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1 Power Technology 203 (2010) Contents lsts avalable at ScenceDrect Power Technology ournal homepage: Drect numercal smulaton o gas sol suspensons at moerate Reynols number: Quantyng the couplng between hyroynamc orces an partcle velocty luctuatons S. Tennet a, R. Garg a, C.M. Hrenya c, R.O. Fox b, S. Subramanam a, a Department o Mechancal Engneerng, Center or Computatonal Thermal-lus Research, Iowa State Unversty, Ames, IA 50011, USA b Department o Chemcal an Bologcal Engneerng, Iowa State Unversty, Ames, IA 50011, USA c Department o Chemcal an Bologcal Engneerng, Unversty o Colorao, Bouler, CO 80309, USA artcle no abstract Avalable onlne 10 Aprl 2010 Keywors: Gas sol suspenson Granular temperature Partcle acceleraton moel Immerse bounary metho Partcle-resolve rect numercal smulaton Prectve evce-level computatonal lu ynamcs (CFD) smulaton o gas sol low s epenent on accurate moels or unclose terms that appear n the average equatons or mass, momentum an energy conservaton. In the multlu theory, the secon moment o partcle velocty represents the strength o partcle velocty luctuatons an s known to play an mportant role n the precton o core-annular low structure n rsers (Hrenya an Snclar, AIChEJ, 43 (4) (1994) [5]). In homogeneous suspensons the evoluton o the secon velocty moment s governe by the partcle acceleraton velocty covarance. Thereore, luctuatons n the hyroynamc orce experence by partcles n a gas sol low aect the evoluton o partcle velocty luctuatons, whch n turn can aect the mean an varance o the hyroynamc orce. Ths couplng has been stue n the lmt o Stokes low by Koch an co-workers usng a combnaton o knetc theory an multpole expanson smulatons. For Reynols numbers beyon the Stokes lmt, rect numercal smulaton s a promsng approach to quanty ths couplng. Here we present rect numercal smulaton (DNS) results or the evoluton o partcle granular temperature an partcle acceleraton varance n reely evolvng homogeneous gas sol suspensons. It s oun that smple extenson o a class o mean partcle acceleraton moels to ther corresponng nstantaneous versons oes not recover the correlaton o partcle acceleraton wth partcle velocty. Ths stuy motvates the evelopment o better nstantaneous partcle acceleraton moels that are able to accurately capture the couplng between partcle acceleraton an velocty Elsever B.V. All rghts reserve. 1. Introucton Gas sol lows are commonly encountere n energy generaton an chemcal processng. The esgn an scale-up o nustral evces motvate a better unerstanng o gas sol low characterstcs an transport phenomena. A unamental unerstanng o gas sol low s ncreasngly relevant wth renewe nterest n zero-carbon an carbon-negatve energy generaton technology such as chemcal loopng combuston. Computatonal lu ynamcs (CFD) smulatons that solve or average equatons o multphase low are beng ncreasngly use n the esgn process because they prove etale normaton about the sol volume racton an phasc mean velocty els n gas sol low [1]. Most CFD coes or evce-level smulatons o gas sol low are base on the Euleran Euleran (EE) multlu approach because these are computatonally less expensve than Lagrangan Euleran Corresponng author. E-mal aress: shankar@astate.eu (S. Subramanam). (LE) smulatons. In the EE multlu approach both the sol an lu phases are treate as nterpenetratng contnua, an averagng technques [2 4] are use to erve the equatons governng the conservaton o average mass an momentum n the lu an partcle phases. Ths results n a closure problem smlar to that encountere n the statstcal theory o sngle-phase turbulence because the averagng proceure results n unclose terms that nee to be moele. For nstance, the mean momentum conservaton equaton n the partcle phase requres closure o the average lu partcle nteracton orce (mean rag orce) an the average stress n the sol partcle phase. Accurate moels or these unclose terms are neee or prectve CFD smulaton o gas sol low. As wth all statstcal closures, an mportant moelng queston s the aequacy o the mathematcal representaton to capture physcal phenomena o engneerng relevance. For nstance, t s now establshe that the precton o core-annular structure n rser lows requres solvng the transport equaton or the partcle granular temperature or pseuo-thermal energy [5]. Ths norms us that a closure at the level o mean quanttes s not aequate to prect mportant low characterstcs such as core-annular structure, but a /$ see ront matter 2010 Elsever B.V. All rghts reserve. o: /.powtec

2 58 S. Tennet et al. / Power Technology 203 (2010) closure at the level o secon moments s necessary. However, t s not clear that closure at the level o the secon moments s sucent or prectve CFD smulaton that wll acltate esgn an scale-up. Closure at the level o thr-orer moments has been attempte by some researchers [6,7]. An alternatve approach to the closure o moment transport equatons s to conser the evoluton o the one-partcle strbuton uncton. Just as closure at the level o the transport equaton or the probablty ensty uncton (PDF) n sngle-phase turbulent reactve low mples a closure or all moment equatons, smlarly a knetc equaton that acheves a closure or the one-partcle strbuton uncton n knetc theory mples a closure or all moment equatons. In partcular, a closure at the one-partcle strbuton level automatcally mples closure o the mean momentum an partcle velocty secon moment equatons. Furthermore, closures at the one-partcle strbuton level are guarantee to satsy realzablty crtera, whereas specal care s neee to ensure the same n the case o moment closures. These conseratons motvate the evelopment o moels or the unclose terms n the transport equaton or the onepartcle strbuton uncton corresponng to gas sol low. Whle there s conserable work on knetc theory o granular lows where the nteracton wth ambent lu s neglecte, the knetc theory o gas sol low s stll beng evelope. For low Reynols number low n the Stokes regme, Koch an co-workers [8,9] evelope a knetc theory closure wth a moel or the contonal partcle acceleraton that accounts or the presence o ambent lu n the term transportng the strbuton uncton n velocty space. Ths theoretcal ramework allows us to conser two couple eects: () the eect o partcle velocty luctuatons on the mean rag, an () the eect o luctuatng partcle acceleraton on partcle velocty luctuatons or granular temperature. Wyle et al. [10] stue the eect o partcle velocty varance on the mean rag or the lmtng case o hgh Stokes number where the partcles move uner elastc collsons but are unaecte by hyroynamc orces. They showe that partcle velocty luctuatons o not aect the mean rag n Stokes low. Ths result s not surprsng because n Stokes low the partcle acceleraton s a lnear uncton o nstantaneous partcle velocty. However, at moerate mean slp Reynols numbers the rag law s nonlnear an Wyle et al. [10] showe that partcle velocty luctuatons o aect the mean partcle acceleraton. They propose a moe rag law n terms o volume racton ϕ, mean low Reynols number Re m an Reynols number base on partcle granular temperature Re T. The ocus o ths paper s on the secon eect: the eect o luctuatng hyroynamc orces on granular temperature. For statstcally homogeneous gas sol lows, the correlaton between the partcle luctuatng velocty an ts acceleraton luctuaton etermnes the evoluton o the partcle velocty secon moment. In the lmtng case o Stokes low, Koch [8,9] analyze the granular temperature, whch s the trace o the partcle velocty secon moment, an ecompose the partcle acceleraton velocty covarance as the sum o source an snk contrbutons. Partcle granular temperature ecreases ue to nelastc collsons an vscous nteractons wth the ambent lu, an these eects are represente by the snk term. I partcle collsons are elastc or low past xe partcle assembles s consere, then the granular temperature ecreases only ue to vscous nteractons wth the ambent lu. In the Stokes low regme the snk term smply relaxes the granular temperature to zero on the vscous relaxaton tme scale. In Koch's ecomposton o the acceleraton velocty covarance nto source an snk terms [9], the source term ue to hyroynamc nteractons wth neghborng partcles can balance the snk term leang to a steay state granular temperature n stable homogeneous suspensons. For moerate Reynols number, there s no unque ecomposton o the partcle acceleraton velocty covarance as the sum o source an snk contrbutons. The source term n the granular temperature equaton plays an mportant role n sustanng a nonzero value o the granular temperature. In ts absence the granular temperature n a homogeneous suspenson woul smply ecay to zero, leang to an nnte Mach number n the partcle phase. Not only s ths problematc rom a numercal stanpont or CFD smulatons, but t s also unphyscal over a we range o mean low Reynols number an volume racton. The orgn o the source term les n the hyroynamc nteractons that each partcle experences wth ts neghbors, an the range o ths nteracton epens on the mean low Reynols number an the sol volume racton. It s well known that a sphere sementng n a lu can have a ratng eect on ts neghbors an raw them nto ts wake. The rat, kss an tumble phenomena are well ocumente n [11]. These physcal mechansms can manest as a source n partcle velocty luctuatons by changng each partcle's velocty. Ths eect s quante through DNS o reely evolvng suspensons n ths work. Although Koch's analyss s useul n the Stokes low regme, t s cult to exten the analyss to moerate Reynols number cases. At moerate Reynols number, DNS oers a promsng approach to quanty unclose terms n the transport equatons or partcle velocty moments, or the transport equaton or the one-partcle strbuton uncton. Ths naturally leas to an evaluaton o exstng moels. We use DNS o gas sol low at moerate Reynols number to evaluate a class o acceleraton moels. The results ncate the nee or mprove nstantaneous partcle acceleraton moels that are capable o capturng the couplng between partcle velocty luctuatons an hyroynamc orces n gas sol low. The next secton escrbes pertnent etals o the statstcal moelng approach that motvate ths stuy. Ths s ollowe by a escrpton o the Partcle-resolve Uncontamnate-lu Reconclable Immerse Bounary Metho (PUReIBM) that s use to perorm DNS o gas sol low. Then the smulaton etals or xe partcle assembles an reely movng suspensons are presente. Results that quanty the couplng are reporte, an a class o partcle acceleraton moels s evaluate. Fnally, the conclusons o ths stuy are summarze. 2. Statstcal moels The average equatons or mean momentum conservaton an transport o the secon moment o partcle velocty n the multlu theory can be erve usng ether the Euleran Euleran or Lagrangan Euleran approach. A comprehensve summary o the relatons between the moment equatons obtane rom these statstcal approaches can be oun n [12]. Here we choose the Lagrangan Euleran approach wth the one-partcle strbuton uncton as our startng pont because t naturally leas to an explct connecton wth the moment equatons One-partcle strbuton uncton The one-partcle strbuton uncton, whch s the number ensty o partcles n an approprately ene phase space, s the unamental quantty o nterest n the knetc theory o granular an multphase low [8,14 17]. It s also reerre to as the roplet strbuton uncton n spray theory [18]. For monosperse partcles the strbuton uncton (x, v, t) s ene n a poston velocty space, an evolves by the ollowng transport equaton: t + x ðv Þ + v ð Ax; v; t Þ = coll ; where x an v enote the graent operators n the poston an velocty space, respectvely, an ḟ coll s the collsonal term that can epen on hgher-orer statstcs. A closure moel or the collsonal term results n a knetc equaton. Ths well-known equaton has been extensvely stue n the context o granular lows where collsons ð1þ

3 S. Tennet et al. / Power Technology 203 (2010) are nelastc. Extensons to non-lute cases that ollow the Enskog approach have also been pursue. The ocus n the knetc theory o granular low s on obtanng close-orm solutons [19], or consttutve relatons [17,20 23], startng rom a knetc equaton. Most o these stues rely on the Chapman Enskog expanson about a normal soluton n terms o a nonunormty parameter that s essentally the Knusen number. The prncpal erence between the knetc theory o gases an the knetc theory o gas sol low s that n the latter, the contonal partcle acceleraton term A x, v; t appears nse the velocty ervatve n the velocty transport term because partcle rag epens on partcle velocty through slp wth respect to the lu. Ths epenence o partcle acceleraton on partcle velocty n Eq. (1) results n the correlaton o A an v that etermnes the evoluton o the secon moment o partcle velocty, an ts trace, the partcle granular temperature. In the transport equaton or the strbuton uncton (c. Eq. (1)), A x, v; t represents the average partcle acceleraton contonal on poston x an velocty v. For the spatally homogeneous case wth monosperse partcles t can be nterprete as the average acceleraton experence by a partcle wth velocty v. The averagng operator represents ntegraton over all hgherorer multpartcle strbuton unctons [8,15] that can be ene on the bass o the ensemble o partcles wth poston an velocty {X (n) (t), V (n) (t), n =1,, N}. In partcular, the contonal acceleraton A x, v; t s obtane by ntegratng out ts epenence on the two-partcle ensty (par correlaton uncton). In other wors, the contonal acceleraton A x, v; t s not completely etermne by the partcle velocty, but may be aecte by the presence o neghbor partcles. The statstcal escrpton o multpartcle nteractons s not contane n the one-partcle strbuton uncton. Subramanam [16] notes that when the gas phase s represente by Reynols-average els, a class o moels or the unclose contonal acceleraton term A can be wrtten as: n Ax; v; t = A T Q g x; t ð Þ o ; q ð ðx; v; tþþ; x; v; ; t ; ð2þ where { Q g (x, t) } represents a set o average els rom the gasphase soluton (such as the mean gas velocty an turbulent knetc energy), an q() s any smply compute uncton o the strbuton uncton. The ellpss enotes the epenence on statstcal quanttes that are not represente n the strbuton uncton, e.g., epenence on hgher-orer multpartcle statstcs, or lu-phase statstcs not represente n { Q g (x, t) }. Recall that the physcal orgns o the source term n the granular temperature equaton le n the hyroynamc nteractons wth neghbor partcles an lu-phase velocty luctuatons. The statstcs o neghbor partcles are not contane n (x, v, t). I the mplementaton o the multlu theory accounts or lu-phase velocty luctuatons, then ths epenence can be ncorporate n the acceleraton moel o Eq. (2). However, many mplementatons o the multlu theory o not account or lu-phase velocty luctuatons. As note earler, closure o the transport equaton or the strbuton uncton (c. Eq. (1)) mples closure or all moment equatons. In the ollowng, the mple closure or the mean an secon moment o partcle velocty s examne Moment equatons The average equatons or mean momentum conservaton an transport o the secon moment o partcle velocty mple by Eq. (1) are erve usng the usual proceure to erve hyroynamc equatons n knetc theory, except or the act that the velocty epenence n the contonal acceleraton results n an atonal term n the secon moment equaton [8,12]. Here these equatons are scusse n the context o moelng the contonal acceleraton A x, v; t to capture the couplng between partcle velocty luctuatons an hyroynamc orce. Snce the DNS results we present n ths stuy are or xe partcles or or those unergong elastc collsons, the moment equatons are presente or the case o elastc collsons only Mean partcle velocty The mean momentum conservaton equaton wrtten n nex notaton s t p ϕ v + x p ϕ v v k = p ϕ A k x p ϕ v v k ; ð3þ k where p s the partcle ensty, ϕ s the sol volume racton gven by ϕ=nπ p 3 /6, where n s the number ensty o the partcles an p s the partcle ameter. For gas sol low, the mean partcle acceleraton A ue to the lu partcle rag orce s an unclose term n Eq. (3). In EE multlu theory, the mean partcle acceleraton A s moele usng a rag law as A = β W ; where W = v u () s the mean slp velocty between the sol an lu phases. In ths enton, u () an v are the lu an sol phase-average veloctes, respectvely. For an solate partcle n Stokes low, β s a constant equal to 3πμ p, where μ s the ynamc vscosty o the lu. The Reynols number base on the mean slp velocty between the lu an partculate phase quantes the relatve mportance o lu nerta, an s ene as Re m = ð1 ϕþ v u ð Þ p ; ð5þ μ where s the ensty o the lu. When the Reynols number base on the mean slp Re m s moerate (Re m N1), β s a uncton o the mean slp velocty between the partcle an the lu phase,.e. β=β( W ), an the rag s no longer lnearly epenent on the mean slp velocty. Typcal rag laws or gas sol low [24 26] characterze the epenence o lu partcle rag orce on the mean slp Reynols number an sol volume racton. These are obtane by a combnaton o ttng expermental ata an usng sem-analytcal approaches n lmtng cases. More recently, rect numercal smulaton o low past homogeneous xe partcle assembles has been use to euce rag laws ([27 29]) escrbng the epenence on mean slp Reynols number an sol volume racton. In the mean partcle velocty evoluton equaton, the last term on the rght han se o Eq. (3) s the transport o partcle Reynols stress arsng rom correlaton o partcle velocty luctuatons. Partcle velocty luctuatons are ene about the mean velocty as v = v v ; an the partcle granular temperature 1 that characterzes the strength o these luctuatons s T = 1 3 v v : Ths term s calculate by solvng a transport equaton or the partcle velocty covarance. 1 Note that we o not stngush between partcle velocty luctuatons arsng rom collsons an other sources, as suggeste by Breault et al. [13]. Our enton s consstent wth the stanar enton n knetc theory o granular an gas sol low, an t s also the enton aopte n the two-lu theory. ð4þ ð6þ ð7þ

4 60 S. Tennet et al. / Power Technology 203 (2010) Transport o partcle velocty covarance The evoluton equaton or the secon moment o velocty wrtten n nex notaton s [8,9,12] t p ϕ v v + x k p ϕ v v v k = x p ϕ v v v k k! p ϕ v v k v x k + v v k v x k + p ϕ A v + A v For statstcally homogeneous gas sol low wth no mean velocty graents the transport, proucton, an trple-velocty correlaton terms rop out an Eq. (8) reuces to t p ϕ v v : ð8þ = p ϕ A v + A v ; ð9þ showng that the partcle velocty covarance evolves accorng to the partcle acceleraton velocty covarance (luctuatons n the acceleraton are ene about the mean acceleraton,.e. A =A A.) Ths equaton shows how luctuatons n the hyroynamc orces aect the partcle velocty covarance. Contractng the nces n Eq. (9) results n the evoluton o partcle granular temperature or a statstcally homogeneous gas sol low: T t = 2 3 A v : ð10þ In the above equaton, the trace o the partcle acceleraton velocty covarance A v can be ether a postve or negatve quantty, an hence t can act as a source or a snk o granular temperature Mean an luctuatng partcle acceleraton From ths scusson o moment equatons we see that the mean acceleraton aects mean momentum, an luctuatons n acceleraton correlate wth luctuatng velocty to act as a source or snk term n the granular temperature equaton. In the ollowng, we relate the mean acceleraton an acceleraton luctuatons to the one-partcle strbuton uncton. The mean acceleraton A s obtane as the ntegral o the contonal expectaton o partcle acceleraton over velocty space: A ðx; tþ = 1 nðx; tþ ½vŠ Ax; v; t ðx; v; tþv; ð11þ an ths leas to the expresson F p = p ϕ A or the lu partcle rag (per unt volume) n the mean partcle momentum equaton. The expresson or the mean acceleraton s useul because t tells us how the velocty epenence n the contonal acceleraton can aect the mean rag through the strbuton uncton. The one-partcle strbuton uncton can be ecompose [16] nto the prouct o a number ensty n (x, t) an a velocty probablty ensty uncton V c (v; x, t): ðx; v; t Þ = nðx; tþ c Vðv; x; tþ: ð12þ Thereore, changes n the strbuton an level o partcle velocty luctuatons are characterze by the partcle velocty probablty ensty uncton V c (v; x, t), an these aect the mean rag through Eq. (11). In the knetc theory escrpton o gas sol low usng the onepartcle strbuton uncton, the luctuatng acceleraton s smply the erence between the contonal an uncontonal mean: A = A v A. Usng ths enton, the partcle acceleraton velocty covarance can be wrtten n terms o the one-partcle strbuton uncton as As note earler, luctuatons n partcle acceleraton can arse rom partcle velocty luctuatons, hyroynamc nteractons wth neghbor partcles, an lu-phase velocty luctuatons. Whle Eq. (13) explctly accounts or the eect o partcle velocty luctuatons, the other eects must be ncorporate n the moel or the contonal partcle acceleraton Moelng the contonal partcle acceleraton A straghtorwar extenson o the mean partcle acceleraton moel gven by Eq. (4) to ts contonal counterpart s A = βw = β v u ðþ ; ð14þ where A represents a moel (c. Eq. (2)) or the contonal partcle acceleraton A v, anw s the nstantaneous slp velocty. Here we have wrtten the nstantaneous slp velocty as the erence between the nstantaneous partcle velocty an the mean lu velocty, rather than as the erence between the nstantaneous veloctes n each phase,.e. W = v u. Ths s because n CFD moels base on the multlu theory there s no representaton o the nstantaneous gas-phase velocty an the gas-phase motons are represente only by the mean gas velocty. Although ths smple moel results n the same mean rag as n Eq. (4), ts mple closure or the acceleraton velocty covarance n the granular temperature equaton results n only a snk o granular temperature. Ths s because the smple extenson n Eq. (14) oes not represent the eects o neghborng partcles or luctuatons n the lu velocty relatve to ts mean. For Stokes low, Koch [8] erve an analytcal closure or the source term n the granular temperature equaton (c. Eq. (10)) usng a knetc equaton applcable to a lute monosperse gas sol suspenson wth hgh partcle nerta. He ene the nstantaneous slp velocty as W=v u (), where u () s the lu velocty exclung the rect eect o the th partcle (but nclung the sturbance eects o all the other partcles). Ths enton o the slp velocty gves rse to a source term n the granular temperature equaton. Lnearty o the governng equatons n the Stokes low lmt an the assumpton o a lute suspenson allowe the ervaton o an explct expresson or u () an the source term. For moerately ense suspensons, the assumptons mae by Koch [8] n the knetc theory approach are not val an hence Koch an Sangan [9] use a semanalytcal approach that use multpole expanson smulatons to erve an expresson or the source o granular temperature n the Stokes low lmt. In Secton 2.3, we revew the closures or the source term gven by [8] an [9] n the Stokes low lmt. Developng smlar closures or A x, v; t an the source term at moerate Reynols numbers s cult because the governng Naver Stokes equatons are nonlnear. In Secton 3, we present a rect numercal smulaton methoology base on PUReIBM as a promsng approach to evelop closures or the source an snk terms n the granular temperature equaton at moerate Reynols numbers Closure or hgh Stokes number partcles unergong elastc collsons n Stokes low In a hgh Stokes number suspenson the partcle veloctes are not sgncantly aecte by hyroynamc orces. For a lute suspenson o very massve partcles (hgh Stokes number) unergong perectly elastc collsons n Stokes low, Koch [8] showe that the steay state partcle velocty strbuton n the knetc theory escrpton s Maxwellan. 2 Thereore, n ths lmt the partcle velocty covarance A v = 1 n ½vŠ A v A gv ð v; t Þv: ð13þ 2 Later Koch an Sangan [9] use an approxmate multpole metho to show that even or ense suspensons o elastc partcles n Stokes low, the velocty strbuton s Maxwellan.

5 S. Tennet et al. / Power Technology 203 (2010) tensor s sotropc an ts evoluton can be smply escrbe by the granular temperature evoluton equaton Dlute suspensons o perectly elastc partcles For a lute homogeneous suspenson o hghly massve an perectly elastc monosperse partcles n Stokes low, the evoluton equaton o the granular temperature erve by Koch [8] s T t = 2R τ T + 2S I 3 : ð15þ The rst term on the rght han se o Eq. (15) s the snk o partcle granular temperature p ue to vscous sspaton. In ths term, R =1+3ϕ 1 = 2 = 2 s the mensonless partcle momentum relaxaton rate an τ=m/(6πμ a) s the characterstc tme scale over whch the velocty o a partcle o mass m an raus a relaxes ue to vscous orces. The secon term on the rght han se o Eq. (15) s the source ue to hyroynamc nteractons. In the lute lmt, the expresson or ths source term s S I = apt W 2 = 2π 1 = 2 τ 2 T 1 = 2 : ð16þ The source term n the lute lmt s enote S I to stngush t rom the source term S II at hgher volume ractons that are scusse n the ollowng secton Moerately ense to ense suspensons o perectly elastc partcles Koch an Sangan [9] use the multpole expanson metho to evaluate the source term ue to hyroynamc orces or ense homogeneous suspensons o massve elastc partcles n Stokes low. In ther smulaton the partcles move as a granular gas an ther moton s not aecte by the ntersttal lu. The evoluton equaton or the granular temperature s wrtten as T t = 2R ss ϕ τ ð Þ T + 2S II 3 : ð17þ For the snk term ue to vscous sspaton (rst term on the rght han se o Eq. (17)), the expresson or the mensonless sspaton rate R ss(ϕ) as a uncton o volume racton gven by [30] s use. The source term n granular temperature (secon term on the rght han se o Eq. (17)) s expresse as an ntegral o the temporal autocorrelaton o the orce experence by the partcles. The nal expresson or the source term gven by [9] s S II = a τ 2 W 2 T 1 = 2 S ðϕþ ð18þ where S (ϕ) s the mensonless source term. Expressons or the mensonless sspaton rate an the mensonless source as a uncton o the volume racton can be oun n [9]. 3. Drect numercal smulaton approach Here we escrbe a DNS approach base on the Partcle-resolve Uncontamnate-lu Reconclable Immerse Bounary Metho (PUReIBM) that s use to solve or low past arbtrary arrangements o sol sphercal partcles. Two types o smulaton results are presente: () or xe partcle assembles, an () or reely movng suspensons. The hyroynamc solver that s common to both types o smulatons s rst escrbe. Then the soluton approach or xe partcle assembles s outlne. Ths s ollowe by a escrpton o the smulatons o reely evolvng suspensons where the postons an veloctes o the partcles evolve uner the acton o hyroynamc an collsonal orces Hyroynamc solver PUReIBM s a partcle-resolve rect numercal smulaton approach or gas sol low where the contnuum Naver Stokes equatons wth no-slp an no-penetraton bounary contons on each partcle's surace are solve usng a orcng term that s ae to the momentum equaton. The salent eatures that stngush PUReIBM rom other mmerse bounary metho approaches are as ollows: 1. Uncontamnate lu: In PUReIBM the mmerse bounary (IB) orcng s solely restrcte to those gr ponts that le n the sol phase, an thereore the low soluton n the lu phase s uncontamnate by the IB orcng. Consequently the velocty an pressure n the lu phase s a soluton to the unmoe Naver Stokes equatons (n contrast to IB mplementatons that smear the IB orcng on to gr ponts n the lu phase aonng sol bounares, resultng n soluton els that o not correspon to unmoe Naver Stokes equatons). 2. Reconclable: In PUReIBM the hyroynamc orce experence by a partcle s compute rectly rom the stress tensor at the partcle surace that s obtane rom ths uncontamnate-lu low soluton (n contrast to IB mplementatons that calculate the hyroynamc orce rom the IB orcng el). Ths eature o PUReIBM enables us to rectly compare the DNS soluton wth any ranom-el theory o multphase low. In partcular, or statstcally homogeneous suspensons t s shown by Garg et al. [29] that the volume-average hyroynamc orce exerte on the partcles by the lu s compute rom a PUReIBM smulaton, t s a consstent numercal calculaton o the average nterphase momentum transer term τ n (s) δ(x x (I) ) n the two-lu theory [3]. Ths reconcles DNS results wth multphase low theory. Owng to these specc avantages, t s shown elsewhere [29,31] that PUReIBM s a numercally convergent an accurate partcleresolve DNS metho or gas sols low. Its perormance has been valate n a comprehensve sute o tests: () Stokes low past smple cubc (SC) an ace centere cubc (FCC) arrangements (rangng rom lute to close-packe lmt) wth the bounary-ntegral metho o [32], () Stokes low past ranom arrays o monosperse spheres wth LBM smulatons o [33] () moerate to hgh Reynols numbers (Re m 300) n SC an FCC arrangements wth LBM smulatons o [34] an (v) hgh Reynols number low past ranom arrays o monosperse spheres wth ANSYS-FLUENT CFD package. It has also been extene to stuy passve scalar transport, an valate or heat transer rom a sngle solate sphere [31]. The numercal scheme use n PUReIBM s a prmtve-varable, pseuo-spectral metho, usng a Crank Ncolson scheme or the vscous terms, an an Aams Bashorth scheme or the convectve terms. A ractonal tme-steppng metho that s base on Km an Mon's approach [35] s use to avance the velocty els n tme. The prncpal avantage o the PUReIBM approach s that t enables the use o regular Cartesan grs to solve or low past arbtrarly shape movng boes wthout the nee or costly remeshng. It also conserably smples parallelzaton o the low solver as compare to unstructure boy-tte grs Fxe partcle assembles The partcle conguraton or DNS o low past xe assembles s generate by rst allowng partcles to attan a ranom spatal arrangement through elastc collsons. A homogeneous conguraton o non-overlappng spheres corresponng to the spece sol volume racton s generate wth partcle centers on a lattce, an partcles are assgne a Maxwellan velocty strbuton. Partcles are allowe to equlbrate uner purely elastc collsons (n the absence o any ntersttal lu) to generate a homogeneous partcle conguraton or the DNS low solver. Ensemble-average low

6 62 S. Tennet et al. / Power Technology 203 (2010) statstcs are obtane by averagng over multple nepenent smulatons (MIS) perorme wth several such conguratons. Each statstcally entcal conguraton correspons to the same average sol volume racton an par correlaton (macrostate), but ers n the specc arrangement o partcles (mcrostates). The PUReIBM smulaton methoology an etals o the computaton o the mean acceleraton (or mean rag) or a xe partcle assembly are escrbe by Garg et al. [29] Freely evolvng suspensons Numercal smulatons [36] o reely evolvng suspensons have been perorme to stuy the sementaton o monosperse partcles uner gravty n the presence o a lu. Smulatons o reely sementng suspensons are carre out n peroc omans such that the mpose pressure graent n the lu balances the weght o the partcles. In sementaton calculatons the steay mean low Reynols number attans a unque value that epens on the problem parameters (lu an partcle enstes, sol volume racton an the value o acceleraton ue to gravty), an ths value s not known a pror. In the present stuy we seek to smulate reely evolvng partcle suspensons at arbtrary mean slp Reynols numbers whle mantanng the sol/lu ensty rato an sol volume racton at xe values. We also want to specy the mean low Reynols number as nput to the smulaton. Ths can be accomplshe by specyng a mean pressure graent that oes not exactly balance the weght o the partcles, but exerts the requste boy orce to mantan the esre slp velocty between the partcles an lu. However, now both the mean partcle velocty an the mean lu velocty change n tme because there s no steay soluton n the laboratory rame to the mean momentum balance n each phase. Note that even though the mean phasc veloctes are evolvng n tme, ther erence the mean slp velocty attans a steay value. The culty n smulatng ths low setup n the laboratory rame wth peroc bounary contons s that the contnuous ncrease n lu an partcle veloctes places unnecessary restrctons on the tme step through the Courant conton. To crcumvent ths problem we evelope a erent smulaton setup that perorms the DNS n an acceleratng reerence rame such that the partcles have a zero mean velocty wth respect to the computatonal gr. The equatons o moton are solve n an acceleratng rame o reerence that moves wth the mean velocty o the partcles. In ths rame, the partcles execute only luctuatng moton. In our setup, partcles on average o not low n or out o the computatonal oman, thereby mantanng a reasonable tme step that s base on the mean slp velocty. Partcles o low n an out o the oman because o ther luctuatng velocty. The avantage o our setup s that the esre mean low Reynols number s spece as an nput parameter, an we are able to solve the problem wth reasonable tme steps that resolve the low. Detals o the equatons solve n the acceleratng reerence rame are gven n Appenx A. In the reely evolvng DNS, each partcle moves wth an acceleraton that arses rom hyroynamc an collsonal orces. The partcles are represente n a Lagrangan rame o reerence at tme t by {X () (t), V () (t) =1,, N p }, where X () (t) enotes the th partcle's poston an V () (t) enotes ts translatonal velocty. The poston an translatonal velocty o the th partcle evolve accorng to Newton's laws as: X ðþ ðþ t = V ðþ ðþ; t t m V ðþ t N p = B + F ðþ ðþ+ t =1 F c ðþ; t ð19þ ð20þ where B s any external boy orce (zero n the smulatons shown here), F () s the hyroynamc orce (rom pressure an vscous stress that s calculate rom the velocty an pressure els at the partcle surace) an F c s the contact orce on the th partcle as a result o collson wth th partcle. Partcle partcle nteractons are treate usng sot-sphere collsons base on a sprng-ashpot contact mechancs moel that was orgnally propose by Cunall an Strack [37]. The avantage o usng sot-sphere collsons s that the smulatons can be extene to hgher volume ractons because enurng multpartcle contacts are taken nto account. In the sotsphere approach, the contact mechancs between two overlappng partcles s moele by a system o sprngs an ashpots n both normal an tangental rectons. The sprng causes collng partcles to reboun, an the ashpot mmcs the sspaton o knetc energy ue to nelastc collsons. The sprng stness coecents n the tangental an normal rectons are k t an k n, respectvely. Smlarly, the ashpot ampng coecents n the tangental an normal rectons are η t an η n, respectvely. The sprng stness an ashpot ampng coecents are relate to the coecent o resttuton an the coecent o rcton (see [38] or etals o the mplementaton). The partcles consere n ths stuy are assume to be perectly elastc an rctonless. Snce the partcles are perectly elastc, the ampng orce arsng rom the ashpot s zero. The tangental component o the contact orce s zero or rctonless partcles. S Thereore, only the normal component o the sprng orce F n contrbutes to the contact orce F c at tme t: F c ðþ= t F S nðþ: t ð21þ S At the ntaton o contact, the normal sprng orce F n s equal to k n δ, where δ s the overlap between the partcles compute usng the relaton δ = p X ðþ X ðþ : ð22þ A tme hstory o the sprng orces s mantane once the contact ntates. At any tme urng the contact, the normal sprng orce s gven by F S nðt + ΔtÞ = F S nðþ k t n V n Δt; ð23þ where V n s the relatve velocty n the normal recton (ene below) that s compute usng h V n = V ðþ V ðþ ˆr ˆr : ð24þ The normal vector r s the unt vector along the lne o contact pontng rom partcle to partcle. The governng equatons o moton that are solve n the lu, an the etals o the computaton o the hyroynamc orce actng on the partcles are scusse n Appenx A. A homogeneous partcle conguraton s generate n the same way as or the xe partcle assembles by equlbratng an ensemble o partcles unergong elastc collsons n the absence o ntersttal lu. Followng the smulaton methoology o [29], a steay low at the esre mean low Reynols number s rst establshe or ths xe partcle assembly. Once the mean lu partcle rag experence by ths xe partcle assembly reaches a steay state, the partcles are release at tme t=0 or the reely evolvng DNS smulaton. The partcles are avance on a tme step Δt coll that s etermne by the sprng stness an the ashpot coecents. The low els are upate on a tme step Δt lu, whch ensures that both the convectve an vscous tme scales are well resolve. At the start o a low tme step the orces actng on the partcles are compute base on the low els obtane at the en o the prevous low tme step. I Δt coll s

7 S. Tennet et al. / Power Technology 203 (2010) smaller than Δt lu the partcles are steppe by Δt coll untl the en o the low tme step, otherwse both the partcles an the lu are steppe by Δt coll. The smulaton s contnue untl the granular temperature reaches a steay state. 4. Results We rst present results rom a valaton test or xe partcle assembles. We then quanty partcle acceleraton an ts couplng to luctuatons n the partcle velocty n low past xe partcle assembles as well as reely movng suspensons Fxe partcle assembles Smulatons wth xe partcle postons an veloctes are representatve o physcal lu-partcle systems n whch the partcle veloctes o not change sgncantly over characterstc lu tme scales (the relevant scale here beng the tme to transt a characterstc length scale such as the partcle ameter at the mean slp velocty). Ths s true or hgh Stokes number (gas sol) suspensons. Smulatons o low past xe partcle assembles are less computatonally emanng than reely evolvng suspensons, an are useul or parametrc stues (varaton o mean low Reynols number an mean sol volume racton). Ths approach has been extensvely use to euce computatonal rag laws or homogeneous gas sol (hgh Stokes number) suspensons by many researchers [27,28,33,34,39]. Here we use ths test to compare PUReIBM DNS results wth exstng LBM-base rag correlatons. The mean rag obtane rom PUReIBM DNS s compare wth the LBM-base rag correlaton o [34] n Fg. 1. The normalze mean lu partcle orce F s ene as F = 3πμ p W ð25þ where s the average lu partcle orce per partcle. The PUReIBM DNS results show an excellent match wth the rag correlaton o [34]. The valaton test shown here s perorme wth all the partcles at rest, so the luctuatons n partcle velocty are zero. I a ranom velocty s assgne to each partcle n the xe be accorng to a Maxwellan strbuton corresponng to a spece value o the partcle granular temperature, then the xe be smulaton can be consere an nstantaneous snapshot o a reely evolvng suspenson. O course n a reely evolvng suspenson the ynamc response o the partcles to the hyroynamc orces wll aect the partcle velocty luctuatons, an ths s not capture by the xe be smulaton. Nevertheless, ths stll allows us to conser the eect o partcle velocty luctuatons on the hyroynamc orces, albet n a lmte sense. The magntue o partcle velocty luctuatons s characterze by enng a Reynols number base on the granular temperature Re T as: Re T = p T 1 = 2 μ : ð26þ In Fg. 2 we plot the streamwse component o luctuatng acceleraton A x or each partcle versus ts luctuaton n the streamwse velocty component v x or Re m =20 an Re T =16 at a sol volume racton o 0.2. The rst observaton s that A x an v x are negatvely correlate. Ths s to be expecte because as seen rom the schematc o the low setup n Fg. 3, a postve luctuaton n partcle velocty results n a lower slp velocty that correspons to a lower rag value because o the relaton A W or solate partcles. Ths manests as a negatve luctuaton n partcle acceleraton. However, the secon nterestng observaton rom the scatter plot n Fg. 2 s that some postve luctuatons n velocty actually result n postve luctuatons n the acceleraton. In other wors, the presence o neghbor partcles an the resultng hyroynamc nteractons can occasonally volate the A W relaton or solate partcles. Also the lu velocty n the proxmty o the partcle can be sgncantly erent rom the mean lu velocty, an the enton o the nstantaneous slp as W=v u () may not accurately represent the nstantaneous slp velocty. The ont statstcs o partcle acceleraton an partcle velocty represent the couplng between hyroynamc orces an partcle velocty luctuatons. In partcular, the acceleraton velocty covarance s mportant or accurate precton o the partcle granular temperature evoluton. We now nvestgate the prectons or ont partcle acceleraton velocty statstcs usng a smple moel (ths moel s use n other works such as [10] to prect the eect o partcle velocty luctuatons Fg. 1. The comparson o the mean rag obtane rom PUReIBM smulatons wth the rag correlaton reporte by [34] at a sol volume racton o 0.2 or the baselne case o zero partcle velocty luctuatons. Fg. 2. Scatter plot o streamwse component o luctuatng acceleraton versus the streamwse component o luctuatng velocty. Square symbols ( ) show luctuatons n the partcle acceleraton obtane rom DNS usng PUReIBM smulatons, whle upper trangles ( ) show luctuatons n the partcle acceleraton precte by smple extenson o a mean rag law to ts nstantaneous counterpart.

8 64 S. Tennet et al. / Power Technology 203 (2010) Fg. 3. Schematc o the low setup. The mean velocty o the lu phase u () s recte along the postve x axs as shown. The mean velocty v o the partcles s zero an so the mean slp velocty W = v u () s along the negatve x axs. The sol partcle shown n ths gure has a postve velocty luctuaton v along the postve x axs. The schematc llustrates that a postve luctuaton about the mean velocty o the partcles mples a reuce nstantaneous slp velocty, v u () between the partcle an the lu. on mean rag). The nstantaneous counterpart o the acceleraton moel escrbe n Eq. (14), A = βw; s use to compute the nstantaneous partcle acceleraton or each partcle velocty value n the DNS. In ths moel β s taken rom the rag correlaton propose by Hll et al. [34]. The acceleraton velocty scatter plot obtane rom ths moel s also shown n Fg. 2 (upper trangles). One can see that ths smple extenson o the mean acceleraton moel oes not recover the scatter obtane n the DNS, but nstea t prects a sgncantly erent ont statstcal behavor. The ata ponts n quarants Q1 an Q3 that are oun n the scatter plot rom DNS are totally absent n the moel. Clearly ths comparson ponts to the nee or an mprove moel or the contonal partcle acceleraton n the velocty transport term n the evoluton equaton or the one-partcle strbuton uncton n the knetc theory o multphase low. Whle useul normaton regarng nstantaneous partcle acceleraton velocty ont statstcs can be extracte rom xe partcle smulatons, they are naequate to characterze the temporal evoluton o the partcle granular temperature. For ths purpose we perorm DNS o reely evolvng suspensons Freely movng suspensons DNS o a reely evolvng suspenson n peroc oman s perorme or a volume racton o ϕ =0.2. Unlke sementaton stues where the mean slp velocty s lmte by the settlng veloctyothepartclesnsuspenson,herewesolvetheequatons o moton n an acceleratng rame o reerence so that arbtrary mean low Reynols numbers Re m can be smulate. A value o Re m =20 s chosen or the smulatons reporte here, whch s well outse the Stokes regme. Three erent partcle to lu ensty ratos ( p / =10, 100 an 1000) are use to analyze the ynamcs o the system. Frst we examne the mean lu partcle rag n the reely evolvng suspenson or erent values o the partcle to lu ensty rato. The tme evoluton o the normalze rag F (c. Eq. (25)) s shown n Fg. 4(a). Fg. 4(a) shows that the mean rag n the suspenson or a partcle to lu ensty rato o 1000 vares slowly n tme when compare to the other two cases. Ths s because the partcle conguraton changes very slowly ue to hgh nerta o the partcles. Thus, when compare to the other two ensty ratos, the behavor o ths system s expecte to be much closer to that o a xe Fg. 4. (a) Shows the evoluton o the normalze mean rag at a volume racton o 0.2 an a mean low Reynols number o 20 or three erent partcle to lu ensty ratos: p / =10 (re), 100 (blue), an 1000 (purple). The black sol lne ncates the rag n a statc be at the same mean low Reynols number an volume racton. The ashe lnes represent 95% conence lmts on the mean rag or the statc be. (b) Shows the evoluton o the stanar evaton o luctuatons n the partcle acceleraton relatve to the mean rag at a volume racton o 0.2 an a mean low Reynols number o 20 or erent ensty ratos. In ths plot, ata or p / =10 are shown on the rght han se y-axs. The stanar evaton n the acceleraton obtane or a xe be s be. However, even the case wth ensty rato o 1000 s not entcal to a xe be wth zero partcle velocty luctuatons because o the changng partcle conguraton, nonzero partcle velocty luctuatons an the eect o ae mass n the hyroynamc orce. Nevertheless, t s clear that as the ensty rato ncreases the mean rag experence by the partcles n a reely evolvng suspenson s better approxmate by the corresponng xe be smulaton. The luctuatons n the partcle acceleraton play a very mportant role n the ynamcs o the suspenson as scusse earler. In Fg. 4 (b), the level o acceleraton luctuatons σ A relatve to the mean acceleraton s plotte wth tme or the three ensty ratos. It can be seen that the partcle acceleraton luctuatons are almost constant or

9 S. Tennet et al. / Power Technology 203 (2010) the suspenson wth the hghest ensty rato o The steay value o σ A / A or the case wth hghest ensty rato s very close to that obtane rom a xe assembly o partcles at the same volume racton o 0.2 an mean low Reynols number o 20. The plot o acceleraton luctuatons n Fg. 4(b) has several sgncant mplcatons. Frst o all, t tells us that the steay state value o σ A / A n reely evolvng suspensons s not neglgble. Thereore, luctuatng hyroynamc orces (relatve to the mean rag) are mportant not ust n the Stokes regme, but at moerate Reynols numbers also. Seconly, t norms us that the level o acceleraton luctuatons n reely evolvng suspensons s not very erent rom that n xe partcle assembles. Ths partally ustes the calculaton o ont acceleraton velocty statstcs rom xe partcle assembles an ther comparson wth a smple moel that was presente earler. The thr nerence we raw rom Fg. 4(b) s that the nstantaneous partcle acceleraton moel must represent the ncreasng level o temporal varatons n luctuatng hyroynamc orce that accompany a ecrease n partcle to lu ensty rato. We now quanty the eect o the luctuatons n the hyroynamc orce on partcle velocty luctuatons n reely evolvng suspensons. The evoluton o granular temperature or the three erent partcle to lu ensty rato values that are consere s shown n Fg. 5. Detals o the estmaton o granular temperature rom DNS o reely evolvng suspensons are gven n Appenx B. As expecte, the lower ensty rato cases attan a hgher steay granular temperature, an the rate at whch the steay value s reache s nversely proportonal to the partcle to lu ensty rato. The value o the scale granular temperature s relatvely low when compare wth the turbulence ntensty n sngle-phase turbulence. It ncates a hgh Mach number n the partcle phase (on the orer o 100 or a scale granular temperature o 10 4 ). Ths ncates that the partcles n the gas sol suspenson are not omnate by collsons lke molecular gases at STP, but rather they are closer to a supercoole state. For comparson, the values o granular temperature n Stokes low as estmate by the theory o Koch an Sangan [9] are 2 to 3 orers o magntue smaller than the DNS results shown here or a mean low Reynols number o Conclusons Fg. 5. Evoluton o the partcle granular temperature at a volume racton o 0.2 an a mean low Reynols number o 20 or erent ensty ratos. The couplng between hyroynamc orces an partcle velocty luctuatons n gas sol suspensons at moerate Reynols number s stue usng rect numercal smulaton o reely evolvng suspensons that mposes no-slp an no-penetraton bounary contons on the surace o each partcle. The DNS results show that luctuatons n partcle acceleraton are sgncant at moerate Reynols numbers. The stanar evaton n acceleraton relatve to the mean acceleraton ranges rom 0.2 to 0.4 epenng on the partcle to lu ensty rato. Ths extens current unerstanng o ths couplng that has been extensvely stue by Koch an co-workers n the lmt o Stokes low. Another key nng that emerges rom ths work s that the steay state granular temperature rom DNS o reely evolvng suspensons at Re m =20 s two to three orers o magntue larger than that precte by the theory o Koch an Sangan [9] or Stokes low. A smple extenson o rag laws or mean partcle acceleraton (base on the mean slp velocty) to moel the nstantaneous partcle acceleraton oes not recover the correct acceleraton velocty covarance that s obtane rom DNS. Ths work motvates the evelopment o better moels or nstantaneous partcle acceleraton that are capable o accurately representng the couplng between hyroynamc orces an partcle velocty luctuatons. Nomenclature one-partcle strbuton uncton (s 3 /m 6 ) ḟ coll source o the one-partcle strbuton uncton ue to partcle collsons (s 2 /m 6 ) v sample space varable or velocty o the partcle (m/s) x poston vector (m) x, y, z components o the poston vector x (m) v x, v y, v z components o the velocty vector v (m/s) x graent operator n poston space gven by x + y + v k z graent operator n velocty space gven by vx tme (s) + + k vy vz t,, k unt vectors n the x, y an z rectons respectvely A x, v; t Contonal expectaton o partcle acceleraton (m/s 2 ) p(x, t) lu pressure el (N/m 2 ) A uncontonal expectaton o partcle acceleraton (m/s 2 ) average lu partcle orce per partcle (N) v average partcle velocty (m/s) F p mean lu partcle rag (N) g ν mean pressure graent n the acceleratng rame (N/m 3 ) u () phasc average lu velocty (m/s) u (s) phasc average sol velocty (m/s) W mean slp velocty between the sol an the lu phases (m/s) β nterphase momentum transer coecent (s 1 ) x poston vector n the acceleratng rame (m) B external boy orce (N) W nstantaneous partcle slp velocty (m/s) V () velocty vector o the th partcle (m/s) X () poston vector o the th partcle (m) {T} granular temperature estmate rom DNS (m 2 /s 2 ) {u (s) } mean sols velocty estmate rom DNS (m/s) Δt tme step n the acceleratng rame (s) Δt coll tme step use to resolve partcle partcle collsons (s) Δt lu tme step use to resolve low el (s) δ overlap between the partcles an (m) n +1 g ν mean pressure graent at (n +1)th tme step n the acceleratng rame (N/m 3 ) u () esre mean lu velocty n the acceleratng rame (m/s) u () n+1 mean lu velocty at (n +1)th tme step n the acceleratng rame (m/s) u (s) n+1 mean sol velocty at (n+1)th tme step n the acceleratng rame (m/s) u () n mean lu velocty at nth tme step n the acceleratng rame (m/s) u (s) n mean sol velocty at nth tme step n the acceleratng rame (m/s) total rag orce actng on the sol partcles at nth tme step F Dn

10 66 S. Tennet et al. / Power Technology 203 (2010) n +1 A η n (n) V s c F n the acceleratng rame (N) rame acceleraton at (n+1)th tme step (m/s 2 ) ashpot ampng coecent n the normal recton use n the sot-sphere collson moel (N s/m) η t ashpot ampng coecent n the tangental recton use n the sot-sphere collson moel (N s/m) S F n normal component o the sprng orce between partcles an that arses n the sot-sphere collson moel (N) () F total rag orce actng on the th partcle (N) r unt vector along the lne o contact pontng rom partcle to partcle regon occupe by the nth partcle contact orce on the th partcle ue to collson wth th partcle (N) V regon o the physcal oman V regon occupe by the lu phase V s regon occupe by the sol phase Re T Reynols number base on the partcle granular temperature μ ynamc vscosty o the lu (N s/m 2 ) ν knematc vscosty o the lu (m 2 /s) total rag orce actng on the sol partcles (N) F D g pressure graent n the acceleratng rame (N/m 3 ) g luctuatng pressure graent n the acceleratng rame (N/ m 3 ) S convectve term o the Naver Stokes equatons n the acceleratng rame (m/s 2 ) t tme n the acceleratng rame (s) V bounary o the peroc box V s nterace between the sol an the lu phases (n) V s surace o the nth partcle ϕ sol volume racton Re m Reynols number base on the mean slp velocty thermoynamc ensty o the lu (kg/m 3 ) p thermoynamc ensty o the partcles (kg/m 3 ) σ A stanar evaton n the partcle acceleratons (m/s 2 ) τ vscous relaxaton tme scale (s) V n relatve velocty between the partcles an n the normal recton (m/s) p partcle ameter (m) A nntesmal area element on the surace o the sphere (m 2 ) F normalze mean lu partcle orce per partcle c v velocty probablty ensty uncton (s 3 /m 3 ) k n sprng stness coecent n the normal recton use n the sot-sphere collson moel (N/m) k t sprng stness coecent n the tangental recton use n the sot-sphere collson moel (N/m) m mass o the partcle (kg) n number ensty (1/m 3 ) N p number o partcles n the oman R mensonless partcle momentum relaxaton rate use by [8] R ss mensonless sspaton rate use by [30] S* mensonless source o granular temperature use by [8] an [30] S I source o granular energy n the lute volume racton lmt erve by [8] (m 2 /s 3 ) S II source o granular temperature n the moerate volume racton lmt gven by [30] (m 2 /s 3 ) T partcle granular temperature (m 2 /s 2 ) V volume o the physcal oman (m 3 ) V volume o the regon occupe by lu (m 3 ) V s volume o the regon occupe by the sol phase (m 3 ) u (x, t ) lu velocty el n the acceleratng rame (m/s) A partcle acceleraton luctuatons (m/s 2 ) A* moele nstantaneous partcle acceleraton (m/s 2 ) A rame acceleraton (m/s 2 ) n (n) unt normal vector pontng outwar rom the surace o the nth partcle n (s) Unt normal vector pontng outwar rom the surace o the sol u(x, t) lu velocty el n the laboratory rame (m/s) u () lu velocty exclung the rect eect o the th partcle use by [8] (m/s) v partcle velocty luctuatons (m/s) v (n) luctuatng velocty o the nth partcle (m/s) V rame velocty (m/s) ψ luctuatng pressure (N/m 2 ) Acronyms CFD computatonal lu ynamcs DNS rect numercal smulaton EE Euleran Euleran IB mmerse bounary IBM mmerse bounary metho LE Lagrangan Euleran MIS multple nepenent smulatons PDF probablty ensty uncton PUReIBM Partcle-resolve Uncontamnate-lu Reconclable Immerse Bounary Metho Acknowlegment Ths work was supporte by a Department o Energy grant DE- FC26-07NT43098 through the Natonal Energy Technology Laboratory. Appenx A. Equatons o moton n an acceleratng rame o reerence Conser a two-phase low n a nte low volume V n physcal space as an ensemble o sphercal partcles as shown n Fg. 6.Attmet,thenth partcle s characterze by ts poston vector X (n) (t) an ts velocty vector V (n) (t). A Lagrangan escrpton s use or the partcles an an Euleran escrpton s use or escrbng the moton o the lu. Denotng the velocty an pressure els o the lu by u(x, t), p(x, t) respectvely, the governng equatons o moton or the lu phase n a reerence rame E xe n space (laboratory rame) are: u x =0; ða:1þ Fg. 6. Schematc o the physcal oman. Hatche lnes represent the volume V occupe by the lu phase, sol ll represents the volume V s occupe by the sol partcle, an V =V vv s s the total volume. V an V s represent, respectvely, the areas o the computatonal box an the sol partcle.