Smulaton of a Shp wth Partally Flled Tanks Rollng n Waves by Applyng Movng Partcle Sem-Implct Method Jen-Shang Kouh Department of Engneerng Scence and Ocean Engneerng, Natonal Tawan Unversty, Tape, Tawan, R.O.C. kouhsh@ntu.edu.tw Hung-Pn Chen 1, Chun-Chung Chang 2, Yen-Jen Chen 3 Abstract - Free surface behavor s a common phenomenon n the nature. It combnes spray creaton, large deformaton and breakup of water. Due to ts unfxed computatonal boundary, the boundary poston needs to be solved as a porton of result. However, t s a challenge to gve suffcent spatal resolutons along the tme-dependently changng free surface. In ths paper, a mesh free method, movng partcle sem-mplct method (MPS), based on the partcle mechancs wll be used and appled to analyze the phenomena of a shp wth partally flled tanks rollng n regular waves. Thereby, not only the external flow around the hull but also the free surface flow nsde partally flled tanks can be smulated together by usng a sngle algorthm based on ths method. The nfluences of the sloshng effect resulted from the free surface flow to the rollng of a shp wll be dscussed n ths paper. Some case studes wll be chosen for demonstratng results thanks to the advantage of ths method. Key Words- Free surface flow, Mesh-free method, Movng partcle sem-mplct method, Sloshng effect. INTRODUCTION Numercal smulaton s dffcult to calculate flud fragmentaton on the free surface. Another dffculty of numercal smulaton exsts n handlng complex geometry. However, n the feld of naval archtecture, the topc of smulatng free surface has been concerned all along. In general, MAC method (Marker and Cell) [1] and VOF method (Volume of Flud) [2] are ntroduced popularly n solvng the free surface problem. Recently, the meshfree methods have been put emphass on smulatng free surface. In ths paper, a meshfree method, movng partcle semmplct method (MPS) [3], based on the partcle mechancs wll be ntroduced and appled to analyze the unsteady free surface flow. The MPS has the capablty to analyze more complex geometry and physcs than grd methods. In the method, t s unnecessary to generate mesh and the numercal dffuson doesn t need to be taken nto account, because the convecton s embedded n the moton of partcles. The accuracy of result can be controlled by the number of partcles. The nformaton of computatonal doman can be obtaned more detal by means of puttng more partcles. The smulaton of two-dmensonal flud fragmentaton s presented n ths paper. NUMERICAL METHOD The movng partcle sem-mplct method (MPS) was a meshless method for ncompressble flow wth free surfaces. It was advocated by Koshzuka et.al [4] n 1995. The basc concept comes from solvng the dscrete partcles dstrbuted n the computatonal and boundary doman. Each partcle has correspondng nformaton of coordnate, velocty and pressure. The kernel functon s employed to smulate a partcle nteracts wth ts neghborng partcles. By tracng the moton of every partcle n Lagrangan coordnate, the nformaton of partcles s obtaned then. I. Governng Equatons Governng equatons are the mass, momentum conservaton equatons for Newtonan, ncompressble flows as follows: ρ (1) t Du 1 (2) P + f Dt ρ Equaton (1) s mass conservaton and (2) s momentum conservaton wth convecton terms. The left hand sde of (2) denotes the convecton terms n Lagrangan tme dfferentaton and descrbes the partcle moton. The rght hand sde conssts of pressure gradent and force terms. All dfferental terms are dscretzed to smulate the nteracton between partcles. II. Kernel Functons The contnuous flow s supposed to consst of partcles wth coordnate, velocty and pressure nformaton. A partcle nteracts wth ts neghborng partcles n a specfc area. The kernel functon ω(r) s employed to descrbe the nteracton 1 Hung-Pn Chen, Department of Engneerng Scence and Ocean Engneerng, Natonal Tawan Unversty, Tape, Tawan, R952526@ntu.edu.tw 2 Chun-Chung Chang, Department of Engneerng Scence and Ocean Engneerng, Natonal Tawan Unversty, Tape, Tawan, R9352521@ntu.edu.tw 3 Yen-Jen Chen, Department of Multmeda and Game Scence, Lunghwa Unversty of Scence and Technology, Taoyuan, Tawan, clve_chen@yahoo.com Combra, Portugal September 3 7, 27 Internatonal Conference on Engneerng Educaton ICEE 27
between partcles. The kernel functon employed n ths research s as follows: governng equaton. Because of (8) s not senstve to absolute pressure, t can be arranged as follows: re 1 ω( r) r ( r < r ) ( r r ) e e (3) ' d φ φ φ 2 n r r ( r r ) ω( r r ) (9) In (3), r s a dstance between two partcles and r e s the radus of the nteracton area. The nteracton can be neglected at r r e. However, the kernel functon s nfnty at r. It means the overlap between partcles can be avoded [5]. III. Partcle Number Densty Partcle number densty n s defned by (4) where partcle s located at coordnate r. ( r r ) n ω (4) Equaton (4) can be expressed as the contrbuton from partcle tself. It also denotes the concentrated extent of neghborng partcles around partcle. The number of partcles n a unt volume can be wrtten as the relaton between n and N : N V n ω( r) dv The domnator of (5) s the ntegral of the kernel functon n the whole regon. ρ mn V mn ω( r) dv Equaton (6) shows that the flud densty s proportonal to the partcle number densty where m denotes mass of a partcle. For ncompressble flow, the partcle number densty keeps a constant as n IV. Modelng of Gradent ( r r ) (5) (6) ω (7) A gradent vector between two partcles and can be defned as (ψ -ψ )(r -r )/ r -r 2. Therefore, the gradent vector at r s obtaned by a weghted average of these vectors. d φ φ φ 2 n r r ( r r ) ω( r r ) where d s the number of space dmensons. In ths paper, (9) s used nstead of (8). The modelng of gradent s ntroduced to compute the pressure gradent term n (8) ψ ' s computed from (1) V. Modelng of Laplacan ' φ φ mn( ) (1) The operator 2 φ s obtaned by ntroducng the dffuson equaton. The modelng of Laplacan s presented as follows: d n λ 2 2 φ λ ω ( φ φ ) ω( r r ) 2 ( r r ) r r ω( r r ) (11) (12) where λ s ntroduced so that the varance ncrease s equal to the analytcal soluton. VI. Boundary Condton Actual boundary condtons for the MPS method are smple handlng than tradtonal numercal method. Because there are no partcles outsde of the free surface, the partcle number densty on the free surface s smaller than n the lqud. If a partcle satsfes n * < β n (13) It wll be defned as free surface. Thus, t s not necessary to defne free surface specfcally. Here β s a parameter whch s smaller than 1. In ths paper, β s chosen to be.97. VII. Calculaton Parameters The dstance between two neghborng partcles l o s.8 m. The densty of lqud ρ s 1 kg.m 3 and the gravty g s 9.8 m/s 2. Koshzuka [4] advocated r e 2.1lo when calculatng the partcle number densty and the modelng of gradent. On the other hand, use r e 4.lo for the modelng of Laplacan. Each tme step s consdered accordng to the Courant equaton. α l 3 t mn,1. 1 (14) vmax α s chosen as.1 and v max s the temporal maxmum velocty of partcles. Combra, Portugal September 3 7, 27 Internatonal Conference on Engneerng Educaton ICEE 27
. sec. CALCULATION RESULTS Calculatons of a shp rollng n waves are presented. The effects of vscosty and surface tenson are neglected. The total number of partcles used for the smulaton s about 7,. The ntal partcle confguraton s shown n Fgure 1. The breadth of the tank s.133 m, the depth s.73 m and the draft s.42 m. The waves are generated by a pston type wave generator. The perod of wave maker s.35 sec. Four dfferent cases as below are calculated. Wthout floodng water n the shp. Wth floodng water n the shp. Wthout floodng water n the shp and wthout stablzer. Wth floodng water n the shp and wth stablzer. 2. m 1. sec. 2. sec. 3. sec. 4. sec..16 m 5. sec. FIGURE 1 GEOMETRY CONFIGURATION FOR COMPUTATION. FIGURE 4 SIMULATION OF FREE MOTIONS OF A SHIP WITHOUT FLOODING WATER IN WAVES. FIGURE 2 A SHIP WITHOUT STABILIZER. FIGURE 3 A SHIP WITH STABILIZER. I. Wthout Stablzer Theoretcally, the free-surface effect may nduce a large roll angle and then reduce the stablty of the shp. In order to nvestgatng the free-surface effect when the water s partally flled n the tank, a tank wthout stablzer s selected for numercal smulaton. Fgure 4 and 6 llustrates the condton of a shp wth and wthout floodng water n waves. Fgure 5 and 7 demonstrates the tme hstores of the sway and heave moton of the shp. FIGURE 5 THE FREE MOTIONS OF A SHIP WITHOUT FLOODING WATER. Combra, Portugal September 3 7, 27 Internatonal Conference on Engneerng Educaton ICEE 27
. sec.. sec. 1. sec. 1. sec. 2. sec. 2. sec. 3. sec. 3. sec. 4. sec. 4. sec. 5. sec. 5. sec. FIGURE 6 SIMULATION OF FREE MOTIONS OF A SHIP WITH FLOODING WATER IN WAVES. FIGURE 8 SIMULATION OF FREE MOTIONS OF A SHIP WITH STABILIZER IN WAVES. FIGURE 7 THE FREE MOTIONS OF A SHIP WITH FLOODING WATER. II. Wth Stablzer Smulate the effects of the stablzer about the former two cases. FIGURE 9 THE FREE MOTIONS OF A SHIP WITH STABILIZER (WITHOUT FLOODING WATER). Combra, Portugal September 3 7, 27 Internatonal Conference on Engneerng Educaton ICEE 27
. sec. 1. sec. 2. sec. 3. sec. 4. sec. III. Results The fgures below show the change of rollng angle and the angular velocty, the strength of waves, of the shp. From fgure 12 and 13, t s the dfference between the shp wth and wthout floodng water. The results show the nner freesurface effect nfluence the stablzaton of a shp serously. If the waves become volent, the shp may have a rsk to capsze. In addton, fgure 14 and 15 tell the dfference between nstallng the stablzer or not when the shp s wthout floodng water. Fgure 16 and 17 demonstrate the dfference between nstallng the stablzer or not when the shp s wth floodng water. From fgure 14 and 16, the results show the average rollng frequency whch s decreased when the stablzer nstalled on the shp. From fgure 15 and 17, the max ntensty of waves was decreased by the stablzer. Therefore, t s obvously that the stablzer can reduce the rollng frequency of the shp and the ntensty of waves actng on the shp. 5. sec. FIGURE 1 SIMULATION OF FREE MOTIONS OF A SHIP WITH FLOODING WATER AND STABILIZER IN WAVES. FIGURE 12 THE EFFECT OF THE INTERNAL FREE SURFACE FLOW. FIGURE 13 THE EFFECT OF THE INTERNAL FREE SURFACE DUE TO FLOODING WATER. FIGURE 11 THE FREE MOTIONS OF A SHIP WITH STABILIZER (WITH FLOODING WATER). FIGURE 14 THE EFFECT OF THE STABILIZER (WITHOUT FLOODING WATER). Combra, Portugal September 3 7, 27 Internatonal Conference on Engneerng Educaton ICEE 27
At present, the smulaton does not nclude the surface tenson and the volent wave generaton. Further mprovements are necessary for computng many other flow problems wth multphase phenomenon. ACKNOWLEDGMENT FIGURE 15 THE EFFECT OF THE STABILIZER (WITHOUT FLOODING WATER). The authors would lke to thank the Natonal Scence Councl R.O.C. for the fnal support of ths research under the grant NSC 94-2611-E-2-16. REFERENCES [1] Harlow F. H., Welch J. E., "Numercal calculaton of tme dependent vscous ncompressble flow wth the free surface", The Physcs of Fluds, Vol. 8, 1965, pp. 2182-2189. [2] Hrt C. W., Ncholls B. D., "Volume of flud (VOF) method for dynamcs of free boundares", Journal of Computatonal Physcs, Vol.39, 1981, pp. 21-221. [3] Koshzuka S., Oka Y., "Movng partcle sem-mplct method for fragmentaton of ncompressble flud", Nuclear Scence and Engneerng, Vol. 123, 1996, pp. 421-434. FIGURE 16 THE EFFECT OF THE STABILIZER (WITH FLOODING WATER). [4] Koshzuka S., Tamako H. and Oka Y., "A partcle method for ncompressble vscous flow wth flud fragmentaton", Comput. Flud Dyn. J. Vol. 4, 1995, pp. 29 46. [5] Koshzuka S., Oka Y., "Numercal analyss of breakng waves usng the movng partcle sem-mplct method", Int. J. Numer. Meth. Fluds, Vol. 26, 1998, pp. 751 769. FIGURE 17 THE EFFECT OF THE STABILIZER (WITH FLOODING WATER). CONCLUSIONS Ths paper presents 2D results from the numercal smulaton of a shp rollng n waves. The nternal flow of floodng water nsde the shp and the external flow about the shp can be calculated smultaneously. In contrast to the tradtonal mesh methods, the MPS method provdes obvous advantage for dealng wth free-surface flow. Wthout strvng for extra arrangement for the movng grds and usng any complcated algorthms, the smulaton can be carred out drectly. Not only the tme and effort n grd generaton can be saved, but also the phenomenon of the free-surface flow can be fully captured. It can be seen that floodng water n the shp can produce a strongly negatve effect on the rollng moton of the shp. Therefore, t s necessary to avod the effect of freesurface flow for safety consderaton. The stablzer can be used to prolong the rollng perod effectvely by referrng to the numercal results shown above. Combra, Portugal September 3 7, 27 Internatonal Conference on Engneerng Educaton ICEE 27