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

365

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:

∫

⋅×=

S

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

x

, M

y

, M

z

– 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

x

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.