ICCM215, 14-17 th July, Auckland, NZ Numercal Study of Interacton between Waves and Floatng Body by MPS Method Y.L. Zhang 1,2, Z.Y. Tang 1,2, D.C. Wan 1,2 * 1. State Key Laboratory of Ocean Engneerng, School of Naval Archtecture, Ocean and Cvl Engneerng, Shangha Jao Tong Unversty, 2. Collaboratve Innovaton Center for Advanced Shp and Deep-Sea Exploraton, Shangha 224, Chna *Correspondng author: dcwan@sjtu.edu.cn Abstract In the present study, nteracton between regular waves and free roll moton of a 2D floatng body s nvestgated by our n-house partcle solver MLPartcle-SJTU based on modfed Movng Partcle Sem-Implct (MPS) method. Numercal wave tank s developed to calculate the nteracton between waves and floatng body, ncludng wave-maker module and free roll moton module. The comparson between the numercal wave elevaton and analytcal soluton shows that the MLPartcle-SJTU can provde acceptable accuracy of wave makng. Roll moton and force actng on the floatng body n waves are n good agreement wth expermental results. Profles of the wave surface surroundng floatng body are presented. Keywords: Partcle method; MPS (Movng Partcle Sem-Implct); Wave Floatng body Interacton; Wave makng; Roll moton Introducton Recent years, a varety of floatng structures, such as shps, offshore platforms, floatng-breakwater, fsh-farms, floatng-arports, play a crucal role n coastal and ocean engneerng. It s common for floatng structures to suffer from loadngs under waves, and responses of these structures mounted n ocean or coastal envronments have sgnfcant relaton to the wave mpacts. The nteracton between free-surface waves and floatng body s one of the key aspects n shp desgn or offshore structure desgn to ncrease performance and effcency. In the past decades, both theory and expermental analyses methods have been used by many researchers to nvestgate the nteracton problem. The early establshed theoretcal methods are manly based on potental flow theores and lmted to solve the moton of floatng body wth smple shape. Chahne, et al. (1999) developed a free surface hydrodynamcs code based on threedmensonal Boundary Element Method and then they modeled the nonlnear evoluton of waves as they progress along a shallow slopng bottom n the presence of a floatng body that s free to rotate and translate. Ba and Eatock Taylor (26) studed the radaton and dffracton problem of vertcal crcular cylnders n a fully nonlnear numercal wave tank based on the boundary element method (BEM). You and Faltnsen (212) developed a 3D fully nonlnear tme-doman Rankne source code. A numercal wave tank wth a pston wave maker and a numercal dampng zone s appled to smulate the nteracton between moored floatng bodes and waves. Jung et al. (24a) expermentally studed waves mpactng on a fxed rectangular structure. PIV technque s used to obtan the mean velocty and turbulence propertes of water around structure. The generaton and evoluton of vortexes of a barge n beam sea condton s smulated. Subsequently, Jung et al. (24b, 25) nvestgated the two-dmensonal flow characterstcs of nteractons between waves 1
and freely rollng rectangular structures. Results between the roll moton and the fxed condton were compared. Ren et al. (215) studed the motons of a freely floatng body under nonlnear waves. Besdes, a wde varety of nonlnear numercal models based on the NS equatons n tme doman have been developed to study the nteracton problem. The fnte dfference method or the fnte volume method (FVM) s typcally used for spatal dscretzaton. And varous technques are used for nterface capturng, such as the Level Set method and the Volume of Flud method. In Boo s work (22), a numercal tank was constructed, the lnear and nonlnear rregular wave dffracton forces actng on a submerged structure was predcted. In L s work (21), a 2-D numercal regular wave tank was bult, whch manly based on the spatally averaged Naver- Stokes equatons and the k-e model was used to smulate the turbulence of flow. The fully nonlnear wave-body nteractons between a surface percng body n fnte water depth and flat/slop bottom topography were also nvestgated. Ye et al. (212) constructed a three-dmensonal numercal wave tank wth a newly developed solver naoe-foam-sjtu based on the open source code lbrary OpenFOAM, and a S- 175 contaner shp salng n regular headng waves was smulated. Zha, et al. (213) studed the moton responses of heave and ptch of a shp n dfferent wave condtons. Numercally smulaton of the moton response of a moored floatng per n regular waves was descrbed n Lu and Wan (213). The above researches are based on Euleran methods and grds are necessary for spatal dscretzaton. It s dffcult and naccurate to obtan free surface wth large deformaton. Recently, Lagrangan partcle methods draw much attenton of researchers and are seen as promsng numercal approaches for free surface flows. For example, Movng Partcle Sem-mplct (MPS), orgnally proposed by Koshzuka and Oka (1996) for ncompressble flow. In the present study, a partcle solver, MLPartcle-SJTU based on modfed Movng Partcle Sem-Implct (MPS) method, s used for all smulaton works. Some mproved schemes are used n ths solver to suppress numercal unphyscal pressure oscllaton usually observed n tradtonal MPS method. These mprovements nclude: (1) momentum conservatve pressure gradent model; (2) modfed kernel functon [Zhang et al., 211b]; (3) mxed sourced term method for Posson equaton of pressure [Tanaka et al., 21]; (4) surface detecton method based on asymmetry of neghbor partcles [Zhang et al., 211a]. The MLPartcle-SJTU was appled n many large free-surface deformaton problems, such as dam breakng flow [Zhang, et al., (211c, 214)], lqud sloshng n LNG tank [Zhang, et al., 212; YANG, et al.,214], Floatng Body Interactng wth Soltary Wave [Zhang, et al., 211b]. Ths paper s organzed as follow: Frstly, the MPS method for ncompressble flud s descrbed. Numercal approach to solve the moton of floatng body s ntroduced. Then, numercal wave tank s developed to calculate the nteracton between waves and floatng body, ncludng wave-maker module and free roll moton module. Tme hstory of wave propagaton s measured and compared wth the analytcal soluton to valdate the accuracy of wave makng. At last, roll moton and force actng on the floatng body n waves s calculated and compared wth expermental results. Profles of the wave surface surroundng floatng body are also presented. Numercal Scheme Governng Equatons 2
Governng equatons are the contnuum equaton and the momentum equaton. These equatons for ncompressble vscous flud are represented as: V= (1) DV Dt 1 P ρ 2 = + ν + V g (2) where V s the velocty vector,t s the tme, ρ s the densty,p s the pressure,ν s the knematc vscosty, g s the gravty acceleraton. Partcle Interacton Models Kernel Functon In partcle method, governng equatons are transformed to the equatons of partcle nteractons. The partcle nteractons are based on the kernel functon. In tradtonal MPS method, the kernel functon s expressed as follow (Koshzuka, 1996): re 1 r< r Wr () = r 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 affect the computatonal stablty. To overcome ths, an mproved kernel functon s used n ths paper (Zhang and Wan, 211b): Wr () = re.85r+.15r e r e r 1 r< r The above kernel functon has a smlar form wth the orgnal kernel functon Eq. (3), but wthout sngularty. Gradent Model Gradent operator s modeled as a local weghted average of the gradent vectors between partcle and ts neghborng partcles j: D φj + φ < φ >= ( ) ( ) r 2 j r W rj r n j rj r (5) where φ s an arbtrary scalar functon, D s the number of space dmensons, n s the ntal partcle number densty for ncompressble flow. The partcle number densty n MPS method s defned as: < n> = W( rj r ) j (6) Laplacan Model Laplacan operator s derved by Koshzuka et al. (1998) from the physcal concept of dffuson as: 2 2D < φ >= ( ) ( ) φj φ W rj r n (7) λ j r e e r e (3) (4) λ = j W ( r r ) r r j j j 2 W ( r r ) (8) j 3
where λ s a parameter, ntroduced to keep the varance ncrease equal to that of the analytcal soluton. Both vscous force 2V n Eq. (2) and 2 P n the rght hand sde of the PPE (Eq. 9 and Eq. 1) are dscretzed by Eq. (7). Model of ncompressblty The ncompressble condton n tradtonal MPS method 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. The pressure Posson equaton n tradtonal MPS method s defned as (Koshzuka et al., 1998): * 2 k + 1 ρ < n > n < P >= Δ 2 t n where n * s the partcle number densty n temporal feld. (9) The source term of the Posson equaton n Eq. (9) s solely based on the devaton of the temporal partcle number densty from the ntal value. As the partcle number densty feld s not smooth, the pressure obtaned from Eq. (9) s prone to oscllate n spatal and temporal doman. To suppress such unphyscal oscllaton of pressure, Tanaka and Masunaga (21) proposed a mxed source term for PPE, whch combnes the velocty dvergence and the partcle number densty. The man part of the mxed source term s the velocty dvergence, whle the partcle number densty s used to keep the flud volume constant. Ths mproved PPE s rewrtten by Lee et al. (211) as: 2 k 1 * (1 γ) ρ γ ρ < k n < > + n P >= V 2 (1) Δt Δt n where γ s a blendng parameter wth a value between and 1. The value of γ has large effect on the pressure feld. In partcular, the larger γ produces smoother pressure feld. However, the volume of flud cannot be constant whle γ =. The effects of γ have been nvestgated by Tanaka, et al. (21) and Lee, et al. (211). γ =.1 s used n ths paper. Free Surface boundary condton On the surface partcles, the free surface boundary condtons, ncludng knematc and dynamc boundary condton, are mposed. The knematc condton s drectly satsfed n Lagrangan partcle method, whle the dynamc condton s mplemented by settng zero pressure on the free surface partcles. So the accuracy of surface partcle detecton has sgnfcant effect on pressure feld. Fgure 1. Descrpton of partcle nteracton doman 4
The nteracton doman s truncated n the free surface (Fg. 1), so the partcle number densty near the free surface s lower than that n the nner feld. In tradtonal MPS method, partcle satsfyng (Koshzuka et al., 1998): * < n > < β n (11) s consdered as free surface partcle, where β s a parameter, can be chosen between.8 and.99. The tradtonal detecton method (Eq. 11) s based on the partcle number densty. However, nner partcles wth small partcle number densty may be msjudged as free surface partcles, thus unreal pressure around the msjudged partcles occur. Ths usually causes nonphyscal pressure oscllaton. To mprove the accuracy of surface partcle detecton, we employ a new detecton method n whch a vector functon s defned as follow (Zhang and Wan, 211c): < D 1 F > = ( ) ( ) r j W j n j j r r r r (12) The vector functon F represents the asymmetry of arrangements of neghbor partcles. Partcle satsfyng: < F > > α (13) s consdered as free surface partcle, where α s a parameter, and has a value of.9 F n ths paper, F s the ntal value of F for surface partcle. It should be specally noted that the Eq. (13) s only vald for partcles wth number densty between.8n and.97n snce partcles wth number densty lower than.8n s defntely surface partcles, whle those wth number densty hgher than.97n should get pressure through Posson equaton. Moton of floatng body The moton of the floatng body s governed by the equatons of rgd body dynamcs, followng the Newton's law of moton. The translaton moton of the center of gravty and the rotaton of the rgd body are gven n a smple 2-D framework by dvg M = Mg+ Fflud sold dt (14) dωg IG = Tflud sold dt where M and I G are the mass and the moment of nerta of the floatng body around the center of gravty, respectvely. V G and Ω G are the lnear velocty of the center of gravty and the angular F s the hydrodynamc force actng on the body, T flud sold velocty of the body, respectvely. flud sold s the hydrodynamc torque wth the drecton normal to the plane. Numercal Smulatons Test of wave makng In present work, a pston-type wave generator was ncorporated n the left sde of 2D numercal wave tank. A slop beach was nstalled at the end of the wave tank to absorb waves and avod reflecton. Sketch of the numercal setup s shown n Fg. 2. The numercal wave tank s 5.5 m 5
wdth and 1.5 m heght wth ntal water depth.9 m. Wave condtons used n present numercal test s shown n Table 1, and travellng waves were generated based on lnear wave theory. Table1. Parameters of wave makng Parameters Values Water densty 1(kg/m 3 ) Water heght.9(m) Wave length 1(m) Wave heght.29 Wave perod.8(s) Flud spacng.4 (m) No. of flud partcles 13275 No. of total partcles 13813 Fgure 2. Sketch of the 2-D wave tank Fgure 3. Comparson between numercal wave elevaton and analytcal soluton at locaton 1m from the pston paddle Fg. 3 shows a comparson between numercal wave elevaton and analytcal soluton at the locaton 1 m from the ntal poston of pston paddle. The trend of numercal free surface heght s n agreement wth analytcal soluton except that the former s less smoother than the later. The dfference can be mproved by reducng the partcle space. Smulaton of freely rollng body In ths secton, the roll moton of a 2D floatng rectangular structure n a numercal wave tank was nvestgated n tme doman. The wave generator and wave absorbng manner here are same as that n prevous secton. The wdth and heght of the rectangular floatng body are.3 m and.1 m, respectvely. The structure was nstalled at the pont 1.2 m from the wave maker and.9 m above the bottom of tank, fxed at the center of ts gravty but free n the degree of roll. The ntal geometry and set-up are shown n Fg. 4. 6
Fgure 4. Sketch of the freely rollng body In present smulaton, the dstance between partcles s.4 m, the total number of partcles s 137718 whle the number of flud partcles s 131762. The gravtatonal acceleraton and water densty are 9.8m/s 2 and 1kg/m 3. The knematc vscosty of water s gven by 1.1 1-6 m 2 /s. The tme step sze s.4s and the total computatonal tme s 1s.Waves wth perod of.8 s, was generated n the present study. The wave condtons used n present computaton are same as shown n Table 1. In the free rollng test of rectangular structure, angles of roll moton about the center of gravty were measured when the regular roll moton of rectangular body can be obtaned. Fgure 5. Rollng moton of the rectangular body wthn wave perod (sold lne: result of smulaton; dashed lne: result of JUNG) Fg. 5 shows the nclned angle of the rectangular structure over one perod of the regular wave. Results about the roll moton of floatng body s compared between present smulaton and experment by Jung(24). It can be found that both the pattern of curves and ampltude of roll angles are n good agreement. Fgure 6. Tme hstory of buoyancy restorng moment of the rectangular body wthn wave perod (sold lne: result of smulaton; dashed lne: result of JUNG) Detals about buoyancy restorng moment (M B ) of the freely rollng body should be noteworthy, because the roll moton s closely related wth the change of M B n tme doman. Fg. 6 shows the tme hstory of buoyancy restorng moment n one wave perod. It can be seen that the calculated 7
results of M B agree farly well wth expermental results, though the calculated curve about moment has a lttle nonphyscal fluctuatons. (a) (b) (c) (d) (e) Fgure 7. Rotary postons and wave surfaces around floatng box, (a) t=t, (b) t=t+t/4, (c) t=t+t/2, (d) t=t+3t/4, (e) t=t+t Fg. 7 shows the evoluton process of rotaton of the floatng body. It can be seen that postons of the floatng box s strong nfluenced by the propagaton of ncdent wave through fve snapshots of representatve-nstants (t, t+t/4, t+t/2, t+3t/4 and t+t) n a wave perod. Frstly, the body rotates clockwse untl the value of M B clmbs up to the maxmum wth the comng wave from left. 8
After that, the crest of the wave transfers from left to rght of the floatng box. At the same tme, buoyancy restorng moment of the body decreases. As a result, box rotates ant-clockwse and returns to horzontalty at the nstant of t+t/2. Wth the propagatng of the wave, water surface fallng on the left and rsng on the rght from t+t/2 to t+3t/4. Box keeps on rotatng antclockwse untl the value of M B declnes to the mnmum. Form the nstant of t+3t/4, t begns to rotate clockwse agan, and returns to nearly horzontalty at t+t fnally. The rotaton of floatng body wll repeat wth the next comng wave from left. Conclusons In ths paper, nteracton between regular waves and free roll moton of a 2D floatng body s nvestgated by our n-house partcle solver MLPartcle-SJTU based on modfed Movng Partcle Sem-Implct (MPS) method. Four mprovements, ncludng nonsngular kernel functon, momentum conservatve pressure gradent model, mxed source term for PPE and an accurate surface detecton method, are employed n ths solver. Numercal wave tank s developed to calculate the nteracton between waves and floatng body, ncludng wave-maker module and free roll moton module. The comparson between the numercal wave elevaton and analytcal soluton shows that the MLPartcle-SJTU can provde acceptable accuracy of wave makng. Numercal roll moton and force actng on the floatng body n waves are n good agreement wth expermental results. At last, the evoluton process of rotaton of the floatng body was shown through fve snapshots of representatve-nstants (t, t+t/4, t+t/2, t+3t/4 and t+t) n a wave perod. It can be seen that postons of the floatng box are strong nfluenced by the propagaton of ncdent wave. Accordng to the results present n prevous sectons, the solver can be used to deal wth waves floatng body nteracton problems. Acknowledgement Ths work s supported by Natonal Natural Scence Foundaton of Chna (Grant Nos. 51379125, 5149675, 114329, 5141113131), The Natonal Key Basc Research Development Plan (973 Plan) Project of Chna (Grant No. 213CB3613), Hgh Technology of Marne Research Project of The Mnstry of Industry and Informaton Technology of Chna, Chang Jang Scholars Program (Grant No. T21499) and the Program for Professor of Specal Appontment (Eastern Scholar) at Shangha Insttutons of Hgher Learnng (Grant No. 21322), to whch the authors are most grateful. References Ba, W., and Taylor, R. E. (28) Fully nonlnear smulaton of wave nteracton wth fxed and floatng flared structures, Ocean Engneerng, 36(3), 223-236. Boo, S.Y., (22) Lnear and nonlnear rregular waves and forces n a numercal wave tank, Ocean Engneerng, 29, 475-493. Jung, K.H., Chang, K.A. and Huang, E.T. (24a) Two-dmensonal flow characterstcs of wave nteractons wth a fxed rectangular structure. Ocean Engneerng, 31, 975-998. Jung, K. H. (24b) Expermental study on rectangular barge n beam sea, Ph.D. Texas A&M Unversty. Jung, K.H., Chang, K.A. and Huang, E.T. (25) Two-dmensonal flow characterstcs of wave nteractons wth a free-rollng rectangular structure. Ocean Engneerng, 32(1), 1-2. Kalumuck, K. M., Chahne, G. L., and Goumlevsk, A. G. (1999) BEM modelng of the nteracton between breakng waves and a floatng body n the surf zone, 13th ASCE Engneerng Mechancs, Baltmore, Maryland. Koshzuka, S., and Oka, Y. (1996) Movng-partcle Sem-mplct Method for Fragmentaton of Incompressble Flud, Nuclear Scence and Engneerng, 123, 421-434. Koshzuka, S., Obe, A., and Oka, Y. (1998) Numercal Analyss of Breakng Waves Usng the Movng Partcle Semmplct Method, Internatonal Journal for Numercal Methods n Fluds, 26, 751-769. L, Y., and Ln, M. (21) Wave-body nteractons for a surface-percng body n water of fnte depth, Journal of hydrodynamcs, Ser. B, 22 (6), 745-752. 9
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