�I

Hydraulics Research Wallingford

QUAYSHIP: A computer model of a ship against a quay in the presence of waves

G H Lean BSc ARCS DIC E C Bowers BSc PhD DIC J M A Spencer MA

Report SR 232 February 1990

Registered Office: Hydraulics Research Limited, Wallingford, Oxfordshire OXlO 8BA. Telephone: 0491 35381. Telex: 848552

This report describes work funded by the Department of the Environment under Research Contract PECD 7/6/111 for which the DoE nominated officer was Or R P Thorogood. It is published on behalf of the Department of the Environment but any opinions expressed in this report are not necessarily those of the funding Department. The work was carried out in the Maritime Engineering Department of Hydraulics Research, Wallingford, under the management of Dr S W Huntington.

� Crown copyright 1990

Published by permission of the Controller of Her Majesty's Stationery Office.

QUAYSHIP: A computer model of a ship against a quay in the presence of waves

G H Lean BSc ARCS DIG, E C Bowers BSc PhD DIG, J M A Spencer MA

Report SR 232 February 1990

ABSTRACT

A computer model QUAYSHIP has been developed to describe the linear response to waves of a vessel against a quay. This model complements UNDERKEEL which was developed to describe responses of vessels to waves in the open sea. Both these models are essential building blocks in the development of a computer model of a moored ship in waves which, in turn, is needed to satisfy the requirement for a realistic first estimate of ship responses in feasibility studies of proposed harbour developments.

QUAYSHIP has been applied to the case of a large tanker against a solid vertically faced quay. Hydrodynamic coefficients in sway and heave have been shown to correlate well with results obtained using a 3D source method. The advantage of QUAYSHIP, here, is that much less computation is needed and difficulties representing small clearances with the 3D source method are avoided.

Hydrodynamic coefficients for all the degrees of freedom of vessel movement have been found to be affected by a "manometer" resonance in which flow moves vertically, in the clearance between the vessel and the quay, and horizontally under the keel. Surprisingly, these flows appear to be cancelled to a significant degree by flows due to wave diffraction, leaving vessel responses to waves largely unaffected by the resonance.

Comparisons of the calculated hydrodynamic coefficients with those measured in a physical model show that the effect of the manometer resonance is much less pronounced in the physical model. A simpler model than QUAY SHIP was developed for two dimensional flow, in planes at right angles to the quay, in order to study the effect of friction on the resonance. The friction expected in physical models appears sufficient to explain the differences between potential theory and the physical model. At full scale it appears that some effect of friction remains on the hydrodynamic coefficients. However, the simplified two dimensional model indicates that vessel movements due to waves are unaffected by friction in both the physical model and at full scale : a result that would be consistent with the finding that actual vessel responses appear to be largely unaffected by the manometer resonance.

Recommendations are made for the further research needed into the roll responses calculated in UNDERKEEL and QUAYSHIP, and the added inertia coefficients for berthing vessels. The latter area being of concern in jetty design due to the size of berthing impacts experienced with large modern vessels. The research reported here also indicates that some simplification in modelling linear vessel responses may be possible. This could greatly assist in the subsequent computation of non-linear wave forces acting on a ship moored to a quay. It is recommended, therefore, that simplifications in modelling linear responses be sought.

CONTENTS

1

2

3

4

5

6

7

INTRODUCTION

THEORETICAL CONSIDERATIONS

2. 1 Manometer resonance

RESULTS FOR A VESSEL NEAR A SOLID VERTICAL QUAY

3. 1 Hydrodynamic coefficients 3.2 Vessel response

FRICTION EFFECTS ON THE MANOMETER RESONANCE

4.1 Application to a block ship

CONCLUSIONS

RECOMMENDATIONS

REFERENCES

TABLES

1. Vessel details (loaded) for results in Section 3

FIGURES

1 The six degrees of vessel movement 2 Definition of coordinates 3 Manometric natural mode of motion 4 Vessel lines 5 Added inertia and damping in sway and heave - B. 25m quay

6

7

clearance Added inertia and damping in sway and heave clearance Added inertia and damping in sway and heave clearance

16.5m quay

24. 75m quay

B Added inertia and damping in sway and heave - 33. 0m quay clearance

9 Added inertia and damping in sway and heave clearance

41.25m quay

10 Added inertia and damping in surge predicted by QUAYSHIP 11 Added inertia and damping in sway predicted by QUAYSHIP 12 Added inertia and damping in heave predicted by QUAYSHIP 13 Added inertia and damping in roll predicted by QUAYSHIP 14 Added inertia and damping in pitch predicted by QUAYSHIP 15 Added inertia and damping in yaw predicted by QUAYSHIP 16 Vessel responses - 270° wave direction (beam sea) 17 Vessel responses 285° wave direction 18 Vessel responses - 300° wave direction

Page

1

3

5

7

8 12

14

15

19

22

24

CONTENTS (CONT'D)

FIGURES (CONT'D)

19 Vessel responses 315° wave direction 20 Vessel responses - 330° wave direction 21 2D sway linear potential theory and model friction 22 2D sway - prototype friction 23 Variation of damping coefficient near resonance with incident wave

amplitude a 24 Variation of damping factor near resonance with incident wave

amplitude a

1 INTRODUCTION

A suite of computer models is presently under

development at Hydraulics Research (HR) to satisfy the

requirement for a realistic first estimate of ship

response to wave action. These models, coupled with

the Boussinesq model describing waves in harbours,

will enable more accurate feasibility studies of

proposed harbour developments to be carried out prior

to detailed studies using physical models. The ship

models are vital in feasibility studies due to the

extreme variability of ship response to wave

parameters: direction and period being particularly

important. This variability invalidates judgements of

berth downtime based on wave height alone.

The computer model UNDERKEEL has been developed to a

stage where it will describe the linear response of a

free ship to waves (Ref 1) . UNDERKEEL has been proved

against a physical model of a tanker underway in

random waves and it is used in project work for

initial optimisation of dredged depths of navigation

channels that are exposed to waves (Refs 2 , 3, 4, 5).

As it stands, UNDERKEEL can also be used to describe

the linear response of a ship moored at an open

berth, ie a piled jetty at some distance from any

reflecting boundaries. This is because waves pass

through vertical piles (assuming typical spacing)

without significant reflection thereby simulating the

conditions that apply to a vessel in the open sea.

But the situation of a vessel moored to a quay is

different since both the incident waves and waves

created by movement of the ship will reflect from the

quay to some degree. With a solid vertical quay

constructed of masonry blockwork, or of steel sheet

piling, this reflection will be almost total making

the water flow around the vessel noticeably different

from the case of an open berth.

l

It was thought originally that the computer model

UNDERKEEL could be adapted to the case of a ship

moored to a quay without a great deal of additional

work but this has not proved to be the case. In

developing UNDERKEEL for the open sea situation use

was made of the fact that surge, pitch and heave (see

Fig l) create flows around the vessel that are

symmetrical about the longitudinal axis of the vessel,

and sway, yaw and roll create antisyrnmetric flows.

This feature simplifies the solution of the resulting

equations. With a reflecting boundary nearby the

motion of the vessel in a particular oscillating mode

(surge, sway etc) leads to flows on each side of the

ship which, in contrast to the open sea situation,

cannot be classified as being either completely

symmetric or completely antisyrnmetric. This means

that a single oscillating mode results in a force and

a turning moment on the ship with components in all

three directions. Although the extra coupling was

expected, solving the resulting equations has proved

to be a much greater task than anticipated and a new

model has been developed to satisfy the requirement.

This model has been named QUAYSHIP in recognition of

the fact that flows in the side clearance between the

ship and the quay will play an important part in

controlling vessel response just as flows under the

keel become important with a small underkeel

clearance.

In parallel with this work, the many non-linear wave

forces that can act on a vessel are being computed as

functions of wave frequency making use of equations

derived in an earlier report (Ref 6). This is being

done for the case of a vessel moored at an open berth

where the linear velocity potentials for vessel motion

and wave diffraction from UNDERKEEL are used in the

formulation of the non-linear wave forces. The same

equations given in Reference 6 can be used to compute

2

2 THEORETICAL

CONSIDERATIONS

the non-linear wave forces acting on a ship against a

quay. Only in this case, the linear velocity

potentials used in formulating the non-linear wave

forces must come from QUAYSHIP. Once this stage has

been reached, the mathematical models will be capable

of describing the motions of vessels moored at open

berths, and to berths with a quay face, although only

for the case of linear moorings where a .frequency

space description of vessel motion is valid. Further

work is then needed to represent non-linear wave

forces in a random sea as functions of time. This is

to allow their incorporation into SHIPMOOR (Refs 7 and 8) : a time domain computer model which is needed

to represent important ship motion effects like

subharmonic sway responses due to fenders being

stiffer than mooring lines. It is necessary that the

ultimate model of a moored ship be a time domain model

as only then can the non-linear characteristics of

conventional mooring systems be described.

From the above discussion it can be appreciated that a

vital building block in computing moored ship motions

against a quay is the linear response to wave action

and that forms the subject matter of this report.

Theoretical aspects are discussed in Section 2 and an

application of QUAYSHIP to a large tanker moored to a

solid vertical quay is described in Section 3. The

effect of friction on the flows induced under the keel

of the vessel are investigated in Section 4. Conclusions and recommendations for further research

follow in Sections 5 and 6.

The method used to calculate wave forces follows the

same pattern as before (Ref ll ie for each mode of

vessel motion the flow beneath the hull is simplified

3

and the flow round the sides of the vessel is

represented by suitable distributed sources where

source strengths are determined by the boundary

conditions on the vessel's surface. However, the

potential (Green's function) representing flow from a

source must now also satisfy the boundary conditions

at the quay face. In addition, to calculate the force

on the vessel arising from diffraction, the potential

of the total incident wave system for the vessel must

be defined.

A simple situation for which these potentials can be

easily derived is that of a ship moored at a perfectly

reflecting straight vertical wall in an otherwise open

situation (Fig 2). This can be taken as

representative of a berth inside a harbour. In this

case the velocity component normal to the wall must be

zero and to satisfy this condition the required

Green's function is equal to the sum of the open sea

potentials of the source and its image with respect to

the wall.

We denote the potential of the incident wave by the

real part of,

cosh K (z+c) iK (Ax + py) -iwt cosh Kd

e e

where a, w are amplitude and radian frequency of the

wave, �is the angle of propagation (2rr > � > rr, Fig

2) with

A, P cosiJ, sinll

and K satisfies the usual dispersion relation for

surface waves,

w' Kg tanh Kd

4

2.1 Manometer

resonance

Here d is the water depth and c is the height of the

origin of coordinates above the bed.

The boundary condition at the quay wall requires

a ay

(� 0

+ �.) = 0 at y -e

This is satisfied if the potential of the reflected

wave (� ) is given by, r

� = R iag r w

where

R =

e-2iK�e

cosh K (z+c) iK (Ax - �y) -iwt cosh Kd e e

Then the total wave system incident on the ship is

given by

As explained in the Introduction, motion of the vessel

in a given oscillatory mode will couple with all the

other modes. One consequence is that the 6x6 matrices

representing inertia and damping each have 36 non zero

components instead of having just 18 components as

described in Reference 1 for the ship in an open sea

situation. This results in a more complicated method

of solution for the amplitudes of vessel motion

compared with the open sea case.

One of the most startling differences with the open

sea situation is that added inertias can become

negative for a ship against a quay (Ref 9) . This will

be demonstrated in the subsequent section where

5

results are presented for a tanker moored near a solid

vertical wall. It must be remembered that added

inertia is just a way of describing the component of

water flow around a vessel that is in phase with

vessel acceleration. In an open sea situation an

extra mass of water has to be accelerated with the

vessel so the added inertia is always positive. But

for a ship against a quay, the water trapped in the

clearances between the ship and the quay, and between

the ship and the seabed, can go into resonance with

the result the added inertia changes its sign at the

resonant frequency. In effect, the water trapped in

the clearances acts like a spring. This type of

motion can be described with reference to Figure 3.

We consider oscillations of the water in the

clearances b and 0 with velocities w and v,

respectively, producing an increased elevation � in

the clearance with the quay (b). The vessel's beam

and draft are taken to be B and D, respectively,

Continuity of the flow requires,

wb vO

Momentum equations in the clearances show that the

additional pressure P1 over and above atmospheric

pressure, produced at the surface in the clearance

between the ship and the quay is given by, • • -wD - vB

We can relate this pressure to the increased elevation

� through

6

3 RESULTS FOR A

VESSEL NEAR A

SOLID VERTICAL

QUAY

and express vertical velocity in the form

' w I;

Eliminating v, w and P1 we obtain the following

equation for the vertical movement of the water

surface in the clearance between the vessel and the

quay,

" B I; [D + b t6J + g I;= 0

So the natural "manometer" frequency is given by,

(1)

It can be appreciated that small vessel movements at

this resonant frequency, in heave or sway for example,

will produce large flows in the clearances. Thus,

exactly on resonance the added inertia will vanish and

significant waves will be produced in the open sea

side of the vessel, indicating large damping of the

vessel motion. In practice, though, friction effects

on the flows in the clearances will limit the size of

these effects (Ref 10) and this aspect of behaviour is

considered further in Section 4 of this report.

Here, we describe the application of QUAYSHIP to the

case of a tanker aligned parallel to, but at various

7

3.1 Hydrodynamic

coefficients

distances from, the quay face. The vessel details are

given in Table 1.

This particular situation has been chosen because it

relates to what appears to be the only comprehensive

set of data in the literature for a ship near a solid

vertical quay (Ref 9). In this reference, van

Oortmerssen presents both physical model and

theoretical results for some of the hydrodynamic

coefficients of the vessel, ie added inertia and

damping coefficients in sway and heave. The physical

model scale was 1 to 82.5 and his theory made use of

sources distributed over the submerged surface of the

hull with strengths that satisfied the relevant

boundary conditions assuming linear potential flow (3D

source method).

Cross sections of the vessel at 21 e qually spaced

stations along its length starting at the stern, are

presented in Figure 4 : stern sections appear on the

left and bow sections appear on the right. These

vessel characteristics were represented in QUAYSHIP

and hydrodynamic coefficients for all six degrees of

freedom of vessel movement (see Fig 1 for definition

of movements) were calculated along with vessel

responses. To allow for comparison with van

Oortmerssen's results 5 different clearances, between

the straight sided part of the vessel and quay, were

considered (distance b in Fig 3). These ranged from

8.25m up to 41.25m.

We consider first the results for sway and heave

hydrodynamic coefficients. Comparisons with van

Oortmerssen's experimental and theoretical results are

presented in Figures 5 to 9. These hydrodynamic

8

coefficients have been made non-dimensional in the

following way

' A,,

Sway added inertia A,,

' B,,

and damping B,, =

M {g/L

' A,,

Heave added inertia A,, =

' B,,

and damping B,, M

In these definitions, M is the displacement (mass) of '

the vessel, L is the length of the vessel while A22 '

and A 33 are the added inertias in sway and heave, and ' .

B22 and B33 are the damping coefficients for sway and

heave. They are plotted as functions of a

non-dimensional wave frequency w {L/g.

The first point to notice is the correlation obtained

between QUAYSHIP (solid line) and van Oortmerssen's

theoretical results (closely spaced dashed line) in

Figure 6. It is possible to make this comparison only

for a separation distance of 16. 5m from the quay as

these are the only theoretical results given in

Reference 9. The advantage of QUAYSHIP over the 3D

source method used by van Oortmerssen is that much

less computation is needed and difficulties

representing small clearances with the 30 source

method are avoided.

9

It is clear that all the results display the manometer

resonance described in Section 2.1. The added

inertias vanish on resonance and the shift of this

resonance to smaller frequencies, seen in Figures 5

to 9 as the separation distance (b) with the quay

increases, is explained by equation (1). We see this

resonance causes large damping with negative added

inertias for frequencies of movement just above the

resonant frequency and very large positive inertias

for lower frequencies. We expect this resonance to

disappear in the limit of a large clearance with the

quay and such a trend is visible in Figures 5 to 9

where the large positive inertias, seen with a quay

separation distance of 8. 25m, become less pronounced

with a separation distance of 41.25m. It is

remarkable, though, how important the effect of the

resonance remains even when the separation distance is

of the order of the beam of the vessel. In practical

situations, where a vessel is moored against a

vertical quay, the separation distance will be of the

order of a few metres at most, ie the order of the

limit of compression of the fenders. In such

situations the manometer resonance will be even more

pronounced. For the ship being studied here, a

clearance with the quay of 2m results in resonance

occurring at a period of some 13 seconds (equation

(1)) which is well within the range of possible wave

periods. However, the important parameter for a

moored ship is the final vessel response and the

complexity of this response means that one particular

factor, like the manometer resonance, is not

necessarily dominant (see Section 3.2).

The other obvious point emerging from Figs 5 to 9 is

the apparent overestimation, by both QUAYSHIP and van

Oortmerssen's 3D source method, of the resonance

effects present in the physical model results. This

is most pronounced at the smaller quay clearance of

10

8.25m (Fig 5). But the difference between results

from QUAYSHIP and results from van Oortmerssen's

experiments appear minimal at the largest quay

clearance of 41.25m (Fig 9). It will be shown in the

next section that these differences between theory and

experiment can be explained by friction effects on the

flow under the keel. Resonance enhances these flows

with a small quay clearance eg 8.25m, making the flows

more sensitive to any resistance due to friction. As

resonance effects decrease with an increasing quay

clearance eg 41.25m, friction can be expected to

become less important so a theory without friction

effects can be expected to apply.

For completeness, we present non-dimensional

hydrodynamic coefficients from QUAYSHIP for all six

degrees of freedom in Figures 10 to 15. Surge added . '

inertia A11 and damping B11 are divided by the same

factors as the sway and heave coefficients ie

' A,,

A11 M

' B,,

B" �

M Jg/L

The added inertia and damping coefficients for roll I I ! I I t

CA44, B44) pitch (Ass. Bssl and yaw CA66, B66) have

been made non-dimensional in the following way,

A ii

A ii

' Aii

= --. M L

i 4' 5' 6

11

3. 2 Vessel responses

Unfortunately, there are no results presented in Ref 9 for surge, roll, pitch and yaw hydrodynamic

coefficients for the vessel near a solid vertical

quay. But it is clear that these additional vessel

movements exhibit the "manometer" resonance already

identified in sway and heave. There also appears to

be some "structure" at higher frequencies than the

manometer resonance in some of the coefficients, eg

A66 in Figure 15. This correlates with a half

wavelength resonance across the gap between the side

of the vessel and the side of the quay. Such effects

appear negligible in sway and heave and may well prove

to be unimportant in practice.

We present responses for all six degrees of freedom in

Figures 16 to 20 for wave directions (see angle l'l in

Fig 2) of 270° (beam seas) , 285°, 3oo• 315° and 33o•

respectively. The responses have been made

non-dimensional for surge, sway and heave by dividing

the amplitude of movement by the amplitude of the

incident wave. This explains, for example, why the

heave amplitude tends to double the incident wave

amplitude for low frequencies ie the vessel follows

the vertical movement of the water surface formed by

wave reflection from the quay face (wave antinode) .

The angular movements of roll, pitch and yaw have been

expressed in degrees per metre of incident wave

height.

Perhaps the most startling result is the smoothness of

the response curves at frequencies corresponding to

the manometer resonance. Considering sway in a beam

sea (Fig 16(b) ) and comparing it with the behaviour of

the sway added inertia and damping coefficients

(Fig 11) we see little sign of an effect on resonance.

For example, the response in Figure 16(b) with a quay

clearance of 8.25m is relatively smooth from low

12

frequencies right up to a value of 3 for the

non-dimensional frequency. Figure 11 shows large

changes in the hydrodynamic coefficients over this

frequency range for a quay clearance of 8.25m due to

the manometer resonance which occurs at a frequency of

about 1.7. This sort of result indicates that flows

induced under the keel by wave diffraction must be

counteracting the manometer resonant flows induced by

movement of the vessel. Such cancelling effects also

appear operative in surge (Fig 16(a) ), heave (Fig

16(c) ) roll (Fig 16(d) ) pitch (Fig 16(e) ) and yaw (Fig

16(f) ) .

There also appears to be an anomalously large heave

response in Figure 16(c) at a frequency of about 2 for

a quay clearance of 4 1.25m. This appears to be

driven, at least in part, by coupling of a very large

roll response at this quay clearance (see Fig 16(d) )

into heave. The roll response is very high in this

case because the centre line of the vessel happens to

lie close to a node which produces a large roll couple

close to the resonant roll period of the vessel. In

practice viscous damping will lead to a much reduced

roll response which in turn means the heave response

predicted by QUAYSHIP is exaggerated. This problem

can only be overcome by representing viscous damping

in the computer model.

Returning to sway responses in a beam sea ie

Figure 16 (b), we see a maximum in the response occurs

at a non-dimensional frequency that varies from about

3. 5 for a quay clearance of 8. 25m to about 2 for a

quay clearance of 4 1. 25m. These frequencies are also

consistent with a node (place of largest horizontal

water particle movement) forming along the centre line

of the ship due to wave reflection from the quay face.

One might expect to find the largest sway response at

such frequencies as the vessel will experience the

13

4 FRICTION EFFECTS

ON THE MANOMETER

RESONANCE

largest sway force in that situation, leaving aside

forces due to wave diffraction around the vessel.

The other feature to notice in the sway responses

(Fig 16(b)) is the behaviour in the limit as wave

frequency tends to zero. The response appears to

increase with the separation distance from the quay.

This feature can be explained using a simple two

dimensional model which can also be used to study the

effect of friction on the manometer resonance. These

aspects are considered in the next section.

QUAYSHIP has been shown to be effective in Section 3

through comparison with theoretical and experimental

results obtained by van Oortmerssen (Ref 9). The

comparison with physical model experiments has also

shown that potential theory appears to exaggerate a

manometer resonance that effects the vessel's

hydrodynamic coefficients. Here we use a simplified

model to investigate the effect of friction on the

flows caused by that resonance.

The method of solution employed in QUAYSHIP allows for

flow normal to the quay in the space between the side

of the vessel and the quay (side clearance). In

practice, however, the distance of moored vessels from

the quay is often small implying that flow normal to

quay can be neglected. In this case the flow parallel

to the quay satisfies a two dimensional surface wave

equation and is readily soluble once the boundary

conditions at the ends of the vessel are defined.

However, for the purposes of investigating the effect

of friction we assume the vessel is infinitely long so

that flow along the quay is neglected. The problem

then becomes two dimensional with flow only occurring

14

4.1 Application to a

block ship

in planes perpendicular to the quay. For an

infinitely long ship with a rectangular (block) cross

section the flow will be vertical in the side

clearance and horizontal in the underkeel clearance

(see Fig 3). Given these conditions it is possible to

derive a linear relation between the potential and the

fluid velocity on the open sea side of the underkeel

clearance space. In this case, the potential on the

open sea side of the block ship can be derived using a

similar treatment to that adopted in UNDERKEEL for a

vessel in the open sea.

Here, we apply the simplified two dimensional model

described above to the situation portrayed i n Fig 3

with the following parameters

6 B b D; 0.067. D; 2.5,

D 0.287

The resulting added inertia and damping coefficients

in sway are shown in Figures 21 and 22 along with the

sway response to waves at normal incidence (beam sea)

calculated in the absence of vessel heave. As the

vessel length is infinite, parameters are expressed

relative to the vessel's beam. Thus, sway added • •

inertia (A22) and damping (B2 2 ) have been made

non-dimensional using,

• B.,

B., ; ---­

pBD {giB

15

and the wave frequency is made non-dimensional using,

w iB g

In Figure 21 there are two sets of results. One is

derived using linear potential theory, which also

forms the basis of QUAYSHIP and the 3D source method

employed by van Oortmerssen. In this case damping is

due solely to waves radiating energy away from the

vessel on the open sea side of the block ship. It can

be verified that the manometer resonance occurs at a

frequency consistent with equation (1). The second

set of results in Figure 21 is derived allowing for

friction effects one expects to find in a 1 to lOO scale physical model.

A corresponding set of results is presented in

Figure 22 allowing for the friction effects one

expects to find in prototype, ie for a real ship.

Figure 21 shows the sort of difference found between

linear potential theory and the physical model results

of van Oortmerssen where sway hydrodynamic

coefficients on resonance in the experiments were

about half of those calculated (see Fig 6(a)). It can

also be seen, by comparing the hydrodynamic

coefficients in Figure 22 with the linear potential

theory results in Figure 21, that friction effects at

full scale, ie for real ships, have a smaller effect

on the manometer resonance.

Two friction effects have been represented. One is

the head loss in the underkeel flow (v) due to viscous

shear (•) on the boundary surfaces. As in steady

flow, this energy loss is assumed to depend on whether

the flow is laminar or turbulent. The controlling

factor is the Reynold's number (R ) e

16

For laminar flow

For turbulent flow

' ; pv'

'; pv'

when R < 1000 e

0.0336/ , , , R

e

when R > 1000 e

Here, p is the density of water and the Reynolds

number is defined by,

R e

= � "

where " is the kinematic coefficient of viscosity.

The pressure loss over the underkeel clearance is then

given by

IJ.p I = 2B ';

p 0 p

The second friction effect arises from eddy losses in

the flow as it separates at entry to, and exit from,

the underkeel clearance during the oscillation. As in

steady flow, the sum of entry and exit losses are

taken to be proportional to the square of velocity,

where � is a coefficient obtained from experiment (� 1.44, Ref 10). In fully turbulent flow the head

losses will be approximately proportional to the

square of the flow velocity under the keel. In these

cases a linearised friction coefficient proportional

to velocity is defined to allow solution of the

equations. This is why the results are sensitive to

the amplitude of the incident wave (Figs 21 and 22).

17

The behaviour of the damping coefficient B11 near

resonance is shown in greater detail in Figure 23. We again see that the linear potential theory result

is roughly halved in magnitude due to the friction

effects expected in a 1 to lOO scale physical model.

It is also apparent that the damping in the physical

model is not very amplitude dependent. This is

because the flow expected under the keel is in the

laminar range where energy losses are more nearly

proportional to velocity, rather than the square of

velocity. In full scale, the flow is expected to be

turbulent making damping more amplitude dependent,

although the friction effect is noticeably smaller

than in the model.

It might be thought strange that including the effect

of friction actually reduces the damping coefficient

B,, (Fig 23). However, the effective damping factor

can be considered to be 822/ll +A,, I after allowing

for added inertia, ie dividing the damping coefficient

by the displacement mass plus the added mass. This

damping factor is shown in Figure 24 for linear

potential theory and full scale conditions on the

left-hand side and for a 1 to 100 scale physical model

on the right hand side. It can be seen the effective

damping factor is indeed increased when friction is

included and that the largest increase occurs in the 1 to lOO scale physical model (note the logarithmic

vertical scale).

Finally, we return to the sway responses shown in

Figures 21 and 22. The smooth behaviour of the sway

response C, over the frequency range encompassing the

rapid ranges in the hydrodynamic coefficients (due to

the manometer resonance) is again apparent, as noted

in Section 3.2 for the three dimensional model

QUAYSHIP It is also apparent that in the 2D model at

least, the addition of friction has little effect on

18

5 CONCLUSIONS

sway responses. This is reassuring for physical model

work using models of moored ships. However, there is

little doubt that added inertia effects, so important

in controlling the berthing impacts of large vessels,

would not be well represented in physical models. In

the case of berthing, mathematical simulations

incorporating the friction effects expected at full

scale, are needed for accurate results.

To verify the magnitude of the sway response it is

possible to use elementary methods to solve the linear

potential equations for the case of long waves

(w = 0) . This leads to an estimate for the sway

amplitude �. which is related to the side clearance

b : an effect noted earlier in Fig 16(b) with

SIDEKEEL.

This equation yields a value of 0.58 for the block

ship in question which agrees with the result at w 0 given for potential theory in Figure 21. This

response appears to be maintained at higher

frequencies and the observation that friction seems to

have a minimal effect on the response is again

consistent with the idea that flows under the keel due

to wave diffraction are tending to cancel the flows

due to the manometer resonance : if the resultant flow

is small the importance of friction on that flow will

clearly be diminished.

l. QUAYSHIP has been developed to describe the

linear response of a vessel moored against a quay

and subjected to wave action, a situation typical

of a berth inside a harbour.

19

2. Correlation between QUAYSHIP and a different

method of computing responses (Ref 9) has been

obtained for hydrodynamic coefficients in sway

and heave for the case of a tanker at a distance

of 16.Sm from a vertically faced quay (Fig 6).

The advantage of QUAYSHIP is that simplifying

assumptions about the flow under the keel enable

results to be obtained with much less computation

and difficulties representing small clearances

are avoided.

3. In comparisons of results obtained using linear

potential theory (QUAYSHIP and the 3D source

method used in Ref 9) with physical model results

it is clear that the effects of a manometer

resonance on vessel hydrodynamic coefficients are

much less pronounced in the physical model

(Fig 6). On resonance large flows occur under

the keel of the vessel and energy losses due to

friction effects appear important.

4. A two dimensional model (infinitely long ship)

has been developed to study the effects of

friction on the flow under the keel. Head losses

have been represented due both to viscous shear

on the boundary surfaces and to eddying as the

flow separates at entry to, and exit from, the

underkeel clearance. This work indicates that

friction effects in physical models will be much

greater than for real ships but that the

turbulent flow conditions expected at full scale

mean that real ship hydrodynamic coefficients

will also be affected by friction on the flow

under the keel.

5. In spite of finding significant friction effects

on hydrodynamic coefficients, the two dimensional

model result for the sway response to wave

20

action appears independent of friction. This

suggests that flows under the keel due to wave

diffraction are tending to cancel flows

associated with the manometer resonance. If the

resultant flow is small it can be appreciated

that friction effects will become less

important.

6. The above result is encouraging for physical

models of harbours that use models of moored

ships to judge berth downtime. It suggests that

horizontal vessel movements should be well

represented in the physical model even though

hydrodynamic coefficients are likely to be

affected by unrealistic friction effects on flows

under the keel.

7. The conclusion for vessels berthing in the

absence of waves is different. Here the

coefficients of added mass and damping in sway

are important factors controlling berthing

impacts and the importance of friction in

controlling added mass and damping shows that

they would be poorly represented in a physical

model due to scaling problems. In this case a

mathematical simulation model that made use of

the hydrodynamic coefficients, calculated

allowing for full scale friction effects, is

needed to investigate berthing vessels.

8. Horizontal vessel responses obtained with the

fully three dimensional QUAYSHIP also imply that

flows under the keel due to wave diffraction tend

to cancel flows due to the manometer resonance.

This suggests that conclusions drawn about

friction from the two dimensional model should

apply in three dimensions as well.

2 1

6 RECOMMENDATIONS

9. It is anticipated that differences observed in

Reference 1 between UNDERKEEL and physical model

results for vessel roll in the open sea apply

equally to the roll of a vessel moored against a

quay. Indeed, the very large resonant roll

responses shown in Figures 16(d), 17(d), 18(d)

19(d) and 20(d) are likely to be damped by the

sort of friction effects described in conclusion

4 above. This in turn should reduce the

anomalous heave responses observed at some wave

frequencies. As these friction effects will be

important at full scale, the resonant roll

responses from QUAYSHIP must be considered

overestimates as found with UNDERKEEL (Ref 1).

The work described in this report has demonstrated a

clear need to develop a greater understanding of

viscous damping effects. Resonant roll responses are

known to be sensitive to such damping, making the

linear potential theory used in UNDERKEEL and QUAYSHIP

yield overestimates of roll. And, the work on vessel

sway described here has shown viscous damping to be

important in controlling added mass and damping which,

in turn, will affect loads on jetties due to berthing

ships.

It is recommended, therefore, that further research be

carried out into viscous damping for the specific

situation of a vessel with a small underkeel

clearance. This with a view to improving the accuracy

of vessel roll predictions in UNDERKEEL and QUAYSHIP

as well as improving added inertia predictions for

horizontal vessel movements near a solid quay. The

improved estimates of added inertia and damping should

be used to develop accurate vessel berthing simulators

as an aid to jetty design : an area of concern at

present due to the size of berthing impacts

experienced with large modern vessels.

22

The finding that flows induced by wave diffraction

tend to cancel the (resonant manometer) flows

associated with vessel motion suggests that some

simplification in modelling linear vessel responses

may be possible. As this could assist greatly in

reducing the amount of subsequent computation needed

to calculate non-linear wave forces, for the case of a

ship moored to a quay, it is recommended that research

be carried out to seek any simplifications that may be

possible.

23

7 REFERENCES

1. Lean G H, Bowers E C and Spencer J M A.

"UNDERKEEL : a computer model for vertical ship

movement in waves when the underkee1 clearance is

small" HR Report SR 160, 1988.

2. Hydraulics Research. "Underkeel allowance for

deep draughted vessels in the Dover Strait.

Phase l, preliminary study to establish critical

parameters. Phase 2, final estimates for a safe

allowance" Report Nos EX 1309, 1985 and EX 1432,

1986.

3. Hydraulics Research. "Use of the approach

channel to Port Muhammad bin Qasim, Pakistan, by

Panamax container vessels" Report No EX 1841,

1988.

4. Hydraulics Research. "Mathematical modelling of

a vessel's vertical motion in waves in shallow

water" Report No EX 1897, 1989.

5. Bowers E C. "Safe underkeel allowances for

vessels in restricted depths" Chapter 8, Recent

advances in Hydraulic Physical Modelling. NATO

ASI Series E Vol 165, Lisbon, Portugal, 1989.

6, Bowers E C. "The mathematical formulation of

non-linear wave forces on ships" HR Report

SR 193 , 1989.

7. Spencer J M A and Bowers E C. ''Mathematical

modelling of moored ships in the time domain" HR

Report SR 103, 1986.

8. Spencer J M A "Mathematical sirnulations of a ship

moored in waves and of the effects of a passing

vessel on a moored ship" HR Report SR 145, 1988.

24

9. Oortmerssen G van. "The motions of a moored ship

in waves". Netherlands Ship Model Basin Pub No

510, Wageningen, Holland.

10. Fontijn H L. "On the prediction of fender forces

at berthing structures Part 2 : Ship berthing

related to fender structure." Advances in

Berthing and Mooring of Ships and Offshore

Structures. NATO ASI Series E Vol 146,

Trondheim, Norway, 1987.

25

TABLE

TABLE 1 Vessel details (loaded) for results in Section 3

Length between perpendiculars

Beam

Draught

Displacement

Distance of centre of gravity forward of mid point

Height of centre of gravity above keel

Transverse radius of gyration

Roll metacentric height

Longitudinal radius of gyration

Water depth (underkeel clearance � 20% of draught)

310m

47.2m

18.9m

235,000m3

6.61m

13. 32m

17.0m

5. 78m

77 .47m

22.68m

FIGURES.

Fig. 1

"' "'

M "' ru > "' ru :X:

The six degrees of vessel movement.

0 cr. X

m'

I I

-

m' Reflected wave

Fig 2 Definition of coordinates.

Image

y

b

e b

angle [3 with Ox defines incident wave direction

Cl

>

..c

Fig 3 Ma-nomet ric n a t ural mode of motion.

0 I

F========-===J� r;?. CO

-,-

Fig 4 Vessel lines.

2Q ,

15

' " s

•• JO

25 I B,

o I 0.0

0. 5

0. s

l.O 1.5

1\ I I I \ I \

!

' / - � ' " ! J \ \ •. ' j I .,

va� Oort:oerssen shIp wIt-h

Distance from quay 8.25m.

0\JAYSHIP.

quay

5.0

van Oortmerssen sh!p • .. :!th quay

Distance from quay 8.25m.

-����--��- OUAYSHlP.

.) \ '

" ' �_::_'-'' ---�.:_ -=-=·-=-=-���---�-

2' s :3. c 3.5 4. 0 4. s 5. :.1 wl (L/gl

5.5

;. 5

Fig S(a) Added inertia and damping in sway-8.25m quay clearance.

5

-5

·"T -15!1 -20

-25

so�-1 45 + I

o.o

0.5 1.0

o.s \.0

I I ' 1,5 I

1.5 '·'

van Oor-tmerssen sh 1 p v 1 th quay Distance from quay 8.25m.

------ CLAYSHiP,

van Oort:nerssen shl.c vlth quay D: st a nee f,�om quay :_:. 25m.

2.5 3.D �.5 .J (l/gl

'· 0 4.5 5.0 s.s

Fig S{b) Added inertia and damping in heave-8.25m quay clearance.

20

IS

10

5 "'"

A,

0

-5

-10

-15

20

15

5

0 .J .... o.s l.O

I)

1.5 2.0

van Oortmerssen shlp 1.1lth quay

Distance from quay l6.5Cm.

-�------- OUAYSHIP. � -- -------- "'�n Oorf'"orssun oxpof'!lltOI"'t. ---- - - ----- v;,n Oo,...�fl\orn!Hirt �huv••y,

van Oort merssen sh 1 p 1.1 l th quay

Otstance from quay 16.50m.

---- �· �-� QU.t,YSH1P. ---------- - ¥,.... 0cf't!II\II<'I>SOn U>o:p&t'[•r�<•nt,

- ---------- van Ocrt•W'l!l!ofl th&O"f'•

2.5 3.0 ml (L/g)

3.5

Fig 6(a} Added inertia and damping '" sway-16.5m quay clearance.

30-

25 + ! 20 �

tS

10

5 AJ3

0 0.5

-5

'

�::r -25 1

60� ! 55-'-

so j 45 40 t 35 j_

3:!:) ' 50

20

15

"

5

0 o.o o.s !,0 1.5 2.0

2.5

van Cc!'"'tmerssen sh!p v! th quay

8Ist:an.ce from

3.0 wJ ll/gl

3.5

quay 16.50m.

4.0 4,5 5.0

van Oortmerssen ship with quay

Distance fr8� quay l6.50m.

-------- QUAYSHiP.

s.s

--------- -- v::m Gort-rsson experl-nt. --------- "¥1 Oort-rJtsen 1:haory.

2.5, 3.D 3.5 wJ IL/gl

4.C 4.5 s.o s.s

Fig 6(bl Added inertia and damping in heave-16.5m quay clearance.

20 ...,.-J i

] I 30+ J

822 20

15

I 0-,-

J I o.c 0.5 1.0

' i.S 2.0

van Oortmerssen ship v,th quay

0 I stance f�o:n quay 24. 75m.

van Oor<::merssan shrp \.ll'th quay

Distance f�o� quay 24. 75m.

---·--- - Q<JJ.l'SKIP.

2.5 3.0 3.5 wl (Lfgi

\. 0 '· 5 s.o 5.5

Fig 7(a) Added inertia and damping 1n sway-24.75m quay clearance.

\ van Cortmerssen ship l.l'lth quay

0 I stance ;:rom .quay 24. 75:n.

QUAYSHiP. \ 1')

\'

vary Ourtii")''$S"$"'1 &>GJ61'"l..,ont.

I c 1 I I 1

\ I I \ / \

I

o+-�-+����������-�---+-�� '\/' '·'

0.5 -5

-10

-15+

35

25

20 ;\ '/ \ \ I• I , ·/ \\ \ \

\

15

10

s

0.0 o.s 1.0 1.5 2.0

'· 5 3.0

wl ll/gl 3.5 '· 0 '· s 5.0

van Oort"r:'!erssen ship lo'lth quay

Olstance from .quay 24.?Sm.

-------- OUAYSH1P.

2.5 3.0 3.5

wl ll/gl 4.0 4.5 5.0

5.5

s.s

Fig 7(b) Added inertia and damping in heave-24..75m quay clearance.

! 5 j_ I

!0

2.0 -5 -!0

-IS-

40

35 J

B, 2St 20-

van Oortmerssen ship vlth quay O:stance from qu-ay 33.00:11,

--·······-� OtJ,i,YSH!P.

2.5 3.0 .1 (L/gl

3. 5 4.0 .\o.S 5.0 s. s

van Oor-:merssan ship vlth qua;/ O!stance from quay 33.00m.

OU,i,YSHIP, v�� Oo�•�a�$$Gn u�porlmant.

Fig 8(a) Added inertia and damping in sway-33.00m quay clearance.

:I 20 � I :5�

10

5 A,

833

0 -5

-!0 -j S

-20

_,J

601 55 i

50 �J., I "+ I ::!+

?;a

25

20.

o.o

I \ \ I

\ ,,

0.5

o.s

I ' '

/

1.0 1.5 2.0

van Oortmerssen sh!p vlt� qvay 0 l stance from quay 33. 001'1.

OUAYSH:P. - - - - - - - - � � - Y&fl Oor�Mm"$frlir'l ""P"PlfO<int.

2.5 3.0 3.5 .1 (L/9)

•• 0 '·' 5.0

van Gortmerssen ship �lth quoy

01 st-ance fr-om quay 33. OOm.

-------- QtJAYSH(P,

2.5 3.0 3.5 wl IL/gl

<.0 4.5 5.0

5.5

5.5

Fig 8(b) Added inertia and damping in heave-33.00m quay clearance.

-S ..l.

-10+ _j

0,0 o. 5 \,0 1.5 2.0

van Oort�erssen shlp vlth quay

Ols�ance fro� quay 4l.2Sm.

van Oorc�erssen ship v1th quay

C1stance from cuay 41.25m.

-� ------- QIJAYSHJP •

• - - - - - -- van Oortlllllll"t:$QI'\ "J(flfii"III<$">L

2.S !:,0 wf IL/gl

3.5 �. 0 '· s 5. 0 s. s

Fig 9(a) Added inertia and damping m sway-4l25m quay clearance.

'"

25

20 \ I

:s

10

s+ A,

0

-5

-10

-15

-20

-25

'0

ss

50

45

"

35

B, 30

25

20

IS

10

5

0. 0.0 0.5 1.0 l.S

2.0

2.0

van Oortmerssen ship u!th quay

0Ls�ance frorr. quay 41.25m.

------

-----

2.5 3,{) wl \l/gl

3,5 4.0

QW,YShiP.

4.5 s.o

van Oortmerssen ship ��t� q�ay

Jlstance from quay 41 . 25m .

OU,.,YSHIP.

s.s

van Oor�marssan axpariMon�.

3.5 4.0 4. 5 s.o s.s

Fig 9(b) Added inert ia and damping 1n heave-41.25m quay clearance.

C.25 T

I cue + I I

0 I S� 0. ! 0 .1.

O.:JS +

van Oc�:MerssBn sh r � v l �h quay , , ��-........ _____ _ ···����- .. 1'. ; .. 8. 25m. from quay.

2 4 . 75m. from quay. - - - - - - - - - - - - - - - - - - - - - - - - - - · · r L, l . 25m. from quay.

,-, ' '

'

' A, : j \ ... \

Fig 10

- - - � - ; - : , , L 0.00 �--+--- +---'-c-'>+-,.L\r+-----,'-+�� ":-"-"-=::::�==±,. =t==r-:-:-�--T'-::< . s ; .o� : ; . � 1 s.o 1 s.s 0.$ 1 . 0

- 0 . os

-0 . 1 5

-0.20

•. so T 0. 4St 0. �0 -!-

0.:35

0.30

0.20

/ 0. 1 S

0. ! ::0

o.os

o.oo 0.0 o.s L O

/\ I l

) \ \ \

' \ ' \

\ \ \ ... .. '/

"'-�oc-· /

1 . 5 2. a

,' ' ' t ' ' ' ' ' ' ' "

•

' ' " " '

van Ool"'tr.'lers:s:en sh i p v ! t h quay .. -� 8 . 2Sm. fl'"'om quay.

- - - - - - � 24, ?Sm. from quay. - - - - - - - - - - - - - - - - - - - - - - - - � - - - - - 4 1 , 25m. from quay .

2.5 3.0 �I (L/gi

'· 5 4.0 '· 5 5,0 5.5

Added inertia and damping in surge predicted by QUAYSHIP,

I S�

-5

- 1 0

-15

40

35

30

25 ,. B,

20

I S

10

s

0

o.o o.s 1 . 0

l\ I I ; I I

\

I . S 2.0

van Oortmersson sh i p � l t h quay ---- B� 25m. from quay .

2 4 . 75m. from quay . · - - - - - - - - - - - - - - - 4 1 . 25m. from quay.

van Oor7mersse.r; sh i p !.I I th quay ------- 8. 25m. from quay.

- -- - - - - - - - - - - - 24. 75m. fr-om quay. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4 1 . 2Sm . f�om quay.

2.5 3.0 3.5 ol (L/g)

4.0 '· 5 5.0 s.s

Fig 11 Added iner tia a n d damp ing m s w a y predicted by QUAYSHIP.

'"

20

15

10

s 0.5

-S

- 1 5 -201 -25

20

I S

)

, . , I 1

I '• 'I ' ' ' </

1

1

van C!or't mer ss en s h I p v 1 t h quay ����- ---�-- 8 . 25m. from quay.

- - - - - - - -· - 2 4 . 75m. from quay, - � - - - - - - - - - - - - - - - - - - � - - - - - - � - - 4 i . 25m. from quay.

3.5 '· 0 4.S s.o s.s

van Oort merssen sh t p v t t h quay --------- 8. 25m. from qua y . - - - - - - - - - - - - 24 . ?Sm. from quay .

- - - - - - - - - - - - - - - - - - - - - - � - - - - - - 4 1 . 2Sm. from qu:a y .

s o J--+--�\ �, '�- �\���==�����=,-�---,���

0.0 o.s 1 . 0 1 . 5 2 .0 2.5 3.0 3 .5 rol (L/gl

4.0 '· 5 s.o 5.5

F i g 1 2 Added inertia and damping i n heave predicted by QUAYSHIP.

0. 0050 -;-

0. 0045 7

0.0040 t 0. 0035 ,..;_

0. 0030

0.0025

0.0015

0.0010

0. 0005 . ! o. 0000 -� -0.0005 1 -O.OQ!Q

,A I ; / ��

� ;:_ ::- � \ \ ' ' . .

'-F - - � .- -

van Oortn;erssen sh ' p v ! t h q:.Jay

--�-------- 8 . 25m. from quay. - - - - - - - · � 2 4 . ?Sm. from quay.

- - - - - - -- - " " - - - - - - - - -- 4 l . 2Sm. from quay.

' / ,I I

I, - --+--+ --+--1 . 5 v2.0 0.5 2.5 3. 0

w! IL/gl 3.5 4.0 4.5 5.0

van Oor•t n;erssen sh i p v l t:h quay

s.s

-----·------- 8 . 25m. from quay� - - - - - · - � - - - - - - - - 2 4 . ?5m. from quay. - � - - - - - - - - - - - - - - - - - - - - - - - - - - - - � 1 , 25rr.. from guay.

Fig 13 Added inert ia and damping m roll predicted by QUAYSHIP.

L a : 1 . 6 I • ; 1 . 0 0. 4 0 . 2 -o. o l

0 . 0

/\ I \ I I I \ I I I \ I I I I

I I i I / I I

I I

I \ \ - ' \ \

\ \ \ ' ',-,., \ ...

van Oortmerssan s'o : p .... 1 th quay

�----�----- --- 8. 25m� from quay, 24. 7Sm. from quay.

- - - -- - � � - - - - - - - - - - · - - - - - � � - - - - 4 1 � 25m. from guay.

-- .

van Oortmerssen sh i p .,.. l t h quay

5.5

--��----------- 6 . 25m. from quay. - - -- - -· � - - 24. 75m� from quay. - - - - - - - - - - · - · · - - - - - - - - - - - - - - - - 4 1 . 2Sm. from quay.

/

/-- \ \ - - �' - .,._.. �_ - - - c::..:c -_... _

-<""" ! .. ------+ 0.5 1 . 0 1 . 5 2 . 0 2.5 l.O 3.5

ffi1 (L/gl 4 . 0

Fig 14 Added inertia a n d damping m p itch predic t ed by QUAYSHIP.

0. 6

o. 7

0.6

o.s

0.4

0. '

A c.< . "

o. 1

2.4

'·'

2.0

1 . 8

1 . 0 + ! o. er· 0.6

0.4

0 . 2

0.0 0.5 1 . 0 1 . 5 2.0

van Ocr t rr.erssen sh ! p \.l ! th quay

------------------------ 6. 25m. from quay. ··· - - - - - - - - - - - - - - 24. ?Srr . • from quay.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4 1 . 25rn. from quay.

van Gcrtmersscn sh ! p 1.1 1 th quay

------------ ------------- 8 . 25m. from quay. - - - - - - - - - - - - - - 24. 75m. from quay. - - - - - - - - - - - - - - - - � - - - - - - - - - - - - - 4 1 . 25m. from quay.

2 . 5 3.0 3.5 wl fl/gl

4.C 4.5 5.0 5.5

Fig 1S Added inertia and damping m Yaw predicted by QUAYSHI P .

.., -·

\Cl _. (7\ -Ql -

(I) c ..., \Cl I'D 1 N -.1 0

0

� Ql < I'D c 5' " !?: 0 c , ..., ... I'D • ,..., " - c a· Q 0. • ::J •

� L • cr "' I'D c , Ql <J1 3

VI I'D Ql -.

2 . 4 + Sh i p moored bes i de qu a y . Vav9 d! roct l on 270• . I

2. 2 + I --------------- 8 . 25m. from quay. - - - - - - - - - - - - - 2 4 . ?Sm. from quay. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4 1 . 25m. from quay •

2 . 0

I 1 . 8 +

1 . 6 +

\ . 4 +

1 . 2 \ ·� 1 . 0 + \ 0 . 8 \�

\� \\' ' \ \' 0 . 6 t \ .

\\' 0, 4 t

' '\ ' ' 0 2 \, � .. -

"'""'""';;;.;;f" ..... ....:';.:·:.;-�-��--.:;-:;...::.,,_.,.1 �--"-�:1 ' T ', "" .' ', -' �- - ""'" · - · r · - - ' -�� �;:..-�=--�-,- . I' , ��- - - _.1 - I 0 , 0 . I I

0 . 0 0 . 5 1 . 5 2 . 5 4 . 0 5 , 0 5. 5 3 . 0 �J IL/gi

3 . 5 4 . 5 1 . 0 2 . 0

-n l.Cl

_.. 0.

I -CT -

tn I � QJ

"< 1

I N -a 0

• � I QJ < ro c

0 !:!-: � u ..., c ro , n �

_,.. • c:r • c ::J c n. - • • CT L ro >-QJ • , 3 "'

VI ro QJ

--.... I

1 0

9

8

7

6

I s

4 I

3

i 2 -��

0 ,

0 . 0

- - -

-

-... ... ... ... ...

' • , '

- - -'

0 . 5 1 . 0

' ' ' '

' '

' ' ' ' ,

' '

1 . 5

' ' ­.... .. .. .... \

' ' I

Sh i p moored bes l quay . Vave d i rect i on 270 • .

-------------- 8. 25m. from guay. - - - - - - - - - - 24. 7Sm. fr-om guay.

- - - - - - - - - - - - - - - · - - - - - - - - - - - - - 4 1 . 25m. from guay.

'-I ' ' 1( \

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Fig 16(c) Heave-270°wave direction (beam sea)

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Sh i p moored bes i de quay . Vave d i rect i on 270 • .

� � � � ��������� 8 . 25m. from guay. - - .. - 24. "?Sm. from guay.

- - - - - - =-- =-- =-- =-- =-- - - - - - - - - - - - - - 4 1 . 2S m . from guay.

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2, ;? ---------------...-- 8. 2Sm. from quay.

� _ _ _ _ _ _ 24. 75m. from quay. ::-_

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_ _ _ _ _ _ _ _ - - - - - - - - - 4 1 . 25m. from quay.

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-----------------:--:- 8. 2Sm. from quay. _ _ _ _ _ 2 4 . 75m. from quay. ::-_ ::-_ _ _ _ _ - - - - - - - - - - - - - - - - - - 4 1 . 2Sm. from quay.

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_ _ _ _ _ _ - - - - - 2 4 . 7Sm. from guay . _ _ -- _ _ _ _ _ _ _ _ _ - - - - - - _ _ _ _ _ _ _ _ - 1! 1 , 25m, from quay .

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--------------- 8. 25m. from guay . � � � - - - 24. 75m, from guay.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4 1 . 25m. from guay.

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quay .

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---------------- B. 25m. from quay. _ _ _ - - - - - - - - - 24. 7Sm. from quay.

- - - - =-- - - - - - - - - - - - - - - - - 4 1 . 2Sm. from quay.

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Sh i p moor ed i de quay. Vava d t ract l on 3QO• ,

----------------:--: 8 . 2 5m. from �uay. 24. 75m. from �uay.

_ _ _ - =-- =-- =---=-- - - - - - - - - - - - - -- - - 4 \ , 2Sm. from quay.

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8. 2Sm. fr-om quay. _ _ _ _ _ - - - 24. 75m. from quay.

_ _ _ - - - - - - - - - - - - - - - - - - - - - - - 4 1 . 25m. from quay,

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8 . 25m. from quay. 2 4 . 75m. fram quay. - - =- - - =-- =-- =-- =-� =--=-- - - - - - - - - - - 4l . 2Sm. from quay.

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------------------------------=- B . 2Sm. from guay. _ _ 24. 75m. from 'l''ay. - -_ ::-_ ::-_ ::-_ ::-_ :·_ _ _ _ _ _ _ 4 1 . 2Sm. from quay •

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------ 8, 25m. fcom quay. - - - - 24. 75m. fcom quay ,

- - - - =-- =-- - - - - - - - - - - - - - - - - - - - - 4 1 . 25m, fcom quay. . . ' ' ' ' : I

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_ 8 . 25rn, from guay . _ _ _ _ _ - - 2tt. 7511'1, from q•Jay. - - - - - - - - - - - - - - _ _ _ 4 1 . 25m. from gtJay • - - - - - - - - - - - - -

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Sh i p moored bes i d e qu a y . Vave d ! ract 1 on 3 1 5 ' . ---------- B . 25m. from guay.

_ _ _ - - - _ - 24. 75m. fcom quay . -

- - - - - - - - - - - - - • 4 \ . 25m, fcom qua y . - - - - - - - - -

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Vave d ! ract l on 3 1 5 • .

---- 8 . 25m. from guay. - - - - - - - - - - - · - - 2 4 . ?Sm. from guay.

- - - - - - - - - - - - - . . - - - - - - - - - - - 4 1 . 25m. from guay.

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-----------------------�-=-=�� 8. 25m. from qvay. 2 4 . 7Sm. from quay.

. _ . _ _ _ _ - - - - _ _ _ _ _ _ _ _ - - - - - - - - - - - 4 1 . 25m, from quay •

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4 . 0 4 . 5 s . o 5 . 5

::!! !.Cl ...... '-0 � ..... -

-< 11.1 :( I w ......

U"l 0

:( 11.1 < fl') a. -· ..., fl') M ....,.. c.r ::I

ri " <i "' • :B c 0 � u c , � • " c 0 0. • • c , 0 >-

2 . 4

1 :. : t 1 . 8 1' ! . 6

1 . 4 1 1 . 2 1 1 . 0 t

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t 0 . 6

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:::t o . o

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Sh i p moored bes i de quay . Vave d t �ect l on 3 i S• .

--------------:-�:--�· 8. 25rn. from quay. -

_ __ 2 4 . 75m. from quay. · - - - - - - - -_ _ 4 1 . 25m . from quay. - � - - - - - - - - - - - - - - - - - - - -

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if}

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0 . 0

- -----r 0 . 5 I . 0

Sh i p moored bes i de qua y . Vavo d ! ract l cn 330· �

--------------- 8. 25rn. from quay. _ _ - _ - - - - - - - - - - - 2l,, 75m. from guay. - - - - - - - - - - - - · - - - - - - - - - - - - - - - - - 41 . 2Sm. from quay .

\ ...._

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\\ '-

-----

---------- � , . I ,\ - .._ � ' , I \ '-. / / \ I \

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. - - - � - - - - - - - ("� I I , 1 . 5 2 , 0 2 . 5 3 . 0

ol (L/ gl 3 . 5 t, , Q t, . S 5 . 0 5 . 5

., us· N C> I -c:r -Vl l( DJ

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6

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Sh t o moored t de guay. Vavs d i rect i on 330 ' .

--�� B. 25m. from guay • �

_ _ _ •. 24 , 75m. from guay. - - - - -

_ _ _ _ - Id , 25m. from guay. - - - � - - - - � - - - - - - - - - - - - -

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+

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Sh i p moored bes i de qua y . Vava d r rsct 1 on 330 ' .

----------------- 8 . 25•. from quay . _ _ - - _ - - - - 24. 75m. from quay . - - - - - - - - - - - - - � - - - � - - 't 1 . 25rn. from quay.

I I ' \ :_ ; ).y-�,, - :::_'----- - - --�� 1-,�- �--· --- +

2 . 0 2 . 5 3. 0

wl (L/gl 3. 5 4 . 0 4. 5 5 . 0 5 . 5

., 10 N I 0 � c. -� 0 � � I

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::!

:1 22

20

t 1 a T 1 6 +

i 1 4 t 1 2 �-1 0 +

i

: r : t 0 -

0 . 0

Sh i p moored bes l quay . Vave d t rect r on 330 ' .

8, 25m. from q�ay. _ ,, _ _ _ _ 24. 75m. from q�ay. =-- =-- - - - - =-- =-- - - - ::-_ _ _ _ _ _ 4 1 . 25m. from q�ay.

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oi

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Fig 20 (e} Pitch-330° wave direction

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1 . 8

1 . 6

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o. o L� -�-� � �--�-------1

0 . 0 0. 5 I . 0 1 . 5 2 . 0

Sh i p moored bes i de qua y . Vave d t ract l on 330 · � -------------------- ------------- 8. 25m. from guay.

24. 75m. from gu�y. _ _ _ _ ::-_ ::,--_ ::-_ :::-_::,--_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4 1 . 25m. from quay .

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5 . 0

----1 5 . 5

11 1.0 N I �

N I CJ Vl :( w

'< I ' :::J ro w -,

""Cl 0 _,_ m :::J _,_ w �

_,_ ::::r m 0 -,

'< w :::J Cl. 3 0 Cl. ro ..., -, n _,_ 0 :::J

1201

100 � 80 -l

60

40

ro t: 20 � ! 0 . 2 j

o I � - 2 0 <t

-4

-60

-80

-10

/ I I I

w f Big'

0 .4 0 .6 0 . 8 1 .0

I

Linear p o tent ial theory 0 with model fri c t i on at scale 1 : 100

100 .• ( incident wave a mp=0.01ml

80

Cl c: '5.60 E ro "'Cl -..;.o

N CO

2 0

o l Jl � � w v Big'

0 .2 0 .4 0 . 6 0 .8 fo f2 2

().) Vl c: 1 0 0.. lll ().) I '- w l Big ' ;:... 0 ro � 0 0 .2 0 . 4 0 . 6 0 .8 1 .0 1 .2 lll

" LC N N

N 0 Vl � a;

'< "'0 .., 0 .... 0 ....

'< "'0 ro ..... .., n .... 0 ::::J

120•

1 0 0 •

80 •

60 •

4 0 -

20 -� "' "t: 0 ()J

.� "0 OJ -20-

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-60"

- SO·

-100-

(

0 .2 0.4 Oi6

0

100

8 0 .

g' 60 • '15.. E .§ 4 0 • N N c:o

w v B/g' 20

0 .8 1 p 0 � """ 2 .

()J Vl 1 • c 0 Cl. Vl ()J <... ,..., 0 "' 0 ): Vl

With prot otype fric t ion (i nc ident wave. amp::1m )

� w v B/g' d A

o'.2 0.4 o'.6 0 .8 1:o 1�2 J

/ 0. ,.. w / B/g'

. o'.4 o'. 6

• 1:0 1:2

I 0.2 0 .8

10

80

� CO � c QJ 60 ·a .._ .._ <1) 0 u

en c D. E ro

-o

� 40 � (/)

20

0 0.40 0.42 0 .44 0.46

w v Big '

- Linear potential theory.

Prototype frict ion -e- a::;;1m ....._ a=2m

Model frict ion(1:100)

+ a=0.01m _._ a=0.02m

0.48 0 .50

F ig 2 3 Var ia t ion of damping coeff icie n t near resonance w i th i n c i dent wave a mpli t ude (a ) .

� + N N

<t

200

100

80

60

40

20

::::- 10 N

eo"' 8 '-

£. 6 u (!J

g' 4 '0. E (!J

D

2

1

0.8

0 .6

__ Linear potential t heory

Prototype -e- a=1m -.�.-- a=2m

Physical model(1:100)

• a=0.01m • a=0.02m

0 .4 L.,----.------,.-----1-,-----,.---,----0.46 .44 .45 0.46 0.44 0.45

w .J Big' w .J Big

'

Fig 24 Variat ion of damping fac tor near resonance with i nc ident w ave ampl itude (a).