NUMERICAL STUDY OF FORCED FLOW IN A BACK-STEP CHANNEL THROUGH POROUS LAYER

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1 Proceedngs of N h ASME Naonal ea ransfer onference Psbrgh, Pennslvana, 0 h - nd Ags, 000 N000-8 NUMERIAL SUDY OF FORED FLOW IN A BAK-SEP ANNEL ROUG POROUS LAYER Edward. han, M.A.Sc. anddae Len Fe-Sang, Asssan Professor M. Mchael Yovanovch, Professor Deparmen of Mechancal Engneerng Unvers of Waerloo, Waerloo, Onaro. NL 3G. anada. ABSRA In hs wor, we presen a nmercal solon for lamnar and rblen convecve hea ransfer n a bacward-facng sep channel hrogh a poros nser. In addon o he Naver- Soes eqaon for he fld regon, Brnman-Forchhemereended Darc s eqaon s nrodced no he nmercal solver o model he poros medm. In he rblen flow scenaros, we se a wo-eqaon - model wh wall fncon for boh he fld regon and he poros medm. he resls obaned from he presen sd concr wh esng benchmars. For he rblen flow resls, s observed ha he amon of flow ressance offered b he poros nser s more heavl dependen on s wdh han he permeabl. INRODUION he sd of flow n poros meda has receved he aenon of man nqrng mnds for more han a cenr. Darc [5] began he whole movemen wh hs hen-emprcal relaonshp. Devaons de o form drag were addressed and recfed b Forchhemer [6], whle Brnman [4] eended Darc s model for hgh-poros meda. Followng Woodng [0], who nrodced convecve effecs no he poros meda model, he governng eqaons somewha resemble he Naver-Soes eqaons. Becermann e al. [3] were amongs he frs o solve hese eqaons, sng convenonal FD echnqes, n wo dmensons. her model cold a leas predc he rends wh reasonable accrac. Vafa and en [7] noed a dscrepanc n Woodng s eqaons. Whle he Darc and Forchhemer erms were nrodced based on a volme-averaged (seepage) veloc, he convecve and Brnman componens were nspred b an nsananeos fld (pore) veloc. he also made he approprae adsmens for he wo veloc scales, based on he poros of he poros medm. Whaer [9] laer confrmed he observaon n hs formal and rgoros dervaon of Darc s law and Brnman s eenson from he Naver- Soes eqaons alone. hogh rblen modellng s sll a sbec of debae, s clear, from wors sch as Ward [8] and MacDonald e. al. [], ha rblence ess n poros meda flow. Asde from s academc vale, here are also man praccal applcaons, from combson o elecronc coolng. Mch of he research n hs area s sll n s nfanc, however. Whle here are man renowned rblence models, onl a handfl accon for he presence of poros meda. hs movaes he presen nvesgaon, allowng s o eplo oppornes for frher mprovemen. he rblen model of neres s a - model based on he formlaons b Lee and owell [0] and Anohe and Lage []. I s hen ncorporaed no he SREAM code appled o a rblen flow over a bacward-facng sep []. De o he lac of epermenal daa for a rblen flow hrogh a bacsep channel wh poros nser, valdaons were esablshed hrogh lamnar es cases from Garlng [7], Le and Mon [9], and Marn e al. [3], Per` e al. [6], and Zhang and Zhao []. he rsworhness of he presen predcons s hen nferred from hese resls. NOMENLAURE a Wdh of poros nser [m] µ onsan for rblen vscos [] oprgh 000 b ASME

2 onsan for rblen dsspaon rae prodcon [] onsan for rblen dsspaon rae desrcon [] Da Darc nmber based on channel hegh [] F Forchhemer nmber [] egh of channel [m], aresan ensor ndces[] rblen nec energ [m s - ] P Fld pressre [Pa] p rblen mean fld pressre [Pa] p rblen fld pressre flcaon [Pa] Pr Prandl nmber [] R α Rao beween effecve and fld α [] R Rao beween effecve and fld [] Re Renolds nmber based on channel hegh [] me [s] U Fld / seepage veloc [m s - ] rblen mean fld / seepage veloc [m s - ] rblen fld / seepage veloc flcaon [m s - ], aresan co-ordnaes [m] Gree Smbols Dsspaon rae of rblen nec energ [m s -3 ] Poros meda poros [] Poros meda permeabl [m ] Fld nemac vscos [m s - ] eff Effecve nemac vscos [m s - ] Edd vscos [m s - ] ρ Fld dens [g m -3 ] rblen Prandl nmber for [] rblen Prandl nmber for [] Flow Geomer he problem nder nvesgaon consss of a channel wh a bacward-facng sep a he enr regon (see Fgre.) he hegh of he bac-sep s half of ha of he channel and eends o wo channel heghs before opens o he flow. he lengh of he fll channel s long enogh o le he flow become fll developed a he ole. A poros nser mmedael follows he bac-sep o provde some ressance o he flow. / a 30 Fgre : Se-p for a bacward-facng sep channel wh a poros nser afer he sep. Nmercal resls for sch flow geomer, wh and who he poros nser, are avalable from Garlng [7], Marn e al. [3], and Zhao and Zhang [] for lamnar flow. Per` e al. [6] also performed an eensve nmercal sd for he same case ndergong rblence. Governng Eqaons Lamnar Flow For compleeness, below are he governng eqaons for ncompressble flow n he fld regon: U () 0 U U P U () U ρ In he poros meda regon, however, we shall consder Darc s Law [5], wh Brnman s [4] and Forchhemer s [6] eensons. Frher, b analog wh he Naver-Soes eqaon, convecve erms are also added o he poros meda flow eqaons. I s noeworh o menon ha whle he convecve erms, as well as Brnman s eenson, are based on he fld (pore) veloc alone, Darc s Law and he Forchhemer erm are derved sng a volme-averaged (seepage) veloc. Vafa and en [7] proposed he relaonshp below beween he wo veloc scales: U, U, (3) pore seepage In order o manan conssenc wh he fld regon, he governng eqaons n he poros meda regon are based on he seepage veloc whch s denoed as U (nsead of U,seepage ) from now on. ence, we have he followng governng eqaons for he poros meda regon: U (4) 0 U U P U F (5) U R U U U ρ where R. eff hogh he presenaon of he lamnar flow eqaons s elemenar, he form he bass for her rblen conerpars. Governng Eqaons rblence n Fld Regon We wll now consder he rblence modellng n he fld regon. he - model s chosen here de o s low compaonal reqremens. Le s frs decompose he rblen veloc and pressre felds as means and flcaons, ha s: U, P p p (6) Ne, we perform me-averagng on eqaons () and (). hs gves rse o he Renolds sress erms, whch are modelled oprgh 000 b ASME

3 3 oprgh 000 b ASME sng he concep of edd vscos. he reslng eqaons become: 0 (7) ( ) p ρ 3 (8) where s nown as he edd vscos. Meanwhle, represens he rblen nec energ, and s s dsspaon rae. he are defned as follows: µ,, (9) Sbseqenl, we reqre ranspor eqaons for and o close he problem, whch can be derved from eqaons () and (8). hrogh approprae modellng, he eqaons become: (0) () wh he sngle-vale consans below, as proposed b Lander and Spaldng [8]: 0.09 µ, 44., 9., 0., 3. () Governng Eqaons rblence n Poros Regon he man sse srrondng rblence modellng n poros meda flow s he local me-averagng process. Whle Whaer [9] derved Darc s Law and Brnman s eenson for an represenave elemenar volme, whch nfers he possbl of local me-averagng, he same canno be sad for he Forchhemer erm, as represens form drag []. In parclar, he reamen of he U U erm n he Forchhemer erm s he sbec of neres. Anohe and Lage [] perform a local me average on he Forchhemer erm self, whch laer gves rse o s nconclsve conrbon o he and eqaons. Lee and owell [0], on he oher hand, sggesed sng onl he mean veloc n he sqare-roo erm, ha s, U U. Owed o s emprcal orgn, s nceran wheher or no he me-average process has been mplcl performed on he Forchhemer s erm. ence, he Lee and owell s approach was adoped. hs resls n he followng governng eqaons for rblen flow n poros meda: 0 (3) ( ) F R p ρ 3 (4) F R (5) F R (6) Anohe and Lage [] arged ha he pore sze lmael resrans he sze of he larges edd whn he poros mar. Assmng ha he poros maeral s soropc, hs favors he hpohess of soropc rblence. hs srenghens he sppor n sng he edd vscos model. Inal, Bondar, & Inerface ondons he veloc dsrbon a he channel nle s chosen sch ha he enrance average veloc s. ang no accon he bac-sep, he followng nle profles s vald beween / and : nle, ; rblen ; Lamnar ; 4 7 (7) For he fld flow, he no-slp bondar condons are mposed n boh he channel walls and he sep. Fnall, he flow a he nerface beween he fld and he poros regon ms be connos. he presence of he Brnman s eenson and he convecve erms n he poros meda eqaon elmnaes he need for mposng an eplc nerface condon n hs case [3, 4], snce hese erms also garanee conn of he nerfacal shear sress. Nmercal Solon Mehod he presen flow solver s based on he SREAM code [], desgned for a general non-orhogonal grd wh he SIMPLE algorhm [5] for pressre correcon. Modfcaons

4 were made o accommodae rblence and poros meda models. Wall fncons were sed o appromae he near-wall behavor a hgh Renolds nmber, whls macroscopc conservaon prncples are observed a he channel ole. he second-order lnear pwnd dfference (LUDS) scheme was sed for appromang he advecon. he compaonal doman for he channel s covered b conrol volmes. he nmercal solon s deemed convergen when he magnde of he accmlaed resdal goes below 0-3. Lamnar Flow Resls here are hree benchmars avalable for he bac sep flow: Le and Mon [9], Garlng [7], and Zhang and Zhao []. he frs wo cases onl nclde fld flow, and he las one ncldes effecs of a poros nser a varos hcness and permeabl. he followng parameers were sed hrogho: Re 800, 0.97, F 0.,. Frs, we consder he reaachmen lengh who he poros nser a dfferen Renolds nmbers ( Re ). he resls are n ecellen agreemen wh he resls of Le and Mon [9] (Fgre ). Noe ha Le and Mon sed a Renolds nmber based on he mamm veloc (.e., ma 3 ) and he approprae adsmen has been made here. Ne, le s concenrae on he case where Re 800, where dealed resls ma be obaned from Garlng [7]. From Fgre 3 below, he presen resls predc he general flow paerns ver well. Garlng, however, repored ha he reaachmen lengh of he lower-wall bbble s 6., whls s onl 5.65 n he presen sd, whch agrees well wh Le and Mon's predcon [9]. Neverheless, a horzonal veloc profle a 7 and 5 showed ver good agreemen beween he wo sdes. Fgre 3: omparson of sreamraces for Re 800 beween Garlng, op, and presen sd, boom. o conclde he valdaon, we have also compared or resls wh hose of Zhang and Zhao []. From Marn e al [3], s of praccal neres o opmse he wdh of he poros nser n order o redce or even elmnae he recrclaon zone. hs, n rn, opmses beween mproved hea ransfer characerscs and head loss. In hs case, however, he re-crclaon zone serves as a benchmar for he presen flow solver. Presened here are onl cases where he recrclaon zone s promnenl feared, all a Re 800. he presen resls, shown n Fgre 4, Fgre 5 and Fgre 6, are n good agreemen wh he benchmars. / / avg 7, Garlng (990) 5, Garlng (990) 7, Presen Sd 5, Presen Sd Fgre 4: omparson of sreamwse veloc a / 7 and / 5 wh Garlng. Reaachmen Lengh ( ) Le and Mon (994) Presen Sd Re Fgre : Reaachmen lengh as a fncon of Renolds nmber for lamnar flows. Fgre 5: omparson of sreamraces for Da 0 - (op) and 0-3 (boom) beween Zhang and Zhao and presen sd for poros nser wdh a oprgh 000 b ASME

5 Fgre 9: Sreamraces nder a fll rblen flow regme a Da (a) 0 - and (b) 0-3 wh a poros nser wdh of a 0.. Fgre 6: omparson of sreamraces for Da 0 - (op) and 0-3 (boom) beween Zhang and Zhao and presen sd for poros nser wdh a 0.. rblen Flow Resls For rblen flow, we se Re o whle he oher parameers reman nchanged. For valdaon, we oban a converged solon who he poros nser and compare he resls wh Per` e al. [6], who also emploed he hgh- Renolds nmber rblen model wh wall fncons for her flow calclaon. he have repored a grd-ndependen reaachmen lengh of Or programme, on he oher hand, repored 3.6, nder he same flow condons. Gven he presen grd dsrbon, boh resls are n ver good agreemen. Fgre 7 shows he correspondng sreamraces. Fgre 7: Sreamraces nder a fll rblen flow regme wh no poros nser. he re-aachmen lengh s 3.6. Aferwards, we rensae he poros nser and eece he for cases, as we have prevosl, nder a rblen flow regme. he resls are llsraed n Fgre 8 and Fgre 9. I s noeworh o menon ha, for cases where he poros nser s hn (a 0.), he re-aachmen lengh changes ver lle, regardless he permeabl of he poros nser. he nflence of he permeabl becomes more apparen when he poros nser becomes mch hcer. he effecs, however, are no sll as far-reachng as n he lamnar case. Fgre 8: Sreamraces nder a fll rblen flow regme a Da (a) 0 - and (b) 0-3 wh a poros nser wdh of a 0.. Smmar and onclson A rblen model sng wall fncon has been developed o accon for flow hrogh a poros medm. he copled eqaons are sccessvel solved sng he SIMPLE algorhm. he resls are hen compared wh esng cases for lamnar and rblen flow. he lamnar flow resls agree eremel well wh all he presenl ced cases. As epeced, he poros nser sccessfll dampens he flow, greal shrnng he recrclaon zone. he effec s boh mmedae and noable, eher n he nser s permeabl or hcness. When he flow s fll rblen, however, he amon of ressance offered b he nser s more heavl dependen on he hcness han permeabl. In parclar, he poros nser has vrall no nflence on he flow f s hn and hghl permeable. Fre wor wll be focsed on dervng he same governng eqaons for he low-renolds nmber rblence model, as well as more eensve esng on a vare of maerals. ea ransfer resls are crrenl beng prodced. AKNOWLEDGEMENS I wold le o eend m grade o he Naral Scences and Engneerng Research oncl for s fnancal sppor. Also, man hans go o Dr. José L. Lage, from whom I drew nsghs and nspraon whle nderang hs proec. Fnall, I wsh o ae hs opporn o epress m admraon o r. Marn Ebl, who has pansangl locaed Forchhemer s orgnal paper [6] from across he Alanc. I has been a grea pleasre sdng sch a classc wor. REFERENES. Anohe, B.V., and Lage, J.L.. A General wo-eqaon Macroscopc rblence model for Incompressble Flow n Poros Meda. Inernaonal Jornal of ea and Mass ransfer, 40, pp , Bear, J.. Dnamcs of Flds n Poros Meda, Amercan Elsever, Becermann,., Vsana, R., and Ramadhan, S.. Naral onvecon n Vercal Enclosres onanng 5 oprgh 000 b ASME

6 Smlaneosl Fld and Poros Laers. Jornal of Fld Mechancs, 86, pp 57-84, Brnman,... alclaons of he Flow of eerogeneos Mre hrogh Poros Meda. Appled Scence Research,, pp 8-86, Darc,.. Les Fonanes Pblqes de la Vlle de Don, Dalmon, Pars, Forchhemer, P.. Wasserbewegng drch Boden. (Par ) Zeschrf des Verenes Descher Ingenere, 45, pp , Garlng, D.K.. A es Problem for Oflow Bondar ondons Flow over a Bacward-Facng Sep. Inernaonal Jornal for Nmercal Mehods n Flds,, pp , Lander, B.E., and Spaldng, D.B.. he Nmercal compaon of rblen Flows. omper Mehods n Appled Mechancs and Engneerng, 3, pp 69-89, Le,., and Mon, P.. Drec Nmercal smlaon of rblen Flow over a Bacward-Facng Sep, Repor F- 58, Sanford Unvers, Lee, K., and owell, J.R.. Forced onvecve and Radave ransfer whn a ghl Poros Laer Eposed o a rblen Eernal Flow Feld. Proceedngs of he ASME-JSME hermal Engneerng Jon onference,, pp , Len, F.-S., and Leschzner, M.A.. Assessmen of rblence-ranspor Models ncldng Non-Lnear RNG Edd-Vscos Formlaon and Second-Momen losre for Flow over a Bacward-Facng Sep. omper and Flds, 3, pp , MacDonald, I.F., El-Saed, M.S., Mow, K., and Dllen, F.A.L.. Flow hrogh Poros Meda Ergn Eqaon Revsed. Indsral Engneerng and hemcal Fndamenals, 8, pp 99-08, Marn, A.R., Salel,., and Sh, W.. ea ransfer Enhancemen wh Poros Insers n Recrclang Flows. ASME Jornal of ea ransfer, 0, pp , Neld, D.A., and Bean, A.. onvecon n Poros Meda, Sprnger-Verlag, Paanar, S.. Nmercal ea ransfer and Fld Flow, emsphere, Per`, M., Rüger, M., and Scheerer, G.. A Fne Volme Mlgrd Mehod for alclang rblen Flows. Sevenh Smposm on rblen Shear Flows, pp , s -3 rd Ags, Vafa, K., and en,.-l.. Bondar and Inera Effecs on Flow and ea ransfer n Poros Meda. Inernaonal Jornal of ea and Mass ransfer, 4, pp 95-03, Ward, J... rblen Flow n Poros Meda. Proceedng of he ASE Jornal of dralcs, 90, Y5, pp -, Whaer, S.. Flow n Poros Meda I: A heorecal Dervaon of Darc s Law. ranspor n Poros Meda,, pp 3-5, Woodng, R.A. Sead Sae Free hermal onvecon of Lqd n a Saraed Permeable Medm. Jornal of Fld Mechancs,, pp 73-85, Zhang, B., and Zhao, Y.. Nmercal smlaons of Fld Flow and onvecon ea ransfer hrogh Fld/Poros Laers. AIAA 33 rd hermophscs onference, AIAA , oprgh 000 b ASME