EFFECT OF HEAT FLUX RATIO FROM BOTH SIDE-WALLS ON THERMAL- FLUID FLOW IN CHANNEL

Transcription

1 8h AIAA/ASME Jon Thermophyscs and Hea Transfer Conference 4-6 June 00, S. Lous, Mssour AIAA EFFECT OF HEAT FLUX RATIO FROM BOTH SIDE-WALLS ON THERMAL- FLUID FLOW IN CHANNEL SHUICHI TORII Deparmen of Mechancal Engneerng, Kagoshma Unversy, Kagoshma , Japan Tel: Fax: E-mal: WEN-JEI YANG Mechancal Engneerng and Appled Mechancs, The Unversy of Mchgan, Ann Arbor, Mchgan 48109, U.S.A. Tel: Fax: E-mal: ABSTRACT Numercal analyss s performed o nvesgae ranspor phenomena n channel flows under unform heang from boh sde-walls. Emphass s placed on an effec of hea flux rao from boh sdes on he velocy and hermal felds. The woequaon hea-ransfer model s employed o deermne hermal eddy dffusvy. I s found ha () under srong heang from boh walls, lamnarzaon,.e., a subsanal deeroraon n hea ransfer performance occurs as n he crcular ube flow case, () durng he lamnarzaon process, boh he velocy and emperaure gradens n he vcny of he heaed walls decrease along he flow, resulng n a subsanal aenuaon n boh he urbulen knec energy and he emperaure varance over he enre channel cross secon and () n conras, lamnarzaon s suppressed n he presence of onesde-heang, because urbulen knec energy s produced n he vcny of he oher nsulaed wall. Therefore, an occurrence of lamnarzaon n he channel s affeced by he rao of hea flux from boh sde-walls. cp Cµ, C1, C Cλ, CP1, CP NOMENCLATURE specfc hea a consan pressure, J/(kgK) emprcal consans of k-ε model urbulence model consans for emperaure feld CD1, CD urbulence model consans for emperaure feld h hea ransfer coeffcen, W/m K H channel hegh, m f frcon coeffcen fµ, f1, f model funcons of k-ε model fλ, fp1, fp urbulence model funcons of emperaure feld fd1, fd urbulence model funcons of emperaure feld g acceleraon of gravy, m/s G mass flux of gas flow, kg/(m s) Gr Grashof number, gqwh 4 /(ν λt)n H channel hegh, m k urbulen knec energy, m /s ML he number of mesh N hea flux rao, q w /q w1 Nu Nussel number, Hh/λ P me-averaged pressure, Pa Pr Prandl number Pr urbulen Prandl number q w1, q w hea fluxes a y=0 and H, respecvely, W/m q + w dmensonless hea flux parameer, Eq. (14) Copyrgh 00 The Amercan Insue of Aeronaucs and Asronaucs, Inc. All rghs reserved. 1 Copyrgh 00 by he, Inc. All rghs reserved.

2 Re Reynolds number, umh/ν R urbulen Reynolds number, k /(εν) Rτ dmensonless dsance, y + S Sanon number, qw/(ρcpum(tw-tb)) T me-averaged emperaure, K flucuang emperaure componen, K * frcon emperaure, qw/(ρcpu*), K emperaure varance, K U, V me-averaged velocy componens n axal, and normal-wall drecon, respecvely, m/s U, u me-averaged and flucuang velocy componens n he x drecons, m/s um mean velocy over channel cross secon, m/s u, v, w flucuang velocy componens n axal, wall-normal and angenal drecons, respecvely, m/s u* frcon velocy, m/s u + dmensonless velocy, U/u * uu j Reynolds sress, m /s - u urbulen hea flux, mk/s x axal coordnae, m x coordnaes, m y wall-normal coordnae, m y + dmensonless dsance, u*δ/ν Greek Leers α hermal dffusvy, m /s ρ densy, kg/m 3 δ dsance from wall, m ε urbulen energy dsspaon rae, m /s 3 ε dsspaon rae of, K/s λ, λ molecular and urbulen hermal conducves, respecvely, W/(Km) µ, µ molecular and urbulen vscoses, respecvely, Pa sec ν flud knemac vscosy, m /s σk, σε, σh, σφ urbulence model consans for dffuson of k, ε, and ε, respecvely θ angenal drecon θ + dmensonless emperaure, + T T c θ = Tw Tc Subscrps c mnmum or nsulaed wall nle nle max maxmum w wall Superscrps - me-averaged value INTRODUCTION When a gas n a crcular ppe s heaed wh exremely hgh hea flux, he flow may be lamnarzed; ha s, a ranson from urbulen o lamnar flows occurs a a hgher Reynolds number han he usual crcal value,.e., Re=,300. Ths phenomenon s referred o as lamnarzaon. Boh he crera for s occurrence and s hea ransfer characerscs have been repored by several nvesgaors [1-6]. In order o nvesgae an effec of passage geomery on an occurrence of he lamnarzng gas flow, Tor e al. [7] and Fuj e al. [8] deal wh he hermal-flud ranspor phenomena n concenrc annul under hgh hea flux heang. They dsclosed ha () when he gas flow s srongly heaed wh he same hea flux level from nner and ouer ube walls, he local hea ransfer coeffcens on boh walls approach he lamnar values along he flow; ha s, he lamnarzaon akes place; () he exsng crera of lamnarzaon for crcular ube flows can be appled o annular flows as well f he occurrence of lamnarzaon s esmaed usng a dmensonless hea flux parameer q + w; bu () annular flows heaed srongly from only one sde are less vulnerable o lamnarzaon even f he usual crera are sasfed. The purpose of he presen sudy s o nvesgae hermalflud flow ranspor phenomena n a channel n whch boh walls are ndvdually heaed wh dfferen hea fluxes. The -ε hea ransfer model proposed by Tor and Yang [9] and he k-ε urbulence model of Tor e al. [10] are employed o reveal he mechansm of hea ranspor phenomena. The urbulen hermal conducvy λ s deermned usng he emperaure varance and he dsspaon rae of emperaure flucuaons ε, ogeher wh k and ε. Emphass s placed on he effec of hea flux rao from boh sdes on he velocy and hermal felds, based on he numercal resuls,.e., he urbulen knec energy, emperaure varance, velocy, and emperaure profles.

3 GOVERNING EQUATIONS AND NUMERICAL PROCEDURES Consderaon s gven o a seady urbulen flow n a srongly heaed channel. The physcal confguraon and he coordnae sysem are shown n Fg. 1. In hs analyss, he dependence of gas properes on emperaure, as well as changes n gas densy, mus be aken no accoun [11]. The connuy, momenum and energy for an ncompressble flud are represened n he ensor form as: Connuy equaon: ρu = 0 (1) x Momenum equaons: ρu U j x Energy equaon: ρ P µ U U = + + x x x j x j j T = λ cpρ c p U T x x x u j ρuu j ) (3) These equaons follow from he dervaon process proposed by Schlchng [11]. Here, he urbulen flucuaons of λ, µ and cp hrough emperaure flucuaon have been dscouned. The erm for body force n he momenum equaon s also neglgble, because a small dameer ube was employed and hroughou he calculaon, he buoyancy parameer Gr/Ren was less han 0.1 so ha forced convecon may be expeced o domnae. The Reynolds sress - ρ u u j usng he Boussnesq approxmaon as: ρ = δ ρ µ U U j uu j j k + 3 x j x n Eq. () s obaned (4) Here, he urbulen vscosy µ s expressed n erms of he urbulen knec energy k and s dsspaon rae ε hrough he Kolmogorov-Prandl's relaon [1] as y x Heang or Insulang Flow Heang H Fgure 1 A schemac of he physcal sysem and coordnaes k µ = Cµ f µ ρ ε (5) Cµ and fµ are a model consan and a model funcon, respecvely. Tor e al. [10] developed a low Reynolds number verson of he k-ε urbulence model capable of reproducng he ranson from urbulen o lamnar flows orgnally developed by Nagano and Hshda [13]. The same model [10] s employed here. Boh ranspor equaons read µ ρ µ ρ ρε µ U k k U k j = j x j + uu x j σ k x j x ( j x ) j (6) ε µ ε ρ µ ρ ε U j C f j x j x j σ ε x j k uu U = x j (7) ρ ε ν µ U C f + ( 1 f µ k )( x ) j The emprcal consans and model funcons n Eqs. (5), (6) and (7) are summarzed n reference [10]. In he presen sudy, he k-ε model for he velocy feld s appled o analyze he srongly heaed gas flows n a channel because can predc lamnarzaon n a saonary ppe wh hgh flux heang [10]. The urbulen hea flux c p ρ u n Eq. (3) can be expressed n he followng form: T cpρu = λ x = C f c p k k ρ T λ λ ε ε x (8) where Cλ s a model consan and fλ s a model funcon. In order o oban and ε n Eq. (9), Tor and Yang [9] proposed he modfed wo-equaon hea-ransfer model capable of expressng he hea-ransfer characerscs n he lamnar and ranson regons, whose model s orgnally developed by Nagano and Km [14]. The ranspor equaons for and ε are expressed as 3

4 λ λ ρu j = x j + x j cp σ hc p x j T λ ρu j ρ ε x j cp x j (9) and ε λ λ ε ε λ T ρu j = p p x j x + j c p σ φ c + C f 1 1, p j p x k c x j ε µ ρ ε ρ ε ε α U + C p f λ T p C D1 f D1 C D f D + ( 1 f k x j k c )( p x ) j (10) respecvely. The emprcal consans and model funcons n Eqs. (8), (9) and (10) are lsed n reference [9]. A se of governng equaons s solved usng he conrol volume fne-dfference procedure developed by Paankar [15]. The power-law scheme for he convecon-dffuson formulaon s employed o lnk he convecon-dffuson erms. Snce all urbulen quanes as well as he meaveraged sreamwse velocy vary rapdly n he near-wall regon, he sze of nonunform cross-sream grds ncreases wh a geomerc rao from he wall owards he cener lne. The maxmum conrol volume sze near he cener lne s always kep a less han 1% of channel hegh. To ensure he accuracy of calculaed resuls, a leas wo conrol volumes are locaed n he vscous sublayer. Throughou he numercal calculaons, he number of conrol volumes s properly seleced beween 7 and 98 o oban a grd-ndependen soluon, resulng n no apprecable dfference beween he numercal resuls wh dfferen grd spacng, as menoned below. The dscrezed equaons are solved from he nle n he downsream drecon by means of a marchng procedure, snce hese equaons are parabolc. The maxmum sep-sze n he sreamwse drecon s lmed o fve mes he mnmum sze n he wall-normal drecon of he conrol volume. A each axal locaon, he hermal properes for conrol volumes are deermned from he axal pressure and emperaure usng a numercal code of reference [16]. The hydrodynamcally fully-developed sohermal flow s assumed a he sarng pon of he heang secon. The followng boundary condons are used a he walls: y=0: U = k = ε = = ε = 0, q w T 1 = (consan hea flux) y λ w y=h: U = k = ε = = ε = 0, q w T = (consan hea flux) y λ w Based on he above boundary condons, he compuaons are processed n he followng order: 1. The nal values of U, k, ε, and ε are specfed and assgned a consan axal pressure graden. Here, he values of U, k and ε n he hydrodynamcally fully-developed sohermal channel flow are employed as he nal values.. The equaons of U, k, ε, T, and ε are solved usng he boundary condons gven here. 3. Sep s repeaed unl he creron of convergence s sasfed. I s se a M M 1 φ φ 4 max M 1 10 φ (11) max for all varables φ (U, k, ε, T, and ε ). The superscrps M and M-1 n Eq. (11) ndcae wo successve eraons, whle he subscrp "max" refers o a maxmum value over he enre feld of eraons. 4. New values of U, k, ε, T, and ε are calculaed by correcng he axal pressure graden. 5. Seps -4 are repeaed unl he conservaon of he sreamwse flow rae s sasfed under he creron U cp dy U n dy 5 U n dy 10 (1) and he convergen values of U, k, ε, T, and ε are evaluaed. Here, Ucp s he axal velocy under he correcon process and Un s ha a he nle of he channel. 6. Seps -5 are repeaed unl x reaches he desred lengh from he nle. In he presen sudy, he nondmensonal hea flux parameer q + w s employed o ndcae he magnude of hea flux a he channel wall. Ths parameer, orgnally proposed by Nemra e al. [17] for deermnng hermal ranspor phenomena n concenrc annular gas flows, s defned as d n qn + d ou qou 1 q + w =, (13) dn + dou ( Gc pt) nle where dn and dou are, respecvely, he nner and ouer ube dameers of he annulus, and qn and qou correspond o hea fluxes on he nner and ouer walls of he annulus. When applyng Eq. (13) o wo-dmensonal channel, s reduced as 4

5 q + w = q = w1 q + q w1 w ( 1 + N) 1 ( Gc pt) nle 1 ( Gc pt) nle. (14) The ranges of he parameers are he nondmensonal hea flux parameer q + w<0.009; he nle Reynolds number,.e., he Reynolds number a he onse of heang Ren=8,500; hea flux rao of boh sde walls N=0-1 and he nle gas (nrogen) emperaure Tn=73 K. Smulaons wh grds of varous degrees of coarseness are conduced o deermne he requred resoluon for grdndependen soluons. Throughou he numercal calculaons, he number of conrol volumes, ML, s properly seleced beween 7 and 98 over he cross-secon of he concenrc annulus. Consequenly, here was only a slghly apprecable dfferences, 0.5%, beween numercal resuls wh dfferen radal grd spacng. (a) boh-sded heang RESULTS AND DISCUSSION Fgure llusraes he local hea-ransfer coeffcens n srongly heaed channel flows n he form of Sanon number S versus Reynolds number Re, wh q + w as he parameer. Fgures (a) and (b) correspond o he resuls for boh-sded and one-sded heang,.e., N=1 and 0, respecvely. Theorecal soluons for urbulen and lamnar hea ransfer n he hermally and hydrodynamcally fully-developed channel flows [18] are supermposed n he fgure n a sold sragh lnes. In Fg., a reducon n he Reynolds number sgnfes a change n he locaon along he channel, because he Reynolds number decreases from he nle wh he axal dsance resulng from an ncrease n he molecular vscosy by heang. The numercal resuls for boh cases show ha he local Sanon numbers a q + w=0.005 frs decrease n he hermal enrance regon, hen ncrease, approachng he urbulen correlaon furher downsream. Ths suggess ha no lamnarzaon wll occur. On he conrary, as he flow goes downsream, he predced Sanon numbers a q + w = depar from he urbulen hea-ransfer correlaon and approach he lamnar correlaon, as shown n Fg. (a). Bankson [1] poned ou ha he subsanal reducon n S along he flow s due o he occurrence of lamnarzaon. In oher words, f he channel s srongly heaed from boh walls, he flud flow s lamnarzed as n he crcular ube flow case. (b) one-sded heang Fgure Predced local Sanon number wh Reynolds number as a funcon of nondmensonal hea flux parameers. However, n Fg. (b) he local Sanon number a q + w= decreases n he hermal enrance regon, hen recovers along he flow and evenually approaches he urbulen correlaon equaon furher downsream. Ths ranspor phenomenon provdes a srkng conras o he bohsded heang case a he correspondng dmensonless hea 5

6 flux. When hea flux on a sngle heang wall becomes hgher, he predced local Sanon number, as seen n Fg. (b), approaches he urbulen correlaon downsream, even a q + w=0.0090,.e., a level ha under wo-sded heang compleely lamnarzes he ube flow. In oher words, f a channel s heaed exclusvely from only one wall, he flud flow can no be lamnarzed and s a srkng conras o he ube flow case. Ths behavor n he channel flow s he same as he hermal-flud ranspor characerscs n he annul heaed srongly from only one sde, as menoned n nroducon. The occurrence of lamnarzaon s hus clearly affeced by q + w and N. An aemp s made o explore he hea and flud flow mechansms for wo-sded and boh-sded heang, based on he numercal resuls a q + w=0.0043,.e., urbulen knec energy, emperaure varance, velocy and emperaure profles. Fgure 3 llusraes he wall-normal dsrbuons of he me-averaged sreamwse velocy U/Umax a hree dfferen axal locaons: x/h=0, 60, and 10. Fgures 3(a) and (b) show he numercal resuls for N=1 and 0, respecvely. The velocy U s normalzed by he maxmum value Umax a each axal locaon. The lamnar flow profle s supermposed n he fgure as a sold lne, for comparson. In Fg. 3(a) a subsanal reducon of he velocy graden akes place n he flow drecon and he velocy profle approaches lamnar one n he downsream regon. In conras, Fg. 3(b) shows ha he velocy gradens a he walls are slghly dmnshed along he flow, parcularly n he vcny of he heang wall, and he velocy profle s subsanally dfferen from he lamnar one n he flow drecon. The correspondng sreamwse varaons of he urbulen knec energy k for N=1 and 0 are llusraed n Fgs. 4(a) and (b), respecvely. Here, he numercal resuls are normalzed by a square of he wall frcon velocy a he onse of heang u*. I s observed n Fg. 4(a) ha he urbulen knec energy level for boh-sded heang,.e., N=1 s exremely aenuaed over he whole channel cross secon n he flow drecon due o hgh flux heang. Ths sreamwse behavor s n accordance wh ha of he velocy dsrbuon n Fg. 3(a). The subsanal aenuaon n boh velocy and urbulen knec energy s he same as he flow characerscs n he lamnarzng flow n boh he heaed ube [9, 10] and he annul heaed from boh nner and ouer ube walls [7]. 6 (a) (b) Fgure 3 Sreamwse varaon of me-averaged velocy profles n a srongly heaed flow wh dfferen axal locaons, (a) boh-sded heang and (b) one-sded heang. However, numercal resuls for N=0 show ha as he flow progresses, apprecable urbulen knec energy sll remans n he velocy feld, parcularly near he nsulaed wall, as seen n Fg. 4(b). Ths behavor corresponds o ha of he velocy dsrbuon n Fg. 3(b) and s smlar o ha n he annular flow heaed from one-seded wall. Tha s, when he

7 channel flow s srongly heaed from one wall, he urbulen knec energy s severely dmnshed n he vcny of he heang wall, whle s nensfed near he oppose wall along he flow. All hese resuls conssenly show ha () f he flow s srongly heaed from boh sde walls of he channel, lamnarzaon occurs, and () he rend owards lamnarzaon from he srongly heaed wall s always suppressed by he urbulen knec energy produced n he regon near he nsulaed wall, where hea flux s added o he flow from one wall only. (a) (a) (b) Fgure 5 Sreamwse varaon of me-averaged emperaure profles n a srongly heaed flow wh dfferen axal locaons, (a) boh-sded heang and (b) one-sded heang. (b) Fgure 4 Sreamwse varaon of urbulen knec energy profles n a srongly heaed flow wh dfferen axal locaons, (a) boh-sded heang and (b) one-sded heang. Fgures 5(a) and (b) show sreamwse varaons n he me-averaged emperaure profle θ + for N=1 and 0, respecvely. Numercal resuls are obaned a dfferen axal locaons: x/h=0, 60, and 10. The subsanal reducon n he emperaure graden for N=1 occurs a he wall along he flow (Fg. 5(a)), whle numercal resuls for N=0 reveal only a 7

8 slgh reducon n he emperaure graden a he heaed wall n he flow drecon (Fg. 5(b)). The correspondng radal dsrbuons of he emperaure varance for N=1 and 0, are llusraed n Fgs. 6(a) and (b), respecvely. Here, he emperaure varance s dvded by he square of he frcon emperaure * a each axal locaon. As he flow moves, here s a subsanal reducon n for N=1 over he whole cross-secon of he channel, as seen n Fg. 6(a). Ths behavor mples aenuaon n he emperaure flucuaons n he hermal feld. For N=0, as he flow moves along he channel, he emperaure varance level s somewha dmnshed n he vcny of he heaed channel wall because of a decrease n he emperaure graden near ha wall, whle s nensfed a he oher wall. In oher words, apprecable emperaure flucuaons reman n he hermal feld when he flow s heaed from one wall only. (a) (b) Fgure 6 Sreamwse varaon of emperaure varance profles n a srongly heaed flow wh dfferen axal locaons, (a) boh-sded heang and (b) one-sded heang. Snce he eddy dffusvy concep s employed o deermne he urbulen hea flux - c p ρ v n Eq. (3), λ s drecly relaed o k, ε, and ε as depced n Eq. (8). Hence, subsanal reducons n he urbulen knec energy and emperaure varance resul n aenuaon n he Sanon number, as shown n Fg. (a). One may hus conclude ha a flow n a channel heaed wh unform wall hea flux from boh walls s lamnarzed as n he ube and concenrc annular flows. In conras, a sreamwse deeroraon of he Sanon number n Fg. (b) s suppressed by he presence of he urbulen knec energy produced n he vcny of he nsulaed wall, even f he hea flux parameer sasfes he lamnarzaon creron for crcular ube flows. Tha s, he rend owards lamnarzng he flow s always suppressed f he hea flux s added o he flow from only one wall, even a levels ha cause lamnarzaon n a crcular ube flow. Nex s o sudy he effec of N on an occurrence of he lamnarzaon of he flow n he wo-dmensonal channel. Frs of all, condons should be specfed under whch he flow s ceranly lamnarzed. Tor e al. [10] esablshed he creron for he lamnarzng flow n a ube wh hgh hea flux usng he k-ε urbulence model. Tha s, lamnarzaon occurs when he calculaed urbulen knec energy a he locaon 150 dameers downsream from he nle becomes lower han one-enh of he nle value. The same dea, n whch he creron s for he urbulen knec energy a x/h=150 o be lower han one-enh of s nle value, s adoped n he presen sudy. Ths s because he sreamwse varaon of a urbulen knec energy n he lamnarzng flow, as depced n Fg. 4(a), s smlar o ha n he srongly heaed ube case [10]. The predced creron for he lamnarzaon of a wo-dmensonal channel flow s depced n Fg. 7, n he form of q + w versus N. Here, he exsng crera for he crcular ube and he predced crera for he crcular and annular ube flows [7, 10] are supermposed n he fgure for comparson. I s observed ha he Predced creron a N=1 s smlar o he crcular and annular ube cases. One may hus conclude ha a flow n a wodmensonal channel heaed wh unform wall hea flux from boh sded walls s lamnarzed a he same heang level as he crcular and annular ube flow cases, whle he creron s ncreased wh an decrease n N. In oher words, f a channel s heaed a he dfferen hea fluxes from boh walls, he flud flow can no be lamnarzed even a a heang level ha he ube and annular flows compleely cause lamnarzaon. 8

9 4. The occurrence of lamnarzaon s affeced by he hea flux rao of boh sde-walls. Fgure 7 Lamnarzaon map on N-q + w plo SUMMARY k-ε -ε model has been employed o numercally nvesgae flud flow and hea ransfer n a channel heaed wh unform hea flux from boh sde-walls. Consderaon s gven o he effecs of N on he occurrence of lamnarzaon. The resuls are summarzed as follows: 1. If he channel s smulaneously heaed from boh walls wh hgh unform hea flux, a subsanal reducon of he local Sanon number causes lamnarzaon along he flow. Therefore, he flud flow n he channel s lamnarzed, jus as n he ube flow case.. When lamnarzaon akes place, he velocy and emperaure gradens n he vcny of he channel wall decrease along he flow, resulng n a subsanal aenuaon n boh he urbulen knec energy and he emperaure varance over he enre channel cross secon. Consequenly, he urbulen hea flux s dmnshed by a decrease n he urbulen knec energy and emperaure varance over he channel cross secon, resulng n he deeroraon of hea-ransfer performance. 3. If he channel s heaed from only one sde wall, subsanal reducon of he local Sanon number s suppressed, resulng n no lamnarzaon. Ths behavor s he same as ha n an annulus heaed wh an only one wall. Ths s because he rend owards lamnarzaon s always suppressed by he urbulen knec energy produced n he regon near he nsulaed wall. REFERENCES 1. C. A. Bankson, The Transon from Turbulen o Lamnar Gas Flow n a Heaed Ppe, Trans. ASME, Ser. C, vol. 9, no. 4, pp , C. W. Coon and H. C. Perkns, Transon from he Turbulen o he Lamnar Regme for Inernal Convecve Flow wh Large Propery Varaons, Trans. ASME, Ser. C, vol. 9, no. 3, pp , D. M. McElgo, C. M. Coon and H. C. Perkns, Relamnarzaon n Tubes, J. Hea & Mass Transf., vol. 13, no., pp , K. R. Perkns, K. W. Schade and D. M. McElgo, Heaed Lamnarzng Gas Flow n a Square Duc, J. Hea and Mass Transfer, vol. 16, no. 3, pp , Y. Mor and K. Waanabe, Reducon n Heaed Transfer Performance due o Hgh Hea Flux, Trans. Jpn. Soc. Mech. Eng., (n Japanese), vol. 45, no. 397, B, pp , M. Ogawa, H. Kawamura, T. Takzuka and N. Akno, Expermenal Sudy on Lamnarzaon of Srongly Heaed Gas Flow n Vercal Crcular Tube, J. Aomc Energy Soc. Jpn., (n Japanese), vol. 4, no. 1, pp , S. Tor, A. Shmzu, S. Hasegawa and N. Kusama, Lamnarzaon of Srongly Heaed Annular Gas Flows, JSME In. J., Ser. II, vol. 34, no., pp , S. Fuj, N. Akno, M. Hshda, H Kawamura and K. Sanokawa, Numercal Sudes on Lamnarzaon of Heaed Turbulen Gas Flow n Annular Duc, J. Aomc Energy Soc. Jpn., (n Japanese), vol. 33, no. 1, pp , S. Tor and W.-J. Yang, Lamnarzaon of Turbulen Gas Flows nsde a Srongly Heaed Tube, In. J. Hea Mass Transfer, vol. 40, no. 13, pp , S. Tor, A. Shmzu, S. Hasegawa and M. Hgasa, Lamnarzaon of Srongly Heaed Gas Flows n a Crcular Tube (Numercal Analyss by Means of a Modfed k-ε Model), JSME In. J., Ser. II, vol. 33, no. 3, pp , H. Schlghng, Boundary-Layer Theory, 6h edon, McGraw-Hll,

10 1. W. Rod, Examples of Turbulence Models for Incompressble Flows, AIAA, J., vol. 0, pp , Y. Nagano and M. Hshda, Improved form of he k-ε Model for Wall Turbulen Shear Flows, Trans. of ASME, Ser. D, vol. 109, pp , Y. Nagano and C. Km, A Two-Equaon Model for Hea Transpor n Wall Turbulen Shear Flows, J. Hea Transfer, vol. 110, pp , S. V. Paankar, Numercal Hea Transfer and Flud Flow, Hemsphere, Washngon. D. C., Propah Group, Propah: a program package for hermophyscal propery, verson 4.1, M. A. Nemra, J. A. Vlemas and V. M. Smons, Hea Transfer o Turbulen Flow of Gases wh Varable Physcal Properes n Annul (Correlaon of Expermenal Resuls), Hea Transfer - Sove Research, vol. 1, pp , W. M. Kays and M. E. Crawford, Convecve Hea and Mass Transfer, nd ed. MacGraw-Hll, New York,

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