EFFECTS OF WALL CURVATURE ON TURBULENT HEAT TRANSFER IN CURVED PIPE FLOW

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1 ROE g Poiie Fance EFFECS OF WLL CURVURE ON URULEN HE RNSFER IN CURVED PIPE FLOW Cangwoo Kang and Kyng-Soo Yang Depamen of Mecanical Engineeing Ina Univeiy Inceon 4-7 Koea (EL) (FX) ( ) kyang@ina.ac.k SRC Diec nmeical imlaion (DNS) of flly-developed blen cved pipe flow a been pefomed o inveigae e effec of wall cvae on blen flow and ea anfe. We conide a flly developed blen cved pipe flow wi axially nifom wall ea flx. e Reynold nmbe nde conideaion i Re τ = baed on e mean ficion velociy and e pipe adi and e Pandl nmbe i.7. e mean velociy pofile and blen ineniie obained fom e peen DNS ae in good ageemen wi e pevio nmeical and expeimenal el cenly available. e mean qaniie and vaio blence aiic ae peened fo flow and empeae field. INRODUCION blen pipe flow a been aacing eeace' aenion fo decade de o i ig applicabiliy o many engineeing device c a ea excange cemical eaco powe-plan piping yem o name a few. In paicla blen caaceiic of flid flow and ea anfe ave been inenively died becae complee ndeanding of em can lead o a ignifican impovemen of anpo efficiency in c device o o an effecive way of edcing flow-acceleaed eoioncooion (Enaye e al. 98; Sdo e al. 998 ). lage volme of wok on blen aig-pipe flow can be fond in e lieae inclding nmeical a well a expeimenal die. Mo of e nmeical inveigaion wee caied o by Diec Nmeical Simlaion (DNS) o Lage Eddy Simlaion (LES) and ome eliable daa bae wee obained (Unge and Fiedic 99; Eggel e al. 994; den oonde and Niewad 997; Wagne e al. ; Feiz e al. 3; W and Moin 8). Howeve a wa no e cae fo blen cved-pipe flow even og cved pa ae widely ed in a moden piping yem o ave pace and alo o enance cala anpo via e econday flow inced by e wall cvae. In e pa die on cved-pipe flow wee mainly done by expeimen (dle 934; Io 99; Moi and Nakayama 96) o by eoeical analyi (Collin and Denni 97; Wang 98; Denni and Ng 98; Gemano ). One can efe o ege e al. (983) and Io (987) fo an exenive eview on e bjec. Recenly moe aenion a been paid o blen flow in a cved pipe and deailed expeimenal meaemen (Webe and Hmpey ) a well a fll ee-dimenional (3D) imlaion ing DNS/LES (oema and Niewad 996; oema 997) ave been pefomed. Hül e al. (999) died e effec of wall cvae on lamina pipe flow by ing 3D imlaion and beqenly Hül and Fiedic ( ) compaed ei DNS el on e aveage velociy and velociy flcaion in e flly developed cvedpipe flow a Re τ =3 wi oe in e aig-pipe flow. fa a ea/ma anfe in cved-pipe flow i concened eeac a been foced on e effec of e econday flow indced by e wall cvae on e cala anpo. Moi and Nakayama ( ) eoeically comped eiance coefficien and ea anfe coefficien in lamina and blen cved-pipe flow and owed a ei el ae well conien wi e expeimenal meaemen (Pa 947; Mainelli 947; Seban and McLaglin 963; Roge and Mayew 964). Mo of e following nmeical inveigaion (kiyama and Ceng 97; Kalb and Seade ; Paanka e al. 974; Yao and ege 978; Zapyanov e al. 98; Pa and Yao 98) oweve mainly died coelaion beween Re (o P) and ea anfe ae. o e ao' be knowledge indep die on blence aiic of nea-wall cala flcaion ae ae. ey ae believed o ignificanly affec local ea/ma anfe ae on e pipe wall. e aim of e cen inveigaion ae o elcidae e neawall caaceiic of blen ea anfe in fllydeveloped cved-pipe flow by ing DNS and alo o idenify e cvae effec on e momenm and cala anpo by compaing o el wi oe of blen aig-pipe flow. GOVERNING EQUIONS e coodinae yem ed in e cen dy a ooidal yem i peened in Fig.. ggeed by Wang (98) and Gemano (98 989) an oogonal cvilinea coodinae yem i inodced in conjncion wi a Caeian coodinae yem. y ing e

2 g Poiie Fance ROE cvilinea coodinae fo e adial diecion θ fo e cicmfeenial diecion and fo e axial diecion a Caeian poiion veco x can be expeed a x=p O =R() + co θ N() + in θ () () wee R epeen a local poin on e cene cve. N and ae e angenial nomal and binomal ni veco. Uing e elaion d dr d d N N N d d () wee κ i e cvae e oogonal meic of i yem i given by co d d d dx dx (3) wee d dθ d ae e infinieimal incemen in e adial cicmfeenial and axial diecion epecively. Wi i meic one can obain e cale faco θ and. (acelo 97) = θ = = + κ coθ (4) Wi ee cale faco we can deive e coniniy momenm and enegy eqaion in a conevaive fom. e govening incompeible coniniy momenm and enegy eqaion ned o o be a follow (oema 997; Kalb and Seade 97) : Coniniy eqaion: () -momenm: co p co (6) θ-momenm: in p in (7) -momenm: in co p Fige. Coodinae yem. in co (8) Enegy eqaion: (9) Hee ρ p ν α epeen deniy pee kinemaic vicoiy and emal diffiviy of e flid epecively. τ ij i e ymmeic vico e eno wi e following componen in co () NUMERICL MEHODS ND OUNDRY CONDIION e govening eqaion (Eq. () ~ (9)) wee diceized by ing a finie volme meod. ll e pyical vaiable wee non-dimenionalized wi e pipe adi (a) mean ficion velociy ( τ ) and mean ficion empeae ( τ =q w α/ τ ). Hee q w i e wall ea flx a i conan. e mean ficion velociy ( τ ) i defined a follow (oema 996; Hül and Fiedic ) : / d () econd-ode cenal diffeence ceme wa ed fo paial diceizaion. ybid ceme wa employed fo ime advancemen. In paicla following kelvoll and Moin (996) we paiioned e -θ domain ino e coe egion ( c ) and e oe egion ( c ). Hee c i e bonday beween e wo egion and we e

3 ( ( ( ( ROE g Poiie Fance. Peen Webe and Hmpey (997) Webeand Hmpey (LDV 993) wee D and Δ epeen e pipe diamee (a) and a diance along e pipe ceneline epecively. U / U b /a Fige. Mean axial velociy pofile along a oizonal line fo Re b =48 κ=/8.. w - / w - m Peen Kalb and Seade (97) / a - Fige 3. empeae pofile fo K= κ=/. P=.7. c =.a. In e coe egion bo e convecive and diffion em ae implicily ime-advanced only in e cicmfeenial diecion wile e oe em ae explicily advanced. On e oe and in e oe egion e convecive and diffion em ae implicily eaed only in e adial diecion and e oe em ae explicily advanced. In bo egion e Cank-Nicolon meod wa employed a e implici ceme wile a id-ode Rnge-Ka meod wa ed a e explici ceme. o decople e coniniy and momenm eqaion a facional ep meod wa ed (Kim and Moin 98). e no-lip condiion wa impoed on e pipe wall wile flow wa amed o be peiodic in e axial diecion. Fo e empeae bonday condiion ea flx (q w ) wa e conan on e pipe wall and i wa amed a e empeae field i peiodic in e axial diecion (Kalb and Seade 97; Paanka e al. 977). e flow i diven in e axial diecion by a mean pee gadien Δp/Δ wic m balance e vico ficion along e pipe wall and can be eimaed fom a imple foce eqilibim (oema and Niewad 996) p 4 D () COMPUIONL DEILS e pipe leng along e axial coodinae i aken o be a. e Reynold nmbe (Re τ ) conideed ee i baed on e pipe adi (a) and e mean ficion velociy ( τ ) weea e Pandl nmbe (P) i.7. e cvae (κ) i /8.. e nmeical eolion wa deemined by a gid efinemen dy. e nmbe of gid cell ed in e cen inveigaion wa 96() 9(θ) 6(). nifom gid wa ed in θ and diecion. In e wall-nomal adial diecion gid poin wee cleed cloe o e wall. (oema 997; Hül and Fiedic ) VLIDION o validae o code o el ae compaed again oe of oe ao cenly available. In Fig. e mean axial velociy pofile along a wall-nomal adial line on e ymmeic plane i peened wen e Reynold nmbe (Re b ) eqal 48 wi κ=/8.. Hee e Reynold nmbe Re b (=U b D/ν) i defined baed on e axial mean blk velociy (U b ) and e pipe diamee (D). e expeimenal el (LDV) of Webe and Hmpey (993) and e DNS el of Webe and Hmpey (997) ae alo own ogee. O el i in good ageemen wi ei. Fig. 3 peen e nondimenionalized empeae pofile along e line - and - fo K= κ=/. P=.7 in compaion wi e nmeical imlaion of Kalb and Seade (97). Hee m and w epeen mean blk empeae and wall empeae epecively. e Dean nmbe (K=Re b ) i e a K=. e wo el ae in good ageemen. I i alo een a e empeae pofile on - vey mc eemble e axial velociy pofile own in Fig.. RESULS Fig. 4 ow e mean axial velociy pofile along e oizonal line (-) and veical line (-) fo Re τ = κ=/8.. De o e cvae effec e peak i ifed owad e oe wall (/a=) along - weea e pofile i ymmeic wi epec o /a= along -. I i alo een a bo e mean axial velociy and i gadien in e immediae viciniy of e oe wall ae lage an oe vey nea e inne wall (/a=-). Howeve e pofile of mean axial velociy along e veical line (-) i ymmeic wi epec o e cene plane and a e maximm poin nea e oe a well a e inne wall. Fige peen e pofile of nondimenionalized mean empeae( w / ) on e oizonal line (-) and on e veical line (-). Hee w denoe wall empeae. e mean empeae diibion on e wo line cloely follow oe of e mean axial velociy componen (Fig. 4). I i een in Fig. a e mean empeae gadien on e oe wall (/a=) i eepe an a on e inne wall (/a=-) on e 3

4 ROE g Poiie Fance (a) - - / + U 8 U /. /a (b) Fige 4. Pofile of mean axial velociy componen Reτ= κ=/8.. / / (c) / /a Fige. Mean empeae pofile Reτ= κ=/8. P=.7. oizonal line (-). i indicae fom e viewpoin of local ea anfe ae a e cvae enance ea anfe on e oe wall b a i no e cae on e inne wall. On e oe and fo e cved pipe flow e mean empeae i almo conan along e cenal pa of e veical line (-). Diibion of mean velociy and empeae field ae depiced on an -θ plane in Fig. 6. Cono of e axial velociy componen ae peened in Fig. 6(a). e ig-velociy egion i ifed owad e oe wall de o e cvae b ill ymmeic wi epec o e oizonal line. I i alo noiced a in e cenal egion e cono ae almo pependicla o e oizonal line. Cono of nondimenionalized mean empeae ae own on a -θ plane in Fig. 6(b). alo een in Fig. e ig-empeae egion i ifed owad e oe wall de o e cvae effec and e mean empeae i almo conan in e veical diecion in e cenal egion. e econday flow ae clealy een in Fig. 6(c)6(d) a peen e velociy veco and eamline epecively on an -θ plane. e cenifgal foce i mainly balanced by e pee gadien. Since e axial momenm nea e pipe wall i vey mall e balance i boken eling in movemen of flid paicle owad e inne wall. Coneqenly flid paicle nea e inne 3/ (d) / 3/ Fige 6. Seconday flow in an -θ plane; (a) mean axial velociy cono (b) nondimenionalized mean empeae cono (c) velociy veco plo (d) eamline. 4

5 ROE g Poiie Fance ' m ecion and e ig-velociy egion i ifed owad e oe wall. Femoe e mean axial velociy componen i almo conan in e veical diecion in e cenal egion of e pipe. e mean empeae diibion on a co-ecion i emakably imila o a of e mean axial velociy componen. Rm of empeae flcaion i lage in e cenal egion a well a nea e ppe and lowe wall /a Fige 7. Pofile of oo-mean-qae of e empeae flcaion Re τ = κ=/8. P=.7. / ' m 3/ Fige 8. Cono of m of e empeae flcaion in an -θ plane Re τ = κ=/8. P=.7. wall ae diven owad e oe wall de o coniniy foming a pai of cone-oaing and ymmeic econday flow. eefoe a agnaion poin i fomed a eac end of e oizonal line (-) and ignifican cicmfeenial velociy i indced nea e pipe wall. Fige 7 ow pofile of e empeae flcaion m along e oizonal line (-) and veical line (-). Like e mean empeae pofile on - in Fig. bo m vale and i gadien ae lage nea e oe wall (/a=) an nea e inne wall (/a=-). On e veical line (-) bo m vale and i gadien ae lage nea bo end. Fige 8 peen cono of empeae flcaion m in an -θ plane confiming a m i lage in e cenal egion a well a nea e ppe and lowe wall. CONCLUSION Diec nmeical imlaion of flly-developed blen cved pipe flow a been pefomed o dy e effec of e pipe cvae on e caaceiic of flow ce and ea anfe. e paamee wee e a Re τ = P=.7 and κ=/8.. e cvae indce a pai of cone-oaing econday flow on a co CKNOWLEDGEMEN i wok wa ppoed by e Naional Reeac Fondaion of Koea (NRF) gan fnded by e Koea govenmen (MES) (No. R39). REFERENCES dle M. 934 "Sömng in gekümmen Roen" Zeicif füangewande Maemaik nd Mecanik Vol. 4 pp kiyama M. and Ceng K. C. 97 "onday voiciy meod fo lamina foced convecion ea anfe in cved pipe" In. J. Hea Ma anfe Vol. 4 pp kelvoll K. and Moin P. 996 "n efficien meod fo empoal inegaion of e Navie-Soke eqaion in confined axiymmeic geomeie" J. Comp. Py. Vol. pp acelo G. K. 97 n Inodcion o Flid Mecanic ppendix Cambidge Univeiy Pe. ege S.. albo L. and Yao L. S. 983 "Flow in cved pipe" nn. Rev. Flid Mec. Vol. pp oema. J. and Niewad F.. M. 996 "Lage-eddy imlaion of blen flow in a cved pipe" SME J. Flid Eng. Vol. 8 pp oema. J. 997 Elecomagneic effec in cylindical pipe flow P.D. ei Delf Univeiy Pe. Collin W. M. and Denni S. C. R. 97 "e eady moion of a vico flid in a cved be" Q. J. Mec. ppl. Ma. Vol. 8 pp Denni S. C. R. and Ng M. 98 "Dal olion fo eady lamina flow og a cved be" Q. J. Mec. ppl. Ma. Vol. 3 pp den oonde J. M. J. and Niewad F.. M. 997 "Reynold nmbe effec in a blen pipe flow fo low o modeae Re" Py. Flid Vol. 9() pp Eggel J. G. M. Unge F. Wei M. H. Weeweel J. dian R. J. Fiedic R. and Niewad F.. M. 994 "Flly developed blen pipe flow : a compaion beween diec nmeical imlaion an expeimen" J. Flid Mec. Vol. 68 pp Enaye M. M. Gibon M. M. aylo M. K. P. and Yianneki M. 98 "Lae-Dopple meaemen of lamina and blen flow in a pipe bend" In. J. Hea Flid Flow Vol. 3(4) pp Feiz.. Old-Roi M. and Laia G. 3 "Lage eddy imlaion of blen flow in a oaing pipe" In. J. Hea Flid Flow Vol. 4(3) pp Gemano M. 98 "On e effec of oion on a elical pipe flow" J. Flid Mec. Vol. pp. -8.

6 ROE g Poiie Fance Gemano M. 989 "e Dean eqaion exended o a elical pipe flow" J. Flid Mec. Vol. 3 pp Hül. J. Wagne C. and Fiedic R. 999 "Navie-Soke olion of lamina flow baed on oogonal elical coodinae" In. J. Nme. Me. Flid Vol. 9 pp Hül. J. and Fiedic R. "Inflence of cvae and oion on blen flow in elically coiled pipe" In. J. Hea Flid Flow Vol. pp Hül. J. and Fiedic R. "Diec nmeical imlaion of blen flow in cved and elically coiled pipe" Comp. Flid Vol. 3 pp Io H. 99 "Ficion faco fo blen flow in cved pipe" J. aic Eng. Vol. 8 pp Io H. 987 "Flow in cved pipe" JSME In. J. Vol. 3 pp Kalb C. E. and Seade J. D. 97 "Hea and ma anfe penomena fo vico flow in cved cicla be" In. J. Hea Ma anfe Vol. pp Kalb C. E. and Seade J. D. 974 "Flly developed vico-flow ea anfe in cved cicla be wi nifom wall empeae" ICE J. Vol. () pp Kim J. and Moin P. 98 "pplicaion of a facional-ep meod o incompeible Navie-Soke eqaion" J. Comp. Py. Vol. 9 pp Mainelli R. C. 947 "Hea anfe o molen meal" an. m. Soc. Mec. Eng. Vol. 69 pp Moi Y. and Nakayama W. 96 "Sdy on foced convecive ea anfe in cved pipe ( Repo Lamina egion)" In. J. Hea Ma anfe Vol. 8 pp Moi Y. and Nakayama W. 967 "Sdy on foced convecive ea anfe in cved pipe (nd Repo blen egion)" In. J. Hea Ma anfe Vol. pp Moi Y. and Nakayama W. 967 "Sdy on foced convecive ea anfe in cved pipe (3d Repo eoeical analyi nde e condiion of nifom wall empeae and pacical fomlae)" In. J. Hea Ma anfe Vol. pp Paanka S. V. Paap V. S. and Spalding D "Pedicion of lamina flow and ea anfe in elically coiled pipe" J. Flid Mec. Vol. 6 pp Paanka S. V. Li C. H. and Spaow E. M. 977 "Flly developed flow and ea anfe in dc aving eamwie-peiodic vaiaion of co-ecional aea" SME J. Hea anfe Vol. 99 pp Pa N. H. 947 "e ea anfe in a eacion ank cooled by mean of a coil" an. In. Cem. Eng. Vol. pp Pa J. and Yao L. S. 98 "Nmeical olion fo flly developed flow in eaed cved be" J. Flid Mec. Vol. 3 pp. 3-. Roge G. F. C. and Mayew Y. R. 964 "Hea anfe and pee lo in elically coiled be wi blen flow" In. J. Hea Ma anfe Vol. 7() pp Seban R.. and McLaglin E. F. 963 "Hea anfe in be coil wi lamina and blen flow" In. J. Hea Ma anfe Vol. 6() pp Sdo K. Smida M. and Hibaa H. 998 "Expeimenal inveigaion on blen flow in a cicla-ecioned 9-degee bend" Exp. Flid Vol. pp Sdo K. Smida M. and Hibaa H. "Expeimenal inveigaion on blen flow og a cicla-ecioned 8 bend" Exp. Flid Vol. 8 pp. -7. Unge F. and Fiedic R. 99 "Lage eddy imlaion of flly-developed blen pipe flow" Poc. 8 Symp. on blen Sea Flow Münic Gemany. Wagne C. Hül. J. and Fiedic R. "Low-Reynold-nmbe effec deived fom diec nmeical imlaion of blen pipe flow" Comp. Flid Vol. 3 pp Wang C. Y. 98 "On e low-reynold-nmbe flow in a elical pipe" J. Flid Mec. Vol. 8 pp Webe D. R. and Hmpey J.. C. 993 "Expeimenal obevaion of flow inabiliy in a elical coil" SME J. Flid Eng. Vol. pp Webe D. R. and Hmpey J.. C. 997 "aveling wave inabiliy in elical coil flow" Py. Flid Vol. 9() pp W X. and Moin P. 8 " diec nmeical imlaion dy on e mean velociy caaceiic in blen pipe flow" J. Flid Mec. Vol. 68 pp. 8-. Yao L. S. and ege S "Flow in eaed cved pipe" J. Flid Mec. Vol. 88 pp Zapyanov Z. Ciov C. and oev E. 98 "Flly developed lamina flow and ea anfe in cved be" In. J. Hea Ma anfe Vol. 3 pp

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