Hydrodynamic load models of slender marine structures

ThGHNISCHE utIvZITT

bum vo

3_oS

AhIef

Meketweg.2, 2828 CD Deift

Tel.: 015- 786373 -Fax: 015-781838

HYDRODYNANIC LOAD MODELS OF SLENDER MARINE STRUCTURES

by

Erling Huse

MARINTEK, Norway

Summary of WEGEMT lecture, Trondheim 1991

The Morison equation is the basis of all commonly used

engineering calculations of hydrodynainic ioad

on slender

marine structures.

It is simple, and it is efficient in terms

of computer capacity requirement.

However, when used with

constant coefficients it represents a severe

over-simplification of the true nature of the flow around such

structures

The choice of coefficients thus tends to become

somewhat arbitrary, and the reliability of the results

correspondingly poor.

(For more information, see Appendix 1)

On the other hand, a very extensive effort has been spent in

recent years on developing vortex methods, i.e. procedures

based on numerical modeling of the individual vortices being

shed from the structure..

This represents a much more detailed

and accurate representation of the real flow.

Considerable

progress has been achieved by this line of approach.

However,

such methods tend to become very expensive in terms of

computer capacity reqirernents. Furthermore, present-day

understanding of the physical process by which the individual

vortices transform into a turbulent wake. field is limited and

hard to modél numerically.

So far the application of such

methods has been limited to very simple geometries, and only

to research activities.

Extensive application of such, methods

in engineering calculations can at the best be considered as

music of a distant future.

(Appendix 2 shows a survey of the

very latest literature on the subject)

Between the above twö extremes with respect to complexity

there is the possibility of describing the wake, not by

individual vortices, but by a turbulent shear flow field.

The

momentum of the wake immediately behind the body is determined

by the instantaneous drag force on the body.

Further

development of the wake in time and space. takes place

according to relatively simple mathematical formulae.

The

instantaneous drag, force. on the body is calculated. by the

Morison formula, using the drag coefficient in stationary

flow, but correcting. the irif low. velOcity for the wake

generated by all previous oscillations of the body.

This line

of approach has proved very fruitful in describing and

predicting the drag force on different types of bodies.

A.

-518 wegemtl.rep

oov mMaioJ

a

e ib

upon the drag coefficient. (or more

_-,

details, see Appendix 3)

A simple but efficient improvement

is to substitute the

quadratic

drag force in the Morison formula by the sum of a

linear plus a quadratic force.

In this way one can by a set

of constant coefficients obtain a reasonably

correct drag

APPENDIX i

L4,

£.

, $

'E-

J,,

(

,

7

VISCOUS WAVE LOADS AND

DAMPING

Viscous flow phenomena are of importance in several problems related to

wave loads on ships and offshore structures. Examples are wave loads on

jackets, risers, tethers and pipelines, roll damping of ships and barges,

slow-drift oscillation damping of moored structures in irregular sea

and

wind, anchor line damping, and 'ringing' or 'springing' damping of

TLPs.

The main factors influencing the flow are:

Reynolds number Rn = CD/v (U = characteristic free stream

velocity, D = characteristic length of the body, y = kinematic

viscosity coefficient

,

Roughness number = k/D (k = characteristic cross-sectional

dimension of the roughness on the body surface),

KeuleganCarpenter number KC) (= U

T1

D for ambient

oscillatory planar flow with velocity U sirn.(2t/T)t +

E'

past

a ftxed bodv,

Relative current number (=UJU when the current velocity

U is in the same direction as the oscillatory flow velocity

sin((2r/ T)t

+ E)),

Body form,

Free-surfisce effects,

Sea-floor effects,

Nature of ambient flow relative to the structure's orientation,

Reduced velocity (UR = U/fD) for an elastically mounted

cylinder with natural frequencyf).

Sometimes /3 = Rn/KC = D2/(vT) is also used to characterize the flow.

Detailed discussions of many of the factors mentioned above can

be

found in Sarpkaya & Isàacson (1981). For harmonically oscillating

flow

around a fixed circular cylinder of diameter D, we may write

KG =

22tA/D, where A is the amplitude of oscillation of the fluid far away

from

the body. We then see that KG expresses the distance a free stream

fluid

particle moves relative to the body diameter.

MORISON'S EQUATION

Morison's equation (Morison et al. 1950) is often used to calculate wave

loads on circular cylindrical structural members of flxd offshore

' iuu. V it Vt

LOADS

ANI) DAMI'! NG

structures when viscous forces matter. Morison's equation tells us that

the horizontal force dF on a strip of length dz of a vertical rigid circülar

cylinder (see Fig. 3.14) can be written as

írD

dF=p 4dzCAfaI+CDDdzIuIu

(7.1)

Positive force direction is in the wave propagation direction.

p is the

mass density of the water, D is the cylinder diameter, u and a are the

horizontal undisturbed fluid velocity and acceleration at the midpoint of

the strip. The mass and drag coefficients CM and CD have to be

empirically determined and are dependent on the parameters mentioned

in the beginning of the chapter.

Considering deep water regular sinusoidal incident waves (see Table

2.1) and assuming CM and C1- to be constant with depth (which might

not be realistic), we may easily show that the mass-force decays with

depth like

The drag force decays like e4

and is even more

concentrated in the free-surface zone. When there is a wave node at the

cylinder axis the mass-force will have a maximum absolute value and the

drag-force will then be zero. The drag-force on a submerged strip will

have a maximum absolute value when there is a wave crest or a wave

trough at the cylinder axis. If viscous effects are negligible, it is possible

to show analytically that Morison's equation is the correct asymptotic

solution for large AID-values (see discussion of equation (3.34'). The

C-value should then be two for a circular cross-section.

If fluid

acceleration can be neglected, Morison's equation

is

a

reasonable

empirical formulation for the time average force. Typical CD- and

C-vaIues for transcritical flow past a smooth circular cylinder at

KC> 40 are 0.7 and 1.8. A roughness number

kID

of 0.02 can

represent more than 100°o increase in CD relative to the CD-value for a

smooth circular cylinder (Sarpkaya, 1985). This means the effect of

roughness is more significant in oscillatory ambient flow than in steady

incident flow.

The application of Morison's equation

in the

free-surface zone

requires accurate estimates of the undisturbed velocity distribution under

a wave crest. The prediction of the velocity distribution based on linear

wave theory was discussed in connection with Fig. 2.2. A straightforward

application of Morison's equation implies that the absolute value of the

force per unit length is largest at the free-surface. This is unphysical

since the pressure is constant on the free-surface. This means the force

per unit length has to go to zero at the free-surface. It should also be

noted that the position of the free-surface at the cylinder is affected by

wave run up on the upstream side of the cylinder and a wave depression

on the downstream side. The vertical position of the maximum absolute

value of the local force has to be experimentally determined. The order

MORISON'S EQUATION

225

of magnitude of the position may be 25% of the wave amplitude down in

the fluid from the free-surface. We should note that Morison'sequatiqfl,

cannot predict at all the oscillatory forces due to voñex shedding in.

the

lift direction, i.e. forces orthogonal to the wave propagation direction and..

in the cross-sectional plane.

Morison's equation can be modified

cylinder. Consider a vertical cylinder

body motion of a strip of length dz by

hydrodynamic force on the cylinder as

dF = PCDD dz(u i1) lu

-+pC.fdzaI-p(CM-1)dZI

(7.2)

Dot stands for time derivative. Both u and a1 are position-dependent. It

should be noted that the inertia term does not depend on the relative

acceleration term. The reason for this can be found by studying the

analytical solution in the potential flow case.

The FroudeKrilod force

(i.e.

undisturbed wave pressure force) results

in a horizontal force

pL-rD-/4; dz a1 which is independent

of the rigid body motion (See

chapter 3'. The

CD-

and C%-values in equation (7.2) are not

necessarily

the same as in equation (7.1).

Morison's equation can also be applied to inclined members. To

demonstrate this let us consider a cylinder inclined in a plane parallel to

the wave propagation direction. The approach would be to decompose

the undisturbed velocity and

acceleration into components normal to the

cylinder axis and components parallel to

the cylinder axis. and then use

Morison's equation with normal components of velocity and acceleration.

The force direction vill be normal to the cylinder axis. In the potential

flow case, it can be proven that this is the correct expression. In the

viscous case it means that we use the 'cross-flow' principle (see chapter

6). Actually what we are proposing to do for an inclined cylinder is not

different from the vertical cylinder case. In the latter case there is also an

undisturbed tangential velocity and acceleration component in the fluid.

The effect of this was neglected in

equation (7. 1).

When the cylinder axis is not in the

plane of the wave propagation

direction, there exist different possibilities for formulations. Let us

illustrate this using a submerged horizontal circular cylinder in waves

where the wave propagation direction is orthogonal to the cylinder axis.

A straightforward generalization

of Morison's equation would be to write

the horizontal and vertical force on a

strip of length dy as

dF1=p

dyCMaI+CDDdyu(u2+W2)

dF = p

dy GMa3 +

CDD

dy w(u2 + w2)

(7.4)

in the case of a moving tircular

and denote the horizontal rigid

We can write the horizontal

226

_{VISCOUS WAVE LOADS AND DAMPING}

Here w and a3 are vertical undisturbed

_{fluid velocity and acce1eration}

components at the midpoint of the strip. Chaplin (1988) has shown that

improved correlation with

_{experiments can be obtained. if djfferent}

_CM

and CD coefficients

are

assigned

to

the

horizontal and

vertical

components .

When waves and currents are acting simultaneously, the combined

effect should be considered. The normal approach is to add vectorially

the wave-induced velocity and the current velocity in the velocity term of

the Morison's equation. As mentioned in the beginning of the chapter,

one should be aware that CM- and CD-values

_{are also influenced by the}

presence of a current.

The design wave approach

is

often

used

_{in combination with}

Morison's equation. This

_{means one analyses the load effect of}

_{a regular}

wave system. Often non-linear wave theories

_{are used. A short}

term-statistical approach

_{may also be applied (Vinje, 1980). However,}

_more

has to be learned about CM- and CD-Values in this context. For instance,

is it right to use just

_{one CM- and one CD-value to represent a sea state?}

If so, is it then sufficient to use a characteristic KeuleganCarpenter

number, Reynolds number, roughness ratio

_{etc. to estimate the CM- and}

CD-value? The answer is probably

_{yes if the characteristic KC-number}

is large and the flow is not in a critical flow regime. What we mean by a

large characteristic KG-number has

_{to be precisely defined. In the}

_case

of extreme wave loads

_{on a strip of a vertical circular cylinder close to the}

free surface 7rHmax/D is

_{a characteristic KG-number, where Hmax is the}

most probable largest wave height. If 7tHmaJD >

_{40 we could consider}

the KG-number to be large. A typical

_{CD-value for a smooth surface is}

then 0.7. It is of course easy to criticize Morison's equation. However,

there is nothing better from a practical point of view. The reason for this

is the ver complicated flow picture that

_{occurs for separated flow around}

marine structures.

Many attempts have been made to solve separated flow around marine

structures numerically. Examples of methods used

_are

Single vortex method (Brown & Michael, 1955; Faltinsen &

Sortland, 1987)

Vortex sheet model (Faltinsen & Pettersen,

1987)

Discrete vortex method (Sarpkava & Shoaff,

1979)

Combination of Ghorin's method and vortex-in-cell method

(Chorin, 1973; Smith & Stansby, 1988)

NavierStokes solvers (Lecointe & Piquet,

1985)

A general description of the

_{state of the art is that the methods}

_are

generally limited to two-dimensional flow,

that

the methods have

documented satisfactory

_{agreement in some cases, but that they}

_are

presently not robust enough to be applied with

_{confidence in}

_omp1etely

new problems, where there is no guidance from experiments. Due

_to

MARINT EK

IIYDRODYNAMIC DATA

2-D circular cylinder

Flow regimes and drag in steady flow

Page:

3.

Fig.:

Des. 1990

100 80 60 40 co 20 10 8 6 4 2 0.8 0.6 0.4 02 0.7

Fig. 3.la. Incompressible flow regimes and their consequences.

io-'

2 6 6 8100 2 4

8iv 2

4

6 62 2

Fig. 3.lb. Drag coefficient for circulai cylinders as a function of Reynolds number Schlicht-ing 1968).

i;

Hl

_Li1 I

o

H

I

I!!

..

0.7 0.3 7.0 M,J2WN 7.9 4.ezeisoe;

j,

4Z.0 800

Jx.0.

¡ .Tflçjda,ef to"b o .3.0 r I I

X-

_V

Fig. 1.4. Drag anefficient for circular cylinders as a function of the Reynoldsnumber

Sarpkaya &

Isaacsofl

Schlichting

(1968)

A

Si.crtticol

B

Critical

C Suoercrlticol D

Post-sercrlticai

Bou.mdory loyer

Sa tian

Shear loyer near separation Strouhal matter Woke Anoroxicinte Re range turbulent laninar about l2deg. transition transition turbulent 120 - 130 deg. lOEltinar sepa-ration, bubble turbulent reattacFsnent 0.35 - 0.'i5 5x105

-

3x106 turbulent about 120 deg. turbulent about 0.29 3x106 lOEninar S

0.212_j1

Re60 lninor

'503

vortex street Re 5000 ,5 < 2x1u transition

ioic

2x105 to 5x1

A

BFC--jD

(1981)

100

C

10 0.1 4 444 4.

I

Il

t

till

I

iii!

if1

I

III

III,

IIl

-...\

III

I 101 100 101 102 _{iO iO}

Re

lO ₁₀₆ iQ

-.68,2 468,2

MARINTEK

e

L.ii.ur separation

Li. ar ....

Turh,jttflt 5e,.lrat tun

HYDRODYNAMIC DATA

2-D circular cylinder

Flow regimes

Laminar urt sees

Luotnur separat icr R ¿tta.hm.nt

i.nc separat ¡on

Cross 3-dlmensuonaiacv gen.'rs tur5oi,gt was. tUflut..;iL vart ..r,. J.fl.nst ream

Rd

R=10-40

L, bscreases with R

R3O-50

Unstable downstream

R50-15O

Castie.,! won kSrmin rane.

Isneral wrbir :enr i.Jk. down .1 ream

transcrit ¡cal

risi..

- P3x1O6

Fig. i. Sketches of the flow past a circular cylinder for various

ranges of Reynolds nûsober.

R:300-2x1 O

wbcritca1 rector

Cru ccii regime

P2-6x105

Sipercrttical reas..

RxO5-3x1O6

Page: 3.3.

Fig.:

1990

Ver].y (1980)

MARINTEK

Figure 14. Drag coefficient ut çy1indei (19.a) having various degrees of surface

(sand-grain size

k as against diameter "d.

HYDRODYNAMIC DATA

2-D circular cylinder

Drag in steady flow

Effect of roughness

REYNOtDS 5UER. e

Fig. 35. Drag coefficient for rougicylinders as a function of Reynolds number.

C0=1 .19

C114

CD.'lO4 1-03

C.106

Hoerner

(1958)

Sarpkaya &

Isaacson

(1981)

Hoe me r

(1958)

MARINTEK

[l1LïTii

2

-

è-\

.._.L

..- .-,- -.

4

---t î

.

.6 --i .-,-.

'

.4 -

-.

..L_---

i-.

'i

-t-

.t-(a) r/b0 0.02!

HYDRODYNAMIC DATA

2-D cylinders, different shapes

Drag in steady flow

Nominal

size. inches

o12

L o

4

3

/

---Ret 3

(b) ,/b:0.I67

.1

234 68!O

23

Reynolds numbe'. R

(c) r/LtO.333

Figure 8Variation

of d,ag coefficient

and Slrc'uhol number

..ith ReiOWS

,vmber for

me I:! fineness

ratio rectangular

cylinders.

Page: .3.'

Fig.:

Des. 1990

DelaflY &

Sorensen

(1953)

I

MARINTEK

1.5 I.0 I I

1111F

i I i I

11111

_Cd

_.

se » .5..' 2=497

"e.f

a_L

3830e GG»

_t

JI!eS15.'S l_?

.

8 L a 3 M - 1107

:

La 34 . L

tat

tt1985

hi'

3123 0.5 K 04 I I i I

III

I i I i

liti

0.5 4

HYDRODYNAMIC DATA

Fig. 3.17. Cd versus K for various values of the frequency parameter (Sarpkaya 1976a).

3.0 2.0 a

-a

¶"r

atL

óaI107

a

J:-'" ...

S

...

. : Là S L S5 IO 20 30

Fig. 3.18. Cm versus K for various values of the frequency parameter tSa.rpkaya 1976a).

2-D circular cylinder

Drag and mass coefficients

Oscillatory flow

so ioo 150 200

Fig. 3.21. Cd versus Reynolds number for various values of K (Sarpkaya 1976a).

Page: ..

/ i

Fig.:

Des. 1990

Sarpkaya &

Isaacson

(1981)

3 4 5 10 20 30 40 50 100 'So 3.0 2.0 LS 1.0

MARINT EK

e., e., I.. i.' e. IO Io a'

C.,

I I t 0.0

HYDRODYNAMIC DATA

2-D circular cylinder

Drag and mass coefficients

Oscillatory flow

Effect of roughness

F.. 3.28e. Cd oera,,o Re for oecythdcea. K 20 iSaspkaye 1976b).

a_000_ _-3

z-.-....

0' 0.0

Fig. 3.28b. C., .eraua Re (nr 000g)e cybodon. K 20 tSezpkaye ¡976b1.

i/On/tOO

7

'a

0.3 o.s e

a., .

Fig. 329e. ( ,enea Re for tou0 C)tifldOfI. K Iflfl eSt,pka 96u

8.010

t I t

s Io ti

FIg. 3.29b. C., reno. R. for rough cytjn&rs. IC lOO (Seopkeya 1976b).

Page: 3.iz

Fig.:

Des. 1990

Sarpkaya &

Isaacson

(1981)

i

il

0.4 I

liii

I i I I

jill

i 3 4 5 6 7

8 9 lO

20 30 40 50 100 150 200

KC

Fig. 6.16.

CA, vs. KC number for various values of RL'

(or ß

Re/KC) for smooth

circular cylinder [Sarpkaya

(1976)]

I I ¡ J i I I- ¡ I = 497

3

...

RezlO =10

Th

20

-:::-

-_ .30

-C 2 ...

\

.

N

N

p784

N

1107

=

-_a80

70 10

.

1985 _{..1 50}

-I I , I I I I

I

I i 10 15 20 30 ¿0 50 100 150 200

KC

Fig. 6.17.

CD vs. KC number for various values of

Re

(or ß =

Re/KC)

for smooth

circular cylinder [Sarpkaya

(1976)]

6.4. HydrodYflami

Coefficients

60 80 2

05

Re, xIO

Fig. 6.20.

Lift coefficient vs.

Reynolds numbcr for various

values of KC for smooth

circular cylinder [SarpkaYa

(1976)]

201

0.5

0.'

-0.3 I I I I I 25 3 4 5 o 7 8 9 3.0 _I I I

II

2.0 1.5 3.0 2.0 1.5 CM 1.0 0.5 C0 1.0

345

Io

is

6.4 Hydrodynamic

Coefficients

CD

4

3

2

i

o

i

BEARMANI et OU 1984)

TANAKA, Ct ol. (1983)

10

20

30

40

50

6Ò

70

80

KC

FIg, 6.23.

Mean values of C%f and CD vs.

KC forsmoöthçylinderS of square section

[Bearman, er al. (1984)

and Tanaka,

t al. (1983)]

3.0

2.0

1.0

o

Ocean Test

Structure Doto

GULF 0F MXIC0

Mean volue

fòt

,"ròuçhenad member

-

Mean value

tôr

clean member

-I- I

o

20

30

40

70

KC.

Fig 6.40.

Mean CD valües vs. KC

for smöoth and rough cylinders from Qcean.

Test Structure Data [Heideman.

et aL (1979)]

203

CN4

MARINTEK

HYDRODYNAPVHC DATA

2-D circular cylinders

Mass-, drag- and lift coefficients

Oscillatory flow

Interaction and shielding effects

Fig. 6.36.

CM, C0. and CL values vs. KC and spacing parameters for the

shaded

cylinder in a vertical live-cylinder array in tandem in a wave tank

Chakrabarti (1987)

The lift coefficient, CL, refers to the transverse

force.

Page:

_3.2Ç

Fig.:

Des. 1990

I-::=

N

0t1

0?1QT

j

-

-.----e-S.e0L 110 33 00 200 ''Q

e--

--\

r

* o 'o 20 40 50 60

I

u o 3 o -k o 3 2 o o 20 30 40

o

60 (cc

APPENDIX 2

8

NUMERICAL

MODELLING

OF

FLOW

AROUNÒ CYLINDERS

81

.nthuctión

Hydrodynamic forces due to Unsteady vortex shedding, play

an important role

in

_{prediting the hydrodynamic fotces}

on

riser

tubes and mOOring

lines,

wave excitation

and

studying the problems associated with the resulting

motions

of offshore structUres in waves.

An extensive

review

of

both the numerical methods.

thainly

_discrete

vottex

methods available

or

Under

development

an4

itnerestiñg

experimental

studies were

given iñ the previous

reports of the Ocean Engineering

Committee.

However, there is still a long way to go before xiumerical

modelling could be

implemented

in the design añd

analysis of oceañ structures.

In this report an. áttCmpt is made to moñitor the research

in numerical modelling of separated flow

_{arOund cylinders}

and to give an

ovetview of some

of the interesting

experimental

results which can be used to validate

numerical methods.

81

Numerical Simulation

f

Separated Flow Around

Cylinders

8.2.1

General.

_{During the last five years. or so there}

has been a considerable increase in the number of

uper

conputers, which played a vital role

_{in the development}

of Computational.

luid

Dynamics (CFD)

Nowadays,

besides

discrete

vortex

methods,

direct

methods. and

turbulent

model

methods

are

available to solve

Navier-Stokes (NS) equations.

While Considerable progress has been madó ¡n

this. fièld,

the numerical tools developed have

_{not been suitable for}

use in analysing practical. design problems.

_{Therefore it}

ro

is important that the researchers in this field should give

emphasis to simplifying these numerical techniques so that

they can be applied to design problems.

8.2.2

Discrete Vortex Method.

The previous report

of the Ocean Engineering Committee introduced a detailed

description

of

the discrete vortex

methods

which

are

referred

to as

the:

point vortex

method, shear layer

method,

vortex blob method, and Chorin's scheine which

can be applied to steady and oscillating

flow.

There

are

also

other methods, which include

viscous diffusion

technique,

determination

of

point

of

separation

using

Schlichting's

unsteady

boundary

layer

theory,

and

the

vortex-in-cell method.

Tiernroth

[8.11 applied the

Chorin's scheme

to simulate

flow over a circular cylinder in a wave field

with

current.

Smith and Stansby

[8.2],

Hansen et al

[8.3,8.4]

and Stansby and Smith

[8.5] also applied

the

Chorin's method

in

order

to

determine the separation

point.

Smith and Startsby [8.21 treated the impulsively

started

flow

around

a circular cylÍnder by using

the

so-called

vortex-in-cell

method,

that

is, the discrete

vortex method. Hansen

et

al

[8.3,8.4]

presented

a

numerical method to calculate the integrated hydrodynamic

loading and structural response

interaction

of

marine

risers.

The numerical method

is the discrete vortex

method

based on operator

splitting, random

walk and

vortex-in-cell method.

Stansby and Smith (8.5] treated

flow around a cylinder by the random vortex method.

Sakata et al

[8.6] used the vortex distribution method for

rectangular and circular cylinders in a uniform flow with

varied angle of attack

to

predict CD and CL and the

Strouhal number.

The predictions agreed well with the

corresponding experimental, results.

Dahong and Xuegeng

[8.7] used the same method to simulate flow around a

circular cylinder with a mirror image.

Both of tiese

investigations solved

the

boundary

layer

equation

to

determine the separation point.

1Jan [8.8] applied the thin shear layer method to simulate

separated

flow around

a

fiat

plate and bilge

keels in

oscillating flow.

van der Vegt [8.9] presented a formulation based on the

3-D

discrete

vortex

method.

He

rust

derived

a

Lagrangian formulation of the inviscid vorticity transport

equation which was then converted

into

a Hamiltonian

description.

8.2.3

The

Direct

Simulation

Method.

As

the

Reynolds numbers increase

it

becomes difficult to apply

finite

difference

or

finite

element

methods

to

the

simulation of separated flow around cylinders.

Kawamura [8.10]

explained

the main reasons

for this as

follows.

The

Non-linear

terms

in

the

NS

equation

generate

components of short wave length disturbance.

These

components of short wave length are unlikely to diffuse.

However, the components whose wave lengths are shorter

than

the

length

of

the

grid

of

finite

difference

are

apparently

interpreted

as

being components with

longer

wave lengths in

the

numerical

procedure.

This

phenomenon

is

called

"aliasing

error";

which leads to instability

of

numerical

calculations and

gives diverging

solutions.

Therefore, in order to make the numerical

scheme

stable,

it is always necessary to

eliminate

the

components whose wave lengths are shorter than the grid

length of finite difference.

It is

possible to

realise this

by introducing an appropriate differentiation with

respect

to

the non-linear terms uduldx, vdu/dy, etc.

in the NS

equation.

Kawamura [8.10] applied

this

method to separated flow

around a

circular cylinder

in uniform flow. In this

study the Reynolds numbers varied from 2000 to 40000

and the finite

difference

solutions were

carried

out

utilising 80 x 80 grids.

Baba

and

Miyata [8.11]

applied

the

finite

difference

method to separated flow around a circular cylinder in

oscillating flow.

In this study the Reynolds number was

1000 and the Keulegan-Carpenter number was 5.

The

results were

in good agreement with

the' corresponding

results of flow visualisation.

Murashige

t aI [8.12] and

Kinoshita et

al

[8.131 applied the same method to the

flow around an oscillating circular cylinder using the body

fitted co-ordinates, the moving mesh' technique

and the

marker-and-cell method at the Reynolds number of 10000

498

and the

_{Keulegan-Carpenter numbers of 5,}

_{7 and}

_10.

The computed. results simulate the effect of the

Keulegan-Carpenter number on

the

flow field

with an excellent

agreement

with

the

corresponding

measured

data.

Lecointe and Piguet [8.14,8.15]. Graham and

_Djahansouzi

[8.161 and Braza and Minh [8.171 also applied the finite

difference method to simulate flow around

_cylinders.

Instead of the finite difference technique, the FEM (finite

element

method)

cari also be

applied

provided

the

Reynolds number is less than 1000.

This condition has

to be satisfied

for

achieving

subie

calculations.

Triantafyllou and Karniadakis [8.18] combined the spectral

element

method

with

the

finite

element

method

and

calculated

the

forces on a

_{vibrating cylinder in steady}

cross-flow.

In this case the Reynolds n'imber

_{was 100.}

In

the

case

_{of flow around a submerged cylinder with}

free surface, Miyata et al [8.19] and Esposito et al [8.20]

successfully used the finite

difference

method

and

calculated breaking waves.

8.2.4

_{Turbulent Model Method.}

_{Important information}

about

turbulent

flow may be

the

global average

charactenjsti of flow rather than

the local

instantaneous

information.

_{Therefore the velocity and}

_pressure

_field

are divided into

an average and

flucttiating

part and,

moreover, the fluctuating term (so called

_{Reynolds stress)}

is obtained by averaging the NS equation - the equation

to model

turbulent

flow is then derived. Various

equations to model turbulent flow are derived according to

averaging of time or space.

The time

averaging model

is also called

the turbulent

transport model or K-E model.

_{The turbulent flow in}

boundary layer

is well

_{modelled so}

_far,

_{and numerical}

results. for flow

around

a

wing section

are

in gOod

agreement

with the

corresponding

experimental

results.

Murakami and Mochida [8.211 applied this method to

flow-around

a

rectangular

cylinder in uniform flow and

simulated wind flow around a building.

_{This method}

_{. is} a

practical

enginering

technique.

rather

than

a

pure

theoretical method,

since it requires 5 to 10 empirical

parameters for its formulation.

_{This K-E model has not}

yet been applied to oscillating flow problems.

The space

averaging method, which

is also called

the

Large

Eddy

Ssmulation (LES)

method,

requires one

parameter

only

for

the

_modelling

of

flow. The.

fundamental equation reduces -to

the NS equation when

the length of the grid of the finite element

_{tends to zero.}

This

method

attracted

attention

recently for the

fundamental research on turbulent flow.

_{Horiuti [8.22]}

applied this LES method to turbulent

_{flow simulation of}

inner duct flow between

_{paraïlel plates.}

_{The results}

show the transient - process from laminar to turbulent flow

and the fully turbulent condition in which

the Reynolds

number is 27500.

The LES method has not yet been

_{applied to outer flow}

such as

separated

flow around

cylinders in uniform or

oscillating flow.

8.2.5

_{Approximate Solution of NS Equation.}

Beasho

[8.23] proposed an approximate solution method in which

the. non-linear terms of NS equation

were considered

as

the non-homogeneous term of the

Oseen equation.

This

method

is

applied

to oscillating

flow around a

circular cylinder.

_{The predicted added mass coefficient for the}

Keulegan-Carpenter number Kc less

_{than 12 agrees}

_well

with the

corresponding

_experimental

_{results, but the}

predicted drag coefficent

_{is in}

_{good agreement with}

_the

measurements only

in

_{the lower Kc number}

_{range, less}

than 1.5. Moshkin _{et al [8.24] obtained numerical}

results from the NS equation

_{assuming flow of very low}

Reynolds number, that is, Stokes

system.

8.2.6

Practical

Methods, Huse and Muren [8.25]

presented and evaluated a principle

_{for interpreting drag}

forces

on

bodies in oscillatory flow, based on

the

assumption

that

the

drag

coefficient

is. the

same in

oscillatory flow as:

in' stationary

flow, provided wake velocities

are

_{properly corrected}

_for.

_{GOod correlation}

with

experimental

data

for

rectangular

sections was

obtained over a Keulegan-carpenter

_{number. range from}

less than unity and up to infinity.

For circular' cylinders

a _reasonably

_correct-

_trend

_was

achieved: - at high

completely when the Keulegan-CarPenter number range

is

below 20.

8.2.7

Future

Development.

The

progress

of

development

of

numerical models

for

flow

around

cylinders is almost

the same as

it was

in ITTC 87.

The number of examples

giving numerical results

is still

limited.

A thorough validation over a wide range of

values of Rn and Kc

has yet to be carried out.

No

simulation method has been

developed for the separated

flow around cylinders in combined current,

oscillating and

wave flow conditions.

No numerical

results have yet

( been

reported on the effect of

surface

roughness of a

cylinder.

8.3

Experimental Studies

Refined

experimental

studies

8.3.1

General.

concerning

pressure

measurements and flow

visualisatiOn

on cylinders in

different flow conditions

have been carried

out for the validation

of numerical modelling

techniques.

in

addition

tO measurementS for

the

determination

forces

acting on full-scale

structures. However,

number of papers

published is still limited. Significant

developments

in

experimental

methods and

increase

experimental measurements

based on these methods

expected in the coming years.

t8.3.2

Review of Recent Experimental Work.

Experi-ments

involving flow visualization were

reported

by

Rockwell et al [8.261, Grass

et al [8.27], Teng

and Na.th

[8.22] and Ongoren and

Rockwell [8.29,8.301.

Onoren

and

Rockwell [8.29] conducted

the

flow visualization

experimentS of

controlled

oscillations in

a

direction

transverse to the

incident flow around circular,

triangular

and square cylinders.

In these experiments the

Reynolds

number range

of 584-1300 was rather

low.

They

observed the characteristics

of

the

phase of the

shed

vortex with respect to

the cylinder displacement.

which

depend on the cylinder

geometry.

These results

are

very useful for the validation of the CFD.

of the

in

[8.32], Wolfram and TheophanatoS [8.41] and Wolfram et

are

al 8.42].

Pressure

measurements

were

conducted

by Borthwick

(8.31).

Chaplin

[8.32] and Kato and

Ohmatsu (8.33).

ICato and Ohniatsu In Ref. (8.331 presents the results

of

an experimental investigation of hydrodynaifliC forces

and

pressure distribution acting on a

vertical circular cylinder

oscillating

at low frequencies

in still

water.

These

forced

oscillation tests

can

be

used

to

validate

the

coresponding CFD.

The investigators

of Ref.

(8.331

obtained the

following

results:

The flow around an

oscillating

cylinder

is

3-D,

and

the

in-line

force

coefficients vary

along

the

direction

of

the

axis.

Therefore, the Morison

formula should be interpreted as

the one that holds for forces

averaged along the direction

of an axis.

The lift force oscillates irregularly with the

double frequency of the oscillation.

The lift force can

be described by a

deterministic dynamical system

with a

small degree of freedom and it suggests that the system

of the lift force may be

transposed to a chaotic system.

The range of the Reynolds

number in

this investigation

was 14400 and

the

Keulegan-CarPenter number varied

from I to 15.

Large scale

experiments were conducted by Humphries

and

Walker(8.34], Grass

et

al

[8.27),

Kasahara

et

al

[8.35). Nath

[8.36],

HumphrieS [8.37], Bearman

[8.38),

Blick

and

Klopman

[8.39].

Justesen

[8.40),

Chaplin

9.

THEORETICAL

METHODS

FOR

PREDICTING NON-UNEAR

PHENOMENA

9.1

Introduction

In the last decade an

increasing interest bas been

shown

by hydrodynamics

researchers in the

investigation of the

influence

of

non-linear

effects

on

the

behaviour

of

offshore structures.

These efforts were first devoted tO the study

of vessels

which showed, in moderate

sea conditionS,

resonances out

of the typical range

of wave periods.

Classic examples.

are the slow-drift

horizontal motions of moored vessels,

the long-period vertical motions of

floating platforms with

small water-plane area, and the higil-frequency

springing

forces on Tension Leg

Platforms (TLP5).

[7.8]

Demirbilek,

z.,

Moe,

G.

and

Yttervoil,

p.0.: aorjn3

FoñflUla Relative Velocity VS

lndepefldeslt Flow

Fields

Formulations

for

a Case gepresentiflg

Fluid Damping".

Symposium on Offshore

Mechan1C and ArctiC Engineering.

vol. lI, p. 25, 1987..

(7.91

Huse, E.

and Muren,

P.: "Drag

in OscillatorY

Flow interpreted from Wake Con

ideratiofls".

Paper No.

5373. Offshore Technology Conference,

HoustOn, 1987.

[7.101

FaltinSen, O.M. and

Sortland,

B.: "Slow

Drift

Eddy-ma

Damping

of

a

Ship",

Applied

Ocean

Research, vol. 9, nO.

1, JaflUaT)V 1987. [7.111

Cotter,

D.C. and Chakrabarti,

S.K.:

"Effect

of

Current and

Waves on

the Damping

Coefficient of

a

Moored Tanker".

Paper No. 6138,

Offshore Technology

onfereflCei HoustOn, 1989.

[7.12]

Standing,

R.G.,

Brendling,

W.J.

and Wilson,

D.: "Recent Developments

in

the Analysis of Wave

Drift

Forces, Low-Frequency Damping änd Response".

Paper

No. 5456,

Offshore

Technology

Conference,,

Houston,

1987.

17.131

Hearn,

G.E.,

Tong.

K.C. and Lau,

S.M.:

"Sensitivity of Wave Drift Damping Coefficient Predictions

to

the Hydrodynamic Analysis

Models Used in the Added

Resistance

Gradient Method".

Symposium on Offshore

Mechanics and Arctic Eiigineering,

vol.

11, p. 213, 1987.

[7.14]

Hearn.

G.E..

Tong,

K.C. and

Ramzan,

F.A.: "Wave Drift Damping

Coefficient

Predictions and

Their

Influence

on

the

Motions

of

Moored Semi-SubmerSibles".

Paper No. 5455.

Offshore Technology Conference, Hòuston, 1987.

[7.151

Ream, G.E.. Lau, S.M.

and Tongs K.C.: "Wave

Drift Damping

Influences

Upon

the

Time

Domain

imulatións

of Moored

Structures".

Paper

No. 5632,

Offshore Technology Conference, HoustOn, 1988.

[7.16] Heamn,

G.E.

and

Tong,

K.C.. "A

Comparative

Study

of

Experimentally

Measured and

Theoretically

Predicted Wave Drift Damping

Coefficient.s".

Paper No.

6136, Offshore Technology

Conference, Houston, 1989.

[7.17] Kaplan, P. and SankarariaraYaflan,

K.:

'Prediction

of

Low

Frequency

Wave

Damping:

A Simplified

Approach".

Paper

No. 6135,

Offshore

Technology

Conference, Houston, 1989.

[7.181 Wichers,

J.W.E.: "WaveCurrent

Interaction

Effects on Moored Tankers in

High Seas",

Paper No.

5631, Offshore Technology

Conference, Houston, 1988.

[7.19] Saito, K.

and Takagi.

M.: "On

the

Increased

Damping' for a Moored

Semi_Submersible Platform During

Low-FrequencY

Motion

in. Waves". Symposium

on

Offshore,

Mechanics

and. Arctic

Engineeering,

vo.

Il,

p. 147.

1988.

(7.20] IGnoshita,

T.

and' Takaiwa,

K.: "Slow

Motion

Forced Oscillation Tests in Waves of Floating Bodies

-Wave Drift Damping and

Added Mass". Symposium on

Offshore Mechanics and

Arctic

Engineering,

vol. II, p.

139, 1988.

[7.21]

HelvaciOglu, I.H. and Incecik,

A.: Second-Order

and

Short'CreSt1'

Wave

Effects

in

Predicting

the

Behaviour

of

a

Compliant

Structure",

Symposium on

Offshore Mechanics and Arctic Engineering, vol.

II, The

Hague, 1989.

(7.22]

Wichers,

.I.W.E. : "A

Simulation

Model

for

a

Single

Point

Moored

Tanker",

Publication

No. 797,

MARIN, Wageningen, 1988.

[7.23]

Pinkster,

J.A.

and

Wichers,

.LE,W.: "The

Statistical

Properties.

of

Low-Frequency

Motions

of

Non-Linearly

Moored

Tankers"

Paper

No. 5457,

Offshore Technology Conference, Houston, 1987.

[7.241

Tèla, Y.S.D. et al: "A

Rational Dynamic Ma1YÇ

Procedure

for

Turrent Mooring Systems".

Paper

No.

5527, Offshore Technology

Conference, Houston. 1987.

[7.25]

Ruse, E.. Yu, B.K. and

Oydvin. A.: "Transient

Rig Motions

and

Mooring Line

Tensions

Alter

Line

Breakage".

Symposium Ofl

Offshore MecbaiüCs and

Arctic

Engineering, vol. II, p. 367.

1987.

(7.26]

Zhao,

R. and Faltinsen,

O.M.: "A Comparative

Study of Theoretical Models for Slòwdrift Sway Motion

of

a Marine Structure".

Symposium on Offshore

Mechanics

and Arctic Engineering. vol.

II, p. 153, 1988.

(7.271

Marchand,

E.

le.

Berhault,

C.

and

Mohn,

B.: "Low Frequency Heave Damping of Semi-Submersible

Platforms:

Some Experimental Results".

Conference on

Behaviour

of

Offshore

Structures,

vol.

Ii,

Trondheim,

1988.

[7.28] Huse,

E.: "Motion

Reduction of Offshore

Structures

by

Damping Tanks".

3rd

Symposium on

Practical Design of Ships and

Mobile Units, Trondheiin,

1987.

[7.29]

Gabrielsen,

P.

and

BjordaL.

B.N.: "Exact

Installation of 'Concrete Platforms Over Pre-drilled Wells".

Paper

No. 5604.

Offshore

Technology

Conference,

Houston, 1987.

(7.30] Lokna,

T.

and

Nyhus,

K.A.: "Damping

and

Natural Frequencies During Towout and Installation of the

Guilfaks

'A'

Platform".

Paper

No. 5786,

Offshore

Technology Conference. Houston,

1988.

[7.311

Pinkster,

J.: "Aspects

'

of

Positioning Large

Tankers.

Offshore".

4th

Conference

on Floating

Production Systems, 1BC Tecithical Services Ltd., London.

December 1988.

[8.11

Tieniroth, E.C.: "Simulation of the' Viscous Flow

Over

a

Cylinder

in

a

Wave Field

16th ONR

Symposium. .1986.

(8.2]

Smith,

P.A.

and

Statisby,

P.K.:

"Impulsively'

Started Flow Around a Circular Cylinder by the Vortex

Method".

J.Flùid Mech., vol. 194, pp

45-77, 1988.

(8.31

Hansen, 1LT., Skornedal,. N.G.

and Vada, T: "A

Method

for

Computation

of Integrated

Vortex

Induced'

Fluid Loading and. Response- Interaction of Mar ne Risers

in Weave and Current". hit!

BOSS COnference, 1988..

18.41

Hansen.

H.T..

Skomedal,

N.G.

T.: "Computation of Vortex Inzced Fluid

Response

Interaction

of Marine Risers".

Symposium. The Hague. 1989.

[8.51

Stansby, P.K. and Smith, P.A.:

"Flow Around a

Cylinder by the Random Vortex

Method".

8th OMAE

Symposium, The Hague, 1989.

[8.61

Sakata,

H.,

Machi, T.

and

Inarnuro,

T.: "A

Numerical Analysis

of Flow Aroun4

Structùres

by

the

Discrete Vortex Method",

Tech.

Repoñ

of Mitsubishi

Heavy Industries, vol. 20, no. 2, 1983.

[8.7]

Dahong. O. and Xuegeng, W.:

"Vortex Shedding

from a Horizontal Circular Cylinder in Oscillatory Flow".

6th OMAE Symposium. Houston, 1987.

Numerical Study

of

Two-Flow Plast Bluff Bodies

at

Applied Ocean Research,

vol. [8.81 Lian,

W.: "A

Dimensional

Separated

Môderate KC-Numbers".

10, no 3, 1988.

ES.91

van der Vegt. J.J.W.: "A

Variauonally Optimised

Vortex Tracing

Algorithm for Three=DimeflSiOflal Flows

Around Solid

Bodies". Doctoral Thesis

of The Delft

Technical University. 1988.

[8.10] Kawamura,

T.: "Numerical

Study

of

High

Reynolds Number Flów Around a Circular

Cylinder".

PhD Thesis of University of Tokyo.

1983.

[8.11] Baba, . and Miyata,

H.: "Higher-Order

Accurate Difference Solutions of Vortex Generation from

a Circular Cylinder in an

Oscillatory Flow".

Journal of

Computational Physics. no. 69. 1987.

[8.121 Murashige,

S.,

Kinoshita.

T.

and Hinatsu,

M.: "Direçt Calculations of Navier Stokes Equations for

Forces Acting on a Cylinder

in Oscillatory Flow".

8th

OMAE Symposium. The Hague.

1989.

[8.13] Kinoshita,

T.,

Hinatsu, M. and Murashige,

S.: "Simulations

of

Forces Acting

on

a

Cylinder

in

Oscillatory Flow by

Direct

Calculation

of

the

NavirStoke5 Equations".

4th

International

Symposium

on Numerical Ship

HydrodynamicS. 1989.

[8.14] Lecointe, Y. and Piquet.

J.: "Periodic

and

Multiple Periodic Behavior of

Locked-In

Vortex

Shedding".

16th ONR Symposium, 1986.

[8.15]

Lecointe, Y. and Piqüet.

J.: "Computation of the

Wake Past an Oscillating Cylinder".

International BOSS

Conference. 1988.

[8.16]

Graham,

J.M.R.

and

Djahansouzi,

B.:

Hydrodyfla!fliC Damping of

Structural

Elements".

8th

OMAE Symposium. The Hague,

1989;

[8.17)

Braza. M. and

Minh. H.H.: "Dìrect

Numerical

Simulation

of

Certain 2-D Transition

Features

of the

Flow Past a Circular Cylinder".

8th OMAE Symposium,

The Hague, 1989.

[8.18]

TriantafylløU, OS. and Karniadakis, G.E. "Forces

on a

Vibfating Cylinder

in

Steady, Cross-Flow".

8th

OMAE Symposium. The Hague,

1989.

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and

Vada,

Loading and

8th OMAE

[8.19]

Miyata, H., Kajitani, H., Zhu, M., Kawano,

T.

and Takai, M. "Numerical Study of Som

Wave-Breaking.

Problems by a

Finite

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Method".

J.Kansai

Society Naval Architects. Japan, no. 207,

December 1987.

[8.20]

Esposito.

P.O.,

Graziani,

O.

and

Orlandi,

P. "Numerical Solution of Viscous

Flows About Submerged

and

Partly

Submerged

Bodies". 4th

International

Symposium on Numerical Ship-hydrodynamicS,

1989.

[8.211

Murakami. S. and Mochida, A.: "3-D

Numerical

Simulation of Airflow Around a Cübic Model by Means of

the K-c

Model".

Journal

of Wind Engineering and

Industrial AerodynamiCs, 31,

Elsevier Science

Publishers,

1988.

[8.22]

Horiuti,

K. "Comparison

of Conservative and

Rotational Forms in large Eddy Simulation of Turbuler.t

Channel Flow".

Journal Computing Physics, 71, 2, 1987.

[8.23]

Bessho, M.: "Study of Viscous Flow by

Oseen's

Scheme (Fourth Report, Two-Dimensional Oscillating

Flow

Without Uniform Velocity)"

Journal óf Society of Naval

Architects of Japan, 161, 1987.

[8.24]

Moshkin, N.P., Pukhnachov, V.V. and Sennitskii,

V.L.: "Numerical

and Analytical Investigations of a

StauonaTy Flow Past a Self-Pr9pelled

Body".

4th

Intl

Symposium on Numerical ShiphydrodynamiCs. 1989.

[8.25] Huse, E. and

Muren,

P.: "Drag

in Osçillatory Flow

Interpreted

From

Wake ConsideraUônS".

OTC

Paper 5370, Houston. 1987.

[8.26] Rockwell,

D.,

Gurnas,

C..

Kerstens,

P.,

Backestose,

J.,

Ongoren, A.,

Chen.

J.

and Lusseyran,

D. "Computer-Aided

Flow Visualisation". 16th ONR

Symposium, 1986.

[8.27]

Grass,

A.J..

Simons. R.R.

and

Cavanagh.

N.J.: "Vortex-Induced

Velocity

Enhancement

in

the

Wave-Flow

Field Around Cylinders".

6th

OMAE

Symposium. Houston, 1987.

8.28]

Teng. C-D. and Nath, J;H.: "Periodic Waves on

Horizontal Cylinders".

7th OMAE Symposium, Houston

1.988.

[8.29)

Ongoren,

A.

and Rockwell, D: "Flow Structure

From

an

Oscillating Cylinder

Part

1: Mechanisms

of

Phase Shift and Recovery in the Near Wake".

Journal

of Fluid Mechanics, vol. 191.. pp 197-223, 1988.

[8.30]

Ongoren, A. and Rockwell, D.: "Flow Structure

From ari Oscillating Cylinder Part 2: Mode Competition

in the Near Wäke".

Journal of Fluid Mechanics,

vol.

191, pp 225-245, 198S.

[8.31) Borthwick,

A.:: "Orbital. Flow Past a Cylinder

Experimental Work".

6th OMAE Symposium. Houston;

1987.

[8.32]

Chaplin

J.R.: "Loading on a Cylinder in Uniform

OsciUatory Flow

:

Part

I

- Planar

Oscillatory

Flow".

Applied Ocean Research, vol. 10, no. 3. 1988.

(8.331 Kato

S. and Ohrnatsu, S.:

"Hydrodyflarnic Forces

and

pressure

Distributions Ofl a

Vertical

Cylinder

Oscillating in Low

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APPENDIX 3

OTC 5370

Drag in Oscillatory Flow Interpreted From Wake Considerations

by E. Huse,' Marintek NS, and P. Muren, Norwegian Inst. of Technölbgy

Copyright 1987 Offshore Technology Conference

This paper was presented atthe 19th Annual OTC in HÒuston, Texas, April 27-30. 1987. The material is subject to correction by the author. Permission

to copyis restricted to an abstract ot not more than 300 wordS,

MMARY

The paper presents and evaluates a principle for interpreting forces on bodies in oscillatory flow, based on assuming that the drag coefficient

is the same as In stationary flow, provided wake

velocities are prOperly corrected for. The

resulting formulae describe general trends

correctly, and can in some cases also be used for quantitative calculation of oscillatory flow phenomena.

INTRÒOUCTIÓN

For many types of offshore structures drag

for-cesare a major part of the design loads, or they can, be of decisive importance t° the operational

performance of the structures The drag forces can be due to current, to the orbital motion in the waves, tO the oscillatory motions of the structure itself, or to various combinations of these. effects. Until

now normal design procedures have been based on

.l .ulation of drag forces by the wellknown Mori son

rmùlation, using drag coefficients determined

experimentally for each structure, or taken from

empirical data.

The present paper introduces and discusses an alternative way of dealing with drag forces in oscillatory flow. It is based on assuming that the

drag coefficient itself is the. same as in stationary

flow, provided the correct relative velocity is

applied. Thi.s means tát the flow velocity has to be

cOrrected for the effect of the wake generated by the previous ôscillàtions. This correction can be done on the basis of momentum considerations and turbülent mixing theory. Based on these principles

one can now calculate how drag forces in oscillatory

flow are influenced by for instance

Keulegan-Carpenter number, geomètry of motion pattern, effect of superposed currént, effect of interaction between several bodies etc.

'eferences _and illustrations at end. of paper.

SCOPE OF INVESTIGATION

The scope of the present investigation is': To develop analytical foñnulae for practical

calcùlations of drag in oscillatory flow

based on wake considerations, momentum

balánce and turbulent mixing theory. To develop computer programs for more exten-sive calculations.

To investigate ranges of applicability of the

theory by verification against published

experimental data, and by means of new

experiments.

To discuss possible practical applications of thê theory.

STATE-OF-THE-ART SURVEY

In practical design and analysis' of offshore

structures the hydrodynamic fOrces acting on a

well submerged, oscillating body are calculated by

the Morison formula.

F = - pVC _{- ACdjVIV}

where the mass coefficient Cm and the drag

coef-ficient Cd depend on:

- the geometry of the body

- the Reynolds number

The Keulegán-Carpenter nûmber.