International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 4

Number 3

September 2010

363

1 INTRODUCTION

1.1 Sloshing phenomenon as one of factors

influencing safety of a vessel at seaway

The dynamic behavior of a vessel at the sea is

greatly affected by the dynamics of moving masses

existing onboard. The cargo securing procedures

ensure avoiding moving of a loose cargo, but the

liquids contained in partly filled tanks cannot be

avoided at all. Regardless the strength calculation

the effects of sloshing should be also taken into

consideration in the course of vessel’s seakeeping

prediction and her transverse stability assessment.

Liquid sloshing phenomenon is a result of partly

filled tank motions. As a tank moves, it supplies the

energy to induce and sustain the fluid motion

(Akyildiz & Unal 2005). Both the liquid motion and

its effects are called sloshing. The interaction

between the ship’s and tank’s structure and the water

sloshing inside the tank consists in the constant

transmission of energy. As the ship rolls, the walls

of a partly filled tank induce the movement of water.

In such an attitude ship’s seakeeping behavior

which comprises the notion of her stability is one of

the researched key issues leading to the increase in

understanding of the safety qualifying factors.

1.2 Intact ship stability assessment

The accuracy of ship’s transverse stability

assessment is the important factor in the vessel’s

exploitation process. The ship’s loading condition of

insufficient stability may induce a list, a strong heel

and even a capsizing. Contrary to such state, the

excessive stability causes high values of mass forces

acting on cargoes and machineries due to a strong

accelerations. Therefore, any scientific efforts

towards the better ship’s stability evaluation are

worthy to be undertaken. The influence of sloshing

phenomenon on the ship’s stability is one of the

issues to be considered.

The vessel’s stability calculation and evaluation,

made on-board nowadays, is based on the stability

criteria published by the ship’s classification

societies. These criteria are mainly based on the

A749(18) Resolution of International Maritime

Organization. The resolution and their later

amendments are known as the Intact Stability Code.

The criteria qualify the shape of the righting arm

curve. In addition, the weather criterion is to ensure

the sufficient stability of the ship to withstand the

severe wind guests during rolling. Although the

weather criterion is a very simple model of dynamic

ship’s behavior, 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 rest of

criteria are based on the statistics of historical

disasters only (Francescutto 2002).

According to the IMO recommendations the

righting lever curve should be corrected for the

effect of free surfaces of liquids in tanks. The

correction may be done by any of three accepted

methods (IMO 2002):

− correction based on the actual moment of fluid

transfer calculated for each angle of heel;

Dynamic Component of Ship’s Heeling

Moment due to Sloshing vs. IMO IS-Code

Recommendations

P. Krata

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: The comparative study of the dynamic component of heeling moment due to sloshing in ships’

partly filled tanks is presented in the paper. The characteristics of heeling moment are obtained in the course

of experimental tests and numerical simulations. The heeling moment is decomposed and the research is

focused on the dynamic component resulting from liquid movement. The results of the research are compared

to the computations performed in accordance with the IMO IS-

Code recommendations. The need for

amending of the intact ship stability assessment procedure is suggested.

364

− correction based on the moment of inertia of

tank’s horizontal projection (simple pendulum

model);

− correction obtained form the simplified formula

given in the Intact Stability Code.

All of the three mentioned above methods of free

surface correction calculation consider the static

attitude towards the sloshing phenomenon only.

They also do not consider the localization of the tank

within the hull of the ship and the localization of the

rolling axis. The only advantage of current

compulsory corrections is the simplicity of their

calculation.

2 RESEARCH INTO THE PRESSURE

DISTRIBUTION IN A MOVING TANK

2.1 Research assumptions

The scheme of undertaken research comprises

physical model tests and numerical simulations as

well. The admitted assumptions refer to both and

they describe dimensions of the model tank, its

movement geometry and characteristics, tank’s

filling level.

The oscillating movement, which induces the

sloshing phenomenon, is described fair enough by

the harmonic function. The research into the

pressure distribution due to the sloshing was

performed for a variety of the external excitation

parameters. The period of the oscillation varied from

T=2,6 s to T=6,5 s. The lever os, as the distance

between the center of the tank and the rotary motion

axis, was changed from os=-0,718 m to os=0,718 m.

The positive value of os describes the tank’s

localization beneath the rolling axis and the negative

value of os describes the tank’s localization above it.

The amplitude of tank’s rotary motion during the

model tests and numerical simulations, assumed to

be 40º. It reflects the heavy seas conditions and

enables to make the conclusions for worst possible

condition at the sea. The tank filling level assumed

to be 30%, 60% and 90%.

2.2 Experimental investigation

The experimental research into the sloshing

phenomenon was performed in Ship Operation

Department of Gdynia Maritime University. It

enabled to measure the dynamic pressure

distribution on the sidewall 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 were measured and

recorded. To achieve this, the test apparatus was

designed and built (Krata 2006).

The main part of the apparatus is the tank. It is

equipped with pressure transducers and an

inclinometer. The tank is forced to oscillating

movement that excites the water movement inside it.

The dimensions of the model tank are: breath –

1,040 m, length – 0,380 m, depth – 0,505 m.

The assumption of plane tank’s oscillation and

the neglected water viscosity, results the two-

dimensional character of water flow inside the tank

(Warmowska, Jankowski 2005). It allowed

equipping the tank with one set of pressure

transducers, fixed in the middle line of the tank. The

pressure transducers were installed evenly alongside

the vertical wall of the tank and in the roof of the

tank close to the upper corner. The experimental

setup is shown in Figure 1. The schematic plan of

the apparatus is shown in Figure 2.

Figure 1. The experimental setup (the tank placed above the

shaft – one of possible cases)

Figure 2. The scheme of the testing apparatus and the

localization of dynamic pressure gauges named P1 to P6 and

the inclinometer L

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The location of pressure transducers installed in

the front wall of the tank and in its upper corner is

specified in the Table 1. Any further details are

described in (Krata 2006).

Table 1. Geometry of pressure gauges installation

The analog signals received from the sensors

were sampled and transformed into discrete digital

signals by the 12-bit A/D card and then they were

recorded. The maximum working frequency of the

measuring device was 1000 Hz. Thus, the aliasing

distortions of the measured signal were avoided,

because the measuring instruments were much faster

than the required Nyquist rate for the sloshing

phenomenon.

The further digital signal processing was carried

out. The main operation was low pass filtering for

high frequency noise reduction. The filtering

enabled to decompose the recorded digital signal and

emerged the non-impulsive dynamic pressure

component.

2.3 Numerical simulation

The pressure distributions obtained in the course of

the experimental investigation were completed by

the results of numerical simulations. The simulations

of sloshing phenomenon were performed by the

computer program “Tank” by M. Warmowska, used

for the estimation of the dynamic pressure

distribution. The sloshing problem was described by

two-dimensional model. It was also assumed that the

liquid is non-viscid, incompressible, of constant

density. As the flow of the liquid assumed to be

irrotational, the potential theory was used to solve

the sloshing problem (Jankowski, Warmowska

1997).

The numerical simulation of sloshing

phenomenon was performed for the oscillation and

tank’s geometry corresponding with the suitable

geometric parameters of the experimental

investigation. The program allows computing time

history of dynamic pressures in ninety points around

the tank’s model. The control points are situated

along vertical walls, the bottom and the tank’s roof.

The correctness of the simulation results was

verified experimentally (Krata 2006).

3 HEELING MOMENT DUE TO SLOSHING

3.1 Computation of heeling moment

The pressure distribution on the walls of the tank

was obtained in the course of the experimental tests

and numerical simulation. The results of the research

enable to compute a heeling moment due to the

liquid’s sloshing. The heeling moment M was

calculated according to the following formula:

∫

⋅×=

dsp nrM

(1)

where: S – the surface of the tank’s walls; r – the

position vector of the considered point on the tank’s

wall; n – the normal vector; p – the local pressure on

the tank’s wall.

Due to the two-dimensional character of the

considered flow in the tank, the heeling moment is a

vector of a direction perpendicular to the plane of

the tank’s movement. As the transverse stability of a

ship is assumed to be considered, the heeling

moment has one spatial component only, as follows:

[ ]

0,0,,,

xzyx

MMMM ==M

(2)

where: M

, M

– spatial components of M vector,

determined about the x, y and z axis in the reference

system fixed to the vessel.

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 value of M

spatial component. The resultant moment obtained

from the formula (1) represents one time-step only.

The computation of heeling moment should be

performed for at least one period of roll. Thus, the

pressures have to be investigated for at least one

period of ship’s roll as well, but actually they were

obtained for the longer time comprising few rolling

periods. The example of the heeling moment history

graph is presented in Figure 3.

366

Figure 3. The time-domain presentation of the computed

heeling moment due to sloshing

The time domain presentation of the computation

results can be useful when the ship’s rolling is to be

computed on the basis of movement equations. In

such case, the heeling moment due to sloshing is one

of the components of total heeling moment rocking a

vessel at seaway.

3.2 Linearization

The time-domain manner of presentation of the

heeling moment due to sloshing which is shown in

Figure 3 as a moment history graph is not

convenient in respect of traditional ship’s stability

assessment (Krata 2008). Such stability assessment

is not based on the movement equations, but on the

static stability curve (IMO 2002). The curve presents

the righting arm GZ in the angle of heel domain and

the righting arm is reduced by the statically

calculated free surface correction. Therefore, the

most convenient way to present the results of the

heeling moment calculation due to the sloshing of

liquid in a partly filled tank is the angle of heel

domain graph.

The interpretation of the results of heeling

moment computation is much more convenient in

angle of heel domain. The main disadvantage of

such presentation is the hysteresis, which is the

effect of wave type phenomena taking place inside

the moving tank. The disadvantage can be removed

by the linearization process (Krata 2008).

As the main task of the research is more reliable

stability assessment with regard to the sloshing

phenomenon, the linearization should refer to the

ship’s stability criteria, especially the weather

criterion. The area under the GZ curve is qualified

within the weather criterion, which represents the

work of heeling moment due to wind guests when a

ship rolls, so the linearization of the researched

heeling moment should be based on the work of the

moment as well. The linearization method applied to

the heeling moment due to the sloshing of liquids is

based on the formula:

∫∫ ∫

⋅=⋅+⋅

4040

)()(

ϕϕ

ϕϕϕ

dMdMdM

(3)

where: M – heeling moment due to sloshing; M

–

resultant linear heeling moment due to sloshing;

–

angle of ship’s heel;

–angle of heel equal 40°

(given in radians).

The formula (3) ensures equality of works done

by the researched heeling moment and linear heeling

moment due to sloshing. Thus, the method may be

called the equivalent work method. The example of

linear heeling moment due to sloshing is presented

in Figure 4.

Figure 4. The example of the linear heeling moment, for the

filling level 30% and os=-0,718 m.

The linear function of heeling moment can be

determined by the fixing of two in-line points having

the coordinates (

, M). One of them is the point (0,

0) and the second one the point (40°, M

l40

Therefore, the complete description of the linear

heeling moment obtained in the course of the

research may be done by one scalar only, which is

convenient for any further analysis.

3.3 Extraction of dynamical component of the

heeling moment due to liquid sloshing

The moment M heeling a ship in consequence of

liquid existence carried in any partly filled tank, may

be decomposed into two components. One of them is

the moment M

of liquid weight and the second is

the heeling moment M

due to the movement of

fluid inside the tank. The heeling moment due to

liquid sloshing in vessel’s tanks can be described in

every time-step by the formula:

367

RBm

MMM +=

(4)

where: M

– heeling moment due to the weight of

“frozen” liquid in tank; M

– heeling moment due

to the movement of fluid inside partly filled tanks.

The simple sum of moment components

analogous to the formula (4) was applied to the

linear heeling moment M

calculated according to the

formula (3) for all considered cases. Thus, the

component M

of the heeling moment due to

sloshing abstracts the static effect of liquid weight in

ship’s tanks. Such abstraction capacitates to bear

comparison of the performed research results with

the quasi-static heeling moment computed according

IMO IS-Code recommendations.

4 RESULTS OF THE RESEARCH

4.1 Comparison of the research results and IMO

IS-Code recommended computation

The research is focused on the comparative analysis

of the heeling moment components arising from the

liquid movement inside ship’s partly filled tanks.

One of them is obtained in the course of the research

and it reflects the dynamic attitude towards the

sloshing phenomenon. The other is calculated

according the IMO IS-Code recommendations and it

is of quasi-static type. The computation formulas

resulted from IS-Code prescriptions.

The heeling moment M

IMO

due to liquid’s

existence inside any partly filled tank may be

decomposed into two components according to the

formula:

RIMOmIMO

MMM +=

(5)

where: M

– heeling moment due to the weight of

“frozen” liquid in tank; M

IMO

– heeling moment of

the transfer of the liquid’s center of gravity.

The moment M

is taken into consideration in

course of the calculation of the ship’s center of

gravity and it assumes the liquid to be “frozen” at

the angle of heel equal 0°. It is important to notice,

that the M

component is equal in formulas (4) and

(5). Therefore, the remaining components M

and

IMO

of the heeling moment may be compared. The

component M

IMO

of the heeling moment can be

calculated at any of the three accepted method. The

simple pendulum model is considered as safest for

the ship therefore the free surface correction based

on the moment of inertia of tank’s horizontal

projection was applied in the course of the further

comparison.

The quasi-static component M

IMO

of heeling

moment is a function of sine of the angle of heel.

Anyway, it could be compared to the researched

linear component of the heeling moment due to

sloshing for the range of angles of heel where the

sine function may be approximate by linear function

fair enough. The reasonable range of such linear

approximation is about 40°, which shows Figure 5.

Figure 5. Linear approximation of sine function

As the sine function is almost linear up to the

angel of heel 40°, the components M

and M

IMO

the heeling moment may be compared. They both

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°. Thus, the

values M

RB40

and M

IMO40

are analyzed instead of the

moment graphs. The comparison of the M

RB40

values

obtained in the course of the research and the M

IMO40

computed according to the IMO recommendations is

shown in Figure 6.

Figure 6. Non-dimensional component of heeling moment due

to liquid movement in partly filled tanks.

The graphs showing analyzed values of the

component of heeling moments are prepared as non-

dimensional referred to the value M

IMO40

of static

free surface correction. The excitation period T is

referred to the first harmonic natural sloshing period

of a liquid in model tank T

. The scope of T/T

ratios reflects the wide variety of characteristics they

can take place on board of ships at different loading

conditions. The distance os between the center of the

moving tank and the rotary motion axis is referred to

368

the breath of the tank b

. The three graphs marked

30%, 60% and 90% are plotted for the

corresponding three levels of tank filling. The

reference surface marked IMO is plotted for M

IMO40

values calculated according the IMO IS-Code

requirements.

4.2 Analysis of the obtained results

The quasi-static heeling moment component

represented by the free surface correction described

in IMO IS-Code depends on the shape of a partly

filled tank only. Presented results of the research

prove the significant influence of other factors. One

of the most important is the localization of the tank

referred to the vessel’s rolling axis os/b

. The

excitation period referred to the first harmonic

natural sloshing period of a liquid in model tank

seems to be less important. The lowest investigated

values of T/T

ratios can occur for very short ship’s

rolling period typical for extremely stable ships. In

any other cases, the T/T

ratio does not play the

important role.

The graph presented in Figure 6 enables the

identification of potential danger to a vessel caused

by the movement of liquid in partly filled tanks. Any

value of analyzed heeling moment component

higher than the reference level IMO should be

considered as potentially perilous to a vessel because

her transverse stability can be worse than calculated

according to IS-Code recommendations.

The surface plotted for 30% of tank filling

demonstrates that such a low level of filling does not

need to be considered as risky one. The influence of

liquid sloshing is weaker than that taken into

account in the course of standard stability

assessment. The only trespass of the reference IMO

level is noticed for the shortest rolling period, which

can take place in the case of large GM only.

The surface plotted for 60% tank filling level

reveals the fair conformability of the research results

and IS-Code recommendations for partly filled tanks

situated above the ship’s rolling axis and the

considerable transgression for tanks placed below

the rolling axis. The potentially dangerous

underestimation of the liquid sloshing influence on

the ship’s transverse stability occurs for all

researched rolling periods.

The surface plotted for 90% of tank filling prove

the potentially perilous situation, which can take

place for high levels of tank filling. The effect of

liquid sloshing is slightly overrated for partly filled

tanks situated above the ship’s rolling axis when

computed according to IS-Code. Such an effect may

be considerably underestimated for tanks sited

below the rolling axis, for instance double bottom

tanks.

5 CONCLUSIONS

The movement of liquids in partly filled ship’s tanks

affects her stability and therefore it is considered in

course of the stability assessment procedure

according to the IMO recommendations. The results

of the research presented in the paper points that the

very simplified methods recommended by IMO

could be improved and reach better accuracy to meet

the modern requirements of ship’s exploitation.

The presented comparative analysis of the

components of heeling moment reveals some

weaknesses of IS-Code. The use of current IS-Code

recommendations may lead to considerable

underestimation of free surface effect. This results

from the quasi-static attitude towards the sloshing

phenomenon. The analysis proves that the dynamic

movement of liquids in partly filled tanks should not

be neglected. The results of the research can

contribute to the further investigation of the new

formula of free surface correction comprising the

dynamics of sloshing phenomenon.

REFERENCES

Akyildiz H., Unal E., 2005, Experimental investigation of

pressure distribution on a rectangular tank due to the liquid

sloshing, Ocean Engineering, 32, 1503–1516

Francescutto A., 2002, Intact Ship Stability - the Way Ahead,

Proc. 6th International Ship Stability Workshop,

Washington

Intact Stability Code 2002, IMO, London

Jankowski J., Warmowska M., 1997, Development of computer

program describing the flow in partly filled tank, Technical

Report No.27/97, PRS, Gdańsk

Krata P., 2006, The comparative study of numerical simulation

and experimental research into the areas of maximum

dynamic pressure due to sloshing in ship’s double bottom

tank, Proc. XV-th International Scientific and Technical

Conference Nav-Support ‘06, Gdynia

Krata, P. 2008. Linear characteristics of sloshing phenomenon

for the purpose of on-board ship’s stability assessment,

XVIII Krajowa Konferencja Mechaniki Płynów, Jastrzębia

Góra.

Warmowska M., Jankowski J., 2005, Simulation of water

movement in partly filled ship tanks, Proc. HYDRONAV

‘2005, Ostróda