Squat effects of very large container ships sailing in a harbor environment

September 2005 Henk J. de Koning Gans Deift University of Technology Ship Hydromechanics Laboratory

Mekelweg 2, 26282 CD Delft

TUDeIft

DeIft University of Technology

Squat effects of very large container ships

Sailing in a harbor environment

door

Hi. de Koning Gans

Report No. 1451-p

2005

Presented at the International Conference on Port-Maritime Development and Innovation, World Trade Center,

Rotterdam, 5, 6 and 7 September 2005,

ISBN-10: 9080989215 - ISBN-13:9789080989214

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International Conference on Port-Maritime

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5, 6 and 7 September 2005

World Trade Center, Rotterdam The Netherlands

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International Conference on Port-Maritime

Development and innovation

Contents of this CDROM

Monday 5 September 2005

Theme Port related Transport, Handling and Logistics

Improving maritime logistics operations through

_{new technology Olszewski (Nanyang}

Technological University, Singapore)

Strategies of terminal operators in changing environments. P.W. de Langen (Erasmus

University, The Netherlands)

EvaluatIon..ofÇoainer

_{Terminal Simulator Hsu}

Wen jing (Nanyang Technological University, Singapore)

Evaluation of integrated port information system and impact

_{on the port competitiveness N.}

JOlíc (Faculty of transport and engineering, Croatia)

Fioting cranes for the container term mal of the future B.A. Pielage Delft University of

Technology, The Netherlands)

Switching between ports of call in global container networks B. Kuipers (TNO - Mobility

and Logistics, The Netherlands)

Theme Port Planning and Design

Container Terminal Development in Bremerhaven I. Krämer (Bremerports

GmbH&Co.KG, Germany)

e Plannmg and design of the Pasir Panjang Container Terminal, Singapore Mr Loh Yan Hui,

(Surbana International Consultants Singapore)

Maasvlakte 2: A unique project in a unique area T. Vellinga (Port of Rotterdam,

_The

Netherlands)

Supporting port facility design with integrated modeling. N. Lang (Erasmus University, The

Netherlands)

Engineering a new port at Hazira, Gujarat, India; L.A.M. Groenewegen (Delta Marine

Consultants, The Netherlands)

Accessibility of the Rotterdam Port in case of calamities H.J. van Zuylen (Delft University of

Technology, The Netherlands)

The human factor in port masterplanning balancing shareholders' and stakeholders'

interests W. Winkelmans ITMMA, Belgiurn

Second Maasvlakte: Practice planning and design through simulation-gaming I.S. Mayer

Delft University of Technology, The Netherlands)

Theme Port Security

Port Security implementation in Singapore. M. Segar (Maritime and Port Authority,

Singapore)

Securing the supply chain

- EU policy initiatives and industry solutions. R. van Bockel

(European Commission, The Netherlands)

Port Security

- A global challenge W. Ernst (ATLAS Elektronik GmbH, Germany)

Page 2 of'3

Prospering in the security economy L. Smit (TNO I Deloitte,

The Netherlands)

Newte.ch11.0.ogies,concepts.nd pro.ducts in. thefight

_{again.st..earobberyand..piracy. W.}

Smith (TNO; The Netherlands)

Supply chain security: From cost center

_{to opportunity G.Huygen (Siemens Nederland N.Y.,}

The Netherlands)

e standards or.then.nmaiitirne.pajit.otthsuppiy chain P. Mackenbach (DNV)

Developments in US Maritime Security Policy L. Fagan (US Coast Guards, USA)

Tuesday 6 September 2005

Theme Port and Marine Environment

i1easjti1ng. M...A.'s..; Exploring the..institutional, embedding..o..,.managem.ent...pla.ns...reconciling

fisheries and nature protection. F. Neumann (Institute for Infrastructure, Environment and

Innovation; Belgium)

Harbour environmental impact study - Project alternatives comparison criteria G. Perfilo

(Università degli Studi di Napoli Federico II, Italy)

Artificial substrates for aquatic organisms in harbours. G. Bolier

_{(Deift University of}

Technology, The Netherlands)

Habi*a.1.Bankjng ¡n.Eu.rope;Çompensati.o..nthrough natural a.sset.. creation C.Kögler

(Institute for Infrastructure, Environment and Innovation; Belgium)

Combining technical and environmental knowledge by evaluating environmental

_{impacts R.}

Kajatkari (Port of Kotka ltd, Finland)

Rcallime..rnanagejnent..of dredge. ..ope.rtio.. s J.H. Aardoom (Aqua Vision, The Netherlands)

Theme Port Planning and Design

The continuous geotube E.J. Broos (Rotterdam public Works, The Netherlands)

A challenge in port planning and design Towards an Area Planning Studio for the Port of

RotterdamJ. Smits (Port of Rotterdam, The Netherlands)

An Area Planning Studio for the Port of Rotterdam R. Chin (Deift University of

Technology, The Netherlands)

M.o.dellingharbour sedimentation -new paradigm..B. Bhattacharya (Unesco

- ifiE; Delft,

The Netherlands)

Impact of very larg container vessels on port design J.H. Wulder (Maritime simulation

Rotterdam, The Netherlands)

Aninn,ov..1.1v....approach. to.sustainabJe container port development, C. Eikelboom

(DHV,The Netherlands)

Port of Dover Master plan M. Mann ion (Port & Dockyards, Haicrow, United Kingdom)

Masterplan Slufter: past, present, future G. Berger (Port of Rotterdam, The Netherlands)

Single Index Measure of Port Efficiency S.N. Chakrabartty (Indian institute of Port

Management, India)

Theme Port Related Maritime Operations

Hum.an...a1r.aspets .o AIS'in the VTS environment W. Hoebee (Port of Rotterdam, The

Netherlands)

INFO

SEATM: A Regional Maritime Information Infrastructure and the Straitsforecast.

M. Womersley (Info@SEA

,

Singapore)

Studies for the w..iden ing,a.nd deepening of the. entrance channtho theport.

ofiiurban, South

Page 3 of 3

Africa H. Moes (Research Engineers, CSIR, South Africa)

The_impo.rta.nc.of windio.ads...on....Portoperations W. van den Bos Delft University of

Technology, The Netherlands)

Wind loads on ships and complex structures M.J. van Hilten (Dutch Pilots'cooperation, The

Netherlands)

Moo.redshipmolionsdueÁopasiingships W. van der Moten 'Delft University of

Technology, The Netherlands)

.soIn.tion....or survey data

ssin.gand...cart.ograpjiic....pro.duds A. Pirozhnikov

(HydroService AS, Norway)

Current forces on moored ships in complex flow situations O.M. Weiler (Deift Hydraulics,

The Netherlands)

Wednesday 7 September 2005

Economic Aspects

Change..i.n.:th..e.S.h.ppm.gindustryand.. Port Implic.ations...A. Penfold (Oceanshipping Ltd.)

Port authority strategies: a research agenda. L.M. van der Lugt (Erasmus University, The

Netherlands)

Developing..a....1....aritim.c....Se.rv.icec.eiI.*.er in..the..po.rtof Ro.tter

A.W. Veenstra (Erasmus

University, The Netherlands)

A vision on European Sea Port Policy from the industries' perspective. R. Bagchus

DeltaIinqs; the Netherlands)

Theme Port Planning and Design; Quay requirements

Development of port design and construction in the p_ort of Rotterdam J.G. de Gijt

(Rotterdam Public works, The Netherlands)

Meeting 2030's quay requirements G.H. Wijnants (TNO, building research, The

Netherlands)

Maasvlakte 2: from vision to reality G. Kant (Port of Rotterdam, The Netherlands)

Underground Tank Storage H. Pals (Royal Haskoning, The Netherlands)

Theme Port Related Maritime Operations

Comprehensive trafflc .modelling. R. Seignette (Port of Rotterdam, The Netherlands)

Bathymetrk..E.NCsforth.eimprovedNavigttion.in Pons F. Moggert (SevenCs AG &Co.

KG, The Netherlands)

Analytical study of a ship berthing to a quay wall Wang, Qian Xi (Nanyang Technological

University, Singapore)

Squat effects of very large container ships sailing in harbor environment H.J. de Koning

Gans (Delft University of Technology, The Netherlands)

Squat results from calculations, of panel ;méthOds

Summary

Squat is a serious. pröblern when very large container ships are entering harbors with small

underkeel margins1, The squät effects are that the ship acquires sinkage and trim. Due to this

sinkage and trim the keel clêarance decreases drastically. The distance bètween' bottom and ship

becomes very small and measures must be taken to avoid contact of the ship with the bottom

From a study carried out by the TU DeIft It appears that there is one special technique that can

predict the squat effects and this method Is the so-called panel methods This method describes a

non-viscous flow exactly The results should be reliable, because the exact shape of the ship hull is

taken into account The panel method is very convenient for squat calculations For these squat

calculations the free surface effects, which generates the so-called Kelvin wave pattern, do not

have to be taken into account The influence of these waves is very small

So a linearised

calculation is made, which predicts the sinkage and trim By using some physics, the non linear

prediction can be obtáined too from the linear calcUlations.

For a better prediction the non linear method is used.. An iteration process has been made for

panel method by replacing the new ship geometry according to the sinkage and trim determination

from the previous 'squat calculation. This will provide a non-linear effect, which means that squat

becomes reasonably larger and so the squat effects become worse 'than those computed from

linear calculations. This non-linear effect is in 'agreement with model experiments and an' emperical

equation according to Schmiechen. The linear calculations have been carried' out for very large

container ships (Post. Panamax (±6500 TEU), JUMBO (±8500 TEU) and MEGA JUMBO' (±Ï2500.

TEU)) to investigate the squat effects This research gives an indication that the squat effects for

all these ships and the sinkage presents a serious problem Also, when the initial keel clearance is

small, the keel clearance in sailing condition becomes dangerously small Last but not least, also

attention is paid to a MEGA JUMBO for a non linear calculations sailing in a real harbor the

'Yangtseehaven'

It appears that these nonlinear effects have a negative effect on the keel

clearance From the calculations it appears that the predicted sinkage and trim from the panel

method: are ¡n 'good agreement With the emperical 'method, When the non linear correction is

made.

Author(s)

H.J. de Koning 'Gans

Year of publication

2004, © Port Research Centre RotterdamDelft

Project leader

Dr.ir. H.J. de Koning Gans, H.J.déKonlngGans©wbmt.tudelfl.n!, Tel.: 431(0)15-2781852

DeIft Uhiversity of Technology,

Faculty of Mechanical, Maritime and Materials Engineering,

Mekelweg 2, 2628 CD DeIft, The Netherlands.

Secrntariat of the Port Research Centre Rotterdam-DeIft

Pim C.B. de Wit, Port of Rotterdam,

pcb.de.Wit@portofrotterdam .com, Tel.:, +31(0)10-2521892

Rob M. Stikkelman, DeIft University ofTéchnology, TBM/CPIRD,

r.m.stikkelman@tbm.tüdelft.nl, Tel.: +31(0)15-2787236

Port of'

Rotterdam

TU IJeift

SQUAT EH'ECTS OF VERY LARGE CONTAINER SHIPS SAILING IN A HARBOR

ENVIRONMENT

Dr. li. H.J. de Koning Gans Assistant professor

DeIft University of Technology Faculty of Mechanical Engineering and Marine Technology

Ship Hydromechanics Laboratory Mekelweg 2

2628 CD Deift The Netherlands

Phone: +31 152781852 Telefax: +31 15 2781836

email: H.J.deKomngGans @WbMT.TUDeIft.nl

Summary

Squat is a serious problem when very largecontainer shipsareentering harborswith small underkeel margins. The squat effects are that theshipacquiressinkage andtrim. Due to this sinkage and trimthekeel clearance decreases drastically. The distance between bottomandshipbecomes very small and measures must be taken to avoid contact of theship with the bottom. This study looks at three types of flòw models. The:models concerned aretheone-dimensional, thetwo-dimensional and three-dimensional flow model. For each of these modelsthe shipgeometry is simplifiedand theonly two parameters are thecrosssection area distribution and the breadth distributionat the water line.

The one-dimensional flow mode1 developed by Kreitner [17], isa model whereonly the longitudinal velocity component is calculated and this component is constant incrossdirection.

The two-dimensional flow model, developedby Tuck and Taylor [24]4 isa model wherethe longitudinal and the cross velocity components are calculated. The model consistsof a source layer located at thecenter of the ship from bowto stern. The layer is defined from theundisturbed free surface to the bottoni. Actually below the keel also asource layer is defined, which is physically not very realistic. Fromthistheory it is possibleEto obtain a relatively simple equation for the sinkage and trim. The results of this theory are in good agreement With the results obtained from model experiments.

The threedimensional flowmodel, is a model whereall the velocity components arecalculated and is called panel method. The modelconsistsof a source layer located on the wetted surface of the ship hull. This method describes a non-viscous flow exactly. The results should be reliable, becausetheexact shapeof the.ship hull is taken into account. The panel method is very convenient for squat calculations. For these squat

calculations the free surface effects which generates the so called Kelvin wave pattern don t have to be taken into account. The.influences ofthese waves are very small.

Aniterationprocess can be applied to the two dimensional flow model according toTuck andTaylorand the panel method by replacing the new ship geometry according to the sinkage and trim determination fromthe previous squat calculation This will provide a non linear effect which means that squat becomes reasonably larger andso the squat effects become larger than those computed from linear calculations. This non-linear effect isin agreement with model experiments.

Introduction

In this article the squat of very large container ships is investigated. First a literature study has been carried out and after this study further research was to becarriedEout to make a number of calculations and computer simulations to determinethe verticalEsinkage and trim. Both the sinkageand trim moment aredue to fact that ships sail in shallow water. Squat is a serious problem in restricted water. The squat causes thatthe ship and local water level to go down while the ship is sailing in shallow water. The keelclearance will also decrease and the gap between shipandcanaltbottom becomes'very small. The squat effects have to becontrolled to avoid contact between the ship and the bottom When anexistingshipsails in restricted waters two parameters can be controlled, the ship speed and the depth of bottom of the canal. The latterhas to berealised when the harboris dredged. Withtheforth going technique container vessels become larger and larger The authorities think vessels of 12500 TEbJ will be realised, which have an equivalent length of about 400m, breadth 55 -60m and a draught ofi7;0m This typeof containership will becalled MEGA JUMBO (container ship). Toplay an important mie

in futureshipping transportthe Havenbedrijf Rotterdam N.y. decided thatanew harbor environmenthas tobe developed. It iscalled Maasvlakte 2. In this Maasvlakte 2' also basins musttobe designed, which can handle

the very largeMEGA JUMBOs Presentlytheauthorities are designing thenew basinsand they need to know how deep'the harbormust to bedredged. Every extrameterwill producean enormous amount ofextra cosis so

it is very important to knowexactly howdeepthetharbor has to bedesigned. As mentioned above, the squat effects are a very important issueand so a study thereof hadtobe Undertaken. Thetheory desciibedin this report

is based on a literature study.

In thisreport adescription is givenof shallow water phenomena, whichoccur whena very large container ship isentering theharbor. Three typescof squat models are described: the first is the Kreitner method or one dimensional flow, the second isthe method according to Tuck and Taylor which actually describes atwo dimensional flow. Special attention must to bepaid on this method, becausethe results are relatively simple and this method is easy to implement in compüter programs Also it appears that the results are reliable and the theoretical results are comparablewith the experiments. The third method is a three-dimensional flowmodel, which is also described by Tuck and Taylor. Howeverthis method is very complicated and not veryconvenient for implementation.

A better method to predict the three dimensional flow is the panel method The panel method describes a ful!y potential flow aroundship ma harbor. This means that there are no viscous effects and only the inertial forcesaresimulatech Also with thepanel method the free surface effectscan be implèmented. (In practice the ship speed is too slow to makea good free surfaceprediction, but this' it is not a problem, because the free surface effects vriththis low speed have no influence on the squat.) The panel method is the only method which calculates the flow effects in the gap between the keel of the ship and bottom. Becausethis gap is very small high velocity componentsmay occurand also very high pressure variations may occur; which havea big influence on the squat phenomena. Due to these effects panel methods are superior to other methodsand therefore much attention is paid tocthepanel methods.

The last part of the literature study deals with experimental results of model tests Three types of ships are tested (Post Panamax, JUMBO and MEGA JUMBO) and theinfluences of squat are investigated. The studyis worked out for the harbor environment of Hamburg (the river Elbe) and'the dimensions are almost the same as the harborin 'Maasvlakte 2'. So this research willgivean indication of the real squateffects that may expectediñ 'Maasvlakte2'.

Overview of some methods

In the literature study the method of Kreilner and some related methods, and the methods of Tuckand Taylor and' other near related theories, are treated Only the sub critical cases are treated, it is assumed that the container ship will sail in the harbor withlimited speed. The Kreitner method isbased on the assumption that theflow along the ship is a parallel flow and that the amplitude of the velocity is determined by calculating the amount of waterofthecreturn flow and thatthisisequal to the amountof water of the fictive forWard flow of the cross section of the canal. The Method of Tuck and Taylor is based on a potential flow with advanced mathematical methods such as asymptotic expansionand singularity calculations. The two dimensional theory is relative simpleto implement and gives a transparent relation betweenthefôrces and the body shape. Usingthisrelatively simple expression it is relatively easy to predict the sinkage and trim.

Forthe two-dimensional case, the shape of the ship is replaced by a source layer locatedat the centerlineof the ship Remarkably althoughthe theory is very sophisticated they have derived a relatively simple form for the:pressureandby integrating thispressure over the water plane they havederived a suitable expression for the sinkage and trim. Froma geometry study itappears that the influencesofthesinkage from theirtheory is not very sensitive for severalship shapes. Therefore they derivedan empiric eúation for thesinkageas well.

The three dimensionaltheory is based on thepanel rnethod The wetted surface of the body isdefinedby panels and the flow simulation can becalculatedfby using(complicated) singularity integrals. Thismethod gives a prediction ofthecsquat of the real ship hull fòrmand:only viscous effects are not taken into account.

The method ofKieitner, the one-dimensional model

The method of Kreilner was:already developed in 1934 (see [17]). The resUlts ofthisresearch were published in [18] The method of Kreitner is based on the law of conservation of water mass and the Bernouilli equation with the boundary condition at the free surface that thepressureat the water surface iscequal to the pressure of the air. When a ship is sailing in a canal the amount of the 'virtual' water mass going through the cross sectioncarea A0 of thecanal far in frontof the ship Will be UA0. Beside the shipithe same amountof Waterhas to pass the ship. But beside the ship the cross section area where the water can flow through is smaller due to the presence of the

ship. The cross section area then isA-S(S=cross section area of the ship). Sowhen the sameamount of water is flowing as well in front of the ship as besidethe ship the relative longitudinal velocity component increases according to the following equation:

u(AS)_UA0

(I')

Also the Bernouilliequation.is used which givesarelation between thepressure and the velocity of the water.

Using alsothecondition far upstream from the ship, where u =U and the water surface elevation is zero(Ç= O)

besidethe ship, the Bernouilli equation reads as follows:

u2+gÇ=-!-u2 ₍₂₎

Eliminating the variable u from both equations (l)and (2), the followingexpression is thenobtained:

jJ(12_2gÇ(A_s)=UA0

(3) An extension of Kreilner method is that thecross area is not constant but is a function of the longitudinal coordinate x. So it is possible to calculate the water level elevationas a function of xand:at each longitudinal location Introducing the dependency of the longitudinal coordinate not only the sinkage and sinkage force but also trim moment and trim can be calculated

Also fromthe Kreitner method it is possible topredict when a ship sails in a sub- or trans-ctiticalarea. The

trans-critical area has to beomitted. There :iSflO stationary solution and somany waves are generated, costing a

lot of energy. Beside the loss of energy, the hindrance to other:(moored) ship isconsiderable and damagesito the banks may alsooccur.

4.1. Some remarks about the one dimensional method

In this one dimensional method nothing is said about the effectof the keelclearance. Whenseveral ships have the samecrosssectionarea, the breadth and the draught can be different. The Kreitner method does not include this effect. So, for ships with the same crosssectional area, when the initial keel clearance is very small,the

sinkageisequal to thesinkage of ship witha bigger keel clearance. So the influence of the keel clearance is not taken into accoUnt at all. Also these methods (such as Kreitner) can onlybe applied wijh restricted widthof the canal. Whenthe canal becomes very wide (or even infinity)thereturn flow is zeroandnosinkage forcescan be calcUlated.

5. The method of Tuck and Taylor and the near related theories, the twodimensional theory

The solution of the method of Tuck and Taylor is based on a two-dimensional flow model. The model: of the flow is that beside the longitudinal velocity component a cross velocity component exists in y-direction The perturbations of these velocity componentsare of the same order. When the(low order) flow isalready determined the vertical component of the velocity and free surface elevationcan be deterninedandthefree surface elevation is always anorder (ortwo) higher. (A higherorder means that the magnitude will be(much) smaller than that:of a lower order component.)

The potential is built upof a parallel flow ((Ix) and a perturbation Potential Uq'. The velocity vectoriS determined by taking the gradient:

ü=UV(x+ip)

(4)

The boundary conditions haveo:be applied too. One of the boundary conditiOns is thatthêre isno flow going through the hull and canal floor. This means that the normal velocity at the hull (and bottom) is zero. The normal velocity can be obtained by calculation of the normal derivative of the potential. So the no leakage boundary condition is:

a(xqi)

OE

an

The governing function which can express the total flow problem isapotential function andthusfunction_w,

fulfills the following equations:

(i F,)!P1 + ''i»

0 (6)

This eqmatibn(6)is the usual equation for the shallowwater theory. Inithisequation the Froudedepthnumber appears and this number is defined as:

(5)

5.1. Pressure

For the squat calculations the pressure is more important than knowing the real flow. The pressure can be obtained by using the Bernouilliequation. Looking only at the first order term the pressure is equal to:

p(x) = pu

'y= O)

- pU2

2f S'()

d

2ithJl_F,

jx

So the pressure can be obtained directly without solving the potential and the velocity. Aithougha laborious and difficult derivation has been undertaken for determrning and solving the potential for the fluid field the end result is relatively simple. Also itis not necessary to determine the flow to obtain thepressure distribution.

5.2. Sinkage force and trimming moment

Knowing the pressure distribution it is easy to find the sinkage force and trimming moment. The sinkage force can be obtained by only integrating the pressure with respect to the length and the local breadth:

F3= JB(x)p(x)dx=

Th0

fdrJB'(x)S'()loglx_Id

(Il)

L.

2,th/l

- F, L L

2 2 2

When the pressureis multiplied with the arm(the distance between the local pressure and the origin (this is amidships)) thetnmming moment can be found:

I

=

fxB(x)p(x)dx=

U2

Jf(xB(x))'S()logIx_ld

L 2ithJl

- F,

L L

(12)

2 2 2

5.3. Sinkage and trim

The sinkage and trim can be obtained by using Archimedes law. When the sinkage isdefined as s (positive downwards) and the trim is defined as.i (positive ahead), the equilibrium according to Archimedes' law for a trim and sinkage ship is:

q =P1(x,y=O)=

2 2,thJl F,

fS'()logIx_Id

-

L

-i

U (9) (I O) 4

'J

Fh=j=

(7) (gI

Alsotheboundary conditions for equation (6)haveto be applied forthat case. For the squat calculation the first order equation will be treated. Tofind the solUtion of the integral equation (6), this equation can be transformed to an integralequation by using the second Greens identity. This integral equationhas the following physical meaning. A layerof sources isdistributed only in longitudinal direction at the center plane of the ship. In vertical directionthe source strength is kept constant and the height h is assumed from the initial free surface level to the bottom of the canal. These sources will provide that the water will flow around the ship and the particles at the ship hull will follow thisshiphùll. The soUrce strength is then assumed:

cr(x)=-S'(x)

(8)

L L L

PgJ(s + xr)B(x)dr= spg + tpg jxB(x)dx = pg(As + St) = F3

(13)

Forthe keel clearance it does-not matter if the trim is by head or by stern, when the origin of the coordinate -system of the ship is defined amidships. By the same amount of trim (ahead or astern) one part of the keel reaches the same distance closer to the bottom.

6. Empirical equation for-sinkage

Taylor and Tuck have investigated a -large number of types of ships and they looked at the trend of the sinkage. It appeared that the sinkage is not very dependent on the ship form and they derived an empincal equation which is based on the sinkage (see equation-(l I) and (13)). In the lastequation the trim tandtrim moment F5 isomitted. Dueto the fact that the sinkage is -less-dependent on-the form, they introduced-a shape factor, to find the influence of theshape. It appears from many tests as well as calculations that the sinkage is mostly-dependent on the block coefficient and the physics accordingto equation (11). The empirical equation they have-derived reads as follows:

s=2.4?

F,

L

6.1. Some remarks about the two dimensional method

The two dimensional method of Tuck and Taylor consists-of a source layer distributed from the bow to the stem. The vertical distribution- is constant from the (undisturbed) free surface-level to the bottom of the canal. This meansalsothat a source -layerexists below the keel, whichis physically not realistic. Thisapplied source -layer height oUtside the shipandabove the bottomin the vertical plane will be- higher when-the keel-clearance is larger. The only adaptation of the source strength is that the source-strength is linear with the derivative with respect to the x-coordinate of the cross section area When the pressure is obtained for the trim and sinkage calculation, thepressure is determined from the local flow and isdetermined along the water area. Theshape of the shipis-not included as well asthe cross and vertical velocity component. (It can be proved that-the-influence on the pressure of the cross and vertical velocity component is one or two orders higher than the influence of the 5

pgf(s

+ xt)xB(x)dx = spg JxB(x)dx + fpg x2B(x)dx = pg(Ss +

it)-= F5

Inthis equation the following known-quantities appear:A: the water areaor zero orderarea,S: the static moment or first order area and¡y: the moment of inertia or second order area-of the water-plane defined from amidships. Substituting these parameters in-equation(13) the following matrix equation is-obtained:

(A

S'('s"

(F3

'y;1.t)(

F5 ) (14)

This matrix equation can be solved easily, the results are: F3Ïy - F5S

and

for-the sinkage (15)

pg(Al S)

F5AW

F3S

for the trim (16)

pg(AI S)

The forces F3 and F5 are already-obtained by equations (I I) an (.12). The keel clearance isobtained by the subtracting the draught T of the ship and the influence of the sinkage.and trim from the water depthh:

longitudinal velocitycomponent and therefore it can be omitted.) The only twoshapeparametersarethebteadth distribution and the cross-section area distribution.

Also the squat calculations fail when the depth of the canal is very large or infinity. According to the Tuck and Taylor method the source strength becomes zero and so the pressure, trim moment and sinking force become zero too. So the sinkage and trim are zero in deep water. This is not realistic. Also a ship in deepwater has squat effects. It is also clear from equations (lO),(15)and (16) that the squat effects are zero. The depth h becomes

infinity and the depth parameter occursin the denominator of the Froude depth number (equation (6)) and so F5

becomes O. Sotheresults of equations(lO), (15) and (16) are zero. Thus there areno squat effects for deep water according to the Tuck and Taylorequatións.

The integrals for determining the sinkage and trim can be calculated analytically when thebreadth and crosssection area areexpressed aspolynomials. Also a part of thecontribution of the integral is zero when B(x) or S'(x)is zero. So the polynomial seriesexpansion'has onlyto be described from the stern to the location where asthe breadth respectivelycross areasection ¡s constant at the aft ship and for the fore ship the polynomial starts where the fore ship is not constant anymore in breadth respectively cross areasection and ends at thebow of the ship.

When the canal is restricted with a limited width, the method can be adapted by mirroring the source layer located at the center planeof theship in the canal walls. Physically abt of source layer planes will occur, soit looks like many ships sail beside each other in the same direction with the same speed. It is also possible to replace the source layers by a new mathematical function which can be derived asan expression of a sum of infinitely many terms. Thesolution procedure is analogue to the method described here above(see alsó [7] and [20]); Also lift effects can be applied. In the same plane wherethe source layer is placed, a vortex layer can be placed. Again the boundary conditions have to beapplied and frorn.this it is possible toobtain the vortex strength distribution; From this ¡t is possibleto calculate the drift force and moment. From this two dimensional theory the lower order terms of the vortex distribution have noinfluence on the trim.

6.2. Non-linear effects

When the trim and sinkage is calculatedthe ship has a new position in the water (, see Figure 1 NeW situation when a ship sails in squat conditions).

Original position ship

Still water level

z,z

Actual water level

Position of the ship due to squat effects

Figure 1 New situation whenaship sails in squat conditions

Also the increased or decreased local water level is now known. So the two-dimensional theory of Tuck and Taylor can be used again for the new situation It is not needed to use the higher order series expansion of the potential. In fact it is an iterationprocess. After eachcalculation the known quantities can beused again andcan bepùt in the algorithm toobtainthenew sinkage and trim; The iteration goes as follows; First of all the water level is increased with

hW=h+Ç(x)

₍₁₉₎

In which the water level 'x) can be obtained from the pressureaccording to equation (10) Ç(x)= p(x)/(gp). Also

the ship hasa new relativeposition in the water due to sinkage and trim, see also figure 1. Due to this, the cross section area alsochanges. The relative changeof water level along a certain zcoordinate is, due to sinkage:

= s-i-Ç(x)and totrim: ¿S.(x) = tx. Thelocal increaseddisplacedcross section area is then equal to the relative water level of ship multiplied with the local breadth:

tSS(x)=Ç(x)B(x)= (iÇ (x) + ¿Ç, (x))B(x) = (s + tx + Ç(x))B(x) (20)

+ xt + s

h

K=h-T-s-ItI-1

Due to the decreased level of the water the area wherethe cross flow is streaming out becomes smaller (see equation(l8)). The amount of water, which is streaming in:crossdirectionshould be thesame. So the cross

velòcity component is increased with:

(I) vh

h(» (21)

Last but not least also the longitudinal flow velocity relative tothe shipis increased. The relative velocity is:

U(x)=U

+u(x) (22)

The perturbated longitudinal velocity can be calculated with:

ax pU

So these three quantities of equation (20) and the increased parameters of the velocity and decreased waterdepth will change the squat effects. These quantities will change the source strengthaccordingto:equation (7). So the new source strength will be:

a1

-

U(»(x)S(»

(i)

(U+u(x))(S'(x)+(t+Ç'(x))B(x)+s+fx+Ç(x))B(x)

(24)

hW(x) h+Ç(x)

Inthisequation use is made of a Taylor expansion and all higher order terms(2) are ignored. Further use has

beenmade of the fact that the expression ofthe increased longitudinal velocity and increased(decreased)water

leveFcan be expressed as a function of the pressure. (Thereason is, also for the non-linear squat calculations, that itappears it is not ñecessary that theflowhasto be known and that it canonly be expressed withthe pressure distribution.) Alsothe term ofthe changedcross-area S'°(x) hasbeenievalúated. The derivative with respect to x can be simply determined by using equation (23). Theiprocedure of determining the new pressure distribution can be donebyusing the newly calculated a1, Which willbe substituted in equation (l0)for the

pressure and the equatiOns (Il) and (12) forthe sinkage force andtrimming moment. But as it was found from Tuck and Taylor for their empirical formulation of the linear squat, it is expected that thecontributions of the integrals are almost constant for anytype of ship form. So the expected nonlinear squat is proportional to:

socaBT

F,(l+ßF,)

₍₂₅₎

T

_J1-F,

Notethat for the pressure in the newly derivedequations, thepressure distribution according to equation (9) has to be used. Also the last three obtained equations can be solved analytically. For the corrected sinkage and trim, the equations( 15) and (16) can be used again. lt makesnosense for an other iteration again, because other effects will play a roll, which is of the same higher order. These higher order effects are the vertical velocity component of liqúid, the wave pattern andiother second orderperturbations, whichoccur in the velocity components and pressure.

Also in the eqúations (10) to (16) the term 'I(1-Fh2 ) appears in the nominator. This is also a nonlinear term but this isalready present in the original form of the Taylor and Tuck model. Alsothisterm givesa non-linearity to the sinkageand trim. In factthere are two non-linearities. First which appearsdirectly from the shallow water theory and secondlytheextra terms whichare mentioned in equation (25). The latter is not formulated inithe Tuck and Taylor method.

So, also with the two dimensional model of Tuck and Taylor non-linear effects can be predicted. lt is not necessarytoknow the floW aroundthe.ship. Only when the breadth distribution and cross section area distribution are known, the method canpredict thesinkage and trim, including nnlinear effects.

The non-linear effect is important, because the squat effects become larger and this will result in ihe predicted keel clearance being less than fromthe linear model. Thiseffect plays a roll whenthe Froude depth numberbecomes high Fhl.5.So it is important to introdücenon-linear effects for high Froude depth numbers.

6.3. Method Schmiechen

Schmiechen has also investigated the squat phenomena. From towing tank tests, he found that the sinkage due to squat is proportional to the cubic Froude depthnumber: s-Fh3. He also used the Tuck derivations of the velocity-independent part which influences the squat. For this velocity-velocity-independent part the expressionsare thesamefor Tuckas well asSchmiechen. Theoriginal Tuck equation is:

(26)

L

lF2

Fromthe Tuck method it isknown that the: predictions are in verygood agreement in around the value of the Froude depth number F5= 0.7 (This number can alsoapproximatedby ½'I2. Sothesquare of the Froude depth number is equal to Fh2=½.) So, he approaches the Tuck Froude depthnumber dependent part by thecubic function:

/1_F2

=2F

(27)

F .7

The factor 2 is calculated byfiuing in the equation at Froude depth number F,, =0.7. Thedifference of the Tuck and the Schmiechen Froude depth number dependent part is shown in figure 2. So the squat prediction of Schmiechen is (this will be represented by themethod Tuck according to Schmiechen):

s=2.4.?i2F,3

(28)

L L

6.3.1. Discussion about the Schmiechen method

It seemsthat the Schmiechen method gives a goodpredictioniof thesinkage due to squat. Especially by model tests the prediction is indeed close to the function describedby Schmiechen.. But the Froudedepth number dependentpart isan odd function. This isa less realistic. Thiscan be seen asfollows. When a ship is symmetric around amidshipsthe squat should besame in astern and ahead direction. This can be only realized by even functions. Probably the cubic functions approach the results of model: tests, but it has no physical back ground. This meansthat any arbitrary functionior results, which:also fjtsto the Schmiechen function, is a realistic functionor result, which gives a good match with model test results.

64. Method of Tuck using the sinkage from previous calculations

6.4.1. Method one

When the methodof Tuck is used the sinkagecan bepredicted. NowIheprocedure will be repeated again when the shipis located in the new positionincluding the sinkage(and trim). This derivation hasbeendone in section 5.2 and 5.3 the non-linear sinkage and trimispredicted according toequation (15) and (16). Now again the equation can be split in a Froude depthnumber dependent part anda geometrical part. From the linear equation of the new position of the ship, it appears that the siiíkageandtrim is a function of:

F,(1±ßF,)

(29)

ilF,,2

It is expected that thegeometrical part will have a more or less constant value, similar to the shape factor method of Tuck. Similar to the technique which isused by Schmiechen the geometry should have the same contribution and the sinkage is matched or tuned at F,, =WJ20.7. So by thisnew method the sinkage prediction can be calculated by:

cbBT 2F,(l+F)

s =2.4

_____

(30)

L

_3JlF,

To avoid misunderstanding, this prediction method will be calledde Koning Gans' or shortlyHdKG'. Also the Froude depth numberdependent part functionis plotted in figure 2. The newly calculated function approximated almost the Froude depth number dependent part function of Schmiechen. Sothis newly derived function will also givea goodcomparison with the model tests.

Fraude depth number dependent orvelocity dependent part of the different methods is alsopresented in figure 2. Itappearsthat the originalTuck meihodoverestimatethesinkage in comparisontothe Tuck method according to Schmiechen The new method, called method one or method Tuck according tode Koning Gans', has almost the same tendencies as the method of Tuck according to Schmiechen. From Schmiechen method it is known that the predictions of the sinkage are in good agreement with model tests. So the methodTuck according to the KoningGans' shouldhave the samepredictions. Also the Tuck method is presented where this function is asymptotic with the new methodat lowFroude numbers. To:match the shallow waterequation,theshape function or the geometrical part has to be multiplied with 2/3rn In this case the Tuck equation (dashed plotted line) hasthesame tendency at low Fraude depth numbers as the new method.

6.4.2.

Method two

According to Tuck and Taylor the sinkage is predicted as:

s = (1) CbBT F,

L _{\/1_Fh2} (31)

Here a is a constant coefficient and is determined after many experiments. The value of a = 2.4 and corresponds almost exactly with a Froude depth number of Fh = 0 7 Looking at equatIon (25) the ship is immersed with s Relativetothestill water the immersed depth of the ship is now:

T2 =T+st1

(32)

Sinkage predictionsaccording to several methods

w C) C (n V_w cl) ce E

o

z

1.2 0.8 0.4 o o

y=2/3Fn2/'I(l -Fn2) Linear Tuck adapted t9 low Foude depth numbers y=2/3*Fn (1+Fn )/'I(l-Fn ) Non Linear Tuck according to de Koning Gans 2Fn3 Tuck according to Schmiechen Fn2/'I(1-Fn2) Tuck (Original)

/

(33)

9

Fn (Froude Number)

Figure 2 Comparation of several Froude depending part functionstopredict squat.

So theblockage is increased with s1 and thetotal draft is 2)This means.that due to the sinkage itself the

blockage increases. This increased blockage or new total immerged depth 2)_{can be substituted in the original}

depth Tof equation (26). Sothe newly calculated sinkage now is:

s=s

CbB(T+SW) F,

T

_J1.Fh2

Now the first iterative sinkage of equation(26) can be substituted in equation(33). So the nonlinear predicted squat is:

s=s

=afT(l+C!JBT

F, F, 34

T T

jlF,

_JlF,

To avoid misunderstanding, this prediction method wilibe called 'de KoningGans2 or shortly HdKG2. Nowthe geometry coefficient a canbe determined. When equation (34) ismatched with the Tuck equation (26) at the Froude depth number of h= 0.7, it appears that a= 2.16

6,5. Remark on the geometry coefficient

From three dimensional panel methods it appears that the value of a shouldbe a=l.45. Compared to a=2.16 this is a big discrepancy (±40%). Apparently the present of boundary layers in model scale testsand full scale has a

biginfluence on thesinkage. Further investigations.ofthepanel method and the present boundary layersshould lead to a more clear understanding of this issue.

6.6. Discussión

The empirical equation (30) is physically more realistic (rather than the empirical equation(28) of Schmiechen). The Froude depth number dependent part only exists of even functions (andso the squat calculations will give the same sinkage, when the ships sail ahead or when theships sail stern). Furthermore this equation'(30) almost approximated the Schmiechen prediction. Dueto the fact that the squat has the behavior accordingtoequation (30) and the Schmiechen function almost overlapsthis function, the resultsof Schmiechen agree rather well with measurements.

The methods oneandtwo canalso be applied for threedimensional calculations. Thederivationsare similartothe two dimensional method. So when the panel methods are used also the nonlinear methods one or two can be applied. For convenience when usingthe panel:methods, the methods one and twoarealso called HdeKG' andHdeKG2.

7. Panel methods, the three dimensional method

In the last part of the previous century numerical methods;havebeendeveloped to calculate the potential flow, withor without wave pattern around ship hulls. The advantage ofnumerical methods is that the velocities and pressures canbecalculated ateachpoint of the wetted: surface of the ship hull as wellasat the free srface. Also

the forces and momentscan be determined by integrating:thepressure with respect tothe wetted hull. So more

and new insights concerning thebehavior of the flowcan be analyzed. Thetests can becarried out withor

without free surfaces effects. When the geometry or flow direction is changed a quick calculation can be made in order to gain more insight regarding the characteristics of the flow. At the TUDeIft a numerical method based program was developed. Thisprogram is based on a panel method using a Dirichlet boundary condition. The panels are placed on the wetted hull surface and onthe free surfacearound the ship. The program can predict the double body flow when panelsareonly placed on the wetted hull. This program can also be used for wave pattern calculations, but as it will be mentioned insection 7.4, wave pattern calculations arenot important for problemsconcerning sinkage and trim. The model is:based on apotential flow. Some important properties and characteristics of the potential theory are discussed below.

For the simulation of the flow around a floating object a simplified flow model is used. One of them isa

so-called potential flow model. This model is atheoretical flow model and has some important characteristics which canbe used for calculations of the flow. One of thesecharacteristics is that an integral method can be usedfor simulating the flow. Now some of the characteristics thereof will be discussed. The potential flow is assumed to simulate an incompressible and inviscid flow around a profileor body. This meansthat only mass inertia forces are simulated andnodrag can occur. And also thismeansthat there will beno vorticity in the ow. Due to the fact that the flow is incompressible, the flow will be divergence free. The meansthat in the flow no material will becreated nor destroyed and is called the equation.ofconservationof mass. Now the potential flow can be described as a solution of the Laplace equation:

a2 a2

(35)

a.2 ,2 az2

This isa homogeneous and linear differentialequation and so the superimposing property can be used. With this property, it is possible to transform the Laplaceequation into the Fredholm's integral equation with help of Greens theorem.

7.1. Integral equations

For solvingthe potential; the Fredholm integralequationsof a second kind are used. On the surface of the body a source and a dipole layer will be placed Thus the following integral equation is used with the Greens function which represents a physical source:

()

JJ(7(1) + «

<.n.r >

(36)

4JD-fi

In which r is the distance between an arbitrary point and apoint at the surface Q. The surface Q is divided into the wetted surface of the double body Q and the free surface FS. To solve the integral equation the inner

Dirichlet boundary condition isused. The source strengthüat the fixed wetted surface Q 8then is:

11

an (37)

7.2. Boundary conditions

Forsimulating the flow with a potential flow solution a number of boundary conditions have to be obeyed The

boundary conditions have to be obtained at the wetted surface of the ship body, at the free surface and faraway

in front of the ship the wave height and the derivatives thereof will bezero. Whennofree surface flow is obtained, all the boundaryconditions can be applied asno leakage condition or kinematic boundary condition. The flow at the solid body Q will be tangential to the surface. So there is no velocity component ininormal direction Thiscan be written as an inner product:

<Vct.ñ>=O

(38)

7.3. Pressures and forces

The pressure on a determined on any particular point can be determined with the Bernouilli equation:

(39) The constant is determined at infinity far in front of theship where the water is in rest and so the perturbation velocitiesand perturbation potentials are 0. (C=½pU2) The velocities in this equation (38) are determined from equation (35) by taking the gradient of the potential. By integratingthepressures from equation (38)over the ship surface the forces F,..F3 are determined:

=

¡f- pidfZ

for thefirsi 3 components (40)

n

The moments on the ship M1..M F4..F6 are determined by using the cross product of the normal'vectorsand the

distance vector fromthe ship center and these results are integrated over the ship surface:

=

_{¡J- pñxFdí}

for the last 3components (41) To find thesinkage and trim!the equations(l4)and ('15) can be used by using in the values of respectively F3 and F5

7.4. Panel methods for squat calculatiOns

For free surface calculations including the wave pattern the speed of the ship hastobe considerable The most wave pattern prediction programs require a panel density of 12 panels per wavelength and for an accurate

calculation the programs require even a density of 20panels. When a ship sails with a speed Of 10 knots=

5. l44m/s the generated wavelength is:

2

2212 17.Om (42) This means, that the panel length atthefree surface should have a length between l.Om to 2.Om. For a stable and a realistic wave pattern calculation the free surface has to be panelized almost ahalf of the ship length in front

of the ship and a ship length aft of the ship. For a ship of a length of 300m or more at least 500 panelshas tobe

generated in longitudinaldirection. Fora moreaccurate calculation the number of panels should beeven higher than 1000 panels. Also incrossdirection the panel width should bethésamedimension as the panel length. So a lot of panels in width direction haveto be generated too. The panel distribution on the ship hull should have almost thesame measurements as the freesürface panels. This means, thatan enormous number of panels hasto be generated (±106). Further the matrix isa full matrix and it has to be stored inside thecomputer. Thisrequiresa lot of computer memory, which is not available. Also due to the large amount of numbers the inversion process will deliver poor results, dueto abadconditioned matrix (the condition of a matrix is also 'linear withthe number of rows), so that no reliable or unstable resUlts may occur. So fromthispoint of view it is not recommendable to use a free surface calculation.

Fortunately with thisvery low speed of the ship the wave effectsonsquateffectsareneglectable. The wave influence in verticaldirection reduces very rapidly and atone wave lengthunderthefree surface the influence is approximately 0.2% of theeffectat the free surface. So the bottom of the shipdoesnot feel muchof the disturbed pressure of the generated Kelvin waves. Also the amplitude of the wave is very low. So actually aLlow speeds of the container ships the free surface waves or Kelvin waves do not influence the squat. For the panel method this means that the free surfacesdo not have to be modelled.

7.5. Adaptation of the panel method for shallow water

The above methods described that the sinkage is consists of a Froudë depth number dependent part and a geometrical or shape oftheshiphull. So the squat canbe calculated according to s=f (Fh) g (Geom) . So for the linear squat calculationsusing a panel methodthe same principle is used. When onlythepanel method in the original caseis used, the surfaces of the bodies are rigid. Furthermore the free surface is removed and the calculations are done by a mirrored body model where the hull and the bottom is mirrored in the free surface The calculated pressures are proportional to the-squareof the velocity. From this the forces, moments, sinkage and trim are also proportional to the square of the velocity p, F, M, s, t -v2-Fh2. But by using thepanel methodthe shallow waterequation is notbuilt in. Tuck has deriVedthat forshallow vater problems the potential equation has to fulfill the shallow water hydrodynamics So from the shallow water hydrodynamics equation the calculatedipressures, forces and moments are proportional to.p,F, M, s, tFh2/f(l Fh2) .. In this equation and model the flow around the ship is calculated in a situation where the ship remains in the same position To get an improved situation the ship has to be movedinto the neW position and again thesamecalculationshave been doneagain. The adaptation of this process will be described now. The Froude depth numberdependent part of equation(30) is derived whilethiscalculation takes note of the fact thattheship will notremain atthefree surface position, but thattheshipviill movetoanew position which will also give an;extra force. This process also occurs when the calculation is made bypanel methods. So the Froude depth number dependent part function of equation. (30) can also beaplied for panel methods:

p,F,M,s,toc

F(l+F)

(43)

Il

-

Fh2

When theinitial sinkagecalcülationis basedonthe linear theory the squat isproportional to the square of the Froudedepth number, then the most realistic squat prediction of Ïhe panel method for thesinkageand trimgives:

p2 (i) _F2

s,1= f(Fh,Geom) = " (1+s h )g(Geom) (44)

T

8.

The squat results of the linear panel methods

81. The dimension of the coñtainervessels

Thecalculationshave been carried out for threedifferenttypesof container ships and three different canal depths The type of containers ships are respectively the so called the Post Panamax the JUMBO and the

'MEGA JUMBO'. The canal depths are relative to each type of container ship and are respectively 1 .Om, I.5m

and 2 0m deeper then the draught of the container ship So the initial keel clearance (this is the keel clearance

when the ship is in rest)is respectively l.Om, l.5mand2.Om. Thedimensions and the specific shipparameters

are shown in tablé 1.

12

Ship Dim. Post Panamax JUMBO MEGA JUMBO

Container Cap. [TE ii] 6500 8700 12500

Lengtho.a. [m]

3l80

346.0

3820

Length p.p. [m] 3020 330.0 364.9 Breadth B [m] 42.9 42.9

570

Depth H Em] 24.1 27.0 30.0 Draught T [m] 14.0 14.5 1.7.0 Displacement V

[mi

l2 1989 142004 242911 Block coef.Cb

[I

0.6726 0.6918 0.6870 Slenderness coef. A

_{= *}

_[]

_6.089 6.325 5.848 Loc. ofcenterofbounancy [ml -4.50 -4.95 -5.65

Table 1 Dimensions and specific parameters of the tested container ships.

8.2.The sinkage fôrce, trim moment, sinkage and trim

Topredict the sinkage force trim rnoment,sinkage and trim a panel method is Used. The panel method:calculates an inviscous irrotational flow. Soonly the inertial forces of the mass of the water are simulated From;thepanel method calculations, itappears that the sinking force and thetrimming moment become greater when the keel

clearance is smaller. Thismeans that the panel method gives aconsistent result. The reliability of the resultsis

moreconcerning. The keel clearances are very small and the panels of the canal: bottom and theshipbottomare

very close to each other. Actually more panels and also smaller panels have to' be used for proper calculations, but then the number of panels increases too much for a convenient calculation. (The numberof matrix coefficients increases too and more floating points operations have to bedone on the same matrix components. From this the truncation errors of the inversion process will occur and become greater.) For some calculations some other gridshave been distributed to obtain the sensitivity of the panel:distribution. It appears that the truncation error of sinkage force is within 2.5% for every case. The truncation error f the trimming moments

differs more and is about 10%. The moments differ due todifficult flow around the bulb and around the _{stern at}

thelocation of the propeller plane. These are areas where the distance to amidships is very large and havea large influence on the trimming moment. So for a proper calculation also a closer distribution of panels has to be obtained at theseplaces But for squat calculations the interest of howthe flow is streaming around the bulbous bow is less interesting; only the force or pressureisof importance.

9rciUeREFleBmo SrctttraJJoBD

Figure 3 The sinkage of several container ships proportional to the square of the velocity.

are the AtPanara.x 9rKe of the IOEGAJ&JO thnkage of the MEGA J1EIB0

L

a.

lIdee2Om

bideeal.5m

it1dee 10m

defined from amidships

02 Lo. htifIdeeaanaom lrthIdeeraan 15m 10m L

9rIraei tira IEGAJlidB3

-- miSal Keel clearance 20 n Initial Keel clearance 1.5 n

-

Initial Keel clearance 1.0 n

Lta1cthI eedLkmlsl Speed Lirmotal

Figure 4 Thesinkage of several container shipsincluding the shallow water equation and precalculated sinkage

13

Lb E%l (=-l.36) (=-l.50) (=-1.55)

Waterlinearea

_k.

Em2] 10393 11554 16902

Waterlineareacoef. CWL

[j

08022 0.8 161 0.8 126

Loc.of waten. cent. XwL [m] -14.17 -14.72 -16;44

[%] (=-469) (=-4.46) (=-4.5 I) Static moment of W.L.' S0 [m3] -147284 -170073 -277919 Mom, of inertia of W.L.' 1 [m4] 5724252 77479912 13818744 02 E e. 02 03

The results of calculations of the sinkages of the three vessels are shown in figures 3 and 4. The trim is not presented, because from the calculations it appears that the trim angle is almost zero for all cases. The figures

give the sinkage as functions of the velocity (inknots).The first figure 3 shows the sinkages according to the

formulation that the sinkage is only proportional to the square of the velocity or to the square of the Froude depth number. The last figure 4 shows the sinkage according to equation (29), in which the sinkage is corrected for the shallow water theory and the predicted sinkage according to the linear theory (method of de Koning Gans'). At thespeedoflø.Oknotsa marker is set which corresponds with the values where the calculations are carried out.

Further a presentation is given in figure 5. This figure gives the pressure distribution along the hull and bottom of the canal. Also the velocity distribution is given at the bottom of the canal. The first figure gives the whole calculation domain. From this figure it is interpretated that the flow characteristic far from the ship is not so very important. So for the presentation of Figure 5 the most important zone is zoomed in. From the figure, it can be seen that higher pressures occur at the front zone and in the aft zones of the ship. In the middle of the ship

(a) lower pressure zone(s) occur(s). Also it is good toseethat there occur cross flow components. These cross

flow components are calculated with the panel method, but the (2D) theory of Tuck does not include these components. Also the flow in the gap between the bottom and ships keel is not implemented in Tucks method. From this viewpoint the panel method should give better results.

04UlCJlC)

o Cil 0010010 Cil O Ch 00 00 Cil 0 Cil 0010010 Cii O

000000000000 000000O0000O

00000000000

00000000000

Figure 5 Example of the pressure and velocity distribution on the bottom and pressure distribution on the wetted ship hull for the MEGA JUMBO. The unit of the pressure is Pa

8.3. Conclusion

From the experiments it is found that the squat effects of large container ships are considerable and these effects have to be taken seriously. The (initial) keel clearance and ship speed has a enormous effects on the actual keel clearance. Also the dimension of the container ship influences the squat and the trim can be going from by head to by stern. To use a theoretical model the method of Tuck according to Schmiechen is superior to other methods and this method can give a good approximation of the sinkage.

9. Comparing several methods with each other

The results of calculations of the panel method are compared with other prediction methods and are presented in figure 6. The prediction methods are method of Tuck, Tuck according to Schmiechen and Tuck according to de Koning Gans. From this figure it appears that the sinkage of each method is of the same order. Only the Tuck Method gives a sinkage prediction that is too large, as already mentioned by Schmiechen. The emperical

equation made by de Koning 5fl5 almost approaches the method of Schmiechen. The results of the method de

Koning Gans' has a more physical interpretation and might be favorite. The calculations done by the panel method and adapted for the shallow water theory and the linear sinkage also give results which are in good agreement with the methods of Schmiechen and de Koning Gans'. For the Post Panamas and JUMBO the sinkage is less than the method of Schmiechen and de Koning Gans but for the MEGA JUMBO the sinkage is more than the method of Schmiechen and de Koning Gans'. The differences (only a few centimeters) are not

very large. Looking at the dimensions of the ship (length 300+ m) this margin is almost neglectable. The panel method takes into account the real geometry, while the other methods predict the sinkage with the aid of a very simple equation. So the panel method should be favorite. Also the panel method gives the same trend as the other methods. The only uncertainty of the panel methods is that the panels of the bottom of the ship are very close to bottom of the canal. But by doubling the number of panels it appears that the forces are almost the same as with a rougher grid. Further all potential flow models methods do not include viscous effects and are therefore not taken into account. Tuck according to de Koning Gans2 gives almost the same result as the panel methods. The only coefficient in Tuck according to de Koning Gans2 is matched according to the panel methods and this coefficient is less when matched with the Tuck and Taylor method at Froude depth number equal to Fh = 0.7.

(a=l.45

matched with panel method and a=2.16 matched with Tuck method at to Fh = 0.7). Squat prediction of different methods

5

Velocity [knots)

9.1. Conclusion

From all presented results, it is difficult to say which method is the best. Using the Tuck method according to Schmiechen the results are comparable to model tests and the extrapolation from panel methods with the shallow water equation or Tuck according to de Koning Gans gives almost the same results. But compared to the results from model tests the method gives apparently too much sinkage. This is due to the present of the boundary layer. From all the presented results it appears that the tendency is almost the same, but Tuck (the shallow water equation) has a higher value at low Froude numbers. Maybe the Tuck method is correct at low Froude depth numbers, but in this case the shape factor is too large in this region of Froude depth numbers.

10. Conclusions

Squat is a non linear phenomena and when the keel clearance becomes small the squat effects becomes larger. Squat effects can be predicted by several developed methods. These methods predict the sinkage as well as the trim. The methods are:

One dimensional flow model.(Kreitner) Two dimensional flow model.(Tuck and Taylor) Three dimensional flow model (Panel method) Model tests

Full scale tests

Predictions methods: Tuck

Schmipchen HdKG'

HdKG(a2.16) (corrected for Tuck) HdKG (a=1 .45) (corr. for panel meth.)

Panel methods:

Panel Method acc. square iaw Panel Method acc. shallow water eq. Panel Method acc. HdKG'

Panel Method acc. HdKG2

3

10 15

Figure 6 The sinkage of a Mega Jumbo container vessel with an initial keel clearance if 2.0 m. In this figure the sinkage is shown as a function of the velocity according to different methods and panel methods with a different adapted Froude depth number dependent part.

15

Now the most importantand/or useful methods.are discussed.

Theitwo dimensional methodof Tuckand Taylor gives agood approximation of the squat prOblem. AlsO non-linear effects canbe takenilitoaccount. It is not necessary to knowthe flow distribution. The equationof Tuck and Taylorpredictsthe sinkage:as well as the trim, by using integrals, which are directly related to the pressure, compared withother methodsandexperimental data, the method of Tuck and Taylor is very superior. The only problem is that the gap between the hull and bottom of the canal hasnodirect influence on the squat calculations

Thepanel methods which are presently available can give reliableresults for the squat magnitude. When the trimandsinkage are iterated non-linear effects can be taken into account too. With the panel method the flow effect iñ the small gap between the bottom and the keel of theshipcanbe determined. This effect is not present in the method of Tuckand Taylor.

The panel methods with free surface flow are less iineresting because they require alarge number of panels at the freesurface, which gives an unstable solutionor thecalculatiöns will be verytime consuming. The calculations have to be applied in the so-called double body flow, without panels at the free surface. Duetothe small Froude number (based on the ship length)the wave phenomena play a very small role. This meanspanel methodscan be applied for the squat effects and willgive very accurate results.

From the research on the sqUat according to different methods and the (non) linear panel methods some tendencies and conclusions areobtained. It appears that the panel method gives a consistent solution. When the initial keciclearance becomes smaller, then the squat effects become larger. The panel methodsaremore sophisticated than the Tuck and Taylor method, because the panel method also takes the gapbetween ship bottom and canal bottom into account.

The panel method gives less sinkage than the predictions methods of Tuck and Tàylor, Schmiechen and de Koning Gans'. From the literatùres[22], theprediction methodsarecompared with tank results The fact that the panel methods give less sinkage is most probably due to the lack of the presence of a boundary layer. Also truncation errors of the panel method can give small errors.

In full scale conditions the boundary layersare relatively less thick. So the influence on the squat due to the boundary layer is less than that according to model tests. The predicted squat at full scale isprobàbly in between

the results of the panel methods and the prediction methods of Schmiechen and de Koning Gans'.

11. The bibliography

Barrass, C.B.ShipSquat - A Reply',The NavalArchitect, November 1981

Beck, R.F., "Forcesand Moments on a Ship Moving in a Shallow Channel", Journal of Ship research, Volume 21, No 21 June 1977, pp 107-1 19

Beukelman, W.,"De Invloed van Trimopde Richtingsstabiliteit van een RO-RO-Schip in Ondiep Water, Report No. 854-E, Laboratory of Ship Hydromechanics, TUDeIft (Delft University of Technology), Januari 1990, DeIft

Beukelman, W.,"The Influenceof Trim on the Directional Stability of a RO-RO-Ship in ShalloW Water', Report No. 854-E, Laboratory of Ship Hydromechanics, TUDeIft (Delft University of Technology),May 1990, DeIft

BeUkelman, W.,"Hydrodynamic AspectsOf Marine Safety", ReportNo. 921-P, Laboratoryof Ship Hydromechanics, TUDeIft (Delft University of Technology), June 1992, DeIft, First Joint Conferenceon MarmneSafety an Environment, ShipProduction, l-5June 1992, Delft University of Technology [61 Beukelman, W.,"l-Iydrodynamic Aspects of Ship Safety", Report No. 934-P, Laboratory of Ship

Hydromechanics, TUDeIft (Delfi University of Technology), June 1992, DeIft, Transactionsof First International Conference in Commemoration of the 300-th Anniversary of CreatingRussian Fleetby Peter theGreat, 7-14 June 1992, St Petersburg, Russia

Breit, SR.,"The Potentialof a Rankine Source between Parallel Planes and in a Rectangular Cylinder", Journal of Engineering Mathematics 25 pp.1 51-163, Kluwer Academic Publishers, Printed in the Netherlands, 1991

Flügge G. & Uliczka, K.,"Fahrverhalten grosser Coñtainerschiffe inextrem flachem Wasser, Das dynamische Fahrverhalten und die Wechselwirkungen mit der Fahrrinennsohle von sehr grossen Containershiffen unter extremen FlachWasserbedingungen.",

Hansa-Schiffahrt-SchifthaUHafen-138 Jahrgang-200l-Nr 12

Hermans, A.J., "Ship Manoeúvringand Hydrodynamic Forces Acting on Ships in Confined Waters", Proceeding of Symposium on Aspects of Navigability, Delft, 1978