Comparative study of MPS method and level-set method for sloshing flows *

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1 577 14,6(4): DOI: 1.116/S11-658(14)665- Comparatve study of MPS method and level-set method for sloshng flows * ZHANG Yu-xn ( 张雨新 ), WAN De-cheng ( 万德成 ) State Key laboratory of Ocean Engneerng, School of Naval Archtecture, Ocean and Cvl Engneerng, Shangha Jao Tong Unversty, Shangha, Chna, E-mal: ceyuxn@163.com HINO Takanor Graduate school of Engneerng, Yokohama Natonal Unversty, Yokohama, Japan (Receved January 6, 14, Revsed March 1, 14) Abstract: Ths paper presents a comparatve study of a meshless movng partcle sem-mplct (MPS) method and a grd based level-set method n the smulaton of sloshng flows. The numercal schemes of the MPS and level-set methods are outlned and two volent sloshng cases are consdered. The computed results are compared wth the correspondng expermental data for valdaton. The mpact pressure and the deformatons of free surface nduced by sloshng are comparatvely analyzed, and are n good agreement wth expermental ones. Results show that both the MPS and level-set methods are good tools for smulaton of volent sloshng flows. However, the second pressure peaks as well as breakng and splashng of free surface by the MPS method are captured better than by the level-set method. Key words: movng partcle sem-mplct (MPS) method, level-set method, lqud sloshng, mpact pressure, free surface Introducton Lqud sloshng s a knd of nonlnear free surface flows, and usually nvolves some complcated phenomena, such as breakng wave, overturnng of free surfaces and splashng lqud. The mpact pressure nduced by sloshng may damage the structure of the tank, and affect the safety of shp wth lqud cargo nteractng wth waves. Therefore, accurate predcton of mpact pressure nduced by lqud sloshng s of sgnfcant mportance n ocean engneerng. CFD has been proved to be an effectve tool to compute sloshng flows n the last decades [1-6]. The * Project supported by the Natonal Natural Scence Foundaton of Chna (Grant Nos , and 1171), the Natonal Key Basc Research Development of Chna (973 Program, Grant No. 13CB3613), the Hgh Technology of Marne Research Project of the Mnstry of Industry and the Informaton Technology of Chna and the Program for Professor of Specal Appontment (Eastern Scholar) at Shangha Insttutons of Hgher Learnng (Grant No. 13). Bography: ZHANG Yu-xn (1985-), Male, Ph. D. Canddate Correspondng author: WAN De-cheng, E-mal: dcwan@sjtu.edu.cn methods adopted n CFD can be generally dvded nto two categores: the grd-based method and the meshless method. In the framework of grd-based method, the dscretzatons of governng equatons are mplemented based on grd system, and the deformaton of free surface s usually descrbed by an extra functon whch s updated n every tme step, such as volume of flud (VOF) [7,8] and level-set [9,1] methods. Nevertheless, there s no need for grd n usng the meshless method, n whch flud s represented by a set of nteractng partcles. These partcles have physcal propertes, such as mass, momentum, and energy, etc. Motons of partcles are descrbed by the Lagrangan approach. Due to the meshless character and Lagrangan nature, the meshless method has two great advantages: (1) fragment and coalescence of flud can be computed straghtforwardly snce there s no constant topology relatonshp between partcles, () numercal dffuson n the dscretzaton of convecton term s elmnated by use of substantal dervatve n governng equatons. However, the meshless method s of hgh computaton cost and usually suffers from unphyscal pressure oscllaton. There are both the pros and cons between the grd-based method and the meshless method.

2 578 The am of the present paper s to make comparatve nvestgaton on two commonly used methods: a meshless method, the movng partcle sem-mplct (MPS) method [11,1], and a grd-based method, the level-set method, n a sloshng context. The present MPS solver s developed at Shangha Jao Tong Unversty, whch employs four mproved numercal schemes to overcome the unphyscal pressure oscllaton commonly observed n tradtonal MPS method, such as: (1) momentum conservatve gradent model [13], () kernel functon wthout sngularty [1], (3) mxed source term for pressure Posson equaton (PPE) [13,14], (4) accurate free surface detecton method [15]. The effect of these mproved schemes on the predcton of mpact pressure n sloshng wll be analyzed by comparng the results by the present MPS method wth that by tradtonal MPS method. On the other hand, the present level-set method s ncorporated n a grdbased solver,.e. SURF (Soluton algorthm for Unstructured RANS wth FVM) [16], whch s an unstructured fnte volume method (FVM) solver dedcated to smulate free surface flow n marne hydrodynamcs. The code has been developed snce 1994 at the CFD Research Center, Natonal Martme Research Insttute (NMRI) n Japan. By use of unstructured grd, SURF can deal wth complex geometry. A sngle phase level-set s adopted n SURF to make t more effcently solve the flow feld around shp [17]. Ths code has been commercalzed by NMRI themselves and used n many shpyards n Japan. In the followng sectons, the numercal schemes of the MPS and level-set methods are frst outlned, then comparson analyss are carred out aganst lqud sloshng problems. The calculated pressures by tradtonal MPS method and present MPS method are compared to show the mprovement on pressure computaton. In addton, both MPS and level-set results are compared wth experment to valdate both solvers. Dscrepancy of the numercal results by the MPS and level-set methods s dscussed. 1. Movng partcle sem-mplct method 1.1 Governng equatons For ncompressble and vscous flud, governng equatons ncludng mass and momentum conservaton equatons are represented as 1 Dρ = V = (1) ρ Dt tme, V the velocty vector, ρ the densty, p the pressure, ν the knematc vscosty, and g the gravty acceleraton. Thanks to Lagrangan nature, convecton terms do not turn up n the momentum equatons of MPS calculaton, thus avodng numercal dffuson n the dscretzaton of convecton terms. 1. Partcle models Governng equatons are transformed to the equatons of partcle nteractons based on so called partcle models, namely the gradent model and Laplace model. In orgnal MPS method, the gradent operator s modeled as a local weghted average of the gradent vectors between partcle and ts neghborng partcle j, gven as [11] D φ φ < φ > = ( r r ) ( r r ) j j W j n j rj r (3) where φ s an arbtrary scalar functon, D the number of space dmensons, r the coordnate vector, W ( rj r ) the kernel functon, and n the ntal partcle number densty. Equaton 3 s not conservatve from momentum pont of vew snce two solated neghbor partcles wth dfferent pressures wll be accelerated to nfnty. A conservatve form can be obtaned by addng φ nto ts rght sde [13] D φ + φ < φ > = ( r r ) ( r r ) j j W j n j rj r (4) Usng Eq.(4), the contrbuton of to j s equal to that of j to, whle the forces between and j are repulsve. A smlar equaton s used n the SPH method. Note that the pressure gradent computed by Eq.(4) s nonzero even for constant pressure feld when partcles are n dsorder. On the other hand, ths would help partcles keep constant densty. The comparson between Eq.(3) and Eq.(4) has been studed by Tanaka and Masunaga [13] aganst a dam break flow, showng that Eq.(4) can compute pressure wth less oscllaton behavor than Eq.(3). In vew of ths, Eq.(4) s employed n the present MPS computaton. The kernel functon can have varous forms. One of the most commonly used kernel functon n the MPS method s proposed by Koshzuka [11], gven as DV Dt 1 ν V g () ρ = p + + r W( r)= e 1 r, < r < r (5a) e where D/Dt denotes substantal dervatve, t s the W( r )=, re r (5b)

3 579 A drawback of the above kernel functon s that t becomes sngular at r =. Ths may cause unreal pressure between two neghborng partcles wth a small dstance, and then affect the computatonal stablty. To overcome ths, a modfed kernel functon s used n ths paper [1] as follows re W( r)= 1.85 r +.15r, r < re (6a) e W( r )=, re r (6b) The kernel functon n Eq.(6) has smlar shape wth the one n Eq.(5), but wthout sngularty at r =. In the MPS method, the densty of flud s descrbed by the so-called partcle number densty n, defned as [11] < n> = W( rj r ) (7) j The Laplacan operator s derved by Koshzuka [11] from the physcal concept of dffuson, expressed as: D = ( ) ( r r ) (8) < φ > φ j φ W j n λ j λ = j W ( r r ) r r j j j W ( r r ) j (9) where λ s a parameter, ntroduced to keep the varance ncrease equal to that of the analytcal soluton. Both vscous force V n Eq.() and P on the rght hand sde of the Posson pressure equaton (.e., Eq.(1) and Eq.(11)) are dscretzed by usng Eq.(8). 1.3 Model of ncompressblty The ncompressble condton n tradtonal MPS method s represented by keepng the partcle number densty constant. Thus the Posson equaton of pressure (PPE) s expressed as [11] ρ < n t > n n * k +1 < p > = (1) where n s the partcle number densty n the temporal feld. The source term n Eq.(1) s solely based on the devaton of the temporal partcle number densty from the ntal value. Pressure obtaned from Eq.(1) s prone to oscllate n spatal and temporal domans snce the partcle number densty feld s not smooth. To overcome ths, Tanaka and Masunaga [13] proposed a mxed source term for PPE, whch combnes the velocty dvergence and the partcle number densty together. The man part of the mxed source term s the velocty dvergence, whle the partcle number densty s used to keep flud volume constant. Ths mproved PPE s rewrtten by Lee [14] as k k +1 ρ * ρ < n > n < p > = (1 γ) V γ (11) t t n where γ s a blendng parameter wth a value between and 1. The range of.1 γ.5 s better accordng to numercal tests conducted by Lee [14], and smaller γ mples smoother pressure feld. Therefore, n ths paper, γ =.1 s used for all smulatons. 1.4 Free surface detecton Interacton doman s truncated near free surface snce there s no partcle out of doman for sngle phase calculaton. Thus partcles on the free surface have smaller partcle number densty than those under the free surfaces. Consder ths, n tradtonal MPS method, partcles are defned as surface partcles when they satsfy [11] < n>< β n (1) * where β s a parameter wth a value between.8 and.99. For the surface partcles, zero pressure condton s mposed. Therefore, the detecton of surface partcle has sgnfcant effect on the pressure feld. Fg.1 Schematc of tank geometry (m) Table 1 Computatonal parameters Case Exctaton ampltude A (m) Exctaton perod T (s) Case Case Equaton (1) has a low accuracy snce nner partcles wth small partcle number densty may be msjudged as surface partcles. Therefore, unreal pressure

4 58 Fg. Comparson of pressure evoluton at poston P between experment, tradtonal MPS and present MPS results around the msjudged partcles occurs. Ths usually causes unphyscal pressure oscllaton. To avod ths problem, an mproved detecton method s employed n ths paper. Ths mproved detecton method defnes a vector functon as follows [15] D 1 < F > = ( r r ) W ( r ) j j n j r rj (13) Partcles would be judged as surface partcles when the magntudes of ther F satsfy < >> F α F (14) where α s a parameter, and has a value of.9 n ths paper, F s the ntal value of F for surface partcle, also equal to the F of boundary partcles wthout neghborng flud partcle. The vector F represents the asymmetry of neghborng partcles. It has a small value at the core of the doman and a large value near the free surface. It should be ponted out that Eq.(14) s not vald for splashed partcle whch has few neghborng partcles, so t s only used for partcles wth number densty between.8n and.97n. Partcles wth number densty lower than.8n should be surface partcles, whle those wth number densty hgher than.97n should get pressure through the Posson equaton. 1.5 Level-set method n SURF solver In the level set method, the shape of free surface s defned by a mathematcal functon φ (, xyzt,,), whch s a sgned dstance from the nterface,.e., φ n water (15a) φ = on the free surface (15b) φ n ar (15c) Accordng to knematc condton, the flud partcles on a free surface reman on the free surface. Thus the followng mathematcal form s obtaned D φ φ + u φ + v φ + w φ = (16) Dt t x y z 1.6 The localzed level set method A sngle phase localzed verson of level set method [18] whch mproves the effcency of the orgnal level set approach n computaton of free surface shape s employed n SURF. In the localzed level set method, the two parameters γ 1 and γ( < γ1 < γ) are ntroduced. The sgned dstance functon s rewrtten as d(, xyzt,,) and the defnton of the level-set functon s modfed as φ = γ f d > γ (17a) φ = d f d γ (17b) φ = γ f d < γ (17c)

5 581 Fg.3 Comparson of pressure evoluton n Case 1 between experment, MPS and level set Thus, the level set functon s localzed wthn the bandwdth γ from the nterface. The transport equaton (Eq.(16)) s modfed as φ w + C( φ) u φ + v φ + w = t x y z (18) where C( φ ) s the cut-off functon defned as C( φ )=1, φ < γ1 (19a) S( φ)= 1 f φ < (1a) S( φ )= f φ = (1b) S( φ )=1 f φ > (1c) As the soluton of Eq.() converges, φ becomes the dstance functon agan snce φ = 1. ( φ γ ) ( φ+ γ 3 γ ) C( φ)= 1 3 ( γ γ1), γ1 φ γ (19b) C( φ )=, φ > γ (19c) In computaton the update of φ s performed only n the regon of φ γ. 1.7 Re - ntalzaton As the level set functon s no longer a dstance functon after the convecton, a re-ntalzaton process s necessary n level set computaton. In SURF, the re-ntalzaton process s conducted usng the partal dfferental equaton as φ + S( φ )( φ 1) = τ () where τ s the pseudo tme, φ the ntal value and S a sgn functon, defned as Fg.4 The process of wave breakng n Case 1 (MPS result)

6 58 Fg.5 Comparson of pressure tme hstores n Case between experment, MPS and level set S ε In a numercal process, S( φ ) s approxmated as ( φ)= φ φ + ε where ε s a typcal grd spacng. () φ φ + Sε( φ) φ = Sε( φ) τ φ (3) Equaton (3) can be vewed as the convecton equaton for φ wth the convecton velocty beng S ε ( φ ) φ / φ. Fg.7 Detaled comparson of pressure evoluton n a sngle mpact behavor between experment, MPS and level set. Computaton results and dscusson Fg.6 Process of wave breakng n Case (MPS result) Thus Eq.() s rewrtten as.1 Computatonal condtons Computatonal model s a -D rectangular tank, as shown n Fg.1, and ts dmensons are L H =

7 583 Fg.8 Comparson of free surfaces between experment, MPS and level set 1. m.6 m. Four pressure probes were preset on the walls of the tank to record the calculated pressure for comparson wth experment whch was conducted by Shp Research Insttute n Tokyo [19]. The depth of water was.1 m, and the correspondng fllng level was %. The tank was forced to harmonously sway as follows πt x= A sn T (4) where A s the exctaton ampltude, and T s the exctaton perod. As the tank s subjected to smple harmonc moton along ts length, flow n the tank s two-dmensonal n the man tank plane unless the flow becomes nstable [,1]. Two cases wth the same exctaton ampltude A =.6 m but dfferent exctaton perods are consdered. In these cases, no flow nstablty occurs, mplyng -D smulaton s able to predct the flud moton. Computatonal parameters are shown n Table 1. In MPS computaton, the number of partcles used n smulaton s 5 84, among whch 3 98 are flud partcles, and the correspondng partcle ntal spacng s.6 m.. Computatonal results Before dscussng the results of the MPS and level set methods, comparson s frst carred out between tradtonal MPS and the present MPS to show the mprovement on smoothng pressure feld. Here, tradtonal MPS employs Eqs.(3), (5), (1) and (1) n computaton, whle Eqs.(4), (6), (11) and (14) are used n the present MPS computaton. Fgure shows the pressure tme hstores at poston P obtaned by experment, tradtonal MPS and the present MPS computatons. Although the tradtonal MPS s able to predct the mpact phenomena, t cannot predct the value of mpact pressure well. Strong oscllatons wth hgh frequency are observed n tradtonal MPS results. However, the pressure curves n the present MPS results are much smoother, and thus seems more reasonable. The mprovement on pressure predcton s evdent. Fgure 3 compares pressure tme hstores n Case 1 between experment, MPS and level set results. It s seen that the lqud mpact behavor s well predcted n numercal smulatons. The general evolutons of mpact pressures obtaned numercally by the MPS and level set methods agree wth expermental results. Two pressure peaks are observed n each perodc

8 584 Fg.9 Comparson of free surfaces between experment, MPS and level set mpact behavor. The frst one has a larger value and a shorter duraton. It s due to the mpact of lqud on the sde wall of the tank wth a large horzontal velocty. After the frst peak, lqud goes up along the wall, and fnally falls down due to the gravty. Second peak occurs when the fallng lqud hts the underlyng lqud. The frst pressure peak s overestmated by both the MPS and level-set methods compared wth expermental data. The reason for ths s that breakng wave occurs when lqud mpacts on the sde wall (see Fg.4), ar s pocketed by lqud and wall. However, the effect of exstence of ar s not taken nto account n sngle phase smulaton, thus hgher pressure are caused when the ar bubble vanshes. Two phase calculaton may smulate such phenomena better. However, ths part of work s not ncluded n ths paper. There s a pressure oscllaton after the frst pressure peak n the level set results, whch mght be due to numercal error nvolved n treatment of wall boundary. Further mprovement on ths wll be desred, however, t does not affect the general profle of pressure curves. Fgure 5 shows the results of Case. Agan, the agreements between experment and numercal smulatons are satsfactory. In ths case, flow s also volent. Breakng wave s observed. However, t occurs earler than t dd n Case 1, see Fg.6, n whch the wave breaks before t reach the sde wall. The results n that the values of frst pressure peaks at postons P1 and P3 are relatvely smaller than the ones n Case 1, and also mples that exctaton perod has sgnfcant effect on the wave n sloshng flow. The results of MPS and level set are compared n Fg.5, and t s seen that the MPS smulaton shows a better results snce the level set smulaton overestmates the frst pressure peak at postons P1 and P3, and underestmates the second peak at poston P1. A detaled comparson between MPS and levelset s shown n Fg.7, whch llustrates the pressure evolutons n a sngle mpact behavor at poston P1 for the two cases. MPS shows a better predcton on the moment and the value of second pressure peak than level-set compared wth expermental data. The reason for ths mght be that the wave breaks after mpactng on the sde walls of the tank, and MPS s able to compute the breakng free surfaces better than level-set, thus shows a good predcton of the subsequent flow whch s related to the second pressure peak. Fgures 8 and 9 show some snapshots n experment and smulatons for Case 1 and Case, respectvely. It s seen that the flows are volent. Breakng

9 585 free surface and splashng lqud are observed when lqud mpacts on the sde wall of the tank. Compared wth experment, both MPS and level set predct the profles of free surface well. The general moton of lqud s accurately captured n computaton. However, partcle method s more flexble to descrbe the lqud splashng phenomena, and thus the free surfaces n MPS results behave more naturally. 3. Concluson In ths paper, comparatve study of the MPS method and level-set method n the smulaton of sloshng flows was presented. Two volent sloshng flows n -D rectangular tank subjected to sway moton were computed. The mpact pressures obtaned numercally show two peaks durng each perodc mpact behavor, and are n agreement wth expermental obsercatons. However, the second peaks by the MPS method are predcted better than by the level-set method. The computed free surfaces agree well wth expermental ones, suggestng that both the MPS and the level-set methods are able to smulate the large deformaton of free surface n volent sloshng flows. Nevertheless, due to the Lagrangan nature, the MPS method can smulate the breakng wave and the splashng lqud better than the level-set method do. Acknowledgement The support of the Center of Hgh Performance Computng (HPC) of Shangha Jao Tong Unversty for ths work s gratefully acknowledged. References [1] KIM Y. Numercal smulaton of sloshng flows wth mpact load[j]. Appled Ocean Research, 1, 3(1): [] LIU D., LIN P. A numercal study of three-dmensonal lqud sloshng n tanks[j]. Journal of Computatonal Physcs, 8, 7(8): [3] PAN Xu-je, ZHANG Hua-xn and LU Yun-tao. Numercal smulaton of vscous lqud sloshng by movng-partcle sem-mplct method[j]. Journal of Marne Scence and Applcaton, 8, 7(3): [4] HU Chang-hong, YANG Kyung-Kyu and KIM Yonghwan. 3-D numercal smulatons of volent sloshng by CIP-based method[j]. Journal of Hydrodynamcs, 1, (5Suppl.): [5] LI Yu-long, ZHU Ren-chuan and MIAO Guo-png et al. Smulaton of tank sloshng based on OpenFOAM and couplng wth shp motons n tme doman[j]. Journal of Hydrodynamcs, 1, 4(3): [6] WU C. H., FALTINSEN O. M. and CHEN B. F. Numercal study of sloshng lqud n tanks wth baffles by tme-ndependent fnte dfference and fcttous cell method[j]. Computers and Fluds, 1, 63: 9-6. [7] LIU D., LIN P. Three-dmensonal lqud sloshng n a tank wth baffles[j]. Ocean Engneerng, 9, 36(): -1. [8] MING Png-jan, DUAN Wen-yang. Numercal smulaton of sloshng n rectangular tank wth VOF based on unstructured grds[j]. Journal of Hydrodynamcs, 1, (6): [9] YU K. Level-set RANS method for sloshng and green water smulatons[d]. Doctoral Thess, College Staton, Texas, USA: Texas A&M Unversty, 8. [1] CHEN Y. G., DJIDJELI K. and PRICE W. G. Numercal smulaton of lqud sloshng phenomena n partally flled contaners[j]. Computers and Fluds, 9, 38(4): [11] KOSHIZUKA S., NOBE A. and OKA Y. Numercal analyss of breakng waves usng the movng partcle sem-mplct method[j]. Internatonal Journal for Numercal Methods n Fluds, 1998, 6(7): [1] ZHANG Yu-xn, WAN De-cheng. Numercal Smulaton of lqud sloshng n low-fllng tank by MPS[J]. Chnese Journal of Hydrodynamcs, 1, 7(1): 1-17(n Chnese). [13] TANAKA M., MASUNAGA T. Stablzaton and smoothng of pressure n MPS method by quas-compressblty[j]. Journal of Computatonal Physcs, 1, 9(11): [14] LEE B. H., PARK J. C. and KIM M. H. et al. Stepby-step mprovement of MPS method n smulatng volent free-surface motons and mpact-loads[j]. Computer Methods n Appled Mechancs and Engneerng, 11, (9-1): [15] ZHANG Yu-xn, WAN De-cheng. Applcaton of mproved MPS method n sloshng problem[c]. Proceedngs of the 3rd Natonal Congress on Hydrodynamcs and 1th Natonal Congress on Hydrodynamcs. X an, Chna, 11, (n Chnese). [16] WACKERS J., KOREN B. and RAVEN H. et al. Freesurface vscous flow soluton methods for shp hydrodynamcs[j]. Archves of Computatonal Methods n Engneerng, 11, 18(1): [17] HINATSU M., HINO T. Computaton of vscous flows around a wgley hull runnng n ncdent waves by use of unstructured grd method[c]. Proceedngs of the th Internatonal Offshore and Polar Engneerng Conference. Ktakyushu, Japan,. [18] PENG D., MERRIMAN B. and OSHER S. et al. A PDE-based fast local level set method[j]. Journal of Computatonal Physcs, 1999, 155(): [19] HINATSU M., TSUKADA Y. and FUKUSAWA R. et al. Experments of two-phase flows for the jont research[c]. Proceedngs of SRI-TUHH Mn-Workshop on Numercal Smulaton of Two-Phase Flows. Tokyo, Japan, 1. [] COLAGROSSI A., LUGNI C. and GRECO M. et al. Expermental and numercal nvestgaton of D sloshng wth slammng[c]. Proceedngs of 19th Internatonal Workshop on Water Waves and Floatng Bodes. Cortona, Italy, 4. [1] FALTINSEN O. M., ROGNEBAKKE O. F. and TIMOKHA A. N. Resonant three-dmensonal nonlnear sloshng n a square-base basn[j]. Journal of Flud Mechancs, 3, 487: 1-4.