l me= force i s directed upwards. The sign of contribution I1 is a l so upwards i n both cases.

TESTS I N IRREGULAR WAVES

One t e s t was carr ied out with the semi-submersible i n i r regu la r head waves. The wave spectrum i s given i n Fig. 12. The purpose of t h i s t e s t was t o meastire di- r e c t l y the mean and low frequency longitudinal wave d r i f t force i n a r e a l i s t i c sea condition. The measured d r i f t force record was analyzed by means of cross-bi- spec t ra l methods (see r e f . [l41 ) which r e s u l t i n e s t i - mates of the quadratic t r ans fe r function Pi; va l id for regular waves and of the value of the amplztude TI2 of the quadratic t r ans fe r function i n regular wa;ve groups consisting of two regular waves with frequen- c ies (31 and w2, These r e s u l t s are compared with r e s u l t s of computations based on po ten t ia l theory.

The measuring system used f o r these t e s t s is de- scribed extensively i n r e f . [l31 and w i l l be l e f t out of consideration here.

The r e s u l t s of computations and measurements are compared i n Fig. 13. The r e s u l t s show t h e value of the quadratic t r a n s f e r function TI2 t o a base of f u l l scale mean frequency of t h e regular wave components fo r the case t h a t w, = w2, which i s equivalent t o the mean force i n regular waves, and f o r the case of wl - w2 = 0.1 rad./sec. , which i s the case t h a t the wave group frequency and hence the frequency of the second order force equals 0.1 rad./sec. ~ L K L scale. The experimental data is denoted by C.B.S. (cross-bi- spec t ra l analysis r e s u l t s obtained from measured data) . In t h e Fig. 13(b) two computed l i n e s a re s denoted by computed (!Total) and computed without mP8Y The l a t t e r computation i s based on equation (12) excluding the contribution due t o t h e second order po ten t ia l s

d 2 ) and md (2 ) . W

Comparison of r e s u l t s of computations and measure- ments shows t h a t a l so r e s u l t s of t e s t s i n r e a l i s t i c i r regu la r waves tend t o confirm t h a t po ten t ia l theory predicts wave dr i f t forces with reasonable accuracy.

\

CONCLUSIONS

The agreement obtained between r e s u l t s of model t e s t s and computations based on po ten t ia l theory indi- cates t h a t viscous e f fec t s on wave d r i f t forces on semi-submersibles a re small i n conditions of waves only. Not only the horizontal d r i f t forces are rea- sonably well predicted using po ten t ia l theory but a l so t h e v e r t i c a l d r i f t forces.

I n conditions of waves combined with current viscous e f fec t s probably do play a ro le with respect t o the mean and low frequency forces however. This i s deduced from r e s u l t s of model t e s t s , which indicate t h a t the t o t a l mean force i n conditions of waves and cuxrent i s not always equal t o the sum of the mean forces found i n waves only and i n current only. Com- putations based on Morisonls equation carr ied out by P i j f e r s and Brink [ l l ] and F e r r e t t i and Berta [l51 confirm t h i s e f fec t i n qual i ta t ive sense. This sug- gests t h a t there a re interact ion e f fec t s between current and waves which, up t o the present, have not been properly quantified. It has been suggested t h a t wave d r i f t forces a re s ignif icant ly changed due t o the presence of current. On the other hand it is a l so possible t h a t the presence of waves dis turbs the cur- rent flow pat tern conskderably re la t ive t o the s t i l l water case, thus leading t o a change i n the current drag. More research w i l l be needed i n order t o clariSy t h i s s i tuat ion.

NOMENCLATURE -

F / ~ ) , &L2), F?) mean second order d r i f t force i n regular waves i n longitudinal, transverse and v e r t i c a l di rect ions respectively as given i n f igures

-(2) M ( ~ ) , d2) mean second order r o l l , t r i m and M1 , 2

yaw moments respectively due t o drift forces i n regular waves

lJ i n f igures , wave direct ion

(1 <a amplitude of a regular wave

V displaced volume of the s t ructure

g acceleration of gravity

'w1/3 s ignif icant wave height - I mean wave period

REFERENCES

1 . Hooft , J.P. : "Hydrodyn&c aspects of semi-sub- mersible platforms", N .S .M.B. Publication No. 400, 1971.

2. Verhagen, J.B.G. and Van S l u i j s , M.F.: "The low frequency d r i f t i n g force on f loa t ing bodies i n waves", Internat ional Shipbuilding Progress, Vol. 17, No. 188, April 1970.

3. Hsu, F.H. and Blenkaxn, K.A.: "Analysis of peak .

mooring forces caused by slow vessel d r i f t os- c i l l a t i o n s i n random seas", Paper No, 1159, OTC, Houston, 1970.

4. Remery, G.F .M. and Hermans, A. J. : "The slow d r i f t osc i l l a t ions of a moored object i n random seas", Paper No. 1500, OTC, Houston, 1971.

5. Bhattacharyya, Rameswhax: "Dynamics of marine vehicles", Wiley Series on Ocean Engineering, 1978.

6. Kuo, C . , Lee, A. , Welya, Y. and Martin, J.: -

"Semi-submersible i n t a c t s t a b i l i t y - s t a t i c and dynamic assessment and steady tilt i n waves", Paper No. 2976, OTC, Houston, 1977.

7. Van Oortmerssen, G.: "The motions of a moored ship i n waves", N.S.M.B. Publication No. 570, 1976.

8. Arai, S., Nakado, Y. and Takagi, M.: "Study on the motion of a moored vessel among t h e irregu- lar waves", J.S.N.A. Japan, Vol.. 140, 1976.

9. Wichers , J.E .W. : "Slowly osc i l l a t ing mooring forces i n s ingle point mooring systems", Paper No. 28, Vol. 3, BOSS 1979, London.

10. Wahab , R. : "Wave induced motions and d r i f t forces on a f loa t ing structure", Report No. 1865, Netherlands Ship Research Centre, TNO, Delf t , 1974.

11. P i j f e r s , J.G.L. and Brink, A.W. : "Calculated d r i f t forces of two semi-submersible platform types i n regular and i r regu la r waves", Paper No. 2977, OTC, Houston, 1977.

12. Pinkster, J .A.: "Mean and low frequency wave d r i f t i n g forces on f loa t ing structures", Ocean Engineering, October 1979.

13. Pinkster, J .A. : "Low frequency second order wave exciting forces on f loa t ing structures", R.S.M.B. Publication No. 650, 1980.

14. Dalzell , J.F.: "Application of the fundamental polynomial model t o t h e ship added res is tance problem", Eleventh Symposium on Naval Hydrody- namics, University College, London, 1976.

15. F e r r e t t i , C. and Berta, M.: "Viscous e f fec t con- t r ibu t ion t o t h e d r i f t forces on a f loa t ing s t ructure" , Internat ional Symposium on Ocean Engineering and Shiphandling, Gothenburg, 1980.

J

TABLE I

* Distance between base and mean still water surface

Main particulars of the semi-submersible and the horizontal submerged cylinder.

Submexged horizontal cylinder

75.60

8.40

12.60~

4,034

75.0

- TIME in sec.

~ ~ ~ ~ ~ r s i b l e

100.00

76.00

20.00

35,925

40.0

Fig. 1 -.Surge motions of a semi-submersible in irregular head waves.

Unit

m

m

m

m

m

Designation

Length (between perpendiculars)

Breadth

Draft

Displacement vo Lume

Water depth

Fig. 2 - Systems of co-ordinate axes.

Symbol

'PP

B

T

V

Wd

SEMI - SUBMERSIBLE CYLINDER

Fig. 3 - The vessels.

FACET SCHEMATlSATlON CYLINDER

TOTAL 286 FACETS

FACET SCHEMATlSATlON SEMI-SUBMERSIBLE

TOTAL 216 FACETS

Fig. 5 - Facet distribution of the submerged horizontal cylinder.

COMPUTED o A MEASURED

WATER LINE SCHEMATlSATlON

TOTAL 72 ELEMENTS

Fig. 4 - Facet and water line element distribution of the semi-submersible (only one floater shown).

Fig. 6 - Mean longitudinal drift force in regular head waves (180~) on the semi-submersible.

- COMPUTED o a o MEASURED

Fig. 7 - Mean longitudinal and transverse drift forces and mean yaw moment on the semi-submersible in regular bow quartering waves (1350).

COMPUTED 0 A a MEASURED

Fig. 8 - Mean gransverse drift force in regular beam waves (90 ) on the semi-submersible.

Fig. 9 - Mean vertical drift force and trim moment on the cylinder in regular head waves, bow quartering waves, and beam waves.

------- I --- II -.-.- m ------------- m TOTAL

CYLINDER

Fig. 11 - Components of the mean vertical drift force in regular head and beam waves on the cylinder.

Fig. 10 - Components of the mean longitudinal drift force in regular waves on the semi- submersible.

W in rad. sec:'

ig. 12 - Spectrum of irregular waves.

-40- COMPUTED

---- FROM C.B.S.

Fig. 13 - Quadratic transfer function of 'che longitudinal drift force on the semi-submersible in regular waves and regular wave groups.