Motion and Tether Force Prediction for a Deepwater Tension ...€¦ · Motion and Tether Force Prediction for a Deepwater Tension Leg Platform ... Netherlands Ship Model Basin

OTC 4487

Motion and Tether Force Prediction for a Deepwater Tension Leg Platform by W.C. de Boom, J.A. Pinkster, and S.G. Tan, Netherlands Ship Model Basin

Copyright 1983 Offshore Technology Conference

This paper was presented at the 15th Annual OTC in Houston, Texas. May 2-5, 1983 The copy Is restricted to an abstract of not more than 300 words

ABSTRACT

This paper deals with a comparison between the computed and measured motions and t e t h e r forces f o r a deep water TLP. The computations a r e compared with model experiments which have been conducted i n both r egu l a r and i r r e g u l a r waves i n the Seakeeping Labo- r a to ry of the Netherlands Ship Model Basin.

A t t en t i on is Paid t o the P red i c t i on of the sec- and harmonic components of the t e t h e r forces i n reg- u l a r waves and t o the computation i n the time domain of the combined wave frequency and low frequency mo- t i o n s and t e t h e r forces i n i r r e g u l a r seas . The t i m e domain computations have been performed using a s input the wave record measured i n the basin.

Comparison of r e s u l t s of com~u ta t i ons and model t e s t s al low i d e n t i f i c a t i o n of the major cause of high frequency t e t h e r f o r c e f l u c t u a t i o n s and demon- s t r a t e the a p p l i c a b i l i t y of the convolut ion i n t e g r a l method i n time domain ana ly s i s of low frequency TLP motions.

INTRODUCTION

I n designing a tens ion l eg platform knowledge of t he motion behaviour and t e t h e r fo r ce s is impor- t a n t . I n an e a r l i e r OTC-paper by Tan and De Boom [l] a computation method t o p r e d i c t t he f i r s t o rde r p la t form motions and t e t h e r fo r ce s i n t he frequency and time domain and the mean and low frequency excursions i n the ho r i zon t a l plane i n t he frequency domain has been presented. The computations which were based on l i n e a r p o t e n t i a l theory were c a r r i e d out f o r a TLP opera t ing i n 450 m water depth. Model t e s t s were performed i n r egu l a r and i r r e g u l a r waves i n t he Seakeeping Laboratory of the Netherlands Ship Model Basin t o v a l i d a t e t he computation method.

This c o r r e l a t i o n s tudy presented i n r e f . [ l ] demonstrated the genera l a p p l i c a b i l i t y of the compu- t a t i o n method f o r providing t he above mentioned d a t a in the preliminary design stage, even though some non - l i nea r i t i e s i n t he p la t form behaviour were ~ b - served. It was found t h a t second harmonic components i n t he motions i n the v e r t i c a l plane and i n the re- l a t e d t e t h e r loads cannot be neglected.

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

material is subject to correction by the author Permiss~on to

I n t he present paper a comparison between t h e computed and measured second harmonic components i n the tether forces in waves is presented* Moreover, the 'Omputer program was extended as to generate the time traces of the motions and tether loads the low frequency induced by low frequency second order wave d r i f t forces . The a p p l i c a b i l i t y of t h i s numerical method was shown by P inks t e r and Huijsmans [2] f o r a semi-submersible. A s i m i l a r c o r r e l a t i o n is made i n t he present TLP study.

MAIN PARTICULARS OF THE TENSION LEG PLATFORM

For t h i s c o r r e l a t i o n study a TLP was designed having a r e l a t i v e l y simple shape. De t a i l s - such a s bracings - of which no s i g n i f i c a n t e f f e c t was ex- pected on the hydrodynamic behaviour were l e f t ou t . The r e s u l t i n g p la t form cons is ted of a square deck supported by fou r major c i r c u l a r columns, which were in te rconnected by submerged r ec t angu la r pontoons. V e r t i c a l mooring t e t h e r s were a t tached a t the c e n t r e of each of the corner columns. Figure 1 shows the main dimensions of the TLP, whereas i n Table l the p a r t i c u l a r s of t he loading condi t ion and mooring t e t h e r s a r e l i s t e d .

The s e l e c t i o n of the t o t a l system pre-tension and the of the mooring tethers was explained in [ l ] * The periods

the platform were as '"'ge (sway) period = lo6 Heave period = 2.0 S

(pitch) period = YaW period = 86 S

To s tudy t he a p p l i c a b i l i t y of the computation

method a s described i n t h i s paper and i n r e f [ l ] , experiments were mrried Out with a at a Of 75 in the Seakeeping of the Netherlands Ship Model Basin. A sho r t d e s c r i p t i o n of t h i s l abo ra to ry a s wel l a s of the test p repa ra t i ons Was given in ref [ l]

Model t e s t s were c a r r i e d out i n r egu l a r waves of seven f requencies and t h r ee he igh t s and two ,random long-cres ted seas, approaching f ron two df f- f e r e n t d i r e c t i o n s , 180 and 157.5 degrees. The t e s t

condi t ions a r e o u t l i n e d i n Table 2 i n which t h e wave s p e c t r a l d e n s i t y f u n c t i o n s of t h e random s e a s a r e a l s o shown.

During t h e t e s t s t h e fo l lowing q u a n t i t i e s were measured: - Three t r a n s l a t i o n a l motions a t t h e c e n t r e of t h e

deck, by means of a c o n t a c t l e s s o p t i c a l t rack ing- system.

- Three a n g u l a r motions of t h e TLP, by means of a r o l l / p i t c h gyroscope and a gyroscope f o r yaw.

- Forces i n t h e f o u r mooring t e t h e r s , by means of ring-shaped s t ra in -gauge f o r c e t ransducers .

- R e l a t i v e motion w i t h r e s p e c t t o t h e wave s u r f a c e a t t h r e e l o c a t i o n s , by means of r e s i s t a n c e wi re type wave probes, f i x e d below t h e model deck.

- Wave h e i g h t , by means of r e s i s t a n c e w i r e type wave probe.

The s i g n a l s , which were recorded by a PDP 11/34 d a t a c o l l e c t i n g system were sub jec ted t o a computer a n a l y s i s . The r e g u l a r wave t e s t s i g n a l s were ana- lyzed harmonical ly, y i e l d i n g mean, f i r s t and second harmonic ampli tudes and phase ang les . The i r r e g u l a r s i g n a l s were s u b j e c t e d t o a s t a t i s t i c a l a n a l y s i s y i e l d i n g va lues f o r mean, s tandard d e v i a t i o n , max- imum and s i g n i f i c a n t ampli tudes e t c . Also a f r e - quency domain a n a l y s i s has been a p p l i e d t o t h e ir- r e g u l a r s i g n a l s t o d e r i v e t h e t r a n s f e r f u n c t i o n s .

COMPUTATIONS

Ge~era!: To p r e d i c t t h e behaviour of t h e TLP des ign a s

model t e s t e d a n e x t e n s i v e computat ional program was c a r r i e d o u t u s i n g N.S.M.B.'s computer programs DIFFRAC, DRIFTP and MOORSIM. DIFFRAC is a program based on 3-D p o t e n t i a l theory. This program gener- a t e s f i r s t o r d e r wave loads on t h e TLP, added mass and damping c o e f f i c i e n t s , f i r s t o r d e r motion response f u n c t i o n s and d e t a i l s of t h e flow about t h e body necessary f o r t h e computation of second o r d e r wave loads . The f a c e t schemat iza t ion of t h e under wate r h u l l of t h e p la t fo rm used f o r t h e computations w i t h DIFFRAC is shown i n F igure 2. DRIFTP is a program hic., computes second order wave loads on the body by t h e d i r e c t p r e s s u r e i n t e g r a t i o n method u s i n g d e t a i l e d in format ion on t h e f low genera ted by DIFFRAC. The second o r d e r wave loads a r e computed i n t h e frequency domain i n t h e form of q u a d r a t i c t r a n s f e r f u n c t i o n s . The computations cover both low frequency second o r d e r wave d r i f t f o r c e s and h igh frequency second o r d e r f o r c e s . The program MOORSIM computes t h e motions and mooring f o r c e s on moored v e s s e l s i n i r r e g u l a r waves i n t h e time domain us ing r e t a r d a t i o n f u n c t i o n s t o determine t h e hydrodynamic r e a c t i o n f o r c e s and, f o r t h e p r e s e n t c a s e , convolu t ion i n t e g r a l s t o compute t h e f i r s t o r d e r wave l o a d s and low frequency d r i f t f o r c e s . Non- l i n e a r r e s t o r i n g f o r c e s and v i scous damping e f f e c t s a r e included.

The t h e o r e t i c a l background t o t h e programs DIFFRAC and MOORSIM a r e t r e a t e d i n r e f . [ 3 ] w h i l e the direct integration method fo r second or-er wave d r i f t f o r c e s is d i scussed i n r e f . [ 4 ] .

E_re_suencn-_domaLn--c0-rn~uta_ti0-n~ The computations s t a r t e d with t h e de te rmina t ion

of f i r s t o r d e r wave e x c i t e d f o r c e s on t h e p l a t f o r m and t h e frequency dependent added mass and p o t e n t i a l

378

damping c o e f f i c i e n t s f o r a number of r e g u l a r wave f requenc ies . Then t h e f i r s t o r d e r motions of t h e p l a t f o r m and t h e t e t h e r f o r c e s were d e r i v e d by solv- ing the linearized equations Of motion.

Computations of second o r d e r wave e x c i t e d f o r c e s , a r e based on t h e d i r e c t i n t e g r a t i o n method, s e e r e f . [ 4 ] . This means t h a t t h e s e f o r c e s are found from t h e second o r d e r term i n t h e fo l lowing expres- s i o n f o r t h e hydrodynamic f o r c e : - F = - p.N.dS . . . . . . . . . . . . . . . . ( l )

S Up t o t h e p r e s e n t on ly t h e low frequency wave

d r i f t f o r c e component has been considered. S ince t h e second harmonic component of t h e t e t h e r f o r t e is a l - so der ived from equa t ion (1) we w i l l b r i e f l y d i s c u s s t h e background and d e r i v a t i o n of t h e frequency domain quadratic transfer function*

Following t h e development g iven i n r e f . 141 we o b t a i n t h e fo l lowing express ion f o r t h e second o r d e r wave f o r c e from equa t ion (1):

2 F(') = - 1 jpg5t1) .n d& + d l ) x (M.$')) +

WL g

( 1 ) 2 ( 2 ) M ( 2 ) ) + - !I {-jPIVo 1 - So t dt

. . . . . . . . . . (2)

Equation ( 2 ) can be used t o compute q u a d r a t i c t r a n s - f e r f u n c t i o n f o r t h e low frequency wave d r i f t and t h e second harmonic f o r c e s on t h e body.

To i l l u s t r a t e t h e procedure we view t h e g e n e r a l in Consider the incident

waves as the sum Of a large number Of regular wave :

N S ( t ) = 'i (wit C i ) * (3)

i= l

The s q u a r e of t h e wave e l e v a t i o n record becomes:

N N - e .I} + i 2 ( t ) = e 1 ~ G . s . [ c o s { ( w ~ - u j ) t +

-7 i-l j31 l J

. . . . . . + cos { (wi t w . ) t + ( E ~ + E ~ ) } ] ( 4 ) J

Equation ( 4 ) shows that the square of the wave 'Ontains both difference and sum frequency

A f i r s t o r d e r o s c i l l a t o r y q u a n t i t y U(') such a s a motion o r p r e s s u r e g r a d i e n t can be w r i t t e n a s f o l - lows :

N . . . . ( 5 ) i= l

( 2 ) contains products and Of pro- duc ts of f i r s t o r d e r q u a n t i t i e s . S u b s t i t u t i o n of ex- p r e s s i o n s such a s equa t ion (5) i n equa t ion (2) f i - n a l l y y i e l d s t h e fo l lowing type of equa t ion :

. . . . . . . . . F(')(,) = F(')-(t) + F('It(t) (6)

where:

N N F(')'(t) C .Z L . C . P- COS { (U -u ) t + (Ei-Ej)}

i-l j-1 i j i j

N N + 2 C C . Q ; ~ s i n { (U a . ) L + ( ~ ~ - 5 ~ ) )

i=1 j=1 " i J

. . . . . . . . . . (7) + N N

F(') ( t ) = c 'OS i(ldiwj)t+('itc_j)l i=l j=l i j i j

N N + C C C Q+ s in{(w+w ) ~ + ( E + E ) }

i=1 j=1 i j i j i j 4 - j

. . . . . . . . . . (8)

Equation (7 ) g ives t he low frequency wave d r i f t f o r ce while equat ion (8) g ives the -sum -freq$ency secon$ order force . The c o e f f i c i e n t s P. . , Q P and Q are quadratic transfer func t$~ns i&ichijare deriv&d from equation ( 2 ) , Comparison of- equations (7) and (8) wi th equat ion (4) shows t h a t Pi and Q give the inhphase and quadrature of tkd low frequency fo rce r e l a t i v e t o the low frequen-

a r t of the square of the wave e l eva t i on while 'and Q g ive t he corresponding sum frequency

Z ~ ~ p o n e n t i ! As an example showing the form of P and Q , t he r e s u l t s f o r the cont r ibut ions due t o t he f i r s t pa r t of the right-hand s i d e of equation (2) a r e given: -

P = J (R).C: ( R ) . i j ~ m i j

cos { E ~ (R)} nl(R).dR . . . . . . ( 9 ) i j

- = - t ~ g 6 : (RI.6: (E).

i j s i n (R)} nl(R).dR (10) . . . . .

i j

cos (R)+Er (R)} nl(R).dR . . . . . . (11) i j

e qij = - tpgc; ( % ) . C ' (L ) .

I WL i s i n (&)+cr (R)} nl(R)dR . . . . . . (12)

i j

i n which the s u b s c r i p t I ind i ca t e s t h a t equations ( 9 ) through (12) g ive the con t r i bu t ion due t o the f i r s t p a r t of the right-hand s i d e of equation (2) . Expressions can a l s o be given f o r o the r contr ibu- t i ons due t o t he remaining p a r t s of equation (2) . This w i l l not be t r e a t e d f u r t h e r here except f o r t he l a s t p a r t o qua t ion (2) involving the second order p o t e n t i a l (t2?. This is a complicated con t r i bu t ion involving the s o l u t i o n of p o t e n t i a l s with a non- l i n e a r f r e e su r f ace boundary condi t ion , s e e r e f . [4] . For the p r e sen t , t h i s con t r i bu t i on is approximated by assuming t h a t the dominant e f f e c t i n t h i s con t r i bu t i on is due t o the d i f f r a c t i o n , by the body, of t he second o rde r p o t e n t i a l a s soc i a t ed with t he undisturbed incoming waves. Furthermore, it is as- sumed t h a t t he second o rde r fo r ce a s soc i a t ed with sum o r d i f f e r ence f requencies i n the incoming waves can be determined by t ransformat ion of the f i r s t o rder wave loads found i n waves with wave numbers

379

equal t o sum re spec t i ve ly t o d i f f e r ence wave numbers of the incoming waves, s e e r e f . [ 41 .

The a p p l i c a b i l i t y of t h i s approach f o r sum f re- quency fo rce components due t o the second o rde r po- t e n t i a l i s subs t an t i a t ed t o a l a r g e degree by con- c lu s ions reached i n r e f . [5 ] regarding the r e l a t i v e importance of sum frequency second o rde r wave f o r c e components on a v e r t i c a l cyc l inder .

The values of the various con t r i bu t ions t o the sum and d i f f e r ence frequency quad ra t i c t r a n s f e r func t ions were computed using N.S.W.B.'s three-di- mensional d i f f r a c t i o n program DIFFRAC and the asso- c i a t e d post-processing program DRIFTP.

G~meurarlons-in-rlme--i~m_aI~ Time domain computations were c a r r i e d out of

t he platform motions and t e t h e r fo r ce s i n i r r e g u l a r waves under the inf luence of f i r s t o rde r wave load and low frequency second order d r i f t forces . The e f - f e c t s of second order sum frequency wave loads were not included i n the time domain computaions. S imi la r techniques a s appl ied f o r the second o rde r low f r e - quency wave d r i f t can, however, be used 'Or sum frequency second order forces'

Time domain s imula t ions were c a r r i e d ou t using t he program MOORSIM which so lves t he equat ions of motion using a s input s t o r e d records of the wave loads. These loads inc lude both f i r s t o rde r and low frequency second order components.

F i r s t and second order wave loads were computed f o r the undisturbed i r r e g u l a r wave t r a i n s measured i n t he bas in u s i n t he func t i ona l polynomial approach, s e e r e f . [ 6 f :

F ( t ) = + . . . . . . . . . . . . . . (13)

with: -P . . . . . . . . . F(1) = I ~ ( t - r ) g ( l ) ( r ) dr (14)

4

$" $"

F(2) , I I ~ ( t - r , ) r ( t - r ~ ) g ( ~ ) ( ~ ~ , T ~ ) d~ dr i m l m

1 2 . . . . . . . . . ( l 5 )

i n which r ;( t ) is the measured wave e l eva t i on record. The f i r s t and second order kerne ls a r e found from:

a

8(1)(T) = X I f (1) . . . . . . . 1 e - i ~ ~ dw (16) 0

W m

g ( 2 ) ( ~ l , T ~ ) = - l l 2n2 0 0

-i(w T -u T ) e dui du2 . . . (17)

I n equation (16) and (17) .€( ' ) (U) and ~ ( ~ ) ( o ~ , u ~ ) a r e complex f i r s t r e spec t i ve ly second order t r a n s f e r func t ions obtained based on DIFFRAC and DRIFTP computa t ions .

The convolut ions a r e computed f o r a l l s i x fo r ce f o r the time span Of the wave record

and stored*

I n t he s imula t ions computation t he fol lowing coupled equations of motion a s formulated by Cumins [7 ] a r e solved:

f

6 t

+ t j ) X j + J % j ( t - ~ ) - g j ( ~ ) d ~ + j= 1 -00

+ Ckj.X Fk ( t ) . . . . . . . . . . . . . . . . (18)

The frequency independent coe f f i c i en t s of iner - t i a and the r e t a rda t ion functions can be computed from the ve loc i ty po ten t i a l . By s u b s t i t u t i o n of a harmonic motion i n t o the time domain equations and comparison with t he frequency domain equations Ogi lv ie [8] has derived the r e l a t i onsh ip between the time domain and frequency domain quan t i t i e s :

m 1

ak j = %j - o I Rkj(t) s i n u t d t . . . . b (19) 0

00

b k j = , i % j ( t ) c o s o t d t . . . . . . (20)

From these r e l a t i onsh ips the r e t a rda t ion func- t i o n and the matr ix of added i n e r t i a c o e f f i c i e n t may be found by inverse Pourier-transformation:

00

2 Rkj = F I b (U) cos u t h . (21)

k j m 1

= ' %j(T) Sin W ' T . . (22) 0

Once the system of coupled d i f f e r e n t i a l equa- t i ons is obtained, an a r b i t r a r i l y i n time varying loading such as wave exci ted forces , cur rent forces , non-potential f l u i d r eac t ive forces and non-linear mooring forces may be incorporated a s ex t e rna l forde cont r ibut ions . I n t h i s case the ex t e rna l loads a r e given by equation (13).

The solution may be approximated by numerical methods such a s the f i n i t e difference technique. Knowing the displacement and i ts t h e . de r iva t ives until a c e r t a i n time, the s imu la t ion may be contin- ued with a small time s t ep predic t ing the ve loc i ty from the acce l e ra t i on and the known time h i s t o r i e s by use of time s e r i e s expansions. The new pos i t i on may then be predic ted by numerical i n t eg ra t ion of v e l o c i t i e s .

Results of the time domain simulat ions a r e t he time records of the s i x motions of t he TLP and the forces i n t he t e the r s . Since the time domain simula- t i ons were ca r r i ed out using the t im record of the same wave a s used for the model t e s t s , r e su l t s of computations and measurements can be compared d i - rec t l y a s time records.

Tether fo rces i n frequency domain

I n t h i s paper some r e s u l t s of computations of the sum frequency second order forces a r e presented. These a r e given f o r the case t h a t the incoming waves a r e regular with a frequency u i , amplitude 5 i and phase angle Ei. I n such cases the sum frequency wave fo rce of equation (8) reduces to :

2 + + 5i Qii sin 2(u i + - E i ) (23)

which shows t h a t the sum frequency i s i n f a c t double t he frequency of t he wave (second harmonic). Equa- t i o n (23) may a l s o be w r i t t e n a s :

U ) + 2 + 4- F ( t ) Ci Tii c0"2(uit + Ei) + E . .} (24) I J

+ Tii = Amplitude of the sum quadra t ic t r a n s f e r funct ion

+2 2 3 = {.Pii + Qii} . . . . . . . . . . . . . . (25)

E+ = phase angle i j +

Qii - - a r c t a n (-1 . . . . . . . . . . . . . (26) + I n t h i s paper r e s h t s w i l l be given on the sum f r e - quency components of che t e t h e r f o r t e s i n r egu la r waves. These were computed based on the frequency domain so lu t ion of t he l i nea r i zed equations of mo- t i o n f o r the case t h a t the R P is exc i t ed by the second order sum frequency wave forces . The equation of motion included the ( l i nea r i zed ) t e t h e r r e s to r ing f o r c e s . P o r t h e s u m f r e q u e n c y v a l u e s h i g h e r t h ~ t 1.1 rad/s , t he added mass and damping c o e f f i c i e n t s of the TLP were estimated. From the s o l u t i o n of t he equations of motion the response funct ions of the t e t h e r forces could be obtained. Since the sum f r e - quency wave exc i t i ng forces were described by qua- d r a t i c t r a n s f e r functions and the equations of no- t i o n were l i n e a r , the t e t h e r forces could a l s o be described i n terms of quadra t ic t r a n s f e r funct ions .

COMPARISON OF MODEL TEST RESULTS AND COMPUTATIONS

Gepera_l

I n r e f . [ l] frequency response funct ions of the f i r s t order motions and t e t h e r fo rces obtained from model t e s t s have been compared with computed respon- s e funct ions . These t r a n s f e r functions compared w e l l a l though f o r t he motions i n the v e r t i c a l lane and

the tether solUe non-1ineari- t i e s and high frequency ComPonentS were observed.

Figure 3 shows some examples of the computed and measured amplitude response functions.

A good s i m i l a r i t y was a l s o found between the computed and measured response funct ion of t he mean wave d r i f t forces i n surge, sway and yaw d i r e c t i o n , s ee r e f . [ l ] .

The low frequency motion response, which was only est imated On bas i s of frequency domain consi- de ra t ions i n ref. [ l ] can i n more d e t a i l be judged

the present time domain work*

Time domain records

I n the Figures 4 and 5 time t r aces a r e shown of measured and computed motions i n the ho r i zon ta l plane and t e t h e r forces . The motions i n the v e r t i c a l plane have been computed, because of t h e i r importance f o r the t e t h e r forces , but they a r e not presented s i n c e t h e i r magnitude is so smal l t h a t comparison with measured motions is hardly possible.

As explained i n the previous s e c t i o n "Computa- t ions" the computed time t r aces a r e generated by means of computed records of both f i r s t and second o rde r wave loads.

Figure 4 concerns the r e s u l t s f o r wave spectrum I ( s i g n i f i c a n t wave height = 9.8 m, mean period = 13.7 S ) approaching from ahead. Figure 5 shows the records f o r the same sea s t a t e but approaching form 157.5 degrees, which y i e ld s an asymmetric loading

f o r the TLP.

For the low frequency dominated ho r i zon ta l mo- t i ons the low frequency damping desc r ip t ion is an important item.

I n the program MOORSIM a non-potential f l u i d damping fo rce has been incorporated a s an ex t e rna l fo rce cont r ibut ion . This damping force has been ob- ta ined from mul t ip l i ca t ion of a non-linear damping c o e f f i c i e n t times the instantaneous ve loc i ty squared.

The damping c o e f f i c i e n t , which has been estimated from:

has been checked with the r e s u l t s of a decay t e s t with t he model of the TLP.

The records i n Figure 4 show a reasonable good comparison between the measured and computed surge motion. The shape of the time t r aces is very s i m i - l a r . The low frequency extremes, however a r e s l i g h t - l y underestimated by the computations.

Although the magnitude of t he low frequency surge is s e n s i t i v e t o the applied non-linear damping f a c t o r , a cons iderable reduction of t h i s f a c t o r y i e ld s computed low frequency amplitudes which a r e s t i l l somewhat smal ler than the measured ones. This leads t o the conclusion t h a t the second order low frequency wave forces remain s l i g h t l y below the measured d r i f t forces . The same e f f e c t has been observed i n - a - s i m i l a r s tudy f o r a semi-submersible, s e e ref . [ 2 ] . The s i m i l a r i t y between the measured and computed t e t h e r forces is q u i t e good.

t h e p i t ch motions of the TLP and not with such components i n the heave motions. The computed r e s u l t s a r e dominated by the f i r s t component of equation (2) which the cont r ibut ion of the r e l a t i v e wave eleva- t i o n around the water l ine t o t he wave exc i t i ng forces t he TLP.

The computed amplitudes of the t e t h e r forces show rapid v a r i a t i o n with the second harmonic f r c - quency. This is i n p a r t due to the va r i a t i ons i n the second harmonic wave exc i t i ng forces and i n p a r t due t o the frequency response c h a r a c t e r i s t i c s of t he TLP when exci ted by the second harmonic forces . The measured values of the forces i n t e t h e r s No. 1 and No. 2 show a l a r g e r s c a t t e r than the corresponding d a t a f o r t e the r s No. 3 and No.4. No c l e a r reason can be given f o r t h i s phenomenon a t t h i s time. I n genera l , t he computed and measured da t a show b e t t e r agreement f o r t e t h e r s No. 1 and No. 2 than f o r t e t h e r s No. 3 and No. 4. I n the l a t t e r case, the measured da t a a r e genera l ly somewhat higher than the computed values while i n the f i r s t case the d i f f e r ences tend to even out over the frequency range. There a r e s eve ra l pos- s i b l e causes f o r the d i f f e r ences between measurements and computations. Where poss ib le , es t imates have been made of such e f f e c t s . A b r i e f review w i l l be given here. Since the TLP is a t tached t o the s ea f l o o r by t e the r s of almost constant length , t he ves- s e l descr ibes a sphe r i ca l path. When the ves se l i s o s c i l l a t i n g i n the surge mode i n regular head seas , t h i s r e s u l t s i n second harmonic c e n t r i f u g a l fo rces which form pa r t of the t e t h e r loads. From the measured surge motions, these loads were est imated t o account f o r l e s s than 5% of t he t o t a l harmonic loads. The regular wave generated i n the bas in con- t a i n s second harmonic components which a r e p a r t l y

The same remarks as made fo r the case in head seas a r e va l id f o r the records of the wave condi t ion from a heading of 157,5 degrees, see Figure It appears t h a t the yaw motion p red ic t i on is l e s s accu- r a t e than the surge and sway motion computations. The v e r t i c a l s c a l e of the yaw record, however, shows t h a t the asymmetry of t he sub jec t square TLP design i n t he waves approaching from 157.5 degrees is s o small t h a t yaw motions were of minor importance.

Second harmonic t e t h e r fo rces

The case considered concerns the amplitudes of the forces in the f o r e and aft p a i r s of te thers of the TLP in regular waves from 180 degrees (head waves). In t h i s (symmetrical) s i t u a t i o n the computed forces in both forward tethers is equal. The same s i t u a t i o n e x i s t s f o r the a f t t e the r s .

The measured values of the amplitudes of t he sum frequency components of the t e t h e r forces were found by Four ier ana lys i s of the fo rce records measured i n regular wave t e s t s . The quadra t ic t r a n s f e r funct ions T.. were subsequently computed by d iv id ing t h e second %armonic fo rce amplitudes by the square of the f i r s t harmonic of the inc ident regular waves.

The computed and measured values of T+ are shown i n Figure 6 t o a base of the frequency t h e i nc iden t regular waves.

The computed values of the quadra t ic t r a n s f e r fut ict ion a r e almost t he same f o r the f o r e and a f t p a i r s of t e the r s . A de t a i l ed examination of the computed r e s u l t s showed t h a t t he f o r e and a f t second harmonic components a r e almost exac t ly 180 degrees out of phase. This means t h a t t he t e t h e r forces a r e mainly associa ted with second harmonic components i n

38

bound second harmonics a n i p a r t l y f r e e second hari mOnics' The effects the bound harmonic components a r e included i n the t e t h e r fo rce computations' At the waterdepth considered these

Orce contributions were

The e f f e c t of the f r e e second harmonic components i n the incoming waves has been est imated by computing the wave loads on the TLP f o t he t o t a l second harmonic components i n the regular waves on bas i s of l i n e a r wave theory. The e f f e c t on the t e t h e r fo rces due t o these second harmonic fo rces

be such that the in the Figure is affected*

The above mentioned e f f e c t s could be est imated. Other phenomena fo r which estimates can be given a t t h i s time, a r e f o r ins tance , flow sepa ra t ion ef- f e c t s due to v i s c o s i t y , wave and i n e r t i a e f f e c t s on the t e t h e r s , the inf luence of the mean o f f s e t of the TLP due t o the mean second order d r i f t f o r c s . From the r e s u l t s obtained it can, however, be concluded t h a t computations based on p o t e n t i a l theory g ive a f a i r es t imate of the magnitude of the second hamon- i c t e t h e r fo rce components which can be used i n t h e prel iminary design s tage . From the r e s u l t s i t is a l - so concluded t h a t , al though viscous e f f e c t s may play a r o l e i n t he second harmonic Loads and a s a r e s u l t ques t ions a r e r a i s ed regarding s c a l e e f f e c t s i n mo- d e l t e s t s , the q u a l i t y of the c o r r e l a t i o n is such t h a t second harmonic e f f e c t s i n model t e s t s cannot be ignored a s being wholly determined by such s c a l e e f f e c t .

CONCLUSIONS

The good genera l s i m i l a r i t y between the computed and measured time t r aces of the motions i n the

1

hor i zon ta l plane and the t e t h e r forces show the ap- p l i c a b i l i t y of the computation methods i n t he pre- l iminary design s tage .

Some discrepancies a r e found a t t he extreme low frequency motions due to a s l i g h t underestimation of the wave d r i f t f o rces by the computations. Fur ther i nves t iga t ions a r e being made t o determine the phenomena causing these d i f ferences .

The magnitude of t he measured second harmonic fo rce amplitudes i n t he mooring t e the r s of a TLP is predic ted reasonably w e l l by t he presented computation method, based on p o t e n t i a l theory.

NOMENCLATURE

A j = flow obs t ruc t ing a r ea of the s t ruc -

t u r e f o r flow i n j-th d i r e c t i o n

= frequency dependent added mass CO- a k j e f f i c i e n t i n t he k-th mode due t o motion i n t he j-th mode

Bj = quadra t i c damping c o e f f i c i e n t

b k ~ = frequency dependent Sinear damping c o e f f i c i e n t i n the k-th mode due t o motion i n the j-th mode

= matr ix wi th r e s to r ing c o e f f i c i e n t s C k ~

C~ - drag coe f f i c i en t - F = fo rce vec to r

3 2 ) = fo rce i n k-th mode

= second order fo rce vec to r

f ( l ) ( w ) - complex f i r s t order t r a n s f e r funct ion

g ( 2 ) ( t > = f i r s t order impulse response funct ion

g ( 2 ) ( t l , t 2 ) = quadra t ic impulse response funct ion

g - acce l e ra t i on due t o g rav i ty

dR = length element on the wa te r l i ne

= mass matr ix

= frequency independent added mass - N = outward poin t ing normal u n i t vec tor

of a su r f ace element dS, i n an earth- bound system of axes -

n = outward poin t ing normal un i t vec tor of a su r f ace element dS, i n a body- f ixed system of axes

nl(f i) d i r e c t i o n cosine of a length element dL i n longi tudina l d i r e c t i o n -

P i j

P in-phase low frequency pa r t of t he quadra t ic t r a n s f e r funct ion

P+ = in-phase sum frequency pa r t of t h e i j

quadra t ic t r a n s f e r funct ion

P = pressure der ivable from Bernoul l i ' s equation

Q; - quadrature low frequency p a r t of the

quadra t ic t r a n s f e r funct ion +

quadrature sum frequency p a r t of t h e Qij quadra t ic t r a n s f e r funct ion

Rk j = matr ix of r e t a rda t ion funct ions

S instantaneous wetted su r f ace

dS = su r f ace element P

S mean wetted su r f ace

8 2 ) ( ~ l , w 2 ) - complex quadra t ic t r a n s f e r funct ion

'ij = amplitude of t h e quadra t ic t r a n s f e r

funct ion = time

$1) = f i r s t order o s c i l l a t o r y component of g the motions of the cen t r e of g rav i ty

+ l ) = f i r s t order motion of a s e r f a c e e le- ment dS

$1) - f i r s t order angular motion vec tor

E i - phase angle of i - th component:

E = random phase uniformly d i s t r i b u t e d -i

over 0 - 2rr

E r = phase angle of the r e l a t i v e wave ele- i vat ion a t point 9. r e l a t e d t o t he un-

dis turbed wave c r e s t passfng t h e cen t r e of gravi ty

c i = amplitude of t he i - t h r egu la r wave component

F ( t ) = time dependent wave e l e v a t i o n

C; (a> - t r a n s f e r funct ion of the amplitude of i the f i r s t order r e l a t i v e wave eleva-

t i o n a t poin t L i n t he wa te r l i ne

P = s p e c i f i c dens i ty of sea water

T Y T ~ Y T ~ = time s h i f t s

@ ( 1) = f i r s t order ve loc i ty p o t e n t i a l in- cluding con t r ibu t ions from the incoming waves, d i f f r a c t i o n and body motions

,(U = second order d i f f r a c t i o n p o t e n t i a l h )

@ W = second order "undisturbed wave" po-

t e n t i a l

i = frequency of i - t h component

REFERENCES

[ l ] Tan, S.G. and De Boom, W.C. : h he wave induced motions of a tens ion l e g platform i n deep water" Proc. Thi r teenth Annual Off shore Technology Con- ference , Houston 1981, paper No. 4074.

[ 2 ] Pinks ter , J.A. and Huijsmans, R.H.M.: "The low frequency motions of a semi-submersible +n waves", Proc. Conference on Behaviour of Off- shore S t ruc tu re s , Boston 1982.

[ 3 ] Oortmerssen, G, van: "The motions of a moored sh ip i n waves" N.S.M.B. Publ ica t ion No. 510, 1976.

[4] Pinks t e r , J .A. : "Low frequency second order wave exci ted forces on f l o a t i n g s t ruc tures" . N.S.M.B. Publ ica t ion No. 650, 1980.

[ 51 fin-Chu Chen and Hudspeth, R.T. : "Non-linear d i f f r a c t i o n by eigen funct ion expansions". Journal of the waterway, po r t , coas t a l and ocean d iv i s ion , Proceedings ASCE, Vol. 108, No. WW3, August 1982.

[ 6 ] Dalze l l , J.F. : "Application of t h e func t iona l polynomial model t o t he sh ip added r e s i s t ance problem". Eleventh symposium on Naval Hydrodynamics, Uni- v e r s i t y co l lege , London 1976.

pp P P

[ 7 ] Cummins, W.E. : "The impulse response func t ion and sh ip motions". D.T.M.B. Report 1661, Washington D.C. 1962.

[8] Ogi lv ie , T.F. : "Recent progress towards the understanding and p red i c t i on of sh ip mot ioh" . F i f t h Symposium on Naval Hydrodynamics, ,Bergen 1964. 8

l

l 8

l 8 8

i l l

l

8

TABLE 1-PARTICULARS OF TENSION LEG PLATFORM l , l

I 8

l

! l

!

i l

i l l

8

1 !

! l 8

1 i 1 I

l l , 8

l

l

1 !

l

l 8

l l 1 8

l i i l

l

1 l l ,

U n i t

m

m

m

m

t o n s

t o n s

t o n s

m

m

2 ton . sec . /m

2 t0n.sec .m

2 t0n.sec .m

2 t0n . sec .m

m

m

ton/m

ton.m/rad.

Value

86.25

8.44

67.50

35.00

54500

40500

14000

6.00

6.00

4130

8.4*106

8.4*106

1.0*107

38.00

415

8.2*10

l.53tlo8

D e s c r i p t i o n

Spac ing between column c e n t r e

l i n e s

Column r a d i u s

T o t a l p l a t f o r m h e i g h t

O p e r a t i n g d r a f t

Displacement

P l a t f o r m w e i g h t

T o t a l sys tem p r e - t e n s i o n i n

t e t h e r s

Longitudinal m e t a c e n t r i c h e i g h t

T r a n s v e r s e m e t a c e n t r i c h e i g h t

P l a t f o r m mass

R o l l moment o f i n e r t i a

P i t c h moment o f i n e r t i a

Yaw moment o f i n e r t i a

V e r t i c a l p o s i t i o n o f C.O.G.

above k e e l

Length o f t h e mooring t e t h e r s

V e r t i c a l s t i f f n e s s o f t h e

combined t e t h e r s

R o l l / P i t c h r e s t o r i n g f o r c e c o d -

f i c i e n t s d u e t o t e t h e r s

Symbol

D

R

H

T

A

W

P

GM1 .- GMt

M

=X

I Y

= z - KG

L

TS Z

T S +/@

TABLE 2-TEST CONDITIONS

Fig. 2-Facet element schematization for computations.

86.251~1 I

Fig. l-General arrangement of TLP.

Regula r

P e r i o d

T i n

s e c .

6.0

6 . 5

7.5

9 . 0

9 . 0

9.0

11 .O

1 5 . 0

15.0

15 .0

21 .0

waves

Height

2Ca i n

m

4 .0

4 .0

4 ' 0

4.0

6 . 0

8.0

8.0

4.0

8 . 0

1 2 . 0

8.0

I r r e g u l a r s e a s

Des igna t ion

Spectrum I

Spectrum I1

Wave s p e c t r a l d e n s i t y f u n c t i o n s : 150

.. 100

n N

E - 3 U

$A W 50

0 o a5 1 .o 1 5

w(rad/sec)

Mean p e r i o d

F i n

s e c .

13 .7

16.7

Sign . h e i g h t

m 'w1/3 in

9.8

1 8 . l

---- COMPUTED ------m---- MEASURED IN IRREGULAR SEAS - WAVE SPECTRUM I

MEASURED IN IhREGULAR SEAS - WAVE SPECTRUM It O A O MEASURED IN REGULAR WAVES - ASCENDING WAVE AMPLITUDE b INDICATION OF WAVE DIRECTION

SURGE I TETHER FORCE F1 l

TETHER FORCE F1

F1

Fig. 3-Amplitude response functions-computed and measured.

MEASURED h ----- COMPUTED

YAW ( d t g 1

seconds r , , , , , , , , 0 10 ,m

Fig. Sa-Computed and measured time traces of horizontal motions in irregular seas approaching from 157.5'.

MEASURED W - - - - -- - - ---- COMPUTED

seconds ,* . . . 0 I 1m

Fig. 5b-Computed and measured time traces of tether forces in irregular seas approaching from 157.5'.

COMPUTED T , ~ = @ A . F F : o A o MEASURED I N REGULAR WAVES - ASCENDING WAVE AMPLITUDE

=G4 INDICATION OF WAVE DIRECTION

2 W;( r a d / s e c 1

Fig. 6-Computed and measured second harmonic tether forces in regular head waves.

TETHER FORCE

0

F4

8 f \ / -\ I I

U

\

Documents

Motion and Tether Force Prediction for a Deepwater Tension ...€¦ · Motion and Tether Force Prediction for a Deepwater Tension Leg Platform ... Netherlands Ship Model Basin