The motions of a ship on a sloped seabed

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ABSTRACT

In standard diffraction theory it is assumed that the water depth is constant and that the seabed is infinitely large. To account for a local varying bathymetry in shallow water (as it

can occur for offshore LNG terminals)

it is sometimes considered to introduce a second fixed body on the seabed representing this bathymetry in diffraction theory. Based on the results presented in this paper it can be concluded that this is (without special measures) not possible. The refraction and interference effects are too strong and affect the wave exciting forces on the LNG carrier in an incorrect way. A large size of

the second body and smoother edges of this body do not

improve the situation. However, a second body in diffraction theory, when chosen properly with respect to size and shape, can contribute to the correct calculation of the added mass and damping of vessels on sloped seabeds as this varies with the

local water depth over the length of the vessel. This will

clearly affect the motion response of the vessel. This can be seen for instance

in the pitch-heave coupling. This will

influence the motions of the ship in waves, as well as the resulting drift forces and related mooring loads.

INTRODUCTION

Recent experience with the development of offshore LNG terminals has shown that the issues related to shallow water hydrodynamics are at least of similar complexity as the ones in (ultra) deep water developments:

In nearshore wave dynamics many different phenomena play a role, such as dispersion, diffraction, refraction, shoaling, reflection, nonlinear wave-wave interaction, wave-current interaction, wave breaking and bottom friction.

The local bathymetry affects the waveloads on (and motions of) moored structures. As can be seen in Figures 1 and 2, both existing LNG jetties and new LNG mooring systems are in nearshore conditions where the local seabed can vary significantly.

Proceedings of OMAE 2006: 25 International Conference on Offshore Mechanics and Arctic Engineering 4-9 June, 2006, Hamburg, Germany

OMAE2006-92321

THE MOTIONS OF A SHIP ON A SLOPED SEABED

Bas Buchner

Manager Offshore, Maritime Research Institute Netherlands (MARIN) Visiting Professor, University of Newcastle upon Tyne

Haagsteeg 2 / P.O. Box 28

6700 AA Wageningen, The Netherlands

b.buchner@MARIN.NL

Deift University of Technology

Ship Hydromechanics Laboratory

Library

Mekelweg 2, 2628 CD Deift

Low frequency wave effects such as set-down and

shoaling can result in significant excitation.

Streamlined LNG carrier hulls have a very low

damping against low frequency motions. The combination of excitation and low damping can result in significant resonant motions and related mooring

loads.

Neglect of these important issues in shallow water

motion and mooring prediction methods could result

in

problems for new offshore (LNG) terminals. The combined input from offshore hydrodynamics and coastal engineering is considered to be vital to solve these issues.

Figure 1: Existing jetty-type moorings of LNG carriers in nearshore conditions

Figure2: New type moorings of LNG carriers are

alsoconsidered in shallow waters

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The low frequency wave effects and resulting low

frequency motions of moored LNG carriers in shallow water are discussed extensively by Naciri et al (2004), Van Dijk et al (2005) and Voogt et al (2005). Wave propagation over a non-constant seabed has been a topic in coastal engineering for

many years already, see for instance Mei et al (2005). In

recent years very good results are achieved for complex

bathymetries and coastal geometries with several methods using the Boussinesq equations, see for instance Borsboom et al (2000) and Madsen et al (2002). Bingham (2000) used Boussinesq-type wave modelling to calculate the motions of moored vessels in a de-coup led sense:

Wave forcing: high-order Boussinesq theory

Wave-body interaction: diffraction theory I panel method

Ship response: time-domain equations of motion At the moment this type of method is being extended into a more coupled solution of the wave forcing and wave-body interaction. The present paper is part of this development

and focuses on the local interaction between the varying

seabed and the LNG carrier in diffraction theory: added mass, damping and wave forces. Even if the wave exciting forces are calculated with Boussinesq type models (Bingham, 2000), in the added mass and damping the local bathymetry effects (varying over the length of the vessel) should be taken into

account.

In standard diffraction theory it is assumed that the water depth is constant and that the seabed is infinitely large. To account for a local varying bathymetry, it

is an option to

introduce a second fixed body on the seabed representing this bathymetry, as shown schematically in Figure 3. Referring to Figures 1 and 2 this is a realistic situation for LNG terminals.

'T".1T'

T"1T

-Figure 3: Modelling a sloped seabed schematically as a second body in diffraction theory

Teigen (2005) used the same methodology to investigate the motion of a spread moored barge close to an underwater ridge of limited size. He focussed on the general effect of water depth on added mass and damping and the effect of the underwater ridge on the wave field. The present paper focuses more on the situation where the vessel is actually above the varying seabottom, so that the added mass and damping are also directly affected by the bathymetry.

Although the sloping seabed can be very large in reality (up to the coastline), it is not possible to model this completely in the diffraction theory, as this would result in extremely long computational times (and diffraction theory would not be able to simulate non-linear effects such as wave breaking on a

beach or wave reflection on a breakwater). Consequently the paper discusses the following questions:

- Is it possible to model a sloping seabed as a second body in diffraction theory?

- What are the effects of the local bathymetry on the added mass, damping and wave forces on the vessel above it?

- How large should the second body (simulating the bathymetry) be and are there special requirements with respect to its shape?

The paper focuses on the first order loads and motions, second order effects will be part of future investigations. METHODOLOGY

As part of this study, both calculations and model tests were carried out.

Model tests

A 1:20 sloped bottom situation was tested in the MARIN Offshore Basin, using a wedge-type wooden slope of 550 by 550 m full scale. With this slope the water depth gradually decreased from 28 m at the bow to 15 m at the stern of the 274 m long LNG carrier (draft 11 m). The normal water depth was 35m. An overview of this set-up and the moored LNG carrier is shown in Figure 4.

\I

4 -4. -.

-± ii V

---Figure 4: The modelling of a sloped seabed in the

basin with a wooden wedge-shaped slope and the LNG

carrier moored to an open jetty.

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Only head wave conditions were tested in this set-up. Diffraction calculations

Two body diffraction analysis was carried out with the MAR[N program DIFFRAC. As a start, the 550 x _{550 m slope}

was _{calculated and checked against the model tests, as shown}

in Figure 5.

Figure 5: The modelling of a sloped seabed in linear diffraction analysis as a second body with a narrow wedge-shaped slope of 550 by 550m.

Originally it was believed that the dimensions of the 550m by 550m slope would be large enough to avoid end effects. However, it was discovered that the edge effects on the sides and end of the slope were much more important than

originally expected, see the next section. As a result of that, the following furtherconditions were simulated:

- _{Much wider slope of 550 by 1650 m, see Figure 6.}

1 =1 -100 100 300 100 200 300 200 -100 -500

Figure 6: The situation with a much wider slope of 1650 by 550 m

Sloped_{edges on the side and end of the slope, both} for the 550 by 550 m and of 550 by 1650 m sloped,

see Figure 7.

-200 ₃₀₀ -400

Figure 7: Sloped edges for the slope of 550 by 550 m For reference, also calculations with 15m and 28m constant depth (without the slope) were carried out.

RESULTS

Based on existing literature in this field (Mei et al,

2005), the following effects were expected during the tests and simulations:

- Increased wave height on slope as the total wave energy remains constant while progressing into

shallow water

- _{A shortening of the waves according to (with h as the}

water depth):

x=ijji

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- Effects on the added mass anddamping

- Refraction effects at the side of the slope

All these effects were observed, but the relative importance of certain aspectswasa surprise, as will be shown below. Figure 8 shows the heave and pitch motion RAOs for the model test with the 550 m wide slope, the diffraction analysis

with the same (narrow') slope and the diffraction analysis with a constant depth of 1 5m (similar to the water depth at the stern of the LNG carrier).

2 1.8 5 1.6 5 1.4 .5 1.2

01

0.8 0.6 0.4 0.2 0 1.4 1.2 .5 0.8 0 0.6 0.4 0.2 0 heave nal'row a-- heae I Sm -+-- heae test -..- pitch narr a-- pitch 1 5m f--pitch test

Figure 8: Heave and pitch motion RAOs for the

400 model tests and the diffraction analysis with the 550m

('narrow') slope and diffraction analysis with a constant depth of 15m.

The following is observed:

- The measured and simulated motion response with the slope is significantly higher than with a constant

depth of 15m.

- There are clear 'wiggles' in _{RAOs with the slope,}

especially in the heave response

0 0.5 15

wave frequency in rad/s

0 0.5 1

wave frequency in radIs

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The model tests

show the same trend as

the

simulations with the slope: a higher response and wiggles in the RAO, although less pronounced and at slightly different periods than simulated. This can be due to slightly different boundary conditions: the sides of the slopes were different in the tests (open) and diffraction analysis (closed).

These differences cannot be explained just by the difference in water depth, see Figure 9. This Figure shows the calculated heave and pitch response for the constant depth of 15m and 1 8m compared to the results for the slope. In shallow water

the peak in the response shifts as expected to the longer

periods as a result of the shorter waves and increased added mass, but the sloped case is completely out of the range of both constant depth calculations.

2 1.8 E 1.6 E 1.4 .E 1.2

01

0.8 0.6 0.4 0.2 0

wave frequency in red/s

-.-

heave narrow -.- hea.e 28m a---heae 15m -.- pitch narrow - pitch 28m a-- pitch 1 5m 0.00 0.50 1.00 1.50 FREQUENCY CRD/SJ 0.00 0.50 1.00 1.50 F'EQUNC1 CRAD/SJ

Figure 10: Heave wave exciting force in kN for the narrow slope (above) and the constant depth of 15m (below).

Figure 11: 3D snapshot of the wave developing over the slope. The size of the visualised wave field is IlOOm x

IlOOm.

Figure 12 presents two interesting types of snapshots from this visualisation (wave period 15.4s):

- A view in perspective looking towards the incoming wave

- A top view with the incoming waves coming from the right, with the wave crests in white and wave troughs in blue

These Figures show a strong refraction from the sides of

the slope, towards the centre where the LNG carrier

is

positioned.

0 0.5 1 1.5 2

Figure 9: Heave and pitch motion RAOs for the diffraction analysis with the 550m ('narrow') slope and constant depths of ISm and 28m.

It was expected that the wiggles in the motion response were a result of varying wave exciting forces for different wave frequencies. This is confirmed in Figure 10, where the heave exciting forces are shown for the slope and the constant depth of 1 5m. The same peaks and troughs are observed in the exciting forces as in the motion response.

To better understand these variations

in

the wave

excitation, the disturbed wave field was visualised (incoming and diffractedlrefracted

wave). A snapshot

of such

a

visualisation is shown in Figure 11 with the slope and the wave, but without the LNG carrier present. The size of the visualised wave field is IlOOmxl lOOm

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Figure 12: Snapshots of a 15.4s wave refracting and interfering on the narrow slope (crests are white, troughs are blue).

Initially it was expected that a wider slope of 1650 m would solve this problem. However, as can be seen in Figure 13, this was not the case. The motion response increased and the wiggles in the response became even larger, especially for pitch.

Figure 14 shows the related snapshot of the wave field, again for a wave period of 15.4s (0.4 rad/s). The following is observed:

Overall the visualised wave field (a I lOOm wide part at the centre of the 1650m wide slope is presented) is long crested

On top of that there are clear transverse waves, perpendicular to the main wave direction

Shortening of the waves with the water depth, both in the main wave direction and the transverse direction. It was concluded that the refraction at the edges of the slope (not in the snapshot) resulted in real transverse waves further on the slope. As the slope is symmetric, these refracted and sometimes almost transverse waves travel onto the slope from both sides. The end result is a complex interference pattern on the slope, affecting clearly the wave exciting forces on the LNG carrier on the slope.

2 1.8 E 1.6 E 1.4 . 1.2 o 1 08 0.6 0.4 0.2 0 0 E C, C 0 1.4 1.2 0.8 0.6 0.4 0.2 0

IA'

Iu

'ii

UiI?i

a 0.5 1 wave frequency in rad/s

Figure 13: Heave and pitch motion RAOs for the diffraction analysis with the 550m ('narrow') and 1650m ('wide') slope and a constant depth of 15m.

15 -.--hea,e narrow heae wide iheae 15m -.- pitch narrow pitch wide pitch 1 Sm

Figure 14: Snapshots of a 15.4s wave refracting and interfering on the wide slope. The slope width of 1650 m is larger than the visualised wave field of 1 lOOm, the length of the slope is indicated with black lines.

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E

C

0

I

Of course

this interference pattern was strongly

dependent on the wave period, see for example Figure 15, where snapshots for the periods of 31.4s (0.2 rad/s) and 1 0.47s (0.6 radls) are shown.

Figure 15: Snapshots of a 31.4s/O.2 radls wave (top) and 1O.47s/O.6 rad/s wave (below) showing the different interference patterns.

This is quantified in more detail in Figure 16, where the wave elevation RAO is shown halfway the wide slope of 1650 m (for wave frequencies from 0.1 to 0.6 radls) at 21.5 m water depth. For an RAO of 1.0 the local wave height on the slope is equal to the incoming wave height in front of the slope.

£1

i1iYk

w

-550 .450 .350 -250 -150 -50 50 150 250 350 450 550

Distance from centre of slope in m (on mid of slope)

Figure 16: The wave elevation RAO over the width of the slope, halfway up the wide slope of 1650 m for wave frequencies from 0.1 to 0.6 radls.

It can be concluded that the interference pattern is very much dependent on the wave frequency. One should also

0.1

0.2

0.3

-.- 0.4 -.- 0.5 0.6

realise that this is a 2D cross section. This RAO will change over the length of the slope, due to the changing water depth and as well as the progressing waves from the sides. Humps and hollows are observed in the RAO, but there is no standing wave: the RAO does not go down to zero. Figure 17 confirms this complexity. This Figure shows the wave elevation RAO over the centre of the slope from deep water on the left to

shallow water on the right (until the end of the slope at

+275m). This Figure reveals another effect: the discontinuity at the end of the slope from shallow to deep water also reflects the waves back onto the slope.

A.

Distance from mid of slope in m (over centrelinel

Figure 17: The wave elevation RAO over the length of the slope, over the centreline of the wide slope of 1650 m for wave frequencies from 0.1 to 0.6 radls.

To check the effect of the strong discontinuity at the side and end of the slope (from the slope directly vertical to the 'seabed' in the diffraction domain), additional calculations were carried out with sloped sides and a sloped end of the overall slope, as was shown in Figure 7. As can be seen in the snapshots in Figure 18 (narrow slope of 550 m) and Figure 19 (wide slope of I 650m) this was not the case. The refraction

seemed to be even stronger and the

interference more complex.

Considering the questions raised in the beginning of this

paper, we have to conclude that

it is (without special measures) not possible to model a sloping seabed as a second body in linear diffraction theory. The refraction and interference effects are too strong and affect the wave exciting forces on the LNG carrier in an incorrect way. A large size of

the second body or smoother edges of this body do not

improve the situation.

So far the investigations focussed on the complete

motion response and the wave exciting forces.

It

is the

question whether the second body would allow the calculation of the added mass and damping for a ship above a sloping seabed, as improved input for the Boussinesq type models as for instance developed by Bingham (2000).

As a first step in this investigation, Figure 20 presents the heave added mass for the constant water depths of 1 Sm and 28m. As concluded by Teigen (2005), we see that the added mass increases rapidly with decreasing water depth.

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Figure 18: Snapshots of a 15.4s wave refracting and interfering on the narrow slope with sloped edges. The size of the flat part of the slope is indicated with the square, the sloped sides and end extend beside that.

Figure 19: Snapshots of a 15.4s wave refracting and interfering on the wide slope with sloped edges.

c

0,00

0.50

1.00

1.50

PRQUNCY

PAD/SJ CE

0.00

0.50

1.00 1.50 PREQUENCY (RAD/SJ

Figure 20: Heave added mass for the 15 m and 28 m

constant depths in t.

In Figure 21 the heave added mass is now shown for the narrow (550m) and wide (1650m) slopes. It can be seen that the added mass for the narrow slope is relatively close to the constant depth situation. However, the wide slope results show again large wiggles and even a negative added mass. Although the wave pattern of the radiated waves as a result of the heave

motion of the LNG carrier above the slope could not be

visualised, it is assumed that the waves radiating away from the slope are reflected partially at the discontinuities at the sides and end of the slope, as was also observed in Figure 17

for the waves over the slope. This can result in 'trapped'

waves on the slope, a typical cause of negative added mass effects in diffraction theory (similar to the effect of the waves between the floaters of a catamaran).

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U,

D

V) U,

D

c

c\1

V

F'RQLENCY [RD/SJ

Figure 21: Heave added mass for the narrow slope (above) and wide slope (below) in t.

This effect can be reduced when the discontinuity at the edges of the slope is reduced by introducing slopes instead of vertical sides at the sides and end of the main slope (Figure 7). The resulting added mass for the narrow and wide slopes is shown in Figure 22. The curves are much smoother.

This makes clear that a second body in diffraction

theory, when chosen properly with respect to size and shape, can contribute to the correct calculation of the motions of vessels on sloped seabeds.

As an example of that, Figure 23 shows the added mass distributed over the length of the LNG carrier. The vessel was divided in

10 sections and the heave added mass was

integrated over the elements on these sections. The Figure shows the result for constant depth of 15m and 28m, as well as the result on the narrow slope (wave frequency 0.5 rad/s).

0 LI,

D

U,

0

10000 20000 4, 30000 40000 50000 60000 70000

0 [.1)0

0

0 FQUNCY Cfl0/S]

Figure 22: Heave added mass for the narrow slope (above) and wide slope (below) with the sloped sides and end in t.

0

wd 15 wd 28 m narrow slope

0

0 0,00

0.50

1.00

1.

(\1

0

D

0.00

E1.5D

1.00

1. PfUENCY CPiD/5J

FRfOLJN0' tRn/5)

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This Figure makes clear that the added mass is quite constant over the length of the vessel with a constant water depth. Again the much larger added mass in shallow water is clear. On the sloped bottom the added mass varies clearly over the length of the LNG carrier, making the stern in shallow water much 'heavier' dynamically than in a constant water

depth. This will clearly affect the motion response of the

vessel. This can be seen for instance in the pitch-heave

coupling (A35 and A53) as presented in Figure 24.

CD CD CD L? Lfl CD C)

0.00

0.50

1.00

1.50

FREQUENCY CRD/SJ 0.00 0.50 1.00 1.50 FREUENC1 1RH0/5J

Figure 24: Pitch-heave coupling coefficients (A35 and A53) for the constant depth of iSm (above) and the narrow slope.

The figure shows a much stronger and complex

pitch-heave/heave-pitch coupling than with a constant water depth: on a sloped seabed a heaving vessel will influence the pitch

motions (and the other way around). This will affect the

motions of the ship in waves, as well as the resulting drift forces and related mooring loads.

CONCLUSIONS

Based on the results presented in this paper, it can be concluded that:

- Without special measures it is not possible to model a sloping seabed as a second body in diffraction theory. The refraction and interference effects are too strong

and affect the wave exciting forces on the LNG

carrier in an incorrect way. A large size of the second body and smoother edges of this body do not improve the situation. This higher excitation results in

significantly higher motion response than realistic (depending on the wave period).

A second body in diffraction theory, when chosen properly with respect to size and shape, can

contribute to the correct calculation of the added

mass and damping of vessels on sloped seabeds as this varies over the local water depth over the length

of the vessel. This will clearly affect the motion

response of the vessel. This can be seen for instance in the pitch-heave coupling. These effects will be

dependent on the size of the vessel and the local

waterdepth (local water depth to draft ratio).

The calculated added mass and damping for a ship

above a sloping seabed can be used as improved

input for the Boussinesq type models as for instance developed by Bingham (2000).

ACKNOWLEDGEMENTS

Shell Global Solutions and Delta Marine Consultants (DMC) are acknowledged for their permission to use the LNG carrier model test results in the present paper.

REFERENCES

Mei, C.C., Stiassnie, M. and Yue, D.K.-P, Theory and Applications of Ocean Surface Waves", World Scientific, 2005, Singapore.

Dijk, R.R.T. van, "Shallow Water Testing of an SPM Moored LNG Carrier", Shallow water hydrodynamics seminar, February 1st & 2nd, 2005, Wageningen, the Netherlands.

Voogt, A.J., T. Bunnik and R.J.M. Huijsmans, "Validation of an analytical method to calculate wave

setdown on

current", OMAE2005-67436, Halkidiki,

Greece.

Naciri, M., B. Buchner, 1. Bunnik, R. Huijsmans & J. Andrews, "Low Frequency Motions of LNG carriers in shallow water", Proceedings of 2004 Offshore Mechanics & Attic Engineering Conference. Paper

OMAE2004-5 1169.

Borsboom, M.J.A., N. Doom, J. Groeneweg and M.R.A.

van Gent (2000). "A Boussinesq type model

that conserves both mass and momentum". In Proc. 27th mt. Conf. on Coastal Eng., ASCE, Sydney, 148-161.

Bingham, H.B., "A hybrid Boussinesq panel method for predicting

the motion of a moored ship",

Coastal Engineering 40 (2000) 2 1-38.

Madsen, P.A., Bingham, H.B. and Liu, H., "A new

Boussinesq method for

fully nonlinear waves from shallow to deep water", J. Fluid Mech. 462 (2002) 1-30. Teigen, P., "Motion Response of a Spread Moored Barge over a Sloping Bottom", ISOPE 2005, Seoul