DELFRAC, a 3-d difraction programm. Wave drift forces in shallow water

WAVE DRIFT FORCES IN SHALLOW WATER J.A.Pmnkster R.H..M. Huijsniìns Repoi±.No 924-P - :BOSS'92 7-10 Júly 1992

Deift University of Technology Ship Hydromechanics Laboratory

Mekelweg2

2628 CD Deift The Netherlands

C/D

BEHAVIOUR OF OFFSHORE

STRUCTURES

PROCEEDINGS OF THE SIXTH INTERNATIONAL

CONFERENCE

1992

VOLUME TWO

EDITED BY

MINO O H PATEL

AND

ROBERT GI'BBINS

BPP TECHNICAL SERVICES LTD

LONDON

Proceedings of the Sixth. International Conference on the behaviour of offshore structures, 7 - 10 July 1992 held at Imperial C'llegeof Science, Technology and Medicine, London.

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Behaviour of Offshore Structures

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SESSION D5

WAVE DRIFT FORCES IN SHALLOW WATER

J.A. Pinkster Technical University of Deif t

R.H.M. Huijsmans Maritime Research Institute Netherlands

SUMMARY

Wave drift forces on a tanker in shallow water have been mea-sured in regular and irregular waves. The results of measure-ments are compared with computations based on

three-dimen-sional poteittial theory. Results of computations and

measure-ments show the effect of the second order part of the wave

potential associated with the 'set-down' phenomenon. in order

to measure the low frequency part of the second order wave

forces, a special system for

restraining

the model of the

vessel was developed. This system incorporated active

posi-tion control deìices working partly on the basis of 'f

eed-back' of the position error of the model and partly on the

basis of the applicailion of a 'wave-feed-forward' control

signal generated from real-time measurements of the relative

wave elevation around the model. The time records of the mean

and low frequency forces on the vessel were analysed by

bi-spectral análys:is techniques to obtain frequency domain

qua-dratic transfer functions of the second order forces. Atten-tion was also paid to low frequency phenomena in the

undis-turbed irregular waves such as wave grouping and wave

set-down. Measured data on both phenomena were compared with the-oretical predictions based on the random wave model and on

potential theory respectively. i INTRODUCTION

Wave drift forces have been the subject of many studies

throughout the last 20 years, see for instance, Pinkster [1], Daizell [2], Van Oortmerssen [3], Bowers (4j and.Wichers [5). It is generally agreed that, at least for large monohuil

ves-sels, the mean and low frequency wave drift forces are

domi-nated by pressure forces which may be predicted reasonably accurately using computation methods based on potential

the-ory. Results of comparisons of computations and measurements

of the mean second order forces on vessels of various shapes ranging from monöhull forms such as tankers and barges to

more slender hull forms such as semi-submersibies have been

given, among others by Faltinsen and Michelsen (6], Pinkster

(i), Pinkster and Huijsmans [7], Van Oortmerssen [3] and Wichers (5].

Prom theoretical äonsiderations, as pointed out for

instance

by Bowers (4], and from results of computations based on

three-dimensional potential theory it can be shown that mean

and low frequency forces of vessels moored in shallow water will be higher than in deeper water. These forces will also contain

significant

effects from pressure contributions which, although---±n--princ iple ai-so -present -i-n--- deeper water can in suchcases generally be neglected. The increase in the

-1-mean forces in shallow water relative to the forces in deeper water can in part be explained by the decrease in the wave

length in shallow water for the same wave frequency and in

part by the modification of the vessel motions when consider-ing low draft/water depth ratios.

In shallow water the irregular incoming waves exhibit the wave set-down phenomenon. This non-linear effect appears as

long waves bound to the incoming short waves. Set-down wave elevations are related to second order pressures in the wave field which in shallow water is dominated by second order potential effects. The incoming irregular waves are charac-terized by wave grouping which is a term describing the fact

that the waves contained in the train display amplitudes

which are relatively slowly varying in time and space, thus giving the impression that waves progress in almost distinct

groups. The long waves associated with wave set-down and

bound to the short waves generally exhibit wave troughs where

the wave group amplitudes are largest and wave, crests where

wave group amplitudes are smallest. See for instance Fig. i. Based on potential theory, it can be shown that the set-down effect is strongly increased in shallow water. It can also be shown that the set-down phenomenon does not contribute to the mean value of the second order forces but only to the slowly

varying components.

Figure 1. Set-down components in an irregular wave in shallow. water.

The computations of the low frequency forces only take into account the bound second order potential effects and not the effects due to the free long waves which are a basin-bound effect, see (4). In the test program described in this paper, attention was paid to these effects by straightforward

spec-tral analysis of the wave elevation records in order to

de-tect the amount of low frequency activity in the wave eleva-tian record and by analysing the measured undisturbed irreg-ular wave trains using bi-spectral analysis methods in order to obtain the quadratic transfer function of the second order

wave elevation record. This could be compared with

corre-sponding results of potential theory computations.

The measured undisturbed wave records were analysed to yield spectra and distributions of the wave envelope square process which is re1ated-to the: wave group phenomen'&; the results c-ould be compared_with_predictions hased_cii random wave

-2-ory. By and large, such predictions of wave grouping seem to conform well to full scale data.

As stated above, such effects on the low frequency second

order wave drift forces are predicted to be large in shallow water but little experimental evidence has been gathered to substantiate such predictions. One of the reasons for this lack of data is the fact that experiments which have to be

carried out to measure the slowly varying second order wave exciting forces on a moored vessel require sophisticated mea-suring systems in order to measure the forces as correctly as possible. Also special signal analyses have to be carried out

to identify the quadratic transfer function associated with the wave forces.

In order to measure the low frequency second order wave

forces on a model in irregular waves it is necessary to

em-ploy a dynamic system of restraint which allows the vessel to carry out freely the wave frequency motion components

associ-ated with the short waves in the spectrum, while at the same time restricting fully motions at low motion frequencies

as-sociated with the slowly varying second order wave forces. The f irst requirement must be met since the second order wave

force itself is influenced by the wave frequency motion

com-ponents. The second requirement is due to the fact that model motions in the frequency range of the force of interest will result in dynamic amplification effects in the measured

force. Both requirements have been taken into account in

se-lecting the system of restraint of the mödel.

For tests in which the mean force in regular or irregular

waves is the only component of interest, a simple soft spring restraining system may suffice. For tests in irregular waves where the non-zero frequency low frequency components were of interest an elaborate dynamic system of restraint using both feed-back and feed-forward control signals was developed.

Results of tests in irregular waves are the time records of

the restraining forces of the model and of the associated

irregular wave elevation.

In order to extract the frequency domain quadratic transfer function of the mean and slowly varying forces use was made

of bi-spectral analysis techniques. In this case a program

was developed based on the original work by Daizell [2]. In order to be able to identify, with sufficiently reliable

re-sults, the quadratic transfer function for the low frequency

forces, _{relatively long time records of the forces measured}

under stationary irregular wave conditions are required.

Re-sults of calculations of the quadratic transfer functions of

the forces based on the use of a three-dimensional dif

frac-tian program DIFFRAC developed at MARIN were compared with

results obtained from model tests in regular and irregular

waves.

The program DIFFRAC was developed by Van Oortmerssen [3] and

f:or the computation of

-3-ing drift forces using the pressure integration method. The

computational procedure makesuse of an approximation for the force contributions associated with the wave set-down

phenom-enon, also called the second order potential contribution,

which takes into account the so-called diffraction terms but neglects terms arising from interactions of incoming,

dif-fracted and radiated first order waves. The method has

re-cently been compared with more complete formulations and

other approximations by Kim and Yue [8] with respect to the

second order wave forces on a vertical cylinder and

previous-ly by Benschop, Huijsmans and Hermans [9] with respect to the

forces on the same tanker. The method, although incomplete

insofar as that a number of contributions are not included,

appeared to be adequate for determining the low frequency

force components due to the second _{order potential terni. It}

was pointed out by Huijsmans, [10] and [li], that. the

approx-imation, which is based on the transformation of the first

order force in a wave with the same wave number _{as that} asso-ciated with the set-down wave, is exact, with respect to the

Froude-Krylof f and diffraction components in the limiting

case of small water depth..

In this paper, after a brief introduction to the mathematical description of the wave drift forces, the model tests will be treated. This part includes discussion regarding the adjusted regular _{and irregular waves, the model test set-up, the test} procedure and the measurements. Following _{this the}

bi-spec-tral analysis method will be discussed. Results of quadratic transfer functions for the second order wave set-down

ob-tained using bi-spectral analysis will be compared with

pre-dictions based on potential theory. Wave envelope _{spectra and'} distribution functions will be compared to random wave theory

results. Finally, the results of three-dimensional potential

theory computations regarding the quadratic transfer _function

for the mean and low frequency second order horizontal drift

forces will be compa'red with the measured d'ata.

2 _{THEORY. ON SECOND ORDER WAVE DRIFT FORCES: THE PRESSURE}

INTEGRATION METHOD

In the following a brief summary of the main points of the

computation method for mean and low frequency horizontal

drift forces is given.. A more complete review of the theory

may be found in ref. (1] or in [5]. The theory is developed in accordance with perturbation theory methods. All

quanti-ties related to the flow, i.e. the potential, the _fluid

pres-sure, the body motions, etc. are expanded in a power series

of a small parameter, in this case typically the wave slope. By grouping all components of quantities such as wave forces,

etc. which are found by integration _{of the hydrodynamic and} hydrostatic pressure over the actual wetted surface, into

terms which are the coefficients of the powers of the chosen

small parameter, the expressions for first and second order

wave forces can be derived. The results _{for the first order}

forces correspond with ,the _{well-known expressions and need}

töt b repeaTted here. _{' Thepsi'on 'for 'the second order} force is as follows:

-4-(2) F1 + F3 + F. + F5 ½.p.g

r

?:4')

i dl .W'L i - ½p i;v (1) 2 dS

so

f

-

p( V$)ii dS

so

F4, = F5

= - f

dS

so

in which:

= fIrst order relative wave elevation at the waterline

dl length element waterline

(1)

total first order potential incl:uding incoming wave, diffracted wave and radiated' wave effects

dS surface element of the mean wetted surface of the body = unit normai vector positive into the fkúid domain

R1

first order.rotational vector of the body

i = firkt. order acceleration vector of the body

second order velocity pötential including effects from

incoming waves, diffracted and radiated waves and ali

interactions of first order potential contributions.

Based on eq.. (1), the low frequency second order drift forces

in irregular waves can be derived.. We assume that the

irregu-lar wave elevation can be described as follows: N

= E A. cos(A3t

-1=1 1

in which:

= i-th frequency

= amplitude of i-th frequency component derived from the wave spectrum

t.

= random phase angle

= wave elevation in a point.

Based' on eqs. (1) and' (2 ), it can be shown that the, second order wave force is of the following form,:

F2(t)

= E E

A1A{P1 cos((-)t

(c_c))

+

+ s.in((cA31-3)t +

(cc)j}

(3)

in which:

P. = quadratic transfer function of the part of the force

which is in phase with the wave envelope square process Ql. = quadratic transfer function of the out-of-phase part of

the force.

As can be seen from eq. (3), the quadratic transfer function is a function of two wave frequencies and physically repre-sents the in-phase and out-of-phase parts of the second order force in a regular wave group consisting of two regular waves

with frequencies c and . respectively. The frequency of the

wave group corresponds tOJ'CA3_)..

J

Eqo (i) is used as the basis for evaluating the components of

the quadratic transfer function for the drift forces using

three-dimensional potential theory based computational

meth-ods, see for instance ref. [1]. When applied to conventional

H hull forms, such as tankers, it is generally found that the

mean and low frequency horizontal drift force components are dominated by the first term of eq. (1) which is due to the

relative wave elevation. The second term generally

counter-acts the first term. The remaining terms involving products of first order quantities may be large at certain wave f

re-quenciesi.f rnotións are large. The last term' involving the

second order potential requires special treatment if it is to. be solved correctly, see ref. [8). A number of approximations, have been developed in order to evaluate this term which, ex-cept in special cases, such as in very shallow water, or for

deep floating, structures, is generally not large. The reader

is referred to the relevant literature f or möre details

re-garding the theoretical and computational procedures.

3 _{THE MODEL BASIN}

Ali model tests were carried out in the Wave and Current

Ba-sin of MARIN at Wageningen. This basin measures 60 x 40 rn. The water depth is adjustable from zero to Ï.1 m. For the

model tests, which were carried out at a scale of i to 82.5 ,

water depths were adjusted corresponding to 22.68. m and 30.20

rn. The basin is equipped with paddle-type _{wavema'kers along}

two sides of the basin and opposite to the wavemakers with wave damping beaches with adjustable slope in order to mini-mize wave reflections at ali water depths. With a view to generating sufficiently long time records of the measured

forces, all model tests in irregular waves were carried out

with a duration of 6 hours full scale. All results given with respect to the adjusted waves were. also derived on the basis of 6 hour records.

3.1 The Ship Model.

The model tests were carried out with an i to 82.5 scale

mod-el. _{of a 20.0 kDWT tanker in fully loaded condition.. )The main}

particulars of the vessel are given in Table I. A body plan is given in Fig. 2. Prior to testing the displacement,

posi-tion of the centre of gravity and the radii, of gyration in

air for the yaw and pitch motion were adj'usted. The roll

pe-nod was adjusted with the model in the water.

Table i. Main particulars of the 200 kDW.T tanker

A P.

'Figure 2. The 200 kDWT: tanker.

General arrangement

st, lo _F.P.

Body plan

7-Designation _{.Symbol Unit} Magnitude

Length between perpendiculars m 309.90 r

Bre:adth ' B m 4.7 .17

Draft T rn 18.90

Displacement weight

Centre of buoyancy forward of

section 10 . ton's m 240,697 6 61 r

Centre of gravity above keel _KG m 13.32

Metacentric height r GM m 5 78

Longitudinal radius of gyration k m

7747

r

Transverse radius. f gyration' k m

r

17.00:

3.2 Restraininq System

Two systems of restraint were used for the present model test

program. For tests in regular waves use was made of a simple

soft-spring system which consisted o,f lon steel wires

incor-porating linear springs attached at one end to fixed points in the basin and at the model to force transducers located on the deck of the vessel. The set-up is shown in Fig. 3.

Figure 3. The soft-spring mooring system and the location of

force transducers.

For tests in irregular waves the model was restrained by

means of a dynamic mooring system. The purpose of this type

of mooring system is to allow the vessel to carry out freely

the motions at wave frequency while restraining, as fully as possible, motion components at the f req:uency of the second

order wave drift forces as indicated in the introduction.

The

restraining

system consisted of controlled servo-units

which applied forces to the ves:sel in surge, sway and yaw

direction through steel wires attached to force transducers

located on the deck of the vessel. The servo-units were acti,-va.ted by a feed-back control ioop based on the measured

in-stantaneous

position of the vessel. The position of the

ves-sel was measured using. an optical tracking system based on a

point light source midships and on a heading gyroscope

lo-cated in the model. The characteristics of the feed-back

control were of the proportional-differential type. In order

to augment the position-keeping capability o:f the system of

restraint, additional servo-units fed by a feed-forward

con-trol loop, based on a real-time estimate of the low frequency

longitudinal and transverse drift forces and: yaw drift moment werè al.so applied.

B:ased ori an analysis of the components of the wave drift

forces on the same tanker using the direct pressure

integra-tion method, Pinkster [1] showed that the major part of the second order

&s dtbT the contribution

assuc±ated withthe

squ-are o-f--t-he--re-l-a-t-i--ve wa-vee-l--e-va-t-i on

around the waterline o.f the vessel. By arraying eight wave

-8-probes around the vessel, this contribution to the mean and

low frequency second order force could be evaluated in real time during the model tests. By applying appropriate gain

factors determined by trial and error methods, the f

eed-forward and the feed-back loops could be optimized to reduce

as much as possible the low frequency horizontal motions of

the model iñ irregular waves. This ensured that the measured

force signals were as little affected by dynamic

magnifica-tion effects as possible. In Fig. 4 a sketch is given of the principle of the dynamic system of restraint. A sketch of the set-up of the model is given in Fig. 5.

Wave-fèed-forwa rd control system Servo-un i t Servo-uni t Force 45.4.L45.4 L 74.3 Vessel Way e s Drift forces Çeed- bac k control system Horizontal motions

Figure 4. Block diagram of the dynamic system of restraint.

Feed-forward contro

S.t St. 6 St. 14 St. 19

Position controller (feed-back.)

Force transducer

Wave probes for posi tion

control system Universal joint

Force transducer

Position controller (feed-back) Feed-forward cntrol

- -Dmens.ons inni(ful i sca]e,)

-Figure 5. Set-up of the dynamic system of restraint.

-9-*

The low frequency moti:ons. that still were present, despite

the control algorithm, are used for the corrections of the low frequency wave drift forces as measured by the control

algorithm. This was established using Newton's law with esti-mated added mass and damping. This _{correction procedure has} been used with several settings of the control system and

gave the same answers. In this way the low frequency residue motions are incorporated into, the wave drift forces. For the

higher sea states this procedúre may lead to inaccurate esti-mates due to the presence 'of wave drift, force damping

ef-fects.

4 _{ANALYSiS OF MEASURED. FORCE TIME HISTORIES}

The measured time traces o.f the. low frequency wave drift

forces requires an analysis that reflects the nature of the

wave drift forces; i.e. the transfer function of the wave

drift forces is, quadratic in nature, so the analysis should be able to identify this quadratic relationship.

In the analysis of linear processes it. is custom to use

spec-tral analysis type of techniques. A natural extension to

qua-dratic type of system's is the Cross-Bi-Spectral techniques

(CBS), which originate from the Volterra series expansion

techniques to describe non-linear systems.

A full review of these techniques is given by Brillinger

(12]. In this section we will explain the details of analysis

for the wave drift force time trace analysis..

In quadratic theory the CBS representation is used.:

Sxxy(L)'iCA)2) =

H(11c2) S(i) S(c2)

. (4.)

As can be seen t:he Quadratic Transfer Function H(1,c2) (QTF) and the CBS now depend on frequency pairs instead of one f

re.-quency as in the linear case. The process of identifyIng the

QTF is based on identifying

the

CBS. In order to use this

spectral analysis techniqué we consider the fact that in

es-sence the QTF of the wave drift force is o.f low frequency

type. This essentially means that the axis on which the QTF

of the wave drift forces is described is dependent on the sum

and difference frequencies.,

=

I

(5)

= 1

-

(6)

The CBS is estimated using the Fourier transform of the cross

covariance The cross covariance is determined from::

Rxxy(tiIt2') E(x(t.-r1) x(t-r2),(y(t)-)j

(7)

The CBS- is

-lo-(9)

S 8

f4Tc±p) Is.

O

L&)) S(+p)'di

= f f

exp -i(21r1+22t2)dr1dr (8)

Possible analysis errors when identifying the QTF from the CBS lies in the fact that the quadra;tic nature is not fully

reflected. This means that to ensure a quadratic relationship we impose the following extra condition:

H (21 122

=

Hz (21,22)

In this equation the denominator refleçts the QTF of the in-put-input squared reation. in theôry this QTF should be

identical to one. However, in determining this QTF of the

input-input squared relation from measured wave records one

often sees a deviation from one..

Another feature of these analysis techniques has to be re-viewed. Since the NyquiCt frequency from the wave drift force time traces is high NYQQ = NYQ2 = it/6t, we are faced with a

1 2

resolution problem in the frequency range of interest. The

number of lags for the spectral analysis must be very high to

ensure enough resolution. This however imposes a restriction

on the statistical confidence bands of the estimated Q.TF at the varies frequencies.

In order to overcome this dilemma a reduction of the Nyquist frequency along the dif;ference frequency axis must be ob-tained.. Fortunately this is possible by selection of a filter

scheme on products of the input in the estimation of the

cross covariance function

5 WAVES

5.1 Wave Set-Down

Examples of regular waves adjusted at water depths

corre-sponding to 22.68 m and 30.20 m are shown in Fig. 6. It will be noted that, as may be expected, the lower wave frequencies contain larger higher harmonic components at the lower water

depths. Spectra of the irregular waves are shown in Fig. 7.

These are based on time records with a duration of 6 hours

full scale. Excerpts from the time records of the wave

eleva-tions are given in Fig. 8. The time records also show the

low-pass filtered records showing the wave down. The set-down is seen in the low frequency tail of the wave. spectra. in the figures showing the wave spectra, the theoretically

predicted spectrum of the wave set-down in the low frequency range is also shown. The prediction is based on the following

in which: (.A) p T(U.,.+p) S, (ca) SsD(.p.) i ..00

w0.3

O -1.00 2.. 00

w=0.4

O -2.. 00 2.00 w = 0.:5 0 -2.00 = = wave frequency

= frequency shift set-down wave frequency = quadratic transfer function for wave set-down

= wave spectrum (high frequency part) = spectrum of wave set-down.

0.3 0 -1...00--Water depth 30.20 rn

ÀAAAAÁ

vvwwvw

V

A A A AA A A A

,vvvvv vvv

Water depth 22.68 rn

1.00-AAL ALA

VV VVV

I I I Seconds O 50 100

Figure 6. Regular waves in shallow water.

For the derivation of eq. (10), it is necessary to invoke the

random wave assumption, i.e.. it must be assumed that the

com-ponent waves in the wave train are independent random

pro-cesses. Correspondence of measured and computed spectra

therefore also has a bearing on the validity o:f the random

wave assumption with respect to the. adjusted waves. The

qua-dratic transfer function expresses the amplitude of the set-down ve eI:vationiwa wavfield-consthsting-of -two-reg.u-lar

waves, with frequencies and w+pThis is also called a regu_

-12-lar wave group. p expresses the frequency difference between the two regular wave components of the regular wave group and

this corresponds w:ith the frequency of the envelope process

of the group. ca+p/2 expresses the mean frequency of the

regu-lar wave group. The quadratic transfer

function

is derived

using potential theory based on the following expression for

the. second order wave elevation:

= - (2) _{- - = o (11)}

in which:

= second order potential = first order potential

= first order wave elevation.

The comparison of the computed and measured low frequency

parts of the wave spectra givenin Fig. 7 shows that these

are predicted reasonably well based on knowledge of the high frequency part of the wave spectrum and the quadratic

trans-fer function based on potential theory. The low frequency tail of the measured wave spectra do not show any significant

peaking; the presence of which might suggest seiches or

standing waves in the basin.

By considering the wave set-down as a quadratic function of

the high frequency part of the wave spectrum, the quadratic transfer function of the set-down may also be obtained by cross-bi-spectral analysis methods considering thewave f re-quency components of the measured wave record as input and

the low frequency part of the measured wave record as output.

This separation of the time record of the measured wave ele-vation is affected by low-pass filtering to obtain the low frequency part. Subtracting this part f:rom the time record of the total wave elevation record yields the high frequency part. The principle of this is illustrated in Fig. 9. Results

of cross-bi-spectral analysis of the undisturbed wave

eleva-tion records are shown in Figs. 10 and li. In these figures

the amplitude of the quadratic transfer function is shown to

a base of the frequency difference between the two: component

waves of a regular wave group. Each of the small figures ap-plies to the stated value of the mean frequency of the com-ponent waves. The lines given in the figures correspond to

results obtained from tests in irregular waves. The dots

given in the figures correspond to the potential theory based values of

¡T(ci+p)I.

Comparison of the results shows that

potential theory results correspond well with results of

measurements and that the wave set-down phenomena is signifi-cantly affected by changes in the water depth when

consider-ing low water depth values. From the comparison indications were also obtained regarding the applicability of the cross-bi-spectral analysis technique. Good comparisons were

ob-tained for mean frequencies which are close to the peak f re-q.uency of the wave spectra considered. At mean frequencies

furthe from the pe&cfrequency of the consideredwave

spectrumt-he comparison becomes worse. The limits, with

-13-spec.t to the mean frequency at which cross-bi-spectral anal-ysis could still give reasonabLe results, were established

from the analysis of the. wave set-down for the various wave

spectra. These limits were also used when analysing, the wave

drift force records from tests in the same waves.

Measured Theøre.tica Water depth 30.20 rn 3 L. 1 .0 0.5 1.0 tu lin radis Water depth 22.68 rn 1.5 10

Figure 7. Spectra of waves, and set-down

r

LL_J

'0.5 w 1 OE in rad/s

14-WAVE SPECTRUM Measured: 5.89 m, = 11.17 s SET-DOWN WAVE SPECTRUM Measured!: H 2.77 m, T1. 9.73 s 0. l'O 3 'J SET-DOWN 0-_ WAVE SPECTRUM Measured:: H5 5.89 rn, T 11.45 s

025\

0; w 0

IOì

WAVE SPECTRUM 1easured: H5 = 2. 76 rn, T1 9.74 s 0.25 3 SET-DOWN w c'j E

r

L lo r-. io

L

0.5 10

i .00 O -1.00 2.00 o -2.00 2.00 o -2.. 00-Water depth 30.20 m H = 2.77 m, T1 9.73 s Hs = 5.89 m,. T1 = 11.17 s Water depth 22.68 m = 2.76 m, T = 9.74 s = 5.89m, T1 = 11.45 s k £ '. _L L . I V---

, ,

''

r ry

., Seconds t 1 I I I I j J I I t 0 5:0 100

Figure 8. Irregular wave trains and set-down in shallow wa-ter.

S (w)

Wave

set-down Wave

spec trum s pec.t rum

(j)

I. 1

Figure 9 Set-down epaated f r&n the high fe encyjart of

the wave spectrum.

-15-.0..05 c.'j E E. C (.'J .0.05 O O O O

i

Theoretical 1rreguar waves: H: = 2.77 rn, T1, = 9.73 s 5.89 rn, T = 11.17 s = 0,43 e

f

= 0.59 0.05 0.76 0.25 0 Water depth 30.20 rn = Q.49 = 0.65 0.81 in radis

Figure. 10. Quadratic transfer

function

of the wave set-down

at 30.20 m water depth. 0.25 0 0.54 0.70 0.25 0.25

i

-16-0.25 0.25 0 0.25 0. 25 0.25 0 = 0.86

i

i

i

Water depth 22.68 m

in radis

Figure 11. Quadrati.c transfer function of the wave set-down

at 22.68 rn water depth. 0.1 O 0.1 C'.' E -S.-E C C'.' s Theoretical Irregular waves: r 2.76 H5, = 5.89 m, m,

I

O s r ₉₇₄ = 11.45 = 0.43 (w1 'w2 = 0.49 = 0.54

,

--.___._..- --& .5.. s .'_S-

ó-.!

_{.5. 5.5__}

.

o 0.25 0.25 _0.25 = 0.59 r 0.65 r 0.70 s. o. 0.25 0 0.25 0.25 0.76 = 0.81 _{= 0.86} O O 0.25 O 0.25 _0.25

6 6

Derived theoretically based on spectrum of measured wave Derived from low frequency part of squared wave record

Water depth 30.20 m 0.25 0.,50 u 'in radis 150 10, 50 .0 Water depth 22.68, m

Figure 12. Spectra o.f wave envelope square.

18.-Irregular waves.: H5 = 2.77 m, T1 = 9.7,3 s H _HIrregular waves:5,89 m, T1 = 11.17 s Irregular waves: H5 = 2.76 m, T 9.74 s Irregular waves:_{5.89 m, 11.45 s} -.5' 0.25 0.50 0.75 i,n rad/s 0.25 0.50 0.75 O 0.25 0.50 0.75 0.75 O 150 t00 50

1 .0

-

0,.1

C.-.'

Q-Derived theoretically based on spectrum of measured wave Derived from. l:ow frequency part of squared wave reccrd

Water depth 30.20 rn 0.1 0.,01 0.00.1 0.0001 Water depth 22.68 rn 0. 1 0.01 0.001 0.001 0.0001 0 2 5 5.0, 7.5 A2, 22 in rn2 A2, 22 in m2

Figure 13. Distribution function of wave envelope square. Irregular waves:

=589rn,=1117s

30 Irregular waves: H5 = 2.77 _{m, 1,1} = 9.73 s Irregular waves = 2.76 rn, T1 9.74 s Frregular waves: H5 5.89 rn,, T1 = 11.45 s

F02030

2, 5 5.0 7., 1.0

-

0..1 c'J Q-0.01 o. 0.001

Grouping in irregular waves can be evaluated through analysis of the wave envelope. Since wave drift forces _{are a quadratic}

function of the wave elevations it is appropriate to analyse

the properties of the square of the wave elevation. In

parti-cular, since we are. interested in the low frequencies of the

drift _{forces we restrict ourselves to the, low frequency part}

of the square of the wave elevation. This is equivalent to the square of the wave, envelope and thus contains information on the wave grouping. By squaring _{and low-pass filtering of} the wave elevation records, time records of the wave envelope

square _{are obtained which. may be subjected to spectral}

anal-ysis and of which the distribution may be determined.

Con-versely, based on knowiedg.e of the wave spectrum (from the

measured elevation record) and the random wave assumption,

expressions can also be given for the expectation of the wave

envelope square spectrum and the distribution of the wave envel.ope square process.. These are as follows:

SA2 = 8 $ S,() S(w+p)dw (12) p(A2) = . exp - . (13) in which: w = wave frequency

S(w). = spectral density of high frequency waves p = frequency shift = envelope square frequency.

In. _{Fig. .12 comparisons are given of envelope square spectra}

as _{derived directly from the wave elevation time records and} as _{derived from eq. (10) using the spectrum of the high f}

re-quency wave components.

Comparisons of the distributions o.f the envelope square

rec-ords with the results of eq. (12). are given in Fig. 13. The

good comparison indicates that the adjusted irregular waves conformed well with the random wave assumption.

6 RESULT;S OF MODEL _{TES.TS IN REGULAR WAVES}

6.1 _{Mean Drift Force.s in Head Waves}

The results on the mean surge drift force in regular head

waves are shown in Fig. 14. for both water depths tested. The

results of computations are also given. In generai the

corre-lation between measurements and computations is satisfactory.

in shallower water t:he scatter in the experimental data is

somewhat larger than is usually found when testing in deeper

water. An unusual feature in the mean longitudinal drift

force is the high peak at a wave frequency of about 0.5

radis. This peak increases considerably when the w.ter depth

is _{decreas.ed from 30.2.0 m to 22.68 m. This behavioùr is also} followed by the computed data.

t, in radis

Theoretical

.A o o Regular waves: 2a = 2 rn:, 4 m, 6 m

'Figure 14. Mean surge drift force in head waves f or the fully loaded condition.

6.2 Low Frequency Surge Forces

These 'results are shown in Figs. 15 and 16 in the form of the amplitude of the quadratic' transfer function to a base of the

'frequency difference which corresponds to the frequency

of the force componen. Êach sub-figure is valid for a con-stant valu'e of the mean frequency of the components of a

reg-ular wave group. The values for zero difference frequency

correspond to the. mean drift force in regtilar waves. Two

tests were carried out in irregular waves at each of the two

water depths. In Fig. '15. the results of the test in irregular waves with the lowest wave height were omitted at a mean .f

re-quency of 0.43 rad/s. Results from the. .est in the highest

wave havebeen omitted for a mean frequency of 0.68 rad/s. These results were considered unreliable since the wave spec'

tra did not contain sufficient 'energy at these frequencies.

This leads to large inaccuracies in the cross-bi-spectral analysis results.

Results of computations show fair agreement. with the results

of cross-bi-spectral analysis of the measured data from the

model tests. The computed and the measured low frequency

surge force show a large peak at a difference f requency of

about 0.1 to 0.1.5 rad/s at the lower mean frequencies.

In-spection of the computed results shows that this peak is due

to the second order potential contribution to the drift.

force. As indicated previously this' is associated with the

phenomenon of wave set-down. Comparison of the results for

22.68 mand 30.2Om shows that the f orce amplitudes are about a f actOr twà ffi'heTrãt' the"

lÖ'wr"watêr'-in agreement with these-nsitivity of the set-down for de-creasing. water depth as seen in Figs. 10 and il.

-21-Water depth 22.68 m o o D

io

Water depth 30.20 m

DO

4j'

30 30 c.'J E 20. 20 -S 4-4-, C' c'J S-> 10 ILi O O O - ' 0.5 '1.0 1.5 0 5 1.0 1 5 (i in radis

O O 5O =0.43 50

.

-..--.

/

/

/

/

/

/

--,

= 0.59 0.25 0 Water depth 30.20 m Theoretical Irregü1ar waves: 2.77 m, T = T1 = 11.17 s = 0.49 s

-/

/

N

/ / .5 = 0.65 0.25 in radis = 0.54 = 0.70 0.25 s s

Figure 15. Quadratic transfer function of the surge drift

force in head waves for the fully loaded condition

at 30.20 m water depth.

O 0 25

100 100 o o = 0.43

.

= 0.59 0.25 0.25

N

Water depth 22.68 rn Theoretical Irregular waves: H = 2.76 m, T1 = 9.74 s 5.89 m, T1 '= fl .45 s = 0.65 0.25 = 0.81 in radis

Figure 16. Quadratic transfer function of the surge drift

force in head waves for the fully loaded condition

at 22.68 m water depth. = 0.54 = 0.70 0. 25 o 0.25 0 0.25 0 0.25 O 0.25 = 0.86

.

The orre1ation between the measured data for the different

tests and the results of computations is such that it is

concluded that this contribution is significant. It is also

noted that the quadratic transfer function for the low f

re-quency force with a (difference.) frere-quency of about 0.1 rad/s is considerably higher than the value for a difference

fre-quency of zero at the same mean frequency. This indicates that when low frequency force components at such frequencies

are of interest, as could be the case with the vessel moored

to a jetty which results in relatively high natural f

requen-cies for the horizontal modes of motion, the mean drift force transfer function severely underestimates the true value. This means that approximations such as proposed by Newman

(13) or Pinkster (i] must be viewed very carefully before being applied for the analysis of relatively stiff shallbw water mooring systems which induce high natural frequencies

in the horizontal motions of the moored vessel combined with sea conditions with relatively long wave periods.

7 FINAL REMARKS

In this paper attention has been paid to mean and low f

re-quency horizontal wave drift forces on a loaded tanker in shallow water conditions. It has been shown on the basis of both the results of computations and the results of model

tests in irregular waves that decreasing the water depth

results in an increase in the magnitude of the wave drift

forces. The. effect on the low frequency drift force

compo-nents of the contribution associated with the wave set-down

phenomenon' has also been demonstrated both from the. results.

of model tests and computations.

It has been found that although drift forces are predicted

reasonably well based on potential theory methods, the

scatter in the measured data on the drift forces tends to be

larger in shallow water than in deeper water. This may be

associated with the occurrence of larger higher harmonic

com-ponents in the incoming undisturbed waves which are present

in shallow water. It has been found that in the present.test program no undue effect of free long waves ön the. measured low frequency forces could be discerned even though these

waves are predicted by potential theory to be of the same

order as the 'bound long waves (set-down waves). This may be

due to the fact that in the basin under consideration the

wave paddies, where the free long waves are supposed to

originate, in order to compensate the bound long waves, are in fact, situated in an. area where. the water depth is consider-ably larger than in the. test basin proper where the

measure.-ments took place.

The model tests aimed at measuring the low f requency force

components were carried out using a novel dynamic system of restraint aimed at minimizing the low frequency motion

re-sponse of the otherwise soft_moored vessel. In this system

use was made of a feed-back loop and a feed-forward loop in

order ta-cont-rol--the dynamic -system - o-f rest-ra-i-n-t.

-24-forward loop was based on the application of real-time eval-ua.tion of the relative wave elevation contribution to the

drift force. The results of measurements and the subsequently applied cross-bi-spectral analysis technique are encouraging and will be developed further as standard tools for

experi-mental investigation of low frequency wave drift fcrces. ACKNOWLEDGEMENT

The data presented in this paper have been made available from an extensive Joint industry Research Program carried out

by' MARIN on behalf of a group of cömpanies engaged in

off-shore activities. REFERENCES

[i.] Pinkster, JA., Low frequency' second order wave exciting

' forces on floating structures, PhD thesis, Technical

University of De'lft, 1980.

DaizeIl, D.F., Application of the fundamental polynomial model' to the ship added resistance problem, 11th Sympo-sium on Naval Hydrodynamics, University College, London,

'

1976.

Oortmerssen, G.. van, The motions of a moored ship in

'

waves, PhD thesis, Technical University of Deift, 1976.

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

subject to short seas, Proceedings of t'he Royal

insti-tute of Naval Architecture, 1975.

Wichers, J.E.W., A simulation model for a single point

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