Mathematical modelling and calculation of hydrodynamic coefficients of sem-submersible platfroms

Lab. y. Scheepsbouwkunde

Technische Hogeschool

Deift

I-JFI--1pr I

lldILL i r'i

Pr'1D

i_cijL-i- i

0F

)FDD'íNhcNI I C CC)FF :1: cl I ENIT

DF

Et"I I 9NIEFf I

LFd F°Ljro-Iy

SV.

Spamov1

FSIIC, Seskeepjng Oud tlanueuvrobiljtg Uivisicyn

Verne,

Bulgaria

STRACT

For investigation of floating structures'

(lpn3T12ics o

the

last years great attention is

paid to the mare

jrecise estimation of all kindn of forces

which

are

ocluded in the evaluation of rections. ft is well known

:nt the

calculation of hydrodyrraznic coefficients

is

a step for reaching

this exactness. ru Ihis paper are

'own a

mathematical model and ressi ts from

the

deter-nation of

added mans and damping coefficients

for

'misubmersíile platforms, by

accounting for the

inter-vtjon between elements.

'riTRODUCT ION

The great

variety of floating structures

put

a

oestion tor creatiol, of mathematical solutions

of

oundary

value problems

and Its applicottoni for

cor-responding

real forms of a

wide range

ol

offshore

truc tures.Of ten a netjen,atical

recdoilirng im not a

sin-ie

adding

of quantities for any elements of structures

nd

it is necessary io perform a mure detailed

mues-tgalion

for a

good approximation of physical

model

ipecial attention will be paid to

mathematical modelling

and calculation of hydrodynamic

coefficicnts.F'or this

purpose s trip theory, boundary integral method and dis

-crete vortex method will be used.The

precise estimation

of aided mass and damping coefitients by

Mankind

rela-tion is connected with the exact determinarela-tion

of

difi-raction forces.

MATHEMATICAL MODELLING OF HYDRODYNAIIIC COEFFICIENTS ON SEMI s1JBMERsIØLE PLATFORNS

Elements of semisuhrnnersible platforms which are urrda

the free surface are columns, pontoons

and braces. They

are

elements in a mathematical modelling of

hydrodyna-mic coefficients too. For this purpose

the

following

assumptions are made:

- braces damping is

neglected,

- columns damping in surge

motion is neglected,

- braces added mass is

independent of the frequency of

motion when Lbr/dbr

- columns added mass inn surge motion is independent

of

the frequency of motion,

- the interdependence between pontoons is neglected when

CLRNS/B, b. 2

(Figs.

2,3

1

- the

interdependence between the colamos is neglected,

- the most important influence

between elements is

be-tween the colamos and the pontoon connected

under

the

free

surface,

- for stating smalL slenderness

in surge motion,

the

Pabst

checking formula is used.

i -

emeet-ch Scientist

It is well knio.-n that the matrix of added mass and

dans-ping coeffitients for semisubmersible catamaran

plat-forms onnly B coeflicients are nonzero.

¡A O O O

/ O

₀₂₂

0

024

jO

O

033

0 0 i 0 044 O A O O O A55 Q51 O O O O

where from Greens formula

All = 051i

024 = 042

Bu

O O B22 O O O 1342

B,

O O o

o.

1313 O-' O O

and

_is

B51n 24

That means that we must estimate

only these B coef

-tic ie,mts.

'line specific configuration of floatingstructures requi

res to pay a speciOl

attentionn to the hydrodynamnic

ccci-fiiicnls of linear motions A11, 1311,A22,B22,A33,333,

hecuase the corresponding coeffitients

of angular

moti-ons A44,844.A55,inSS,Ab6, BSb,AlS,BlS.A24,BlI

are their

funct ions.

Using the above-mentioned assumptions

and

applying

moment transformation, we obtain the

following

expres-sion

for

a

nnnathenn,atical model of hydrodynanrmic

coef-ficients on sesisubmersible platforms n

O 24 O 344 O O

ARCHIEF

12APR. 1988

)ÇV

SMS

¡

4'6, Vztbt

O o O O A66

I

O O O O O O O B55 O O B

M)

333 = B

+ B3

044 = A

(A'3)+04 (A2)+A

1A3)+04 (04)

B44

+ A'(0) +

A51(A33) = +

066

1l6(l32) + A(A)

B61 =

+ B(B)

JMENCLATURE

- column diameter

L,

- brace diameter

- column irft

- length of pontoon

- length of brace

- number

of braces

- umher of columns

- number of pontoons

- Pabst checkinG formula

A11 =

c

₊ + B11 1111 A22 A _{+ + 22} 22

= B

+

033 = A

+ 33

+ A3

r

n24

B4(B2)

A15 =

and B22 is l.ow frequency damping coefficient, which is

the result of sy,msetric5l vortex shedding around an

oscillating pontoon.

CALCULATION OF HYDRODYNAMIC COEFFICIENTS

I

this chapter we shall discuss Formulas for

esti-mation of

added

mass arid damping coefficients.

Greater

attention

B22, A33,

is pair: to the linear ones - A11, B11,

A22,

B33.

pid02

A" Nf

4

dz

- Npf An5dqp

11

OLb

Npf

J25Tdh

NJ

B11m

In Fig.5 a comparison between experimental and

nume-rical c1eulation is shown. The differences between than

are insignificant. The influence between the elements

is neglected.

A sore interest Log problem J a the calculation

of

A22 and 22 The attempts to calculate A22 by the for

-mula

A22

A2

+

A2 + A2

shows that the results are greater than experimental

data. Because of that, instead of AC2, A

is used for

thin

part of column-pontoon (Fig. 1.

Followir.q ($). lt is found:

A2 = N

f

A2

f2(x)dx

Qd

2) A

= Nkf)A2-A2)q22dx

y

In

an

anaLogou way are obtained

2'

.Function t2Cx)

shows the interference between pontoons under the

free sorfacefFig 2).

On the basis of serial model rests by wide variety of

geometric paraiseters the cheking relation is found 111/

E-22

C

where T is draft of p1atfrnisHis bight

of pontoon.

This kind of obtained correcting coefficients

allows us to use strip theory for estimation of . -.

A(B) .

The difficulties and expensiveness of

experi-mental

works

impose

to search an

ability to

calculate

these ccrrecting coefficients. The differences between

a slender body with two-dimensional form and

the real construction from columns and pontoons

is the attacked area in sway oscillations. Inertial and

drag forces are proportional to this area. Using

only

geometric parameters, we find

qr q(THpd)

Parameters T,)) ,d estimate the calculating area S =T.).

and the real

'ara S

1i .L0.5N .d (T-H )

and

.2

p

cc

p

relation of these areas

(5L

_fl

(31

= _d

2

/

The comparison

between

q2 and

shown in Fig.

4. The application of these correction coefficients

(Fig.5 C shows a good agreement between experimental

data mrd calculation with q22

These results allow to econorsjse model test and

in-crease the accuracy of calculation, using formule ( .3)

Using

the same procedure, we find

) q3

q33(T.L,U, B0)

Using results from Fig. ₃ and formula )4 C, we

sake a comparison between experimental and numerical

data and, as is well seen in Fig. ₆ the agreement

between

the

theoretical calculation 4the correcting

functions f3 and q33 is good.

As shown in (22), these coefficients A22, 822,A33,

533

have a

main influence on all others except A24, 24 A15, B. At BSHC (e)this problem for

deterrni-nation of cross-coupling coefficients is experimentally

investigated. Now numerical calculations are performed

on the basis uf d De .Jongs workl2/

und

the

comparison

between frequency independent and dependent calculations

nd results from experimental tests are made on Fig.7

The solution ai amerjeal results from the calcultjon

of A . ,b. . _{are presented in apendices A and B.}

t 1.1

CONCLUSION

The results of this detailed modelling,calculatioflS

and

compensons with the experiments are the following

-a new mathematical model for estimation of

hydro-dynamic coffifents of semisubmersible platforms; n numerical procedute for precise calculation of

added nuns and damping coefficients by accounting the

interaction between elements;

-osing s discrete vortex method for approximate es-timation of sway damping coeffitients in low KC num-bers

The above mentioned results allow economy

of a dreat

deal of experimental investigations and decrease of

computer consuming lisle , which es in correspondance

with ITTC recommendations.

SUBSCR IPTS

b - index of brace

e - index of colman

cg - index of

column-pontoon

p -

pon

toon

s - index for strip coefficients

BEFERENCES

Bearman Y.W., Graham J.M.K., Odsaju E.D.,A MOdel

Equation for the Transverse Forces on Cylinders

in

Oscillatory Flows", Appl.Ocean Res.

1986,vol.6,No. 3.

De Jong D., "Computation of Hvdrcsdynamic

Coeffi-clerics of Oscillating Cylinders' , TB Delft,Report 210,

i 973.

Frank W .," Oscillation of Cylinders in or belca

Free Surtace of Deep Water Fluids' , DTNSRDC, 1967,Re

2375.

Graham .J.M., "The

Forces on Sharp Edged Cylinders

in Oscillatory Flow at Low Keulegar-Carpenter Nom

-bers", .JFM,

vol.97, Part 1, 1980.

Ikeda Y., Tanaka N., "On Viscous Drag of

Oscil

-lating Bluff bodies", XIII

S)435H, Vanna,

1983.

Keulegan G., Carpenter L.,'Forces

on Cylinders

and Plates in an Oscillation Fluid"

Journal Res,

of

the National Bureau of Standards, vol.60, 1958.

Spassov S. "Calculation of Wave Loads on the

Semi-submersibles", XIV SMSSH, yama, 1985.

Spassov S.," Investigation of

Dynamics of

Floating

Structures", Internal Report BSHC,

1985.

Spassov

s.,

'Application

f Two Linear Methods foe Calculation of HydrodynasiC Coefficients", Buig.Acad. Sciences, Sofia, 1981.

Wehausen 3. Laiton C., "Surface Waves"

Springer

Verlag, 1960.

ll.Kishev B.,Spassov S."xpenisental investigationand

analysis of the influence of semisubmersible geomet

ricric parameters i on the hydrodynainic coeffitients"

BSIiC,1984,Internal Report

r

12

-'

0

i

2 E 3

Fig.2

Interference between submerged cylindrical 30

bodies for horizontal motions

f3

Fig. 1 Coordinate system

X

Fig.3 . Interference between submerged cylindrical

bodies for vertical motions

si

0 01 02 03 0 I. J1 LrnJ

Fig.4 Correction formulae for swaying added mass

coef ic len is

10

06

0.25 050 075 00 105 W(rUd/5J

Fig.5

. Comparison between experimental and numerical results for horizontal added nass coefficients

20

- from exp

from (3 o o T

-o o exp. theary etut - - fleory with 007 o

- tbory

Fig.6 . Effect of checking formulae for j33 on the

vertical damping coefficients

1 2 3 E 05 -e---06 o 0. 02 f2 1. 1.6 10 116 12

sI.

Fig.7 Frequency influence on the sway-roll

cross-coupling added sass coefficients

APPENDIX A

DESCRIPTION OFTNE TWO-DIMENSIONAL PROBLEM

Byusing strip theory (W.Frank ['3)), the

three-di-mensional boundary value problem for estimtion of

hyd-rodynainic

coefficients

is

reduced to two-dimensional for

calculation cf

the

velocity

potential

function

tI

rx,z,t)

and

corresponding.

conditions on the

fluid

boundaries. With

the assumption of

small

oscillation.

only the linear frequency response of the fluid to the

disturbance

will be considered.

The velocity potential can be written as

f

(5) (x,z,t)Re j,(x,y)e

jr

cost+45sinu

The motion of any point on the surface of the

cylin-der is expressed by

th

X=X10sint

i2,3,4

The velocity potential as a harmonic function Satin-fies Laplace equation

(7)

_Hxz)=O

The assumption of a slight Ji sturbence on the

free

surface leads to a linearized form of the froc

surface

condition

ZrO

The linearized kinematic condition on the cylinder

contour is given by

ici

_

-Ein

-V0

VW'.fl

at the mean position of the cylinder. tiere Vr5 is the

normal component of the cylinder contour velocity. It

can be readily shown that

(10)

V0X0coS(n,Z)

where

n is an outward

unit normal vector on the cylin

-der contour, con (n,z) means the directionaL cosine

be-tween the normai vector and

the s

direction.

ijurrihee

The .pected decay of the fluid disturbance as z

can he

escribed by

(11)

L2__..

dx

O

_dz

The far-field behaviour o1 as s ,_.should

re-present outgoinq waves, i.e. Sommerfeld s radiation

condition

112)

Due to the symmetry (i 3) or asyrirnetry (i

= 2.4),

there cannot be flow crossing the z-axis.

That

is

(x.z)± 1XZ)

This completes the statement of u proble, the boon-,

dary conditiops of which are from

(5)

to 1(3). The soin-,

tl.on of this bouDdary-value problem will provide the;

sought hydrodynamic quantities such as pressure

distri-bution,

hydrodynamic

_{force, added mass and damping}

co-efficients.

NONES fOAL SOLUTION

The solution of the Velocity pctential4l(x,z)

is

assumed to be represented by a distribution

of

source

singularities over

the immersed contour of the cylind

(p)fQ(S)G(p;s)ds

Sb

where p - (x,z) is the field point,

- c + iQ is source density,

GR GR +

iRs

is Green function,

immersed contour of the cylinder in

The solution for is given, for example Wehausen

1O)am satisfying 3), (4), (7) in terms of a complex

plane z-

X +

iz, I

=

I

+

il

GR

Re 1/2'ií[tog(z-r)- log Cz-).

2

e-1Z_7

_k-s

ds-12'jTe

J

'The kinematic

boundary

coridition(9) is used to

ob-tain the strengths of the sources Q(S) 9iven in

tion

It will he assumed that the contours of the cylinder

Sb can

be

approximated by N

number

of

straight-line

segments, each of which is denoted by

cj.

j=1, u

Thus equation

( I'P

can be written as

i(p)s

_f

Q(s)G(p;s)ds

jal

e

lt is further assumed that the variation of

source strength on each segment is so mrrall

that it can

be treated as constant on each segment. The latter

as-sumption

yieds the expression

QfE(P;s'ldS

Cj

When

the kinematic boundary condition

g.ven by

equation

(9)

is applied in equation (16),

it

follows

that

Q Cn.v)J'G1P;s)ds

X10i.cos.

C

pp,

where is the tangent angle of the contour

cylinder at the point P0. By taking N number

on

the contour Sb and by assuming that these

located at the midpoints of the line segments

cars

be

shown that

(4f)

Qj(fl.V)JGR(PS)dS

:Xj0cosc

(n)

Cj jsl2,.., N i r 2,3.4 of the

of points

points are

Cj.

it

E e exp Lili

-

theory w)h fheory aarrsns ax 75 tas 0:50 3.75 100 675 75 c)radis

The separation of the equations (17) of the

rea)

and

the

imaginary parts yields

Qc I -Q5JtX0 (i)COSj

G.s I

cO

where

j

_{c(flV)JGRc(p;S)dSIpp}

J_(n.v}_Çnc(p;s)dsIpsp

For the estimation of

and

from equations(4)

it is necessary to derive Green's function, which

can

be interpreted as a wave source near a vertical

wall,

where the principal-value integrals were

encountered.

7 e

-ik(Z

J

k-s

dsm

f

ds4ilíe

s,'

a

where

7

indicates that the path of the integration ist

intenã

above the pole at S

K arid the last

term

is

a residue value at the pole. As shown in Fig.

g

the

path of the iriteoral changes to the two different paths,

depending on whether

X -

> O or

X -

3' < O.

These integrale can be reduced to tho

exponential

integral which can be expanded to an infinito series

tcZ)mJ --dt=-'

i.og z

-z

where

= 0.5772, Euler constant.

Then

-is(z-l')

js(z5)

7e

e

ds +

e

-<(z-r)

o

e_k

_{E1(-ik(z-))±i}

-ik)Z--)

ie

for x-O

Substituting

_1j

in (13(

the unknown

coeffi -.

.cients Qcj, Q5, j =

...

,

N of 2N simultaneous

equa-tions can be otained. Then separation of equatiOn (16).

into the real and imaginary parts yields

c(Xo.yo )=

(cf

_ptxo,Yo;s)d5

and

J. G)

Cxo,y0;s)ds)

s(Xo,Yo)

(QcjJ0.(3c(xe,yo:S)ds +

J

GR0(xo,y0;s)ds

The

hydroynamic pressure at the point

(x0,y0)

on

the cylinder is obtained from the linearized

Lagrange

inegral by

P(x0,y0, f)=-

d4 ¡df

cost

snut}

Then,for exasrile, the vertical hydrodynarnic

force

acting on the cylinder can be obtained by

(20) F:- 2fSb p cossCc1s2çi(cos bltJb

cos(n,x )ds

-5Ífl()fJ5c

COS(fl, x)cls)

et

If we

et the total hydrodynamic force be

e<pressed

in the forni

-ij=2,3,4

and substitute

X1(t)X10sinf

(2e)

By equating

(21)

and (20) we find that

r Xo

fcos(n,xk1s

x10

Sb

-

:jc23,4

On the basis of this solution a numerical realization

(9) for

calculation of hydrodyriamic coefficients

of

two-dimensional

forms of floating or submerged

bodies

Is

made. Illustration of the abilities of this

program-able system is given in Figs

9 ,

f, ei..

.

so o SOG so o sii r0

r.K

rK

° - K I z loo

Fig.

_{.. Change of integration path for}

Pe(z -(

u rod/si

Fig. 9. Influence of depth of submergence ori

vertical dan-ping

50 E 35.0 25.0 20.0 t0.0 to 5.0 0

Fíg.(O . Comparison between experimental and theoretical results for floating body

TO10.; B4O.'.3

-

hoary

o exp

Fig.41 . Influence of column diameter on vertical

added mass

APPENDIX B

ESTIMATION OF DAMPINGCOEFFICIENTS IN HORIZONTAL MOTION

we shall discuss

an approach in

the determination of

the vortex part of the

damping coefficient at trans

-verse-horizontal oscillations. At harmonic oscillations

of a prismatic body with arbitrary breadth a vortew

sheet is formed, Whet-i the breadth tends to zero, then

this body degenerates

into

a flat plate. Following (1),

(4), (5), we shall discuss the motions of the flat

plate, substituting

the

vortex sheet by a couple of

s5a-metric discrete vortices, separating from both plate

ends at its motion (Fig. 12).

During the plate motions the phenomenon is of

un-steady nature. There is a vortex formation at each half period of the motion. Nearer orte end of each swing, be-fore the plate stops, the sign of the vorticity density near the edge becomes the opposite to the sign which it

had during the last swing. This means that the

strength of vorticity has a phase difference with

re-spect to the plate motion velocity. At the next swing

the preceding vortex moves in the previous direction,

and the new vortex is generated behind the plato. At

continuous transversehorizontal oscillations the vor -tices seem to flow away from the plate.

Let

us discuss

harmonic horizontal oscillations of

the plate with a speed

Vrsjribut

We shall assunse that the

_{vortex generation field}

_is

represented by two

pairs

of discrete vortices which

lie somewhere on

the

horizontal lines through the

ed-ges. Let the distance between them be X2a

x1 - X0,

and

their strengths r0 and r1 .

At motion to

the right we

assume that r0 is the preceding vortex and r0 = const,

and

r1

is the new, growing with the time vortex. The

Circulation

in the upper plane

has the form

r rcos(wt+E)

where the amplitude is , and the phases .

In the time

i,otE there is Only

one

pair of vortices with inten-'

-sity

r,s r

. After that

r

const, whereas the new

pair

with strength F1 grows until

it

_{reaches its maximum}

value.

ra-2f5

for

fa7-E

At this montent the total speed circulation

rr0+r-r for

tjT-E

From there onwards the picture is an iterative one.

The strength

F1 becomes preceding for sit

s

, and we

do not discuss F0

in the development that follows. To

de-termine the

vortex

strengths intensity,

we shall use the

ondition

of

Kutta-ZhukovSky-Ch2ìplygitl at the plate

ends

¡For

sit

E there are pairs

of

vortices with intensity

-r; The velocity at the ends for z

i orso

O _frornL

z

=

tJPT -

conformal mapping) is equal to

r0

i

i

wt-Q V

+

i'! )

o

Then from the condition of Kutta-Zhukovsky

-Chaplygi

O we obtain

r0m2ivE--2ív

j [x

+1+v" + 1x02+4

j1/2

and for small values of x

this becomes

r0=2Îi

vsinrr

At times such that

dt >-E

a new pair of

vortices

appears and the strength of the growing vortex F1 is

(22)

QV(f)+V0(f)+-(---- ---)o

or

r1=2ii'[V+Vo]

for

small i.

where

V0

is the induced velocity for the preceding vortex with

strength F, const, but with unknown location, so that

V0 is also unknown.

For r1

the

velocity Xl, out of symmetry considera

tions, is

23)

X11/2

LV+V0]

for

-E <L,)t <'ji-E

The value V +V0 is the velocity at' O .z =

1); then

c2) r1r_r0=r[cosR.t+E)_1

After

substituting )2'i) into

(22) ,

we obtain

-r1

itcos(t+E1-1J

(25)

V4V00

_2iIi

2j'if

Then, substituting

(2g) jito

(23)

and tnteqrating. we

obta in

a/3x1312dr/tE -sin(t+E)I

and for

r c8/3o,Xi1

V

snd from the syomsetry of vortices location between

the

times (0t a-E

and'Jl-E

we obtain

ii [rizVsinE]2

X1 = 3ji/4

(sina)x20

It is clear that the phase difference

between

the

circulation r ana the velocity V is of

particular

impor-tance, bt its accurate determination

is difficult.

Ana-logous to (5), we shall assume that

EmjT/4 o45°

The force obtained applying the

impulse moment

pre-servation law of flew field:

1V) Fm

-kzp'íi--2p -V

We substitute (27) with

the expression.s (2fl and(jÇ)

and

obtain

Fm FLNcosudt+FDpsnt

In this analysiS the force

has components only

from the

basic frequency. To compare the

resulte with

expo-rimezital

invemtgation0 carried out by

Keulegan-Carpen-ter (6) with a flat plate

and at BSHC (6) with a

sub-merged elongated pri30natic

body, we shall present

the

force by transforming the

damping component in

noni

i-near form, corresponding

to the damping due to the

vor-tex generation of an oscillating body

F aCm

2QIì2COSU)t + DpX202

22t

where

for the flat plate

Sbal

FDP1/2PCDSbV2

'yo

perform this equalization, it is

necessary

to

approxmite linearly the damping

force daring

harmonic

oscillations. The basic assumption is that the real nont

linear force and its equivalent

linear damping

force

emit the saine energy quantity for the same period.

In

this way we can obtain the damping coefficient

Bz

ßP0tenti0(2

+

Fig.

13 shows a cosparison between the

experimen-tally and the theoretically determined

damping coeff

i-'ciant cf arm elnogated prismatic body - a

pontoon.

It

can be

pointed out that for 4SKC

12 the

coinci

-Idences

are completely satisfactory

(a symmetric

wake

pettern) .

We

must emphasize that for

floating

faci-lities 'this zone is of the

greatest practical

impor-tance at

long-period oscillatiomie, where EC

I

The modelling of the damping

coefficient for

semi-submersibles at transverse-horizontal

oscillations

is

'O

illustrated

in Fig. 14. The thus

determined value

B22

affects also the damping coefficient

at roll

inotion

(B44

,

though not so strongly, since

there the wave

ge-neraticn

induced by the vertical motion

exerts

the

I

basic influence:

4Nnf(Z'

Hn)2dX

20

The

influence of

B22

on the motion

characteristics

is illmsstrated in Fig.

15. '

Fig. 12. Coordinate system for investigaj0 of

_fiai

plate harmonic oscillati0115

snuit

9-X0

C

rs

V3 Ti exp theory X t n I lo

-20 31) 40

/

IO!

/

_/

'I

to.1'

/

¡e:,"

/

₄

/

IO /

,I

50/ 40 /

_,/

20 0.20 050 0.75 UIdt,I ¶25

Fig. 13

Effect of amplitude of motion on the low

frequency swaying damping coefficients

Iheory exp

-B:

O

X

- u ¶rod)s(

Fig. 14. Modelling of the swaying damping

coeffi-e lcoeffi-ents

i:

r

X or r3O0 theorr without B22 with 622 X20O,1On,

00 lOo ISO 000 530 000 OSO &.a(rruJfSl

%O oso Sos os So JI o?

Fig. 15. Effect of low frequency damping on the swaying

amp1itude of motion