Sloshing simulation of a tank oscillating towards multiple degrees of freedom by particle method

7 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 8 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. 10 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 11 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 12 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) 13 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) 14 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. 15 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