FINITE ELEMENT ANALYSIS OF SLOSHING IN LIQUID-FILLED CONTAINERS

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1 FINIE ELEMEN ANALYSIS OF SLOSHING IN LIQUID-FILLED CONAINERS Mustafa Arafa Lcturr, Dpartmnt of Mchanical Dsign and Production Enginring, Cairo Univrsity, Cairo, Egypt ABSRAC h focus of th prsnt papr is on th dvlopmnt of a finit lmnt formulation to invstigat th sloshing of liquids in partially filld rigid rctangular tanks undrgoing bas xcitation. h liquid domain is discrtizd into two-dimnsional four-nod rctangular lmnts with th liquid vlocity potntial rprsnting th nodal dgrs of frdom. Liquid sloshing ffcts inducd by both stady-stat harmonic and arbitrary horizontal bas xcitation ar invstigatd in trms of th slosh frquncis, liquid vlocity fild, fr surfac displacmnt and hydrodynamic forcs acting on th tank walls. h modl is mployd to study th ffcts of insrting a bottom-mountd vrtical rigid baffl, as wll as sid-mountd horizontal baffls that ar wholly immrsd in th liquid rgion, in an attmpt to invstigat thir viability in acting as slosh supprssion dvics. KEYWORDS Sloshing, finit lmnt analysis, partially filld tanks, fluid structur intraction.. INRODUCION Sloshing is known as th oscillation of th fr surfac of a liquid in an xtrnally xcitd containr. Partially filld liquid tanks undrgoing acclratd motions ar suscptibl to sloshinducd loads, which may affct th dynamic bhavior of th liquid-rtaining structur, and may vn b svr nough to caus structural damag. Sloshing of liquids in moving containrs is of practical concrn in many nginring applications, such as ful sloshing in aircraft tanks as a rsult of sharp flight manuvrs, liquid sloshing in larg storag tanks du to arthquaks and fluid oscillation in tankr trucks travling on highways. An xtnsiv rsarch has bn conductd in th past four dcads to invstigat th motion of liquids in rigid and dformabl containrs. Early simulations of th liquid sloshing problm rlid upon constructing mchanical analogis that compris pndulums or spring mass lmnts whos paramtrs ar dsignd to simulat th rsultant dynamic prssur loads impartd on a tank during sloshing for various tank gomtris and fluid charactristics []. For a comprhnsiv rviw of th phnomnon of sloshing, including analytical prdictions and xprimntal obsrvations, th radr is rfrrd to th work of Abramson []. Mor rcnt studis aim at dvloping numrical modls to invstigat th fluid structur intraction of th liquid-coupld systm [3-7]. Svral mans ar adoptd in practic to ovrcom th undsirabl ffcts of sloshing [8, 9]. A common tchniqu is to provid th 793

2 tank with baffls or sparators in ordr to rduc th bulk motion of th containd liquid during sloshing and also to introduc som nrgy dissipation du to flow sparation ffcts as th liquid oscillats past ths baffls or othr obstacls in th tank [0-5]. his papr prsnts a finit lmnt formulation to study th sloshing of liquids in xtrnally xcitd rigid rctangular tanks. h analysis aims at studying th dynamic bhavior of partially filld liquid-rtaining structurs with and without rigid baffls that ar placd within th liquid rgion. h formulation rlis upon discrtizing th liquid domain into twodimnsional four-nod lmnts, with th liquid vlocity potntial bing th nodal dgrs of frdom. Fluid structur intraction is includd in th modl to coupl th liquid motion with th rigid tank walls to nsur continuity of liquid and structural motion at th liquid tank intrfac. Liquid sloshing ffcts inducd by both stady-stat harmonic and arbitrary horizontal bas xcitation ar invstigatd in trms of th slosh frquncis, liquid vlocity fild, fr surfac displacmnt and hydrodynamic forcs acting on th tank walls. Effcts of introducing a bottom-mountd vrtical baffl, as wll as sid-mountd horizontal baffls that ar wholly immrsd in th liquid domain ar invstigatd. h rmaindr of this papr is dividd into four sctions. First, a statmnt of th problm, togthr with th basic govrning quations, is givn. Nxt, th finit lmnt modl that is dvlopd to study sloshing of liquids in rctangular rigid containrs is prsntd. h final two sctions prsnt numrical xampls, conclusions and aras whr futur rsarch could b dirctd.. PROBLEM SAEMEN AND GOVERNING EQUAIONS Figur shows a schmatic diagram of th liquid tank systm undr considration. A rigid rctangular tank of lngth L is partially filld with liquid to a hight H. Owing to th twodimnsional gomtry of th problm, a Cartsian coordinat systm is mployd to dscrib th position of any point blonging to th liquid domain. h fr surfac displacmnt, masurd from th undisturbd liquid surfac at quilibrium, is dnotd by η ( x, t). Effcts of th liquid comprssibility, viscosity and surfac tnsion ar nglctd in th prsnt study. h tank is also quippd with a vrtical bottom-mountd baffl of hight h, placd at a distanc l from th lft tank wall, as indicatd. h baffl is assumd to b thin, rigid and wholly immrsd in th liquid rgion. y η(x,t) Fr surfac H l Rigid tank h x L Fig.. Rigid rctangular tank and coordinat systm. h quation govrning th irrotational motion of an inviscid incomprssibl fluid is givn in trms of th vlocity potntial φ ( x, yt, ) as: 794

3 r h liquid vlocity vxyt (,, ) φ = 0 () rlats to th vlocity potntial through: r v = φ () For th cas of fr vibrations of th liquid in a fixd rigid containr without baffls, th solution of Eq. () in th rgion occupid by th liquid must satisfy th propr boundary conditions. Prscribd normal vlocitis ar imposd on th fluid particls adjacnt to th rigid tank surfacs. hus at th rigid tank bottom, y=0, th liquid vlocity in th vrtical dirction vanishs: φ ( x,0, t) = 0 (3) y At th rigid tank walls, x = 0 and x = L, th liquid vlocity in th horizontal dirction vanishs: φ φ (0, yt, ) = 0, ( Lyt,, ) = 0 (4, 5) x x At th liquid fr surfac, assuming small-amplitud motion, th boundary conditions ar dfind aftr Haroun and Housnr [6] as: φ η φ ( xht,, ) = ( xt, ), ρ ( xht,, ) + ρgη( xt, ) = 0 (6,7) y t t his formulation can b mployd to invstigat th bhavior of th coupld liquid rigid tank surfac-wav systm, and forms th basis of th finit lmnt modl dscribd in th following sction. 3. FINIE ELEMEN FORMULAION h finit lmnt modl dvlopd in this work rlis upon discrtizing th liquid domain into four-nodd rctangular lmnts, with th liquid vlocity potntial bing th only dgr of frdom at ach nod. h vlocity potntial at any point within th liquid lmnt is intrpolatd by: φ( x, y) = α+ αx+ α3y+ α4xy (8) Accordingly, th vlocity potntial at any point in th fluid lmnt can b xprssd in trms of th nodal dgrs of frdom as: φ( xy, ) = Nxy (, ) φ (9) { }{ } N x y is th vctor of shap functions and { } whr { (, )} φ is th nodal dgrs of frdom vctor. As th liquid undr considration is incomprssibl, th potntial nrgy of a liquid lmnt consists only of a gravitational potntial nrgy trm: a U = ρbg η ( x, t) dx (0) 0 whr ρ is th liquid dnsity, b is th out-of-plan width of th tank, g is th gravitational acclration, a is th lmnt lngth, and th intgration is carrid out ovr all lmnts blonging to th fr surfac. h kintic nrgy of a liquid lmnt is xprssd as: = ρ ( φ) dv () V whr V is th lmnt volum. Imposing th shap functions as dfind in Eq. (9) and substituting into Eq. () yilds: 795

4 d a = ρb φ N N + N N dxdy φ { } { } { } { } { } x x y y { } 0 0 whr th subscripts x and y dnot partial diffrntiation with th rspctiv variabl, and d is th lmnt hight. Accordingly, th stiffnss matrix of th liquid lmnt is givn by: d a K ρb N N N N dxdy { x} { x} { y} { y} = From Eqs. (7) and (9), th fr surfac displacmnt can b xprssd in trms of th liquid vlocity potntial as: η( x) = { N( x, d) }{& φ } (4) g whr th tim dpndnc is droppd for brvity. Hnc th liquid lmnt potntial nrgy is xprssd as: a ρb U = {& φ } { N( x, d) } { N( x, d) }{ } g & φ dx (5) 0 from which th liquid lmnt mass matrix is givn by: a ρb M = { N( x, d) } { N( x, d) } dx g (6) 0 In ordr to includ th rigid nclosur in th prsnt finit lmnt formulation, thr springsupportd pistons ar attachd to th liquid domain, as dpictd in Fig.. Mass and stiffnss paramtrs of th additional mass spring systms ar slctd to nsur th walls of th containr ar practically rigid and possss natural frquncis that ar apprciably highr than th frquncy rang of intrst which includs th liquid slosh frquncis. Normal displacmnts of all liquid particls lying on th tank boundaris ar coupld to th displacmnts wi, wii, wii I of th pistons. In ordr to impos this condition, th work don by th liquid prssur forcs on th spring-supportd pistons can b xprssd by: W = ρ & φ w da ρ & φ w da ρ & φ w da (7) d I I I II II II III III III AI AII AIII whr th intgrals ar carrid out along all th fluid structur intrfac aras, commonly known as th wttd aras, dsignatd A, A, A. I II III () (3) w III w I m m k k w II k m Rigid pistons Fig.. Modling of th tank walls as rigid spring-supportd pistons. Insrting th shap functions as dfind in Eq. (9) and substituting into Eq. (7) yilds: 796

5 whr { }{ φ } { }{ φ } { }{ } W = w Ω & w Ω & w Ω & φ (8) d I I II II III III d a d N( a, y) bdy, N( x,0) bdx, N(0, y) bdy { Ω I} = ρ { } { Ω II} = ρ { } { Ω III} = ρ { } ar th fluid structur coupling vctors. h lmnt quations of motion ar thn obtaind by using Lagrang s quation. Upon assmbly, th quations of motion for th ntir liquidcoupld systm ar xprssd as: [ M] ρ{ Ω } { } { } { } [ ] { } { 0 I ρ ΩII ρ Ω p& K p } III 0 m 0 0 w { I} ρk 0 0 ρ & I Ω wi 0 + = 0 0 ρm 0 w { II} 0 ρk 0 (9) & Ω II wii ρm w Ω { III } 0 0 ρk & III wiii 0 p is th vctor of liquid nodal prssurs. Elmnts of th liquid mass matrix whr { } [ M ] hav contributions only from th nods blonging to th fr surfac. Hnc it bcoms appropriat to sparat th dgrs of frdom corrsponding to nods lying on th fr surfac from th total dgrs of frdom. In this way, th quations of motion of th liquid domain ar xprssd as: M 0 { p } K ff K fr f { p } ff f { If } wi { IIIf } w & Ω & + Ω & III + = ρ (0) 0 0 { p& r} Krf [ Krr ] { pr} { Ω Ir} w& I + { Ω IIr} w& II + { ΩIIIr} w& III whr th mass and stiffnss matrics ar accordingly partitiond, and subscripts f and r ar introducd to idntify ntitis prtaining to th fr surfac and thos blonging lswhr in th liquid rgion, rspctivly. h vctor of liquid prssurs at nods not blonging to th fr surfac, { p r }, can thn b liminatd from th scond st of quations in (0). Aftr som algbraic manipulation, th rsulting quations of motion can b xprssd in th form: M ff { M} { M3} { M } 4 { p& } f [ K] { p } f {} 0 0 m { } 0 0 m3 m K k f 4 w& I ρ w ρ I I 0 m { K3} 0 k 0 f 3 m33 m + = 34 w ρ () & II w ρ II II 0 m { K4} 0 0 k f 4 m43 m 44 w ρ III w ρ & III III Solution of th ignvalu problm thus ntails handling rducd-siz matrics, as only th dgrs of frdom of th fr surfac nods nd to b considrd, along with thr additional displacmnts dscribing th motion of th rigid tank walls. Modling of th baffl is accountd for by introducing a fourth mass spring lmnt, togthr with an additional displacmnt, wiv ( t). h modl is thn usd to dtrmin th liquid slosh frquncis in fixd tanks, as wll as th forcd rspons of th systm undr tank wall movmnt. 4. NUMERICAL RESULS AND DISCUSSION h finit lmnt formulation dscribd in th prvious sction is mployd to study th fr and forcd sloshing charactristics of watr ( ρ =000 kg/m 3 ) in a rigid rctangular tank. h width b is takn as unity in all cass. h liquid rgion is dividd into 0 by 0 lmnts. h liquid natural frquncis and mod shaps ar obtaind for a fixd tank, both with and without a rigid baffl, placd at various locations, and xtnding vrtically to diffrnt 797

6 hights. Harmonic bas xcitation is thn applid horizontally to th tank rigid walls, and th rsultant hydrodynamic forcs xrtd on th tank walls ar obtaind. h modl is thn xtndd to handl sid-mountd horizontal baffls and arbitrary bas xcitation schms. Exampl : Slosh frquncis in a fixd tank. As a first illustrativ xampl to vrify th accuracy of th prsnt formulation, th liquid slosh frquncis ar calculatd for a rctangular tank with L=m, H=m. abl lists th first fiv natural frquncis, in comparison with th analytical solution prsntd by Abramson []. h rsults ar shown to b in good agrmnt, with a prcntag rror blow % up to th fifth mod. Highr frquncis can b prdictd mor accuratly by incrasing th numbr of lmnts. Figur 3 shows th liquid vlocity fild in th unbaffld tank at th first two slosh mods, as obtaind from th prsnt finit lmnt analysis. h liquid vlocity vctors shown ar th avrag lmnt vlocitis valuatd at th cntr of ach lmnt usd in th msh. Liquid particls adjacnt to boundaris of th tank ar shown to possss vlocity vctors that ar paralll to th boundary surfacs. abl. Liquid slosh frquncis in a fixd tank (L=m, H=m) Mod Natural frquncis [rad/s] Prsnt Abramson [] (a) (b) Fig. 3. Liquid vlocity fild at th (a) first and (b) scond slosh mod. Exampl : Horizontally xcitd tank. h tank of th prvious xampl is now subjctd to a prscribd stady-stat harmonic iωt iωt horizontal bas xcitation of th form wi() t = W, wiii() t =± W. h lft tank wall can b mad to mov ithr in-phas or out-of-phas with th opposing wall, as st by th rlativ sign of wiii ( t). Motion of th tank bottom is st qual to zro. im rspons is obtaind by numrical intgration of th quations of motion using th Nwmark schm. Figur 4 shows th non-dimnsional tim history plot of th fr surfac displacmnt, valuatd for liquid particls lying at th lft wall of th tank, undr th two bas xcitation frquncis 798

7 ω ω =. and ω ω = 0.999, ω bing th fundamntal slosh frquncy. h non- g dimnsional tim is xprssd as t whil th fr surfac displacmnt is xprssd in a H η(0, t) dimnsionlss form by. h amplitud of bas motion is takn as.86 mm. Rsults of W th prsnt analysis ar almost in xact agrmnt with thos prsntd by Wu t al. [4]. h plots also match similar prdictions rportd by Frandsn and Borthwick [7]. Fig. 4. im rspons of fr surfac displacmnt at lft wall (L= m, H= m, W=.86 mm)., ω ω =.;, ω ω = Exampl 3: Sloshing in a baffld tank. h abov formulation is xtndd to includ th ffcts of providing th tank with a rigid bottom-mountd vrtical baffl that is wholly submrgd in th liquid. In this cas, th fluid structur intraction btwn th baffl and nighboring liquid ntitis is handld by dfining Ω to nforc coupling of th liquid and baffl displacmnts at a fourth coupling vctor { IV } th liquid baffl intrfac. In a sns, th baffl acts to split part of th liquid domain into nighboring rgions, and consquntly, it bcoms ncssary to introduc additional liquid nods into th finit lmnt modl in ordr to hav liquid nods lying on both sids of th baffl, as indicatd in Fig.. h numbr of lmnts, though, for both th unbaffld and baffld tanks is th sam. h tank considrd in this xampl has th dimnsions L=30m and H=5m, to nabl comparison with th rsults rportd by Choun and Yun [3]. Figur 5 shows th variation of th first thr slosh frquncis vrsus baffl hight, xprssd in a non-dimnsional form through division by th liquid dpth. h rsults, which corrspond to a baffl that is placd midway along th tank lngth, ar compard with 799

8 prdictions by Choun and Yun [3] and Evans and McIvr [5], who invstigatd th ffcts of a vrtical baffl on th ignfrquncis of fluid in a rctangular containr using th linarizd thory of watr wavs. h gnral trnd is a dcras in slosh frquncis with incrasing baffl hight for th first and third mods. h sam pattrn was obtaind in th abov two rfrncs, with rsults of th prsnt analysis matching mor closly thos of Evans and McIvr [5] for th fundamntal slosh frquncy. Fig. 5. Variation of slosh frquncis with baffl hight (L=30 m, H=5 m, l/l=0.5)., ω ;, ω ;, ω 3 ;, Choun and Yun [3]; Evans and McIvr [5]. Figur 6 shows th liquid vlocity fild at th first two slosh mods throughout a tank which is providd with a baffl, placd midway along th tank lngth and submrgd by on-half th liquid hight. Liquid motion at th tip of th baffl faturs a slight swirl at th first mod. (a) (b) Fig. 6. Vlocity fild in a baffld tank at th (a) first, and (b) scond slosh mod (l/l=0.5, h/h=0.5). 800

9 Exampl 4: ank with horizontal baffls. A tank with horizontal baffls is now invstigatd. h configuration is shown in Fig. 7. wo idntical rigid baffls ar fittd to th opposit sid walls of th tank. h analysis follows dirctly from th formulation prsntd abov. y H η(x,t) B/ Baffl Rigid tank L Fr surfac Fig. 7. Rctangular tank with horizontal baffls. Figur 8 shows th liquid vlocity fild at th first two slosh mods throughout th tank. h horizontal baffls ar placd at a hight of 70% of th total liquid dpth, masurd from th tank bottom, and th combind lngth of th two baffls covrs 60% of th tank lngth. Liquid motion bnath th horizontal baffls is significantly rducd. c x (b) Fig. 8. Liquid vlocity fild in a tank with horizontal baffls at th (a) first, and (b) scond slosh mod (B/L=0.6, c/h=0.7). Figur 9 shows th variation of th fundamntal slosh frquncy with baffl location for various baffl sizs. Baffls xtnding ovr longr spans and placd closr to th liquid fr surfac hav a gratr influnc on lowring th fundamntal slosh frquncy. h limiting cas is whn th two baffls ar placd at th fr surfac and xtndd horizontally to mt on anothr, ntirly trapping th liquid, in which cas th liquid bhavs lik a rigid body as if it wr frozn. 80

10 Fig. 9. Fundamntal slosh frquncy vrsus baffl location for various baffl sizs. (L=30 m, H=5 m). Exampl 5: Arbitrary xcitation. In this xampl, th tank is subjctd to an arbitrary input xcitation, and a comparison of th tank bhavior for various baffl configurations is mad. h chosn input bas motion is a rlaxd stp input, in which th tank displacmnt during ris tim is givn by: W s πt () wi () t = wiii () t = cos t whr W s and t r dnot th stady-stat displacmnt and ris tim, rspctivly. Figur 0 shows th tim rspons of th rsultant hydrodynamic forcs acting on th tank walls for th cass of an unbaffld tank, a tank providd with a vrtical baffl, and on with horizontal baffls, all valuatd for W s =m and t r =5s. For th particular configurations chosn, it is shown that significant rduction (almost 50%) in th inducd hydrodynamic forcs can b attaind by introducing horizontal baffls in th tank dsign. A 30% rduction in forcs is achivd with a singl vrtical baffl. r Fig. 0. Rsultant hydrodynamic forc acting on tank walls (L=30 m, H=5m, W s =m, t r =5s)., unbaffld tank;, tank with a vrtical baffl (l/l=0.5, h/h=0.6);, tank with horizontal baffls (B/L=0.6, c/h=0.7). 80

11 5. CONCLUSIONS his papr prsntd a finit lmnt formulation of th fluid structur intraction problm to study th sloshing bhavior of liquids in rigid rctangular tanks. Liquid sloshing ffcts inducd by horizontal bas xcitation ar invstigatd in trms of th slosh frquncis, liquid vlocity fild, fr surfac displacmnt and hydrodynamic forcs acting on th tank walls. Rsults of th prsnt work compar quit favorably with stablishd prdictions rportd in th litratur. h modl is mployd to study th ffcts of insrting both vrtical and horizontal rigid baffls within th liquid domain on th slosh frquncis and fr surfac motion during forcd vibration, in an attmpt to invstigat thir viability in acting as slosh supprssion dvics. Numrical simulations indicat that significant rduction in th hydrodynamic forcs is achivd for baffld tanks subjctd to various xtrnal xcitation schms. REFERENCES. Housnr, G.W., Dynamic Prssurs on Acclratd Fluid Containrs, Bulltin of th Sismological Socity of Amrica, Vol. 47, pp. 5-35, Abramson, H.N., h Dynamic Bhavior of Liquids in Moving Containrs, NASA SP- 06, Washington, D.C., 966, updatd by Dodg, F.., Southwst Rsarch Institut, Frandsn, J.B., Sloshing Motions in Excitd anks, Journal of Computational Physics, Vol. 96, Issu, 53-87, Wu, G.X., Ma, Q.W. and aylor, R.E., Numrical Simulation of Sloshing Wavs in a 3D ank basd on a Finit Elmnt Mthod, Applid Ocan Rsarch, 0, , Pal, N.C., Bhattacharyya, S.K. and Sinha, P.K., Non-linar Coupld Slosh Dynamics of Liquid-filld Laminatd Composit Containrs: A wo Dimnsional Finit Elmnt Approach, Journal of Sound and Vibration, 6(4), , Çlbi, M.S. and Akyildiz, H., Nonlinar Modlling of Liquid Sloshing in a Moving Rctangular ank, Ocan Enginring, Vol. 9, Issu, , Frandsn, J.B. and Borthwick, A.G.L., Fr and Forcd Sloshing Motions in a -D Numrical Wav ank, Procdings of OMAE 0: h st Intrnational Confrnc on Offshor Mchanics and Arctic Enginring, Jun 3-8, Oslo, Norway, Propllant Slosh Loads, NASA SP-8009, August Slosh Supprssion, NASA SP-803, May Warnitchai, P. and Pinkaw,., Modlling of Liquid Sloshing in Rctangular anks with Flow-Dampning Dvics, Enginring Structurs, Vol. 0, No. 7, , Cho, J.R. and L, H.W., Numrical Study on Liquid Sloshing in Baffld ank by Nonlinar Finit Elmnt Mthod, Computr Mthods in Applid Mchanics and Enginring, Vol. 93, , Cho, J.R., L, H.W. and Ha, S.Y., Finit Elmnt Analysis of Rsonant Sloshing Rspons in -D Baffld ank, Journal of Sound and Vibration, in prss. 3. Choun, Y-S., and Yun, C-B., Sloshing Charactristics in Rctangular anks with a Submrgd Block, Computrs and Structurs, Vol. 6, No. 3, pp , Biswal, K.C., Bhattacharyya, S.K. and Sinha, P.K., Dynamic Charactristics of Liquid Filld Rctangular ank with Baffls, h Institution of Enginrs (India), Vol. 84, 45-48, Evans, D.V. and McIvr, P., Rsonant Frquncis in a Containr with a Vrtical Baffl, Journal of Fluid Mchanics, Vol. 75, , Haroun, M.A. and Housnr, G.W., Dynamic Charactristics of Liquid Storag anks, Journal of Enginring Mchanics, ASCE, Vol. 08, No. EM5, ,

Dynamic Modelling of Hoisting Steel Wire Rope. Da-zhi CAO, Wen-zheng DU, Bao-zhu MA *

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