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ARTICLE IN PRESS
0029-8018/$ - see
doi:10.1016/j.oc
�CorrespondiE-mail addre
Ocean Engineering 34 (2007) 10–20
www.elsevier.com/locate/oceaneng
The effects of LNG-tank sloshing on theglobal motions of LNG carriers
S.J. Leea, M.H. Kima,�, D.H. Leea, J.W. Kimb, Y.H. Kimc
aCivil Engineering Department, Texas A&M University, College Station, TX 77843-3136, USAbTechnical Department, American Bureau of Shipping, Houston, TX, USA
cDepartment of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, Korea
Received 26 September 2005; accepted 22 February 2006
Available online 21 July 2006
Abstract
The coupling and interactions between ship motion and inner-tank sloshing are investigated by a time-domain simulation scheme. For
the time-domain simulation, the hydrodynamic coefficients and wave forces are obtained by a potential-thoery-based three-dimensional
(3D) diffraction/radiation panel program in frequency domain. Then, the corresponding simulations of motions in time domain are
carried out using convolution integral. The liquid sloshing in a tank is simulated in time domain by a Navier–Stokes solver. A finite
difference method with SURF scheme is applied for the direct simulation of liquid sloshing. The computed sloshing force and moment
are then applied as external excitations to the ship motion. The calculated ship motion is in turn inputted as the excitation for liquid
sloshing, which is repeated for the ensuing time steps. For comparison, we independently developed a coupling scheme in the frequency
domain using a sloshing code based on the linear potential theory. The hydrodynamic coefficients of the inner tanks are also obtained by
a 3D panel program. The developed schemes are applied to a barge-type FPSO hull equipped with two partially filled tanks. The time-
domain simulation results show similar trend when compared with MARIN’s experimental results. The most pronounced coupling
effects are the shift or split of peak-motion frequencies. It is also found that the pattern of coupling effects between vessel motion and
liquid sloshing appreciably changes with filling level. The independent frequency-domain coupled analysis also shows the observed
phenomena.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Sloshing and motion interactions; Filling level vs. sloshing effect; Coupled dynamic analysis; Navier–Stokes solver with SOLA scheme; Peak
frequency shift and split; Time-domain simulation; Frequency-domain coupled analysis
1. Introduction
In the conventional ship-motion analysis, the effects ofinner free surface and its sloshing inside the liquidcontainer are usually ignored. Recent experimental andnumerical study has shown that the coupling effect betweenliquid cargo sloshing and LNG ship motion is significant atpartial filling conditions. This is of a great concern to theLNG FPSO/FSRU operation in the production site andoffloading operation of LNG carriers close to LNGterminal, which is particularly so as the size of LNGcarriers significantly increases with greater demand.
front matter r 2006 Elsevier Ltd. All rights reserved.
eaneng.2006.02.007
ng author. Tel.: +1979 847 8710; fax: +1 979 862 8162.
ss: [email protected] (M.H. Kim).
The response of a LNG carrier during offloadingoperation is one of the crucial factors to the safety andoperability of offshore LNG terminals. Nevertheless, theinfluence of the time-varying liquid cargo and its sloshingon global tanker motions for various loading conditionshas rarely been investigated. In the present study, thecoupling effects between the vessel motions and liquidsloshing in partially filled condition are investigated in bothfrequency and time domain.The coupling between ship motion and sloshing has been
studied by Molin et al. (2002), Malenica et al. (2003), Kimet al. (2003) and Newman (2005) based on linear potentialtheory in the frequency domain. In the present study, atime-domain simulation method with nonlinear viscoussloshing calculation is adopted. All the hydrodynamic
ARTICLE IN PRESSS.J. Lee et al. / Ocean Engineering 34 (2007) 10–20 11
coefficients of ship motion are calculated from three-dimensional panel method and they are incorporated in thetime-domain equation through Kramers Kronig relation.Since the nonlinear viscous sloshing calculation is used, thefree-surface motion inside the liquid tank is not necessarilysmall. On the other hand, the independently developedlinear potential theory in frequency domain assumes smallmotions of liquid sloshing. When the liquid sloshingmotion is mild, both approaches should produce similarcoupling effects unless viscous effects are important. It isreported in Bass et al. (1985) and Lee et al. (2005) that theviscous effect on sloshing motion is not appreciable.
When solving the motion of LNG carrier, linearpotential theory is used under the assumption of small-amplitude ship and wave motions. It is well known that thelinear diffraction–radiation potential theory reproduces thevessel motions pretty well except the roll. The hulldiffraction/radiation problem in the frequency domain issolved by a three-dimensional panel program WAMIT(Lee, 1995). Then, the vessel motions are simulated in timedomain using the inverse Fourier transformation of thefrequency-domain results (Kim et al., 1999). In the time-domain vessel-motion simulation, the nonlinear hullviscous damping can be included.
A FDM-based sloshing analysis program, ABSLO3D,has been used for the numerical simulation of liquidmotion inside the tank including impact pressure (Kim,2001). When the fluid motion is so violent that the tankboundaries are exposed to impact loads, some localphysical phenomena are extremely difficult to analyze.For example, splash and wave breaking are typicalphenomena in violent flow, but too much effort is neededfor such a case (Kim, 2001). The primary concern ofABSLO3D is the global fluid motion causing nonbreakingor nonsplash loads on the tank wall, therefore, such localphenomena with strong nonlinearity are not considered.
The ship and liquid-cargo motions are coupled by thekinematic and dynamic relations in that the vessel motionwill excite the tank sloshing, while the sloshing-inducedload will in turn influence vessel motions. The calculated
-100-50
050
100
-20-10
010
Fig. 1. Grid generation of hull for WA
ship motions with or without liquid sloshing are comparedwith the model test results. The model test was conductedby MARIN as a part of SALT JIP (Gaillarde et al., 2004).The numerical results generally compare well with themodel test.
2. Frequency domain formulation
2.1. Ship motion
In this study, a panel-based three-dimensional (3D)diffraction/radiation program, WAMIT, is used to obtainhydrodynamic coefficients and linear/drift wave forces. Thedetailed mathematical background can be found in Lee(1995). Fig. 1 represents the mesh generated for the presentstudy.The results of the diffraction/radiation analysis include
added mass, radiation-damping coefficients, responseamplitude operators (RAOs), and wave-force lineartransfer functions (LTFs), and drift forces. The waterdepth for the present study is infinite. The measured andpredicted RAOs for 1351 wave heading are shown in Fig. 2.Since potential theory is used, the roll amplitude is over-predicted near resonance. Other than that, the agreementbetween the prediction and measurement is good.
2.2. Sloshing
When considering the dynamic effects of sloshingphenomenon, the inertia force is more important thandamping or restoring forces. In this regard, the added massof sloshing fluid is shown in Fig. 4. WAMIT was also usedin the calculation of the added mass of sloshing fluid. Fig. 3shows the grid generation for sloshing tanks at the fillinglevel of 37%. The total number of panels used in the case is600 for both 18% and 37% fill ratios.Fig. 4 shows the sway-added mass calculated by
WAMIT for two different fill ratios. At each filling level,a resonance peak frequency is observed. The calculated
-100-50
0
50
10
MIT (number of panels ¼ 2300).
ARTICLE IN PRESS
0
Frequency (rad/sec)
0
0.4
0.8
1.2
Sur
ge R
AO
(m
/m)
0 0.4 1.6
Frequency (rad/sec)
0
0.4
0.8
1.2
1.6
2
Rol
l RA
O (d
eg/m
)Heading Angle= 135 deg.WAMITExperiment
0 0.4 0.8
Frequency (rad/sec)
0
Sw
ay R
AO
(m/m
)
00
0.8
Pitc
h R
AO
(deg
/m)
00
0.4
0.8
1.2
Hea
ve R
AO
(m/m
)
00
1
2
3
Yaw
RA
O (d
eg/m
)
1.2
0.8
0.4
Frequency (rad/sec)
0.4 0.8 1.2 1.6
Frequency (rad/sec)
0.4 0.8 1.2 1.6
Frequency (rad/sec)
0.4 0.8 1.2 1.6
0.2
0.4
0.6
1.20.81.61.20.80.4
1.61.2
Fig. 2. Measured and predicted motion RAOs for 1351 wave heading.
Fig. 3. Grid generation for sloshing tanks (filling level: 37%).
S.J. Lee et al. / Ocean Engineering 34 (2007) 10–2012
ARTICLE IN PRESS
0.3 0.6 0.7
6E+011
Sw
ay a
dded
mas
s [k
g]
18%37%
4E+011
2E+011
0E+000
-2E+011
-4E+011
-6E+011
-8E+0110.0 0.1 0.2 0.4
Freq [rad/s]
0.5 0.8 0.9 1.0 1.1 1.2
Filling level
Fig. 4. Sway-added mass of sloshing fluid by WAMIT.
S.J. Lee et al. / Ocean Engineering 34 (2007) 10–20 13
resonance frequency is well matched against analytic valuesof sloshing resonance frequency.
2.3. Coupling two problems in frequency domain
Under the assumption of small-amplitude ship andliquid motions, they can be coupled in the frequencydomain based on linearized potential flow theory. Theequation of motion is given by
½Mij þmijðoÞ�q2~xqt2þ ½bijðoÞ�
q~xqtþ ½cijðoÞ�~x ¼ ~F ðtÞ, (1)
where Mij and mij(o) are ship’s real mass and addedmass matrices, bij(o) is radiation damping matrix, andcij(o) is restoring matrix. In roll, viscous effect may beimportant. In such a case, viscous effects can be includedby adding linear equivalent damping coefficient b�44ðoÞ tob44ðoÞ:
b�44ðoÞ ¼ 2gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM44c44
p, (2)
where g is the damping ratio which means the systemdamping divided by critical damping. The body motionand force vectors can be written as
~x ¼ Re xj;0eiot� �
,
~F ðtÞ ¼ Re F j;0eiot
� �. ð3Þ
The coupling of ship motion and liquid sloshing can beinvestigated by adding the hydrodynamic force vectors ofsloshing fluid to the right-hand-side of Eq. (1):
½Mij þmijðoÞ�q2~xqt2þ ½bijðoÞ�
q~xqtþ ½cijðoÞ�~x ¼ ~F ðtÞ þ ~F
sðtÞ.
(4)
~FsðtÞ in Eq. (4) represents the force vector due to liquid
sloshing. We only consider the inertia force of the sloshing
since there is no radiation damping for the internalproblem:
~FsðtÞ ¼ ms
ijðoÞ~€xþ csij
~x, (5)
where msijðoÞ is sloshing fluid’s added mass.
The hydrostatic effect of sloshing fluid can be includedas restoring force correction due to inner free surface, asshown in
csij ¼ I srsg, (6)
where Is is the 2nd moment of inner free surface withrespect to the axis of rotational motion, rs the density ofinner fluid, and g the gravitational acceleration.The resulting coupled equation of motion can be written
as
� Mij þmijðoÞ �msijðoÞ
n oo2 þ iobijðoÞ
h
þ cijðoÞ � csijðoÞn oi
xj;0 ¼ F j;0. ð7Þ
3. Time-domain formulation
3.1. Ship motion
All the hydrodynamic coefficients were first calculated inthe frequency domain, and then, the corresponding forceswere converted to those for time domain using two-termVolterra expansion including convolution integral (Kimand Yue, 1991), as shown in Eqs. (8) and (9):
FR ¼ �mð1Þ€z�Z t
�1
Rðt� tÞ_zdt, (8)
where the convolution integral represents the memoryeffects of the wave force on the platform from the wavesgenerated by platform motion prior to time t. R(t) is called
ARTICLE IN PRESS
Table 1
Principal particulars of FPSO (bare hull) and mooring system
Description Magnitude
Length between perpendicular 285.0m
Breadth 63.0m
Draught 13.0m
Displacement volume 220,017.6m3
Displacement mass in seawater 225.518.0 ton
Longitudinal COG 142.26m
Transverse metacentric height 15.30m
Vertical center of gravity 16.71m
Vertical center of buoyancy 6.596m
Transverse metacenter above base line 32.01m
Mass radius of gyration around X-axis 19.49m
Mass radius of gyration around Y-axis 78.42m
Mass radius of gyration around Z-axis 71.25m
Mooring stiffness
Surge 6.50� 105 kN/m
Sway 2.43� 106 kN/m
Yaw 1.76� 108 kN/rad
Table 2
Characteristics of sloshing tanks
Designation Magnitude
AFT TANK no.4 (inner dimensions given)
Tank aft from aft perpendicular 61.08m
Tank bottom from keel line 3.3m
Tank length 49.68m
S.J. Lee et al. / Ocean Engineering 34 (2007) 10–2014
retardation function and is related to the frequency-domain solution of the radiation problem. The formulafor R(t) is given by
RðtÞ ¼2
p
Z 10
bðoÞ cosðotÞdo, (9)
where b(o) is the radiation/wave damping coefficients atrespective frequencies. The term mð1Þ in Eq. (8) is theadded mass of the body at infinite frequency. The infiniteadded mass coefficients can be obtained from
mð1Þ ¼ maðoÞ þZ 10
RðtÞsin ot
odt, (10)
where maðoÞ is the added mass at frequency o. Then thetotal potential hydrodynamic force can be obtained bythe summation of incident wave force, added mass, andradiation damping forces.
The second-order diffraction/radiation computation fora 3D body is computationally very intensive. Therefore,many researchers avoid such a complex procedure andhave instead used simpler approach called Newman’sapproximation, i.e., the off-diagonal components of thesecond-order difference-frequency QTFs are approximatedby their diagonal values (mean drift forces and moments).This approximation is valid when the system’s naturalfrequencies are very small like the horizontal motions ofthe present problem.
3.2. Sloshing problem
To analyze the liquid sloshing inside a partially-filledtank under forced excitation, two coordinate systems areemployed, as shown in Fig. 5. A tank-fixed coordinate isdefined at the center of the tank bottom, rotating withrespect to point G. Another Cartesian coordinate system(X, Y, Z) is defined at the origin G, and it has thetranslational motion with velocity ~U . Assuming incom-
η (x,y.t)
zx
yVN
rG
X
Y
Z
Fig. 5. Coordinate system of sloshing analysis program.
pressible fluid, the equations governing the flow inside thetank are the continuity and Navier–Stokes equations,
r �~u ¼ 0, (11)
D~u
Dt¼ �
1
rrpþ nr2~uþ ~F , (12)
where u*
is the velocity vector ðux; uy; uzÞ, defined in thetank-fixed coordinates. The symbols r; n; p; F
*are the
Tank breadth 46.92m
Tank height 32.23m
FORWARD TANK no.2 (inner dimensions given)
Tank aft from aft perpendicular 209.54m
Tank bottom from keel line 3.3m
Tank length 56.616m
Tank breadth 46.92m
Tank height 32.23m
Table 3
Simulation environment
Wind N/A
Current N/A
Wave Heading 90 deg.
Significant height 5.0m
Peak period 12 s
g of JONSWAP spectrum 3.3
ARTICLE IN PRESSS.J. Lee et al. / Ocean Engineering 34 (2007) 10–20 15
liquid density, kinematic viscosity, pressure, and externalforce vectors, respectively. Moreover, D/Dt indicates thematerial derivative.
The external force consists of the gravitational force,translational and rotational inertia forces, i.e. F
*takes the
following form:
F* ¼ g*�
dU*
dt�
dO*
dt� ðr
*�R
*Þ
� 2O*�dðr*�R
*Þ
dt� O
*�fO
*�ðr
*�R
*Þg, ð13Þ
where g* and O*
are the gravitational vector and rotationalvelocity vector. In addition, r* and R
*are the position
vectors of the considered point and the origin G. Thesecond term of the right-hand side is the translationalinertia, while the third, fourth, and fifth terms are due tothe rotational motions, which are the angular acceleration,Coriolis and centrifugal forces. It should be noticed thatthese forces are defined with respect to the tank-fixedcoordinate system.
On the free surface boundary, both the kinematic anddynamic conditions should be satisfied:
D~rfDt¼ ~uf , (14)
pf ¼ patm, (15)
where the subscript f means the values on free surface andpatm is the atmospheric or ullage pressure inside of tank.Besides, a proper condition is necessary on the tank wallsand internal members.
The present study focuses on a simplified sloshingproblem without highly violent liquid motions includingsplash and breaking. As well known, the sloshing flow canbecome strongly nonlinear, particularly near the resonance
0-6
-4
-2
0
2
4
6
Wav
e he
ight
[m]
00
2
4
6
8
10
Spe
ctra
l den
sity
(W
ave)
400 800
Tim
Freque0.60.40.2
Fig. 6. Wave time series and spectr
frequencies. Such strong nonlinearity includes wave break-ing, particle splash, jet flow, and impact occurrence. Takinginto account all these complicated local phenomena isextremely difficult, and such local flows may not be ofprimary concerns in the viewpoint of engineering applica-tion. In this regard, the free surface boundary is assumedto be a single-valued function. Then the kinematic free-surface boundary condition can be written as follows:
qZqtþ u* �rZ ¼ 0, (16)
where Z indicates the free-surface elevation.
3.3. Coupling ship motions and sloshing in time domain
The coupling between tank sloshing and ship motion canbe done by adding sloshing force vector in the right-handside of Eq. (1) as follows:
~F ðtÞ ¼ ~F extðtÞ þ ~F sloshðtÞ. (17)
~F extðtÞ is the external excitation force vector on hull surfaceby waves and hydrodynamic reactions, while ~F sloshðtÞ is thesloshing-induced force acting internally on the tank. Themass matrix [Mij] in Eq. (1) represents the total ship massincluding fluid mass inside the tank. The mass andhydrostatic matrices are modified for different volume ofliquid. Since the inertia force as a rigid fluid mass isincluded in the sloshing program, we need to cancel out itseffect by adding the fluid mass inertia in the right-hand sideof Eq. (1):
~F sloshðtÞ ¼ ~F intðtÞ þ ½mtan k;ij�q2~xqt2
. (18)
~F intðtÞ is the force vector from the sloshing programincluding hydrostatic and dynamic forces by fluid motions.
Simulated wave
Simulated wave
Measured wave
e [sec]
1200 1600 2000
1.41.2ncy [rad/sec]
0.8 1
al density (Hs ¼ 5.0m, g ¼ 3:3).
ARTICLE IN PRESSS.J. Lee et al. / Ocean Engineering 34 (2007) 10–2016
4. Simulation results
In the present study, the developed coupled-dynamicsprogram is verified by comparing the simulated results withMARIN’s FPSO experiment including sloshing phenom-ena. Table 1 presents the principal particulars of the FPSO.In the experiments, two separated sloshing tanks wereequipped on FPSO. These two tanks are located at foreand apt parts of FPSO and their characteristics aredescribed in Table 2. The experimental environments arepresented in Table 3.
As was done in MARIN’s experiment, the draft of thevessel is maintained to be constant for different fill ratios
0 400-8
-4
0
4
8
12
Sw
ay [m
]
00
40
80
Spe
ctra
l den
sity
(S
way
)
0
0
Rol
l [ra
d]
0Spe
ctra
l den
sity
(R
oll)
0 0.2 0.4
Freque
0.6
Tim
800400
0.08
0.04
-0.04
-0.08
0.006
0.004
0.002
Tim
800
160
120
Freque
0.4 0.60.2
Fig. 7. Simulated and experimen
by counter de-ballasting. In the numerical simulation, theplate-drag elements are used to represent the hull dampingfor surge-sway and roll motions.The inclusion of viscous damping is particularly
important for roll motions. Linear and quadratic rolldamping model is used in the present calculation and therespective coefficients were obtained from the free-decaytests in calm-water test. The measured damping values arefurther tuned to represent their increase due to waves andinner liquid.In the present sloshing calculation, we considered only
one equivalent tank since our study is limited to only beam-wave condition, in which the phase of the two tanks in
Calculation
1
Calculation
Experiment
Calculation
0.8
ncy[rad/sec]
1 1.2 1.4
200016001200
e [sec]
e [sec]
1200 1600 2000
1.41.20.8
ncy[rad/sec]
Calculation
Experiment
tal results of 0% filling level.
ARTICLE IN PRESSS.J. Lee et al. / Ocean Engineering 34 (2007) 10–20 17
sway and roll is the same. Therefore, the sloshing forcesfrom one tank are multiplied by relevant factors consider-ing the dimensional ratio of the volume of two tanks.
Fig. 6 shows the simulation results of incident wave field.Fig. 7 shows sway and roll time series and spectra for 0%filling level. Both simulated time histories and spectra showgood agreement with experimental results. Figs. 8 and 9show the sway and roll time series and spectra for 18% and37% fill ratios, which include tank-sloshing effects on shipmotions. The most important coupling effect is the shift ofresonance peaks in roll. Particularly for 37% fill ratio, thesingle peak is split into two separated smaller peaks both inexperiment and simulation. The secondary peak is relatedto the natural frequency of the lowest tank-sloshing mode
0 400-8
-4
0
4
8
12
Sw
ay [m
]
00
40
80
Spe
ctra
l den
sity
(S
way
)
0
0
Rol
l [ra
d]
0Spe
ctra
l den
sity
(R
oll)
0 0.2 0.4
Freque
0.6
Tim
800400
0.08
0.04
-0.04
-0.08
0.006
0.004
0.002
Tim
800
160
120
Freque
0.4 0.60.2
Fig. 8. Simulated and experiment
(see Table 4). In the case of 18% fill ratio, the hull-rollnatural frequency coincides with the lowest sloshingnatural frequency, and thus the split of resonance peaksdoes not happen.Both sway and roll amplitudes tend to decrease as filling
level increases. The observed phenomenon is related to thefact that the inner water tank may be effective in reducingthe vibration of a tall building caused by earthquake. Thestatistics of Figs. 6–9 are summarized in Table 5. Thestatistics show good agreement between numerical predic-tion and measured data.The coupling effects between hull motion and liquid
cargo can be seen more clearly in the comparison of rollRAOs for different fill ratios, as in Fig. 10(a–c) i.e. (a) was
Calculation
1
Calculation
Experiment
Calculation
0.8
Calculation
Experiment
ncy[rad/sec]
1 1.2 1.4
200016001200
e [sec]
e [sec]
1200 1600 2000
1.41.20.8
ncy[rad/sec]
al results of 18% filling level.
ARTICLE IN PRESS
0 400-8
-4
0
4
8
12
Sw
ay [m
]
Calculation
0 10
40
80
Spe
ctra
l den
sity
(S
way
)
Calculation
Experiment
0
0
Rol
l [ra
d]
Calculation
0.80S
pect
ral d
ensi
ty (
Rol
l)
Calculation
Experiment
0 0.2 0.4
Frequency[rad/sec]
0.6 1 1.2 1.4
200016001200
Time [sec]
800400
0.08
0.04
-0.04
-0.08
0.006
0.004
0.002
Time [sec]
800 1200 1600 2000
160
120
1.41.20.8
Frequency[rad/sec]
0.4 0.60.2
Fig. 9. Simulated and experimental results of 37% filling level.
Table 4
Natural frequencies of FPSO and tank sloshing
Natural frequencies (rad/s)
Bare hull Sway : 0.06
Roll: 0.50
Sloshing fluid FL: 18% 0.49(1st), 1.31(2nd)
FL: 37% 0.66(1st), 1.55(2nd)
S.J. Lee et al. / Ocean Engineering 34 (2007) 10–2018
obtained from irregular-wave model test, (b) was obtainedfrom time-domain simulations, and (c) was obtained fromfrequency-domain calculation. As was already pointed out
in Fig. 9, we can clearly see double peaks in the roll RAOof 37% fill ratio. The roll RAO’s generally decrease with fillratio in the range of 0.4–0.7(rad/s) where significant waveenergy exists. This is why the 37% case produced thesmallest roll motion. Fig. 10(c) shows similar trend butresonance peaks are significantly over-predicted becauseviscous effects are not included.Finally, let us consider the simplest correction method
through mass–stiffness adjustment. The mass correction isthe change of liquid mass, mass moment of inertia, andvertical center of gravity due to additional liquid cargo(this effect is minimized in MARIN’s experiment byadjusting ballast). The stiffness correction is the loss of
ARTICLE IN PRESS
Table 5
Statistics of time series
Sway Roll
Exp. Cal. Exp. Cal
Filling level : 0%
Mean 0.151E+01 0.176E+01 0.940E�03 0.344E�04
Std. dev. 0.328E+01 0.309E+01 0.252E�01 0.224E�01
Max. 0.110E+02 0.111E+02 0.690E�01 0.683E�01
Min. -0.724E+01 �0.508E+01 �0.731E�01 �0.707E�01
Filling level : 18%
Mean 0.128E+01 0.178E+01 �0.426E�03 0.214E�04
Std. dev. 0.287E+01 0.341E+01 0.257E�01 0.207E�01
Max. 0.883E+01 0.121E+02 0.767E�01 0.601E�01
Min. �0.569E+01 �0.650E+01 �0.847E�01 �0.584E�01
Filling level : 37%
Mean 0.111E+01 0.133E+01 �0.123E�02 0.144E�04
Std. dev. 0.260E+01 0.296E+01 0.126E�01 0.123E�01
Max. 0.798E+01 0.986E+01 0.377E�01 0.378E�01
Min. �0.559E+01 �0.597E+01 �0.415E�01 �0.392E�01
Wave elevation
Exp. Cal.
Mean 0.474E�01 �0.133E�02
Std. dev. 0.122E+01 0.124E+01
Max. 0.437E+01 0.397E+01
Min. �0.407E+01 �0.373E+01
0.7 1
Freq [rad/sec]
0
1
2
3
Rol
l RA
O [d
eg/m
]
10
1
2
3
Rol
l RA
O [d
eg/m
]
10
1
2
3
Rol
l RA
O [d
eg/m
]
10
1
2
3
Rol
l RA
O [d
eg/m
]
Freq [rad/sec]1.21.10.90.80.70.60.50.4
Freq [rad/sec]
0.9 1.1 1.20.80.70.60.50.4
1.21.10.90.80.60.50.4
Freq [rad/sec]
1.1 1.20.90.80.70.60.50.4
Frequency domain
FL : 0%
Stiffness correction only
FL : 0%FL : 18%FL : 37%
Frequency domain
FL : 0%FL : 18%FL : 37%
Time domain
FL : 0%FL : 18%FL : 37%
Experiments by MARIN
(a)
(b)
(c)
(d)
Fig. 10. Comparison of roll RAOs. (a) Experiments by MARIN, (b) from
time-domain simulation, (c) from frequency-domain calculation, and (d)
by simple approximate method through mass–stiffness correction.
S.J. Lee et al. / Ocean Engineering 34 (2007) 10–20 19
roll-pitch hydrostatic restoring coefficients due to thepresence of inner free surface, which is given by Eq. (6).From Eq. (6), the inner-free-surface restoring correction isaffected only by the density of inner fluid and the 2ndmoment of inner free surface, not by the filling level ofliquid cargo. Therefore, the stiffness correction givesidentical results for different filling levels. Fig. 10(d) showsthe result of the simple mass–stiffness correction methodcompared with the case without cargo liquid. The rollnatural frequency is shifted lower due to the decrease ofroll restoring stiffness. This example illustrates that thesimple correction method cannot reproduce the complexdynamic and coupling effects by liquid sloshing.
5. Conclusions
The interaction effects between ship motion and inner-tank liquid sloshing are investigated by newly developedtime-domain and frequency-domain simulation programs.For time-domain simulations, both ship-motion programand inner-tank-sloshing program are independently devel-oped. In the ship-motion program, the hydrodynamiccoefficients including wave forces and drift forces wereobtained from a three-dimensional (3D) panel-baseddiffraction/radiation program. The time-domain sloshingprogram is based on the Navier–Stokes equation solverincluding SOLA scheme method for free surface. Duringthe time marching, the tank-sloshing program is coupledwith the vessel-motion program so that the influence oftank sloshing on vessel motions can be assessed. On the
other hand, the frequency-domain analysis was done byapplying the 3D panel method both for interior andexterior problems. The inner-tank-sloshing effect is con-sidered by importing the added mass of sloshing fluid andthe hydrostatic correction of inner free surface. Althoughthe frequency-domain analysis is based on linear potentialtheory, the results reproduce the qualitative trend of theeffect of inner-tank sloshing on ship motion. When theresonance sloshing frequency is away from the motionnatural frequency, the liquid cargo can be a good vibrationabsorber.
ARTICLE IN PRESSS.J. Lee et al. / Ocean Engineering 34 (2007) 10–2020
More findings include:
�
The peak frequency of roll motions can be shifted due tothe tank-sloshing effect. The secondary peak appearsnear the sloshing natural frequency and its effectsincrease as filling ratio increases, which can be observedin both numerical and experimental results. � The simple mass–stiffness correction method canroughly include the shift of roll natural frequency butcannot reproduce the complex dynamic and couplingeffects by liquid sloshing.
� The roll-motion amplitudes and RAOs are reduced withhigher fill ratio, which means that we may have anti-rolling effects due to tank sloshing when properlydesigned.
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