Finite element method simulation of turbulent wavy core-annular flows using a k-ω turbulence model method

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1 Last updated Decembe 6, 00. To be submtted to Int. J. Multpase flow. Fnte element metod smulaton of tubulent way coe-annula flows usng a - tubulence model metod T. Ko*, H. G. Co **, R. Ba* and D. D. Josep* *Depatment of Aeospace Engneeng and Mecancs, nesty of Mnnesota, Mnneapols, MN 55454, SA ** Scool of Mecancal and Aeospace Engneeng, Seoul Natonal nesty, Seoul 5-74, Sout Koea Abstact A numecal smulaton of way coe flow was caed out by Ba, Kela and Josep. (996). Tey calculated te ntefacal wae sape fo lamna flow. In ou pesent smulaton, te SST (sea stess tanspot) tubulence model s used to sole te tubulent netc enegy and dsspaton ate equatons and a splttng metod s used to sole Nae-Stoes equatons fo te wae sape, pessue gadent and te pofles of elocty and pessue n tubulent way coe flows. Te waelengt deceases wt Reynolds numbe R and wt te olume ato η. Te pessue gadent nceases wt Reynolds numbe R and wt te olume ato η. Hg pessues ae geneated at a stagnaton pont leadng to wae steepenng, wle low pessues ae geneated at a eattacment pont. Te computed wae sapes and fctonal losses ae n satsfactoy ageement wt epements and geatly mpoe on peous esults.. Intoducton Wate lubcated ppelnng s a metod of tanspotng gly scous fluds at low cost. A scous flud foms a coe suounded and lubcated by a wate annulus. Te wate educes te sea stess on te wall of te ppe. An mpotant sees of epements on te wate lubcated ppelnng was caed out by Russell and Cales (959), Russell, Hodgson and Goe (959), Cales (960) and patculaly by Cales, Goe and

2 Hodgson (96). Glass (96) and otes found tat te lowest pessue gadent was aceed wen te nput ato of wate to ol was between 30% and 40%. Ote epements on wate lubcated tanspotaton n ozontal ppes wee epoted by Sten (978) and Olemans, Ooms, Wu and Duyestn (985). Ooms, Segal, Van de Wees, Meeoff and Olemans (984) and Olemans and Ooms (986) ted to use lubcaton teoy to analyze te case of a ey scous, way eccentc coe-annula flow fo lamna case. Tey sowed tat t geneated a buoyant foce popotonal to te fst powe of te elocty to balance te gaty. In te study, te sape and ampltude of te wae must be gen as empcal nputs. Olemans, Ooms Wu and Duÿestn (987) also ted to etend te esults to te tubulent case but agan te teoy eques te nputs of wae foms and waelengts, wc ae ade to dentfy n te tubulent flows. Te tubulence model undepedcts te aaton of te pessue gadent wt te elocty of coe, een toug wae ampltudes and waelengts obseed toug te epement ae used as nput data. Bentwc (964) studed fo te lamna case wt te basc coe-annula model wc te geneatos of te coe and annulus ae goously paallel and te coss sectons of te ppe and coe ae ccula saped, but te centes of tem do not concde. He soled te Posson equaton tat goens n te eccentc coe flow model wt a Foue sees n bpola coodnates. Howee, e dd not use s soluton to ealuate te fcton facto o old-up ato. Huang, Cstodoulou and Josep (993) studed lamna and tubulent coe-annula flow to assess te effects of eccentcty and te olume ato on te fcton facto and oldup ato. Tey used a model of coe-annula flow n wc te ol coe s a pefect cylnde wt geneatos paallel to te ppe wall, but off-cente and adopted a standad -ε model wt a low Reynolds numbe capablty fo tubulent case. Ba, Kela and Josep (996) aceed te mpotant numecal calculaton of te ntefacal wae sape fo lamna flow. Te waelengt and ampltude wee calculated by solng te nomal stess balance on te nteface of te wae. Te deeloped waes

3 ae asymmetc wt steep slopes nea te g-pessue egon at te font face of te wae cest and sallowe slopes nea te low-pessue egon at te lee sde of te cest. In ts pape, we etend te numecal smulaton of asymmetc lamna coe-annula flow to tubulent case by adoptng te SST (sea stess tanspot) - model poposed by Mente (994). Te waelengt and ampltude ae obtaned by usng te nomal stess balance followng Ba, Kela and Josep (996). A splttng metod wt lnea equalode fnte element metod poposed by Co, Co and Yoo (997) s used as a soluton algotm and a consstent SPG (steamlne upwnd Peto-Galen) deeloped by Boos and Huges (98) s adopted as a stablzng tecnque fo conecton domnated flows. sng ts code, te sape of nteface, pessue dstbuton and seconday flow motons ae analyzed fo tubulent coe-annula flow.. Goenng Equaton Consde two concentc mmscble fluds flowng down an nfnte ozontal ppelne. We assume tat te coe s asymmetc wt ntefacal waes tat ae peodc along te flow decton. Patana et al. (977) decomposed te pessue n peodc fully deeloped flows as P (, ) β p(, ), (-) wee β s a mean pessue gadent and p(,) epesents te peodc pat of te wole pessue P and beaes n peodc fason fom module to module. Te tem β accommodates te geneal pessue dop along te flow decton. Te contnuty equaton and Nae-Stoe equaton fo te unsteady ncompessble flow n cylndcal coodnates can be wtten as follows: Contnuty Equaton ( u) ( ) 0, (-) 3

4 4 -momentum u p u u u t u eff β u eff 3, (-3) -momentum u p u t eff eff eff 3. (-4) Te tubulent netc enegy equaton and te dsspaton ate equaton ae obtaned fom Mente s sea stess tanspot model (F. R. Mente 994). Te SST model utlzes te ognal - model of Wlco n te nne egon of te bounday laye and swtces to te standad -ε model n te oute egon of te bounday laye and n fee sea flows. In te dsspaton ate equaton, te functon F s desgned to be one n te nea wall egon (actatng te ognal model) and zeo away fom te suface (actatng te tansfomed model). Ten te tubulent netc enegy and te dsspaton ate equaton modfed by SST model ae wtten as Tubulent Knetc enegy ( ) P u t T σ ( ) β σ T *, (-5) Dsspaton ate equaton ( ) β γ σ t T P u t ( ) ( ) F T σ σ ( ) F σ, (-6)

5 5 wee u u P T, eff T. Let φ epesent a constant n te ognal - model (σ, ), φ a constant n te tansfomed -ε model (σ, ). Te coespondng constant φ of te new model (σ, ) s gen as follows: ) ( Φ Φ Φ F F. (-7) All constants, as well as te functon F, ae gen n te Append A. 3. Numecal Metod 3.. Fou-step factonal metod Te fully mplct fou-step factonal fnte element metod (Co et al. 997) s used to ntegate n tme te contnuty equaton Eq. (-) and te momentum equatons Eq. (- 3) and (-4). In ts appoac, pessue s decoupled fom tose of conecton, dffuson and ote etenal foces. Te fully mplct fou-step factonal metod s wtten as follows: ( ) n n n n n n S p t τ τ ˆ ˆ ˆ ˆ (3-) n p t ˆ * (3-) n t p * (3-3) n n p t * (3-4) wee t s te tme ncement, ˆ and * ae ntemedate eloctes, and supescpt n denotes te tme leel. In te pocedue, te ntemedate elocty does not necessaly

6 satsfy te contnuty equaton. At te fst step, te ntemedate ˆ s obtaned by usng eloctes, pessue and etenal foces calculated on te peous tme step. Hence, at te net step, te ntemedate elocty s coected by te pessue and te pessue s obtaned fom te contnuty constant. Te fact tat te pessue s decoupled fom te elocty n te factonal step metod was utlzed n te fnte element analyss of te ncompessble Nae-Stoes equatons by seeal eseaces, wo poed tat te factonal step metod can be successfully appled to te fnte element analyss. Ts appoac s moe accuate tan te SIMPLE algotm based fnte element metod fo te same gd, because te factonal step metod does not nclude any appomaton pocedue. Ts compason s found n H. G. Co, H. Co and Yoo (997). 3.. Galen fnte element dscetzaton Te tanspot equatons of momentum, tubulent netc enegy and dsspaton ae dscetzed usng a consstent steamlne upwnd Peto-Galen metod and te pessue equaton usng a Galen metod. Te wea fomulatons of te goenng equaton ae deed by multply tem by a coespondng wegtng functon and ntegatng oe te spatal doman of a poblem. Te wea fomulatons of goenng equaton can be wtten as follows: Fnd (, t) H, (, t) H and (, t) H suc tat Ω Ω p S w w dω dγ 0 τ t w Γ w H, w 0, Γ b, (3-5) Γ * w w dω dγ 0 P β t w Γ w H, w 0, b Γ Γ, (3-6) 6

7 7 ( ) Ω Ω d T w F P w σ β γ 0 d Γ Γ w t, w H 0 Γ w, b Γ. (3-7) Consdeng tat Galen metod coesponds to a cental fnte dffeence fomulaton, a conecton-domnated poblem can not be soled effectely wtout usng a ey dense gd. One of te metods to oecome ts poblem s te SPG sceme n wc te wegtng functon s dffeent fom te tal functon. Te SPG fomulaton fo momentum equaton s stated as follows: Fnd H t ), ( suc tat,,, Γ Ω Γ Ω d w t d w S p w σ 0 ~,,, Ω Ω e e N e d S p p σ, H w, 0 Γ w, b Γ, (3-8) wee w c p, ~ s a petubaton wegtng functon. Ts s deed fom a tensoal atfcal dffuson n a multdmensonal space wc acts along te flow decton. In te pesent study, te petubaton wegtng functon s appled to all tems n te momentum equaton by te consstent SPG metod. Te coeffcents of te petubaton wegtng functon ae gen n te Append B. Te Posson type pessue equaton s obtaned fom te contnuty constant. sng te degence teoem, te wea fomulaton of contnuty equaton can be epessed as follows: Γ Ω Γ Ω d n w d w n n,. (3-9) Ten, nsetng Eq. (3-4) nto te left and sde of Eq. (3-9), te followng Posson type pessue equaton s obtaned as follows:

8 t Ω w, p n, * n dω w dω w n dγ. (3-0) Ω, Γ Wen used wt a factonal 4-step metod, te pessue equaton Eq. (3-0) as an adantage n teatng te outflow bounday condton, because te unnown appomated by te nown ˆ toug Eqs. (3-) and (3-4). n s well Te soluton at steady state s sougt toug tme macng of te coespondng unsteady goenng equaton. Teefoe, te alue of / t can be consdeed as an netal elaaton facto of steady SIMPLE algotm. (S. V. Patana 980) 3.3 Calculaton of te ntefacal wae sape Fo way coe-annula flow smulatons wt gly scous fluds, te scosty of te coe lqud s muc geate tan tat of annula lqud. Te flow of ol can be egaded as a ceepng moton on te fowad moton of a gd coe. Te slow seconday moton sould not ae a geat effect on te oeall dynamcs. Teefoe, te coe flow s assumed to be sold wt standng waes on te nteface. Te nomal stess equlbum condton on te nteface s wtten as follows: ( P Hσ ) n D n 0, (3-) and te sea stess equlbum condton s wtten as follows: t D n 0, (3-) wee D T, () (), te subscpt and ndcate te coe and annulus espectely, H s te sum of te pncpal cuatue, σ s te coeffcent of ntefacal tenson, nn s te unt nomal ecto fom lqud to and t s te unt tangent ecto. Te scous pat of te nomal stess condton Eq. (3-) on te nteface anses: 8

9 n n n D n 0, (3-3) n n snce s s n n 0, (3-4) and on te nteface s 0. (3-5) s Teefoe, te nomal stess condton on te nteface can be ewtten as follows: P Hσ. (3-6) Te peodc pessue equaton Eq. (-) s ewtten fo eac coe and annulus as follows P β C p P β p(, y) C p (3-7) (3-8) wee C p s constant. Hence, te nomal stess balance at te nteface Eq. (3-6) s epessed n Ba et al (996) as follows: σ df f d d f σ d df d 3 P P P, (3-9) wee f f() s te egt of nteface fom cente lne. In ode to sole ou way coe-annula poblem fo a gen wae speed, we must compute β. te pessue dop n one wae s defne as βl, wee L s lengt of one wae. Te βl may be epessed by te sea stess actng on te wall as follows: L 0 β LA F πr τ d, (3-0) wee A s te aea of te coss secton of ppe. w 9

10 4. Numecal Results 4.. Valdaton of tubulent code 4... Te fully deeloped ppe flow To efy te pesent tubulent code, fully deeloped Poseulle flow n ppe s calculated at aous Reynolds numbes 00 ~ 40,000. Te Reynolds numbe s defned D as Re, wee s te mean elocty and D s te damete of te ppe. Te lengt of calculaton doman s long enoug to get te fully deeloped pofle fo te elocty, pessue and netc enegy. Te bounday condtons at sold sufaces ae gen as follows: u υ 0, 6 β ( y, (4-) ) wee y s te dstance to te net gd pont away fom te wall. Te Neumann u bounday condton 0 s appled fo u, and and te Dclet bounday condton 0 s appled at et. Te bounday condtons at nlet ae gen as follows: n u n, υ 0,, 0 t, t. (4-) L Te fcton facto s computed fo aous Reynolds numbes fom 00 ~ 40,000 n te fully deeloped ppe flow and sown on Fgue 4-. Note tat te pesent numecal metod ges accuate esults fo te tubulent flow as well as fo te lamna flow. Fo lamna flow, te computed esults of fcton facto satsfy te Hagen-Poseulle equaton λ 64/Re. Te esults of tubulent fcton facto ae epesented by Blasus coelaton λ 0.36/Re 0.5. Fgues 4- (a) and (b) sow te elocty pofle and te pofle of tubulent netc enegy of a fully deeloped ppe flow fo Re 40,000. Te esults obtaned fom te pesent tubulent code ae compaed wt te esults of Wlco s ognal - model and 0

11 te epemental data of Laufe (95) fo Reynolds numbe based on ppe damete and aeage elocty of 40,000. Numecal Result λ 0. λ 64/Re λ 0.36/Re Re Fgue 4-. Te fcton facto s. Reynolds numbe n te fully deeloped ppe flow Hagen-Poseulle equaton fo lamna flow; Blasus coelaton fo tubulent flow. Fom te compason wt te esult of Wlco s - model and epemental data fo te elocty pofle and te tubulent netc enegy pofle, we can see tat tat te pesent code usng te Mente s SST model and te SPG metod ges moe accuate esult tan te ognal - model n fully deeloped ppe flow. Te pofles obtaned fom ou smulaton at egon away fom te wall ae moe accuate alues tan nea te wall.

12 Y/ R Laufe's epemental data Ou calculaton Wlco's -w Model / ma Fgue 4- (a). Te compason of computed and measued elocty pofle n te tubulent ppe flow at Re 40,000. Ou esult s close to te epemental data tan Wlco s - model, patculaly n te egon away fom te wall. Y / R 0.5 Laufe's epemental data Ou calculaton Wlco's -w model / u τ Fgue 4- (b). Compason of computed and measued pofle of tubulent netc enegy n te tubulent ppe flow at Re 40,000. R s a adus of ppe and u τ. τ w

13 4... Te way coe-annula flow Wt te assumptons and equatons descbed n te secton 3.3, te wae sape of coe flow and te pofle of annula flow ae computed fo gen wae speed c, wate flow ate Q w and aeage ol adus R. In ou pesent smulaton, te wall moes wt coe elocty c opposte te coe; n ts fame te coe s standng. Ten, we calculate te flow pofles n te annula egon. Te bounday condtons at wall ae gen as follows: u c, 0, 6, (4-3) υ β ( y) and te ones at te nteface ae gen as a statonay sold bounday condton Eq. (4-). Befoe te flow feld n te annulus s calculated, we assume a fee suface sape aound a gen aeage coe adus R. Dung eac teaton of te flow-feld, te pessue gadent β s adusted to satsfy te foce balance n one waelengt. sng te pessue on te suface obtaned at te peous step, te sape of suface s calculated by solng te nomal stess condton Eq. (3-9). Te waelengt s adusted n eey teaton n ode to get a conegng suface sape fo te gen aeage coe adus R and wate flow ate Q w. Te new nteface detemned at te peous step s used n computng te flow feld agan. Ten, tese steps ae epeated untl te solutons ae coneged. Te mes and bounday condtons fo te elocty ae sown n Fg We defne te Reynolds numbe as [see Aney, Ba, Gueaa & Josep et al. (993) fo a dscusson of R ] VD R 4 4 [ η ( m ) ] Re[ η ] ( m ), (4-4) Q Q R wee V s te mean elocty, η s te coe facton and πr R scosty ato. m s te 3

14 u c, 0.0 Wate Ol u 0.0, 0.0 Ol Fgue 4-3. Te mes of calculaton doman as te body ftted and stuctued mes and te bounday condtons at te wall and nteface ae gen as te sold bounday condton. Te wall moes wt coe elocty c and te coe s standng. Fgue 4-4 sows te elocty pofle n te annulus fo lamna flow and tubulent flow. Wen te wall moes and te flow s den by pessue, te elocty pofle on te cest of wae fo R 6700 (Fg. 4-4 (b)) as te typcal S-sape epected fo tubulent Couette flow. Fo R 000 (Fg. 4-4(a)), te pofle as te full sape epected fo lamna Couette flow wt a poste pessue gadent..3.3 (cm).. (cm) (cm) (a) R (cm) (b) R 6700 Fgue 4-4. Te compason of te elocty pofle of annula flow fo lamna flow ( R 000) and tubulent flow (R 6700). Hee, we ntoduce te old-up ato as a dmensonless paamete fo coe-annula flow. Te old-up ato s defned as te ato Q /Q of olume flow ate to te ato V /V of olume n te ppe Q Q Q Q c, (4-5) V V R R R c ( ) 4

15 Q wee c s te wae speed fo gd coe flow and πr c Q s te aeage π ( R R ) elocty of annula flow. Te dmensonless nput paametes R, and η eplace te dmensonal nput paametes Q w, c and R n te pesent smulaton. Te defnton of te dmensonless paametes L*, p* and β* ae defned as follows L* dmensonless wae lengt L / R, (4-6) βr β* dmensonless pessue gadent, (4-7) p p* {p(f(),)-p (f(l),l)} R. (4-8) To aldate te pesent code fo coe-annula flow, ou esult fo lamna code usng te fnte element metod s compaed wt te esult fom Ba et al.(996) based on te fnte dffeence metod. Te waelengt and pessue gadent s calculated fo te case wt.4 and η 0.8. To compae te waelengt and pessue gadent obtaned by solng te tubulent equatons fo g Reynolds numbe wt esults obtaned fom te typcal dect numecal smulaton wtout usng any tubulent model, te smulatons ae done by two appoaces wt same gd ponts. Fgues 4-5 and 4-6 sow ow te waelengt and te pessue gadent ay wt R fo fed.4 and η 0.8. Te waelengt deceases wt R and te pessue gadent nceases wt R fo fed and η. Fom tese fgues, we can note tat, fo g Reynolds numbes, te esults obtaned fom te tubulent two-equaton model (dotted lne) dffe geatly fom a dect numecal smulaton of lamna flow usng same gd ponts. Fo ge Reynolds numbes, te waelengt s sote and te pessue gadent s lage fo te tubulent code tan fo te lamna code. Te pesent lamna code ges esults close to te data fom Ba et al. (996). 5

16 4 Lamna code Tubulent code Ba et al. (996) 3 L* Fgue 4-5. Te dmensonless wae lengt L* s. Reynolds numbe R at.4 and η 0.8 R.5E06 Lamna code Tubulent code Ba et al. (996).0E06 β 5.0E05 0.0E Fgue 4-6. Te dmensonless pessue gadent β* s. Reynolds numbe R at.4 and η 0.8 R 6

17 Table 4. compaes te computed alue of te waelengt wt measued data fom epements fo aous R and a fed.4 and η It s appaent tat te tubulent code ges moe accuate esults at g Reynolds numbes. In te epement fo ozontal way coe flow, te nput s Q and Q and te alues of R, and η fo aous Q and Q ae obtaned fom mage pocessng and Eq. (4-5). Te moes of coeannula flow ae ecoded by usng a Koda EtaPo EM g-speed deo camea tat taes at ate up 000 fames pe second. Te moes ae played and analyzed fame by fame wt a compute. Table 4.. Compason of measued and computed alues of waelengt at.4 and η Reynolds numbe Dmensonless Epements Waelengt Lamna code ( L*) Tubulent code Eo (%) Lamna code Tubulent code Een toug te computed alues fom tubulent code and measued alues of te waelengt ae not te same, we can see te same tend n bot; te waelengt s a deceasng functon of R. Te alues obtaned fom te computaton usng te tubulent model ae close to te alues of epement tan te alues fom te lamna models. Wle te eo fo lamna code nceases to aound 50% fo ge Reynolds numbe, te eo fo te tubulent code deceases to less tan 8%. We can nfe tat te eo comes fom te assumptons as gd coe and a-symmetc coe flow wt zeo densty dffeence between coe and annulus. In te epemental measuement, te coe s slgtly lgte tan te annulus and as a lmted scosty een f t s so lage tan one of wate. Teefoe, te coe flow s off-cente and defomable. 7

18 4.. Numecal esults We study ow te waelengt, pessue gadent, pessue dstbuton on te nteface and wae sape ay wt R and η. In te etcal ppelne studed n te epement of Ba, Cen and Josep (99), te old-up ato s about.39 ndependent of te nput flow ate Q and Q. In te numecal smulaton of te way coe flow fo lamna flow by Ba, Kela and Josep (996), tey computed many esults fo.4. We also compute fo fed.4 to compae ou esults wt tes. In ou computaton, we coose te actual pyscal paamete n way coe flow of wate n a one nc damete ppe, 0.0 pose, g/cm 3 and σ 6 dyne/cm..3.. (cm) (cm) (a) R (cm) (cm) (b) R 5000 Fgue 4-7. Te compason of te pessue pofle and steamlne of annula flow fo lamna flow ( R 000) and tubulent flow ( R 5000). Te da colo ndcates a low pessue and te lgt one ndcates a g pessue. 8

19 Fgue 4-7 sows te seconday motons and pessue dstbuton fo R 000 and R Te flow n te annulus s composed of a stagt flow and an eddy. Te low pessue at te bac of te cest of wae s assocated wt te eattacment pont, te g pessue at te font of te cest of wae s elated to a stagnaton pont. We note tat te aea of te eddy fo tubulent flow s smalle tan fo lamna flow due to nceasng of te momentum tansfe fom man flow n tubulent flow. Fgue 4-8 sows te waelengt and te wae sape fo aous alues of R at.4 and η 0.8. Wen te Reynolds numbe nceases, te wae steepens at te font of te cest and te waelengt s deceased. 0.5 R 300 R 700 R 4000 R 6700 R 9400 R Fgue 4-8. Te wae sape fo dffeent Reynolds numbes at.4 and η 0.8. Fgues 4-5 and 4-6 sow ow te waelengt and te pessue gadent ay wt R fo any fed old-up ato and olume ato. Fgue 4-9 sows te pessue dstbuton on te ol-wate nteface. Te poste pessue pea appeas at te stagnaton pont and te negate pessue pea at te eattacment pont. Fo 0.3 < */L* < 0.7, te pessue on te nteface nceases wt R fo g R wle fo low R te pessue does not cange. Fgue 4-0 sows ow te waelengt and pessue gadent ay wt η fo fed.49 and c 5 cm/sec. Note tat te waelengt deceases wt η and te pessue gadent nceases wt η. 9

20 .5E06 Re 300 Re 3300 Re E06 p* 5.0E05 0.0E E05 X* / L* Fgue 4-9. Te dmensonless pessue on te nteface at R 300, 3300 and Te oldup ato s fed at.4 and te olume ato η s fed at E04 Dmensonless waelengt Pessue gadent 3.5E04.5 L* 3.0E04 β.5e η.0e04 Fgue 4-0. Te dmensonless waelengt L* and pessue gadent β s. η at.49 and te wae speed c 5 cm/sec. 0

21 To aldate te tubulent code fo coe-annula tubulent ppe flow, te pessue gadent β fo aous dametes of ppe was computed and compaed wt te Blasus fomula fo tubulent ppe flow. Fom te elatonsp between te pessue gadent β and te sea stess τ w on te ppe walls, Josep, Ba, Mata, Suy and Gant (998) obtaned an epesson fo te pessue gadent β{kpa m - 7 ], n tems of te t powe 4 of te elocty [m s - 5 ] to te t powe of te ppe adus Ro [m], namely β K (4-9) 5 4 Ro Ro wee s an unnown constant ( 0.36 fo wate alone) Damete.7 cm Damete.54 cm Damete 5.08 cm Blasus coelaton β V.75 /R V R fo aous ppe adus at te coe- Fgue 4-. Te pessue gadent β s. annula tubulent ppe flow. In ou smulaton, ppe adus R o s eplaced by R and elocty by te mean elocty V. Tese cues fo aous ppe adus (0.5, 0.5 and ) ae sown n fgue

22 4-. Tese cues collapse to a sngle cue wt te alue 0.37 wc s almost same wt te Blasus alue fo wate alone ( 0.36). Fom ts esult, we can wte te elaton of te pessue gadent fo coe-annula flow wt an nfntely scous coe and fo wate alone flow as follow β CAF wt gd coe.0 * β wate alone. (4-0) Te pessue gadents fo epemental coe-annula flow wt defomable coe s bgge tan calculated pessue gadent wt gd coe. It can be sad tat way coe flow of an nfntely scous coe n tubulent wate can be tanspoted as ceaply as wate alone. Inceased costs due to seconday motons n te scous coe and to foulng of te ppe wall ae pesently unde study. 5. Concluson Te pesent code usng te sea stess tanspot model and te steamlne upwnd Peto-Galen metod ges moe accuate esults tan te ognal - model n fully deeloped tubulent ppe flow. Fo te tubulent way coe-annula flow wt fed old-up ato and olume ato, te waelengt obtaned fom te tubulent code s close tan te lamna code to te alues of epement. Te pessue gadent n tubulent flow nceases moe saply wt Reynolds numbe tan te pessue gadent computed wt te lamna code. Fo te tubulent coe-annula flow, te aea of te eddy n te annulus s smalle tan fo te lamna flow due to nceasng of te momentum tansfe fom man flow. Te wae steepens moe at te font of te cest and te waelengt deceases moe tan n lamna flow. Wle te waelengt deceases wt Reynolds numbe at fed old-up ato and olume ato, and te waelengt deceases wt olume ato at fed old-up ato and te wae speed, te pessue gadent as an opposte tend fo te same cases.

23 Te pessue gadent calculated fom tubulent code satsfes wt Blasus fomula n te tubulent coe-annula flow wt a gd coe; te pessue gadent s a lnea functon of te 4 7 t powe of te mean elocty to te 4 5 t powe of te ppe adus and te constant s close to te alue fo wate alone tubulent ppe flow. Acnowledgements Ts wo was suppoted by te DOE (Engneeng Reseac Pogam of te Depatment of Basc Enegy Scences), te NSF/CTS unde Gant Oppotuntes fo Academc Lasons wt Industy and te Mnnesota Supecompute Insttute. Append A. Te coeffcents of Sea-Stess Tanspot Model Te constants of set (φ ) ae σ 0.85, σ w 0.5, β 0.075, a 0.3, κ 0.4, γ * * β β σ κ β (A-) Te constants of set (φ ) ae σ.0, σ w.856, β 0.088, β 0.09 γ * * β β σ κ β (A-) F s gen by F tan(ag 4 ) 500 4σ ag mn ma ; ; 0.09y y CD y (A-3) (A-4) wee y s te dstance to te net suface and Cd w s te poste poton of te cossdffuson tem of Eq. (-6) 0 CD ma σ ; 0. (A-5) And te eddy scosty s defned as 3

24 t a ma( a ; Ω F ) (A-6) wee Ω s te absolute alue of te otcty. F s gen by F tan(ag ) ag ma 0.09y 500 ; y (A-7) (A-8) Append B. Te coeffcents of te petubaton wegtng functon c z u e, (B-) u e wee z cot(pe) /Pe mn[, Pe/3] Pe element Peclet numbe ( u e e ) ν u e elocty at an element cente u -component elocty at an element cente e element caactestc lengt [ ] Refeence Aney, M.S., Ba, R., Gueaa, E., Josep, D.D., Lu, K., 993. Fcton facto and oldup studes fo lubcated ppelnng-i. Epement and coelatons. Int. J. Mult. Flow 9, Ba, R., Cen, K., Josep, D.D., 99. Lubcated ppelnng: stablty of coe-annula flow. Pat 5: epements and compason wt teoy, J. Flud Mec. 40, Ba, R., Kela, K., Josep, D.D.,996. Dect smulaton of ntefacal waes n a g scosty ato and asymmetc coe-annula flow. J. Flud Mec. 37, -34. Ba, R., Josep, D.D., 000. Steady flow and ntefacal sapes of a gly scous dspesed pase. Int. J. Mult. Flow 6, Bentwc, M Two-pase scous aal flow n a ppe. J. Bas. Engng,

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26 Olemans, R.V.A., Ooms, G., 986. Coe-annula flow of ol and wate toug a ppelne. In: Hewtt, G.F., Delaye, J.M., Zube, N. (Eds.), Multpase Scence and Tecnology, Vol.. Hemspee Publsng Copoaton. Ooms, G., Segal, A., Van de Wees, A.J., Meeoff, R., Olemans, R.V.A., 984. A teoetcal model fo coe-annula flow of a ey scous ol coe and a wate annulus toug a ozontal ppe. Int, J. Mult. Flow 0, Olemans, R.V.A., Ooms, G., Wu, H.L., Duÿestn, A., 987. Coe-annula ol/wate flow: te tubulent-lubcatng-flm model and measuements n a 5cm ppe loop. Int. J. Multpase Flow. 3, -3. Patana, S.V., Lu, C., Spaow, E.M., 977. Fully deeloped flow and eat tansfe n ducts ang steamwse-peodc aatons of coss-sectonal aea. Tans. ASME J. Heat Tansfe. 99, Patana, S.V., 980. Numecal Heat Tansfe and Flud Flow. Hemspee Publsng Copoaton. Pezos, L., Cen, K., Josep, D.D., 989. Lubcated ppelnng: stablty of coeannula flow. J. Flud Mec. 0, Russell, T.W.F., Cales, M.E., 959. Te effect of te less scous lqud n te lamna flow of two mmscble lquds. Can. J. Cem. Eng. 39, 8-4. Russell, T.W.F., Hodgson, G.W., Goe, G.W., 959. Hozontal ppelne flow of mtues of ol and wate. Can. J. Cem. Eng. 37, 9. Wlco, D.C., 988. Reassessment of te sacle-detemnng equaton fo adanced tubulence models. AIAA Jounal. Vol. 6, Wlco, D.C., 998. Tubulence modelng fo CFD, nd ed. DCW Industes, Inc. 6