Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

47, 2, pp. 307-320, Warsaw 2009

LINEAR CHARACTERISTICS OF THE SLOSHING

PHENOMENON FOR THE PURPOSE OF ON-BOARD SHIP’S

STABILITY ASSESSMENT

Przemysław Krata

Gdynia Maritime University, Department of Ship Operation, Navigational Faculty, Gdynia, Poland

e-mail: inhalt@klif.am.gdynia.pl

The paper presents results of experimental research and numerical simu-
lations of the sloshingphenomenon.The researchwas focused on compu-
tation of the heelingmoment affecting stability of a vessel.The proposed
linearisation enables application of the results to the assessment of the
ship’s stability.The dependence of the heelingmoment upon localisation
of a tank in the ship’s hull is analysed. The heeling moment obtained
in the course of the research was compared to the moment computed
in accordance with the Intact Stability Code requirements. The study
may be a contribution to themore sophisticated estimation of the ship’s
stability than it is achieved nowadays.

Key words: sloshing, free surface of liquids, ship’s stability

1. Introduction

Dynamic behaviour of a vessel at sea is greatly affected by the dynamics of
movingmasses carriedonboard.Thecargo securingprocedures ensureavoiding
motion of a loose cargo, but liquids contained in partly filled tanks cannot be
avoided at all. The modelling of the interaction taking place between water
sloshing inside ship’s tanks and the tank’s structure is very important for the
sake of safety of the sea transportation system, human life and environment.
The effects of sloshing should be taken into consideration not only during
strength calculations but in the course of the vessel’s sea keeping prediction
and transverse stability assessment as well.

The accuracy of the ship’s transverse stability assessment is an important
factor in thevessel’s exploitation process.The ship’s loading condition of insuf-
ficient stability may induce list, strong heel and even capsizing. On the other



308 P. Krata

hand, excessive stability causes high values ofmass forces acting on cargo and
machinery due to strong accelerations. Therefore, any scientific efforts towards
better evaluation of the ship’s stability are worthy undertaking. The influence
of the sloshing phenomenon on the ship’s stability is one of the issues to be
considered.

2. Ship’s stability assessment

The stability of a ship is defined as a feature of counteracting against external
heeling forces and moments, which should enable a vessel to realize main
tasks in the course of its exploitation (Dudziak, 1988). A complete description
of the stability may be obtained by solving equations of the ship’s motion.
The most convenient attitude towards this task is to assume that there is a
symmetrical mass distribution and steady values of moments of mass inertia.
The equations of motion, in relation to the center of gravity G, take form
of six differential equations (Dudziak, 1988). The solving of such generally
formulated equations of motion is impossible at the present state of the art.
By neglecting couplings for the sake of simplicity, the ship’s rolling is often
analysed by a single degree-of-freedom system (Senjanovic et al., 1997). The
governing differential equation of motion, as the result of the equilibrium of
moments, is:

I4ϕ̈+D4(ϕ̇)+R4(ϕ) =M4(t) (2.1)

where
I4 – moment of inertia of a ship and addedmasses [kgm

2],
D4 – dampingmoment [Nm],
R4 – restoring moment [Nm],
M4 – excitation moment [Nm],
ϕ – the angle of heel [rad].

The resultant excitation momentM(t) consists of as many components as
many influencing factors swing the ship. Themain components are waves and
wind. Anyway, when the analysis of the ship’s rolling comprises the effects
of liquid sloshing in partly filled tanks, then the moment of the liquid-ship
interaction has to be included as a component of M(t) in a/m formula (2.1).
In spite of the simplified attitude towards the ship’s rolling, the solution

to equation (2.1) is still too complicated to be used in the course of on-board
stability calculations made by cargo officers during their everyday practice.
Calculation of the vessel’s stability and evaluation made on-board nowa-

days, is not based on the ship’s equations of motion, but on the stability



Linear characteristics of the sloshing phenomenon... 309

criteria published by the ship’s classification society. These criteria aremainly
based on A749(18) Resolution of International Maritime Organization. The
resolution and its later amendments are known as the Intact Stability Code
(IMO, 2002).
The criteria quality the shape of the righting arm curve, especially the

minimum value of initial metacentric height, the maximum righting arm and
the angle of this maximum, the area under the righting arm curve calculated
within the prescribed range of the angle of heel. In addition, the weather
criterion, as an attempt at dynamic stability calculation, is to ensure the
sufficient stability of a ship to withstand severe wind gusts during rolling. But
theweather criterion is a very simplemodel of ship’s dynamicbehaviourwhere
the static stability curve is used.Anyway, theweather criterion is the only one
that is partly based on themodel of the heeling phenomenon and not only on
statistical data, while the other criteria are based on the statistics of historical
disasters only (Francescutto, 2002).
According to the IMO regulations, the righting arm curve should be cor-

rected due to the effect of free surfaces of liquids in tanks. The correctionmay
be done by any of the three accepted methods (IMO, 2002):

1) correction based on the actual moment of fluid transfer calculated for
each angle of heel;

2) correctionbasedon themomentof inertia of tank’s horizontal projection;

3) correction based on summation of Msf values for all tanks taken in-
to consideration, where the moment Msf should be obtained form the
simplified formula given in the Intact Stability Code.

All of the three mentioned above methods of calculation of free surface
correction consider the static attitude towards the sloshing phenomenon only.
They do not consider the localisation of a tank within the hull of a ship
and localisation of the rolling axis. The only advantage of current compulsory
corrections is the simplicity of calculation.

3. Study of pressure distribution in a moving tank

The liquid sloshing phenomenon is the result of motion of a partly filled tank.
As the tank moves, it supplies the energy to induce and sustain fluid motion
(Akyildiz and Unal, 2005). Both the liquid motion and its effects are called
sloshing. The interaction between the ship’s and tank’s structure and water
sloshing inside the tank consists in steady transmission of energy. As the ship



310 P. Krata

rolls, the walls of the partly filled tank induce movement of water. Then,
the water presses against the opposite tank’s wall and returns the energy to
the ship, taking simultaneously the next portion and enabling the counter-
direction movement. The mass and energy are conserved within the cycle.
The rolling characteristic of a vessel at sea is affected by the movement of
liquids in tanks (Kim, 2002).

An experimental research on the sloshing phenomenon was carried out at
theShipOperationDepartment ofGdyniaMaritimeUniversity. It enabled one
to measure the dynamic pressure distribution on the side wall of the model
tank and in its upper corner (Krata, 2006). The experimental investigation on
the pressure distribution due to sloshing required the arousing of the sloshing
phenomenon. After that, the dynamic pressure time history in selected spots
was measured and recorded. To achieve this, the test apparatus was designed
and built (Krata, 2006).

The main part of the apparatus is a tank. It is equipped with pressure
transducers and an inclinometer. The tank is forced to oscillate which excites
water movement inside it. The dimensions of the model tank are breadth –
1.040m, length – 0.380m, depth – 0.505m.The tankwas hanged on a shaft by
bearings and forced to oscillate by a drive mechanism. The drive mechanism
is based on an electric motor, transmission reducing revolution velocity and
a crank mechanism. The view of the testing apparatus and localisation of
dynamic pressure sensors is shown in Fig.1.

Fig. 1. Experimental setup

Assuming plane tank oscillations and negligible water viscosity, one obse-
rves a two-dimensional character of the considered liquid flow inside the tank



Linear characteristics of the sloshing phenomenon... 311

(Warmowska and Jankowski, 2005). It allows one to equip the tank with one
set of pressure transducers only, fixed in themiddle line of the tank. The pres-
sure transducers are equally spaced along the vertical wall of the tank and in
the roof of the tank close to the upper corner.

The oscillatingmotion which induces sloshing is described accurately eno-
ugh by a harmonic function. The experimental research on the pressure di-
stribution due to sloshing was performed for a variety of external excitation
parameters. The period of the oscillation varied from T =2.6s to T =6.5s.
The lever os, as the distance between the center of the tank and the rotary
motion axis, was changed from os=−0.718m to os=0.718m. The positive
value of os describes the tank localisation beneath the shaft (modelling the
rolling axis) and the negative value of os describes the localisation above the
shaft. The amplitude of the tank rotary motion during the tests was assumed
to be 40◦. It reflects heavy sea conditions and enables one to draw conclusions
for theworst possible condition at sea (Francescutto andContento, 1999). The
tank filling level was assumed to be 30%, 60% and 90%.

The pressure signal, measured by the transducers, consists of two compo-
nents.Oneof them is called non-impulsivedynamic pressure and the other one
impulsive pressure, or impact pressure (Akyildiz and Unal, 2005). The non-
impulsive dynamic pressure is slowly varying. It is a result of global motion of
the liquid in the tank and it affects the transverse stability of the ship.

All signals received fromthe sensorswereverifiedandthemeasuring instru-
ments were calibrated. The gain coefficient and shift coefficient were determi-
ned for every pressure sensor and the inclinometer. The calibration procedure
allowed one to deem the experimentalmeasurements to be correct and reliable
(Krata, 2006).

Theanalog signals received fromthe sensorswere sampledand transformed
into discrete digital signals by a 12-bit A/D card and then theywere recorded
in text format files. Themaximumworking frequency of themeasuring device
was 1000Hz. Thus, the aliasing distortions of the measured signal were avo-
ided, because the measuring instruments were much faster than the required
Nyquist rate for the sloshing phenomenon (Zieliński, 2002). A further digital
signal processing was carried out. The main operation was low pass filtering
for high frequency noise reduction. The filtering enabled one to decompose
the recorded digital signal and emerged the non-impulsive dynamic pressure
component (Zieliński, 2002).

The pressure distributions obtained in the course of experimental investi-
gation were supplemented by the results of numerical simulations. The simu-
lations of the sloshing phenomenonwere performed by the computer program



312 P. Krata

”Tank” by M. Warmowska, used for the estimation of the dynamic pressu-
re distribution. The sloshing problem was described by the two-dimensional
model. It was also assumed that the liquid is inviscid, incompressible, and of
constant density. As the flow of the liquid was assumed to be irrotational,
the potential theory was used to solve the sloshing problem (Jankowski and
Warmowska, 1997).
The numerical simulation of the sloshing phenomenon was performed for

oscillation and tank’s geometry corresponding with suitable geometric para-
meters of the experimental investigation. The height of the water level varied
from 30% to 90% of the maximum tank’s height. The program allowed one
to compute the time history of dynamic pressures in ninety points around the
tank’s model. The control points were situated along vertical walls, the bot-
tom and the tank’s roof. The correctness of the simulation results was verified
experimentally (Krata, 2006).

4. Linearisation of the heeling moment due to sloshing

The pressure distribution on the walls of the tank was obtained in the course
of experimental tests and numerical simulations. The results of the research
enable one to compute the heeling moment due to sloshing. The heeling mo-
ment M [Nm] was calculated according to the following formula

M =

∫

S

r×np ds (4.1)

where
S – surface of the tank walls [m2],
r – position vector of the considered point on the tank wall [m],
n – normal vector [–],
p – local pressure on the tank wall [Pa].
Due to the two-dimensional character of the considered flow in the tank,

the heeling moment is a vector of the perpendicular direction to the plane of
tank’s motion. As the transverse stability of a ship is considered, the heeling
moment has one component only, as follows

M = [Mx,My,Mz] = [Mx,0,0] (4.2)

where Mx, My, Mz are the spatial components of M vector, determined in
relation to the x, y and z axis, respectively, in the reference system fixed to
the vessel.



Linear characteristics of the sloshing phenomenon... 313

As the direction of the heeling moment is fixed and steady in the time
domain, the heeling moment due to sloshing may be described by the Mx
component. The resultantmoment obtained from formula (4.1) represents one
time-step only.Computationof theheelingmoment shouldbeperformed for at
least one period of roll. Thus, the pressures have to be investigated for at least
one period of the ship’s roll as if in fact they were obtained for a longer time
comprising few rolling periods. An example of the heeling moment changes is
shown in Fig.2.

Fig. 2. Time history of the heeling moment due to sloshing for the filling level 30%
and os=0.359m

The time domain presentation of the computation results can be useful
when the ship’s rolling is computed according to formula (2.1). In such a
case, the heeling moment due to sloshing is one of the components of the
total heelingmoment swinging a vessel at sea. However, the assessment of her
stability is not based on the equations of motion, but on the static stability
curve (IMO, 2002). The curve presents the righting arm GZ versus the angle
of heel, and the righting arm is reducedby the statically computed free surface
correction. Therefore, the most convenient way to present the results of the
heeling moment calculation due to dynamic sloshing of a liquid in a partly
filled tank, is to plot a graph of the angle of heel domain. An example of
such a graph is presented in Fig.3. The heeling moment was time-domain
calculated, but it is plotted as a function of the angle of heel.

The interpretation of the results of the heeling moment is much easier in
the domain of the angle of heel. Themain disadvantage of the graph shown in
Fig.3 is the hysteresis resulting fromwave typephenomena taking place inside



314 P. Krata

Fig. 3. Heeling moment vs. angle of hed for the filling level 30% and os=−0.718m

themoving tank. This disadvantage can be eliminated by linearisation. As the
main task of the research is amore reliable stability assessmentwith regard to
the sloshing phenomenon, the linearisation should refer to the ship’s stability
criteria, especially the weather criterion. The area under the GZ curve is
regulatedwithin theweather criterion,which represents theworkof theheeling
moment due to wind gusts when the ship rolls. Hence the linearisation of the
examined heeling moment should be based on the work of the moment as
well. The linearisation method applied to the heelingmoment due to sloshing
of liquids is based on the formula

ϕ40∫

0

M(ϕ) dϕ+

0∫

ϕ40

M(ϕ) dϕ=2

ϕ40∫

0

Ml dϕ (4.3)

where
M – heeling moment due to sloshing [Nm],
Ml – resultant linear heeling moment due to sloshing [Nm],
ϕ – the angle of ship’s heel [rad],
ϕ40 – the angle of heel equal 40

◦ [rad].
An example of the linear heeling moment due to sloshing is presented in

Fig.4.
The linear function of the heelingmoment can bedetermined by fixing two

in-line points having the coordinates (ϕ,M). One of them is the point (0,0)
and the other one thepoint (40◦,Ml40).Therefore, the complete description of
the linear heelingmomentmay be done by one scalar only. This is value Ml40



Linear characteristics of the sloshing phenomenon... 315

Fig. 4. Linear heeling moment for the filling level 30% and os=−0.718m

of the linear heeling moment due to sloshing for the angle of heel equal 40◦

and obtained from formula (4.3).

5. Characteristics of the linear heeling moment due to sloshing

The linear heeling moment due to sloshing taking place in partly filled tanks
in ships depends on some parameters. One of the most important seems to
be the localisation of the tank with respect to the vessel’s rolling axis. The
comparison of the linear heeling moment due to sloshing and the quasi-static
moment of free surface correction may be a point of interest as well.

Themoment of heeling of a ship, as a consequence of the liquid carried in
any partly filled tank, has two components. One of them is the moment Mm
of the liquid weight and the other is the heeling moment MT due to transfer
of the fluid calculated for each angle of heel. The total heelingmoment due to
liquids contained in the vessel’s tanks can be described from the static point
of view by the formula

Mstat =Mm+MT (5.1)

where
Mm – static heeling moment due to weight of ”frozen” liquid in a

tank [Nm],
MT – static heeling moment of the transfer of the liquid’s center of

gravity [Nm].



316 P. Krata

The moment Mm is taken into consideration while calculating the ship’s
center of gravity, and it assumes the liquid to be ”frozen” at the angle of heel
equal to 0◦. The other component MT concerns the shift of the liquid’s center
of gravity only, and can be calculated in accordance with the Intact Stability
Code requirements (IMO, 2002). Both a/m moments are functions of sine of
the angle of heel. Anyway, the resultant heeling moment can be compared
with the linear heeling moment due to sloshing for angles of heel where the
sine function may be approximated accurately enough by the linear function.
The reasonable range of such a linear approximation is about 40◦, which is
shown in Fig.5.

Fig. 5. Linear approximation of the sine function

As the sine function is almost linear up to the angle of heel 40◦, the
static heelingmoment Mstat computed according to the Intact Stability Code
requirements may be compared to the linear heeling moment Ml obtained in
the research program. Both moments Mstat and Ml have the zero values for
the zero angle of heel, so their comparison may be done as the comparison of
their values for the angle of heel equal 40◦.
The graphs showing theheelingmoments are preparedas non-dimensional.

The excitation period T is referred to the first harmonic natural sloshing
period of the liquid in themodel tank Tw. The scope of T/Tw ratios reflects a
widevariety of characteristics that can take place on boardof ships in different
loading conditions. The distance os between the center of the moving tank
and the axis of rotary motion is referred to the breadth of the tank bz. And
the values of the heeling moments are referred to the moment Mf.s.40 of the
static free surface correction computed according to the Intact Stability Code
requirements. The characteristics of the linear heelingmoment due to sloshing
for different localisations of the tank are shown in Figs.6, 7 and 8.
The analysis of graphs of the heelingmoment obtained for different locali-

sations of the partly filled tank reveals a distinguishable relationship between
them. The values of the linear heeling moment Ml40 due to sloshing at the



Linear characteristics of the sloshing phenomenon... 317

Fig. 6. Non-dimensional linear heelingmoment at the angle of heel 40◦ for the filling
level 30% and different localisation of the partly filled tank

Fig. 7. Non-dimensional linear heelingmoment at the angle of heel 40◦ for the filling
level 60% and different localisation of the partly filled tank

Fig. 8. Non-dimensional linear heelingmoment at the angle of heel 40◦ for the filling
level 90% and different localisation of the partly filled tank



318 P. Krata

investigated angle of heel 40◦ referred to the roll axis localisation have charac-
teristics very close to the linear ones. Their linear correlation coefficient R2

exceeds 0.95. The value of the static heelingmoment for 40◦ angle of heel, the
computation ofwhichwasbasedon the free surface correction, is linear aswell.
But this correction does not consider the dynamics of liquid sloshing in the
tanks. A considerable difference is noted between the results of investigations
and simple static calculations required to be done in the course of the ship’s
stability assessment on board of a vessel.

6. Conclusions

The research of the influence of the sloshing phenomenon on the heeling mo-
ment is based on a more sophisticated method than that used nowadays ac-
cording to the Intact Stability Code. It takes into consideration the dynamics
of the sloshing liquids, and the only disadvantage is the hysteresis. This pro-
blem was solved by linearisation performed according to proposed formula
(4.3). The obtained linear heeling moments represent the equivalent work of
the heelingmoment, which is the key issue in theweather criterion of stability.

Motion of liquids in partly filled ship’s tanks affects her stability and,
therefore, it is to be taken into account in the course of the stability assessment
in accordance with the IMO recommendations. The research presented in the
paperpoints out that the very simplifiedmethods recommendedby IMOcould
be improved and reach better accuracy tomeetmodern requirements of ships
exploitation. The analysis of the heeling moment characteristics reveals their
almost lineardependenceon the localisation of the tank, although theobtained
lines significantly vary from the IMO recommended ones. As a result of the
discrepancies, the stability of a vessel may be less or more affected by the
liquid sloshing in partly filled tanks. This may be dangerous to the vessel or
– on the other hand – restrict its ability to carry cargo. Both situations could
be avoided by the use of more accurate dynamic free surface corrections.

The results of the research and the proposed method can be the basis for
further investigation on the new formula of free surface correction. As long
as the prescriptive attitude towards the stability criteria will be in force, the
dynamic free surface correction could be used. It should consider the locali-
sation of partly filled tanks referred to the axis of roll. This would be a step
ahead towards the increase in accuracy of the ship’s stability estimation,which
should allow one to improve the safety of the ship.



Linear characteristics of the sloshing phenomenon... 319

References

1. Akyildiz H., Unal E., 2005, Experimental investigation of pressure distribu-
tion on a rectangular tank due to the liquid sloshing, Ocean Engineering, 32,
1503-1516

2. Dudziak J., 1988,Teoria okrętu,WydawnictwoMorskie, Gdańsk [in Polish]

3. Francescutto A., 2002, Intact ship stability – the way ahead, Proc. 6th
International Ship Stability Workshop, Washington

4. Francescutto A., Contento G., 1999, Bifurcations in ship rolling: experi-
mental results and parameter identification technique,Ocean Engineering, 26,
1095-1123

5. Intact Stability Code 2002, IMO, London

6. Jankowski J.,WarmowskaM., 1997,Developmentof computerprogramde-
scribing the flow inpartly filled tank,Technical Report No.27/97, PRS,Gdańsk

7. Kim Y., 2002, A numerical study on sloshing flows coupled with ship motion
– the anti-rolling tank problem, Journal of Ship Research, 46, 52-62

8. Kim Y., Nam B.W., Kim D.W., Kim Y.S., 2007, Study on coupling effects
of shipmotion and sloshing,Ocean Engineering, 34, 2176-2187

9. Krata P., 2006, The comparative study of numerical simulation and experi-
mental research into the areas ofmaximumdynamic pressure due to sloshing in
ship’s double bottom tank,Proc. XV-th International Scientific and Technical
Conference Nav-Support ’06, Gdynia

10. Senjanovic I., Parunov J., Cipric G., 1997, Safety analysis of ship rolling
in rough sea,Chaos, Solitons and Fractals, 8, 4, 659-680

11. Warmowska M., Jankowski J., 2005, Simulation of water movement in
partly filled ship tanks,Proc. HYDRONAV ’2005, Ostróda

12. ZielińskiP.T., 2002,Od teorii do cyfrowego przetwarzania sygnałów,Wydział
EAIiE AGH, Kraków [in Polish]

Linearyzacja w opisie zjawiska sloshingu na potrzeby eksploatacyjnej

oceny stateczności statku

Streszczenie

Artykuł prezentuje wyniki badań eksperymentalnych i symulacji numerycznych
ruchu cieczy w niepełnych zbiornikach statku. Istostą badań było wyznaczenie mo-
mentu przechylającego statekwskutek ruchu cieczy w zbiornikach, cowpływa na sta-
teczność poprzeczną ststku. Zaproponowanametoda linearyzacjimomentu umożliwia



320 P. Krata

implementację wyników badań przedmiotowego zjawiska do praktyki oceny statecz-
ności poprzecznej statku.
Szczególną uwagę zwrócono na zależność pomiędzy momentem przechylającym

statek a lokalizacją niepełnego zbiornika względem osi kołysań bocznych. Zarazem
wyznaczany moment przechylający został porównany do quasi-statycznego momen-
tu wyliczonego zgodnie z zaleceniami Kodeksu ISC. Przeprowadzone badania mogą
stanowić element poprawywiarygodności oceny stateczności statku dokonywanej ru-
tynowo przez oficerów ładunkowych.

Manuscript received July 10, 2008; accepted for print November 18, 2008