Multiphase MPS Method for Two-Layer-Liquid Sloshing Flows in Oil-Water Separators

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1 Proceedngs of the Twenty-eghth (08 Internatonal Ocean and Polar Engneerng Conference Sapporo, Japan, June 0-5, 08 Copyrght 08 by the Internatonal Socety of Offshore and Polar Engneers (ISOPE ISBN ; ISSN Multphase MPS Method for Two-Layer-Lqud Sloshng Flows n Ol-Water Separators Xao Wen, Decheng Wan *, Gang Chen Collaboratve Innovaton Center for Advanced Shp and Deep-Sea Exploraton, State Key Laboratory of Ocean Engneerng, School of Naval Archtecture, Ocean and Cvl Engneerng, Shangha Jao Tong Unversty, Shangha, Chna * Correspondng author ABSTRACT The am of ths paper s the nvestgaton of the two-layer-lqud sloshng phenomena nsde a rgd tank usng a new multphase method developed based on the meshless MPS method. The MPS method s partcularly sutable for smulatng the deformatons of two-phase nterface n multphase flows. To extend the MPS method to a multphase system, specal nterface treatments are appled for the nterface partcles, ncludng densty smoothng technque, nterpartcle vscosty model and surface tenson model. The capabltes of present mult-phase MPS method to treat ths two-layer sloshng problem are assessed through comparsons wth expermental data n lterature, and the results show that the present method s able to capture the deformatons of two-phase nterface at dfferent locatons. The pressure feld obtaned by present method s smooth and the mpact pressure at the nner walls of lqud tank shows a reasonable agreement wth experment. KEY WORDS: Mult-phase MPS; two-layer-lqud sloshng; olwater separators; nterface treatments; surface tenson. INTRODUCTION On offshore structures, such as FPSOs and ol producton platforms, ol-water separators are necessary for the ntal processng of crude ol. However, volent mult-layer-lqud sloshng phenomena may occur when the separators are excted by the large oscllatons of offshore structures. On the one hand, the mult-layer-lqud sloshng nduces the undesrable moton of the two-phase nterface and reduces the processng effcency of the ol-water separators. On the other hand, the sloshng nduced mpact pressure can brng about crtcal damages to the ol-water separators. To fgure out potental hazards and provde gudance for desgnng more effcent separators, accurate predcton method of the moton of nterfaces and mpact pressure s necessary to check that no undesrable resonances take place at the desgn stage. In general, sloshng wth only a sngle layer flud has been the man focus of most studes reported, whle the mult-layer-lqud sloshng phenomena have rarely been concerned. Xue et al. (03 developed a lqud sloshng expermental rg, to study the layered lqud sloshng n a rectangular tank partally flled wth water and ol. In ther work, the pressure dstrbuton on the tank walls and the nterfacal wave dsplacement are estmated by changng external exctaton frequency of the shakng table. La Rocca et al. (005 performed a theoretcal and expermental nvestgaton on the sloshng of a two-lqud system wth both two-phase nterface and free surface. Scortno et al. (009 nvestgated the sloshng of a layered flud system usng both a new Hamltonan mathematcal model and new laboratory experments, and good agreements between mathematcal predctons and laboratory measurements were found n all the analyzed cases. Moln et al. (0 reported sloshng experments wth a rectangular tank flled wth three fluds of dfferent denstes, then an analytcal model s proposed based on lnearzed potental flow theory and compared wth a fully nonlnear CFD code wth VOF trackng of the nterfaces. However, the potental flow theory and VOF method employed n the above mentoned studes become less effectve wth the ncrease of the sloshng ntensty. To overcome ths lmtaton, a multphase method s developed based on the meshless MPS method and appled to the numercal smulaton of the two-layer-lqud sloshng phenomena n ol-water Separators. In recent few years, Lagrangan meshless methods are popular for smulatons of volent flows wth large deformaton of free surface. One of the most commonly used meshless methods s the MPS method proposed by Koshzuka and Oka (996 for fully ncompressble flows. However, the orgnal MPS method appears to be less stable and shows extra pressure oscllaton. To mprove the performance of MPS method, numerous studes have been carred out for optmzaton of source term n the Pressure Posson Equaton (Tanaka and Masunaga, 00; Khayyer and Gotoh, 0, gradent model (Koshzuka et al., 998; Tsuruta et al., 03, Laplacan model (Khayyer and Gotoh, 00; Khayyer and Gotoh, 0; Ikar et al., 05, collson model (Lee et al., 03, boundary condtons (Lee et al., 0 and detecton of free surface partcles (Khayyer et al., 009; Zhang and Wan, 0. In general, Gotoh and Khayyer (06 revews the latest achevements made n partcle method, n partcular the projecton-based ones. Wth these mprovements, the MPS method can be successfully employed to 859

2 deal wth sloshng problems wth a sngle layer flud. However, to deal wth the two-layer-lqud sloshng problems whch are the focus of ths paper, there are stll a lot of works needed to be done to extend the sngle phase MPS method to multphase flows. For ths purpose, Shakbaena and Jn (0 developed a mult-vscosty model based on a weakly compressble MPS method to smulate the Raylegh Taylor and Kelvn Helmholtz nstabltes. Khayyer and Gotoh (03 presents an enhanced stablzed MPS method for smulaton of multphase flows characterzed by hgh densty ratos, whch benefts from four prevously developed schemes (Khayyer and Gotoh, 0 and a new densty smoothng scheme. Km et al. (04 nvestgated the multlqud-sloshng smulaton by developng a new multfludlayer MPS method, n whch buoyancy correcton and surface tenson models are ncluded. Duan et al. (07 developed two multphase MPS method, the MMPS-HD and MMPS-CA method. In MMPS-HD method, harmonc mean densty s utlzed to avod extremely large acceleraton at nterface. In MMPS-CA method, new stable formulatons are developed to keep acceleraton contnuous at nterface. In ths paper, we developed a multphase method based on the MPS method. In present method, the multphase system s treated as the mult-densty and mult-vscosty flud. The nterpartcle vscosty defned by the harmonc mean vscosty of two partcles s frstly adopted to consder the nteracton between dfferent phases. Then the densty smoothng technque proposed by Shakbaena and Jn (0 s employed to reduce pressure dscontnuty crossng the nterface and obtan the contnuous acceleraton and velocty felds. The nfluence of surface tenson force on the nterface s consdered through a contoured contnuum surface force (CCSF model proposed by Duan et al. (05. The present multphase MPS s then employed to smulate the twolayer-lqud sloshng n a rgd tank and verfed through comparsons wth experment by Xue et al. (03 for two-layer-lqud sloshng. In general, the pressure dstrbuton on the tank walls and the nterfacal wave dsplacement obtaned from numercal smulaton and experment show good agreement, valdatng the sutablty of the present multphase MPS method for ndustral and engneerng problems ncludng two-layer-lqud sloshng. NUMERICAL METHOD Governng Equatons In the present multphase MPS method, the multflud system s treated as a sngle flud wth mult-densty and mult-vscosty. Therefore, the governng equaton for all fluds s unform, expressed by contnuty and momentum equatons. They can be wrtten as: Dρ = ρ( u = 0 Dt ( Du ρ = P + ( μ u + g+ F s Dt ( where ρ and μ are respectvely the densty and dynamc vscosty, u s the flow velocty, P s the pressure, g s a constant gravtatonal acceleraton, F s s the nterface tenson force between dfferent phases. The dfference n governng equaton for dfferent fluds s manly reflected n the value of ρ and μ. Compared wth grd-based method, the Lagrangan descrpton of governng equaton effectvely avods the numercal dsperson nduced by convectve acceleraton term. Partcle Interacton Models In MPS method, the governng equaton s dscretzed through a set of dsordered partcles n space. And partcle nteracton models based on kernel functon are used to replace all terms of dfferental operators n the rght hand of governng equaton. In ths paper, kernel functon suggested by Zhang and Wan (0 s used, expressed as: ( W r re (0 r < re = 0.85r+ 0.5re 0 ( re r where r s the dstance between two partcles and r e s the radus of the partcle nteracton. In smulaton of multphase flows, the mathematcal dscontnuty of densty at two-phase nterface causes a dscontnuous acceleraton feld and accordngly numercal nstabltes. To deal wth ths dscontnuty, a densty smoothng technque s carred out for partcles near nterface, based on a smple spatal averagng as follow: ρ = j j (, r ( j, re ρ W r j j e W r The smoothng scheme above s ncluded n the multphase method based on WC-MPS formulatons proposed by Shakbaena and Jn (0, and s mproved to frst-order accurate through a frst order accurate Taylor seres expanson by Khayyer and Gotoh (03. In ths paper, the lower order accurate scheme s adopted because that the densty dfference between two layers fluds s not very sgnfcant. For multphase wth large densty rato, such as ar-water flows, hgher order curate scheme may be more essental. The gradent model represents a local weghted average of the gradent vectors between partcle and ts neghborng partcle j. In the case of pressure gradent, the modfed gradent model proposed by Koshzuka et al. (998 s adopted, wrtten as: ( Pj P,mn d P = W e ( r r (5 n, 0 j j e j rj where denotes kernel approxmaton operator, P,mn s the mnmal pressure among neghborng partcles of partcle, d s the number of space dmenson, ej = r j / rj s the unt vector wth a 0 drecton from partcle to partcle j. n s the partcle number densty at ntal arrangement, the partcle number densty s defned as: (, n = W r r (6 j e j Compared wth the orgnal gradent model (Koshzuka and Oka, 996, the modfed gradent model above ensures that the pressure forces between any two partcles are purely repulsve forces, whch s an mportant factor for numercal stablty. In the meantme, n (3 (4 860

3 consderaton of great nfluence of the gradent model on the conservaton of energy, a well-known gradent correcton (Khayyer et al., 07 s recommended for the problems senstve to energy conservaton. The dvergence model of velocty s gven by: ( j u d u = u e ( (7 n, 0 jw rj re j rj In present multphase MPS method, vscosty dscontnuty cross nterface s dealt wth by an nterpartcle vscosty between partcles belongng to dfferent phases. Accordng to the numercal test of Shakbaena and Jn (0, the mult-vscosty model wth harmonc mean nterpartcle vscosty s adopted: d μμ ( μ u = ( u j W( rj, re n u (8 j 0 λ j μ + μj where λ s appled to make the ncrease of varance equal to that of the analytcal soluton: λ = j j ( j, e rj ( j, re W r r W r Interface Tenson Model For multphase flow, nterface tenson s mportant force actng on the nterface. To mpose the force to the nterface partcles, a contoured contnuum surface force (CCFS model developed by Duan et al. (05 has been adopted, where nterface tenson force s calculated as: F = σκ C (0 s where σ s the surface tenson coeffcent, κ s the nterface curvature, C s a color functon defned as: 0 partcle belongs to the specfed phase C = ( partcle belongs to the other phase To calculate the local curvature of partcle, frstly, the smoothed color functon f at an arbtrary locaton near the partcle s obtaned through a smoothng process based on the Gaussan kernel functon: CGr j (, r j 9r f ( x, y =, G( r, r = exp Gr (, r r r j j s 9 j j s j s π s s where r s represents the smooth radus. (9 ( Then, f s expanded at pont based on the Taylor seres expanson: f xy f x y f x x f y y f x x 3 + fxy, ( x x ( y y+ fyy, ( y y + O( rs (, = (, + ( + ( + ( x, y, xx, (3 where the subscrpts are the frst- and the second-order partal 3 dervatves of f wth respect to x or y at partcle. O( r s s the hgh order error term whch s proven to have no nfluence on the accuracy of curvature calculaton and can be omtted (Duan et al., 05. The local contour functon of f passng through partcle satsfyng the followng: (, (, f x y = f x y (4 Thus the local contour at partcle can be obtaned by combnaton of Eq. 3 and Eq. 4, expressed as: fx, ( x x + fy, ( y y + fxx, ( x x + fxy, ( x x ( y y + fyy, ( y y = 0 (5 Fnally, the curvature of partcle s analytcally calculated as follow: y f f f f f f f κ = = '' x, y, xy, x, yy, y, xx, 3/ 3/ ' ( + y ( fx, + fy, Model of Incompressblty (6 The ncompressblty condton n MPS method s presented by a constant partcle number densty. Each tme step conssts of two stages, explct stage and mplct stage. In the explct stage, temporal velocty of partcles s calculated explctly based on gravty, vscosty and nterface tenson terms. In the second mplct stage, the temporal velocty s projected nto a dvergence-free velocty feld accordng to pressure. The pressure feld s mplctly calculated by solvng a Pressure Posson Equaton (PPE. The Pressure Posson Equaton used n ths paper s the mproved one wth mxed source term, wrtten as: k 0 k ρ * ρ n n + P = ( γ u γ (7 0 Δt Δt n where γ s a relaxaton coeffcent. Accordng to the numercal tests by Lee et al. (0, γ = 0.0 s used n ths paper to reduce error of the numercal pressure. 86

4 Boundary Condtons In MPS method, the knematc and dynamc free surface boundary condton s mposed on the free surface partcles. The knematc boundary condton s automatcally satsfed n Lagrangan method, whle the dynamc free surface boundary condton s mplemented by settng zero pressure on the free surface partcles. To mpose the dynamc free surface boundary condton, free surface partcles should be detected frst. To mprove the accuracy, we employ an mproved detecton method called ASA (method the Assessment of free-surface based on nearly Symmetrc Arrangement of non-free-surface partcles. Ths detecton method s orgnally proposed by Khayyer et al. (009, and has been proven to be effectve n a great many applcatons (Zhang and Wan, 0; Tang et al., 06a; Tang et al., 06b. In ths method, a functon based on the asymmetrc arrangement of neghborng partcles of the center partcle s defned as: F d r = (, j W rj re n j rj Partcles satsfyng 0 (8 F > 0.9 F (9 0 are judged as free surface partcles, where F s the value of F for free surface partcles at ntal arrangement. As for wall boundary condton, no-slp condton s mposed by ntroducng movng dummy partcles. At each tme step, locatons and veloctes of the dummy partcles are rearranged accordng to correspondng flud partcles near the wall. The locatons of dummy partcles are symmetrcal to correspondng flud partcles about the wall, and ther veloctes are decded as follow: u n = ( u u n, u t = ( u u t, (0 dum w dum w where n and t are respectvely the normal and tangental vectors to the wall. The subscrpts dum, and w represent the dummy, correspondng flud and wall partcles. Table. Densty, dynamc vscosty, and surface tenson coeffcents for the studed two fluds Surface Dynamc Densty tenson vscosty coeffcent Water 000 kg/m m /s N/m Desel ol (0# 846. kg/m m /s N/m NUMERICAL SIMULATION In present study, the two-layer-lqud sloshng phenomena are numercally smulated usng the above multphase MPS method. For further verfcaton, all the smulaton condtons are set equal to the two-layer-lqud sloshng experment conducted by Xue et al. (03, n whch a rgd rectangular tank wth a length of 0.57 m s nstalled on a shake table. In ths study, the shakng table s drven by a wave-maker for a sway moton wth a sngle degree of freedom, whch can be gven as the followng equaton: x( t = αcosωt ( where α = 0.0 m represents the ampltude of sway moton, ω s the external exctaton frequency of the forced moton. As shown n Fg., the rectangular tank s partally flled wth two mmscble fluds, water and 0# desel ol, the physcal propertes of whch are lsted n Table. The water layer wth a thckness of 0. m and the ol layer wth a thckness of 0.05 m locate at the bottom and top of the tank, respectvely. In present numercal smulaton, the ntal partcle dstance s 0.00 m and a total of 674 partcles are used, n whch 450 partcles for water layer, 7075 partcles for ol layer, and the others for sold wall boundary. Three wave probes are arranged to record the nterfacal wave elevaton at dfferent locatons. And three dynamc pressure probes are employed to record the dynamc pressure at the rght wall of the tank. Wave probe Wave probe Wave probe m Ol Water 0.77 m 0.05 m 0. m 0.57 m Fg. Model for two-layer-lqud sloshng 0.0 m 0.04 m 0.04 m 0.05 m Comparson of Numercal Results wth Experment In ths secton, the sway moton of the tank wth an external exctaton frequency of 7.57 rad/s s consdered, and the nterface wave elevaton and dynamc pressure varaton are smulated wth the multphase MPS method. Qualtatve and quanttatve comparsons are carred out between numercal results and experment. Fg. (a and Fg. (b exhbt the comparson of layered sloshng pctures respectvely obtaned by present multphase MPS method and experment at dfferent tmes. They generally show good agreement of both the profles of surface and two-phase nterface. The newly developed method can accurately capture the hydrodynamc characterstcs of the two-layer-lqud sloshng. Fg. (c and Fg. (d are respectvely the pressure fled and densty feld (expressed as mass of each partcle obtaned by present multphase MPS method. The pressure feld s smooth and specally, no obvous pressure dscontnuty s observed across the nterface. Due to the adopton of densty smoothng scheme, a transton regon exsts n P3 P P 86

5 (a Snapshots of experment by Xue et al. (03 (b Snapshots of present multphase MPS Water Ol (c Pressure feld Fg. Comparson of MPS results wth the experment, t = 0.0 s,. s,.65 s,. s,.55 s and 3.5 s Fg. 3 Transton regon n present multphase MPS method 863 (d Densty feld

6 Fg. 4 Comparson of nterface elevaton between present multphase MPS and the experment/smulaton by Xue et al. (03. Fg. 5 Convergence test for dfferent ntal partcal dstance, dp=0.00 m, 0.00 m, and m. the densty feld of Fg. (d and keep the densty change across the nterface contnuous. Fg. 4 plots the ol-water nterface elevatons at mddle-tank ( η, left tank ( η and rght tank ( η 3. In ths studed case, the nterface elevaton at md-tank s obvously smaller than the left and rght tank. The nterface elevatons at left and rght tank are ant-symmetrc and beatng phenomena appear. The multphase MPS results agree well wth the IVOF based smulaton and experment by Xue et al. (03, especally wth the former. Besdes, a convergence test s carred out n Fg.5, n whch the elevaton at left tank wall ( η 3 calculated wth dfferent ntal partcle dstances (0.00 m, 0.00 m, m s compared. It shows that the convergence s acheved for the cases of 0.00 m and 0.00 m. In consderaton of computaton tme, the adopton of 0.00 m s enough for ths computaton. Fg. 6 shows the comparson of the dynamc pressure dstrbuton at rght tank. Smlar wth the nterface elevatons, excellent agreement s acheved agan between the multphase MPS results and IVOF based smulaton, gnorng the slght pressure oscllatons whch s common for partcle method. The largest dscrepancy takes place n the regon where the dynamc pressure s relatvely small. Ths may be nduced by the less effectveness of pressure sensor when the dynamc pressure s small. 864

7 Fg. 6 Comparson of dynamc pressure between present multphase MPS and the experment/smulaton by Xue et al. (03. CONCLUSIONS In ths study, the nvestgaton of the two-layer-lqud sloshng phenomena nsde a rgd tank are carred out by usng a new multphase method developed based on the meshless MPS method. The new multphase MPS method treats the multflud system as the multdensty and mult-vscosty flud, thus only a sngle set of equatons needs to be solved for all phases. Besdes, extra densty smoothng technque, nterpartcle vscosty model and surface tenson model are ncluded n the present method for nterface partcles. The new method s verfed aganst the two-layer-lqud sloshng experment by Xue et al. (03 and ther numercal smulaton based on IVOF method. In general, both the qualtatve comparson of layer sloshng pctures and quanttatve comparsons of nterface elevaton and dynamc pressure show good agreement. ACKNOWLEDGEMENTS Ths work s supported by the Natonal Natural Scence Foundaton of Chna ( , 43009, , Chang Jang Scholars Program (T04099, Shangha Excellent Academc Leaders Program (7XD40300, Program for Professor of Specal Appontment (Eastern Scholar at Shangha Insttutons of Hgher Learnng (030, Innovatve Specal Project of Numercal Tank of Mnstry of Industry and Informaton Technology of Chna (06-3/09 and Lloyd s Regster Foundaton for doctoral student, to whch the authors are most grateful. REFERENCES Duan, G, Koshzuka, S and Chen, B (05. A Contoured Contnuum Surface Force Model for Partcle Methods, J Comput P3333hys, 98, Duan, G, Chen, B, Koshzuka, S and Xang, H (07. Stable Multphase Movng Partcle Sem-mplct Method for Incompressble Interfacal Flow, Comput Methods Appl Mech Engrg, 38, Gotoh, H and Khayyer, A (06. Current Achevements and Future Perspectves for Projecton-based Partcle Methods wth Applcatons n Ocean Engneerng, J Ocean Eng Mar Energy,, Ikar, H, Khayyer, A and Gotoh, H (05. Corrected Hgher Order Laplacan for Enhancement of Pressure Calculaton by Projectonbased Partcle Methods wth Applcatons n Ocean Engneerng, J Ocean Eng Mar Energy, (4, Khayyer, A and Gotoh, H (008. Development of CMPS Method for Accurate Water-Surface Trackng n Breakng Waves, Coastal Eng, 50(, Khayyer, A, Gotoh, H and Shao, S (009. Enhanced Predctons of Wave Impact Pressure by Improved Incompressble SPH Methods, Appl Ocean Res, 3, 3. Khayyer, A and Gotoh, H (00. A Hgher Order Laplacan Model for Enhancement and Stablzaton of Pressure Calculaton by the MPS Method, Appl Ocean Res, 3, 4 3. Khayyer, A and Gotoh, H (0. Enhancement of Stablty and Accuracy of the Movng Partcle Sem-mplct Method, J Comput Phys, 30(8, Khayyer, A and Gotoh, H (0. A 3D Hgher Order Laplacan Model 865

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