Drift forces in directional seas

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Laboratorium voor Schaepc.hydrornechanica Archief Mekafweg 2, 2528 CD Deift Tel.: 015- 783373 - Fax:015- 781035 Z50545 - Pubi. MARINTEC DRIFT FORCES IN DIRECTIONAL SEAS by: J.A. Pinkster

Paper to be presented at: MARINTEC CHINA '85

Shanghai, December 2-8, 1985.

-1--DRIFT FORCES IN DIRECTIONAL SEAS

by: J.A. Pinkster

Head Project Department/Ocean Engineering Division Netherlands Ship Model Basin

The Ede/Wageningen Laboratories of

1. INTRODUCTION

The analysis of the behaviour of vessels _{moored at sea}

is generally based _{on measurements from}

model tests in

ir-regular uni-directional _{waves carried out in}

suitable model facilities.

Data obtained from _{such model tests has in the past}

proved to be indispensable _{in the design of}

offshore

float-ing structures. Although, _{at all times, the}

conditions of

the model tests represent _{a simplified reality, the}

on-site

performance of full scale _{structures bears witness}

to the

general validity of _{such model test data as a sound basis}

for _{judging the performance of}

a particular design with

regard to both the motion _{behaviour and the loads}

on the structures.

Notwithstanding, however, the _{generally accepted}

valid-ity of model testing under such simplified _{conditions, there}

is _{a need to investigate, in more detail,}

the effect of

directional spreading _{of irregular waves, as}

it occurs in reality, on the loads and _{motions of floating offshore}

structures. See, for instance, _{ref. [i].}

One method of obtaining _{such information is to conduct}

model tests in basins _{fitted with wave}

generators which have

the capability to generate _{irregular directional waves. Such}

model tests will produce quantitative data, based ori a

phys-ical reality, of the _{effects of directional waves. [i], [2}

-3-Another method of _{obtaining data}

on the effects of

multi-directional _{seas on the behaviour}

of floating

struc-tures is based on

theoretical computations; [4]. _{It will be}

clear that efforts

should be made to compare results of such

computations with model _{test results. This}

is _necessary

since significant _{physical effects}

peculiar to directional

seas may be present which _{are not accounted for in}

the theo-retical approach.

For irregular uni-directional _seas,

computer programs

exist which _{can, with reasonable}

accuracy, predict the

be-haviour of a moored _{vessel in both the}

frequency and the time domain; [5] and [6].

This type of _{computer program can be used}

to assess

certain aspects associated with multi-directional _seas

pro-vided a _{realistic formulation}

can be given for the

wave

loads, both oscillatory, _{first order wave loads}

and mean and

low frequency second order drift forces in _irregular

direc-tional waves, [7].

In this study attention _{is paid to such formulations}

for the mean and slowly _{varying drift forces.}

Expressions

are given for the wave drift _{forces for the case that the}

undisturbed incident directional _{wave field is described by}

a Fourier series with random phase _angles.

Based on such descriptions _{for the second order wave}

loads, _{frequency domain results}

are obtained in terms of

expressions further insight can be obtained _{regarding the}

effects of directional seas.

Attention is also paid to representation of second

order wave loads using the Volterra series _{or Functional}

Polynomial approach. This allows the use of measured random

irregular time records of waves approaching _{the body from}

various directions combined with second order impulse

re-sponse functions in order to generate force records in

multi-directionál waves for the purpose of time domain

simulations.

Numerical results on the wave drift forces in _{a special}

type of directional sea, _{namely, regular cross waves,}

ob-tained using three-dimensional diffraction calculations and

direct integration of second order _{pressure are compared}

with results of model tests.

Results are also given of tests in irregular

uni-direc-tional waves and of tests in irregular _{cross seas on the}

basis of which the validity of the superposition principle

with respect to mean drift forces could be checked. For

b-

-5-2. THE INCIDENT WAVES 2.1. General

In order to derive expressions for the drift _{forces in}

directional seas we will first choose _{an appropriate}

de-scription of the _{irregular directionally spread sea. We}

assume that the wave elevation in a point can be described

by a double Fourier series, which characterizes _{the surface}

elevation as a sum of regular long-crested _{waves from}

vari-aus directions: N M (t) = E E . cos(w.t + e. i

ik

i=l k=l (2.1) in which: = wave frequency = random phase angle

N = number of discrete frequency components

M = number of directional components

= wave amplitude

i = wave frequency index

k = wave direction index.

In a multi-direction sea the amplitudes of the wave

components are found from:

= /2 s (w.»4. ) J L)

ç i (

where:

S(wi,1Pk) = directional wave spectrum = frequency interval

= direction interval.

For the directional spectrum _various

formula-tions can be chosen, [8] and [9].

After chosing the random phase angles, _{a time record of}

the surface elevation can be computed from equation (2.1).

2.2. Evaluation of wave elevation in time

The computation time necessary _{to evaluate the double}

summation can be reduced by replacing _{equation (2.1) by the}

following equivalent expression:

N

(t) E _{(a. cos w.t + B. sin .t)}

i J_ i i 1=1 in which: M a. = E . COS E.

k1

M B. = - E . Sin E. k=l ik

ik

(2 3) (2 4) (2 5)

This shows that by first computing _{the coefficients ai}

and B1 the wave elevation _{at each time step can be found}

average:

S(w.) w = (. cos w.t + . sin

i i i i

2.3. The spectrum of the wave elevation measured in _{a ncint}

Having selected this description for the waves it is of interest to check the properties of such a discrete formula-tion. For this we chose the wave spectrum.

The point spectrum S(w) can be

found from equation

(2.3) by taking the mean square, or variance, of the wave

elevation component with freauency w as follows:

(2.6)

where the overbar indicates that the mean value should be

taken.

Given a particular realization of (t), i.e. a fixed

set of phase angles £.k, equation (2.6) results in:

which, taking into account equations (2.4) and (2.5),

re-suits in:

M M

S(w.) _{= ½} z z _{cos(c.k -} . . . . (2.8)

i

k1 £=l

This shows that the spectral density of a particular reali-zation is a function of the random phase angles E.k and

Averaging equation (2.8) over all realizations (all

possible Lik and values) gives the following ensemble

M

2

E{S(w.)

SCi)] _{= ½} E

ik

i

_k1

This result implies that with respect to the spectral density the wave elevation formulation given by equation

(2.1) is non-ergodic, i.e. time averages are not equal to ensemble averages. A consequence of the non-ergodicity of

such formulations is, for instance, that the wave spectral

density, for any particular realization, becomes dependent on the location in the horizontal plane; [lo].

Ta}ing into account equation (2.2), equation (2.9)

becomes:

M

E[S(u.)

w] =

E S

i'k

i

_k1

Eliminating Lw on both sides and passing from a discrete to a continuous formulation results in:

27r =

f

d

= S(w.)

i

_o

(2 9)

(2 10)

(2.11)

which shows that the point spectral density of a realization

of the type given by equation (2.1) is only equal to the

input point spectrum in the ensemble average sense. A more detailed analysis of the properties of the double Fourier

series description in which also the phenomenon of wave

grouping, which is of importance for the low freauency drift forces, is t&ken into account, has been given in [li] and [12].

-9-3. SECOND ORDER WTE DRIFT FORCES 3.1. General

Second order wave forces acting on vessels or

struc-tures in waves can be computed based on direct integration

of second order pressures and forces, [13]

In vector notation the general expression for the sec-ond order wave forces is as follows:

(1)2 = _{- f ½pg n dl - 5f {- ½ V} (1) 2 -

_ndS+

r WL S o - 5f {_

((1) v('))}

dS + X ₍ (1)) ± t g S o - If {- + dS s o (3 1) in which: 1)

= first order relative wave elevation around the water-line WL

(l) _{= first order velocity potential including effects of}

incoming waves, diffracted waves and waves generated by the body motions

mass matrix of the body in vacuum

first order oscillatory linear motlon vector

first order oscillatory angular motion vector of the body

outward pointing normal vector to the hull mean wetted hull surface

M =

(1)

_X = = = S0 =

(2)

= second order "incoming wave2' potential

(2)

second order "diffraction" potential.

3.2. Realizations of drift force records in time domáin

Equation (3.1) shows that the second order force

con-sists of terms involving products of first order quntities or integrals of products of first order quantities.

Using the discrete formulation of equation (2.1) for

the waves it follows that the second order force in irreg-ular directional seas can be written as:

N N M M

F2(t)

= E E E _ç. P. cos{(w

- wjt +

j

ik

i J

i=l j=l

:k=l £=l

+ (E.

_-ik

)}+

N N M M + E E E E ik sin (. - (Ü .)t +i _J i=1 j=l k1 £=l

in which Pjj and are the in-phase and quadrature

quadratic transfer functions derivable based on equation

(3.1).

For instance, for the quadratic transfer function due

to the first contribution for equation (3.1) we may write, in vector notation:

-11-pijki

= -

f

/4

Y

Y

cos(E

- E

_{) (3.}

r. r. r. r.

WL 1k j9. lic ji

0ijki =

f

1/4Pg

Y

Y

Sin(E - Er. ) dl . . (3.4)

WL ik ji _1k ji

in which:

= first order transfer function for the relative _wave

r. ik

elevation for wave frequency w _{and wave direction}

Er = phase angle of relative wave elevation

j.k

= direction cosine of the waterline element _{dl with}

components n1, fl2, n3

i, j = wave frequency indices

k, £ = wave direction indices.

Each of the components of equation (3.1) _{contribute to}

the total in-phase and quadrature _{quadratic transfer}

func-tions. This will not be treated further _{here. It should be}

mentioned that evaluation of these transfer functions can be

made through the application of 3-D diffraction computer

programs, [13].

The contribution due to the non-linear potentials (2)

and (2) _{are approximated using the method given in [13]}

making use of the non-linear second order potential for the

undisturbed directional waves given in [14] . For irregular

long-crested waves the validity of this approach was

recent-ly demonstrated by comparison of results obtained using this

approximation and results based on a more exact solution

We will assume hereafter that the quadratic transfer

functions include contributions from all _{components of}

equa-tion (3.1). It should be noted that through regrouping of

terms such as given in equations (3.3)

_{and (L4) certain}

syrrinietry relationships will apply to Pjj and These will be given in a later section.

3.3. Mean value of realizations of wave drift force records

The time average drift force is found from equation

(3.21 for 1j: (2) N M M F (t) = Z

f

1ikL cos(E.

- c.) +

i=1 k=l 2. 1 + 0iikZ sin(s. - c. ) -11ç -i2.

It can be shown that the ensemble average of the drift

force is:

N M

E[F2]

= = E E 2 p

j=1 k1

k iikk

which, in continuous form becomes:

2ir = 2

f

f

S

(u,ct) P(w,w,,)

d dw

00

(3 5) (3 6) (3 7)

in which P(w,w,c,cz) is _{the quadratic transfer function of}

the mean drift force in regular waves of frequency w from

-13-3.4. Frepuency domain representation of the _{drift forces:}

Drift force spectrum

It is of interest to obtain information _{on the low}

fre-quency drift forces in irregular waves in the frefre-quency

do-main. To this end we use equation (3.2) _{as starting point to}

obtain the spectral density of the drift forces.

The derivation of the spectral density will not be giv-en here. The procedure is similar to that followed in Sec-tion 2.3 for the wave spectrum. As is the case with the wave

elevation in a point as given by equation (2.1) the second

order force given by equation (3.2) is non-ergodic and for a

particular realization (choice of s) the spectral density

of the force is dependent on the phase angles c.

The average over all realizations (ensemble average) is found to be: 21T 2i

sF() = 8

_f

_J

_f

s (+p,a) d d dw

000

(3 8) in which:

T(±p,w,,$) =p(+,w,cr.,) + iQ(w+i,w,a,)

For the quadratic transfer functions the following sym-metry relationships apply:

p(w--,,,S)=

p(ui,w+7a

(3.9)

(311)

which is the same as:

T(w+p,w,a,)=T*(,w+i,,a)

₍₃₁₂₎

in which * denotes _{the complex conjugate.}

Similar symmetry

equations apply to the discrete quadratic transfer _functions

ijk 0ijk and the amplitude TjjkL.

3.5. Drift forces in irreqular cross seas

We will apply equation (3.8) _{to a simple case to show}

the effect of _{wave directionality on the wave drift force.}

To this end

_{we assume that the irregular}

directional sea

consists of a superposition _{of two irregular long-crested}

uni-directional seas approaching _{from directions and a1}

respectively. This type of sea _{condition is known as a}

cross-sea condition.

_The

_{relevance of the effect of}

such sea conditions on the behaviour of moored vessels has been dem-onstrated in [i].

The directional wave spectrum of irregular long-crested waves from the direction a0 is defined as follows:

S (w,a) = (a-a0) S (w) _(3.13)

a0 _a0

in which the DIRkC delta _{function is defined}

-15-The directional wave spectrum of two long-crested

ir-regular waves from directions a0 and a1 is:

S(w,a)

= S(a-a0) S (w) + 6(a-a1) S (w) . . . (3.16)

a0 ai

in which S (w) and S (w) are ordinary point spectra for

a0

long-crested waves.

The mean drift force in an irregular directional sea is as follows:

2ir

= 2

_f

_f

S (w,a) P(w,w,a,a) da dw

00

Substitution of equation (3.16) and taking into account

equations (3.14) and (3.15) gives:

F F

+F

a0 a1 = 2

_f

S (w) P(w,w,a0,a0) dw + O a0 + 2

_f

S (w) P(w,w,a1,a1) dw o _cil (3 17) (3 18) L = 0 _for ci _{a0 (3} 1 and: 2

f

-)

da = i _{(3 5)} o

This sl-iows _{that the mean drift force of two sea states is}

the sum of the mean forces from the _{individual wave trains.}

The spectrum of the wave drift _{force in an irregular}

sea is: 2ir 2iî = 8

_f

f

_f

_{S(w+p,a) S(w,)jT(w+,w,a,)I2}

000

(3 19) Substitution of (3.16) gives: 2î 2ir

SF()

= 8

f

_J

_f

_{{(a-a0) S (w+.i)}

+ 6(a-a1)

_S

o o o _a1

{6(-a0) S (w) ± _{6(-a1) S} (w)]

a0

which results in:

SF()

= 8 _{J S (w±u) S (w) T(w+,w,a01a0)j2} dw + O _aO

a0

+ 8 _J s

(w) S

₍₎

1T(w+,w,a0,a1)I2

_{dw +} O a0 a1 + 8 _J S

(w+) S

(w) _{T(w+,w,a11a0)l2 dw +} O a1 + 8

_J

S (w+) S _{(w) !T(w+,w,a11a1)2 dw} O a1 a1 (3 21) da dS dw _{(3 20)}

-17-The first and last _{terms of equation (3.21)}

correspond

respectively with the _{drift force}

spectrum from each _{of the}

two sea conditions

separately. This shows _{that the low}

fre-quency second order _{excitation in}

a sea state consisting _of

two long-crested _{irregular seas is}

larger than the _{sum of}

the excitation _{from each sea}

independently. The _{second and}

third terms of _{equation (3.21)}

show the excitation _arising

from interaction _{of both long-crested}

seas. The nature of

these terms also _{shows under which}

conditions such inter-action terms _{may be neglected.}

Consider, for instance, _{that waves from}

direction a0

are relatively short, wind driven seas while _{those from}

di-rection a1 are _{relatively long-period}

swells. _{oth spectra} are shown in Figure 1.

If the spectra _{have little or}

no overlap on the

fre-quency axis, then the _{products of the spectral}

densities in

the second and third _{parts of equation (3.21)}

are small,

even for relatively large _{p values. This}

can have important

consequences for simulation _{computations since such}

combina-tions of wind driven _{and swell seas}

frequently occur. The

impact of this effect _{will be apparent when}

it is realized

that the major part of the low frequency _{response of moored}

vessels is at or _{near the natural frequency.}

These

frequen-cies are generally very low. The _{consequence of this effect}

is that the system only reacts to very low _frequency

compo-nents in the excitation. _{If the above-mentioned}

separation is present then equation (3.21) _{shows that the}

excitation may be assumed to be _{simply the sum of the}

exci-tation arising from each sea independently.

For the case that _{both long-crested irregular wave}

trains are from the same direction , equation (3.21)

becomes:

= 8 _{J {s}

(+i.,a)

(w,)

_{+ S (w+.i,a)}

_s

_(w,a)

₊

o

+S(ui+j,a) S

(u,)+S (w+p,)

_s(w1a)]

which, with ₌

= , is the same as:

sF() = 8

_J

s

(w+M,a) S (w,a)T(w+p,,a,a)[2 dw . . (3.23)

o

in which:

S(w,a) = S

(w,a) + S (w,a) _{(3 24)}

a1

Equation (3.23) is _{again the well known result for}

uni-directional waves.

The aforegoing discussion shows _{that combinations of}

uni-directional seas interact in such _{a way that the}

resul-tant oscillatory part of the drift forces _{is always larger}

-19-than the sum of the drift forces arising from each wave

train independently.

The constant (mean) part of the drift forces as given

by eauation (3.18) does not suffer the same interaction

effects.

The mean force in

two uni-directional irregular

wave trains from directions and is simply:

= 2 ₅ S (w) P(w,w,a0,a0) dw +

O a0

+ 2 ₅ s (w) P(w,w,a1,ci1) dw

O a1

(3 25)

This shows that the mean is equal to the sum of the contri-butions from each wave train independently.

Equation (3.25) is, however, only true in the ensemble

average sense. Time average values of particular

realiza-tions as given by equation (3.5) retain interaction terms due to the fixed choice of the phase angles. The influence of these interaction terms on the time average values re-duces as the number of frequency components is increased,

4. APPLICATION OF THE FUNCTIONAL POLYNOMIAL OR VOLTERRA SERIES TO FIRST AND SECOND ORDER WAVE LOADS IN IRREGULAR DIRECTIONAL SEAS

4.1. General

Within potential theory it is assumed _{that the}

poten-tial describing the flow and all _{quantities derived from the}

potential may be expanded in a convergent power series. For instance, for the wave force:

F =

F(0)

+ EF1

+

£2F(2)

±

. o

(4.1)

in which:

(0)

F _{hydrostatic force}

F1

= first order oscillatory wave loads

F2

second order wav loads, _{the low frequency part of}

which is known as the drift force

C _{= a small parameter (wave steepness).}

In such cases the Volterra series _{expansion is also}

valid, [16] and [17]: + F =

F°

₊

_f

(t-r)

h(1)(T) _{dt +}

±

+

f

f

ç(tt1) (tT2) h(2)(11T2) _{dt1 dt2+} +

_(4.2)

in which:

(t) = wave elevation record

T, _{T1, '2} = time shifts

h(1)(T) _{= first order impulse response function} = second order impulse response function.

4.2. Wave field

We consider the wave field to consist of a

superposi-tion of long-crested irregular _{wave trains from discrete}

wave directions. in which: N -i(w.t + . = Re { e ik i=l

We furthermore suppose that the wave elevations k = 1 . .. M are available as time records.

4.3. Mean and low frequency second order wave drift forces We may rewrite equation (3.2), which gives the low

fre-quency drift force in irregular directional seas, as

fol-lows: M M

F2(t) =

Z Z _Re tF(2)(t)] L

k1 £1

-21--(4 4) (4 5) M (t) = _{(4 3)}

in which: N N -i{ (w. - w .)t

± (E.

- E.

) }

F(t) = E

E T.jkz e

k

-1=1 j=l (4 6)

The equivalent Volterra series expression _{for equation (4.6)}

is as follows:

F(t)

=

f

f

_{k(t_Tl)} (t-T2) h2

kL

(T1T2)

dT1 dT2

- ...

(4 7)

Equation (4.7) will be set equal _{to equation (4.6) in}

order to determine the form of h(T1T2).

_{To this end we}

write for the wave elevations _{k(t_Tl), (t-T2):}

N -i(w.t

+ E.

₎ jw«r i ik i Re _[ Z e _- e . . (4.8) ik i=l N

-i(w.t ±

i

i

. )

jw.T

i

2,

Re [E

e _e j . (4.9)

Products of wave frequency quantities contain both sum

and difference frequencies which are found from:

k(tTltT2)

½Re[k(t_rl)(t_r2) +(t-T1)(t-T2)]

-23-where the *

denotes the complex conjugate. The _(complex)

wave _{levations in the right-hand side of}

equation (4.10) are given by the expression inside the [

j in equation (4.8)

and equation (4.9).

Substitution in equation (4.7), neglecting sum

fre-quency components, yields:

+ +

N N -±{w.- w.)t + (E - E )l

F(t)

_f

_f

½Re[ E E _e i ik -j9.

-=

₁₌₁ ik jL

j(w.T

_{i i}

- W.T

j 2

j h(i1i2) di1 dT2

. . (4.li)

e

In order to determine h(i1i2)

_{we will equate}

_the

coefficients of cos {(w1

- w)t +

-ik

- E.)} and of sin

{(w1

_{- w)t +}

- in equation (4.6) and equation

(4.11). This leads to _{the following expression:}

-- -- i(w.i _- W.T ) Tjjk

= ½ f

_f

h(T1i2)

1 1 j 2 di1 dt2 e (4.12)

from which it follows that:

+ +

-i(.i

- w.t )

e

f Tijk 1 1 2 du. dw.

--

_{i J}

we will use equation (4.13) to determine a symmetry

rela-(2)

tionship for h (T1T2).

According to equation (3.12):

This results in the following symmetry relationship for

(2)

tLL (T1T2):

(2)*

_hjk

(t2T1)

We now return to equation (4.5) and reformulate this as

follows: M M

P2()

= E E

½Re[F(t) + F(t)]

. . . (4.16)

k1 £=1

with: for = 2 for k L _{(4 17)}

Taking into account equation (4.7) and the symmetry re-lationship of equation (4.15) we find:

(4 15)

F2(t)

= M M

±+

(2) = E _{E ½Re{}

_f

_f

k(t_Tl (tt2) hkj (t1t2) dt1 dt2 + + L (t_Tl)k(t_T2) h (t1t2) dt1 dT2] . . (4.18)

Interchanging

t1

and T2 in the last part leadsto:

M M

+±

F2(t)

E E _f

f

ck(t_rl) (t-t2) k=Z £=1 (2)*

½ {(t1t2) + h

(t1t2)] dt1 dT2 M M

++

= E E

f

f

k(t_T1) C2(t--r2) k=2. £=1

- -25-Re

[h2()J

dt1 dT2 ic,. (4.19)

The real part of h(T1t2) is found from equation (4.13).

4.4. Discussion

In equation (4.19) use can be made of arbitrary random records _k(t) and _(t). These may be generated, for instance, based on filtered time records of white noise for which the filters are chosen to suit the required

direction-al spectrum shape as a _{function of directional index k.}

Alternately, for the case of cross seas generated in model

corn-ponents, which can be generated and measured separately, can be used.

This presents the possibility of _{comparing time records}

of computed and measured _{wave drift forces on a vessel in}

long-crested irregular waves from two directions. _{Tests of}

this kind will reveal the basic effects of the influene of

directional spreading of waves, while retaining _the

require-ment that the wave field should be well defined. _{At this}

time this is still a problem with respect to directional

seas generated in model basins using large numbers of wave

5. SOME AS PECT S OF THE RELAT_{IONSHI P BETWEEN WAVE FORCES IN} UNI-Dl RECT 10MAL AND MULTI -DIRECT 10MAL WAVES

5.1. General

In this section the effect of the directional spectrum

S(w,) on the mean and _{spectral density of the low}

frequen-cy drift force is briefly discussed.

Two formulations for the directional _{spectrum are in}

general use. The simplest of these is:

-27-in which:

= frequency dependent point spectral density

f()

= a spreading function.

For the point spectral density any of the current

spec-tral forms, i.e. JONSWAP, Pierson-Moskowitz, _{etc., may be}

chosen. For the spreading function, _{a function which is}

pos-itive and symmetrical about the dominant _{wave direction is}

generally chosen, [8] and []

The spreading function has to be chosen in _compliance

with the following equation:

2i 27r

S ()

=

f

S (w,a) da ₌

_f

S () f(a) da . . . .

(5.2)

o o

or:

2ir

f

f(a) da = 1 _{(5 3)}

o

A slightly more complicated formulation is _{as follows:}

S(w) f(w,)

which yields the following condition:

2Tr

J f(w,a) da = i

0

(5 4)

(5 5)

In this case the spreading function varies with the wave

frequency w. Equation (5.4) allows directional _{spectra with,}

for instance, different dominant wave directions depending

on the wave frequency and also different degrees of direc-tional spreading dependent on the wave frequency.

In order to give an impression of the effect of

direc-tional spreading on second order drift forces use will be

made of equation (5.1).

5.2. Mean and low freouency drift forces

The mean second order wave drift forces in irregular

directional seas are obtained from:

2ir

= 2

_f

_f

S (w,a) P(w,w,cx,a) da dw

Application of equation (5.2) gives:

2ir

= 2

_f

_{S(w) {}

_f

_{f(a) P(w,w,cz,a) da} dw}

. . . (5.7) o o = 2

_f

S(w) P(w) dw o in which: 2iî

P(w)

₌

_f

_{f(a) P(w,w,a,a) da} o

This shows that P(w) can be _{interpreted as a weighted mean}

drift force coefficient.

Another way of presenting equation (5.7) _{is as follows:}

2rr

F ₌

f

f(a) F(a) dOE o

in which:

2

_f

S (w) P(w,w,a,a) dw

o

-29-where F(a) is the mean drift force in _{uni-directional seas}

from direction OE for the spectrum s(w).

The spectrum of the low frequency drift _{force follows}

from:

(5 8)

(5 9)

(5 10)

2iî 2ir

= 8

_f

_f

_f

_S

d dw

000

(5 12)

which, using equation _{(5.2) becomes:}

2î 2u

sF(P) = 8

_f

S(w+) S(w){ f

f

f(a) f(s) 0

00

T(w+p,w,,)l2dd}dw

(5.13) = 8

_f

_s

(w+j) S _{(w) F(w+p,w) dw} . . (5.14) 0 in which: 21T 2iT

F(w+,w)

_{= J}

_f

_{f(a) f(s) 1T(w+,w,a,)j2} d d

00

(5 15) 5.3. Conclusions

In this section some aspects regarding the _relationship

between forces in _{uni-directional and}

multi-directional

waves have been briefly highlighted. _{The results obtained}

using the simplest formulation _{for the directional}

spectrum

show that uni-directional data can, in many _{cases, be easily}

-31-6. MODEL TESTS 6.1. General

In the previous sections _{a general outline of the}

the-ory regarding mean and low frequency second order wave drift

forces has been given. It _{was found that, given a}

descrip-tion of the irregular direcdescrip-tional _{wave field in terms of}

either a set of spectra of long-crested irregular waves from

dis crete directions or _{a continuous directional wave}

spec-trum, the second order drift forces _{can be determined if the}

frequency and wave direction dependent _{quadratic transfer}

functions

ijk and are known.

-It was indicated in Section 3 _{that these could be}

ob-tained based on 3-D diffraction calculations. _{In this}

sec-tion results _{of such calculations will be compared with}

results of model tests for some simple, _{fundamental cases.}

The model tests and calculations have been carried out

for a 200,000 DWT _{fully loaded tanker moored in a water}

depth of 30.24 ni. The main particulars of the vessel _are

given in Table 1. A body plan is shown in Figure 2.

The model tests were carried out at _{a scale of 1:82.5}

in the Wave and Current Laboratory of the _{Netherlands Ship}

Model Basin. This facility measures 60

_{m by 40}

_{ni with a}

variable water depth up to 1 m. The basin is equipped with snake-type wave generators on two sides of the basin. The

wave generators on each side can be driven independently to produce regular and irregular cross-sea conditions.

Model tests were carried out in regular cross-wave

con-ditions and irregular uni-directional _{and cross-sea}

condi-tions. In the following a brief description will be given

with respect to the choice of the _{test conditions, the}

re-suits of tests and the comparison with _{results of}

computa-tions.

6.2. Tests in regular cross waves

According to equation (3.5) the mean force in irregular directional seas is given by:

N N M

F(2)(t) _{= E Z E}

j2 {p11 cos(E. - s ) +

ik

i

i=l k1 £=1

Each of the terms in this equation _{reflects a}

contribu-tion due to interaccontribu-tion of _{a regular wave with frequency Wi}

from direction k with a regular wave with frequency W from

direction L.

Consider the case that we have _{one regular wave with}

frequency w1 from direction i and one regular wave with the same frequency from direction 2.

The mean drift force would then be:

F(2)(t) =

2

- 11 P1111 + l2 ll22 + 1l ç12{P1112 cos(511 - 12 +

+ 0kL

sin(s.

_1k

- C.

)'

-33-+ Q1112 sin( - E12)} +

l2 11{p1121 COSCE12 - cil

Taking into account the symmetry _{relationship of equations}

(3.10) through (3.12) this results in:

F2(t)

2 2

= 1l llll ± l2 ll22 +

+ 2 C11 C12 {p1112

cos(E11 -

_{c12) +}

The first two components of equation (6.3) are the mean

forces due to each regular wave independently. _{The third}

contribution is due to interaction effects _{of the two}

reg-ular waves in the mean drift force. It is seen that this

component is, besides being a function of the quadratic

transfer functions P1112 and

1112' a function of the phase

angles E11 and E12 of the two regular _waves.

The quadratic transfer functions P1111 and P1122 can he

found from tests or computations for regular waves from one direction. For this particular tanker, these uni-directional mean drift force transfer functions have been given in [13].

The purpose of the tests in regular cross _{waves is}

spe-cifically to identify the quadratic transfer functions _P1l12

) ±

+ Q1121 sin(c12 - _{(6 2)}

and Q1112 _{which represent the interaction effects due to the}

simultaneous presence of two regular _waves.

Equation (6.3) shows that, in order _{to identify these}

effects, tests in regular _{cross waves should be carried out}

whereby in each test two regular waves with the same

fre-quency and amplitude are generated. The phase _angle

differ-ence of the components (E:11

-

E:12) _{must be different for}

each test however.

From equation (6.3) _{it is seen that the measured mean}

drift forces on the vessel will _{then contain a mean part and}

a part which is a harmonic function of the phase angle dif-ference

(C11 -

C12).

We have chosen to carry out tests _{in regular cross}

waves with the component waves at right angles to each other and at 45 degrees to the port and starboard bow of the

ves-sel respectively. The set-up is _{shown in Figure 3.}

Regular waves were adjusted separately _{from both sides}

of the basin. The frequencies _{were the same and the wave}

amplitudes _{(first harmonic component of the wave elevation}

record) measured at the mean position of the centre of

grav-ity of the vessel were almost the same. A review of the

reg-ular waves adjusted from each side of the basin is given in Table 2.

During the tests _{in regular cross waves, the drive}

shafts of the wave generators _{on both sides of the basin}

-35-.

adjustment of the phase angle _{difference - £12) from}

one test to the next and at the same time _{assuring that both}

wave generators were running at the _{same frequency.}

The model of the tanicer _{was moored in a soft spring}

mooring system. The longitudinal _{and transverse forces}

ex-erted by the mooring system _{on the vessel were measured by}

means of force transducers situated at the fore and _aft

con-nection points of the mooring system. The set-up is shown in Figure 3.

Series of tests were carried out in regular cross waves

of a given frequency and amplitude _{whereby the phase angle}

difference _{- £12) between the component regular waves}

was changed by 60 degrees for each consecutive test. _{In each}

test the mooring forces were measured. The time _{average of}

the mooring forces yielded the mean longitudinal and

trans-verse drift forces and mean yaw moment as _{a function of the}

phase difference.

The results of measurements are shown in Figures 4

through 7 in terms of the mean forces and yaw moment to a

base of the phase angle between the wave _{generators. Also}

included is the amplitude of the first harmonic of the

re-sultant undisturbed wave elevation measured at the centre of gravity of the vessel.

The results shown in these figures confirm that the

mean forces contain a constant part and a part which varies

periodically with the phase angle difference of the _wave

Due to the symmetry of _{the test set-up,}

i.e. regular

waves of the same frequency and _{approximately the same}

am-plitude approaching from 135° _{and 225°, the mean values of}

the transverse force F _{and the yaw moment}

should be

equal to zero leaving _{only the periodic}

component of the

mean force and moment. Due to slight _{differences in wave}

amplitudes and possibly due _{to slight error in model}

align-ment this is not quite the _{case. The periodic parts of these}

mean forces are, however, dominant.

The mean longitudinal force _{F contains a mean}

compo-nent corresponding to the sum of the mean forces due to _the

component regular waves and the _{additional component which}

is periodic with the phase difference between the wave

gen-erators.

When comparing the mean forces and yaw moment as a

function of the wave generator phase angle difference, it is

seen that except for tests at _{wave frequency 0.267 rad/s,}

the mean transverse force is in phase with the mean yaw

mo-ment. At 0.267 rad/s, _{a phase di'fference of about 1800 is}

seen between these quantities. In geñeral, the mean

longitu-dinal force F is about 90 _{out-of-phase with F}

_{and M.}

This is in agreement with equation _{(6.3) when the symmetry}

of the test set-up and _{wave conditions are taken into}

ac-count.

In a fully symmetrical _{case in regular cross waves,}

-37-zero, the wave elevation pattern consists of a square

pat-tern of three-dimensional peaks and troughs travelling in a

direction parallel to the longitudinal axis of the vessel

with the maximum peaks and troughs moving in a line exactly on the centre line of the vessel.

In this case the mean transverse force F and ya

mo-ment _{are equal to zero. Assuming that the component wave}

amplitudes are equal, this implies that for the transverse

force and yaw moment, the value of

1112 in equation (6.3)

is equal to zero. Based on similar reasoning it can be shown

that for the mean longitudinal force Fi the value of

is equal to zero.

Consequently, according to equation (6.3) the varying

part of F

will be 900 out-of-phase with respect to the

varying parts of F and M1. The periodic part of the

varia-tions in the mean value of the forces to a base of the wave generator phase difference are governed by the third

compo-nent and fourth compocompo-nent for and for F1 M1J) respectively

in equation (6.3).

In order to compare

the results of model tests in

regular cross waves with the results of computations, the

following analysis was applied to the measured data:

From the results given in Figures 4 through 7, the am-plitude of the first harmonic of the mean forces and moment

and their respective phase angles relative to the origin

thus obtained were divided by 2 _{21 i and}

2 being the

amplitudes of the component irregular waves given in Table 2 for the various frequencies.

For the surge force F, the computed values cf the

qua-dratic transfer functions and are given in Table

3 and in Table 4 _{for the case that the direction index k}

corresponds to the waves from 135° and the index £

corre-sponds to waves from 225°.

All combinations have been given. In this paper use is

only made of results for the case ij. The cases when ij correspond to regular cross waves with non-equal

frequen-cies.

For comparison purposes, the quadratic transfer

func-tions for the surge force are also given for the case that

both waves come from the same direction, i.e. for the case

that k is equal to L. These results are given in Tables 5

and 6.

The transfer functions were then compared with the

com-puted values of the amplitude T1112 for F, F and M1,. The

results of the comparisons are shown in Figures 8 through lo.

The phase angles, _CFç of the forces and moment

rela-tive to the point at which the amplitude of the undisturbed

regular cross waves is at a maximum are also compared with

-39-In general, _{the amplitudes of the quadratic transfer}

functions T1112 are reasonably well predicted by the

compu-tations as are the phase angles relative to the undisturbed

regular cross waves. On the whole, the _{agreement is}

consid-cred to be satisfactory.

6.3. Tests in irrecular _{cross seas}

In the previous section model tests in _{regular cross}

waves were described. _{It was seen that, dependent on the}

relative phase angles of the component _{regular waves, mean}

drift forces could be higher _{or lower than the total mean}

drift force due to the sum of the drift forces _{from each}

wave component independently.

In irregular seas all relative phase angles between

wave components from different directions are equally prob-able. The result of this is that the total mean drift forces

in irregular seas is simply the _{sum of contributions from}

all wave components. This is

expressed in equation (3.7). for

the general case of directionally spread seas and in

equa-tion (3.18) for the case of irregular cross seas consisting

of a superposition of two long-crested irregular wave

trains.

On the other hand, if we consider that in an irregular directional sea the relative phase angles between wave

com-ponents from different directions are continually changing

reflect the additional low frequency force components due to

interaction _{effects predicted by, for instance,}

equation

(3.21).

Tests in irregular cross _{seas were carried Out to check}

whether the theoretical prediction is _{borne out by}

experi-mental findings with respect to the _{mean forces.}

The model set-up was the same as _{used for tests in}

regular cross waves _{(see Figure 3), i.e. the model was}

moored in a soft spring system and _{irregular long-crested}

seas were approaching the vessel from 135° or 225° _{or both.}

In order to check the validity of _{the superposition}

principle with respect to the mean drift _{forces, tests were}

carried out in three phases, i.e. _{(1) one test to measure}

mean drift forces in irregular seas from 135°, _{(2) one test}

in waves from 2250 and (3) _{one test in irregular cross seas}

consisting of a superposition of both irregular _{wave trains.}

Before treating the test results, it is of _{interest to}

check the superposition principle with _{respect to the}

undis-turbed wave elevation at the centre of gravity _{of the}

ves-sel.

The spectra of the long-crested, uni-directional _wave

trains generated from either of the two basin sides are

shown in Figure 11. In Figure 12 the calculated sum of the

individual spectra are compared with the _{spectra obtained}

from the wave elevation records measured in irregular cross

-41-using inderendent random wave generator control signals. The

agreement shows that the superposition principle _{holds very}

well for the undisturbed _{wave trains.}

The test duration for tests in irregular _seas

corre-sponded to 90 minutes full scale.

The mean wave drift forces _{on the tanker measured in}

the various irregular long-crested, uni-directional seas

along with calculated data are given in Table 7.

In general the measured and calculated _{mean surge and}

sway forces and are in good agreement. For spectrum 1,

the calculated mean surge force is _{some 30% below the}

mea-sured value. The reason for this difference is not clear at

this time. _{In general differences in the forces are less}

than 10%.

The difference between the calculated and measured mean

yaw moments appear to be larger. It should be remembered, however, that the mean yaw moment is derived from the

dif-ference in the mean transverse forces measured fore and aft.

The moment arm amounted to about 320 m for the full scale vessel. The mean yaw moments are therefore rather small in

terms of transverse forces applied at the fore and aft ends. In Table 8 the measured mean forces in irregular cross seas are presented together with calculated data obtained by adding the mean values measured in irregular uni-directional seas, and calculated data obtained by adding calculated data for irregular uni-directional seas.

The calculated mean forces _{for irregular}

uni-direc-tional seas were obtained based _{on equation (3.17), using}

the measured uni-directionl wave spectra presented in Figure

11. The transfer function

P(w,U3,,)

_{was given again, based}

on 3-D diffraction calculations and _{on equation (3.1). See,}

for example, Table 5 for _{surge force F.}

The results shown in Table 8, in general, show that the

superposition principle holds _{very well for the mean wave}

drift forces in irregular _{cross seas.}

In general, the data calculated based on 3-D

diffrac-tion theory also are in _{good agreement except for those}

tests involving spectrum 1. In those cases, diffraction cal-culations underestimate the mean surge force to some

X

extent. This is in agreement with the results given in Table

-43-7. SUMMARY AND CONCLUS IONS

In this paper, _{some aspects of the general theory}

re-garding mean and low frequency second order drift forces in

irregular directional seas were discussed. It was shown

that, _{given the bi-directional and bi-frequency dependent}

quadratic transfer function for the _{wave drift forces1 the}

mean forces and spectral density of the slowly varying part

of the forces could be computed using _{the directional wave}

spectrum as description for the sea condition.

This frequency domain representation, although very useful, is _{not entirely complete however, since strictly}

speaking information should also be given on the

distribu-tion of low frequency forces. The _{general problem of the}

distribution of the second order drift forces in irregular

long-crested seas has been treated among others in ref. [18] and [19], to which the reader is referred.

Time domain representations of drift forces in

direc-tional seas through direct summation of Fourier components

or through the application of the second order term of a

Functional Polynomial have been discussed. In both cases it

was assumed that the wave field consists of a sum of

long-crested irregular wave trains from a number of discrete

directions. The Functional Polynomial allows a deterministic

comparison to be made between measured and computed drift

force records in irregular cross seas.

This

comparison has

addressed in more detail in future _research.

The results of model tests with a 200,000 DWT tanker in regular cross waves and irregular cross seas have confirmed

theoretical predictions with respect to the applicability _of

the superposition principle for _{the mean wave drift forces}

in irregular seas while it has _{also been shown that}

addi-tional drift forces occur through _{the simultaneous presence}

of two regular wave fields. The _{latter effect can be used to}

clarify the theoretically predicted _{increase in low}

frequen-cy excitation due to the interaction of waves approaching a vessel from different directions.

Results have been presented of the comparisons _between

drift forces obtained from experiments in regular cross

waves and in irregular cross seas and obtained _from

3-dimen-sional diffraction theory calculations. In general, the

computed and measured data _{are in good agreement. The}

com-parison has, for the present, been restricted _{to the mean}

drift forces. In a future phase it _{is envisaged to carry out}

experiments and calculations to check _{the accuracy of}

pre-dictions with respect to the slowly _{varying part of the}

-45-REFERENCE S

Grancini, G., lovenitti, L.M. and Pastore, _{P.: "Moored}

tanker behaviour in crossed _{sea. Field experiences and}

model tests". Symposium on Description and Modelling of

Directional Seas, Technical University of Denmark,

Copenhagen, 1984.

Marol, P., _{Römeljng, J.U. and Sand, S.E.:}

"Bi-articu-lated mooring tower tested in directional waves".

Symposium on Description and Modelling of Directional

Seas, Technical University of Denmark, Copenhagen, 1984.

Teigen, P.S.: "The response of a TLP in short-crested

waves". Paper No. _{OTC 4642, Offshore Technology}

Con-ference, Houston, 1983.

Marthinsen, T.: _{"The effect of short-crested seas on}

second order forces and motions". International Work-shop on Ship and Platform Motions, Berkeley, 1983.

Van Oortrnerssen, G., Pin]cster, J.A. and Van den Boom, H.J.J.: "Computer simulations as an aid for offshore

operations". WEMT, Paris, 1984.

Wichers, J.E.W. and Van den Boom, H.J.J.: "Simulation

of the behaviour of SPM-moored vessel in irregular

waves, wind and current". Deep Offshore Technology,

Malta, 1983.

Mohn, B. and Fauveau, V.: "Effect of wave-directional-ity on second-order loads induced by set-down". Applied

Mitsuyasu, .: _{"Directional spectra of}

ocean waves in

generation areas. _{Conference on Directional Wave}

Spec-tra Applications, Berkeley, _1981.

Hasselman, K., _Dunckel, M. _{and Ewing, J.A.:}

"Direc-tional wave spectra observed during JONSWAP 1973".

Journal of Physical Oceanography, 8, 1264-1280, _1980.

Lambrakos, K.F.: _{"Marine pipeline dynamic response to}

waves from directional wave sepctra". _Ocean

Engineer-ing, Vol. 9, No. 4, 1982.

Tucker, M.J., Challenor, P.G. and _{Carter, D.J.T.:}

"Numerical simulation of a random sea: _{a common error}

and its effect _{upon wave group statistics".}

Applied

Ocean Research, Vol. 6, No. 2, _1984.

Pinkster, J.A.: _{"Numerical modelling of directional}

seas". Symposium on _{Description and Modelling of}

Directional Seas. _{Technical University of}

Denmark, Copenhagen, 1984.

Pinkster, J.A.: _{"Low frequency second order wave}

ex-citing forces _{on floating structures".}

N.S.M.B.

Publi-cation No. 650, Wageningen, _1980.

Bowers, E.C.: "Long period oscillation of moored ships

subject to long waves". _{R.I.N.A., 1975.}

Benschop, A.: _{"The contribution of the second order}

potential to low frequency _{second order wave exciting}

forces on vessels". Department _{of Mathematics,}

-47-Dalzell, J.F.: "Application of the Fundamental

Poly-nomial model to the ship added resistance problem".

Eleventh Symposium on Naval Hydrodynamics, _University

College, London, 1976.

Gao, H. and Gu, M. _{"Determination of non-linear drift}

force quadratic transfer function and synthesis of

drift force time history". Marintec Offshore China Con-ference, Shanghai, 1983.

Vinje, T.: _{"On the statistical distribution of second}

order forces and motions". International Shipbuilding

Progress, March, 1983.

Langley, R.S.: "The statistics of second order _wave

LIST OF TABLES AND FIGURES

Table i : Main particulars and stability data

of loaded 200,000 DWI' tanker

Amplitude of first harmonic of adjusted _regular

uni-directional wave components

Quadratic transfer function _{for the}

longi-tudinal force in cross seas Quadratic transfer function

0ijk for the longi-tudinal force in cross seas

Quadratic transfer function

ijkk for the

longi-tudinal force in long-crested seas

Quadratic transfer function _{for the}

longi-tudinal force in long-crested seas

Measured and calculated mean drift forces and

moment in irregular uni-directional _seas

Measured and calculated mean drift forces and moment in irregular cross seas

Schematic representation of wind sea and swell

spectrum

Body plan of 200,000 DWT tanker

Schematic representation of set-up of tanker model in Wave and Current Laboratory

Measured mean drift forces and yaw moment and

undisturbed wave height in regular _{cross waves.}

Wave frequency 0.267 rad/s Table 2 Table 3 : Table 4 Table 5 : Table 6 : Table 7 : Table 8 : Figure 1 : Figure 2 : Figure 3 Figure 4

-49-Measured mean drift forces and yaw moment and undisturbed wave height in regular cross waves. Wave frequency 0.443 radIs

Measured mean drift forces and yaw moment and undisturbed wave height in regular cross waves. Wave frequency 0.713 rad/s

Measured mean drift forces and yaw moment and undisturbed wave height in regular cross waves. Wave frequency 0.887 rad/s

Amplitude and phase angle of mean longitudinal drift force due to interaction effects in

regu-lar cross waves

Amplitude and phase angle of mean transverse drift force due to interaction effects in regu-lar cross waves

Amplitude and phase angle of mean yaw drift moment due to interaction effects in regular cross waves

Spectra of adjusted uni-directional irregular

seas

Spectra of irregular cross seas: Comparison of measured results with results calculated from

superposition of uni-directional irregular seas

Figure 5 : Figure 6 : Figure 7 : Figure 8 : Figure 9 Figure 10 : Figure 11 : Figure 12 :

Table i

Main particulars and stability data of loaded 200,000 DWT tanker

Designation _{Symbol Unit}

Magnitude

Length between perpendiculars _m

310.00 Breadth _{B m} 47.17 Depth _{H m} 29.60 Draft fore TF m 18.90 Draft mean TM m 18.90 Draft aft TA m 18.90 Displacement weight _{tf 240,697} Block coefficient CB - _0.850

Midship section ceofficient

CM - 0.995

Waterplane coefficient _{C - 0.868}

Centre of buoyancy forward _of

section 10 FE m 6.61

Centre of gravity above keel _m

13.32

Metacentric height _{GM m 5.78}

Radius of gyration in air:

- transverse _m 14.77 - longitudinal rn 77.47 - vertical m 79.30

-51-Table 2

Amplitude of first harmonic of _adjusted

regular uni-directional wave components

Wave frequency

Wave from _{Wave from}

in rad/s

east side _{south side} Amplitude of first harmonic

mm

(2250) _(135°) 0.267 _{1 .90 i . 94} 0.443 1 . 88 _{i . 90} 0.713 _{i . 99 2.02} 0.887 1.82 1 .87

Table 3

Quadratic transfer function

for the longitudinal force

in cross seas + Wi + 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0 -7 -11 -9 0 0 0 0 0 0.3 -7 1 -4 -8 -5 0 0 0 0 0.4 -11 -4 -5 -10 -6 5 0 0 0 0.5 -9 -8 -10 -11 -7 6 8 0 0 0.6 0 -5 -6 -7 -9 -2 8 5 0 0.7 0 0 5 6 -2 -10 -1 10 12 0.8 0 0 0 8 8 -1 -7 4 15 0.9 0 0 0 0 5 10 4 -6 -4 1.0 0 0 0 0 0 12 15 -4 -25

Table 4

Quadratic transfer function

for the longitudinal force

in cross seas 0.4 -38 17 0

E

E E E -1 ijk 0.5 0.6 0.7 0.8 0.9 1.0 -29 0 0 0 0 0 -26 -21 0 0 0 0 -15 -21 -19 0 0 0 = 135° 6 o = 225° i. + 0.2 0.3 Wi + 0.2 0 -31 0.3 30 0 0.4 38 17 o.s 29

Table 5

Quadratic transfer function

ijkk

for the longitudinal

force in long-crested seas w. + WI 4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0 -23 -51 0 0 0 0 0 0 0 -4 -21 -26 0 0 0 0 -23 -4 -10 -14 -17 -7 0 0 0 -51 -21 -14 -20 -12 -6 2 0 0 0 -26 -17 -12 -13 -6 -4 7 0 0 0 -7 -6 -6 -12 -8 0 20 0 0 0 2 -4 -8 -10 -3 12 0 0 0 0 7 0 -3 -10 -11 0 0 0 0 0 20 12 -11 -31

Table 6

Quadratic transfer function

for the longitudinal force

in long-crested seas + Wi + 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0 -146 -166 -116 0 0 0 0 0 0.3 146 0 -81 -90 -45 0 0 0 0 0.4 166 81 0 -50 -50 -17 0 0 0 = 135° 0.5 116 90 50 0 -35 -37 4 0 0 0.6 0 45 50 35 0 -32 -19 23 0 = 135° 0.7 0 0 17 37 32 0 -29 -4 28 0.9 0 0 0 -4 19 29 0 -27 1 in 0.9 0 0 0 0 -23 4 27 0 -30 1.0 0 0 0 0 0 -28 -1 30 0

Table 7

Measured and calculated

mean drift forces and

moment in irregular uni-directional seas Spectrum 1 Spectrum 2 Spectrum 3 M e a n (225°) (225°) (135°) forces Unit and flea- Calcu- Mea- Calcu- Mea- Calcu-morne nt sured lated sured lated sured lated y tf -75.7 -53.0 -16.9 -16.6 -16.9 -18.1 tf -286.5 -293.2 -98.2 -107.5 93.5 112.6 tfm -4685 -3489 -582 -471 1608 600

Table 8

Measured and calculated mean drift forces and moment in irregular cross seas

Calculated Calculated Calculated Calculated Measured Measured (1) (2) (1) (2) tf -98.6 -71.1 -92.6 -84.4 -56.3 -80.6 X tf -195.8 -180.6 -193.0 -277.6 -275.3 -271.1 y tfm -3242 -2889 -3077 -4745 -3234 -3671

Calculated (1): Sum of calculated values given in Table 7 Calculated (2): Sum of measured values given in Table 7

Spectrum 2 (225°) + Spectrum 2 (225°) + + Spectrum 3 (135°) + Spectrum 4 (135°) Mean Spectrum 1 (225°) + Spectrum 1 (225°) + forces + Spectrum 3 (135°) + Spectrum 4 (135°) and Unit moment Calculated Calculated Calculated Calculated Measured (1) (2) Measured (1) (2) tf -32.8 -34.7 -33.8 -18.4 -19.9 -21.8 X tf 0.6 5.1 -4.7 -88.1 -89.6 -82.8 y tfm 708 129 1026 -294 -216 432

w

Figure 1 : Schematic representation of wina sea and swell

BOO'r PLAN

AZT SIDE

SriJTH SIDE

'o

Figure 3 : Schematic representation of set-up of tanker

model in Wave and Current Laboratory

EX

WAVES 225 og

ORCE TRASOUCER WAVES .3S

10400 200

2a

in rn Fx F in ti -200 -400 PHASE -61-20000 10000 0 M in tin, -10000 -20000 120 240 350

ANGLE BETWEEN WAVE GENERATORS

n degrees

Figure 4 : Measured mean drift forces and yaw moment and

undisturbed wave height in regular cross waves. Wave frequency 0.267 rad/s

2a N WAV( FREQUENCY

5

/

O.27 r/

-

-a----.-,-

N F "1!

/

/

/

_-.-*__

0 o

20000 10000 o Mg ñ) UrTI 10000 -400 -20000 0 120 240 360

PHASE ANGLE BETWEEN WAVE GENERATORS

in aegrs

Figure 5 : Measured bean drift forces and yaw moment and

undisturbed wave height in regular cross waves. Wave frequency 0.443 rad/s

a WAVE FRE0UElC' 0443 r/3 ,A

/i

B S--/

\

1/

/

\\

\\//

I, /

/

\.S / / I I / /

/

lo 400 5 200 2Ça in m o o F F ri ti -200

-400

o

P NA SE

-63-120 240 360

ANGLE BETWEEN WAVE GENERATORS

in degreEs 20000 10 000 o M in ttm -10 000 -20 000

Figure 6 : Measured thean drift forces and yaw moment and

undisturbed wave height in regular cross waves. Wave frequency 0.713 rad/s

N

WAVE FREQUEHC'r O713 r/

/

/

\

/

//

//

/

---i---/f

\//

/

/

_// I 400 10 5 200

2a

in ni 0 0 x y ¡n tf -200

10 5 2 o F F u-i tf -200 -400 0 120 240 360

PHASE ANGLE BETWEEN WAVE GENERATORS

in degrees

Figure 7 : Measured bean drift forces and yaw moment and

undisturbed wave height in regular cross waves. Wave frequency 0.887 rad/s

400 .200 0-20000 10000 o M n tim 10000 20000

y

WAVE FRLJECY _--.--,.-. \ 0.657 r/i u'

/

/

\

\

---

-/

20 10 o -65-400 200

Figure 8 : Amplitude and phase angle of mean longitudinal

drift force due to interaction effects in regular cross waves -CALCULATED PIEASURED o o o o s

.

e tS 05 1.0 w 'n

c\J z o 10 50 o 400 200

Figure 9 : Amplitude and phase angle of mean transverse

drift force due to interaction effects in regular cross waves CALCULATED .1EASURED

-

o D Q _o l-5 05 10 W in rad/s

4000 2000 E o I o -67-CALCULATED P4EASURED o o o o o

C..

10 1. W in

Figure 10 Amplitudè and phase angle of mean yaw drift

moment due to interaction effects in regular cross waves 400 200 t, C LL o

7. 3 ac To o 'Q 'j-, o w 3 o

Figure 11 : Spectra f adjusted uni-directional irregular

seas 22 SPWrRT.)-4 i n, s i WAVE SPECTWJ.4 2 22 GÇ lOT ?fl Y, WAVE SPCTRJ..r T35 øn.ç. 315 r,, n, 3 WAVE S T35 wy, rRTWT 05. 1.52e" ti.o s

E0

3

o

-69-- .,

Figure 12 : Spectra f irregular cross seas: Comparison of

measured results wjtl-i results calculated from superposition of uni-directional irregular seS

o o E o 3 2O SPCTPtJ,l SPECTRt4 - CM.OA.ATW 1 (22E 3 (135 O.ç) L5 s--s .çi c, s --VECL»4 scas

SCUM4 (1351 (22E o.c)

. sas e_ns ) Y c.,, Y -te s CTRS4 CTPt).4

k

2 (225 a..( 4 (135 seq.) t 's Y1 ' EPC'Rtft4 2 (225 Ses) SPCTRLI4 3 (13E C..

- -.s'

_{CAC5.L*VW SASS} T. -.'.._.5*5

-

--

CS&.O&ATW 'SS. 33555.3 *