SMOOTHED PARTICLE HYDRODYNAMICS METHOD FOR TWO-DIMENSIONAL STEFAN PROBLEM

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1 The 5th Internatonal Symposum on Computatonal Scences (ISCS) 28 May 2012, Yogyakarta, Indonesa SMOOTHED PARTICLE HYDRODYNAMICS METHOD FOR TWO-DIMENSIONAL STEFAN PROBLEM Dede Tarwd 1,2 1 Graduate School of Natural Scence and Technology, Kanazawa Unversty Kakuma, Kanazawa , Japan 2 Faculty of Mathematcs and Natural Scences, Insttut Teknolog Bandung Jl. Ganesha 10, Bandung 40132, Indonesa emal: dede.tarwd@yahoo.com Astract. Smoothed partcle hydrodynamcs (SPH) s developed for modellng of meltng and soldfcaton. Enthalpy method s used to solve heat conducton equatons whch nvolved movng nterface etween phases. At frst, we study the meltng of floatng ce n the water for twodmensonal system. The ce ojects are assumed as sold partcles floatng n flud partcles. The flud and sold moton are governed y Naver-Stokes equaton and asc rgd dynamcs equaton, respectvely. We also propose a strategy to separate sold partcles due to meltng and soldfcaton. Numercal results are otaned and plotted for several ntal condtons. Keywords: SPH, enthalpy method, Stefan prolem, movng nterface 1. INTRODUCTION Stefan prolem s related to heat transfer prolem nvolvng phase-change such as sold to lqud (meltng) or lqud to sold (soldfcaton). Ths ssue has the attenton from many researchers snce the last few decades. Several methods are used to solve the Stefan prolem oth analytcal and numercal solutons. Due to ts complexty, the analytcal soluton s lmted to one-dmensonal and smple two-dmensonal prolem. The detal of the analytcal and numercal soluton of the one-dmensonal Stefan prolem s dscussed n [1]. As to the numercal soluton, n general, t can e dvded nto two methods, the grd/mesh-ased method and the grdless/meshless-ased method. Fnte dfference, fnte element, and fnte volume method are grd/mesh-ased methods often used n solvng Stefan prolem. Voller and Shada [2] used fnte dfference method va enthalpy formulaton for solvng two-dmensonal Stefan prolem. Moreover, the fnte element method [3-5] and the fnte volume method [6, 7] are wdely employed for solvng oth one-dmensonal and two-dmensonal Stefan prolem. The smoothed partcle hydrodynamcs (SPH) method s a meshless method. Ths method s a partcle-ased nterpolaton whch was orgnally used for modellng astrophyscal prolems. However, ths method s eventually developed n flud dynamcs prolems. In ths paper, we propose SPH method for solvng two-dmensonal Stefan prolem. The SPH method whch s appled n meltng prolem together wth flud flow could e seen n [13]. In other case, Monaghan et al. [8] used SPH method n solvng soldfcaton prolem, ut the flud flow s not consdered n the model. In the present case, we employ SPH method to smulate ce meltng prolem n twodmensonal systems. For the reverse prolem that s water soldfcaton, the mathematcal formulaton s the same ut dfferent n the ntal and oundary condton. In the meltng or soldfcaton case, two regons of the system are consdered, lqud (water) and sold (ce) separated y phase change nterface. Moreover, the nterface s always movng n each tme step. The system leads us to the two-phase Stefan prolem.

2 DEDE TARWIDI Two heat conducton equatons should e solved n sold and lqud regon, ut the oundary of the doman s not known. Our goal s to determne temperature feld T ( x, y, t ) that satsfy a heat conducton equaton nsde sold regon, a heat conducton equaton nsde lqud regon. Also, T ( x, y, t ) must satsfy nterface condton, ntal and oundary condton. 2. GOVERNING EQUATION There are three categores for governng equatons of the system: flud flow, sold moton, and heat transfer (energy equaton). The flud moton s assumed weakly compressle. The governng equatons for weakly compressle flud are the Naver- Stokes and the contnuty equatons as follow: Dv 1 = p + F (1) Dt ρ Dρ ρ Dt = v..(2) where v, ρ, and p are velocty, densty and pressure of the flud, respectvely. Whle F s external force per unt mass whch can e gravtatonal acceleraton or repulsve force per unt mass. For the present case, surface tenson s assumed not to e sgnfcant, so t can e neglected. The equaton of state n smple form s gven y 2 p = c ρ ρ + p..(3) ( ) 0 atm where c s speed of sound, ρ0s densty reference, and patm s atmospherc pressure. The choce of speed of sound s very mportant n the smulaton snce t nfluences the densty varaton. In order to keep densty fluctuatons less than 1%, c = 200gH s chosen where g s gravty acceleraton and H s heght of water level. The governng equaton of heat transfer and sold moton s dscussed further n Secton 3 and Secton 6, respectvely. 3. ENTHALPY METHOD Enthalpy method s usually used n modellng that nvolved phase-change such as meltng and soldfcaton. Ths method refers to reformulate the heat conducton equaton n dfferent phases nto the enthalpy equaton. Also, y usng enthalpy equaton the latent heat (heat of fuson) s accounted n calculatng the temperature feld. In the meltng case, ths latent heat s needed to reak up the ndng n the sold structure. Otherwse, n the soldfcaton case, the latent heat s released. One of the advantages of the enthalpy method s, the heat conducton equaton n sold and lqud regon can e solved wthout need to know the nterface poston and the heat flux n the nterface s automatcally contnuous. Moreover, f we know the relatonshp etween enthalpy and temperature then we can calculate temperature feld. If we assume heat transfer y conducton only, then the enthalpy equaton s H 1 = ( k T )...(4) t ρ

3 SPH METHOD FOR TWO-DIMENSIONAL STEFAN PROBLEM where H s enthalpy, ρ s densty and k s thermal conductvty. The relatonshp etween enthalpy and temperature s gven y [1] H Tm +, H 0 (sold) cs T = Tm, 0 < H < L (nterface) (5) H L Tm +, H L (lqud) cl T m s meltng pont, cs and c l are specfc heat of sold and lqud respectvely, and L s latent heat. 4. SPH FORMULATION The man dea of the SPH method s to represent a functon or ts dervatve nto ntegral representaton and dscretze t nto set of partcles. The detal aout SPH method s dscussed n [14]. The ntegral nterpolant of functon f ( x ) s f ( x) = f ( x ) δ ( x x ) dx...(6) Ω where δ ( x x ) s the Dract delta functon and Ω s the volume of the ntegral that contan x. If we replace the delta functon y smoothng functon or kernel functon, W, we get kernel aproxmaton of functon f ( x ) : f ( x) = f ( x ) W ( x x, h) dx (7) Ω where h s smoothng length that represents nfluence area of kernel functon. We also f x : can get the kernel aproxmaton of the dervatve of functon ( ) f ( ) = f ( ) W (, h) d x x x x x (8) Ω The followng equatons are derved y usng SPH dscretzaton: dv a p + p a (Naver-Stokes equaton ) = m aw ( ra r, h)...(9) dt ρaρ dρa (contnuty equaton) = m ( ra r ) aw ( ra r, h)... (10) dt Equaton (9) and (10) mean that the velocty and densty n a partcle can e otaned y summng up the contruton of the neghorng partcles. The contruton s governed y kernel functon W. Therefore the choce of W wll affect the accuracy of the smulaton. There are many choces of kernel functon, ut for the present smulaton, the followng cuc splne kernel s used: q + 3 q, 0 q < W ( q, h) = 2 4 ( 2 q), 1 q < 2.(11) 7πh 0, otherwse

4 DEDE TARWIDI where q = r r / h. a The heat transfer etween partcles (water, ce, and sold oundary) s represented y the temperature and enthalpy whch are carred y SPH partcles. In the present smulaton there are only two types of heat transfer to e taken nto account of the conducton and convecton. But, the heat transfer y convecton s automatcally fulflled ecause the lqud partcle s always moves accordng to the Naver-Stokes equaton [13]. The heat transfer y conducton n SPH form s gven y [9] dh ( ) ( ) (, ) a m kak ra r aw ra r h = Ta T (12) 2 2 dt ρ ρ k + k r r + η a a a where η = 0.01h. The steps n constructng the SPH heat conducton of equaton (12) s refly dscussed n [11]. Note that y usng equaton (12), n nterface etween sold and lqud phase, the heat flux s automatcally contnuous. Once we otan the enthalpy of the partcles, the temperature partcles and phase-change transton are determned y usng equaton (5). The leap-frog tme steppng s employed for equaton (9) and equaton (10) whle explct Euler tme steppng s used for equaton (12). 5. BOUNDARY TREATMENT In SPH modellng, t s very mportant to treat the partcles that approach wall oundares. In ths smulaton, the wall oundares are assumed as sold oundary partcles. We need these sold oundary partcles to prevent the water partcles penetrate the wall oundares. The water partcles approachng wall oundares wll experence repulsve force comng from sold partcles. The force s experenced y a water partcle j normal to the sold oundary partcle s gven y [10] f = n R j ( y ) P ( x ).(14) where n s normal vector of the sold oundary. Here, R y A q 1 ( ) (1 ) q ( ) 1 = and P( x) = 2 1+ cos( πx p ) where x s dstance of projecton water partcle j on tangent vector of the sold oundary partcle, y s perpendcular dstance of water partcle j from the sold oundary partcle (see Fgure 1), q = y /(2 p) and p s ntal partcle spacng. The choce of partcle spacng of sold oundary partcles depends on the ntal water partcle spacng. If p s too small, then the water partcle wll experence a repulsve force efore t approaches the wall oundary. Whereas f p s too g, some of water partcles may penetrate the oundary wall efore experencng the repulsve force. 6. FLOATING ICE MODEL There are two ways for modelng ce (sold) ojects. Frst, consder ce partcles as f they were water partcles. Hence, oth sold and lqud partcles are solved y usng Naver-Stokes equaton. In ths way the densty of ce partcles s also calculated y usng equaton (2). But, they are not updated n each tme step n order to keep the consstency of the densty of ce. However, t makes prolems ecause ce densty s 917 kg/m 3 and water densty s 1000 kg/m 3 (the dfference s too wde) whle the Naver-Stokes equaton

5 SPH METHOD FOR TWO-DIMENSIONAL STEFAN PROBLEM s desgned for weakly compressle. Consequently, ce partcles wll have low pressure, and t makes water partcles penetrate ce ojects. The second way s to consder ce partcles as sold oundary (wall) partcles. In ths way, If the water partcles approachng ce partcles, they wll experence repulsve force. The prolem s we have to fnd normal vector of ce ojects n each tme step. It s slghtly dffcult snce ce formaton s always changng. In meltng case, t may e easy to fnd normal vector of ce wall ecause at ntal we set ce partcles neatly. But, n soldfcaton case, t s rather dffcult to fnd normal vector of ce wall whch s formed from water partcles. It s ecause the ce partcles formed are unstructured. In the present smulaton, the ce ojects whch float n water are assumed as wall oundary partcles. So that for each water partcle that approaches the ce partcles, t wll experence repulsve force whch s calculated accordng to equaton (14). But, the challenge s n determnng the normal vector of ce wall snce the wall of ce s changng and movng n each tme step. The normal vector of ce partcle due to water partcle j can e determned y projectng vector r j onto the tangent vector of partcle. To fnd tangent vector, t s necessary to fnd two neghorng partcles of partcle says -1 and +1 wth the opposte poston, see Fgure 1. The tangent vector t s r + 1 r 1. Hence the unt normal vector s n = ( r proj r ) / r proj r j t j j t j where projt r j s projecton of vector r j onto tangent vector t. The advantage of ths strategy s that n s always outward normal vector. Fgure 1. Normal vector and tangent vector The repulsve force of sold partcle s calculated y summng up the contruton of surroundng water partcles [15] f = f j (15) j WPs where WPs denotes water partcles. The equatons of moton for ce ojects are derved from the asc rgd dynamcs equaton. The translatonal and rotatonal equatons of sold moton are dv dω M = mf and I = m ( r R0 ) f..(16) dt IPs dt IPs where M, I, V, Ω, and R 0 are mass, moment of nerta, velocty, rotatonal velocty, and center of mass of the ce ojects, respectvely. Here, IPs represents ce partcles. The velocty for each ce partcle s gven y

6 DEDE TARWIDI dr ( ) V Ω r R..(17) dt = + Thus the poston of ce partcles for each tme step can e calculated y ntegratng equaton (17) n tme. An ce oject may e separated nto some ce ojects durng meltng process and also new ce ojects may e formed durng soldfcaton process. To control the moton of these ojects, there should e a strategy to separate the ce partcles nto certan ce ojects, so that equaton (16) can e employed. A smple strategy to separate the sold partcles has een proposed n [12]. Frst, each ce partcle s assgned an ndex from 1 to N, where N s the numer of ce partcles. Second, each ce partcle ndex s updated teratvely y the maxmum ndex of the neghorng partcles. Here, the neghor of a partcle s defned as all partcles wth dstance less than or equal to certan numer. The last, the ce partcles ndex that elong to the same oject wll converge to the maxmum ndex of the ce partcles n that ojects. 7. RESULTS AND DISCUSSION 7.1 Physcal setup of the ce meltng The physcal setup of the system n 2D can e descred as follow. Consder a water tank and two ce cues wth the dmenson of each ce cue s m x m. The dmenson of water tank s 1 m x 1 m, and at frst flled wth water of heght 0.25 m as seen n Fgure 2. The ntal temperature of the frst and the second ce cue are C and -10 0C, respectvely, and the temperature of water s 30 0 C. The outsde oundares of the system are adaatc wth the ntal temperature s C. The heat transfer n the system s assumed to occur only y conducton and convecton. The partcles are used for ths smulaton. All parameters for ths smulaton can e seen n Tale 1. For soldfcaton case, the physcal setup s generally same ut dfferent n ntal and oundary condton. The smulaton result for soldfcaton case may e shown later n other paper. 0 Fgure 2. Physcal setup of the ce meltng Tale 1. Parameter for the smulaton No Parameter Value Unt 1 Specfc heat of ce kj/kg 0 C 2 Specfc heat of water kj/kg 0 C

7 SPH METHOD FOR TWO-DIMENSIONAL STEFAN PROBLEM 3 Conductvty of ce 2.216e-3 kj/ms 0 C 4 Conductvty of water 0.566e-3 kj/ms 0 C 5 Meltng pont 0 0C 6 Latent heat 334 kj/kg 7 Densty of ce 917 kg/m 3 8 Densty of water 1000 kg/m Smulaton results of ce meltng At frst, two ce cues fall freely from heght of 0.25 m aove water surface. These two ce cues then nteract wth the water. The forces actng on the water partcles and ce partcles are calculated y usng equaton (1) and equaton (16) respectvely. To get temperature of the partcles, heat transfer n each partcle s determned y the enthalpy equaton (4). Temperature dstruton of oth sold partcles and lqud partcles for several tme steps can e seen n Fgure 3. In that fgure, the color represents the temperature of partcles n certan poston. Each mage n Fgure 3 represents not only the phase change from sold to lqud, ut also the patterns of convecton flow. The convecton patterns are qute nterestng to e explored further. These patterns are caused y dfferences n densty etween new water partcles and old water partcles. The new water partcles mean ce partcles whch have just transformed to water partcles. In ths smulaton, the ce partcles own have fxed densty that are 917 kg/m 3, whle the water partcles have densty n the range of kg/m 3 (weakly compressle flud). However, after the ce partcles change the phase from sold to lqud, the densty of the new water partcles are changed to 1000 kg/m 3. The new water partcles caused densty dfference qute strkng wth the surroundng water partcles. There wll e movement of the partcles accordng to equaton (1) and (2). Fgure 3. Temperature dstruton for several tme steps

8 DEDE TARWIDI Fgure 4. Temperature vs tme of a sngle partcle of the frst and second ce cue The left graphc n Fgure 4 shows the temperature of a sngle partcle that ntally located on the edge of the frst ce cues. The ntal temperature of the partcle s C, then the temperature s gradually changed to 0 0 C n seconds. However, n ths condton, the state s stll n the sold phase. To change the phase of ths partcle from sold to lqud the energy s needed for amount latent heat (heat of fuson). The latent heat per partcle s m 0L = 2.99 kj. After the partcle state s changed nto lqud the temperature wll e nfluenced y the temperature of the surroundng water partcles. Total tme requred for a sngle partcle n frst ce cue to change the phase from sold to lqud s aout 2.06 seconds. Whle the rght graphc of Fgure 4 shows the temperature of a sngle partcle of the second ce cue that has the ntal temperature of C. For a sngle partcle n the second ce cue, the total tme needed to change the phase s aout 1.96 seconds. Total tme that requred y an ce partcle to melt depends on the mass and latent heat of the partcle. 6. CONCLUSION In ths paper we have shown that an SPH method can e used to smulate Stefan prolem especally n ce meltng prolem. The advantage of the SPH method s all physcal nformaton such as densty, pressure, velocty, and temperature can e stored n an SPH partcle. Wth ths way, t s easy to treat the nteracton etween sold and lqud phase whch s very hard f on the contrary, the mesh-ased method s used. By usng enthalpy method va SPH formulaton, heat conducton equaton n sold and lqud regon can e solved wthout needed to know the nterface poston etween sold and lqud. As though, ths model s new n SPH development, so t needs many mprovements to have more realstc smulaton. Acknowledgements The author would lke to thank Prof. Sero Omata for gvng very helpful comments and suggestons. The author also thanks Dr. Masak Kazama for the useful dscussons. REFERENCES [1] Alexades, V. and Solomon, A. D., 1993, Mathematcal Modelng of Meltng and Freezng Processes, Taylor & Francs, Washngton, DC.

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