Post on 10-Mar-2018
�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( \
' / \ \ '
2 . 0
�\\' 1 / ' ,y· \ - -....
\_ - ·t -- - - - -- - - - .:::. .:::. ::-..::::. .:cc � .::-:. .:: _-: I
2. 5 3 . 0 3 . 5 4. 0 4 . 5
�: 5 . 0 5 . 5
I I ,
I ' I I ' I
� ' -_ _ .... - _ _ ... -..
- - - - - -,-- - -
.... _ _ _ _ -- - I - - - - -
,;
): ( :1 i I '
I I I
' ' \ / •
.>. -' I ' '
� - 1 - - - - - --
0 '
1'-
• ' _,.
Fig 16(c) Heave-270°wave direction (beam sea)
I I
I I
I ' ' ' ' I ,'
\1 : I : •
,1. h' f'
! '
' ' '
I I ' '
• ' 0
en ..n
Vl .;
0 �
tn "'
0 -m ,; ' d .....
a
tn N
0 N
0
en 0
0 0
-n <.0
_. 0'> -- I c. �
::0 0 � � I I N -..J 0
0 I � • DJ '-< • I'll en • c. :!! ..., c I'll 0 n � ..... 0 0 c
, :J ...
- • � c- c I'll 0 "-QJ � • 3 c
� VI � I'll 0 DJ "' -
3o T 28 + 26
24 + 22 20
1 8 + 1 6
1 4
1 2 1 0
8
6
4 '
' I I
I I I I I
I I
/1
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.
• /1 " I I I I I I I >
�
>
I
' ' ' ' ' ' ; I ' ' ' !J l\ i �I \ ' \ : I ', I :
' I I I I 1 1 '
, , :--. ' \ ' ' I ' ' I : I ' ' I
\ \
/1 -
'. "
- ' 2t
0 I �--::-
,, ,"
_ _ _ ,_ �
'-
f_,.,..f< -
',
I
- - -
I I -- -":"--l _ _ .,..____ ·-! ----� l =;. _ .. .. -! ...
..,... - I
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . s 3. 0 3. 5 4 . 0 4 . 5 s . o s . s �1 (L/gl
.,., liS' ..... ""' -RI -
3! -n ::r I N -.1 0 0 • <( '.. 111 • < rn • RI :::? 0.. c =r 0 -RI � u n c - , er � • ::I • c -c- g_ RI 0 • 111 L 3 .r 0 CA "
RI "'-11.1 -
2 . 4 Sh i p moored bes l �uay . Vave d l ract 1 on 270• .
2, ;? ---------------...-- 8. 2Sm. from quay.
� _ _ _ _ _ _ 24. 75m. from quay. ::-_
-
_ _ _ _ _ _ _ _ - - - - - - - - - 4 1 . 25m. from quay.
? . 0
. 8
I . 6 �-1 . 4 T I . 2 t 1 . 0 t
! { \
I \ I \ I I
I \ \
0 . 8 f' 1
0 . 6 tl :
... "' "" .... - \ 1 ' ' I {) . 4 /
1\ l ' I ' i I ' I
t
i I \ I 0 . 2 -_...- I ..., _ _ �-=----=--::-_::-_�-"'!= r·- - -- 'l I ; � .... I . . 0 . 0 I
0 . 0 0 . 5 1 . 0 \ , S 2 . 0 2 . 5 3 . 0 � 1 (L/gl
3. 5 4 . 0 4 . 5 5 . 0 5 . 5
::!1 IQ .... a-� -- 2 . 4 ••. -< QJ 2. 2 l( I N -.J 2 . 0
0 0 e ' " l( 1 . 8
o; • QJ :g < 11) 1 . 6
c c. "-" =r
0 c 11) 1 . 4
.2 n .... • -· • 0 1 . 2
c 0 :::J 0. " � • c:r L 11) 1 . 0 > QJ � ,_ 3 0 . 8
Ill 11) QJ 0 . 6 -
0 . 4
0 . 2
0 . 0
0 . 0
I I
Sh i P moored bes i de quay . \Jave d i rect l on 270 ' .
-----------------:--:- 8. 2Sm. from quay. _ _ _ _ _ 2 4 . 75m. from quay. ::-_ ::-_ _ _ _ _ - - - - - - - - - - - - - - - - - - 4 1 . 2Sm. from quay.
I I I \
\
,- '
� /' \ �/ \
�-·---..,-�-=-=--=�:...�..-- � " . ... ... ... __... ...... -- .:-:. - - - .... - ... J - ' ... - - ��-- -�� -�'-'r='·��':::,""·�==""='='�"'""�"---; 1 ':.-...:: ;-::-::=: 1 '-' ! r-- " - - -e- � � - -�-�- f- · .s 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0
�J IL/gl 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5
., I.CI ..... ....;j
� Ill - I U'l c: ..., l IQ l1) I N l 00 1.11 0 :( r Ill < l1) c 0. 0 :::;· " l1) 0 n
c , � ..... ()' • ::::1 • c 0 0. • • L • 01 L , "'
2. 4
2. 2 .
2 . 0
1 . 8
I . 6 -!-I .J 1 . 2 +
I 1 . 0 t 0 . 8
o . 6 ·•· 0 . 4 I
', I ', \ \ I
\ \ I I I I \ \ I \ I I \ I ' \ ' \ \ \
' \ \ \ '
' \
Sh i p moored bes i de quay . Vave dl roct l on 285' , --------------- B. 25m. from quay.
_ _ _ _ _ _ - - - - - 2 4 . 7Sm. from guay . _ _ -- _ _ _ _ _ _ _ _ _ - - - - - - _ _ _ _ _ _ _ _ - 1! 1 , 25m, from quay .
\ \
\ \ � \ \ � ..- -
, ..... .... .. ... / :: :L --t--- \. \ \ � ' , ' - � - - � -�_,._, ' : - � -r < I
� , __ ' �---= =m=m- ••·· · ··- · ! 1-,�-. .. _, __ ·-+·---�· ' - -(
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2. 5 3 . 0
ol (L/gl 3. 5 4 . 0 4 . 5 5 . 0 s. s
>._ (1) ;J r::r (!)
-o "' ID .,.., _() "'
N -o ID c L 0
0 � 0 0 • E L
a.. " • ..<: > � (jJ "'
0
• >..>.. >..� � • :J :J :J er er er
E E E O O O L L c � �
� . . • E E
""'"' lf""'-· N N • •
- � "' "' .,.
I ' I ' I ,
=
' '
' '
Fig 17(b) Sway-285° wave direction
I I
I
-' ' ' ' /
/ / / /
I I
! I i I I \ I
- -I
/
N
'
J I i 1 : I ' ,.,_ !/ :
1.' 11 + ' It ! ' !,' I ,'
/ ' / :
( ; ' !-
' '
0
tn '"
0 "'
"' _;
0 .;
lfl "'
0 "'
U"\ N
0 "'
0
0 0
;;, ' d --.
•
11 \.Cl
..... -...3 - I 1"1 -:X: 11) Ql < 11) I
N Q) 1.1'1
0 :( w <
a 11) c. - · � 0 c .., J 11) '>-1"1 ::!: • w c 0 a CL ::1 0 � L
• > • • :r:
1 0 T 9 -\-
: I I
6 _ _,_ 5 -
'• 3 I 2 +
' ' ' ' ' ' ' ' < I < I 1 I < I
Sh i p moored bes i de qu ay. Vave d f ract ! on 285 • .
----�---------- 8. 2Sm. fcom quey. - - - - - - - - .. - - - · - - - 24. 75m. fcom quay. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4 1 . 25m. from quay .
< I -- -----._ J I / \ --=-- - -...., r j �'= =-- - - - - - - -. -- - - "'-)�\
" ... ... ... .. � ... ""' J I \
o -1----+-
' "'
- \ ------� ··i
• • - • •
I
', I � - -- - - - - - -- - �.... I
' ' -\ I l \ ,: I \ ' I . I I
0 . 0 0 . 5 1 . 0 1 . 5 2 , 0 2 . 5 3 . 0
ol iL/gl 3 . 5 tf . 0 4 . 5 5 . 0 5 . 5
., 1.0
.... -..1
� I c. -
:::0 0 � �
I N c:o V'l 0 �
� • ' DJ ,; < (]) • I'D :::!
0.. c =r 0 I'D V
0 n c ...... , s· �
0 :::1 " c 0 0.. " • c
--· ··' 0
"'
3o T I 28 +
" t 24
22 +
20
I B + 1 6
" I 1 2
I D
B 6
4
2
0
0 . 0 0 . 5
. ""' - �
1 . 0 1 . 5
' ' I ' ' ' I
} / )I
I : \ , I /
I 1 ' '•
'
/i
. ,. I ' I ' I ' I
' ' '
' ' I ' I
Sh i p moored bes i de quay . Vave d r ract l on 285 • .
--------------- 8. 25m. from guay . � � � - - - 24. 75m, from guay.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4 1 . 25m. from guay.
' ' ' "
./ ; \ \
\ ' ' \ , ... ... , ' ' '
� -f - - - --------+---- --------1 ... :::-:�?--.;.---� -- I I , I
2 . 0 2 . 5 3 . 0
.J (L/gl 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5
J:1 !.Cl _. -.._] � ro - I ""0 - · ..... n I � 1
N I (XI IJI
0 .: � " Ql < ro c.. c 0 - · -""J " ro
• n 0 § ..... �
a· '::J
m � c 0 o. " • L J:: u :: CL
2 . 4
2 . 2
2 . 0
1 . 8
I . 6
' ' ' ' ' '
• , , ' \ ' I ' '
Sh i p moored bes i de Vave d i rect i on 285 � .
quay .
I - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 8. 25m. f�om quay. 24. 7Sm. f�om quay . 4 1 . 25m. loom quay .
' I : r I
�� l ' I I ' \ I•
1 : I
I \ l '
I ' ' ' ' ' ' '
0 . 0+------+------+------+------+------+------+-----�·
0 . 0 0. 5 L O 1 . 5 2 . 0 2 . 5 3. 0 3. 5 4 . 0 4 . 5 5 . 0 5 . 5 .J
::!! \Cl
..... -..J
� -�
-< QJ l: I
N <XI 1./1 • :( .1
" QJ ,; < 01
m ro ;g c. c
0 ::;· V ro u
n § ..... �--· 0 0 :::1 •
c c "-• �
• I L > �
>-
2. 4
2 . 2
2 . 0
I . 8 �-
1 . 6
1 . 4
1 . 2 1 . 0
' ' '
'
' ' ' /
- - -
� /
' /
0 . 6 t• ,/ / / ...----------./ / /
0 . 4 - - \ ,/> / / /
, , � -�/
0 . 8
0 . 0 � 0 . 0 0. 5 1 . 0 1 . 5
,-, ' ' I '
' ' I >
' ' ' ' ' '
/ , , ' _... \ �
Sh i p moored bes i de �uay . Vava d i rect i on 285 " ,
---------------- B. 25m. from quay. _ _ _ - - - - - - - - - 24. 7Sm. from quay.
- - - - =-- - - - - - - - - - - - - - - - - 4 1 . 2Sm. from quay.
1 \ I I
I \ � I
--------..._ �I I
2 . 0
' ' ' .\
2 . 5
\
_ ... - .....
3 . 0
.,J IL/g)
" "
3 . 5
" " "
4 . 0
"
4 . 5
" ' ' ' '
s . o 5 . 5
-n 1.0 ..... CD
2.
4 t I � OJ
�
Vl I 2 . 2 ' c ...,
1.0 2 . 0
fl) I I w 0
1 . 8 0 I 0 �
1 . 6 OJ
c: < u fl) -\ . 4 � a. 0 c � ..., � fl) • 1 . 2 ,.... " ...,.. c 0 0 Q_ 1 . 0 w ::J @ � m 01 0 . 8 c � (f)
0 . 6
0 . 1t -'-
0 . 0 0. 5 1 . 0 1 . 5 2 . 0
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.
2. 5 3 . 0
�1 (L/gl 3 . 5 4 . 0
' ' ' ... ... ... .. - � 4 . 5 5 . 0 5 . 5
"'TI -·
� _.... CID ' : f
- I CT � l/') :£. QJ
8t
'< I UJ 0 0
7
t
Q :£. QJ <
c 6
1
Ill 0 9: � u ..., c Ill " � : l
t"l ....,.. • -· " c c 0 ::iJ a. " • L I "' 3 � � , if>
I 2 + I + I 0
o. o
- - - - .. ..
' ', -
' .... ...... '' ' ' ' ' , ' ' ' , , r
Sh i p moor ed bes i de qua y . Vava d t �ect l on 300 ' ,
8. 2Sm. fr-om quay. _ _ _ _ _ - - - 24. 75m. from quay.
_ _ _ - - - - - - - - - - - - - - - - - - - - - - - 4 1 . 25m. from quay,
�,�, >- --\';\, "-' ' , � ' '
� ' ' ' ' '-
�- --+
0 . 5 l . O . 5 2 . 0 2 . 5 3 . 0 �J [L/gl
3. 5 4 . 0 4 . 5 5 . 0 5 . 5
., 10 ..... 01:1
I -n � :::t: I Ill IIJ < Ill I I w 0 0
0 t: I IIJ < Ill c 0 -Cl. " 0 =r c , Ill '•· n ..... • " a· c 0 {L ::::J • • c • > • • :r:
1 0
9
6
7 -
6
5
4
3
2 I ' ' ' ' ' ' ' '
,-, ' ' ' ' ' ' " ' I \ 1\ \
I ', \ '
Sh i p moored bes i de quay. Vavo d i rect i on 300" .
8 . 25m. from quay. 2 4 . 75m. fram quay. - - =- - - =-- =-- =-- =-� =--=-- - - - - - - - - - - 4l . 2Sm. from quay.
{ \ ... ,\ Ill -
::. - - - - -- -
0 . . --+ - ----1 _ _ '___:, ��--+���""""'=::::t===""!--� -��·---1
5 . 5 o.o 0. 5 1 . 0 I . 5 2. 0 2. 5 3 . 0 •1 ll/gl
3 . 5 4 . 0 4 . 5 5 . 0
., ..c·
_,. = - I c.. -
:;o 0 .... .... I I UJ 0 I 0 0 � E ' QJ ,; < m ro • :9 c.. c =r
� ro � n 0
::!: c j
0 �
:::1 0 " c 0 a. "' 0 L
� �
0 er
30
28 26
24 22 20
I B -•
1 6
1 4 -
i 2 1 0
8 ., 6
4 lj
,.�/. ' .
' ' ' ./ //
2 t /'V" /
I _ - /
0
- - -r "'����
0 . 0 0 . 5 1 . 0 1 . 5 2 . 0
A
��I I '
< I '\ I ' I I' '
i \ \ •I '
,' 1
' I I
Sh i p moored bes i de quay . Vave d i rect i on 300 ' .
------------------------------=- B . 2Sm. from guay. _ _ 24. 75m. from 'l''ay. - -_ ::-_ ::-_ ::-_ ::-_ :·_ _ _ _ _ _ _ 4 1 . 2Sm. from quay •
I I ' 1 I ' '
I I 1 I I I I
l I ' ,1
I I \
I I ' I ,I
\ 1 ',�
1 -\ 1 \_, �-- -
2 . 5 3. 0 �� (L/gl
-,.�---==,��---......,.-�---! 3. 5 4 . 0 4. 5 5. 0 5. 5
.:!! IC .... CX> -11) � I :; , 4
-o -·
2 . 2 .... I ,., =r I
2 . 0 w I 0 0
• 1 . 8
0 I ' � Qj < 11) -- 1 . 6
c a � V 1 . 4 0 ... c 11) :J ,., .... ,, • 1 . 2
- ·
" 0 c :::1 a 0.
1 . 0 " • L
£ u 0 .8 V
a_
0 . 6
0 . it ->-�
0 . 2
0 . 0
0 . 0
?-- ---�
' ' I
� �
Sh i p moored bes i de quay . Uave d i rect i on 300• .
------ 8, 25m. fcom quay. - - - - 24. 75m. fcom quay ,
- - - - =-- =-- - - - - - - - - - - - - - - - - - - - - 4 1 . 25m, fcom quay. . . ' ' ' ' : I
I '
' I ' d l I ' lt ! :
' I ' I I l '
\
/,�- - - -- - "'
/;_:-- --\1
' '
' ' -' ' ' - ' . ' ' . ' ' ' ; I
' 1\ ' J I I
� )--' ... , , \
"
' ' I \.
\. - -
J-0 . 5 1 . 0 1 . 5
; I I ----'-��--+1--2. 0 2 . 5
�J ��1�---;�������������--I �- - - -- - -r--_,__ �-"T M I
\ I -- ·- - �
3 . 0 3. 5 4 . 0 4 . 5 5 . 0 5 . 5
, 1.0 � CXI --
� I -< QJ I .-; I w 0 I 0 0 • .-; "' QJ • < 01 • I'D Cl c. c 0 ..., I'D u 0 n c .... , o·
-0 ::::l " c 0 a_ " 0 '-, • ,..
2 . lt 2. 2 -
2. 0 -·-1 . 8
1 . 6
1 . 4
1 . 2
I . 0
0. 8
0 . 6
0 . 4
0 . 2
' '
• ' • ( ( �•1 1 -/
' • • ' • /
I
• ' ' ' '
'
I /
I
' '
I
' ' ' '
/ /
_. _. - - -
...
o. 0 +---1-----+------, 0. 0 0 . 5 1 . 0 1 . 5
• ' ' '
/ '
' ' ' ' " ' ... ' ' ' ,_ .. r- \�
Sh i p moored bes ! de gua y . Vavo d ! rsc� l on 300' .
_ 8 . 25rn, from guay . _ _ _ _ _ - - 2tt. 7511'1, from q•Jay. - - - - - - - - - - - - - - _ _ _ 4 1 . 25m. from gtJay • - - - - - - - - - - - - -
\
2 . 0
) :I ' I • ' \
\ ' '
/ � " ' ' '
.... ' ' ' " ' ,
2. 5 3 . 0 oi IL/gl
· ' - , " ' ,.... \ ' , ' � !• \ ' I ' , ... - .. .. - - ....
\ ..... '
- �7-f---,�. ' .. ' , _ __ _ __ ... _ "["7'- 1
' , ����- - -�
3 . 5 4 . 0 4 . 5 s . o 5 . 5
'Tl lCl ....
'-0 -ClJ
� I 2 . 4 VI c: ...., I lCl 2. 2 ro I w I 2 . 0 .... U1
0 � I I . 8
ClJ < ro 1 . 6
c. c 0 ...., p ro 0 1 . 4 ,., c 0 ..,.. -c:r • 1 . 2 ::1 " c 0 "-• \ . 0 • L • m L 0 . 8 0 (f)
0 . 6
0 . 4
0 . 2
0 . 0
0 . 0 0 . 5
': \ \ \ \ ' I ' : I ' \ ' ' I ' ' I •
' I ' ' I ' ' I '
' I ' ', I\ ' I ' ' I ' ' \ ' ' \ ' ' I ' ·, \ ', I ', I ' \ ' ' \ '
\I
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 . - - - - - - - - -
'\ \ � """' '? .... ... �-- ' ' '
I I '). /- - - ..... -r - -. .;;, _ _ _ ,.. " .... ::---;;;. ...... :c.. - "' j ---!----+---+ ! · -----'�:::::..0�
! . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5. 5
�1 (L/gl
>--ro ::J a-(J) Ll I ' (f) (J) V, _o
,..., Ll . m c: L 0 • . 0 " 0 0 � E L o_" m
_c > .. c.n "' I '
' ' ' ' ' ' •
Fig 19(b} Sway-315 .. wave direction
� - - -' . ' - - - - '
( I ' ' '
• ' ' ' ' • (
- -....
I ,
I' I ' ' I ' •
I ' � -· ' ' '
/1 :I
/I /I
/; /j ' ' ' I ' /(
I I I I
N 0
U1 Ul
0 .;.
LO ,;
0 ...;
"' N
0 N
0
0 0
o, ' � .., a
. ,..,.. ,..� . � , , :r o- tr <T
E E E O O C L L c � �
� • E E
E VHJ1 L'it-- N N
>-- _ .,. _ (0 <DN _,. :::J er
I ' (J) I -o ' "' ID ,j, ..a ;;) -o c I ' Q) 0 (_ I 0 " 0 0 I ' • E L I ' Q_ , I - 0
.c > N (/) ;:.
I ' I '
m In
Fig 19(c) Heave-315° wave direct ion
' ,..,
111 111
0 IJ1
111 .;.
0 .;.
/, �
ln ....,
' ' ' I 0 ' H- .,; :[I I� -.
a
111 N
�
- - - - - � :: --' - - - - - - I 0
/ ' , N 1 , - -'
;,' !,' If)
1/ !,'
!/ !/ 0 !/
/�/ ( 1: If) � 0
I ' 0 ' 0 N 0
..,., u::J ..... -a � I c.. -
::0 0 -- I I 1..1.1 ..... I Ln 0 � E Ill " < � I'll tJ)
$ c.. � ::;· c I'll 5' n " .... u
o· J :l • • c 0 CL • • L
--' �
0 a
30 Sh i p moored b s s l quay . 28 26
Vave d ! ract l on 3 1 5 • .
---- 8 . 25m. from guay. - - - - - - - - - - - · - - 2 4 . ?Sm. from guay.
- - - - - - - - - - - - - . . - - - - - - - - - - - 4 1 . 25m. from guay.
2 4
2 2
20 + 1\ \ ! {\\ 1 8 1 6
1 4
1 2
1 0
8 6
4
2
0 ·
I I I
' I 1 I
li
/H \'\
/ 1 f\ i ' 1 ' J I 'I I "'I
f.- ' I \ 1 1 I \I -' f' \ f I
-- � "/ \J' \1
� .. --'
.f-- +-----�· · · · ··· \/ �=�:::----0 . 0 0. 5 1 . 0 1 . 5 2 . 0 2 . 5 3. 0
�I IL!gl 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5
., - ·
lCl ..... -a
�
ro I �
::P. .... M ::r l I UJ
.... • 1.11
' 0 • � 01 • QJ
:!l < ro c 0 0. � -· 0 c .., J ro ·� M • .... • - · c 0 0 ::;, 0-• • L .<: 0 � Q.
2. 4
2 . 2
2 . 0
: : : l 1 , 4 ,
I . 2 -·-
I . 0 ·•· "
/ o. a
/ ,., - - - -
0 . 6
v 0. 4
/ / '; / I' ' %" 01' ' v
o . 2 t I 0 . 0 I I
0 . 0 0 . 5 1 . 0
�
1 . 5
� '
' '
Sh i p moored bes i de qua y . Vava di rect i on 3 1 5 · .
-----------------------�-=-=�� 8. 25m. from qvay. 2 4 . 7Sm. from quay.
. _ . _ _ _ _ - - - - _ _ _ _ _ _ _ _ - - - - - - - - - - - 4 1 . 25m, from quay •
I I
' I I
I
I I '
, ' - ,.. . .f ... , \ r\ ' . ' ...... y r \ " - .,. I \
'
2 . 0 2. 5
' \� ' ' ' . ' , \ ' ' '
3 , 0
roJ !L/gl 3 . 5
.. .. - .. .. ,.. .. � ...... . - I .,_..,_ -.1 = - � --l
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
O . A
t 0 . 6
0. 4 t
:::t o . o
' ' ' , "' ""
-.. _
' ' � -' ,/ � ....
' ' / /
'
,' / / .. �--- ' '
' I / '
' I
' _ , - , / I
" ,
/ I \ '/ \ ' \ ,' I
\ '
"" \i �/ ' � � '.I \\
'
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. - � - - - - - - - - - - - - - - - - - - - -
I I I
I
I I
I I I
\'< ... �
.... .... - .... , ..-
-t o . s 1 . 0
' 1 \ I " I \ ' ' ' \ ,..� 1//
" - ""- ' I ).. \ � - ' �-----+------+------;------�-----}--�����- I
, ,\ J �
2 . 5 3 . 0 3 . 5 4. 0 4 . 5 5 . 0 1 . 5 2 . 0 o f IL/gl
5 . 5
., 1.0' N 0 -OJ I 2. 4 -VI I c: 2 . 2 ..,
10 (1) I 2 . 0 t UJ UJ 0 I I . 8 0 � OJ 1 . 6 < c (1) � c. .. 1 . 4 0 - · c .., 0 (1) "-n • 1 . 2 -1--+ " (5' c
0 a_ :::1 " i . O • L • Ol
0 . 6 L 0
if}
o. 6
0 . 4
0 . 2 •. ,_
0 . 0 ·I · · · · ····
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 .
\ ...._
' \
\\ '-
-----
---------- � , . I ,\ - .._ � ' , I \ '-. / / \ I \
', , - - - - - / - .. - ' ·- - - - :;.- ;:- ... ,;: _
... ,... " "'-... ... ""' - - .... _.. ... � "'!:=:?""':=- '"" 1 ! - -
. - - - � - - - - - - - ("� 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
'< I
UJ UJ C>
0 I l( DJ < rD Q. ...., rD 1"'1 .... er :::1
c 0 � " c " � • • c 0 Q. • • L "" � > U1
1 o T
: t 7
6
5 -1-- +-3 M• 2
I -
0
o. o
- - - - ....
-'-0 . 5
" ' ' ' 'I \
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. - - - � - - - - � - - - - - - - - - - - - -
''<
+
' _ _ , - ,
/ ' I ,.."'<; -- -T, ...... ./ I
l . O 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4. 5 s . o 5 . 5
�� (L/g)
, 1.0 N 0
�
n �
:J: I'D QJ < I'D I
w w 0 0 � QJ < I'D c. =r I'D n ..... 5' ::1
2 0 " 0 c , � • • c g_ • • L • � •
:I:
\ 0 _,_
9
B -�··
: l 5
lt �f .•
3
2 + ---.:;:-�-;::;:
.. ... �� " � ' "' , ,
u --1 o . o 0 . 5 1 . 0 1 . 5
,, ' ' ' ' ' ' ' / � \
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
I.J.J I.J.J 0 0 � QJ < I'D c. -· ..., I'D n ...... ()' ::I
-E " . !!, � :g c 0 V 0 § -� • c 0 CL " • c 0 "'
::!
: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.
I I I I
I I
I I
/,J, ;�'\ I \ t t I I ' f ' J ., , \ ' I \ / I
,' I �
/ ' _,.; -.... ' ........, .. ... ..... -:� , _ �=�-�-=�'"-t----1"'-" -.._,_--==::J==:::::::;::� ·-+---- I --
0 . 5 1 . 0 1 . 5 2.0 2.5 3.0 3 . 5 4 . 0 4 , 5 5 . 0 5 . 5
oi
>-. (l) 6-<l)
"0 (f) <J) c, ...0 "'
"' "0 (j) c L 0 0 �
0 0 0 E L o.."'
- � _c > � (J) ::..
· >- >-,... . . O :J :J :J crcr er E E E 0 o O L L L � �
� • E E E tr. lfl lnl'-N N •-"-COf"'-!-4'" I
I I '
I '
I '
I
"' c-i
- - - - - - --
- - - - - - - �
0 c-i
I I
Fig 20 (e} Pitch-330° wave direction
' ' '
' ' '
-
' , ,
,' ,
- - -- -
,,,
0
._, , », � -ID 0
- - -
- - ---- - -
' '
..,.
0
I \ :
' '
' '
' ' ' '
' ' I l
I I I I / ' I
I
, , ' _.
+
j
:J-"' 0 0 0
0 ,.;
V< .;.
0 .;.
ln .,;
0 .,;
0 "'
0 0
.._. a
.,., ..o· N 0 � -- I -< QJ I � I
Ul I Ul 0
0 E �
' QJ • Ol < • Ill :::! a. c =; ·
0 � Ill 0 n c ..... " � ;:r � :::;, " c 0 a. • • L > • ,_
2. 4
2 . 2
2 . 0
1 . 8
1 . 6
1 . ,,
1 . 2
1 . 0
o . a
0 . 6
0 . 4 t 0 . 2 +
,,. ;:::.--,� / · � ...;:, ",� . ' .1 1' 7 \�
\ /.. '0 \ ? '� ' .. ... ?//"' \' ' ' ' ' \ ' ' . - \
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 .
� I \
I
I
\
\
\
; ( J� ,, I \
(/.
�}---
,'
, - ,
' -...., I \
I - - -'
' ' .... ........... ;
' ., ,...,._ .... ,. ... ... , ' I - '
�
2. 5 3. 0
oJ IL/gl 3 . 5 4 . 0 4 . 5
I r I
I I
I I
� � �--
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-
"0 "0 3
� - 4 0 <(
-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).