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ACC JOURNAL 2019, Volume 25, Issue 1 DOI: 10.15240/ ul/004/2019-1-001
SLOSHING SIMULATION OF A TANK OSCILLATING TOWARDS MULTIPLE
DEGREES OF FREEDOM BY PARTICLE METHOD
Shigeyuki Hibi1; Kazuki Yabushi a
Na ional De ense Academy, Depa men o Mechanical Sys ems Enginee ing,
1-10-20 Hashi imizu, Yokosuka, Kanagawa, 239-8686, Japan
e-mail: 1[email p o ec ed]c.jp
Abs ac
The ank sloshing p oblem is e y impo an a design ime in LNG/LPG ships. This p oblem
causes impulsi e loads o ship s uc u es and is o en ea ed as a non-linea one. In o de o
es ima e hese impulsi e loads, p ope ly many s udies ha e been ca ied ou h ough bo h
expe imen al and nume ical app oaches. Impulsi e p essu e on he wall o a ank induced by
o ced mul i-deg ee oscilla ions is ocused in his esea ch. I is shown in he pas au ho s’
expe imen ha o ced mul i-deg ee oscilla ions cause s onge impulsi e p essu e as
compa ed o indi idual oscilla ions. Nume ical analysis by a pa icle me hod based on ini e
olume echnique is in oduced in his s udy o simula e he abo e phenomena. The sugges ed
pa icle me hod is shown o be use ul o simula ing a s ong nonlinea phenomenon. The
au ho s discuss he calcula ed esul s o p essu e ime his o y wi h he expe imen al esul s.
Keywo ds
Sloshing; Impulsi e load and p essu e; Pa icle me hod; Fini e olume me hod.
In oduc ion
Sloshing is a phenomenon in which a liquid su ace luc ua es iolen ly when a ank wi h a
ee su ace is shaken a a cycle close o he na u al pe iod o he ank. La ge impulsi e loads
may be applied o walls o he ank a his ime. In LNG/LPG ships sailing h ough i egula
wa es, la ge impulsi e p essu e may be gene a ed on he wall nea ceiling a ea when liquid
ca go collides o walls o a ca go ank. Then i is impo an o know he luc ua ing p essu e
caused by sloshing when conside ing he s eng h o ank as a s uc u al membe in ship
design.
On he o he hand, i is known ha a ship na iga ing h ough wa es has a coupled mo ion o
wo deg ees o eedom and h ee deg ees o eedom, called longi udinal mo ion and la e al
mo ion, espec i ely. Fo his eason, he hull will luc ua e wi h a phase di e ence in some
axial di ec ion o o a ion di ec ion o hose axes. As a esul , i is possible ha la ge sloshing
may be caused in anks inside he ship.
In he p e ious s udy [1] an appa a us which can oscilla e a ank by o ce was in oduced in
o de o in es iga e impulsi e p essu e on he wall o he ank. This appa a us can oscilla e i
simul aneously owa ds 3 deg ees o eedom (up-down, le - igh and o a ion) wi h each
phase di e ences. Th ough he expe imen s i was shown la ge impulsi e p essu e could be
exci ed unde oscilla ing he ank simul aneously owa ds mul iple deg ees o eedom han
unde single di ec ion oscilla ion and he speci ic phase di e ences o appea he la ges peak
alues o p essu e was iden i ied.
In his s udy he au ho s e i ied he esul s o abo e expe imen s by nume ical simula ion.
The au ho s ha e de eloped a pa icle me hod [2] wi h ini e olume echnique in o de o
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s ably simula e a luid phenomenon wi h s ong nonlinea i y such as sloshing. A i s s a ic
p essu e in a ank was e i ied wi h he pa icle me hod a ying he pa icle numbe . Nex by
compa ing he ime his o y o p essu e wi h he expe imen esul o p e ious s udy, he
e ec i eness o he p oposed pa icle me hod was assu ed.
1 Nume ical Analysis Based on Pa icle Me hod wi h Fini e Volume Technique
In pa icle me hods, luid is modeled as a collec ion o pa icles and disc e ized by
conside ing he in e ac ion be ween pa icles wi hou a calcula ion la ice. The e o e, pa icle
me hods a e sui able o in ense luid analysis wi h s ong nonlinea i y. He e, we explain he
ou line o he pa icle me hod p oposed by us wi h ini e olume echniques.
The undamen al equa ions desc ibed by a luid pa icle coo dina es
ˆ
a e ep esen ed by he
Na ie –S okes equa ions and he con inuous equa ion.
(1)
(2)
He e is he speed o pa icles, p is he p essu e o pa icles,

is he densi y o luid, K is
ou e o ces and

is he kinema ic coe icien o iscoci y o luid. F om Gauss’s heo em
equa ions (1) and (2) a e ans o med in o low equa ions o in eg al shape.
(3)
(4)
He e n is he no mal ec o o a pa icle and V is he olume o a pa icle.
E en ually, he disc e ized equa ions a e de i ed om equa ions (3) and (4) in he p oposed
me hod.
(5)
(6)
He e
9
(7)
Subsc ip i means he pa icle numbe o in e es and subsc ip j means he neighbo ing
pa icle numbe o he pa icle i. Supe sc ip n indica es he ime s ep numbe .
i means posi ion ec o o he i- h pa icle. The pa icles a e assumed o be sphe ical (in 2D
model jus ound) and he in luence o he neighbo ing pa icle j o he pa icle i is ca ied ou
h ough he mic o su ace a ea

Sij exp essed by equa ion (7) acco ding o he concep o he
Fini e Volume Me hod.
R s ands o he adius o a pa icle and sa is ies he ollowing equa ion.
πR2 = D2
(8)
D is he a e age in e -pa icle dis ance and is de e mined by he spa ial esolu ion o he
calcula ion model. This pa ame e co esponds o he smoo hing leng h h in he SPH[3].
Conside ing he e iciency and s abili y o calcula ion, i is conside ed ha neighbo ing
pa icles a om he so called in luence adius e do no a ec . In his s udy he ollowing
allue is adop ed o he pa ame e e.
e = 3.1 D
(9)
The p essu e alue o pa icles a each ime s ep is ob ained by implici ely sol ing he
equa ion (6) using he PCGS me hod The eloci y alue o pa icles is also ob ained om he
equa ion (5) implici ly using he PCGS me hod in case o conside ing iscosi y o luid and
explici ly in case ha iscosi y is no aken in o conside a ion.
2 Expe imen al Appa a us and Condi ion
2.1 Speci ica ion o Oscilla ion Appa a us
The appa a us used in his s udy can oscilla e a ank a ached on he b acke ha monically and
simul aneously owa ds 3 deg ees o eedom wi h each phase di e ences (See Fig. 1 and
Table 1).
The size o ank is 400, 400, 100 (mm) (Heigh , Wid h, Dep h). We de ine he symbols o he
oscilla ion di ec ions o 3 deg ees o eedom as ollows in Fig. 2. Hea e oscilla ion (Up –
down) s ands o Z and Sway oscilla ion (Le – igh ) s ands o X and Roll oscilla ion
(Ro a ion) s ands o

.
A p essu e senso is a ached on he igh side o he ank moun ed on he appa a us. The
posi ion o he senso is 40 mm om he bo om. The p essu e ecei e has ound a ea wi h
diame e o 8mm.
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Fig. 1: Oscilla ion appa a us wi h 3 deg ees o eedom
Tab. 1: Speci ica ion o he appa a us abou oscilla ion
Di ec ion o oscilla ion
Maximum ampli ude
Minimum pe iod
Hea e (Z) (Up-down)
50
mm
1.0 sec.
Sway (X) (Le - igh )
50
mm
1.0 sec.
Roll (

) ( o a ion)
45
deg.
1.0 sec.
2.2 Selec ion o Phase Di e ences du ing Simul aneous Oscilla ion owa ds
2 Deg ees o F eedom
In p e ious s udy [1] we chose 3 pai s o 2 deg ees o eedom om 3 ones. Those will be
subsc ibed as ollows (


X,


Z, X

Z ). The oscilla ion pe iod is uni o mly 1.08 sec. which
is close o he na u al pe iod o he ank in case o he wa e dep h is 60 mm ( ixed). The
oscilla ion ampli udes a e 15 mm o X and Z and 20 deg ees o

.
We measu ed ime his o ies o p essu e on he wall o he ank epea edly while changing he
phase di e ence om 0 deg ee o 360 deg ees. We iden i ied he phase di e ence which
cause he la ges peak alue o p essu e o each oscilla ion pai (


X,


Z, X

Z ). The
esul s a e shown in Table 2.
Tab. 2: Phase di e ence which cause he la ges peak alue
Oscilla ion pai
Phase di e ence


X
15 deg ees


Z
335 deg ees
X

Z
220 deg ees
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3 Nume ical Simula ion and Discussion
3.1 Compa ison o Calcula ion Accu acy o he Pa icle Me hod wi h S a ic
P essu e
A i s we con i m accu acy o he p oposed pa icle me hod by s a ic p essu e. In Fig. 2 a
ank model used in he abo e expe imen is shown. Wa e dep h is 60mm as in he
expe imen . The o al pa icle numbe is 3,700. The a e age in e -pa icle dis ance D is 38
mm. We de ine his model o spa ial esolu ion as he base model and name i X1 model. In
addi ion o his model we p epa e 3 addi ional models named X1/2 (1,500 pa icles), X2
(10,500 pa icles), X4 (33,700 pa icles) model o which D is hal , double and 4 imes each
o he .
Nume ical simula ion was ca ied ou wi h he ime s ep o 0.0001 sec. in all cases. Fig. 3
shows he ime his o y o s a ic p essu e a he bo om o ank. In all calcula ion models we
can see some luc ua ions o he i s ew seconds in simula ion ime and a e ha he
p essu e alues become ai ly s able. These luc ua ions o p essu e occu when ela i e
posi ion o pa icles u ns mo e s able and close om he ini ial pa icle a angemen in a
e agonal la ice.
Fig. 2: Ini ial pa icle a angemen o s a ic p essu e
Table 2 shows Compa ison o s a ic p essu e wi h he heo e ical alue. I can be seen ha he
calcula ion accu acy is imp o ed acco ding o he spa ial esolu ion.
Tab. 3: Compa ison o s a ic p essu e wi h he heo e ical alue
Simula ion esul (Pa)
E o (%)
X1
616
3.36
X2
610
2.35
X4
607
1.85

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Fig. 3: Time his o y o s a ic p essu e a he bo om o ank
3.2 Nume ical Simula ion o Fo ced Oscilla ion owa ds 2 Deg ees o F eedom
Nume ical simula ion was ca ied ou unde he same condi ions as expe imen ob ained in
Sec ion 3.2. Fig. 4 (a) o Fig. 4 (d) show compa ison o he expe imen and simula ion esul s
o ime his o y o p essu e alue a he posi ion whe e he p essu e senso is a ached. The
expe imen al alues a e esul s ob ained h ough a 100Hz low-pass il e and he accu acy o
hose esul s a e e alua ed by a FFT analysis and s a is ical app oaches in he p e ious
pape [1].
F om hese esul s, i can be seen ha he nume ical esul s ha e good ag eemen wi h he
expe imen o each models whe e p essu e peaks a e obse ed. Bu a he ime when he
p essu e senso is exposed by o ced oscilla ion as shown in Fig. 6 (a) and (b), he p essu e
alues by nume ical simula ion a e 0 in any models, whe eas he expe imen esul s show
signi ican alues which is no 0. The eason o his is due o iscosi y and su ace ension in
ac ual wa e . E en i he p essu e senso is exposed, ac ually some wa e is a eling on he
wall su ace o some wa e emains on he senso because o su ace ension. Howe e he
cu en p oposed pa icle me hod does no conside hese e ec s.
0 4 8
560
600
640 X1 (3,700)
X2 (10,500)
X4 (33,700)
p (Pa)
(sec)
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Fig. 4 (a): Time his o y abou


X
oscilla ion.
Fig. 4 (b): Time his o y abou


Z
oscilla ion.
Fig. 4 (c): Time his o y abou X

Z
oscilla ion
Fig. 4 (d): Time his o y abou


X
oscilla ion.
Al hough he peak pe iod and peak alue shown in Figu es a e almos ep oduced, he
simula ion esul s show ha he p essu e alues a e e alua ed o a smalle alue as compa ed
wi h he expe imen al alues in all models. One eason o his is ha ela i ely low peak
alues a e conside ed o be di icul o s and due o nume ical iscosi y. Ano he eason is
belie ed o be due o he ela ion be ween he a ea o he p essu e senso and he special
esolu ion in he simula ion. While he diame e o he p essu e senso is 8 mm, he a e age
in e -pa icle dis ance D is 38 mm in he X1 model and 19 mm in he X2 model. Fu he s udy
will be needed o his p oblem.
4 6 8
0
1000
2000
Exp.
X2 model
X1 model
Exp.
X2 model
X1 model
(s)
p(Pa)
4 5 6
0
2000
4000
6000
X2 model
Exp.
X1 model
X2 model
Exp.
X1 model
p(Pa)
(s)
3 4 5 6
0
1000
2000
Exp.
X2 model
X1 model
p(Pa)
(s)
4 6
0
1000
2000
Exp.
X1 model
Exp.
X1/2 model
X1 model
(s)
p(Pa)
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Fig. 6 (c): Time his o y abou Z

X
oscilla ion.
3 4 5 6
0
1000
2000
Viscosi y
X2 model
X1 model
p(Pa)
(s)
Fig. 6 (b): Time his o y abou


Z
oscilla ion.
4 5 6
0
2000
4000
6000
X2 model
Viscosi y
X1 model
X2 model
Viscosi y
X1 model
p(Pa)
(s)
Fig. 6 (a): Time his o y abou


X
oscilla ion.
4 6 8
0
1000
2000
Viscosi y
X2 model
X1 model
Viscosi y
X2 model
X1 model
(s)
p(Pa)
Fig. 5 (b): Momen o exposu e o he
p essu e senso abou X

Z
oscilla ion.
Fig. 5 (a): Momen o exposu e o he
p essu e senso abou


X
oscilla ion.
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In Fig. 4 (d) he nume ical esul added o he X1/2 model abou


X is shown. E eni he
numbe o pa icles is jus 1,500, i is obse ed ha he accu acy o he analysis is simila .
This shows he supe io i y o he p oposed pa icle me hod.
Fig. 6 (a) o Fig. 6 (c) show compa ison o he expe imen and simula ion esul s conside ing
luid isco si y o ime his o y o p essu e alue a he posi ion whe e he p essu e senso is
a ached.
F om hese esul s, i can be seen ha he e ce ainly is a endency ha he p essu e
luc ua ion becomes loose , such as peak alues bu e en in he case o conside ing he
in luence o he luid iscosi y, he e is no big change in he o e all p essu e luc ua ion. This
implies ha in luence o he luid isco si y is limi ed in he simula ion o in ense lows like
his s udy’s expe imen e en hough explici ly aking in o conside a ion and he su ace
ension and we abili y by ha may be much mo e .impo an o p essu e luc ua ion.
Conclusion
In his s udy he au ho s ca ied ou nume ical simula ion o o ced oscilla ion owa ds
2 deg ees o eedom and compa ed wi h he p e ious expe imen al esul s. Some
conside a ions we e ound abou i as below.
 E ec i eness o he p oposed pa icle me hod is shown h ough he nume ical simula ion
o in ense sloshing phenomena.
 In addi ion o he shape o he p essu e senso , he iscosi y o wa e and he su ace
ension on he wall o ank a e conside ed o be ela ed o he p essu e luc ua ion.
Li e a u e
[1] HIBI, S.: S udy on he impulsi e p essu e o ank oscilla ing by o ce owa ds mul iple
deg ees o eedom. EPJ Web o Con e ences. 2018, Vol. 180, Pape No. 02034.
DOI: 10.1051/epjcon /201818002034
[2] YABUSHITA, K.; HIBI, S: o be pos ed. Jou nal o Ma ine Science and Technology.
[3] MONAGHAN, J. J.: Simula ing F ee Su ace Flows wi h SPH. Jou nal o
Compu a ional Physics. 1994, Vol. 110, Issue 2, pp. 399–406.
Shigeyuki Hibi; Kazuki Yabushi a