535
1
INTRODUCTION
The main commonly discussed features of maritime
transport are usually its safety and effectiveness.
Amongthemtheshipssafetyissuesarecrucialfrom
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
ships
relatedtohersafetyistheirstability.
Ship stability is a term used to describe the
tendencyofashipto returnbacktoherequilibrium
when she is inclined from an upright position [5].
Since the initial position of a ship is not always
uprightone,themore practical
definitionstatesthat
the stability is a feature enabling to perform, when
remaining in determined position, the task she is
constructed for. The complementary definitions lead
topointoutthatthestabilityofashipisanelementof
heroperationalsafetyqualifyingfactors.
The seagoing vessel’s stability calculation and
evaluation
madeonboard nowadays is based onthe
prescriptive stability criteria published by the ship’s
classification societies [5]. These criteria are mainly
based on the A749(18) Resolution of International
MaritimeOrganization.Theresolutionandtheirlater
amendments are known as the Intact Stability Code
[4].
Theshipstabilitycriteriaqualifythe
shapeofthe
rightingarmcurve.Inaddition,theweathercriterion
is to ensure the sufficient stability of a ship to
withstand the severewindguestsduringrolling[4].
Althoughtheweathercriterionreflectsaverysimple
modelofdynamicship’sbehavior,thestaticstability
curve is used. Anyway, the weather
criterion is the
only, which is partly based on the model of heeling
phenomenonnotonlyonthestatisticdata,whilethe
remaining criteria are based on the statistics of
historical disasters only [3]. The modern and still
developing approach towards ship stability
A Method of Assessment of the Liquid Sloshing
Impact on Ship Transverse Stability
P.Krata
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT:Liquidsloshingphenomenontakingplaceinpartlyfilledships’tanksdirectlyaffectsthestability
ofavessel. However,onlystaticcalculations are carriedoutonboard ships nowadays and statictransferof
liquid weight is taken into account in the course of routine stability calculation. The paper is focused on a
dynamicheelingmomentduetoliquidsloshingintanksonboardships.Thesetofnumericalsimulationsof
liquidsloshingtakingplaceinmovingtanksiscarriedout.Therealisticrangeofgeometricparametersistaken
intoaccount.TheconductedCFDsimulationsareexperimentallyverified.Finally,themethodofanassessment
of the liquid sloshing impact on ship transverse stability is worked out. The key point of the method is a
dynamiccoefficientdescribingrelationoftheresearcheddynamicheelingmomentandthequasi ‐staticonein
termsofdynamicstabilityofavesselwhichisrelatedtotheweathercriterionof
shipstabilityassessment.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 8
Number 4
December 2014
DOI:10.12716/1001.08.04.07
536
qualification is an implementation of performance‐
based stability criteria in the future. They are based
mainlyon theriskassessment [5]however,itis still
farfromcommonuseonboardships.
Regardless the approach towards ship stability
evaluation, the physical background of phenomena
takingplaceonboardoughttobetaken
intoaccount.
In case of contemporary prescriptive stability
standards,therightingandheeling 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 [5]. The balance of righting arm
(rightingmoment)andheelingarm(heelingmoment)
shall comprise all significant components of each
momentandamongothers the heelingmomentdue
toliquidsloshinginapartlyfilledmovingtanktoo.
In the light of
ship stability related concepts, the
accuracyofship’stransversestabilityassessmentisan
importantprobleminvesselsoperationprocess.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 carried out
onboard generally on the basis of the IMO IS‐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 any of two
acceptedmethods[4]:
correction based on the actual moment of fluid
transfercalculatedforeachangleofheel;
correctionbasedonthemomentofinertiaoftank’s
horizontalprojection(simplependulummodel).
Figure1.Flatsurfaceofwaterlineand liquid’s freesurface
in partly filled tank (left) and the quasi‐static transfer of
fluidmassduetoshipheeling(right)
Both mentioned above methods of free surface
correction calculation consider the static attitude
towardsthesloshingphenomenononly.Theyalsodo
notconsiderthelocationoftankswithinthehullofa
shipandthelocationoftherollingaxis.However,the
main advantage of currently applied compulsory
correctionsisthe
simplicityoftheircalculation.
Regardlesstheexplicitcomputationalformulafor
free surface correction, the liquid surface is always
assumedflatanddependsonlyonanangleofship’s
heelnottime.Theideaispresentedinthesketch(Fig.
1).
The liquid sloshing phenomenon takes place in
partlyfilledships
tanks.Asatankmoves,itsupplies
energy to induce and sustainthe fluidmotion. Both
the liquid motion and its effects are called sloshing.
The effect of water motion inside the tank is the
pressurefieldonthetank’sstructure.Theinteraction
between ship’s and tank’s structure and the water
sloshing inside the tank consists in the permanent
transmissionofthe energy[1].As the shiprolls, the
wallsofthepartlyfilledtankinducethemovementof
water. Then the water press against the opposite
situated tank’s walls and return the energy to the
ship, then taking the next
portion enabling the
counter‐direction movement. The exemplary general
viewontheliquidsloshingphenomenontakingplace
insideamodeltankswingingduringanexperimental
researchisshowninFig.2.
Figure2.Exemplaryshapeofafreesurfaceofliquidinside
apartlyfilledmodeltank(ownresearch)–itmaybeclearly
seen that the free surface is far different than assumed
withinthequasi‐staticapproach(ref.toFig.1)
Under external large amplitude excitations or an
excitationnearthenatural frequencyofsloshing,the
liquidinsidetankisinviolentoscillationswhichisof
greatpracticalimportanceto thesafetyof the liquid
transport[11].Thecharacteristicsofheelingmoment
due to liquid sloshing depend on a variety of
parameters,
for instance tank’s geometry, its filling
level, location of a tank within the hull of a ship,
rolling period and others. Due to the noticed
nonlinearity a description of the sloshing
phenomenon is relatively complex, however, the
energycharacteristicsseemtoplayanimportantrole,
especiallywhenreferredtothe
shipdynamicstability
requirements.
537
3
CFDMODELINGOFLIQUIDSLOSHING
PHENOMENON
The noticeable weaknesses of the static approach
towardsthefreesurfacecorrectionanditsapplication
in the course of ship stability assessment are
commonly known, therefore, a number of studies is
still undertaking worldwide. Generally the main
stream of researches comprises two corresponding
trends i.e. numerical simulations of liquid sloshing
phenomenon
andexperimentaltests.Bothapproaches
arecoupledandtheycomplementeachother.
At the present state‐of‐the‐art numerical
simulations require an experimental validation
allowingadjustingofasetofinputparameterslikea
computational mesh, a time step, etc. The features
beingusuallyvalidatedaredynamicpressureat
some
selected points and a shape of the free surface of
liquid.
TheresearchprojectcarriedoutintheDepartment
ofShipOperationattheGdyniaMaritimeUniversity
wasfocusedontheshipstabilityissueswithregardto
liquid sloshing phenomenon in partly filled tanks.
The application for this task a
CFD technique of
numerical simulations of liquid sloshing flow is
relativelycost‐effectivetechniqueallowingmanyruns
ofsimulationscoveringthewidescopeofconditions.
Contrary to this, the experimental tests are rather
costlyandtime‐consuming,thereforeeverysinglerun
of experiment needs to be carefully planned and
justified.
The applied numerical solving a flow problem
insidemovingtanksincludesseveralsteps[2]:
creatingacomputationaldomain;
specifyingamathematicalmodel;
specifyinginitialandboundaryconditions;
specifyingacomputationalmesh;
calculations;
visualizationandanalysisofresults.
Allthelisted stepswerecarried out forthemost
typicalrectangularshipstanks.Thesizeandlocation
ofthetankcorrespondwithitscommonlocation[9].
The range of angular motion was set for 40 degrees
and it reflects the very heavy sea conditions in
extremelystormyweather[8](Fig.3).
Figure3.Shiprollingamplitude(rangeofangularmotionof
thetank)applied in thecourseof CFD modelingofliquid
sloshinginconsideredtanks
Thecomputationalmesh appliedinthecourseof
the simulations was hexahedral type and related to
two coupled reference frames, the stationary and a
moving ones. The arrangement of the reference
systemsispresentedinFig.4.
Figure4. Computational mesh and two coupled reference
framessystems
Thethree‐dimensionalsimulationsofthesloshing
phenomenon were carried out by the use of
FlowVisioncodebeinganuniversalandflexibletool
designedformodelingofliquidsdynamics.Thecode
isbasedonthefinitevolumemethod(FVM),anduses
the VOF method for free surface problems [2]. The
RANS
(Reynolds‐averaged Navier–Stokes) equation
isimplementedwhichisthetime‐averagedequations
ofmotionforfluidflow.Simulationofturbulentflows
isbasedontheeddyviscosityconcept.Theturbulent
eddyviscositycontributesthediffusioncoefficientsin
theNavier‐Stokesandconvection‐diffusionequations
[2]. The semi‐empirical
k‐ε model turbulence model
wasapplied.
Theresultofthesimulationcomprisesmainlythe
general flow pattern and the velocity and pressure
fields.Theexemplaryshapeofafreesurfaceisshown
inFig.5.
Figure5.Computedshapeofafreesurfaceinamovingtank
(example)
538
Moreover, the user defined parameter, i.e. the
heelingmomentdueto liquid sloshing inside partly
filled tank which is essential from the conducted
research point of view, was also calculated. The
heelingmoment
Mvectorwascalculatedaccordingto
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
stability of a ship is assumed to be
considered, the
heeling moment may be described by one spatial
componentonly,asfollows[6]:
xyz x
M,M,M M, 0, 0
M (2)
where:
Mx, My, Mz – spatial components of M vector,
determinedaboutthe
x,yandzaxisinthereference
systemrelatedtothevessel.
Forfurtherusethesolenon‐zerocomponent
Mxof
thecomputedheelingmomentduetoliquidsloshing
whichisdescribedbytheformula(2)wasnamedthe
total dynamic moment and marked
MTotal_dyn. Such
heeling moment was the subject for post processing
andreasoning.
4
EXPERIMENTALRESEARCHINMODELSCALE
Althoughthenumericalsimulationofliquidsloshing
inapartlyfilledtankisapowerfultechnique,itstill
requiresanexperimentalverification forsomecases.
Generally, the experiment is commonly found as an
unambiguousprove for thecorrectnessof numerical
computations. Therefore, the experimental research
intothesloshingphenomenonwascarried
outinShip
Operation Department at the Gdynia Maritime
University. It enabled to measure the dynamic
pressure distribution on the side wall of the model
tank and in its upper corner and furthermore, to
recordashapeoffreesurfaceforanyangleoftank’s
tilt. The experimental investigation required
the
arousingofthesloshingphenomenon.Afterthatthe
dynamicpressuretime history inselectedspotswas
measured and recorded. To achieve this the test
apparatuswasdesignedandbuilt.
Themainpartoftheapparatusisatankequipped
with pressure transducers. The tank is forced to
oscillatingmotionby
thehydraulicdrivemechanism,
thusexcitingthe watersloshinginsidethetank.The
dimensionsofthemodeltankare:breath–1,040 m,
length–0,380m,depth–0,505m.Thegeneralviewof
the testing apparatus and the location of dynamic
pressuresensorsisshowninFigure6.
Figure6. Experimental setup: general layout (left) and
pressuregaugearrangement(right)
Thepressuresignal,measuredbythetransducers,
consists of two components. One of them is called
non‐impulsive dynamic pressure and the other one
impulsivepressure,orimpactpressure[1].Thenon‐
impulsive dynamic pressure is slowly varying. It is
theresultoftheglobalmovementofliquidinthetank
anditaffectsthetransversestabilityofaship.
Despite the dynamic pressure distribution the
shape of free surface of liquid sloshing in the tank
was recorded. Theliquiddistribution andavelocity
fieldaregovernedmostlybytheinertiaofliquidmass
and a pressure field. As a consequence,
the correct
modelingofliquid’sfreesurfaceemergesasastrong
proveforthecorrectnessoftheCFD‐basednumerical
simulations of sloshing flows. The exemplary
comparisonoffreesurfacesrecordedinamodeltank
during experiment and computed in the course of
simulationsisshowninFig.7.
539
Figure7.Comparisonofashapeoffreesurface:experiment
(upperphotos)andnumericalsimulations(lowergraphics)
The pressure history in the control points of the
tank obtained in the course of the experiment were
compared to the computed ones. Both experimental
resultsthepressureandthefreesurfacemeetrelevant
results achieved by the use of CFD simulations.
Consequently, the results of simulations were
acknowledgedascorrect
andreliable.
5
CHARACTERISTICSOFDYNAMICHEELING
LEVERDUETOLIQUIDSLOSHINGINTANKS
The result of the numerical simulations of liquid
sloshingwhichismostessentialfromtheconducted
research point of view, is the total dynamic heeling
moment due to the analyzed phenomenon. It is
obtained in time domain, however, since the
momentary angle of ship’s heel is know for every
timestep
ofCFDcomputations,the heelingmoment
maybeplottedversusanangleofheelaswell.Thisis
aconvenientapproachforresultsvisualization.
Regardless the way of display of the heeling
moment, the momentary values of heeling moment
dependsnotonlyonthegeometryofthetank(likein
caseofquasi‐staticapproach)butalsoonthelocation
of the considered tank. This circumstance
significantlyimpedesinferringabouttheinfluenceof
any parameter. The suggestion and inspiration
regarding addressing the described problem can be
found in the static free surface correction and the
contemporarywayofshipstabilitycalculation.
Thetotalstaticheelingmomentduetoapresence
of liquid inside a tank is divided into two
components. One of them reflects the moment of
liquid’sweightwithoutanychangesofshapeduring
shipheeling.Onemayfindsuchliquidasa“frozen”
one or generally as a solid
[7]. This component is
included in standard weights and moments
calculation sheet and it is not revealed as a free
surfaceeffect[4].Thesecondcomponentofthestatic
heelingmomentduetotank’spartlyfillingcomprises
the fluid transfer calculated for each angle of heel.
Thiscomponentrevealsthefree
surfacecorrection.
The similar reasoning may be applied to the
dynamicheelingmomentduetoliquidsloshing.The
totalvalueofthemomentcouldbedecomposedinto
twocomponents.Thefirstonecomprisesthemoment
duetodynamicactionofliquid“frozen”atanangle
ofheelequal0degrees.
Thesecondcomponentofthe
dynamic heeling moment due to liquid sloshing
coversonlythemomentresultingfromlettingfreethe
liquid to slosh inside the tank. Analogously to the
static case, the component containing the moment
comingfromthefrozen‐likeliquidisincludedinthe
weight distribution calculation. And
the remaining
dynamic component of the heeling moment due to
liquidsloshingwhichmaybecalled“thefreefloating
component” is the matter of further consideration.
Thecoreideaofananalogyinthisapproachmaybe
expressedbytheformulas:
TstatFLstatTotal
MMM
__
(3)
FfdynFLdynTotal
MMM
__
(4)
where:
MTotal_stat – total static moment due to a presence of
liquidwithfreesurfaceinatank;
MFL_stat–static heelingmomentdue to theweightof
frozen‐likeliquidinatank;
MT–staticheelingmomentoffluidtransfercalculated
foreachangleofheel;
MTotal_dyn – total dynamic moment due to liquid
sloshinginatank;
MFL_dyn–dynamicheelingmomentduetotheweight
offrozen‐likeliquidinatank;
MFf–freefloatingcomponentofthedynamicmoment
duetoliquidsloshing.
On the basis of the formula (4) the free floating
component of the heeling moment can be extracted.
TheresultofapplieddecompositionisshowninFig.
8.
Figure8.Decompositionftheheelingmomentduetoliquid
sloshing carried out according to the formula (4) –sample
case
According to the formulas (3) and (4) it may be
stated that the frozen‐like mass of liquid in both
equationsreferstoexactlythesamevolumeofliquid
whilethe only differenceisthepace ofshipheeling
(quasi‐static vs. dynamic approach). Thus, the
remainingcomponentofthetotal
heelingmomentin
both approaches may be compared. Such a
comparison carried out for two exemplary cases is
showninFig.9.
540
Figure9. Comparison of Mff and Mstat components of the
heelingmomentduetoliquidmotioninpartlyfilledtanks–
samplecases
The most significant remarks related to the
comparisonof
MffandMstatcomponentsoftheheeling
moment due to liquid motion in partly filled tanks,
refer to two main issues. Firstly, it is possible
emergence of both: quasi‐static approach (e.g. free
surfaceeffect)produces the results more severetoa
shipstabilitythendynamicapproachorreverselythe
dynamic calculation is
more demanding in terms of
shipstabilitythenstaticone.Secondly,thereisalack
of an objective measure enabling clear classification
which of the analyzed moments produces more or
less severe results from the ship stability point of
view.
Addressingtheproblemof comparisonof the
Mff
and
Mstat components of the heeling moment due to
liquid motion, the aggregative variable was worked
out and named the “dynamic coefficient”. This
coefficientreferstothedynamicstabilityofavesselor
inother words to the weather criterion of the intact
shipstabilityassessmentwhichisbasedontheenergy
balancemethodofstabilitycalculation[5].
Inthetypicalcase ofasymmetriclocation ofship
tanks,thedynamiccoefficientneedstobecalculated
separatelyforaportsidetank(coefficient
kdL)anda
starboard one (
kdP respectively). The definitions of
appliedcoefficientsaregivenbyfollowingformulas:
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)
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
(formulas 5 and 6) a set of sample calculations was
carried out. The ship particulars taken into account
reflectroundPanamaxsizeandtheywerefollowing:
breadthB=32m, height H=20m,
elevationofrolling
axisKR=9m.Thelocationofanalyzedsampletanksis
showninFig.10.
Figure10.Arrangementofconsideredtanksinship’shull
Onlyonesideoftheship(e.g.starboardside)was
takenintoconsiderationtoperformthesetofsample
calculationsone breadthofthetank
bz=5mandtwo
possibleheightsoftanks
hz=1,5m andhz=2,5 m.The
dynamic coefficient
kdP was computed which is
presentedinFig.11.
The computation of the dynamic coefficient
kdP
was carried out only for a sample set of possible
cases, thus the quantitative analysis cannot be
representativeforanypotentialsizesandlocationsof
shiptanks.However,theinterpretationoftheresults
isanalogousforanycasesatanyship.
541
Figure11. Correction factor kdp for different location and
heightofconsideredtanks
It ought to be emphasized that the considered
dynamiccoefficientcomprisesonlytheeffectsof the
MffandMstatcomponentsoftheheelingmomentdue
to liquid motion. The value
kdP =1 indicates the
dynamic effect of liquid sloshing equal to the static
oneintermsofshiptransversestability.Thevalue
kdP
=0 denotes a lack of the dynamic effect of liquid
sloshinginapartlyfilledtankwhichispossibledue
tothewavecharacterofsloshingflowandanoticed
phase shift. Such a phenomenon is utilized in anti
rolling devices (flume tanks) and it is also
encounteredduringliquefiedcargo
carriage[10].
6
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.Thenthefurtherprocessingofafreefloating
component enabled implementation of a novel
variablenamed
adynamiccoefficient.Thecoefficient
correspondswiththeenergybalancemethodofship
dynamicstabilitycalculationsandthankstothisitis
compatiblewith theweathercriterionrecommended
bytheIMOIntactStabilityCode.
A set of sample calculation of the dynamic
coefficient was carried out. The results reveal the
possibility of greater impact of sloshing liquid in a
tankthenitisexpectedonthebasisofcontemporary
quasi‐static calculations.However,inmostanalyzed
casestheimpactofliquidsloshingwasfoundsmaller
then it is supposed on the basis of quasi‐static
approach.
Such a conclusion may be
important from the
economical point of view. Ship stability standards
quiteoftenrestrictthecapabilityofavesseltocarryas
much cargo as may be physically loaded. The so
called safety margin is maintained. The common
applicationoftheproposeddynamiccoefficientcould
make the stability requirements less severe
then
nowadays.Itmaybeespeciallyimportantintheage
ofeconomicalcrisisandaworldwidetendencytocost
optimization.Anyextracargocarriedoverthecurrent
restrictionscontributestotheshipoperator’srevenue.
It could be accepted when without any significant
decayofsafetystandardonboard.Thus,arousingofa
discussiononIMOforumseemstobejustified.
ACKNOWLEDGMENT
The research project was funded by the Polish
NationalScienceCentre.
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