Differentiation between the lower and upper parts of columns of semi-submersibles for heave response improvement

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Differentiation Between the Lower and Upper Parts of Columns of Semi-Submersibles

for Heave Response Improvement

Mardel de Conti^, Bernardo de Andrade\ and Lothar Birk^

1 University o f Sao Paulo, Av. P r o f Mello Moraes, 2231 - Sao Paulo (SP) - CEP 05508-900 - Brazil. Corresponding author: nibdconti@usp.br

2 University o f New Orleans, USA

Abstract

Semi-submersible platforms are composed basically o f pontoons and columns. Wave forces and motions can be expressed o n the basis o f first principles for such transparent structures. Numerical results through this method, which follows Morison's approach, are here compared, w i t h reasonable agreement, to experimental results i n a flume. This model exhibits design trends i n accordance ivith methods based on a more robust hydrodynamic estimation, namely d i f f r a c t i o n panel methods. The first principles model is understood to be a valuable tool i n the early stages o f design. The approach here employed demonstrates improvement o f heave response through differentiation between the lower and upper parts o f columns.

Keywords

Semisubmersibles; first principles; design

Nomenclature

A : Wave amphtude A„,,: Waterline area

b„: H a l f transversal distance between the centers o f columns bp: H a l f transversal distance between the centers o f pontoons c: Wave celerity

Cad,y.p: Sectional heave added mass coefficient of pontoons

C,4,, p: Heave added mass coefficient o f columns d: H a l f distance between accelerometers d^: Diameter o f the upper part o f columns

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f: Wave excitation frequency <p : Angle o f roll

g: Gravity H : Draft

h.: Total height o f the columns

h,: Height o f the lower part o f t h e columns, f r o m the center o f the p o n t o o n t o the base o f the u p p e r p a r t o f the column

h„: Height of the upper part o f the columns, from the top o f the lower part to the undisturbed free surface

k: Wave number

1: Subscript referring to the lower part of columns m: Mass o f the p l a t f o r m

m^j^j,: Heave added mass X; Wave length

y : Displacement volume o f each column

y : Displacement volume o f each pontoon (0: Wave excitation circular frequency

CO,: First a n n u l l i n g f r e q u e n c y (forces acting o n p o n t o o n s cancel those acting on columns)

cOj: Second annulhng frequency (half wave length equals the transversal distance between p o n t o o n s and c o l u m n s , the vertical force vanishing f o r beam seas)

co„: Natural circular frequency T: Wave period

oxyz: System fixed w i t h respect to earth; axis oy is vertical, pointing upwards w i t h origin on the undisturbed free surface

O X Y Z : System f i x e d w i t h respect to the body, c o i n c i d e n t w i t h o x y z f o r static e q u i l i b r i u m c o n d i t i o n ; axis O X is l o n g i t u d i n a l and O Z transversal; plane O X Y is the v e r t i c a l - l o n g i t u d i n a l plane o f s y m m e t r y ; p l a n e O Y Z is the v e r t i c a l - t r a n s v e r s a l p l a n e o f symmetry

u: Subscript referring to the upper part o f the columns

X- Wave propagation direction; coincident w i t h z f o r beam

waves

yo: Heave w t h respect to oxyz

1 Introduction

A t least three classes o f problems have been addressed c o n c e r n i n g h y d r o s t a t i c s and h y d r o d y n a m i c s o f s e m i -submersibles: a) stationkeeping, either by m o o r i n g lines or b y dynamic p o s i t i o n i n g systems; b) stability under b o t h i n t a c t a n d damaged c o n d i t i o n s ; c) seakeeping, either regarding survivability under extreme waves or o p e r a b i l i t y under typical waves.

T h e present paper refers to the seakeeping p r o b l e m , specifically regarding the c o n c e p t i o n stage o f the design, w h i c h needs i n s i g h t f o r the d e f i n i t i o n o f parameters to satisfy the i n t e n d e d o p e r a t i o n a l behavior.

One o f the pioneering works i n this area is the proposal o f ( M o r i s o n et al, 1950). I t employs a semi-empirical f o r m u l a to estimate the force on a v e r t i c a l fixed pile. The f o r m u l a has t w o terms, one t a k i n g i n t o account i n e r t i a l effects and the other viscous drag effects. I n a m e t h o d k n o w n as h y d r o d y n a m i c s synthesis, ( H o o f t , 1971) c o n s i s t e n t l y applied this f o r m u l a to a semisubmersible geometry free to oscillate under waves, comprised o f elongated elements, compact t h r e e - d i m e n s i o n a l elements and plane areas.

(Babu and Raja, 1987) applied the h y d r o d y n a m i c synthesis m e t h o d to estimate heave, surge and sway m o t i o n s o f a s e m i s u b m e r s i b l e , w i t h n u m e r i c a l r e s u l t s c o m p a r i n g f a v o r a b l y to experimental data. The authors c o n d u c t e d an i n f l u e n c e s t u d y o f the v a r i a t i o n o f s o m e d e s i g n p a r a m e t e r s o n seakeeping b e h a v i o r . T h e p a r a m e t e r s considered were: d r a f t , ratio between c o l u m n s and t o t a l d i s p l a c e m e n t s , n u m b e r o f c o l u m n s p e r p o n t o o n , l o n g i t u d i n a l distance between f o r e a n d a f t c o l u m n s , a n d transversal distance between pontoons. (Clauss et al, 1992) made a generalization o f M o r i s o n ' s approach as a vector equation f o r members o f a r b i t r a r y o r i e n t a t i o n . ( S ö y l e m e z a n d A t l a r , 2 0 0 0 ) p r e s e n t e d a c o m p a r i s o n b e t w e e n M o r i s o n ' s equation and a 2 D panel m e t h o d a p p l i e d t o a semi-submersible. The panel m e t h o d is based o n the Frank close-fit technique (Frank, 1967). The authors c o m m e n t e d that M o r i s o n ' s approach includes the effect o f viscous forces and the effect o f surge exciting force, n o t considered b y the panel m e t h o d . O n the other h a n d , the 2 - D panel m e t h o d takes i n t o account the c o l u m n - h u l l i n t e r a c t i o n effect a n d p o t e n t i a l d a m p i n g , w h i c h is n o t considered b y M o r i s o n ' s approach. (Sweetman et al, 2002) considered the d y n a m i c air-gap between an effective wave surface a n d the b o t t o m o f a semi-submersible deck. The authors assessed the n u m e r i c a l i m p a c t o f m o d e l i n g second-order d i f f r a c t i o n effects b y c o m p a r i n g the statistical b e h a v i o r o f the free surface estimated b y n u m e r i c a l m e t h o d s w i t h e x p e r i m e n t a l results. D i f f r a c t i o n results were p r e d i c t e d b y an i n d u s t r y - s t a n d a r d s t a t e - o f - t h e - a r t c o m p u t e r p r o g r a m w h i c h applied the second-order panel d i f f r a c t i o n t h e o r y . ( B i r k a n d Clauss, 2 0 0 1 , 2002) c o n s i d e r e d t h e parametric o p t i m i z a t i o n design o f semi-submersible h u l l s

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Differentiation Between the Lower and Upper Parts of Columns of Semi-Submersibles for Heave Response Improvement Mardel de Conti, Bernardo de Andrade, and Lothar Birk

described matiiematically o n the basis o f NURBS ( N o n U n i f o r m R a t i o n a l B - S p l i n e S u r f a c e ) . The a p p r o a c h i n v o l v e d : a) f o r m g e n e r a t i o n ; b ) e s t i m a t i o n o f R A O s (response a m p l i t u d e operators) f o r e x c i t i n g forces a n d motions by a panel method, and associated seakeeping merit; c) m o d e l i n g o f operating conditions by irregular seas; d ) o p t i m i z a t i o n o f the d o w n t i m e related to excessive heave.

(Clauss et a l , 2002) argued that heave, p i t c h , r o l l , and air-gap are r e l e v a n t p a r a m e t e r s t o be c o n s i d e r e d f o r semisubmersibles under centenary seas, b u t suggested the i m p l e m e n t a t i o n o f an "accidental l i m i t state" relative to t h e so c a l l e d r o g u e waves, w h i c h w e r e o b s e r v e d t o effectively occur. The authors applied a n u m e r i c a l panel m e t h o d to this latter p r o b l e m , using a p o t e n t i a l f l o w t i m e d o m a i n approach, i n comparison w i t h a frequency d o m a i n approach, and also w i t h experimental results. (Nakada and Suzuki, 2004) considered a semi-submersible geometry f o r a f l o a t i n g a i r p o r t , f o r m u l a t i n g an o p t i m i z a t i o n p r o b l e m w i t h the objective f u n c t i o n based o n risk a n d considering structural weight, strength and aspects related to f u n c t i o n a l i t y such as m o t i o n s and elastic response. The authors adopted f o r analysis a s i m p l i f i e d m o d e l , based o n the m o t i o n equations f o r a rectangular plate. (Chakrabarti et a l , 2007) analyzed a truss p o n t o o n semi-submersible (TPS) w i t h respect to heave and p i t c h m o t i o n s , c o m p a r i n g results obtained t h r o u g h the linear d i f f r a c t i o n theory and t h r o u g h the s i m p l i f i e d M o r i s o n equation.

I n the present w o r k , a specific stage o f a p p l i c a t i o n was considered, namely the c o n c e p t i o n phase o f the design processes. T h e r e f o r e , a s i m p l i f i e d approach was adopted, enabling i n s i g h t regarding the seakeeping behavior w i t h respect to geometry. N u m e r i c a l results o b t a i n e d t h r o u g h this m e t h o d were compared, w i t h reasonable agreement, to experimental results o b t a i n e d f o r regular waves i n a f l u m e . Some i d e n t i f i e d c h a r a c t e r i s t i c s o f the results presented can be c o n v e n i e n t l y taken as part o f t h e design r e q u i r e m e n t s , i n r e l a t i o n t o t h e d e s i g n p a r a m e t e r s obtained t h r o u g h the first principles model here presented.

T h i s analytical-numerical m o d e l exhibits design trends i n a c c o r d a n c e w i t h m e t h o d s based o n m o r e r o b u s t h y d r o d y n a m i c e s t i m a t i o n , n a m e l y d i f f r a c t i o n p a n e l methods. The first p r i n c i p l e s m o d e l is u n d e r s t o o d to be a valuable t o o l i n the early stages o f design, w h i l e the m o r e r o b u s t methods are necessary to be used i n the stages that f o l l o w the c o n c e p t i o n .

A O Y . O Y

2 Conventions and Geometry

Figure 1 shows the system of coordinates here adopted as well as other conventions.

Fig. 1 Systems of coordinates and other conventions

OXYZ is fixed w i t h respect to the body, and oxyz is fixed i v i t h respect to earth.

B o t h systems c o i n c i d e w h e n the b o d y is i n its m e a n e q u i l i b r i u m position.

Heave, sway and roll motions are defined w i t h respect to oxyz. Incompressible irrotational f l o w and in viscid fluid are assumed, characterizing a potential flow. Surface tension effects are ignored. Deep water is assumed.

The f o r m u l a t i o n o f the linear problem o f oscillating body under incident waves is summarized as:

a) Mass c o n s e r v a t i o n e n f o r c e d t h r o u g h o u t the flow ( n o divergence of the velocity field);

b) M o m e n t u m (or energy) conservation enforced throughout the flow;

c) E q u a l i t y o f air and water pressures o n the free surface (atmospheric pressure taken as u n i f o r m and constant); d) N o split or holes on the free surface ( f l o w velocities

tangential to this surface);

e) N o c r o s s - f l o w o n the s o l i d boundaries ( f l o w velocities tangential to these boundaries);

f ) N o energy coming f r o m i n f i n i t y , except f o r the incident waves

g) Deep waters.

Regular harmonic progressive waves are one o f the possible solutions f o r the above problem i n the absence o f oscillating bodies. Incident regular waves are here taken to progress i n the positive ^-direction.

The specifications below regarding the semi-submersible under regular waves do not restrict the application o f the method, and were adopted only f o r convenience:

• Beam waves incidence, w i t h axis % coincident w i t h axis z; • S e m i - s u b m e r s i b l e w i t h f o r e a n d a f t , as w e l l as l a t e r a l

symmetries;

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Columns divided into upper and lower parts, not necessarily aligned and n o t necessarily w i t h the same diameters; Axes o f pontoons in the same transversal position as axes of lower columns;

Lower columns and pontoons w i t h the same diameter; Gravity center coincident w i t h the origin o f OXYZ.

Experiments

Fig. 3 Model displaced from its rest position Tests o f four semi-submersible models w i t h circular cyUndrical

columns and pontoons were conducted i n a wave flume 24 m long, 1 m wide and circa 1 m deep. A l l models had t w o pontoons and f o u r columns. Three models had columns w i t h different diameters i n their upper and lower parts. Table 1 M a i n n o m i n a l dimensions of the semi-submersible

models Model dp (m) d . (ni) 11 (m) h i (m)

K

(m) (m) bo On) bp (111) 1 0.075 0.075 0.200 0.037 0.125 0.550 0.700 0.700 2 0.050 0.075 0.250 0.075 0.150 0.550 0.680 0.720 3 0.050 0.075 0.300 0.075 0.200 0.550 0.700 0.700 4 0.050 0.075 0.300 0.075 0.200 0.550 0.800 0.800 Regular waves were generated using a plunger-type generator and measured by a capacitance wave probe.

Two sets o f experiments were conducted for the models under beam waves:

1) Measurement of forces by strain-gage load cells at port and starboard f o r models 1,2 and 3, w h i c h were fixed w i t h respect to the flume (see figure 2)

The data were digitally registered and analyzed. F r o m the forces measured o n each side o f the p l a t f o r m , the vertical excitation force and the transversal excitation m o m e n t were calculated f r o m the sum and the difference o f the measured signals, respectively. Regarding heave and r o l l motions, the f o l l o w i n g kinematic relations were considered (see figure 3 ) , being'd' the half distance between the accelerometers:

• Acceleration on one board (A): = - ( p - d - i y o C O S ( p - f ( l - c o s ( p ) g

Acceleration on the other board (B): Yj3 = q p d - f y o c o s i p - f - ( l - c o s ( p ) g

Roll (harmonic approximation): <P =

-Heave (harmonic approximation): I 1

Ö1 coscp

| ( Y A + Y B ) - ( l - c o s 9 ) g

Regular wave frequencies were roughly in the range 2.5 to 15.0 rad/sec.

2) Measurement o f motions by strain-gage accelerometers for models 1 to 4, w h i c h were free to oscillate.

Fig. 2 Measurement of forces: model 2

4 IVIethod Based on First Principles

Semi-submersibles are structures composed o f elongated elements, whose geometry favors a relevant seakeeping characteristic, namely a significant transparency w i t h respect to incident sea waves.

C o n s i d e r e d f i r s t l y , are geometries where c o l u m n s a n d p o n t o o n s axes are at the same transversal p o s i t i o n i n each b o a r d o f t h e s t r u c t u r e , a n d w h e r e a l l e l e m e n t s are c y l i n d r i c a l w i t h u n i f o r m cross sections. One m e c h a n i s m t h r o u g h w h i c h a favorable seakeeping behavior is achieved is t h e o p p o s i t e d i r e c t i o n s o f wave f o r c e s a c t i n g o n p o n t o o n s and c o l u m n s .

This can be explained based o n the f u n d a m e n t a l principle

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tliat wave excitation pressures d i m i n i s h exponentially w i t h i m m e r s i o n . O n p o n t o o n s , w h i c h are t o t a l l y i m m e r s e d , hydrodynamic pressures have a higher intensity o n the upper face than o n the lower face. O n columns, w h i c h cross the free surface, hydrodynamic pressure acts only o n the lower face, being therefore opposite to the resulting force on the pontoons.

This argument is applicable, n o t o n l y when the pressures are negative {under wave h o l l o w s ) , b u t also w h e n they are p o s i t i v e ( u n d e r wave crests). A t a s p e c i f i c e x c i t a t i o n frequency, the forces acting on pontoons cancel those acting on columns. This frequency, hereafter referred to as ' f i r s t a n n u l m e n t frequency', is a very special attribute f o r design purposes, and can be expressed i n terms o f the structure parameters as:

No«-dimciisioiK)l Excil.ifioii Force

V p ( l + c , , . J e 2 ' + 2 V , I

V k h . + Ced,y.c 1 = 0

which, w i t h Cjj y p = 1.0 f o r circular cylinder and e'^'''''^s 1.0 , leads to

1 C.d.y.c + ( l + C a d , y , p )

2 V . (1)

Another important mechanism related to seakeeping behavior is the composition o f forces on either side o f the structure. For beam seas, the relevant d i m e n s i o n is the transversal distance between pontoons. W h e n an odd n u m b e r o f half wave lengths equals this distance, there is an annulment o f the vertical wave excitation forces, and the corresponding r o l l wave excitation m o m e n t assumes a local m a x i m u m . When an even number o f half wave lengths equals the distance between p o n t o o n s , the r o l l wave e x c i t a t i o n m o m e n t vanishes, and the corresponding heave force assumes a local m a x i m u m . The smaller frequency leading to vertical forces annulment, named hereafter "second annulment frequency", may be expressed as:

2b = -X

, 2 _ t g

2b (2)

A n example o f a graphic o f vertical forces versus excitation frequency is presented i n figure 4, where numerical results for model 1 are displayed, noting that the geometr>'of this model satisfies the previous assumed characteristic o f columns and pontoons axes being aligned on each board, a n d all elements being cylindrical w i t h u n i f o r m cross sections.

I t is seen that the t o t a l v e r t i c a l f o r c e results f r o m the composition o f the opposite forces o n the pontoons and o n the columns. The annulments that resulted f r o m the summing of effects on opposite boards are present both i n the pontoons' and i n the columns' components o f the force, as well as i n the total vertical force. The first annulment, however, is o n l y present for the total force, since i t results f r o m the composition o f opposed effects on columns and pontoons.

E.xciüilion F o r c e 1.51

0 5 \

J,

6 1 i \ 2 0 / 2i

F i s q i i c i i c y ( H z J

Fig. 4 Vertical exciting forces for model 1 (continuous thin line: columns; dashed line: pontoons; thick line: total)

Heave R . \ 0

H c i v e p e r W a v e . \ i i i p l i t i i d e s 201

1-^

F r e . n i e i i o j f H z l

Fig. S Heave for model 1

The p l a t f o r m m a y b e seen as a harmonic oscillator subjected to the wave excitation. Figure 5 presents the heave m o t i o n response amplitude operator (heave RAO), which results after t h e c r o s s i n g o f the v e r t i c a l e x c i t a t i o n f o r c e a n d the magnification factor. The heave m o t i o n is here considered to be a single degree o f freedom, this approximation being valid for symmetrical platforms w i t h respect to b o t h boards as well as to the fore and aft bodies.

Comparing the vertical excitation force R A O to the heave m o t i o n RAO, one observes that the annulment frequency coincides f o r b o t h operators, as expected. Also, i t can be observed that the first annulment frequency is close to the peak frequency o f the heave RAO, somewhat higher than the peak frequency. This is n o t a coincidence. I n fact, the peak frequency can be approximated by the natural frequency: „ 2 _ P g A „ i

" ' • ^ • " a d . y

Developing expressions for the mass and added mass, one has:

I - l - C , _{' - ( l + C a d . y . p )} 2V„

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= 1 +

-C.d,y.c + (• + <=ad,y,p) 2V,.

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The above relation is always greater than 1, the amount being calibrated by the following design parameters;

Cnd.y.c c.d.y,p.and—.AstypicallyC.,3y^ >1.0,c„dyp > 1 . 0 V

and - ^ ^ ^ 1 . 0 , the first annulment frequency is above the peak frequency f o r just an amount o f the order o f a tenth. One design approach is to make the peak frequency well below the range o f frequencies o f the sea spectra where energy is significant. Another approach is to make the lowest frequency corresponding to 1/100 of the peak energy of the spectrum still be higher than the peak frequency o f the heave RAO.

Considering a Pierson-Moskowitz spectrum (Pierson and Moskowitz, 1964), this frequency w o u l d be approximately:

s 0.624704 • —

where T» is the sea spectrum peak period.

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The relation bet^veen the 1/100 energy frequency o f the sea and the natural heave frequency is:

'^1/100

:(0,624704.2JI)--

2K

_{l + Cad,y,c + (l + Cad,y,p)} 2V, 6)

Once the 1/100 energyfrequency is kept well above the natu-ral frequency, and above the first annulment frequency, one can expect small heave motions.

Corresponding expressions can be written for the cases where the transversal distances between the upper and lower parts o f columns and between the pontoons are not the same, and where the upper and lower diameters of columns are different. The corresponding a d d i t i o n a l design parameters become important i n these cases.

The f o l l o w i n g design parameters are thus seen to play an important role m t h respect to heave m o t i o n f o r beam seas.

V e r t i c a l E x c i l i i i ^ F o r c e

Noo Duiiiailoo.ll W m c i l Force

\

-0 - ^ ? / .' . . . / _ / / f V

\

-0 - ^ ? / .' . . . / _ / / f

\

-0 - ^ ? / .' . . . / _ / / f

\

^ ? / .' . . . / _ / / f

\

•

/

i

1

ƒ $

\l

\

/ ' 1

•

/

i

1

ƒ $ \ / 1 \ / F i c i H i s n c j i H z i

Fig. 6 Vertical exciting force versus frequency - model 1 (experimental: dots; numerical: continuous line)

V e r t i c n ! E x o i l i n g F o r u d

NiHi DiiueinuHi.ll W i l i o i l Fi'ice

2

t s

\

*

\

• \ • • \t •• • •< • -' 0 0 0 3 1 0 I 5 1 Frequency I H z )

V e r t i c a l E x c i t i n g F o r c e .2

\

v,

,,

• 'V \

•

Frcjiieiicy ( H z l

5 Experimental and Numerical

Results

Figures 6 to 8 present the numerical and experimental results for vertical exciting forces versus frequency for models 1 to 3. Firstly, i t is noted that the experimental and numerical results compared well.

In addition:

a) M o d e l 1 has c o l u m n s w i t h no d i s t i n c t i o n between the upper and lower parts, and the transversal positions o f the axes o f the p o n t o o n s and columns are the same at each board. The ratio between the a n n u l l i n g frequency O), and the n a t u r a l f r e q u e n c y ©„ j . c a n t h u s be e s t i m a t e d b y expression (4). I t depends on the ratio between the volumes o f t h e columns and pontoons t ^ ^ ) on the vertical sectional

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added mass coefficient o f the pontoons (c„^,. p) and on the vertical added mass coefficient o f t h e columns (C.j^,,,). For

model 1, the result is ;1.12 : The first annulling frequency has a special i m p o r t a n c e f o r design purposes since i t is only slightly above the natural frequency (12% i n the case o f model 1), and i t is responsible f o r a significant diminishing o f the heave RAO just above its peak. b) M o d e l 2 has d i f f e r e n t transversal distances between the

upper and lower parts of the columns and the center o f the platform. The transversal distance between the lower parts of the columns is equal to that between the pontoons. Due to the increase i n the p r o p o r t i o n o f vertical forces on the columns i n r e l a t i o n to the forces o n the p o n t o o n s , and because o f the action o f the pressure o n the base o f the upper columns, the component o f the force relative to the pontoons never cancels the one relative to the columns. Consequently, there is no first annulling frequency. c) M o d e l 3 has equal transversal distances between the upper

and lower parts o f the columns, as well as the pontoons, and the center o f the p l a t f o r m . Similarly to model 2, there has been an increase i n the p r o p o r t i o n of vertical forces on columns relatively to the pontoons, because o f the action o f the pressure o n the base o f upper c o l u m n s . Yet, the c o m p o n e n t o f the f o r c e relative to the p o n t o o n s does cancel the one relative to the columns i n a first annulling frequency, w h i c h is observed to be 34% above the natural frequency This p r o p o r t i o n increased i n relation to model 1 due to the enlargement of the upper columns w i t h respect to the lower columns.

A l l the above forces were estimated under the assumption o f the inertia regime disregarding the viscous damping terms. Since the wave height is supposed to be o f the order o f centimeters and the transversal typical dimensions of pontoons and columns are o f the order o f tens o f centimeters, the KC n u m b e r is o f the order o f the m a g n i t u d e o f one t e n t h , justifying the adopted procedure.

Figures 9 to 12 present heave R A O f o r models 1 t o 4.

Heave pei W'ave

H e a \ e R A O VnipliIudcJ (Ul) 1.5

"^{j

0 ^

K

F r e i j u e i i c j i H z l

Fig. 9 RAO heave, model I (experimental: dots;numerical: continuous line)

HeJVtf Wiive .^m-lilihltfi (III

F i e t i i i e i i a l H z l

Fig. 10 RAO heave, model 2 (experimental: dots; numerical: continuous line)

H e a v e R . \ 0

He.ive [-.er W a v e .Aluplitiule-i ( m l

: F i e q a e u e y ( H z )

Fig. 11 RAO heave, model 3 (experimental: dots; numerical: continuous line)

Heave |h;r \V:ne .^ii{ili(iUe:< im>

Fie.pk:iie;(Hzl

Fig. 12 RAO heave, model 4 (expenmental: dots; numerical: continuous line)

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There is a significant difference, other than those already noticed between model 1 and the others. Due to the fact that the first annulling frequency is very close to the peak frequency, the humps to the right o f the peak for model 1 are not as m u c h attenuated by the magnification factor as the corresponding humps f o r the other models.

6 Analysis

Figures 13a and 13b demonstrate the vertical exciting forces RAOs f o r Uvo series o f semi-submersibles generated f r o m model 1.

I n the first series, the ratio between the upper and lower heights of the c o l u m n was kept the same f o r all tests (h„/h, = 1.0), while the ratio o f displacements o f the upper and lower parts of the columns varied (Vc,„/Vo.i= 1.0;1.1;1.2;1.3;1.4). Other m a i n dimensions and the t o t a l displacement were kept constant.

In the second series, the ratio between displacements o f the upper and lower parts of the columns was kept the same f o r all the tests (Vc,u / Vc,i= 1.0), and the ratio betvveen heights varied ( h „ / h i = 0.6;0.7;Ó.8;0.9; 1.0). Other m a i n dimensions and the total displacement were kept constant.

Figures 13a (for the first series) and 13b (for the second series) show that the first annulling frequency displaces to the right and, correspondingly, the humps of the exciting force response diminish when:

• f o r the f i r s t series, the u p p e r v o l u m e o f the c o l u m n s becomes increasingly greater than the lower volume; • f o r the second series, the u p p e r h e i g h t o f the c o l u m n s

become increasingly smaller than the l o w e r height, and correspondingly the upper diameter becomes increasingly greater than the lower.

N o n - d i m e n s i o n n l V e r l i c n l E x c i t a t i o n F o r c e

V'ertK'itl li-KCilalion Force

Fig. 13b Comparison of vertical exciting forces for V c u ' V M ^ 1.0 and h / h , = 0.6;0.7;0.8;0.9;1.0 (from thickest to thinnest)

Figures 14 and 15 demonstrate heave RAOs f o r the same t w o series of semi-submersibles generated f r o m model 1. I n these figures, there are also curves f o r three conditions o f irregular seas according to the Pierson-Moskowitz spectrum, f o r 12 sec, 10 sec and 8 sec peak periods, on a scale o f 1:180 (figure 14) and o f 1:150 (figure 15).

The same tendency regarding the displacement o f the first annulling frequency that was observed regarding vertical forces (figure 13) is reproduced regarding heave (figures 14 and 15). However, due to the attenuation effects related to the dynamics of the semi-submersible m o t i o n , only the first h u m p to the right o f the heave RAO peak presents a significant attenuation. Figures 14a and 14b show that the greater the distance between the sea spectrum peak and the heave RAO peak is, the lesser the benefit of either increasing volumes or diminishing heights of the u p p e r p a r t o f t h e columns with respect to the lower one is. This is valid on the scale o f 1:180 for sea spectrum peaks below 12 sec. However, when the spectrum peak increases, or when the scale decreases (see figure 15), it is possible that a non-neglectful energy o f the spectrum w i l l lie at the heave RAO peak region, where the inverse tendency regarding RAO intensities exists in contrast w i t h the first hump region o f the RAO. Therefore, for certain sufficiently small scales i n combinafion w i t h certain sufficiently high spectrum peaks, i t may even be deleterious to increase volumes or d i m i n i s h heights o f the upper part o f the columns w i t h respect to the lower one.

N o n - d i m c i L s i o i m l \"eftical E x c i t a t i o n F o r c e H e a v e R A O \"erTieat n.\cit.i(Kxi F.x

i ///

'///

#

•1 5 A N : ^ ; ^ r / 1 V — Fre.r>flie\"<H2i

Fig. 13a Comparison o f vertical exciting forces for h y h , = 1.0 and Vc,„/Vc.i= 1.0;1.1;1.2;1.3;1.4 (from thinnest to thickest)

Heave per Ware Amplitudes (m)

0.5 1.0 1.5 l^'^^W

Fig. 14a Companson of heave RAOs for h^/h =1.0 and v,„/Vc 1.0;I.I;I.2;1.3;I.4 (from thinnest to thickest). Also shown irregular sea spectra (ordinates multiplied by 100) for 12, 10 and 8 sec peak periods (dashed, dot-dashed, dotted, respectively); scale 1:180

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Differentiation Between tiie Lower and Upper Parts of Columns of Semi-Submersibles for Heave Response Improvement Mardel de Conti, Bernardo de Andrade, and Lothar Birk

Heav^ per Wave Ajiiplituctes (m)

FrequeiKy (Hz)

Fig. 14b Comparison of heave RAOs for

V^jv,r

' ' u * i '

0.6;0.7;0.8;0.9;1.0 (from thickest to thinnest). Also shown irregular sea spectra (ordinates multiplied by

100) for 12, 10 and 8 sec peak periods (dashed, dot-dashed, doUed, respectively); scale 1:180

ilca\-t per Wave Aniplitu.le3 (m)

—^Frequency ( I l z )

Fig. 15a C o m p a r i s o n o f heave R A O s for h^/h, = 1.0 and V „ / V . , i = 1 0 ; 1.1; 1.2; 1.3; 1.4 (from thinnest to thickest). Also shown irregular sea spectra (ordinates multiplied by 100) for 12, 10 and 8 sec peak periods (dashed, dot-dashed, doUed, respectively); scale 1:150

regarding heave response, to increase the displacement ratio and / or to d i m i n i s h the height ratio bet^veen the upper and the lower parts o f columns. For values greater than one o f the ratio between the first annulhng and the 1/100 sea peak energy frequencies, some specific calcidafions must be conducted. Semi-submersibles of a greater size are seen to be favorable f o r differentiating upper and lower parts o f columns.

R a t i a l i e h v e e i i tlie fir-il a i i n u l i n g a n d (he 1 / 1 0 0 B p e e l n i m frequencie-s 1:3 1 0 1 0 8 1:3 1 0 1 0 8 , — — • 1 , I f ' ' • 1

Z j —

, 1 0 4 -1 1 8 0 0 0 Ihspl.KeilKlil rade

Fig. 16a Comparison of ratio between the firs! annulling and the 1/100 sea peak energy frequencies for h / h , = 1.0 and

V,JV,r

1.0;1.1;1.2;1.3;1.4 (from thinnest to thickest). Lines numbered correspond to 12, 10 and 8 sec peak periods; scale 1:180

R a t i o betAveeir the fll^t aruitiling a n d the 1/100 spectnirn freqilenejes

•4

Fig. ISbComparison of heave R A O s for

S/^Jy^,=

1.0 and hyh, = 0.6; 0.7; 0.8; 0.9; 1.0 (from thickest°io thinnest). Also shown irregular sea spectra (ordinates multiplied by 100) for 12, 10 and 8 sec peak periods (dashed, dot-dashed, dotted, respectively); scale 1:150

Figures 16 ( f o r 1:180 scale) and 17 ( f o r 1:150 scale) display a clearer picture about the influence o f the sea spectrum peak frequency value, the displacement o f height ratios between the upper and the lower parts o f columns, as well as the scale in the first a n n u l l i n g detachment f r o m the significant sea energy region. Here, the lower frequency corresponding to 1 /100 of the sea specftum peak energy is considered. For values smaller than one o f t h e ratio between the first annulling and the 1/100 sea peak energy frequencies, i t is always convenient.

Fig. 16b Comparison of ratio between the first annulling and the 1/100 sea peak energy frequencies for V c / V t i ^ 1.0 and h „ / h | = 0.6;0.7;0.8;0.9;1.0 (from thiSest°'to thinnest). Lines numbered correspond to 1 2 , 10 and 8 sec peak penods; scale 1:180

R n t i o b L a

e t w e e n Uie l i R t nmrultng a n d the 1 / 1 0 0 s p e e t n i m fret) u e n c i e s

t , , —4 3 ^ • , ' b ' f > 1 ' — ^ 0 6 1 •> 1 1 1 6 I 0.4 0.4 0 0 Duplaeeoieiit ratio

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R.-itio i K U v e e i ! tlie f i n . ! a i m i i l i i i c anJ tlie 1/IUÜ s p e e t n i i n f r e q u e n c i e s

OA

02

Fig. 17b Comparison of ratio between the first annulling and the 1/100 sea peak energy frequencies for V c u ' V c i ^ 1.0 and h^/h, = 0.6;0.7;0.8;0.9;1.0 (from thickest'to thinnest). L i n e s numbered correspond to 12, 10 and 8 sec peak periods; scale 1:150

7 Conclusions

The following overall conclusions can be drawn:

a) The numerical results based o n first principles applied for seakeepmgbehaviorofsemi-submersibles present reasonable agreement w i t h experimental results obtained i n a wave flume.

b) Some characteristics of the seakeeping responses, particularly i n t e r e s t i n g f o r the semi-submersibles c o n c e p t i o n , are i d e n t i f i e d and related to design parameters t h r o u g h the first principles model.

c) The first principles model is able to p o i n t trends regarding semi-submersible conception compatible to those obtained w i t h m o r e robust seakeeping estimation methods, e. g., d i f f r a c t i o n panel methods.

d) The first principles m o d e l is understood to be a valuable t o o l i n the early stages o f design, while the more robust methods are necessary i n the f o l l o w i n g stages.

Regarding specifically the design characteristics, it was observed that:

e) For values smaller than one o f the ratio between the first a n n u l l i n g and the 1/100 sea peak energy frequencies, i t is always convenient, regarding heave response, to increase the displacement ratio and / or to diminish the height ratio between the upper and the lower parts o f columns. f ) For values greater than one o f the ratio between the first

a n n u l l i n g and the sea peak energy frequencies, some specific calculations must be conducted i n order to reach a decision o n differentiating the upper and lower parts o f columns.

h) Semi-submersibles of a greater size are seen to be favorable for differentiating upper and lower parts o f columns.

8 References

BABU, K . G.; Raja, G. L . V . , 1987, Hydrodynamic Behaviour o f a Semisubmersible [ o n - l i n e ] . I n s t i t u t e o f Electric and Electronic Engineers (IEEE).

B i R K . L . ; Clauss, G. R, 2001 .Automated H u l l Optimisation o f Offshore Structures Based on Rational Seakeeping Criteria. I n : Proceedings o f the Eleventh (2001) I n t e r n a t i o n a l Offshore and Polar Engineering Conference. Stavanger, Noruega, Jun. 17-22.

BIRK, L . ; Clauss, G. R, 2002, Parametric H u l l Design and A u t o m a t e d O p t i m i z a t i o n o f O f f s h o r e S t r u c t u r e s . I n : I . M . A . M . 2002, Greta, Paper-No. 93.

CHAKRABARTI, S. et a l . , 2007, D e s i g n A n a l y s i s o f a Truss Pontoon Semi-Submersible Concept i n Deep Water. Ocean Engineering, v. 34, p. 621-629.

CLAUSS, G . ; L e h m a n n , £ . ; Ö s t e r g a a r d , C , 1992, O f f s h o r e Structures, Vol.1: Conceptual Design and Hydromechanics. Springer-Verlag, Berlin Heidelberg NewYork.

CLAUSS, G . ; Schmittner, C ; Stutz, K . , 2002, T i m e - d o m a i n Investigation o f a Semisubmersible i n Rogue Waves. I n : Proceedings o f the 21st I n t e r n a t i o n a l C o n f e r e n c e o n O f f s h o r e Mechanics and A r c t i c E n g i n e e r i n g ( O M A E ) , Jun., 23-28. Oslo, N o r w a y

FRANK, W , 1967, Oscillation o f Cylinders i n or below the Free Surface o f Deep Fluids. NSRDC Report, N o . 2357. HOOFT, J. P., 1971, A Mathematical M e t h o d o f D e t e r m i n i n g

Hydrodynamically Induced Forces on a Semisubmersible. In: Transactions o f the SNAME.

MORISON, J. R., O'Brien, M . R, Johnson, J. W , Schaaf, S. A , 1950. T h e Forces Exerted by Surface Waves o n Piles. Petroleum Transactions, A I M E , v. 189, p. 149-157. NAKADA, S.; Suzuki, H . , 2004, O p t i m i z a t i o n o f the D y n a m i c

Response o f a Semi-submersible Type Megafloat [on-line]. Institute o f Electric and Electronic Engineers (IEEE).

P i E R S O N Jr, W. J.; Moskowitz, L., A., 1964, Proposed Spectral

F o r m f o r a F u l l y D e v e l o p e d W i n d Seas Based o n the S i m i l a r i t y T h e o r y o f S. A . K i t a i g o r o d s k i i . J o u r n a l o f Geophysical Research, v. 69, n . 24, p. 5181-5190. SOYLELMEZ, M ; Atiar, M . A., 2000, Comparative Study o f Two

Practical Methods for Estimating the Hydrodynamic Loads and M o t i o n s o f a Semi-Submersible. Journal o f Offshore Mechanics and Arctic Engineering, v. 122, p. 57. SWEETMAN, B . ; Winterstein, S. R; M e l i n g , T. S., 2002, Airgap

P r e d i c t i o n f r o m Second-Order D i f f r a c t i o n and Stokes Theory. I n : I n t e r n a t i o n a l Journal o f O f f s h o r e and Polar Engineering (IJOPE), v. 12, n . 3.