International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 4
Number 3
September 2010
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
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.
correction based on the moment of inertia of
tank’s horizontal projection (simple pendulum
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
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
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
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
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
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).
3.1 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
where: S the surface of the tank’s walls; r the
position vector of the considered point on the tank’s
wall; nthe normal vector; pthe 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:
[ ]
[ ]
where: M
, 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.