International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 2
Number 4
December 2008
389
Model of Interaction of Water and Tank’s
Structure in Sloshing Phenomenon
P. Krata
Gdynia Maritime University, Gdynia, Poland
ABSTRACT: The paper presents the model of interaction between ship’s tank structure and water contained
in the tank. The experimental investigation of sloshing loads in ships’ tank is characterized and the need for
description of mutual influence is underlined. The model of interaction is made up for rectangular ship’s tanks
and it is assumed to be in time-domain and nonlinear. The solution is based on the data obtained by the
experimental research and numerical simulation. The study may be the contribution to more sophisticated
estimation of ship’s stability than it takes place nowadays.
1 INTRODUCTION
The dynamic behavior of the vessel at the sea is
greatly affected by the dynamics of moving masses
existing onboard. The cargo securing procedures
ensure avoiding the loose cargo moving, but the
liquids contained in partly filled tanks cannot be
avoided at all. The modeling of interaction between
water sloshing inside the ship’s tank and the tank’s
structure is very important with regards to the safety
of transportation system, human’s life and
environment. The sloshing loads should be also
taken into consideration in the process of designing
of tank’s structure and ship’s hull structure.
Regardless the strength calculation the effects of
sloshing should be also taken into consideration in
course of vessel’s sea keeping prediction and
transverse stability assessment.
2 EQUATIONS OF SHIP’S MOVEMENT
The most precise analysis of ship’s movements
should be based on the differential equations of
movement. The most convenient attitude towards
this task is to make the assumption of symmetrical
mass distribution and steady values of moments of
mass inertia. These equations referred to the center
of gravity G are following (Dudziak 1988):
GG
G
MPVK
dt
Kd
FP
dt
Pd
=×+×+
=×+
ω
ω
(1)
where:
P
- momentum of the ship and added masses
[kg∙m/s];
G
K
- angular momentum of the ship and
added masses referred to the center of gravity G
[kg∙m
2
/s];
V
- velocity of point G [m/s];
ω
- ship’s
angular velocity [1/s];
F
- resultant external force
[N];
G
M
- resultant moment of external forces about
center of gravity G [N∙m].
When formula (1) is divided into components and
momentum and angular momentum is derived, then
movement complex has the form of six differential
equations (Dudziak 1988):
390
zzxzyxxyyyx
x
zx
z
zz
yzxxzxxzzxz
y
yy
xzxyxyyzzzy
z
zx
x
xx
zxyyx
z
yzxxz
y
xyzzy
x
MIII
dt
d
I
dt
d
I
MIII
dt
d
I
MIII
dt
d
I
dt
d
I
FVVm
dt
dV
m
FVVm
dt
dV
m
FVVm
dt
dV
m
=+−+−
=−−−+
=−−+−
=−+
=−+
=−+
ωωωω
ω
ω
ωωωω
ω
ωωωω
ω
ω
ωω
ωω
ωω
)(
)()(
)(
)(
)(
)(
22
(2)
where: I
ii
- moment of mass inertia about appropriate
axes (i = x, y, z) [kg∙m
2
]; I
ij
- moment of mass
deviation about appropriate axes (i ≠ j) [kg∙m
2
];
m – mass of the ship and added masses [kg]; and the
rest of the symbols like in formula (1).
The solution of such general formulated
movement’s equations is impossible at present state
of the art. By neglected coupling, for the sake of
simplicity, the ship’s rolling is usually analyzed by
the single degree-of-freedom system. The governing
differential equation of motion, as the result of
equilibrium of moments, is (Senjanovic I. &
Parunov J. & Cipric G. 1997):
)()()( tMRDI =++
•••
ϕϕϕ
(3)
where: I - moment of inertia of ship and added
masses [kg∙m
2
]; D - damping moment [N∙m]; R -
restoring moment [N∙m]; M - excitation moment
[N∙m];
ϕ
- angle of heel [rad];
•
ϕ
- angular velocity
of heel [1/s];
••
ϕ
- angular acceleration [1/s
2
].
The resultant excitation moment M(t) consists of
as many components, as many influences swing the
ship. The main components are waves and wind.
Anyway when the prediction of ship rolling should
comprise the effects of water sloshing in partly filled
tanks, then the moment of water-ship interaction
have to be included as a component of M(t) in
formula (3). The moment due to sloshing water
should be obtained in time-domain.
3 CHARACTERISTIC OF SLOSHING
PHENOMENON
Liquid sloshing phenomenon is the result of
partly filled tank motions. As the tank moves, it
supplies energy to induce and sustain the fluid
motion (Akyildiz & Unal 2005). Both: liquid motion
and its effects are called sloshing (Warmowska
2005).
The effect of water motion inside the tank is the
pressure on the tank’s structure. The hypothetical
pressure distribution on the side wall of rectangular
tank is shown in figure 1. The dimensions of the
wall are y by z.
The interaction between ship’s and tank’s
structure and the water sloshing inside the tank
consists in the permanent transmission of the energy.
As the ship rolls, the walls of the partly filled tank
induce the movement of water. Then the water press
against the opposite situated tank’s walls and return
the energy to the ship, taking simultaneously the
next portion enabling the counter-direction
movement. The mass and the energy are conserved
within the cycle.
Fig. 1. Pressure distribution on the side wall of rectangular tank
(source (Krata 2007))
If the influence of sloshing water dynamics is to
be taken into consideration when ship’s sea keeping
predicted, then the moment of sloshing loads have to
be obtained and put into formula (3) as a component
of M(t). The calculation of time-domain history of
heeling moment due to sloshing is based on the
dynamic pressure distribution and its integration.
The pressure distributions have to be determined on
both side walls of the considered tank, on its roof
and the bottom. The assumption of simplified
formula (3) is getting rid of any coupling of
movements, so the time changing pressure
distributions on fore and aft tank’s wall are not
required.
391
4 PRESSURE DISTRIBUTION DUE TO
SLOSHING IN SHIP’S TANK
4.1 „Tank” program - assumptions and simulation
The computer program “Tank”, used for estimation
of dynamic pressure distribution due to sloshing,
is developed in Polish Register of Shipping.
The computation algorithm is based on the Euler
equation (Jankowski & Warmowska 1997). The
sloshing problem was described by two-dimensional
model (Warmowska & Jankowski 2005). It was
also assumed that the liquid is non-viscid,
incompressible, of constant density. The additional
assumption refers to the liquid boundary. It is
assumed that (Warmowska & Jankowski 2005):
− the liquid particles slide on the free surface and
on the wetted surface of the tank walls;
− the particles in the corners are not moving;
− pressure on the liquid free surface is equal to
the atmospheric pressure.
All these assumptions allow for application of the
potential theory to solve the problem (the flow is
irrotational).
The numerical simulation of sloshing
phenomenon, made by “Tank” program, was
performed for the oscillation and tank’s geometry
corresponding with the suitable geometric parameters
of experimental investigation. The angle amplitudes
of tank’s oscillations were 18°, 30° and 40° and the
height of water level in tank varied from 150 mm to
450 mm. The program allows to compute 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.
4.2 Experimental investigations
The experimental investigation on determining the
pressure distribution due to sloshing requires the
generation of sloshing phenomenon. After that the
dynamic pressure time history in selected places is to
be 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 six pressure transducers and one
inclinometer. The tank is forced to oscillating
movements that excite the water movement inside it.
The dimensions of the model tank are following:
− length - 1040 mm;
− width - 380 mm;
− depth - 505 mm.
The tank is hanged on the shaft by the bearings
and forced to the oscillation by the driving
mechanism.
The drive mechanism is based on the electric
motor, the transmission reducing revolution velocity
and the crank mechanism. The view of testing
apparatus and localization of dynamic pressure
sensors is shown in figure 2.
Fig. 2. Picture of the tank and pressure gauges
The oscillating movement which induces the
sloshing phenomenon is described fair enough by the
harmonic function. The amplitude of tank’s rotary
motion assumed to be 18°, 30° and 40°. It reflects
the heavy seas conditions and enables to make the
conclusions for worst possible condition at the sea.
The water depth in tank (tank filling level) assumed
to vary from 50 mm to 450 mm. The period of the
oscillation was equal 2,6 s.
The assumption of plane tank’s oscillation and
the neglected water viscosity, resulted the two-
dimensional character of water flow inside the tank
(Warmowska & Jankowski 2005). It allowed to
equip 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 (5 sensors) and one in the
roof of the tank close to the upper corner.
The pressure signal, measured by the transducer,
consists of two components. One of them is called
non-impulsive dynamic pressure and the second one
impulsive pressure or impact pressure (Akyildiz &
Unal 2005). The non-impulsive dynamic pressure is
slowly varying. It is the result of global movement of
liquid in the tank (CTO 1998). The impact pressure
is usually short lasting, local and may be of very
high value. It is caused by hydraulic jump during the
impact stroke of liquid’s free surface against the
solid surface of the tank construction (wall). The
assumption of the experiment was to measure and
392
record both components of dynamic pressure. The
measure appliances have to be fast enough.
All the signals received from the sensors were
verified and the measuring instruments were
calibrated. All pressure sensors were hydrostatically
calibrated and the gain coefficient and shift
coefficient were determined. The inclinometer was
calibrated by geometric formulas. The calibration
procedure allowed to deem the experimental
measurements to be correct and reliable (Krata
2006).
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 in the text format files.
The maximum working frequency of measuring
device was 1000 Hz. Thus the aliasing distortions of
the signal were avoided, because the measuring
instruments were much faster than the required
Nyquist rate for sloshing phenomenon (Zieliński
2002).
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 recorded digital signal
an emerged the non-impulsive dynamic pressure
component (Zieliński 2002).
The example of pressure distribution estimated
on the basis of experimental investigations is shown
in figure 3.
Fig.3. Estimation of pressure distribution on the side wall of
model tank (source (Krata 2007))
5 MODEL OF INTERACTION OF SLOSHING
WATER AND TANK’S STRUCTURE
5.1 Assumptions for heeling moment calculations
The mathematical model of interaction between
water sloshing inside the ship’s tank and the hull’s
structure is prepared for rectangular tank. It
corresponds with the shape of the tank, which was
used during the numerical simulations described in
point 4.1. and the experimental research described in
4.2. The example of tank’s localization and sloshing
forces is shown in figure 4.
Fig. 4. The arrangement of forces affecting tank’s structure
It is assumed, that the rolling motions of the ship
and the tank take place about the rolling axe which is
perpendicular to the plane of figure 4 and contains
point O. The forces F
1
to F
6
are local values and they
acts on both side walls of the tank and its roof and
bottom, as shown in figure 4.
The local value of the force acting on the tank’s
structure can be obtained from the formula:
dspF ⋅=
(4)
where: F - local force [N]; p - local pressure [Pa]; ds
- infinitesimal segment of tank’s wall area [m
2
].
Taking into consideration the two-dimensional
character of water flow inside the rectangular tank,
described in point 4.2. the force may be calculated
from two formulas:
dzlp
F
zV
⋅⋅=
)(
(5)
dylp
F
yH
⋅⋅=
)(
(6)
where: F
V
- local value of force due to sloshing
acting on vertical side walls of the tank [N]; F
H
-
local value of force due to sloshing acting on
horizontal roof and bottom of the tank [N]; l - length
393
of the tank (along x-dimension) [m]; z - vertical co-
ordinate [m]; y - transverse co-ordinate [m].
The local value of moment of force due to
sloshing is the simple product of multiplying: force
by the lever of acting the force about the rolling axe.
The lever can be defined as the distance between the
line of force acting and the point O, as shown in
figure 5 for horizontal force direction and in figure 6
for vertical force direction.
Fig. 5. The lever of heeling moment of sloshing force acting
on the tank’s side wall
Fig. 6. The lever of heeling moment of sloshing force acting
on the tank’s roof
The localization of side walls of the tank and its
roof and bottom is fixed by the vertical and
transverse co-ordinates. It is assumed that the bottom
of the tank is situated at the height z
min
and its roof at
the height z
max
. The port side wall of the tank has the
transverse co-ordinate y
min
and the starboard side
wall y
max
. The symmetry plain has the y co-ordinate
equal zero. Taking all the assumptions into
consideration, the total values of force moments on
the individual walls numbered 1 to 6 (see Fig. 4) can
be calculated as the following integrals:
∫
∫
∫
∫
∫
∫
⋅⋅⋅=
⋅⋅⋅−=
⋅⋅⋅=
⋅⋅⋅=
⋅⋅⋅−=
⋅⋅⋅−=
max
min
max
min
min
max
max
min
0
)()(66
0
)()(55
)()(44
0
)()(33
0
)()(22
)()(11
y
yy
y
yy
z
z
zz
y
yy
y
yy
z
z
zz
dylrpM
dylrpM
dzlrpM
dylrpM
dylrpM
dzlrpM
(7)
where: M
i
- force moment on i-numbered tank’s wall
[N∙m]; p
i
- local pressure on i-numbered tank’s wall
[Pa]; rest of symbols same as in formula (5) and (6).
The total value of heeling moment due to sloshing
inside the ship’s tank is calculated as the sum:
∑
=
=
6
1i
i
MM
(8)
The heeling moment is obtained from the formula
(8) for one time-step only. The time domain
calculation of heeling moment, which is required to
be put into formula (3) governing ship’s rolling, has
to be performed for at least one rolling period. Thus
the pressures p
1
to p
6
have to be investigated for the
time of one rolling period as well.
5.2 Localization of rolling axe
The rolling motion of the ship’s hull takes place
about the non-steady positioned axe. The area, where
the rolling axe occurs, depends mainly on the shape
of the hull, its damping moments in the water,
character of its external excitation, coupling with
motions in any other degree-of-freedom. In course of
practical computations of ship’s rolling at the sea,
the fixed placement of the roll axe is assumed.
The most often used approximation of the rolling
axe localization is the ship’s centre of gravity G
(Dudziak 1988). The more exact approximation of
the rolling axe localization is described by Balcer
(Balcer 2000). The damping coefficients of roll and
sway were taken into consideration. The values of
added masses were obtained with respect of strip
theory for simplified ship’s shapes. The
computations were made for numerous existing
vessels of different size and shapes (Balcer 2000).
394
The final formula obtained in course of such
reasoning is following:
BzTz
GO
⋅−⋅+⋅= 1,043,057,0
(9)
where: z
O
- height of the rolling axe above the base
line [m]; T - mean draught [m]; z
G
- height of the
centre of gravity above the base line [m]; B - breadth
of the ship [m]. The formula (9) enable to obtain the
height of rolling axe and the axe is situated in the
symmetry plain of the ship, what fix the axe clear-
cut within the hull’s body.
5.3 Results of computation
The computations of time-domain heeling moment
due to sloshing were performed for two localizations
of rolling axe (beneath the tank and above of it) and
three angular amplitudes for each case: 18, 30 and
40 degrees. The pressure distributions on the tank’s
bottom and the roof (except of the upper corner)
were obtained by numerical simulation described in
point 4.1. The distributions of water pressures
affecting the side walls of the tank and its upper
corner were investigated in course of experimental
research described in point 4.2.
The moments of forces acting on any of the tank’s
walls were obtained from the formula (7). The total
value of heeling moment due to sloshing inside the
ship’s tank was calculated from the formula (8). The
example of resultant heeling moment obtained in
time-domain is shown in figure 7. The moment was
calculated for angular amplitude 30°, water depth
400 mm and the rolling axe situated 1220 mm above
the tank’s roof.
Fig. 7. The time-domain heeling moment due to sloshing
The results of presented computations may
comprise a part of compound estimation of
ships rolling movement including the sloshing
phenomenon. Thanks to the time-domain presenta-
tion, presented in the paper, computations may be
also considered regarding to the phase shift of the
heeling moment and the external moment of ship’s
roll excitation.
6 CONCLUSIONS
The model presented in the paper comprise the
complete procedure of the time-domain calculation
of heeling moment due to sloshing in ship’s tanks. It
is based on the investigated sloshing phenomenon.
The experimental research and numerical simulation
were performed.
The model enables the determining of reliable
interaction between water and ship’s hull, including
the influence of tank’s localization. This aspect is
the step ahead in comparison with free surface effect
considered in course of standard procedure of
stability evaluation nowadays. The proposed model
improves the credibility of estimation of the
dynamical free surface effect. It is much more
precise than quasi-static attitudes towards sloshing
phenomenon. The dynamic behavior of the water
sloshing inside the partly filled tank was passed over
so far.
There are available brand new publications
covering the problem of coupling between the
sloshing phenomenon and ship’s movement. The
Polish Register of Shipping developed the computer
program, which enables the time domain calculation
of ship’s rolling when carrying the liquid cargo
(Warmowska & Jankowski 2006). Anyway the
procedure of dynamic heeling moment calculation is
not published and the results of numerical simulation
are showed only. The model used by PRS is based
on the potential theory of fluid dynamics and there
isn’t any possibility to put into the calculation any
experimental data. Moreover the simulations
presented during the National Conference of Fluid
Dynamics in 2006, were performed for one tank’s
filling level only and one loading condition of the
ship. Anyway such publication as mentioned above
shows the pending direction of the research in the
marine industry.
The time domain estimation of heeling moment
due to sloshing can be used in any ship’s movement
equation, for instance like formulas (2) or (3).
It makes the model universal part of time domain
considerations regarding vessel’s sea keeping.
REFERENCES
Akyildiz H., Unal E., „Experimental investigation of pressure
distribution on a rectangular tank due to the liquid
395
sloshing”, Ocean Engineering 32 (2005),
www.sciencedirect.com
Balcer L., “Model matematyczny kołysań statku z biernym
zbiornikiem stabilizacyjnym”, rozprawa doktorska, Wydział
Oceanotechniki i Okrętownictwa Politechniki Gdańskiej,
Gdańsk 2000 (in Polish).
Centrum Techniki Okrętowej, „Porównanie wyników
eksperymentu modelowego z wynikami obliczeń
numerycznych przy użyciu programu LR FLUIDS”, Raport
Techniczny Nr RK-98/T-055, CTO, Ośrodek Mechaniki
Konstrukcji Okrętu, Gdańsk 1998 (in Polish).
Dudziak J., „Teoria okrętu”, Wydawnictwo Morskie, Gdańsk
1988 (in Polish).
Jankowski J., Warmowska M., „Development of computer
program describing the flow in partly filled tank”, PRS,
Technical Report No.27/97, Gdańsk, 1997.
Krata P., „Experimental determination of the areas of maximum
liquid sloshing loads in moving rectangular ship’s double
bottom tank”, Proceedings of the XV-th International
Scientific and Technical Conference “The Role of
Navigation in Support of Human Activity on the Sea”,
Gdynia, Poland 2006 (in Polish).
Krata P., “Determining of maximum water thrust affecting
ship’s tank structure based on the experimental research of
sloshing phenomenon”, Scientific Works of Navigational
Faculty of Gdynia Maritime University No 20, Gdynia 2007
(in Polish).
Senjanovic I., Parunov J., Cipric G., “Safety Analysis of Ship
Rolling in Rough Sea”, Chaos, Solitons and Fractals, Vol.
8, No 4, Elsevier Science Ltd, Great Britain 1997.
Warmowska M., Jankowski J. „Ship motion affected by moving
liquid cargo in holds”, Krajowa Konferencja Mechaniki
Płynów, Bełchatów 2006.
Warmowska M., Jankowski J. „Simulation of water movement
in partly filled ship tanks”, HYDRONAV ‘2005, Ostróda
2005.
Warmowska M., „Określenie parametrów ruchu cieczy w
zbiorniku okrętowym metodą rzutowania z uwzględnieniem
zjawisk nieliniowych - Ocena dokładności opracowanej
metody za pomocą istniejących w literaturze wyników
obliczeń dla problemów szczególnych, posiadanych
wyników eksperymentu; stabilność metody obliczeniowej”,
PB nr 5 T12C 057 24, Politechnika Gdańska, Gdańsk 2005
(in Polish).
Zieliński P.T., „Od teorii do cyfrowego przetwarzania
sygnałów”, Wydział EAIiE AGH, Kraków 2002 (in Polish).
