Numerical investigations of hydrodynamic coefficients and hydrodynamic interaction between two floating structures in waves

Numerical investigations of hydrodynamic

coefficients and hydrodynamic interaction

between two floating structures in waves

Dr. I. Dmitrieva

Report No. 1018 December 1994

TU Deift

Faculty of Mechanical engineering and Marine Technology Ship Hydromechanics Laboratoiy

TUDeift

/

NUMERICAL INVESTIGATIONS OF

HYDRODYNAMIC COEFFICIENTS AND HYDRODYNAMIC

INTERACTION BETWEEN TWO FLOATING STRUCTURES

IN WAVES

Dr. I. Dmitrievcz

S hiphydrodynamics Laboratory

DelfE University 0f Technology

CONTENTS

Introduction.

Descrjption of bodies.

Calculation conditions.

Equations of motion of two separated bodies.

Description of dimensions of output data.

Calculations and conclusions.

References.

INTRODUCTION.

One of the versions of the program DELFRAC that is based on the three-dimensional potential theory is investigated for the case when two structures are

floatmg in waves in each other's vicinity The hydrodynamic coefficients of each 'body and 'hydrodynamic interaction coefficients are calculated for several configurations of bodie& Results are obtained for two closely spaced floating bodies free to move

independently The dimensions of bodies are similar Compansons with experimental and theoretical results of other authors are discussed. Söme conclusions about the use of a 3D diffiaction method and especialy the version of DELFRAC program for

multibodies are given.

DESCRIPTION OF BODIES.

In order to analyse the results of computations and especially the hydrodynamic

interaction between two structures, floating in waves, two bodies have been selected (

according to the results 'of computations and experiments, of G. van Oortmerssen

[1 1)

cylinder;.

.

box.

These bodies are shown in figures 1 - 4 , including the panels. In figure 1 .the meshes of two bodies from the bottom side are shown. The different configurations of these bodies are shown in figures 2 through 4.

In the paper of van Oortmerssen theresults are given in non-dimensional' form. In our investigation we have used' the fill scale which is selected' as 1(00 where appropriate computed. Results' have been made non-dimensional' in the same way as

applied by van Oortmerssen.

The first body is a floating cylinder of radius R = 47.9 m. The draft of the cylinder is 30 m The centre of gravity is located at the waterline of the cyhnder The wetted surface of the body is discretized by 98 panels per quadrant or 392 fOr the

whole body. In general, a use is made of

uadrilateral' panels but in' the bottom triangular panels are applied close to the centre of cylinder as can be seen in figure 1

For the calculation of drift forces 7

waterline elements have been applied per quadrant.

The second body is a rectagular barge of 109.7 m length, 101.4 m breadth with a draft of 30 m. The calculation point is at gravity centre located at the waterline .The wetted surface of box is discretized by 108 numbers quadrilateral panels per quadrant or 432 for the whole body. For calculation of the drift forces 11 waterline

elements per quadrant are applied.

The table below shows the number of panels per wave length at different periods for the two investigated bodies.

Number of panels per wave length for two bodies

Tab. 1.

From this table it can be seen that the mesh refinements for the two bodies are satisfactorily. Even for the highest frequency the number of panels per wave length is sufficient in order to calculate the first order loads and motions accordance with existing practice. However, for the accurate calculation of second order loads these numbers of panels for the highest frequency are not enough. For this case smaller panels are preferable.

In figures 2 - 4 the different configurations of the two bodies are shown Calculations were carried out for three distances between bodies. See table 2.

Distances between bodies

Tab. 2.

CALCULATION CONDITIONS.

In our numerical investigation we obtained results for regular waves. In this

case, the amplitude of wave is 1 m. The circular frequencies of oscillation of the

4 Frequency, rad/sec 0.05 0.3 0.5 0.7 0.75 Period , sec 125.6 20.93 12.56 8.971 8.373 Wave length, m 24609.6 683.4 246.1 125.55 109.37 CyL98*4{110.66m1 Box 1O*4 JI 14m] 23086 26923 6411 74.77

2309

2693 11.78 1374 1OE26

lU?

Separation distance, m 50 100 150

Distance between the centres of bodies, m 152.75 202.75 252.75

incident waves w are varied from 0.05 to 0.75 rad/sec corresponding to wave periods of 125.6 to 8.373 seconds. This range corresponds to the range of frequencies applied by van Oortmerssen [1].

The calculations were cam ed out for one wave direction- 180 degree.

All of results were obtained for a water depth of 220 meters and for one draft

of 30 meters.

The program of investigation of bodies'

interaction

consisted of the

comparisons of the following data:

hydrodynamic coefficients (surge and heave added mass and damping coefficients) calculated for single bodies by using the main DELFRAC program;

the same hydrodynamic coefficients obtained by taking into account the interaction effects and using the version of DELFRAC program for multibodies;

hydrodynamic interaction coefficients which are equal to zero for the case of a

single body.

EQUATIONS OF MOTION OF TWO SEPARATED BODIES.

We describe here the equations of motions for the several bodies to illustrate clearly the mechanism of the interaction between the bodies as well as hydrodynamic

interaction coefficients

According to potential theory the equations of motions can be written as follow; first body 6 i=1 ±

](2)

)((l)

(1)

second body 5

6

i=1

+ [-c2D12 -icoE2

(1)

( M2 + A

(2)) _{- iO) B} (2) ₊

]E (2) +

where the iñertia matrix of body;

A -the dimensional added mass,

B1 - the dimensional damping coefficient;

D - in-phase hydrodynamic interaction coefficients, yielding a force in i-mode due to motion in the j-mode of a neighbour structure;

-out-of-phase hydrodynamic interaction coefficients, yielding a force in

i-mode due to motion in the j-mode of a neighbour structure; X - wave exciting. force;

- complex amplitude of displacement in the j-mode.

The affix (1) or (2) serves to identify the bodies, while the subscripts i and j denote the modes of motion

It is obvious that in the case of single body's motions the coefficients P and E equal to zero. For the case of multybody's motions the symmetry relationships exist, i.e.

(1) (2)

¡J JI ,

= (2)

'J JI

DESCRIPTIONS OF DIMENSIONS OF OUTPUT DATA.

Output files öf all versions of DELFRAC contain the dimensional values In order to compare the results they were made non-dimensional according to the data of van Oortmerssen [1] in the following way:

Non-dimensional wave frequency - = J 1(2) /g

where w is dimensional circular wave frequency;

1(2) is the length of second body (inour case it is a box); g - gravity accéleration.

Non-dimensional hydrodynamic coefficients are given in the table 3.

= _,

_{( 2 )}

6

in which:

p is the mass density of water, g is gravity acceleration;

l' or 1(2) are the diameter ofa cylinder or the length of a box respectively;

1)

or 2) are the volumes of displasement of a first or second bodies respectively

and other values are described earlier in the equations of motions.

CALCULATIONS AND CONCLUSIONS

We obtained

the results of

calculations using the multibody version of DELFRAC . This program is based on the solution of a 3D diffraction problem and

takes into account interaction effects. This can be made by using the velocity potential which consists the contributions from all modes of motion of both bodies and from the

incident and diffìacted wave fields.

As the first step of the investigations we obtained the hydrodynamic forces on two bodies separately using the main 3D difiìaction program ( see description of this program ref. [3 1). These results were compared with the results of van Oortmerssen obtained by using the 3D diffraction code developed in MARIN. All comparisons are graphically represented in figures 5 - 12 in the non-dimensional forms described earlier. These figures clearly demonstrate the quite good correlation between the results for single and rather simple bodies obtained by using the different 3D codes

Non-dimensional hydrodynamic coefficients

Tab. 3

Non-dimensional added mass

_{Ai' I ( p V°)}

I ( p v2)

Non-dimensional damping

Bii'/(pV '

I

g ii'')

B2/(pV2 I g / ¡(2))

Non-dimensional interaction coefficients:

coupled added mass

I ( p y

(2))

coupled damping coefficients E (1) /(pV2 'Ig / 1(2)) /(pV2 I g / 1(2))

7

even for the cases in which different number of panels have been used. The last can be

seen from the table 4.

Comparison of the total number of panels

Tab. 4

In fig. 13 through fig. 20 the results are given of calculations of hydrodynamic coefficients of two bodies - cylinder and box separately for the smallest separation

distance between them, where the effects of interaction are more significant.

Agreement between the results of DELFRAC and of van Oortmerssen is found to be generally very good for all coefficients. However some differences between the computed and measured hydrodynamic coefficients are seen which are difficult to

interpret.

On the same graphs the results for the single bodies are also shown. By comparing the graphs, it can be seen that the interaction effects are more important for

the horizontal mode than for the vertical mode and also for the rather high frequencies. In our calculated case the coefficients diverge above the non-dimensional wave

frequency o' > i

. No effects due to the presence of the neighbouring structure are observed at the lower range of frequencies.

Figures 21 - 28 give an impression of the interaction effects which exist for the multibody case. The hydrodynamic interaction coefficients D and are given in

non-dimensional forms. From the calculations presented in figures 21 through 28 it can be seen that the hydrodynamic interaction coefficients really have the symmetry relationships. These coefficients depend strongly on the presence of the moving neighbouring structure over a wide range of frequencies, especialy the coefficients D1

which are not equal to zero at

all frequencies calculated. Following to van Oortmerssen [ i ], it can be concluded that, where the effect of the neighbouring

structure on the added mass and damping disappears at very low frequencies, this is not the case for the interaction forces. Even at low frequencies, a structure will experience hydrodynamic forces as a resultof the motions of a nearby floating structure. These forces appear to be in phase with the motion of the neighbour, since for frequencies approaching zero the coefficients

tend to zero. For the case of

hydrodynamic interaction forces the correlation between calculated and measured values are reasonable.

The results obtained from the calculations of hydrodynamic coefficients for the second configuration are demonstrated in figures 29 through 36. The same conclusions

8

Type of a body _{DELFRAC van Øortmerssen i:ii}

Cylinder 392 92

can 'be. drawn as in previous case . A significant influence of interaction effects' is observed for the horizontal hydrodynamic coefficients and no influence on the vertical

hydrodynamic coefficients.

The hydrodynamic interaction coefficients for the distance between the centres of bodies 2O275 m are plotted in figures 37 through 40. These coefficients are more influenced by motions of neighbouring structures than the hydrodynamic coefficients

A1 and

The next senes of figures show the hydrodynamic coefficients of the two investigated bodies ( see fig 41 - 48) and the hydrodynamic interaction coefficients ( see fig. 49 - 52) plotted to a base of non-dimensional wave frequencies for the third configuration (252.75 meters between the centres of bodies). In this case the results are quite similar to the previous ones; As expected the interaction forces decrease with increasing distance betweeen the two bodies. For this distance which is 8.425 times to compare the draft of bodies, the interaction still exists however. Of course, it

is interesting to continue, the calculations in order to öbtain 'the results for other bodies' configuration. But the computing cost increases 'significantly.

Computatión times strongly depend on the number of bodies. For the case ola single body (cylinder or box with approximatelly the same total number of panels) the

calculation of forces and motions mcluding drift forces takes 10 minutes per frequency

and per body. For the case of calculations of two bodies' interaction the time increase to one hour. If it is not necessary to calculate the drift forces using the pressure integration method the time of calculatiOn can be decreased

to 20 minutes per

frequency . It is necessary to underline that for the multibody case a computer IBM -. 486 DX with frequency 66 MHz and 32 Mb RAM is used. For calculation' of a single body a computer IBM - 386 DX with frequency 40 MHz and 16 Mb RAM is used.

The main conclusions from our'numerical investigations areas follows:

linear 3D diffractión theory is suitable to predict hydrodynamic interaction effects between floating bodies;

the 'comparisons of computed .and measured values of hydrodynamic coefficients

of the bodies and hydrodynamic interaction coefficients are quite good;

the computed results. obtained using the multibody version of DELFRAC correlate well with the results of other authors

the biggest interaction effects are observed at rather high frequencies and can be

significant .

When two structures have the same size and similar motions

amplitudes, it appears that the hydrodynamic forces due to the motions of the neighbour, can be of the same order of magnitude as the threes, induced by the body's own motions;

the other interesting question is to investigate the inflUence of bo4y's interaction on

drift forces Some examples of such Investigation are found m literature ( see, for instance, [4 j )

REFERENCES.

Oortmerssen, van G. (1979 ) Hydrodynamic Interaction Between Two Structures, Floating in Waves. Second mt. Conf. on Behaviour of Off-Shore Structures, BOSS'79, 28 - 31 August 1979, pp. 339 - 356.

Taylor, R.E., Zietsman, J. (1982 ) Hydrodynamic loading on multi-component bodies. 3rd International Conference BOSS'82, p.p. 424 - 443.

DELFRAC. 3-D potential theory including wave diffraction and drift forces acting on the structures. Description of the program TU Deift, Shiphydrodynamics Laboratory. 1994.

D.M. Ferreira, C.-H. Lee (1994) COmputation of Second-order Mean wave Forces

and Moments in Multibody Interaction. BOSS'94,, 7th International Conference on

the Behaviour of Offshore Srtuctures, vol.2, Hydrodynamics and Cable Dynamics,

Ed. C. Chryssostomidis, MIT, p.p. 303-3 13.

Fig. i

The bottom view of two bodies,

Fig. 2

The distance between the centres of two bodies is 152.75 meters,

the ratio distance / draft is 5.092.

Fig. 3

The distance between the centres of bodies is 202.75 meters,

the ratio distance / draft is 6.758

Fig. 4

The distance between the centres of bodies is 252.75 meters,

the ratio distance / draft is 8.425

.8

.6

.4

.2

o

Horizontal added mass coefficient ofa cylinder with total 392 panels

i t i t t

9--- ---9 Single.body (DELFRAC) G -E) Single body (caicul Oorlrnerssen)

i

O--'

i

III 111,1, I

t ì t I t t j i t i i o .5 1.0 1.5 2.0 2.5 =weSQRT(1O1.4 / g) Fig. 5

1.5

.5

Vertical added mass coefficient of a cylinder with total 392 panels

I

I D---9 Singlebody (DELFRAC)

G E) Single body (calcul. Oortmerssen)

-.

I

.I,,I,,1I

i

i.

I t . i o .5 1.0 1.5 2.0 2.5 = _{.4 / g)} Fig. 6

1.0

-.2

Horizontal damping coefficient of' a cylinder with total' 392 panels

Fig.7

J

G -O Single body (DELF.AC)

9---9 Single body.(calcui.Oortmerssen),

irr

i :1i D

0.0.

-H

0 00 000

o .5 1.0 1.5 2.0 2.5 w' = (O*SQRT(1 01.4 / g)

.4

.2

.1

o

Vertical damping coefficient of a cylinder with total 392 panels

9---EI Single body (DELFRAC)

O -O Single body (calcul. Oortmerssen)

..O--O I t I

L

I t

I

i -I i i i t t i i a-i i i i i i i t .5 1.0 1.5 2.0 2.5 (Ut = (j)*Q .4/ g) Fig. 8

.8

.6

.2

o o

Horizontal added mass coefficient of a box with total 432 panels

.5 1.0

I...I

t t t 1.5 * SQRT(1O1.4/g) Fig. 9 2.0 2.5

G -O Single body (DELFAAC) B---EJ Single body (calcul. Oortmerssen)

I J

'i

J. i T I

I

1.5

.5

o

Vertical Added mass coefficient of. a box with total 432 panels

9---9 Single body, (DELFRAC) G- -E) Single body (calcuL Oortmerssen)

I i C I . . . C C o .5 Lo 1.5 2.0 2.5 co = co* SQRT ( 101 .4Ïg Fig. 10

1.0

.6

.2

-.2

Horizontal damping coefficient of a box with total 432 panels

D---9 Singie body (DELFRAC)

O- -O Single body(caIcuI. 0ortmerssen) j

I

,EO-G.

-o'

¿r

H

:1

0 .5 1.0 1.5 2.0 2.5 = w* SQRT (1Oi.4/g)

Fig. il

.5 .4 .3 .2 .1 o

Vertical damping coefficient of a box with total 432 panels

9---EI Single body (DELFRAC)

0 -ø Single body (caicul. Oortmerssen) I

I

___

i

I

0

I

'I

f

I . . i I . . o .5 1.0 1.5 2.0 2.5 co = co SQRT ( loi .4/g ) Fig. 12

1.00 .75 > .50 .25 o

Horizontal added mass coefficient of a cylinder with total 392 panels

9---9 DELFRA

V---V Calcul Oortmerssen

O Experiment Oortmerssen

--- Single body (DELFR.AC)

o

(O' = (O*sQRT(1o1 .4 1g)

Fig. 13

1.5

O

--- DELFRAC

'---V Calcul Oortmerssen

O Experiment0o,tnerssen

... Single body (DELFRAC)

Vertical added mass coefficient of a cylinder with total 392 panels

= _{.4/ g)}

Fig. 14

1.0

c

o

Horizontal damping coefficient of a cylinder with total 392 panels

---

DELFRAC G -O CaIcul.Oortmerssen D Experiment Oorjmerssen

i---

. SingIebody(DELFRAC) Distane 1.52.75 m

l

-r o D

o

o .5 1.0 1.5 2.0 2.5 O)' = WasQRT(lol.4/g) Fig. 15

.4

0

---

DELFRAC

G -O Calcul. Oortrnerssen

D ExpenmentOortrnerssen

N---N Single body (DELFRAC)

L

F Distance 1i52.7 rn D

ø/N

L

ø1.

E

.1

N.

1

F

f

M Vertical damping coefficient ofa cylinder with total 392 panels

Fig. 16

o .5 1.0 1.5

= SQRT(1:O1.4 f g)

i .0

.8

.4

.2

o

Horizontal added mass coefficient of a box With total 432 panels

Fig. 17

V---V DELFRAC

G -ø CalcuL Oortmerssen.

D ExperimentOornerssen

--- Single body (DEUFRAC)

Distance 1 52.7 _rn 'J ¡

.'

L

M? .5 1.0 1.5 2.0 2.5 (o'= co * SQRi(1Oi.4/g) > .6 a

1.75

1 .50

1.00

Vertical Added mass coefficient of a box with total 432 panels

--- --- -. DELFRAC (3- -0 CalcuI.Oortrnerssen. D Experiment Oortmerssen I' Single'body(DELFRAC) Distance 152.75 m -I i i D

a

'\

Di

o

I

L

. i

o---1:i:1

O _{.5 1.0} 1.5 2.0 2.5 co = w * .SQRT (101 AIg) Fig. 18

1.0.

.8

.2

o

Horizontal damping coefficient of a box with total 432 panels

G--- DELFRAC -O Calcul. Oorüì,erssen D Experiment Oortmerssen SiñgIebody(DELFRAC) I . ! , '

cr

I. O D I Distance 152.75 rn I

J]

:, :.

-'

w i

c\J

I u I D t I I i.

II

í

h'1

-

¡ i ,

.._..,u:L

,

o.p

, , i j I i j O _.5 1.0 1.5 2.0 _2.5 = * SORT (1O1.4/g) Fig. 19

Vertical damping _{coefficient of a box with total 432 panels}

o

_{Distance 152.7 rn} SS

-R-\---f---____-

-±---D

Fig. 20 --- DELFRAC: G -O Calcul. Oortmerssen O _{Experiment.Qortjiierssen}

R---R Single body (DELERAC)

o

2.0 _2.5

I _{. I}

.5 _{1.0 1.5}

.2

ce

o

-.2

-.4

Hydrodynamic interaction coefficients between a cylinder and a box

DELFRAC (cyl)

¿---

DELFRAC(box) G -O 'Calcul Oortmerssen. V ExperimentOortmerssen(cyI.) Experiment 0ortmerssen(box) Distance 1:52.75 rn

a

¿

I'

¡

o .5 1.0 1.5 2.0 2.5 = w SQRT (I12Vg) Fig. 21

.4

2

-.4

-.6

Hydrodynamic interaction coefficients between a cylinder and a box

DELFRAC(cyl.) DELFRAC:(box) Calcul. Oortmerssen Experiment.00rtmerssen (cyI) Experiment Oortmerssen(box)

--- --- --1-O- -O V i L Distarce 152.75 rn

P

L V

,0'

-: o .5 1.0 1.5 2.0 2.5 co' = * SQRT (I12tg) Fig. 22 o w ce w -.2

.2

Hydrodynamic interaction coefficients between a cylinder and a box

j

-

G (box) Oortmerssen (cyl:) (box) O ExperimentOortrnerssen V ExperlmentOortrnerssen

---M---

I DELFRAC(cyl.)DELFAAC -ø Calcul.

I

L

1 i t Distance 152.75 i

Ti

I

i i

-00

i I i H

0

0

.

/

t, i I 1 t i -i i t I , . i! t I I I I o .5 1.o 1.5 2.0 2.5 (i) * SQRT (I(2/g) Fig. 23 Q

t,

Q o

LU V c w .1 o -.2 -.3

Hydrodynamic interaction coefficients between a cylinder and a box

Fig. 24 i ---'DELFRAC(cyL) S---S DELFRAC (box) G -O Calcul. Oortmerssen S Experiment.Oorth,erssen;(box) V ExperimentOornerssen(cyl.) I F Distance 152.75 m V

y

H

vr

...

1

o .5 1.0 1.5 2.0 2.5 w = w* SQRT (I2lg)

.2

o

Hydrodynamic interaction coefficients between a cylinder and a box

R---R DELFRAc (cyl.)

&--- DELFRAC (box)

G -O Calcul. Oortrnerssen

V Expenment Oortmerssen (cyl:) Experiment Oortmerssen (box)

'J

t I

i'

i t t

I ...1T.... I

o .5

to

1.5 2.0 2.5 = O)* SQRT(I(2) 1g) Fig. 25 .1 c1 c

.45

.30

o

Hydrodynamic interaction coefficients between a cylinder and a box

Fig. 26

--- DELFRAC (cyl.)

---

DELFRAC (box)

G -O caIcu. Oortmerssen

D Experiment Oorm,erssen (cyl.) Experiment Oortmerssen (box)

I H Distance 152.75 m -I

D

-

ll*-'

-

LIII

I I I

A

I . -O .5 1.0 1.5 2.0 2.5 (OtO)* SQRT(I2/g)

.4

.2

.1

o

Hydrodynamic interaction coeffidents between a cylinder and a box

Fig. 27 -- A V r r r r J r r r r r r 1--- DELFRAC (cyl.) -- --1- DELFRAC (box) G -ED CaJcuI. Oortmerssen

Experiment'Oortmerssen (box) V ExperimentOornerssen (cyl.) -J Distanc 152 75 m V V

_v

1 I, .5 1.0 1.5 2.0 2.5 O) (* SQR ((2) ¡g)

Hydrodynamic interaction coefficients between a cylinder and a box

Do

D D s .3 .2 s --- DELFRAC.(cyl.) -f -- --+ DELFRAC(box) O -O CaJcuI. Oortmerssen

D Experiment Oortmerssen (cyl.) Experiment Oortmerssen (box)

D Distance 12.75 m

-.5 1.0 1.5 = w* SQRT (1(2) 1g) Fig. 28 2.0 2.5

1.00

.75

.50

o

Horizontal added mass coefficient of a cylinder

- U...U DELFRAC

B---D Calcul.Oortmerssen

V Experiment Oortmerssen

O -E) Single body (caic. Oortmerssen)

-

1

'U

_{IDistance 202.7 m} - L i V

tGO.fx

I i r r r j r I . . . -o .5 1.0 1.5 2.0 2.5

=w SQRT(I/g)

Fig. 29

1.25

.75

Vertical added mass coefficient of a cylinder

u

---

DELFRAC

D---EJ Calcul. Oorbîierssen

V Experiment Oortmerssen

O -E) Singlebody (calc. Oormierssen) Distanàe 202.75 m

0 .5 1.0 1.5 2.0 2.5

o)=o)SQRTQ2/g)

1.5

1.0

O

-.5

Horizontal damping coefficient of a cylinder

I

.4 I

i I

---

DELFRAC

- D---EJ Calcul. Oortmerssen

V Experiment Oortmerssen

- G -O Single body(calc. Oortmerssan)

Distance 202.75 m

I

_VI.

p0.000o

.

-R. ii.

¿Iv

i _i I I I i i i t I t i i ì i t i i I O .5 1.0 1.5 2.0 2.5 = W SORT (1(2) /g) Fig. 31

.4

.2

o

Vertical damping coefficient of a cylinder

R---R DELFRAC

- 9---9 Calcul Oornerssen

V Experiment Oormierssen

- G -ø Single.body (caic. Oortmerssen)

Distance 202.75 m

R

Fig. 32.

o .5 1.0 1.5 2.0 2.5

1.00

.75

::25

o

Horizontal added mass coefficient of a box

---

DELFRAC

9---EJ Calcul.

V Experiment Oortmerssen

G -E) Single body (caic. Oortmerssen)

I r

1

J Distance 202.75 V -m I ;_ J

si

i

E35 I I

_i_V

I I

-_

I

(H

iit.

tS V

-i

.ItII.ÌI-- i

I i i i I , . i o .5 1.0 1. .5 2.0 2.5

o=*sQRT(I(2)/g)

Fig. 33

2.0

1.0

Vertical added mass coefficient of a box

I

V vr ---U DELFRAC G---9 Caicul. Oortrnerssan V ExperirnentOortrnerssen

G -E) Single body (caic. Oortmerssen)

Distance 202.75 m

O .5 1.0 1.5 2.0 2.5

co = w * SQRÏ (1(2) 1g)

1.5 j O -.5 o

I

- DELFRAC 9---9 Calcul. Oorthierssen V Expenment Oortmerssen

0- -ø Single body (caic. Oorthierssen)

Distance 202.75 rn

Horizontal damping coefficient of a box

.5 1.0

(O = w* SQRT (1(2) 1g)

Fig. 35

-. DELFRAC

D Cacu Oortmerssen

V Experiment 0rtrnerssen -O SingIebody (caic. Oortmerssen)

Distance 202.75 rn

.5

Vertical damping coefficient of a box

1.0 L5

= * SORT (j(2) 1g₎ Fig. 36

-f -- --+ DELFRAC (cyl:) ---R DELFRAC (box) O- -O Calcul. Oortmerssen V Experiment Oortmerssen(cyL) S Experiment Oortmerssen(box) Distance 202.75 m ,

I.

I'

¿ti

4

Hydrodynamic interaction coefficients between a cylinder and a box

Fig. 37 O .5 1.0 1.5 2.0 2.5 (0*03* SQRT(I(2)/g) .4 .2 o -.2

.4

o

-.2

Hydrodynamic interaction coefficients between a cylinder and a box

¿----

DELFSAC (cyl)

-I-- -- --I- DELFRAC (box) G -E) CacuI. Oortmerssen

D Experiment Oortrrierssen (cyl.)

V Experiment Oor,erssen (box)

m j . i Distance 202.75 i

V'&LV

D

--:

.1

H

o .5 1.0 1.5. 2.0 2.5 (0=O)* SQRT(I(2)/g) Fig. 38

Hydrodynamic interaction coefficients between a cylinder and a box I

I...

i i

+-

--G V --4- DELFRAC (box) -O Calcul. Oortrnerssen

Experiment Oortmerssen (cyL)

Experiment Oornerssen (box) i

Distaice 202.75 rn i i

9

Q 1 I /1

-4f'--

i

- I

ii

t'l6

I WO4lO

I i+.

I.,

IP ¿'1 i -i i H E

-.I

I I . . . . I . . . . . o 1.0 1.5 2.0 2.5 Û)=o)*SQRT(I(2)/g) Fig. 39 .75 .50 _ .25 w ce LU _o -.25 -.50

.2

.1

o

Hydrodynamic interaction coefficients between a cylinder and a box

V

---

DELFRAC(cyl.)

+- -- + DELFRAC:(box)

D---9 Calcul. Oortmerssen

V Experiment OornArssen(cyl.)

Experiment OorUnerssen (box)

Distance 202.75 m

Fig. 40

.5

to

1.5 2.0 2.5

.8

.6

.4

.2

o

HorizontaIadded mass coefficient of a cylinder

t

'r--i

--- DELFRAC.

D---D.CáJcuL Oortmerssen

V _{'Experiment Oortmerssen}

G -O Single body (caic Oortmerssen) , 252 75 m Distance j V Ir

-4--'i\

t i k1 i i r

'2

- . I I I I

'i'

o .5 1.0. 1!5 2.0 2.5 co c»* 'SORT (1(2)

/)

Fig. 41

1.35

1.20

Vertical added mass coefficient of a cylinder

---

DELFRAC

9---EI Calcul. Oornerssen

V ExpenmentOortmerssen

G -O Single body (calc. Oorm,erssen)

=w SQRT(I2/g)

Fig. 42

.90

.75

I- DELFRAC

D--O Calcul. Oortmerssen

V Experiment Oortmerssen

G -0 Single body(caic. Oortrnerssen)

Horizontal damping coefficient of a cylinder

Distance 252.75 m 0 .5 1.0 1.5 2.0 2.5 w'=w*SQRT(I(2)/g) Fig. 43 V!. V O .. î

.4

o

Vertical damping coefficient of a cylinder

Fig. 44 --- DELFRAC 9---9 Calcul. Oortmerssen. V Experiment Oorinierssen G -O Singlebody (calc..00rtmerssen) i

y

. -. I j Distance 252.75 V m

'-jp

"

V

'.

//

-..,,Ìi.i. i

I I I I I I Ì I o .5 1.0 1.5 2.0 2.5 w=co*SQRTQ(2)/g)

1.00

.75

:t;.25

o

Horizontal added mass coefficient of a box

Fig. 45

.,rrrrrr J

r r r r

J . r r . r r r

r

e--- -. DELFRAC

G -E) Calcul. Oortrnerssen

V Experiment Oortmerssen

B---Ei Singe body (calcOortmerssen)

V Distance 252.75 rji

Th.

I r I I o .5 1.0 1.5 2.0 2.5 W=O) SQRT(I2/g)

1.7

1.5

1.1

.9

Vertical added mass coefficient of a box

S'

H

r r . r r j i r r

.

-- DELFRAC D----9 Caicul. Oorutierssan V ExperimentOortmerssen

O -O Single body (calc. Oortmerssen)

m Distance 252.75 -i.

-.

¡V

S' i o .5 1.0 1.5 2.0 2.5 = w * SORT(1(2) 1g) Fig. 46

1.0

O

Horizontal damping coefficient of a box

Fig. 47

,..J.,,,

. V j I

""UDELFRAC

- 9---E] CacuI.Oortmerssen V Experiment Oortmerssen G -O SingIebody(caJc.Oornerssen)

t

.1

/

;'

I

I

I

Distance 252.75 m

--

__

u,

- i I I I I I I. .5 1.0 1.5 2.0 2.5 (0= (0 SQRT(I2/g)

-. DELFRAC

B ---El Calcul. Oortmerssen

V Experiment .Oortmerssen

G- -O: SingIebody(caJcOortmerssen).

Distance 2S2.75 m

C t

Vertical damping coefficient of a box

Fig. 48

.5 1.0 1.5 2.0 2.5

.4

.2

o

-.2

Hydrodynamic interaction coefficients between a cylinder and a box

---U DELFRAC (cyl.)

(box) Oortmerssen (cyL) - ---- DELFRAC G -E) CalcuI.Oortmerssen; V Experiment - S Experiment:Oortmerssen(box). - Distance 2.52.75 m -F i

'I

--i i I

/

-111.1.1.1

.

/

. I I I .

I111.11

' I

/ I

77

o .5 1..o 1.5 2.0 2.5 (I) =co*SQRT(I(2)/g) Fig. 49.

.6

.2

-.2

-.6

Hydrodynamic interaction coefficients between a cylinder and a box

* -- --I- 'DELFRAC (cyl:)

DELFRAC (box) CaIcuI.0ortmerssen

Experiment,Oortmerssen(cyL)

Pmeh1t ortrflerssetl (box)

¿----O -O V S -Distance 252.75 m I

i-/

,1

/

.iIiiitÌi

r . i i i o .5 1.0 1.5 20 2.5 (j)= wSQRT(I2)/g) Fig. 50

C., C.) .3 .2 0 -.2

Hydrodynamic interaction coefficients between a cylinder and a box

D I

i--.

----4 -- ---- --4-0 D DELFRAC(cyl) DELFRAC (box) -ø Calcul. Oortmerssan ExperlrnenUOornerssen (cyl) ExperimentOor'nerssen(boX) -V m I i Distanqe 252.75 I

:,

:%.

t

o4

I

r

-'

V o .5 1.0 1.5 2.0 2.5

ú)'W SQRT(I2/g)

Fig. 51

.3

.2

o

Hydrodynamic interaction coefficients between a cylinder and a box

I I

--

DELFRAC(cyL

-4-- - - --I- DELFRAC(box)L

o -ø CatcuL Oortmerssen

V Experiment Oortrnerssen (cyl;)

S Experiment Oortmerssen (box)

Distance: 252.75 m

V

Fig. 52

.5 1.0 1.5 2.0 2.5