481
1 INTRODUCTION
The main commonly discussed features of maritime
transport are usually its safety and effectiveness.
Among them the ship safety issues are crucial from
the operational point of view and they can be
considered as one of the most prospective technical
affairs. One of the most critical features of
seagoing
shiprelatedtohersafetyisthetransversestability.
Ship stability is a term used to describe the
tendencyofashiptoreturnback toherequilibrium
when she is inclined from an upright position
(Kobyliński&Kastner2003).Sincetheinitialposition
ofashipis
notalwaysuprightone,themorepractical
definitionstatesthatthestabilityisafeatureenabling
to perform, when remaining in determined position,
the task she is constructed for. The complementary
definitionsleadtopointoutthatthestabilityofaship
is an element of her operational safety qualifying
factors.
Ship stability performance depends on two main
factors – a shape of her hull and a weights
distribution.Thefirstoneisaconstantvalueinshort
and moderate term and can be changed very rarely
during rebuilding of a vessel. However, the weights
distribution changes in every port due
to cargo
operations, bunkering and related to both of them
ballastoperations.
The particular sort of changes in weight
distributiononboardisliquidsloshingtakingplacein
partlyfilledtanks.Movingmassesneedtobeavoided
onboard,thoughitisimpossibletoevadethematall.
Thecargosecuringproceduresensure
alackofloose
cargo onboard but some free surfaces of liquids in
ships’tanksareinevitable.Thecrucialgroupoftanks
onboardshipswhichmaybepartly filled are ballast
tanks. The problem of an assessment of liquid
sloshingeffectisnowadaysmoreimportantthenever
because of the
obligatory ballast water management
requirement. The most common way of maintaining
The Impact of Sloshing Liquids on Ship Stability for
Various Dimensions of Partly Filled Tanks
P.Krata
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Liquidsloshingphenomenontakingplaceinpa rtly filledships’tanksdirectlyaffectsthestability
of a vessel. However, only static calculations are carried out onboard ships nowadays and static transfer of
liquid weight is taken into account in the course of routine stability calculation. The paper is focused on a
dynamicheelingmomentduetoliquidsloshingintanksonboardships.Anumberofnumericalsimulationsof
liquid sloshing taking pla ce in a moving tank is carried out. The wide range of ship’s tanks is taken into
account.TheconductedCFDsimulationsareexperimentallyverified.Finally,themethodofan
assessmentof
theliquidsloshing impactonshiptransversestabilityisworkedout.Thekeypointofthemethodisadynamic
coefficientdescribingrelationoftheresearcheddynamicheelingmomentandthequasi‐staticoneintermsof
dynamicstabilityofavesselwhichisrelatedtotheweathercriterion
ofshipstabilityassessment.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 7
Number 4
December 2013
DOI:10.12716/1001.07.04.02
482
ballast water clean and safe for the overseas
environment is exchanging it during a voyage of a
vessel.Thisoperationcanbedangeroustothevessel
and the fair example of such a danger may be the
capsizing of M/V Cougar Ace. She lost the stability
duringballastwater
exchangingoperationtiltingher
significantlyonheavyswellwhichresultedthecargo
shiftandfinallylayingonherportside(Davis2008).
The seagoing vessel’s stability calculation and
evaluationmade onboard nowadays is based on the
prescriptive stability criteria published by the ship’s
classification societies (Kobyliński & Kastner 2003).
These
criteria are mainly based on the A749(18)
Resolution of International Maritime Organization.
Theresolutionandtheirlateramendmentsareknown
astheIntactStabilityCode(ISC2009).
Theshipstabilitycriteriaqualifytheshapeofthe
rightingarmcurve.Inaddition,theweathercriterion
is to ensure the sufficient stability
of a ship to
withstandtheseverewindguestsduringrolling(ISC
2009).Although the weather criterion reflects a very
simple model of dynamic ship’s behavior, just the
static stability curve is used. Anyway, the weather
criterion is the only, which is partly based on the
model of heeling phenomenon not
only on the
statistic data, while the remaining criteria are based
on the statistics of historical disasters only
(Francescutto2002).Themodernandstilldeveloping
approach towards ship stability qualification is an
implementation of performance‐based stability
criteria in the future. They are based mainly on the
riskassessment(Kobyli ń
ski&Kastner2003)however,
itisstillfarfromcommonuseonboardships.
Regardless the approach towards ship stability
evaluation, the physical background of phenomena
takingplaceonboardoughttobetakenintoaccount.
In case of contemporary prescriptive stability
standards, the righting and heeling arms need to be
obtained
and compared. Then the work of the
righting arm enabling accumulation and then
dissipation of the energy could be compared to the
energyprovidedtotheshipbyexternalforceswhich
is called the energy balance method for dynamic
stabilitycalculation(Kobyli ński &Kastner2003).The
balance of righting
arm (righting moment) and
heeling arm (heeling moment) shall comprise all
significant components of each moment and among
otherstheheelingmomentduetoliquidsloshingina
partlyfilledmovingtanktoo.
In the light of ship stability related concepts, the
accuracyofship’stransversestabilityassessmentisan
important
probleminvesselsoperationprocess.Both
approachestowardsshipstabilityassessmentknown
nowadays call for characteristics of heeling moment
duetoliquidsloshingintanks.Thisneedjustifiesthe
research program focused on the liquid sloshing
phenomenon.
2 FREESURFACEEFFECTANDLIQUID
SLOSHINGPHENOMENON
The intact ship stability assessment
is generally
carriedoutonboardonthebasisoftheIMOIS‐Code.
Thus, the standard stability measures like a
metacentric height, righting arm curve etc. are in
common use. According to the IMO
recommendations the righting arm curve shall be
corrected for the effect of free surfaces of liquids in
tanks. The correction may be done by one of two
acceptedmethods(ISC2009):
correction based on the actual moment of fluid
transfercalculatedforeachangleofheel;
correctionbasedonthemomentofinertiaoftank’s
horizontalprojection(simplependulummodel).
Bothmentionedmethodsoffree
surfacecorrection
calculationconsiderthequasi‐staticattitudetowards
thesloshingphenomenononly.Consequentlytheydo
notconsiderthelocationoftankswithinthehullofa
shipandthelocationoftherollingaxis.However,the
main advantage of currently applied compulsory
correctionsisthesimplicityoftheircalculation.
Regardless
the explicit computational IMO‐
recommendedformulaforfreesurfacecorrection,the
liquid surface is always assumed flat and depends
onlyonanangle of ship’s heel nottime. The ideais
presentedinthesketch(Fig.1).
Figure1. Quasi‐static transfer of fluid mass due to ship
heeling
The liquid sloshing phenomenon takes place in
partlyfilledshipstanks.Asatankmoves,itsupplies
energy to induce and sustain a fluid motion. Under
external large amplitude excitations or an excitation
near the natural frequency of sloshing, the liquid
insidetankisinviolentoscillationswhichisofgreat
practical importance to the safety of the liquid
transport (Zeineb, Chokri, Zouhaier, Khlifa 2010).
Both the liquid motion and its effects are called
sloshing. The interaction between ship’s tank
structureandwatersloshinginsidethetank consists
in the constant transmission of energy (Akyildiz &
Unal2005).Theexemplarygeneral
viewontheliquid
sloshing phenomenon taking place inside a model
tank swinging during an experimental research is
showninFig.2.
Figure2.Exemplaryshapeofafreesurfaceofliquidinside
apartlyfilledmodeltank(ownresearch)–itmaybeclearly
483
seen that the free surface is far different than assumed
withinthequasi‐staticapproach(ref.toFig.1)
The characteristics of heeling moment due to
liquidsloshingdependonavarietyofparameters,for
instancetank’sgeometry,itsfillinglevel,locationofa
tankwithinahullofaship,rollingperiodandothers.
The pre‐study carried out in the course of the
research enabled the classification of
typical shapes
and dimensions ship tanks (Krata, Wawrzyński,
Więckiewicz, Jachowski 2012). The most often
designedshapesareshowninfigures3,4and5.The
double bottom tanks belongs to standard
arrangement of all seagoing ships, while the side
tanksandwingtanksaretypicalforsametypes
only.
Figure3.Typicalsidetanksandwidedoublebottomtanks
Figure4. Division of side tanks and double bottom tanks
typicalforlargeships
Figure5.Wingtanks,bilgetanksanddoublebottomtanks
ofbulkcarriers
Theearlierauthor’sresearches revealsthat liquid
sloshingdynamicsinshipsnarrowsidetankscanbe
neglected actually. The natural period of liquid
sloshing is short enough to justify the quasi static
calculationofthefreesurfaceeffect.Thisapproachis
wellknownandroutinelyappliedinthecourseofthe
stabilityassessment(ISC2009).Aliquidcontainedin
partlyfilledsidetanksremainsinfacthorizontaland
flat within the ship rolling cycle, which is shown in
figure 6. The remaining tanks, e.g. double bottom
tanks and wing tanks need to be the subject to
examineintermsofpossible
sloshingcharacteristics.
Figure6.Thesurfaceofliquidsloshinginpartlyfilledhigh
side tanks (Krata, Wawrzyński, Więckiewicz, Jachowski
2012)
3 SHIPANDTANKGEOMETRYAPPLIEDINTHE
COURSEOFTHERESEARCH
The tanks onboard seagoing ships can be classified
according to their purpose as follows (Krata,
Wawrzyński,Więckiewicz,Jachowski2012):
trimming tanks (fore and after peaks) which are
utilizedveryoftenaspartlyfilledduetothe
need
forprecisetrimmingofaship,sotheballastwater
level is adjusted according to the variable
requirements, thus providing free surface of
liquid;
stability tanks improving ship’s stability
performance due to a decrease in the vertical
center of gravity (usually double bottom tanks
locatedbetween anengine
room and a fore peak
createsthisgroup);quiteoftenthebreadthofthese
tanks equals half breadth of a ship or even
sometimes it equals full ship’s breadth (in ship’s
foreregion)thereforethefreesurfaceofliquidcan
bemassiveinthesetankssogenerallytheyshould
befull
oremptyduringvoyage;
list control tanks (side tanks) which are usually
locatedamidshipsandduetotheirfunctionquite
oftenpartlyfilledwithfreesurface;
strength control tanks utilized to adjust
longitudinal weight distribution (fore and after
peaks, double bottom tanks, side tanks and
sometimes even cargo
holds prepared for
ballasting) which are very often partly filled to
reduce excessive sheering force and bending
momentandroutinelytheyprovidefreesurfaceof
liquid;
specialpurposetankslikeforinstanceanti‐rolling
tanks (flume) or anti‐heeling tanks, which are
usually filled up to the 50% level, providing
free
surface.
Althoughmostofthementionedtanksareusually
filleduptotheirtoporalternatively,theyareempty
during ship voyage, there are some relatively long
periods when they are only partly filled. The most
obviousisballastwaterexchangeappliedtomeetthe
requirements of ballast water
management
instructions providing protection of natural
484
environment. During the time of such operation the
sufficientstabilityofashipoughttobemaintained.
Regardless the exact purpose of ballast tanks
onboard,theirtotalvolumeandresultingfromittotal
weightofballastwaterissignificantwhichisshown
in figure 7. This justifies focusing on this
group of
tanksinthecourseoftheconductedresearch.
0
50
100
150
200
250
0 20000 40000 60000 80000 100000 120000
Size of a ship (displacement when fully loaded [t])
Weight of ballas water / lightship
displacement [%]
bulk
carriers
other ships
Figure7. Relation of ballast water weight to the lightship
disp.(Krata,Wawrzyński,Więckiewicz,Jachowski2012)
Thegraph(Fig.7)revealsthatthetotalweightof
ballast water carried onboard may reach and even
exceed the lightship weight among ships other than
bulk carriers. Moreover, ballast water weight can be
twice a lightship in case of bulk carriers (Krata,
Wawrzyński, Więckiewicz, Jachowski 2012).
Obviously not
all the tanks are partly filled at the
sametimebutsomeofthemcanbesowhichcreates
the need for stability calculations comprising the
phenomenonofmovingliquidsintanks.
InthecourseofthestudyatypicalPanamaxship
was taken into consideration. In case of quasi
static
approach to the free surface effect the location of a
partly filled tank does not play any role. Reversely,
the dynamic approach is related to ship rolling and
the location of considered tank is crucial. Therefore
notonly the dimensions of a model ship need to be
specifiedbuther
rollingaxisaswell.Theparticulars
appliedintheresearcharegiveninthetable1.
Table1.Maindimensionsofconsideredships
_______________________________________________
Shipparticulars[m]
_______________________________________________
breadth32,00
height20,00
elevationofrollingaxis9,00
_______________________________________________
Figure8.Arrangementofconsideredtanksinship’shull
Thelocationofanalyzedsampletanksisshownin
Fig.8.Theheightandbreadthofconsideredtanksare
namedbandh,respectively.
In order to carry out an analysis the considered
modeltank has varyingsizei.e. itsbreadth b ranges
from5to10metersandits
heighthrangesfrom1,5
upto4meters.Thefillinglevelofthetankequals50%
inallconsideredcases.
4 COMPUTATIONOFHEELINGMOMENTDUE
TOLIQUIDSLOSHINGINSHIP’STANK
Theheelingmomentduetoliquidsloshinginapartly
filled tank was computed with the use of
CFD
technique.ThesoftwareFlowVisionwasapplied.The
simulationsofliquidsloshingwerecarriedoutin3D
mode for the most typical rectangular ship ballast
tank.Therollingperiodwasvariableaccordingtothe
research assumptions and the range of angular
motion reflects the very heavy sea conditions in
extremely
stormy weather. The rolling period
dependsonthestabilityperformanceofaconsidered
shipthereforethewiderangeofsuchrollingperiods
istakenintoaccountinthepresentedstudy.
Thecomputational meshappliedin the course of
the simulations was hexahedral type and related to
two coupled reference frames, the
stationary and a
moving ones which is shown in figure 9. The Sub‐
GridGeometryResolution(SGGR)wasappliedwhere
thetri angulatedsurfacesnaturallycutCartesiancells
andreconstructingthefreesurface(FlowVision2010).
485
Figure9. Computational mesh and two coupled reference
framessystems
The SGGR method is intended for an
approximation of curvilinear boundaries on a
hexahedral mesh. The method consists in natural
splitting of the boundary cells by the triangulated
boundarieswhichisshowninfigure10.
Figure10. Sub‐grid resolutionof curvilinearwall
(FlowVision2010)
Thenumberoftheobtainedchildcellsdependson
the geometry peculiarities. The child cells are
arbitrary polyhedrons. The equations of a given
mathematical model are approximated on the
polyhedrons without simplifications. The approach
enablesaccuratecalculationsinacomplexdomainon
areasonablycoarsemesh(FlowVision2010).
TheFlowVisioncode
isbasedonthefinitevolume
method(FVM)andusesthevolumeoffluidmethod
(VOF)forfreesurfaceproblemswhichispresentedin
figure11(FlowVision2010).
Figure11. VOF (Volume Of Fluid) variable is the volume
fractionoffluid1inacell;VOF=1‐thecellcontains only
fluid1;VOF=0‐thecellcontainsonlyfluid2;0<VOF<1the
cellcontainsfluid1andfluid2(FlowVision2010)
High accuracy of computation is achieved by
solving the governing equations in theʹfree surfaceʹ
cells (the cells partly filled with liquid) (FlowVision
2010).TheRANS(Reynolds‐averagedNavier–Stokes)
equation is implemented and the simulation of
turbulent flows is based on the eddy viscosity
concept. The semi‐empirical k‐ε model
turbulence
modelwasapplied.
Figure12. Computed shape of a free surface in a moving
tank(example)
Theresultofthesimulationcomprisesmainlythe
general flow pattern and the velocity and pressure
fields.Theexemplaryshapeofafreesurfaceisshown
inFig.12.
Moreover, the user defined parameter was also
computed, i.e. the heeling moment due to liquid
sloshing inside partly filled tank which is
essential
from the conducted research point of view. The
heelingmomentMvectorwascalculatedaccordingto
thefollowingformula:
S
dsp nrM (1)
where:
S–thewettedsurfaceofthetank’sshall;
r–thepositionvectoroftheconsideredpointonthe
tank’swall;
n–thenormalvector;
p–thelocalpressureonthetank’swall.
Due to the prevailing two‐dimensional character
of the considered flow in
the tank, the heeling
momentisavectorofadirectionperpendiculartothe
plane of the tank’s movement. As the transverse
486
stability of a ship is assumed to be considered, the
heeling moment may be described by one spatial
componentonly,asfollows(Krata2009):

0,0,,,
xzyx
MMMM M
(2)
where:
M
x, My, Mz – spatial components of M vector,
determinedaboutthex,yandzaxisinthereference
systemrelatedtothevessel.
Forfurtherusethesolenon‐zerocomponentM
xof
thecomputedheelingmomentduetoliquidsloshing
whichisdescribedbytheformula(2)wasnamedthe
total dynamic moment and marked M
Total_dyn. Such
heeling moment was the subject for post processing
andreasoning.
5 EXPERIMENTALVERIFICATIONOF
NUMERICALSIMULATIONS
Although the CFD‐based numerical simulation of
liquid sloshing in a partly filled tank is a powerful
technique,itstillrequiresanexperimentalverification
for some cases. Generally, the experiment is
commonly found
as an unambiguous prove for the
correctnessofnumericalcomputations.Therefore,the
experimentalresearchintothesloshingphenomenon
wascarriedoutinShipOperationDepartmentatthe
GdyniaMaritimeUniversity.
Themainpartoftheapparatusisatankequipped
with pressure transducers. The tank is forced to
oscillatingmotion
bythehydraulicdrivemechanism,
thus exciting the water sloshing inside it. The
dimensionsof themodel tank are: breath – 1,040 m,
length–0,380m,depth–0,505m.Thegeneralviewof
theutilizedtestingapparatusisshowninfigure13.
Figure13.Experimentalsetup‐generalarrangement
The experimental setup enabled to measure the
dynamicpressuredistributiononthesidewallofthe
model tank and in its upper corner. Furthermore, it
wasfeasibletorecordashapeoffreesurfaceforany
angleoftank’stilt.
Figure14. Comparison of a shape of free surface:
experiment (lower photos) and numerical simulations
(uppergraphics)
Despite the dynamic pressure distribution the
shape of free surface of liquid sloshing in the tank
was recorded for numerous runs of the experiment.
The liquid distribution and a velocity field are
governedmostlybythe inertia of liquid mass and a
pressure field. As a consequence, the correct
modelingof
liquid’sfreesurfaceemergesasastrong
proveforthecorrectnessoftheCFD‐basednumerical
simulations of sloshing flows. The exemplary
comparisonoffreesurfacesrecordedinamodeltank
during experiment and computed in the course of
simulationsisshowninfigure14.
The pressure history in the
control points of the
tank obtained in the course of the experiment were
compared to the computed ones. Then the shape of
liquid’s free surface was confirmed. Both
experimentalresultsthepressureandthefreesurface
meet relevant results achieved by the use of CFD
simulations.Consequently, theresultsofsimulations
wereacknowledgedascorrectandreliable.
487
6 EVALUATIONOFSLOSHINGLIQUIDIMPACT
ONSHIPSTABILITY
Theobtainedhistoryofheelingmomentduetoliquid
sloshing in tanks was decomposed into two
components.Thefirstonecomprisesthemomentdue
todynamicactionofsolid‐likeliquid(i.e.‘frozen’)at
an angle of heel equal 0 degrees.
The second
component of the dynamic heeling moment due to
liquid sloshing covers only the moment resulting
from letting free the liquid to slosh inside the tank
(Krata, Jachowski, Wawrzyński, Więckiewicz 2013).
All moments (components) are computed about the
shiprollingaxiswhichisfixedatthesymmetry
plane
ofavesselatanelevationgiveninthetable1forthe
consideredship.
The component containing the moment resulting
from the solid‐like liquid is included in the weight
distributioncalculation.And theremaining dynamic
component of the heeling moment due to liquid
sloshing which may be called
‘the free floating
component’isthematterofthispaper.Thecoreidea
ofthisapproachmaybeexpressedbytheformula:
TstatFLstatTotal
MMM
__
(3)
FfdynFLdynTotal
MMM
__
(4)
where:
M
Total_stat – total static moment due to a presence of
liquidwithfreesurfaceinatank;
M
FL_stat– static heeling moment dueto the weight of
frozen‐likeliquidinatank;
M
T–staticheelingmomentoffluidtransfercalculated
foreachangleofheel;
M
Total_dyn – total dynamic moment due to liquid
sloshinginatank;
M
FL_dyn–dynamicheelingmomentduetotheweight
ofsolid‐likeliquidinatank;
M
Ff–freefloatingcomponentofthedynamicmoment
duetoliquidsloshing.
The exemplary result of CFD computations is
presented in figure 15. The total dynamic moment
duetoliquidsloshinginatankM
Total_dynisobtainedin
timedomain.
-300000
-200000
-100000
0
100000
200000
300000
20 40 60 80 100
time [s]
moment [Nm]
Figure15.HistoryofheelingmomentMTotal_dynduetoliquid
sloshingintheconsideredtank–samplecase
Asthemomentaryangleofship’sheelisknowfor
every time step of CFD computations, the heeling
moment may be plotted versus an angle of heel as
well.Thisisaconvenientapproachwhichisshownin
figure16.
-300000
-200000
-100000
0
100000
200000
300000
-50 -40 -30 -20 -10 0 10 20 30 40 50
angle of heel [deg]
moment [Nm]
Figure16. Heeling moment MTotal_dyn due to liquid sloshing
versusanangleofship’sheel–samplecase
Thenextheelingmomentemergingintheformula
(4)isthedynamicheelingmomentduetotheweight
ofsolid‐like liquid ina tank. This momentis shown
foranexemplarycaseinfigure17.
-300000
-200000
-100000
0
100000
200000
300000
20 40 60 80 100
time [s]
moment [Nm]
Figure17. History of the dynamic heeling moment MFL_dyn
duetotheweightofsolid‐likeliquidinatank–samplecase
Similarlytothetotalheelingmoment,thismoment
duetothesolid‐like weight in atank can be plotted
versusandangleofheel–likeinfigure18.
-300000
-200000
-100000
0
100000
200000
300000
-50 -40 -30 -20 -10 0 10 20 30 40 50
angle of heel [deg]
moment [Nm]
Figure18. Dynamic heeling moment MFL_dyn due to the
weightofsolid‐likeliquidinatankversusanangleofship’s
heel–samplecase
Accordingto the formula (4) the core component
oftheheelingmomentduetosloshingisadifference
488
between the total dynamic moment due to liquid
sloshing and the dynamic moment due to solid‐like
weightinatanks.Theresultisshowninfigure19.
Figure19.HistoryofthefreefloatingcomponentMFfofthe
dynamicmomentduetoliquidsloshing–samplecase
However,themostconvenientwayofpresentation
ofthefreefloatingmomentisagraphplottedversus
anangleofship’sheelwhichispresentedinfigure20.
-60000
-40000
-20000
0
20000
40000
60000
-50 -40 -30 -20 -10 0 10 20 30 40 50
angle of heel [deg]
moment Mff [Nm]
Figure20. Free floating component MFf of the dynamic
moment due to liquid sloshing plotted versus an angle of
ship’sheel–samplecase
The key point of the research is an attempt to
evaluate the impact of the heeling moment due to
liquid sloshing in tanks. The most convenient
approachseemstobeasortofcomparisonofthefree
floating component of the dynamic moment due to
liquid sloshing M
Ffand the contemporary utilized
quasi‐staticmomentM
Tduetofluidtransferinatank
(refertoformulas3and4).
Addressingtheproblemof comparisonof theM
ff
and M
T components of the heeling moment due to
liquid motion, the aggregative variable was worked
out and named the “dynamic coefficient”. This
coefficientreferstothedynamicstabilityofavesselor
in other words to the weather criterion of the intact
shipstabilityassessmentwhichisbasedontheenergy
balancemethodofstabilitycalculation(Kobyliński&
Kastner2003).
In the case of asymmetric location of ship tanks,
the dynamic coefficient needs to be calculated
separatelyforaportsidetank(coefficientkd
L)anda
starboard one (kd
P respectively). However, usually
similartanksarelocatedsymmetricallyonbothsides
of a vessel. Moreover, such couples of tank are
usuallyfull,emptyofpartlyfilled.Thus,insuchcases
the dynamic coefficient may be calculated for both
partly filled symmetrical tanks simultaneously
(coefficient kd
sr). The definitions of applied
coefficientsaregivenbyfollowingformulas:


A
A
A
A
T
Ff
dMdM
dMdM
W
W
kd
TT
FfFf
PM
PM
P
0
0
0
0
_
_
(5)


A
A
A
A
T
Ff
dMdM
dMdM
W
W
kd
TT
FfFf
LM
LM
L
0
0
0
0
_
_
(6)
2
LP
sr
kdkd
kd
(7)
where:
PM
Ff
W
_
‐ work of moment
Ff
M
during ship’s
heelingtostarboardside;
PM
T
W
_
‐ work of moment
T
M
during ship’s
heelingtostarboardside;
LM
Ff
W
_
‐ work of moment
Ff
M
during ship’s
heelingtoportside;
LM
T
W
_
‐ work of moment
T
M
during ship’s
heelingtoportside;
‐angleofship’sheel
A
‐amplitudeofrolling;
remainingsymbolslikeinformulas(3)and(4).
On the basis of introduced dynamic coefficients
(formulas5,6and7)asetofsamplecalculationswas
carried out. The ship particulars taken into account
reflect round Panamax size which particulars are
given in the table 1. The
size and location of
consideredtankareshowninfigure8.Theresultse.g.
thevaluesofadynamiccoefficientkd
srversussizeofa
tankarepresentedinfigure21.
Figure21. Correction factor kdsr for different location and
dimensionsofconsideredtanks
489
It ought to be emphasized that the considered
dynamiccoefficientcomprisesonlythe effects ofthe
M
ffandMTcomponentsoftheheelingmomentdueto
liquidmotion.Thevaluekd
sr=1indicatesthedynamic
effect of liquid sloshing equal to the static one in
termsofshiptransversestabilitywhenheavyrolling
onsea waves.Thevaluekd
sr=0 denotesalackofthe
dynamic effect of liquid sloshing in a partly filled
tank which could be possible due to the wave
characterof sloshing flowand anoticed phase shift.
Suchaphenomenonisutilizedinantirollingdevices
(flume tanks) and it is also encountered during
liquefied
cargo carriage (Warmowska, Jankowski
2006).
The characteristics presented in the figure 21
reveals that the value of the considered dynamic
coefficient neither exceeds significantly kd
sr =1 for
doublebottomshiptanksnordropsoftenbelowkd
sr
=0,9.Contrarytothedoublebottomtanklocationthe
upper tanks are more sensitive in the context of
dynamic coefficient variation. The coefficient kd
sr
ranges from 0,6 for the widest and lowest analyzed
tank (b=10 m and h=1,5 m) up to almost 1,3 for the
highest tank. Thus, the actual impact of liquid
sloshingphenomenononshiptransversestabilitycan
benotably lowerthan expected on the basis of IMO
IS‐Codeor
remarkablygreaterforsomecasesaswell.
7 CONCLUSION
The study presented in the paper is focused on the
dynamic effects of liquid sloshing taking place in
partly filled ship tanks. The decomposition of the
dynamicheelingmomentduetoliquidsloshingwas
applied.Thenthefurtherprocessingofa
freefloating
component enabled implementation of a novel
variablenamedadynamiccoefficient. Thecoefficient
correspondswiththeenergybalancemethodofship
dynamicstabilitycalculationsandthankstothisitis
compatiblewith the weather criterion recommended
bytheIMOIntactStabilityCode.
A set of sample calculation of the
dynamic
coefficient was carried out for double bottom tanks
andupperones.Thewiderangeoftank’sdimensions
wastakenintoaccount.
Accordingtotheobtainedresults oftheconducted
research, a significant number of ship tanks
arrangementarousestheliquidsloshingphenomenon
generating less severe impact then assumed in
the
course of contemporary quasi‐static approach.
However, the results reveal also the possibility of
greaterimpact of sloshingliquid in a tank then it is
expected on the basis of contemporary quasi‐static
calculations.
Such a conclusion may be important from the
economical point of view. Ship stability standards
quite
oftenrestrictthecapabilityofavesseltocarryas
much cargo as could be physically loaded. The so
called safety margin is maintained. The common
applicationoftheproposeddynamiccoefficientcould
result in less demanding stability standards still
providingtheassumedlevelofshipsafetyintermsof
hertransversestability.Itmaybeespeciallyimportant
in the age of economical crisis and a worldwide
tendency to cost optimization. Any extra cargo
carriedoverthecurrentrestrictionscontributestothe
ship operator’s revenue. It could be accepted when
without any significant decay of safety standard
onboard.Thus,arousinga
discussiononIMOforum
seemstobejustified.
ACKNOWLEDGMENT
The research project was funded by the Polish
NationalScienceCentre.
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