MPS Simulation of Sloshing Flows in a Tuned Liquid Damper

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1 Proceedngs of the Twenty-seventh (2017) Internatonal Ocean and Polar Engneerng Conference San Francsco, CA, USA, June 25-30, 2017 Copyrght 2017 by the Internatonal Socety of Offshore and Polar Engneers (ISOPE) ISBN ; ISSN MPS Smulaton of Sloshng Flows n a Tuned Lqud Damper Xao Wen, Xang Chen, Decheng Wan * 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 In the present paper, our n-house meshless solver MLPartcle-SJTU, based on the mproved MPS (movng partcle sem-mplct) method, s employed to smulate the lqud sloshng n the tuned lqud damper (TLD). For the valdaton purpose, parameters from experments n prevous lterature are adopted n frst numercal smulaton, the roll angle and wave shapes nsde the TLD show agreement wth the expermental data. Then, the nfluence of varous external exctaton on the dampng characterstc of TLD has been studed through the changng of moton ampltude and frequency of a sldng mass. The numercal results ndcate that the dampng characterstc s more dstnct when the exctaton frequency falls near the natural frequency of the TLD system, and the flow n the TLD turns from travelng wave nto the quas-dam-break flow when the exctaton ampltude ncreases from 50 mm to 200 mm. KEY WORDS: Partcle method; MPS (movng partcle sem-mplct); MLPartcle SJTU solver; TLD (tuned lqud damper); sloshng; roll. INTRODUCTION In ths paper, the sloshng phenomenon n a tuned lqud damper (TLD), whch s manly composed of rectangular tank wth the freedom of roll moton, s numercally studed. In the shp engneerng, the TLD s commonly employed as the ant-roll devce to suppress the roll moton of shp operatng n severe sea. Compared wth actve shp stablzer, the TLD have advantages of smple structure, low cost and easy mantenance, so a lot of attenton has been pad to both expermental (Vera et al., 2010) and numercal researches on the sloshng n TLD (Bulan et al., 2010). The dampng characterstc of TLD manly depends on the sloshng n the tank and the propertes of lqud n t. The physcal behavor of sloshng flow n the partally flled TLD s smlar to shallow water waves (Verhagen and Van, 1965), and shows hgh non-lnear when volent sloshng occurs. Due to the phase lag between the roll moton and the wave movement, the roll moton of tank s damped by the moment nduced by the mpact loads of waves actng on nner wall of tank. One of the oldest papers concerned wth expermental examnaton of flud dynamcs n a movng contaner s probably that by Housner (1963). Souto et al. (2011) desgn a seres of experments to study the sloshng characterstcs of dfferent flud n a free-surface tank. However, only the local flow feld nformaton can be obtaned from experments. To acqure the global nformaton, we need turn to Computatonal Flud Dynamcs (CFD). In recent years, wth the ongong development of CFD, many numercal methods have been appled to the sloshng phenomenon n TLD. Van (2001) smulated the problems of water sloshng n free-surface and U-tube ant-roll tanks wth VOF method, and obtaned results whch are n good agreement wth expermental study, ncludng the water heght, the sway force and roll-moment ampltudes and phases. Bulan and Souto (2010) used SPH method to treat the couplng problems same as the one n ths paper, and ganed the moton response of the TLD n resonance regon and off-resonance regon. However, few researches are conducted wth the movng partcle sem-mplct (MPS) method, whch can handle the steep deformaton of the free surface well and provde hgh precson and good stablty n solvng such a sloshng problem. Ths paper focuses on the smulatons of the sloshng phenomenon wth the movng partcle sem-mplct (MPS) method (Koshzuka and Oka, 1996), whch s a Lagrangan meshless method for ncompressble flows wth a free surface, s dstngushed from SPH method that the pressure feld s obtaned by solvng a Posson pressure equaton at every step (Cummns and Rudman, 1999). Therefore, an explct geometrc compatblty between the mass and the volume can be acheved. Besdes, the ncompressble treatment allows a second semmplct amendment of velocty feld, whch can sgnfcantly mprove the stablty of numercal smulaton. As so far, the MPS method has proven to be vald n lqud sloshng, water-entry flow, dam break flow, flud-structure nteracton and so on. Lee et al. (2011) mproved the accuracy and effcency of MPS method by usng optmal source term, optmal gradent and collson models, and mproved sold-boundary treatment and search of free-surface partcles. Hwang et al. (2016) successfully appled MPS method to the smulaton of sloshng flow nteractng wth elastc baffles by proposng a new scheme, correspondng to flud structure couplng force. The major drawback of 1009

2 2 MPS s that the pressure of free surface must be explctly set to zero at every step. However, the problem has been allevated because a lot of numercal nce work have been carred out to enhance the ablty of the MPS method (Kondo and Koshzuka, 2011; Zhang and Wan, 2012). NUMERICAL SCHEME Governng Equatons The governng equatons n the MPS method nclude the mass and momentum conservaton equatons: 1 Dρ = V= 0 (1) ρ Dt DV 1 = P + ν V + g (2) Dt ρ where ρ denotes the densty, P s the pressure, V s the velocty, g s the gravty acceleraton, ν the knematcs vscosty and t s the flow tme. Partcle Interacton Models In the present work, we adopt the followng modfed kernel functon suggested by Zhang and Wan (2012): re 1 0 r < re W() r = 0.85r+ 0.15re 0 re r Where r = r r denotes the dstance between two partcles, r e s the j supported radus of the partcle nteracton doman. To calculate the weghted average n MPS method, partcle number densty s defned as (Koshzuka et al., 1998): < n> = W( rj r ) (4) j Gradent Model In ths paper, the gradent operator can be dscretzed nto a local weghted average of radal functon as follow (Tanaka M. and Masunaga T, 2010): D P + P < P>= ( r r ) W( r r ) j 0 2 j j n j rj r Where D s the number of space dmenson, r represents coordnate 0 vector of flud partcle, W() r s the kernel functon and n denotes the ntal partcle number densty for ncompressble flow. Laplacan Model Laplacan operator s derved by Koshzuka et al. (1998) from the (3) (5) physcal concept of dffuson as: 2D ( ) ( r r ) (6) 2 < φ >= φ 0 j φ W j n λ j λ = j W ( r r ) r r j 2 j j W ( r r ) j In Eq.6, the parameter λ s ntroduced to keep the ncrease of varance equal to analytcal soluton. Model of Incompressblty In the present work, we adopt a mxed source term for PPE whch s proposed by Tanaka and Masunaga (2010), ths mproved PPE s rewrtten by Lee et al. (2011) as: 2 n 1 ρ * ρ n n < P + >= (1 γ) V γ < > Δt Δt n * Where : γ s a blendng parameter to account for the relatve contrbutons of the two terms. The range of 0.01 γ 0.05 s better accordng to numercal experments conducted by Lee et al. (2011). Free Surface Boundary Condton In the MPS method, the knematc condton s drectly satsfed n Lagrangan partcle method, whle the dynamc condton s mplemented by assgnng zero pressure to the free surface partcles. In tradtonal MPS method, partcle satsfyng (Koshzuka et al., 1998): < n> < β n (9) * 0 s consdered as on the free surface, where β s 0.97 n ths paper. To mprove the accuracy of surface partcle detecton, we adopt a new detecton method n whch a vector functon s defned as follow (Zhang and Wan, 2011): < D 1 F > = ( r r ) W ( r ) 0 j j n j r rj The vector functon F represents the asymmetry of arrangements of neghbor partcles. Thus, partcles satsfyng: (7) (8) (10) < F > > α (11) are consdered as surface partcle, where α s a parameter wth a value 0 0 of 0.9 F n ths paper, F s the ntal value of F for surface partcle. Wall Boundary Condton As shown n Fg. 1, the wall boundary condton s represented by three layer of wall partcles. One layer of wall partcles s set along the sold wall, and nvolved n the pressure calculaton. In order to prevent the flud partcles near the sold wall from been msjudged as free surface partcles, two layer of ghost partcles are ncluded nsde the wall and only nvolved n the calculaton of partcle number densty. In the 1010

3 orgnal MPS method, all the wall partcles are unmovable, so the wall boundary condton s not flexble. 2 [ I 0 + m] ξ m ( t) ϕ + 2mξ ( t) ξ m m ( t) ϕ g SG + m g ξ m ( t) cos( ϕ ) = Qdamp ( t) + Q flud ( t) sn( ϕ) (12) Qdamp ( t) = - K df sgn( ϕ) Bϕ ϕ (13) Fg. 1 Partcles dstrbuton near the wall NUMERICAL SIMULATION Smulaton Model The smulaton model n ths paper s based on the TLD devce used by Souto (2010), dsplayed n Fg. 2. The devce conssts of two parts,.e., a rectangular tank lmted to roll moton and a horzontal lnear gude fxed on the rotaton center. Along the gude a sldng mass moves wth a defned harmonc moton, to excte the roll of the tank. Where, ϕ [rad] s the roll angle, g = [m/s 2 ] s the gravtatonal acceleraton, I 0 = [kg m2 ] s the polar moment of nerta of the rgd system wth respect to the rotaton axs, m = [kg] s the mass of the movng weght, ξ m (t) [m] s the nstantaneous (mposed) poston of the exctaton weght along the lnear gude (tank-fxed reference system), ξ m (t ) [m/s] and ξ m (t ) [m/s 2 ] are the frst and second tme dervatves of ξ m (t ) [m], SG = M R η G = [kg m] s the statc moment of the rgd system wth respect to the rotaton axs, M R [kg] s the total mass of the rgd system, η G [m] s the (sgned) dstance of the center of gravty of the rgd system wth respect to the rotaton axs ( tank-fxed reference system), Qdamp ( t) = - K df sgn( ϕ) Bϕ ϕ [N m] s the assumed form of roll dampng moment comprsng: A dry frcton term K df sgn(ϕ ) wth K df = [N m] beng the dry frcton coeffcent A lnear dampng term B ϕ ϕ wth B ϕ = [N m/ (rad/s)] beng the lnear dampng coeffcent When ϕ n s calculated at n th tme step accordng to Eq. 12, ϕ n+ 1 and ϕ n+1 can be get from the Eq. 14 and Eq. 15. ϕ n+ 1 = ϕ n + ϕ n Δt (14) ϕ n+ 1 = ϕ n + ϕ n Δt (15) (a) Tank (b) Sldng mass and gude Fg. 2 TLD testng devce (Souto, 2010) The wdth of the tank s much smaller than the length and heght, that the physcal model can transformed nto a 2D rectangular tank n numercal smulaton, as shown n Fg. 3. The sze of the tank s 900 mm 508 mm. The water depth s 92 mm, n order to keep the frst sloshng frequency of water equal to the nature frequency of the TLD system. In the MPS method, the TLD system s substtuted by unformly dstrbuted partcles. The ntal dstance between partcles s 4 mm. The total number of partcles s 7584, ncludng 5424 flud partcles and 2160 boundary partcles. The sldng mass s not ncluded n the smulaton model n the form of partcles. Instead, an analytcal model of the TLD system s ntroduced to ncorporate the moton of the sldng mass nto the MPS code. The analytcal model s derved from the equaton of moment of momentum of the tank, wth the values of the parameters n the equaton determned by a seres of tests (Bulan et al., 2010). The analytcal model s descrbed as follow: Fg. 3 Smulaton model Model Valdaton In ths part, numercal smulatons under two dfferent condtons are carred out, to test the feasblty of the mproved MPS model when appled to solve TLD sloshng problem. One condton s A=100 mm, ω = ω0, the other s A=150 mm, ω = 0.9ω 0, where A s the moton ampltude of the mass, ω s the moton frequency of the mass, and ω0 s the natural frequency of the system. The A and ω also 1011

4 represent exctaton ampltude and frequency. Wth defned A and ω, the nstantaneous poston ξ m (t ) and the velocty ξ m (t ) of the sldng mass are obtaned. However, the values of ξ m(t) and ξ m (t) wll not unquely determned by A and ω n ths secton. Dfferent from numercal smulaton, the moton of the sldng mass s techncally dffcult to be purely snusodal n real experments. In fact, the moton curve s found to consst of four parabolc branches n every perod. For better comparson between numercal smulaton and experments, ξ m(t) and ξ m (t) are set to the values get from measurement n experments. Fg. 4 shows the moton response of empty tank n the condton of A=150 mm, ω = 0.9ω0, correspondng to the off-resonance exctaton. Nce beatng phenomenon can be observed because the exctaton frequency s very close to the natural frequency of the TLD system, and the curve obtaned from numercal smulaton match well wth the curve from experments, provng the valdaton of the MPS method. However, there are stll some dfference between numercal and expermental results over tme, n both ampltude and phase. Ths s because the exctaton ampltude and frequency n smulaton s not absolutely consstent wth experments. Fg. 5 Roll angle of TLD, A=150 mm, ω = 0.9ω 0 The two groups of pctures n Fg. 6 and Fg. 7 show the wave shapes n TLD tank at two dfferent nstants n the condton of A=100 mm, ω = 1.0ω 0, correspondng to resonance exctaton. The breakng wave appears because larger moton ampltude oscllaton can be nduced by resonance exctaton n ths case, whch has been found to be an mportant factor of dampng characterstc of TLD (Bouscasse et al., 2014b). The nstantaneous shapes of breakng wave reach a good agreement wth the results ganed from experments and SPH method. (a) Experments Fg. 4 Roll angle of empty tank, A=150 mm, ω = 0.9ω 0 In Fg. 5, due to the lqud dampng nsde the TLD tank, the beatng phenomenon shows up only at the begnnng of roll moton. Then, the roll moton s modulated to a constant ampltude oscllaton, wth an ampltude about 5 degrees, smaller than the peak ampltude of empty tank, about 8 degrees. Ths means that the dampng characterstc of TLD can be captured by the MPS method. Smlar to the roll angle of empty tank, there s some dfference between numercal and expermental results, whch s acceptable. (b) SPH method 1012

5 ξ m ( t) = A ω cos( ωt) (17) (c) MPS method Fg. 6 Wave shapes, t / T 0 = 8.66, A=100 mm, ω = 1.0ω 0 Wth a varety of moton ampltudes and frequences of the sldng mass, dfferent external exctatons are mposed on the TLD system. Table 1 shows the test matrx of numercal smulaton n ths part. Seven moton frequences ( ω = 0.7ω0, ω = 0.8ω 0, ω = 0.9ω 0, ω = 1.0ω 0, ω = 1.1ω 0, ω = 1.2ω 0, ω = 1.3ω 0 ) are respectvely appled to four moton ampltudes (A=50 mm, A=100 mm, A=150 mm, A=200 mm), so there are total 28 cases been smulated n ths part. The nature frequency ω0 and the natural perod T0 of the present TLD system are equal to 3.26 rad/s and s respectvely. Table 1. Test matrx A ( ω /ω0 ) 50 mm 100 mm 150 mm 200 mm /50 0.7/ / / /50 0.8/ / / /50 0.9/ / / /50 1.0/ / / /50 1.1/ / / /50 1.2/ / / /50 1.3/ / /200 (a) Experments (b) SPH method (a) ω = 0.7ω0 (c) MPS Method Fg. 7 Wave shapes, t / T 0 = 8.85, A=100 mm, ω = 1.0ω 0 Dampng Characterstc of TLD The roll moton of TLD n dfferent condtons s smulated to study the dampng characterstc of TLD. The sldng moton of the mass s determned as Eq. 16 and Eq. 17. ξ m ( t) = A sn( ωt) (16) (b) ω = 0.8ω

6 (f) ω = 1.2ω0 (c) ω = 0.9ω0 (g) ω = 1.3ω 0 Fg. 8 Roll angle of empty tank and TLD, A=150 mm (d) ω = 1.0ω0 (e) ω = 1.1ω 0 The response ampltude of empty tank ncreases wth the exctaton frequency gettng closer to the natural frequency of the TLD system. In off-resonance regon ( ω 1.0ω0 ), The beatng phenomenon appears, and the beat perod gets longger near the naturel frequency. In resonance regon ( ω = 1.0ω0 ), the response ampltude keeps ncreasng n a approxmately lnear law throughout the calculaton tme, and fnally, the roll angle wll exceed 180 degrees, whch means the tank wll be overturned.the dampng characterstc of TLD s apparent under varous external exctaton frequency, wth the roll moton of tank vastly suppressed. The tank s lmted to a constant ampltude oscllaton, n both resonance regon and off-resonance regon, and the constant oscllaton s much smaller than the moton ampltude of empty tank under the same exctaton. Although TLD s a passve ant-roll devce, ts dampng effect can be observed at the early stage of the moton, and ts reacton tme s short enough for practcal applcaton. In Fg. 8d, the response ampltude of TLD keep ncreasng n the begnnng of roll moton n resonance regon, but at a much smaller speed than that n empty tank. After 40 seconds, the ampltude become constant. In off-resonance regon, the beatng phenomenon dsappears after the frst beat perod and the roll 1014

7 moton rapdly turns nto a constant ampltude oscllaton. The dampng characterstc of TLD s stronger when the exctaton frequency gettng closer to the natural frequency of the TLD system. In order to characterze the sze of the dampng effect, one varable, ampltude reducton rato, s defned. It s equal to the reduced percentage of maxmum ampltude when the moton of empty tank turns to the constant ampltude oscllaton of TLD. Fg. 9 shows the ampltude reducton rato of TLD under dfferent external exctatons. In dfferent exctaton ampltudes, the same rule s dscovered. Wth the exctaton frequency below the natural frequency of TLD system, the dampng characterstc s not dstnct, even can be gnored n some cases. Along wth the ncrease of the exctaton frequency, the ampltude reducton rato rapdly rses, and reaches to a peak n the resonance regon. Then the dampng characterstc of TLD gradually decreases f the exctaton frequency keep ncreasng. (a) A=50 mm, ω = 1.0ω 0 (b) A=100 mm, ω = 1.0ω 0 Fg. 9 Ampltude reducton rato under dfferent external exctatons Sloshng Wave n TLD Bouscasse (2014a) ponted out that the breakng waves n TLD can sgnfcantly affect the dampng characterstc. In Fg. 10, the sloshng wave shapes of TLD obtaned by the MPS method are shown, whch match well wth the results of experments. The four resonance cases are chosen because the sloshng wave shapes are most evdent n resonance regon. Two nstants are selected for each case, one wth the maxmum angle and one wth a flat angle. In Fg. 10a, the sloshng waves tran from one sde of TLD tank to the other sde, wthout breakng wave. In Fg. 10b, a plunge breakng event appears n the mddle of the tank, and volent mpact on the nner wall happens when the tank rolls to the maxmum angle. In Fg. 10c and 10d, when the TLD rolls to the maxmum angle, the water almost accumulates on one sde of the tank, and the moton of water can be classfed as a quas-dam-break type flow. (c) A=150 mm, ω = 1.0ω

8 (d) A=200 mm, ω = 1.0ω 0 Fg. 10 Wave shapes n TLD, MPS (left), Experments (rght). CONCLUSIONS In ths paper, the n-house solver MLPartcle-SJTU s employed to study the lqud sloshng n TLD system. The solver s developed based on the mproved MPS method and has been appled nto numerous free surface flows n our prevous works. As a result of lqud sloshng, the TLD has excellent dampng characterstc. The dampng effect of TLD largely ncreases when the exctaton ampltude gettng closer to the natural frequency of the TLD system, and s strongest n the resonance cases, n whch beyond 80% of the TLD oscllaton s suppressed. The dampng characterstc belongs to a passve functon, but can come nto effect wthn a short tme after the moton begns, whch s mportant n practcal applcaton of TLD. Wth the exctaton ampltude ncreasng from 50 mm to 200 mm, dfferent knds of sloshng flow can be observed n the TLD, at frst, only a wave trans wthout breakng waves, then a plungng breakng event takes place, fnally a quas-dam-break flow occurs. From comparson wth experments and SPH method, t can been seen that the MPS method s fully capable of capturng the sloshng feature of TLD system. ACKNOWLEDGEMENTS Ths work s supported by the Natonal Natural Scence Foundaton of Chna ( , , , ), Chang Jang Scholars Program (T ), Shangha Excellent Academc Leaders Program (17XD ), Shangha Key Laboratory of Marne Engneerng (K ), Program for Professor of Specal Appontment (Eastern Scholar) at Shangha Insttutons of Hgher Learnng ( ), Innovatve Specal Project of Numercal Tank of Mnstry of Industry and Informaton Technology of Chna( /09) and Lloyd's Regster Foundaton for doctoral student, to whch the authors are most grateful. REFERENCES Bulan, G, Souto-Iglesas, A, Delorme, L, and Bota-Vera, E (2010). 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