Proceedngs of the Eleventh (14) Pacfc/Asa Offshore Mechancs Symposum Shangha, Chna, October 1-16, 14 Copyrght 14 by The Internatonal Socety of Offshore and Polar Engneers ISBN 978 1 88653 9-6: ISSN 1946-4X www.sope.org Numercal Smulatons of 3D Lqud Sloshng Flows by MPS Method Yaqang Yang, Zhenyuan Tang, Decheng Wan * State Key Laboratory of Ocean Engneerng, School of Naval Archtecture, Ocean and Cvl Engneerng, Shangha Jao Tong Unversty, Shangha, Chna * Correspondng author ABSTRACT: In ths paper, a modfed MPS (Movng Partcle Sem- Implct) method s appled nto 3D lqud sloshng to show ts capablty n modelng 3D sloshng flows. Volent sloshng flow n 3D rectangular s nvestgated. Furthermore, mpact pressure value obtaned by D and 3D smulaton s compared, whch ndcates that 3D MPS smulaton s able to produce smoother pressure feld and thus makes better agreement wth experment than D MPS smulaton. In addton, the snapshots of free surface deformaton and velocty feld are compared between experment, D and 3D MPS smulaton. Results show that the detals of free surface deformaton, such as the breakng regon of the wave and the splash of lqud flud can be capszed by 3D MPS method effectvely. KEY WORDS: MPS method; 3D lqud sloshng; rectangular tank; sloshng; mpact pressure. INTRODUCTION Sloshng refers to the movement of lqud nsde a partally flled tank due to external exctatons. The lqud sloshng n LNG (Lquefed Natural Gas) tank s of sgnfcant mportance n marne ndustry. When the ampltude of the shp moton s very large or ts frequency s close to the natural frequency of the lqud tank, volent sloshng flows may appear, exertng strong mpact pressure on the wall of the tank, whch may cause too large deformaton on the structure locally, and affect the stablty of shp globally. As a result, t s essental to predct the mpact pressure accurately to avod severe structural damage. Due to the ever-ncreasng nterests n lqud sloshng, a number of researchers have appled numercal smulaton methods to the sloshng problem. Most of the numercal smulatons are focused on grd-based methods, such as FDM (Fnte Dfference Method, Lee et al., 8), FEM (Fnte Element Method, Wang and Khoo, 5), VOF (Volume of Flud, Lee et al., 7) and so on. In recent years, meshfree methods such as SPH (Smoothed Partcle Hydrodynamcs, Gngold and Monaghan, 1977) and MPS (Movng Partcle Sem-Implct, Koshzuka and Oka, 1996; Koshzuka et al., 1998) have been developed to model flud moton wth large deformaton of free surface. In partcle method, flow s modeled as an assembly of nteractng partcles whch have physcal propertes, such as mass, momentum, and energy, etc. Snce these partcles have no fx topography among each other, meshless methods are more flexble to deal wth the large deformed free surface flows (Zhang and Wan, 11a). There has been some work on lqud sloshng based on partcle method, such as Cu and Lu (9) appled SPH to smulate the sloshng phenomenon n a D tank subject to the moton of surge and ptch, showng a good agreement between numercal smulaton and experment n terms of free surfaces deformaton, but overestmated mpact pressure. Shao et al (1) mpled SPH method to model vscous ncompressble lqud sloshng wth dfferent external exctatons and dfferent structures. The obtaned numercal results ncludng flow pattern, wave heght, pressure feld, and pressure load on sold walls were agreeable wth expermental results. Delorme et al (9) smulated D sloshng and dscussed the nfluence of vscosty and of densty re-ntalzaton on the SPH results, good agreements were obtaned n terms of free surface shape and global dynamcs of the flow between expermental and SPH results. Khayyer and Gotoh (11) smulated D volent sloshng based on mproved MPS method, obtanng smooth mpact pressure wth small oscllaton. Zhang and Wan (1) computed lqud sloshng n D low-fllng tank based on modfed MPS method, n whch good agreement between numercal results and expermental data s obtaned. Although the partcle method can better deal wth the aforementoned problems, t also has some drawbacks, such as spurous fluctuatons n pressure value and hgh computaton cost. To suppress the fluctuaton n flow feld, an mproved MPS method has been adopted n ths paper. The modfed schemes nclude: a mxed source term n pressure Posson equaton (PPE), and a surface detecton wth a hgh accuracy. The mxed source term of PPE conssts of two parts: the dvergence free condton and the partcle number densty condton. Whle the surface detecton method s based on the asymmetry of arrangement of neghbor partcles, smooth pressure feld s obtaned based on the present MPS method (Zhang and Wan, 1). Because of the great amount of computaton and hgher requrement of computatonal stablty n 3D MPS sloshng, fewer 3D lqud smulatons are computed by MPS method than D smulatons. Thus, we utlze the mproved MPS method to smulate D and 3D lqud sloshng n a rectangular tank under translatonal exctaton n ths 19
paper. Volent sloshng flow n 3D rectangular tank s nvestgated. To study the dmensonal effect n MPS method, pressure hstores obtaned by D and 3D MPS are compared wth expermental data. Furthermore, deformaton of free surface and velocty feld are also compared between D smulaton, 3D smulaton and experment. NUMERICAL SCHEME Governng Equatons For ncompressble, vscous flows, the governng equatons are expressed n Lagrangan form as follows: 1 D V= (1) Dt In ths paper, the gradent operator can be dscretzed nto a local weghted average of radal functon as follows: D P P P ( r r) W( r r ) j j j n j rj r where D s the number of space dmensons, r represents coordnate vector of flud partcle, W() r s the kernel functon and n denotes the ntal partcle number densty for ncompressble flow. Eq.5 can not only mprove the stablty of the calculatons but also mantan the momentum conservaton. Laplacan Model The Laplacan operator s formulated as (Koshzuka et al., 1998): (5) DV Dt 1 P V g () D ( ) W ( r r ) (6) j j n j Where denotes the densty, P s the pressure, V s the velocty, g s the gravty acceleraton and the knematcs vscosty. Because MPS method s based on Lagrangan approaches, the tme dfferentaton s represented by substantal dervatve nvolvng advecton terms n both of the equatons. As a result, the numercal dffuson nduced by advecton terms can be elmnated. The parameter s ntroduced as: j W ( r r ) r r j j j W ( r r ) j (7) Partcle nteracton models In MPS method, the governng equatons are transformed nto the equatons of partcle nteracton. The weght functon n the partcle nteracton s defned as kernel functon. Dfferent from the kernel functon n tradtonal MPS method (Koshzuka and Oka, 1996), an mproved kernel functon (Zhang and Wan, 11b) wthout sngualr s adopted to smooth the computatonal results n ths paper: re 1r re Wr.85r.15re re r Where, r rj r, denotes the dstance between two partcles, r e radus of the nfluence area of each partcle. To calculate the weghted average n MPS method, partcle number densty s defned as (Masahro Kondo, 1): n W r rj j (4) Ths value s assumed to be proportonal to the densty, so the partcle number densty can be appled nstead of densty n partcle dscretzaton. As the densty s almost constant n ncompressblty calculaton, we use the constant n nstead of n n the formulaton of weghted average. Gradent Model (3) In Eq.6 the parameter s ntroduced to keep the ncrease of varance equal to that of the analytcal soluton. Model of ncompressblty In tradtonal MPS method, the ncompressble condton s represented by keepng the partcle number densty constant. In each tme step, there are two stages: frst, temporal velocty of partcles s calculated based on vscous and gravtatonal forces, and partcles are moved accordng to the temporal velocty; second, pressure s mplctly calculated by solvng a Posson equaton, and the velocty and poston of partcles are updated accordng to the obtaned pressure. Here we adopt a mxed source term for PPE proposed by Tanaka and Masunaga (1), whch combnes the velocty dvergence and the partcle number densty. Ths mproved PPE s rewrtten by Lee et al. (11) as: k k 1 * n n P (1 ) V t t n where: s a blendng parameter (between.1 and.5) to account for the relatve contrbutons of the two terms. Free Surface boundary condton In the MPS method, the free surface dynamc condton s enforced by assgnng zero pressure for surface partcles. By now, some approaches have been developed to detect the free surface partcles. Koshzuka and Oka (1996) recognzes the surface partcles accordng to the partcle number densty. Tanaka & Masunaga (1) and Lee et al. (11) judge the surface partcle by usng number of neghbor partcles. Khayyer et (8) 193
al. (11) proposed a new crtera based on asymmetry of neghborng partcles n whch partcles are judged as surface partcles accordng to the summaton of x-coordnate or y-coordnate of partcle dstance. In the present study, we employ a detecton method whch s also based on the asymmetry arrangement of neghborng partcles, but use dfferent equatons, amng at descrbng the asymmetry more accurate, as follows (Zhang and Wan, 11c): D 1 F ( r r ) W ( r ) j j n j r rj (1) Where, the vector functon F represents the asymmetry of arrangements of neghbor partcles. Thus, partcles satsfyng: F (11) are consdered as surface partcle, where s a parameter wth a value of.9 F n ths paper, F s the ntal value of F for surface partcle. NUMERICAL SIMULATIONS In ths secton, numercal tests have been conducted for both D and 3D lqud sloshng n rectangular tank under horzontal moton. Fg.1 shows the computatonal model, whch s the same as the expermental model gven by Chang. The length of the tank s L=.79m, ts heght s H=.48m, and ts wdth s W=.48m. The depth of water s d=.144 m, correspondng fllng level s 3%. Sloshng pressure at pont P s measured n the smulaton. Pont P s near the free surface,.1m from the bottom of the tank. The tank s subject to snusodal horzontal exctaton: Fg. 1 Schematc of the tank Snapshots of 3D MPS smulaton n one perod are shown n Fg.. It can be seen that wave propagates n the tank and mpacts on the sde walls and celng of the tank. As the exctaton frequency s equal to resonant frequency, flow s qute volent. Breakng wave and splash water are observed. When the tank reaches ts maxmum dsplacement and starts to return, lqud wth large horzontal velocty mpacts on the sde wall and causes large mpact pressure, seen n Fg. (b). Then an up-shootng jet s formed, whch hts on the top of the tank, resultng n large mpact pressure on the upper corner, see Fg. (c). After that, the jet breaks and falls down due to gravty. The fallng lqud hts on the underlyng lqud, and dsturbs the free surface, as shown n Fg. (d). Though the flow s volent, the present MPS s capable of computng such complcated flows. x Acos( t) (1) Where A s the ampltude of exctaton wth the value of.575m, s exctaton frequency, here =4.49 rad / s, whch s equal to the frst order resonant frequency of flud moton. In D sloshng smulaton, 5944 partcles are used, among whch 4396 are flud partcles. The ntal partcle space s.5 m and the tme -4 step s 1 s. The acceleraton of gravty s g=9.81m/s. The densty 3 of water s =1kg / m. (a) t nt In 3D smulaton, 663458 partcles are used, among whch 4176 are flud partcles. The value of other parameters, such as the ntal partcle, the ntal partcle space, the tme step, the acceleraton of gravty, the densty of water and so on, s the same as the value n D smulaton. It s essental to note that, n ths paper, n ths paper, we only nvestgate the dmensonal effect on the numercal results. Therefore, the ntal partcle space between D MPS and 3D MPS s the same. Effect of the total partcle number wll be dscussed n other papers. (b) t nt.1t 194
(c) t nt.t (g) t nt.7t (d) t nt.45t (h) t nt.9t Fg. Evoluton of the sloshng waves The tme hstory of expermental and computed pressure at pont P are compared n Fg. 3. As can be seen, the global shape of the pressure curve s well reproduced by both D and 3D MPS method, whch ndcates that both D and 3D MPS method can predct the mpact pressure nduced by lqud sloshng n general. (e) t nt.5t (f) t nt.6t From the pressure hstory, we can also observe that the pressure pattern looks lke a typcal church roof profle. As s shown n Fg. (b), when the tank reaches ts maxmum poston on the rght, the momentum drecton of the flud suffers nstantaneous change. As a result, the breakng wave mpnges the sde wall, causng an mpulsve pressure wth a hgh peak value at pont P at t nt.1t. Then, the sloshng flow turns upward along the rght-sde tank wall, resultng relatvely unform pressure that lasts about.4t. Subsequently, the elevated sloshng lqud starts to break and falls from the top wall due to gravty, causng the second mpact on the water lower whch nduces another peak value. After that the water moves toward the left-sde wall, lowerng the water depth near the rght-sde wall to below pont P so the pressure s essentally zero. Fg. 4 shows the detals of mpact pressure evoluton at pont P obtaned by experment, D MPS and 3D MPS method n one perod. It can be seen that the evoluton of pressure obtaned by 3D smulaton shows better agreement wth expermental data. Whle the second peak of the pressure hstory n D smulaton s slghtly hgher than expermental value, whch may be caused by the unphyscal oscllaton n pressure feld n D smulaton. From the comparson, we can also observe that 3D smulaton pressure curve s much smoother. Ths s because each partcle has more neghbor partcles n 3D computaton, leadng to the weghted average value smoother. 195
(b) D MPS results Fg. 3 Comparson of mpact pressure at pont P between experment, D MPS and 3D MPS method (c) 3D MPS results Fg. 5 Comparson of snapshots between experment (upper), D MPS smulaton (mddle) and 3D MPS smulaton (lower) Fg. 4 Enlarged vew of the tme hstory of the pressure at pont P n one perod Fg.5 compares some snapshots of free surface deformaton and velocty feld between experment, D and 3D MPS smulaton. An acceptable agreement can be obtaned between D smulaton, 3D smulaton and expermental results n terms of both free surface shape and velocty feld n sloshng flow on the whole. Thus, the detals of free surface deformaton, such as the breakng regon of the wave and the splash of lqud, can be capszed by 3D MPS method more effectvely than D MPS method. From the dscusson above, we can conclude that the detals of flow sloshng, such as breakng wave, overturned free surface and splashng water, can be affected by the dmensonal effect n MPS smulaton. (a) expermental results For the sloshng smulaton n ths paper, 5944 partcles and 663458 partcles are used n D MPS and 3D MPS method respectvely, whch result n much larger computaton tme 3D MPS method than D MPS method. So D MPS method s more effcent to predct quas-3d sloshng flows n general. However, 3D MPS method s essental to deal wth pure 3D sloshng. Furthermore, 3D MPS method s more approprate than D MPS method when smooth pressure fled or sloshng detals are requred n sloshng problem. CONCLUSIONS In ths paper, the 3D modfed MPS method s utlzed to smulate lqud sloshng n rectangular tank under horzontal exctaton. Compared the results obtaned by D and 3D MPS method, the dmenson effect has been analyzed. For valdaton of the present MPS method, a comparson s made between the computatonal results for D and 3D smulaton and avalable expermental data, for whch favorable agreement s shown n general. The results show that: 3D MPS method can capsze the detals of sloshng flow such as breakng wave and splashng water more effcently. The results of 3D smulaton agree better wth expermental results than D smulaton. For the predcton of mpact pressure, 3D MPS method can predct the mpact pressure much smoother and more accurately than D MPS method. However, because of the less effect of the detals, such as breakng wave, overturned free surface and splashng water, computatonal results between D MPS and 3D MPS have an excellent agreement. Consderng large computaton n 3D smulaton, t s sensble for us to apply D MPS method to predct sloshng problem generally. Thus, t s essental to utlze 3D smulaton n pure 3D flows or other condtons wth requrements of smoother pressure feld and sloshng detals. 196
ACKNOWLEDGEMENTS The authors wsh to thank Kuang-An Chang and Youn Kyung Song for provdng the expermental data of sloshng. Ths work s supported by Natonal Natural Scence Foundaton of Chna (Grant Nos. 5137915, 1171, 5141113131), the Natonal Key Basc Research Development Plan (973 Plan) Project of Chna (Grant No. 13CB3613), Hgh Technology of Marne Research Project of the Mnstry of Industry and Informaton Technology of Chna, the Program for Professor of Specal Appontment (Eastern Scholar) at Shangha Insttutons of Hgher Learnng(Grant No. 13), Center for HPC of Shangha Jao Tong Unversty. REFERENCES Cu, Y and LIU, H (9). "Numercal Smulaton of Sloshng n Two Dmensonal Rectangular Tanks Wth SPH," Annual Journal of Hydraulc Engneerng, JSCE, Vol 53, pp 181-186. Delorme, L, Colagross, A and Souto-Iglesas, A (9). "A set of canoncal problems n sloshng, Part I: Pressure feld n forced roll comparson between expermental results and SPH," Ocean Engneerng, pp 168-178. Gngold, RA and Monaghan, JJ (1977). "Smoothed partcle hydrodynamcs-theory and applcaton to non-sphercal stars," Royal Astronomcal Socety, Vol 181, pp 375-389. G.X. Wu, Q.W. Ma and R. Eatock Taylor (1998). "Numercal smulaton of sloshng waves n a 3D tank based on a fnte element method," Appled Ocean Research, pp 337-355. Kondo, M and Koshzuka, S (8). "Improvement of stablty n movng partcle sem-mplct method," Int J Num Meth Fluds, Vol 65 pp 638-654. Koshzuka, S, and Oka, Y (1996). "Movng-partcle sem-mplct method for fragmentaton of ncompressble flud," N Sc Eng, Vol 13, pp 41-434. Koshzuka, S, Obe, A, and Oka, Y (1998). "Numercal analyss of breakng waves usng the movng partcle sem-mplct method," Int J Num Meth Fluds, Vol 6, pp 751-769. Lee, D.H., Kma, M.H. and Kwon, S.H. (7). "A parametrc senstvty study on LNG tank sloshng loads by numercal smulatons," Ocean Engneerng, Vol 34, pp 3-9. Lee, BH, Park, JC, Km, MH, and Hwang, SC (11). "Step-by-step mprovement of MPS method n smulatng volent free-surface motons and mpact-loads," Computer Meth App Mech Eng, Vol, pp 1113-115. Shao, JR, L, HQ, Lu, GR and Lu, MB (1). "An mproved SPH method for modelng lqud sloshng dynamcs," Computers and Structures, Vol 1-11, pp 18-6. Song, YK, Chang, KA, Ryu, Y and Kwon, SH (13). "Expermental study on flow knematcs and mpact pressure n lqud sloshng," Zachry Department of Cvl Engneerng, Texas A&M Unversty, College Staton, TX, USA. Lee, S.J. M.H. Km, Y.S. Shn and B.K. Km(8). "The Effects of Tank Sloshng on the Coupled Responses of LNG Vessel and Floatng Termnal," Proc 18th Int Offshore and Polar Eng Conf, Lsbon, ISOPE, Vol 3, www.sope.org. Tanaka, Masayuk and Masunaga, Takayuk (1). "Stablzaton and smoothng of pressure n MPS method by Quas-Compressblty," Journal of Computatonal Physcs, Vol 9, pp 479-49. Wang, CZ and Khoo, BC (5). "Fnte Element Analyss Of Two- Dmensonal Nonlnear Sloshng Problems In Random Exctatons," Ocean Engneerng, Vol 3, pp 17-133. Zhang, YX and Wan, DC (1) Comparson Investgatons of Incompressble Vscous Flows by SPH Method and MPS Method, Proceedngs of the 9th Internatonal Conference on Ocean, Offshore and Arctc Engneerng, OMAE1, June 6-11, 1, Shangha, Chna, Paper No OMAE1-843. Zhang, YX and Wan, DC (11a). "Applcaton of MPS n 3D dam breakng flows (n Chnese), " Sc Sn Phys Mech Astron, Vol 41, pp 14 154. Zhang, YX, and Wan, DC (11b). "Apply MPS Method to Smulate Moton of Floatng Body Interactng wth Soltary Wave," Proc 7th Int Workshop Shp Hydr, IWSH, Shangha, Chna, pp 75-79. Zhang, YX, Wan, DC (11c). "Applcaton of mproved MPS method n sloshng problem," Proc. of the 3rd Chnese Symposum on Hydrodynamcs, X an, Chna. Zhang, YX, and Wan, DC (1). "Numercal smulaton of lqud sloshng n low-fllng tank by MPS," Chnese Journal of Hydrodtnamcs, Vol 7, No 1, pp 1-17. 197