ETNA Kent State University

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1 ) Eleconic Tansacions on Neical nalysis. ole 30 pp opyigh 200 Ken Sae Univesiy. ISSN ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER OR LMINR-TURBULENT TRNSITION IN PIPE LOWS HIDESD KND bsac. This aicle descibes he calclaion of he ini ciical ynolds nbe fo laina-blen ansiion in pipe flos. o he conclsions of o pevios epeienal sdy i is clea ha a ansiion occs nea he pipe inle and he ciical ynolds nbe akes he ini vale of abo 2000 in he case of a saigh pipe. Moeove in o pevios calclaions of laina enance pipe flo i as fond ha nea he pipe inle a lage pesse gadien in he adial diecion eiss hich deceases as he ynolds nbe inceases. Ths e have bil a ne ansiion acoodel o deeine sing he effec of he adial pesse gadien. The calclaed esls ee = 370 hen he nbe of adial gid poins = 1 and 2200 hen = 101. Key ods. hydodynaic sabiliy gid efineen heodynaics MS sbjec classificaions. 76E0 6M Inodcion and say. Osbone ynolds fond o ciical ynolds nbes ( ) in pipe flos: of 1230 fo laina o blen flo and of 2030 fo blen o laina flo [16]. Eve since he pioneeing epeienal ok of ynolds (13) he isse of ho and hy he flid flo along a cicla pipe changes fo being laina o blen as he flo ae inceases has iniged physiciss aheaicians and enginees alike [11]. The objecives of his invesigaion ae o deive a acoodel of laina-blen ansiion fo Hagen-Poiseille flo o pipe flo and o calclae he ini vale of he ciical ynolds nbe!"$ hich is in he neighbohood of To dae aeps o heoeically obain vales of have been ndeaken sing sabiliy heoy ih he O-Soefeld eqaion and disbances. Hoeve %&!"$ of appoiaely 2000 has no ye been calclaed. o flo in he enance egion Tasi obained = [21] and Hang and hen obained = 3900 and 3960 ih aisyeic and non-aisyeic disbances especively [ 6]. In he flly developed egion he flo is sable ih espec o boh aisyeic and non-aisyeic disbances [2]. In his sdy e do no fhe conside sch sabiliy heoy. The line of hogh on calclaing fo laina-blen ansiion is folaed on he basis of o epeiens [10] and calclaed esls [ 9]. (1) The ynolds nbe () piaily and geneally affecs since ansiion occs as inceases. Theefoe e s fhe sdy ha faco besides ainly affecs '. (2) The laina-blen ansiion occs nea he pipe inle in he enance egion. I is ipoan o noe ha he flo ay becoe blen long befoe i becoes flly developed [4 20]. (3) We s neically find a ne nknon vaiable hich vaies nea he inle. I is he noal all sengh (NWS); see Sbsecion 3.1. (4) We s neically evalae he effecs of NWS on in his sdy. is fo he viepoin of o epeiens le s conside he poble in oe deail. (*) () is appaenly deeined by he enance shape o he conacion aio (= + )- + ) of he belloh diaee + o pipe diaee +. In os pevios epei-. ceived Janay cceped fo pblicaion Mach Pblished online on Jly coended by. Senge. ope Science and Engineeing Univesiy of iz iz-wakaas kshia 96-0 Japan (kanda@-aiz.ac.jp). 16

2 \ \ I \ ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER 169 Iniial nifo velociy pofile (Uo) Pipe all lly developed paabolic pofile 0 R D eneline Enance lengh ( ) Pipe all Diensionless enance lengh (Le = /(D )) IG elociy developen in enance egion of a saigh cicla pipe. ens pipes ee fied ih pe ohpieces o bellohs a he inle so ha ae igh ene iho disbances. /012"$ of appoiaely 2000 is obained in he case of a saigh cicla pipe. The shap edge of he saigh pipe is no a singla poin in he ansiion since is a sooh fncion of ( ) a (6) The ansiion occs nea he pipe inle. o eaple in he case of a saigh pipe he ansiion occs appoiaely 6 13 diaees donsea a onside a diensionless aial coodinae (4 ); le 76 be he acal aial coodinae hen 49:76 ;+. o he above eaple he ansiion occs a he pipe inle of 4=<?@ BBDE G EDHDI. (*) (7) o ynolds colo-band epeiens (J) ( ) vaies hen = paiclaly hen = If he adial disance of = 1.4 is ansfoed o he aial disance o check he ode of lengh hen fo K< 1.4/2000 = Second le s conside he viepoin of pevios epeienal and neical esls. In he enance egion he velociy pofile changes fo a nifo disibion a he pipe inle o a paabolic one a he enance lengh as shon in ig Geneally hs fa hee ajo vaiables have been sdied [3]: (i) he velociy disibion in all secions (ii) he enance lengh LNM hich is defined as he disance fo he inle o he poin hee he ceneline velociy eaches 99O of he flly developed vale and (iii) he pesse dop P'. The egion fo he inle o L M is called he enance egion and he donsea egion fo L M is called he flly developed egion. L M [1] is epessed as LSRT 6 + U HD G G DIEHG (1.1) WX o (1.1) LYR = a = 300 and LYR akes a consan vale of 0.06 a Z 00. The oal pesse dop P' 4[ fo he pipe inle is epessed as he s of he pesse dop 644 ha old occ if he flo ee flly developed pls he ecess pesse dop 4[ o accon fo he developing egion. P' 4[]9 ]_ 04[[ 1 04`a 4[ is assed o be \ de in (1.2) fo he flly developed egion [1]. o (1.2) \ de is a = 00 and a = \ de is appoiaely he sae a Z 00. (1.2) 2dfgh HEb 4c 04`

3 ETN Ken Sae Univesiy hp://ena.ah.ken.ed 170 H. KND () We discss popeies (a) and (b) of o paaees hich ill enable s o deeine accae vales of : paaee (a) is a consan egadless of fo Z 00 hile paaee (b) vaies invesely as inceases. The inesecion of he lines of he paaees shold indicae a ciical vale; see ig (9) I is clea fo (1.1) ha he velociy disibion and L M in he 4 coodinae ae appoiaely he sae a Z 00. Ths paaee (a) of a consan agnide is se o be he incease in kineic enegy KE on he basis of he velociy developen fo a nifo o a paabolic pofile; see Sbsecion 3.3. oncening oving flid paicles he physical ni of KE is poe i.e. enegy pe second. (10) We fond neically ha a 3k a lage pesse gadien eiss in he noal diecion nea he inle and disappeas as inceases. This noal pesse gadien is cased by NWS. (11) We s evalae he elaionship beeen and he noal pesse gadien o NWS. The fis la of heodynaics (consevaion of enegy) saes ha he incease in he enegy of a aeial egion is he esl of ok and hea ansfes o he egion [12]. If hee is no hea ansfe hen soe ok is done on he flid paicles fo KE. The physical nis of enegy and ok ae he sae. Ths paaee (b) is se o be he poe RW done by NWS. Theefoe he ain eseach sbjec is o neically invesigae he elaionship aong KE RW and. 2. alclaion of adial pesse gadien Govening eqaions. is e conside diensionless vaiables. ll lenghs and velociies ae noalized by he pipe diaee + and he ean velociy ls especively. The pesse is noalized by n B pogl. is based on lq and +. Noe ha he diensionless aial coodinae (= 6 + ) is sed fo calclaion and 4e2W6 + s is sed in o figes and ables hee 6 is he acal aial coodinae. We conside nseady flo of an incopessible Neonian flid ih a consan viscosiy and densiy and e disegad gaviy and eenal foces. We inodce he seafncion and voiciy folas in he o-diensional cylindical coodinaes fo he govening eqaions in ode o avoid he eplici appeaance of he pesse e. ccodingly he velociy fields ae deeined iho any asspions concening he pesse. Sbseqenly he pesse disibion is calclaed sing he vales of he velociy fields. (2.1) The diensionless anspo eqaion fo he voiciy is epessed as v v y z e{ y v y } *~ The Poisson eqaion fo is deived fo he definiion of i.e. y y y (2.2) 1 The aial velociy and adial velociy ae defined as he deivaives of he seafncion i.e. (2.3) _ y Only he angla ( ) coponen of a o-diensional flo field is non-negligible; hs e shall eplace ih û (2.4) Š Œ ]Ž y

4 ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER 171 Pipe inle Pipe all Pipe ole J0 lo J1 J2 j R - R 2 1 eneline 1 i i+1 I0 IG Gid syse in a cicla saigh pipe. The pesse can be calclaed sing he seady-sae fo of he Navie-Sokes (N-S) eqaions. The pesse disibion fo he deivaive is (2.) and ha fo he deivaive is (2.6) Since and ae knon a evey poin fo (2.3) a sooh pesse disibion ha saisfies boh (2.) and (2.6) is calclaed sing Poisson s eqaion (2.7) [1] (2.7) ' In his sdy iniial vales ae obained sing (2.) and hen (2.7) is sed o obain bee solions Neical ehod fo voiciy anspo eqaion. The ecangla gid syse sed hee is scheaically illsaed in ig. 2.1 hee and ae he ai nbes fo aial and adial gid poins especively and Ũ'š W š %_ W and. To obain pecise esls in his sdy e sed a efined aial gid of P!4 G EDE. o calclaions he diensionless aial gid P! 2 P!4: ) is sed. The gid space P! 6 2 P!4Š œ+ ) in acal lengh is consideed. o a pipe of + = 2.6 c and = 2000 he diensionless gid space P!4 = coesponds o P! 6 = 0.02 c in acal lengh: P! 6 = = 0.02; fo = P!4 = coesponds o P! 6 = 0.26 c in acal lengh. To gid syses ae sed: (i) = 1001 = 1 and P!4 = and (ii) = 101. The ai 4 is o nseady pobles geneally an eplici ehod is fase han an iplici ehod in PU ie b lacks calclaion sabiliy. The finie diffeence eqaion fo (2.1) as fis solved by he Gass-Seidel eplici ieaive ehod [7 9] hee calclaion sabiliy as achieved by adding seps 4 in ig..1; see ppendi. This eplici schee hoeve eqied long PU ies o ainain copaional sabiliy. Ne i as ipoved by he

5 y P y ª ETN Ken Sae Univesiy hp://ena.ah.ken.ed 172 H. KND Sa 1) Se iniial and bonday condiions SBž YBž 2) alclae fo Ÿ yn and Eq. (2.) iplici heck Ÿ Bž ž YBž - Ÿ E f Gass-Seidel ynbž SBž ysbž 3) alclae fo and Ÿ Eq. (2.2) heck y Ÿ Bž ž ynbž - Ÿ E - Gass-Seidel 4) heck ynbž - ys D - ) Se iniial vales fo pesse Eq. (2.) 6) alclae bee pesse Eq. (2.7) heck 7Ÿ ž - Ÿ D e Gass-Seidel End IG locha fo iplici ieaion ehod. iplici ehod shon in ig. 2.2 hee Ÿ and Ÿ and is he inde of ieaion [1 19]. In his sdy e se he iplici ehod. The iplici fo fo (2.1) is ien as (2.) YBž Y vyn vybž z f{ ae povisional vales n is he ie sep ys vybž YBž YEž ys YBž «~ This copaional schee involves he oad-tie eneed-space (TS) ehod. he all a hee-poin one-sided appoiaion fo deivaives is sed o ainain secondode accacy. The schee hs has second-ode accacy in space vaiables and fis-ode accacy in ie. onside he iniial seafncion. o (2.3) he iniial condiion fo he seafncion is given by 0 [ T/ P Ž <fy<œ < <k U

6 ² ² 3 ² 3 b b b b b ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER 173 Wihin he bondaies he iniial voiciy is obained by solving (2.2). The velociies and ae se sing (2.3) heneve he seafncion is nely calclaed. The folloing ae he bonday condiions. (2.9) yq 0 (i) he ceneline: y <fs<fṽ. (ii) he inle: y U I T/sP Ž < <k. (iii) he all: GiI ±p fxsp Ž <fs<fṽ. The voiciy bonday condiion a he no-slip alls is deived fo (2.4) as hee-poin one-sided appoiaion fo (2.9) is sed o ainain second-ode accacy (2.10) ² ² ² 0 J P (iv) he ole he linea eapolaion ehod is sed: < <³ G S` 0 P ys² The folloing ae he bonday condiions fo pesse. (v) o he pesse a he ceneline e se he hee-poin finie diffeence fo. Since a 0 0 <Tq<œ U (vi) The pesse a he inle is given as zeo iho he leading edge: < <k (vii) The pesse a he all is deived fo (3.2); see Sbsecion 3.1. o he leading edge ih µ and K pesse gadien is epessed ] J s $ P o he all ih <es<tṽ and º 0 ] 0 ¹ s $ P! < <k G he hee-poin appoiaion is sed fo and and he y ² P! ž º 0» P! (viii) o he oflo bonday condiions he linea eapolaion ehod is sed: 2.3. alclaed esls of adial pesse gadien. The neical calclaions ee caied o fo = and in 2006 on an NE SX-7 specope ih a peak pefoance of.3g-lops/pocesso. Table 2.1 liss gid syses ie sep (P ) nbe of ie seps nil seady sae (T-seps) and PU ies. P as fo T-sep = 0 o and as inceased o o gadally o and iges 2.3 hogh 2.7 sho he calclaed esls of (a) aial pesse dop and (b) pesse disibion in he adial diecion. The elaionship beeen he pesse ( ) and pesse dop (P' ) is P' = 0 - = - hee is zeo a he inle. To veify he accacy of calclaions he esls of calclaions ae copaed ih he sooh cves dan sing Shapio e al. s epeienal esls [17] as shon in ig. 2.3(a) y ²

7 ETN Ken Sae Univesiy hp://ena.ah.ken.ed 174 H. KND TBLE 2.1 Gid syse ie sep ¼¾½ T-seps and PU ies. I0/J0 ¼º½ T-seps PU / h / h / h / h / h / h / h / h / h / h / h / h ceneline epeien all X= X= X= X= Pesse Dop Pesse Donsea Disance ( X) Radis () IG (a) ial pesse dop and (b) pesse disibion in -diecion = hogh 2.7(a) hee he diaond and do sybols denoe he calclaed esls fo pesse a he ceneline ( ) and fo pesse a he all (7 ) especively. Nea he pipe inle he epeienal esls fall beeen and and agee ell ih he coped esls donsea. The ajo conclsions concening he adial pesse disibion ae as follos. Hee (P' P' ) = (7ˆ ). (1) I is clea fo ig. 2.3(a) ha a = 1000 hee is a lage diffeence beeen P' and P' acoss he adis of he pipe a 4= G and ha his diffeence deceases as 4 inceases. (2) I is seen fo igs. 2.3(a) hogh 2.7(a) ha he diffeence (P' - P'7 ) deceases as inceases. = he diffeence eiss a 4K and disappeas donsea. (3) Noe ha P' is lage han P'7. This indicaes ha is loe han 7 as veified in igs. 2.3(b) hogh 2.7(b). This diffeence conadics he esls obained by ohes sing he bonday-laye heoy and i also conadics Benolli s la alhogh Benolli s la does no apply o viscos flo. The adial pesse gadien nea he all and inle is seen o be lage b i deceases nea he ceneline and donsea Radial pesse disibion. Le s conside he qesion heoeically: Which is zeo a he all is highe o in he adial diecion? Since he aial velociy

8 3 ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER ceneline epeien all X= X= X= X= Pesse Dop Pesse Donsea Disance ( X) Radis () IG (a) ial pesse dop and (b) pesse disibion in -diecion = ceneline epeien all X= X= X= X= Pesse Dop Pesse Donsea Disance ( X) Radis () IG. 2.. (a) ial pesse dop and (b) pesse disibion in -diecion = he coponen of velociy can be linealy appoiaed as (2.11) o (2.9) (2.10) and (2.11) he voiciy a he all is siply appoiaed as (2.12) û 0 À Sbsiing (2.12) ino (3.2) (see Sbsecion 3.1) gives (2.13) $Â Á 0 ÃÂ 3 U P ÅÄ Á vû ÃÂ 3 P ž Á `» P!7P < G Ths since ž Æ» in he enance egion he noal pesse gadien a he all becoes negaive. Theefoe i is veified fo (2.13) ha he pesse gadien in he adial diecion is negaive a he all of he enance egion. On he ohe hand in he flly developed egion since ž?» in (2.13) he noal pesse gadien a he all becoes 0 hs aking he pesse disibion nifo in he adial diecion.

9 Á ETN Ken Sae Univesiy hp://ena.ah.ken.ed 176 H. KND ceneline epeien all X= X= X= X= Pesse Dop Pesse Donsea Disance ( X) Radis () IG (a) ial pesse dop and (b) pesse disibion in -diecion = ceneline epeien all X= X= X= X= Pesse Dop Pesse Donsea Disance ( X) Radis () IG (a) ial pesse dop and (b) pesse disibion in -diecion = (2.14) The velociy disibion in he flly developed egion is given by ` lˆ { *W hee l in diensionless fo. Diffeeniaing (2.14) ih espec o gives $ h àh Y b Èj hee he diensionless vale of Á is 0.. Ths he vale of onoonically deceases fo a lage posiive vale a he leading edge o in he flly developed egion. 3. Evalaion of adial pesse gadien. Ç. The diensionless N-S eqaion in veco fo [3] is ien as _ š ÉEÊ-ËEÌ Í'f Œ (3.1) 3.1. Noal pesse gadien a all. Hee e conside he adial pesse gadien Since he velociy veco š o (3.2) àÁ a he all he noal coponen of (3.1) a he all edces ΠàÁ U $ Á

10 ÚÚÚ Á Ø Ž Á Ù ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER 177 Œ lo Wall ÏÑÐ ÒcÓ ÔfÕ Öf ŽÜÛ Ã $ Á Á Ý $ Á eneline lid paicle ih voiciy àÁ IG Diecions of cl of voiciy a all. Noe ha he noal pesse gadien is deived fo he negaive noal coponen of he cl of voiciy a he all hich is heeafe called he noal all sengh NWS. o (3.2) NWS is epessed as (3.3) NWS Þ Î Ã v $ Á J àÁ The folloing chaaceisics of NWS ae consideed. (i) NWS is effecive nea he pipe inle hee he voiciy gadien in he -diecion is lage and deceases invesely ih. In he flly developed egion NWS vanishes since he voiciy a he all is consan and hen he cl of voiciy disappeas. (ii) I is clea fo (3.3) ha NWS cases a pesse gadien in he adial diecion ha is he pesse gadien a he all esls fo he cl of voiciy. NWS and he adial pesse gadien have he sae agnide a he all b ae opposie in diecion. When NWS is dieced fo he all o he ceneline as shon in ig Noe ha NWS cases he flid paicles nea he all o ove oads he ceneline in he noal diecion i.e. i acceleaes he flid paicles in he cenal coe. (iii) When adoping he bonday-laye asspion NWS vanishes since is alays negleced in he asspion Tangenial-voiciy soce sengh. We conside ohe foces a he all in accodance ih Lighhill [13]. oiciy is podced on a solid body o solid all sface and speads fo hee ino he flid. alos all poins on he bonday he voiciy has a nonzeo gadien along he noal. This gadien liplied by (2/) epesens he flo of oal voiciy o of he sface pe ni aea pe ni ie so ha i is he local sengh of he sface disibion of voiciy soces. (1) Tangenial-voiciy soce sengh. The angenial-voiciy soce sengh has a siple elaionship ih pesse gadien. If he sface is aken as k he flo of -voiciy o of i is epessed as ß }à ß á âà _ Ý Hence he angenial-voiciy soce sengh a he all is idenical o he aial coponen of he cl of voiciy liplied by (2/) in agnide b opposie in diecion as shon in ig Noe ha his sengh affecs he pesse gadien in he -diecion b does no affec he pesse gadien in he noal o adial diecion. ŽÜÛk

11 Þ ê Á ê ß Á ê ê ó j ê Û ê ETN Ken Sae Univesiy hp://ena.ah.ken.ed 17 H. KND (2) Noal-voiciy soce sengh. The noal-voiciy soce sengh is diecly deived nde he coniniy condiion on. The coniniy eqaion fo in cylindical coodinaes is Ìäã±å v v æ 7ç U o he above eqaion he noal-voiciy soce sengh is epessed as In o diensions Á ` vû v ß à Û and hei deivaive ih espec o ae all zeo. ccodingly his noal-voiciy soce sengh vanishes in he o-diensional coodinaes and affecs nohing Incease in kineic enegy. In he enance egion he velociy disibion changes fo nifo a he inle o paabolic in he flly developed egion. The agnide of he incease in kineic enegy is consideed belo. (i) he inle he velociy pofile is nifo: ºèl The kineic enegy acoss he inle is given by liplying he fl by is kineic enegy (3.4) é  ëê ëìdí l í ogl [ j o+ l (ii) In he flly developed egion he velociy has a paabolic disibion (2.14). ccodingly he kineic enegy is calclaed as (3.) é  Bê oä$î l Jœ àï ìe b o+ l (iii) The incease in kineic enegy (KE6 ) in he enance egion is obained by sbacing (3.4) fo (3.) hich gives KE6 b o+ l The diension of his incease in kineic enegy is {ñðò ó j ðò o+ l ó ðò j o+ l This ni coesponds o physical poe i.e. enegy pe second. We define a diensionless incease in kineic enegy pe second KE as (3.6) KE Þ j o+ l o+ l This vale of KE = 0.7 is a consan and is independen of ; hs KE saisfies he necessay condiion of paaee (a). RW is siilaly noalized by (1/2)o+ l ; see Sbsecion 4.2. ê b GÍô ó IG

12 ê ó l l ó ETN Ken Sae Univesiy hp://ena.ah.ken.ed 4. Poe done by NWS. LULTION O MINIMUM RITIL REYNOLDS NUMBER aiaion of enhalpy ih pesse. Using he pesse dop ( ` ) in he adial diecion he aon of ok WK done by NWS is consideed. Hee on he basis of heodynaics [12] he vaiaion of enhalpy õ ih pesse a a fied epeae can be obained fo he definiion õ Œlk hee l is he inenal enegy and is he vole. o changes in õ e have P1õö P]lµKP`Í% o os solids and liqids a a consan epeae he inenal enegy ĺ 0 % does no change as ì lø ì e T ì Š hee is he epeae. Since he change in vole is ahe sall nless he changes in pesse ae vey lage he change in enhalpy P1õ esling fo a change in pesse P' can be appoiaed by (4.1) WK kp1õö3èñp' Eqaion (4.1) can be applied o incopessible flo as ell. The ni of %P' is epessed as { í ðò ðò ó ðò í This ni hoeve is eqal o ok in physics and no o poe sch as KE Poe done by NWS. The poe RW done by NWS o he ae of change of he ok WK can be obained by dividing he ok given in (4.1) by peiod P b a his poin he peiod is no ye knon RW WK P Hee conside he diensionless RW. RW is noalized by he sae ehod as KE in (3.6). %P' P Lenghs pesse and ie ae noalized by + (1/2)oGl and + l especively. ccodingly he diensionless RW is epessed as RW 'P' P + í B ùãûqü& pogl ûsý í 2l +þ n EùÃÜûqü& po+ ûsý l hee (6 ) denoes diensional qaniies. The poe RW ill be deeined by he folloing seps. (1) We begin by calclaing he ok fo (4.1) fo he shaded space beeen 0 and?/ in ig. 2.1 hee i is assed ha NWS is effecive fo he all o in he adial diecion becase hee ae fe diffeences in pesse in he adial diecion nea he ceneline as shon in igs. 2.3(b) hogh 2.7(b). Hence he vole ÿ ha NWS affecs is epessed as (4.2) ÿ 2 g î : P ± fxsp Ž ï P!

13 P 3 3 ETN Ken Sae Univesiy hp://ena.ah.ken.ed 10 H. KND Pipe inle Pipe all Pipe ole lo J0 J1 J2 j 1 2 R - R 2 1 eneline 1 i i +1 i +2 IG Gid syse ¼ ¼. Ne he pesse diffeence in he adial diecion is appoiaed by he diffeence beeen 2 and N0 : P' 0 2 ]9 0 2 ˆ1 0g Í Ý ž (4.3) b Œ 0 Ý ž Œ ž (2) The peiod ding hich NWS acs on he flo passing along voiciies a and 0Çk is consideed. The disance beeen and 0ÃWX is P!. The velociies a o poins 0 ˆX and qš / especively ae 0 and ž. ccodingly he povisional peiod P 0 ay be given by dividing he aial gid space P! by he ean velociy a? p[þ P! 0 0 q ž P! ž Hoeve if his povisional P is he coec peiod he folloing inconsisency ill be enconeed. To siple cases ae aken as eaples. is if he gid aspec aio is (a) P!k P as shon in ig. 2.1 he ok WK(a) and he poe RW(a) fo he shaded space beeen p and 7:/ ae epessed as (4.4) WK äg 'P' RW äg 'P' ;'P'7Ç ž (4.) P! ž P Ne if he gid aspec aio is (b) P!] P and ÿ*t š as shon in ig. 4.1 he ok WK(b) in is calclaed by adding he ok in ÿ and in (4.6) WK [ÈÿXP'$g= P' 3ÈñP' hee i is assed ha P' 3 P' 3 P'. Siilaly he poe RW(b) in is calclaed by adding he poe in and in RW&g ÿxp' P' ;ñp'ã ž (4.7) P ž P ž P 3 RW ä

14 P P 3 p /. o sipliciy he above saeen is claified sing (3.2) and (4.9) in discee fo. Seing P! o be " P P' ž P žã P! P! " P " P žã ß žã» $k žã» S žã» (4.9) $ í&íí ñ û ž à 3 "q ž ž " P û P ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER 11 voiciy(i +1 J0) voiciy(i J0) voiciy(i +2 J0) Wall J0 noal all sengh(i ) J1 noal all sengh(i +1) R - eneline i i +1 i +2 ž IG Balance of NWS and pesse a all. ž 3À ž hee 3h. s seen fo (4.4) and (4.6) WK(a) and WK(b) ae he sae. When copaing RW(a) and RW(b) especively calclaed sing (4.) and (4.7) hoeve he poe RW(b) is ice as high as RW(a) alhogh he vole and posiion ae he sae. To avoid his inconsisency & he folloing peiod is eqied fo a geneal gid syse of : P!þW" P (4.) 0aÞ 0 0 hee fo (2.12) œ P 0 ž P. û ž This peiod is based on he folloing asspions. (i) The no-slip condiion a he all eans ha he flid paicles ae no ndegoing anslaion; hoeve hey ae ndegoing a oaion. I can be iagined ha he all consiss of an aay of ables ha ae oaing b eain a he sae locaion a he all [14]. (ii) The oaion of a flid paicle a he all yields a voe and voiciy. Then he cl of voiciy yields NWS fo (3.3). The diaee of he voe of he flid paicle a he all is P. ccodingly NWS is podced pe voe o pe P. ige U I 4.2 shos he balance beeen NWS and pesse a he conac sface of hee i is assed ha he voiciy gadien is linea in a sall space beeen 0 and. Ç:/ û i.e. žã û žã» 3 û žã» û žã» 3 íí&í 3 ž

15 P ETN Ken Sae Univesiy hp://ena.ah.ken.ed 12 H. KND TBLE 4.1 Poe (RW) and evalaion cieia a = 2000 = 1. c1 c2 c3 c4 B Ç ë s B ä /à ë Ç % ±Ü / ± RW !n"$ TBLE 4.2 Poe (RW) and evalaion cieia a = 2000 = 101. c1 c2 c3 c4 ä / % ±Ü / ± RW !n"$ (3) The poe RW(i) fo he vole þ0 2 is deived fo (4.2) (4.3) and (4.). Ths he oal poe RW is (4.10) RW &% ÿ 2 P' Effecive egion of NWS. In ode o calclae he poe RW in (4.10) e s deeine he effecive aial and adial egions of NWS. The effecive egion is deeined by he folloing cieia c1 c4. Hee Z[ 0NZ` 0 SZ`v *0 in he adial diecion. ially: (i) c1 = ( 1v ) (ii) c2 = (v g! )/. Radially: (iii) c3 = ( )/( 1v (iv) c4 =. The calclaed esls of RW ae lised in Tables 4.1 fo = 1 and 4.2 fo = 101. The effecive egions of NWS ae deived sing Tables 4.1 and 4.2 and ig. 2.4 a = 2000.

16 Â Â M j ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER 13 TBLE 4.3 Poe RW and RW (1) The aial effecive lengh of NWS is consideed. o ases No. 4 (c1 = 0.0) and 13 (c2 = 0.0) he effecive aial lenghs ae and o ig. 2.4(a) a X = e can see a sall pesse diffeence ( 1 ) so cieion c2 is jdged o be bee han c1. 4 = = and = c1 = ( a ) = and c2 = ( )/ = = e can see a vey sall pesse diffeence (7S] ). Then 7 = and = ; c1 = 0.01 and c2 = = hee is no pesse diffeence (Ç ). Then7 = and = ; c1 = and c2 = ccodingly he heshold of c2 igh be se a I0=1001 J0=101 I0=1001 J0=1 KE=0.7 Poe (RW) ynolds nbe () IG Poe (RW) vs. (2) The adial effecive lengh is consideed. o ases No. 23 and 26 (c3 = 1.00) in Table 4.2 he pesse is consan fo = 0 (ceneline) o = 0.30 and fo = 0 o = 0.0 especively. o ases No. 23 hogh 2 c3 = 1.00 a = a = 0.31 and 0.90 a = 0.32 especively. ccodingly he heshold of c3 igh be se a 0.9. (3) The voiciy is ansfeed fo nea he all ino he cenal coe by NWS. o ases No. 16 and 32 he peneaed adial lengh of 9»(' is = 0.2 fo = 1 and 0.33 fo = 101. Since he vale of RW is vey sall sch as and 0.07 especively e canno se c4 as a cieion fo deeining he effecive egion of NWS alclaion of ini. The calclaed RW vales ae lised in Table 4.3 and ploed agains in ig. 4.3 hee he aseisks and dos denoe he calclaed esls fo = 1 and = 101 especively. The ini ciical ynolds nbe 012"$ is calclaed via linea inepolaion eploying he vales of RW fo = 2000 and 3000 EE BDE BDE 012"$ `@ `@ ED RW ss 012"$ DE EED RW s GÍô Iä

17 ETN Ken Sae Univesiy hp://ena.ah.ken.ed 14 H. KND We hs obained!"$ of 370 hen = 1 and 2200 hen = 101. onclsions. concepal acoodel as bil o deeine /01"$ fo pipe flos on he basis of he esls of o epeiens and pevios calclaions. The calclaed esls ee /01"$ = 370 hen = 1 and 2200 hen = 101. The odel is based on NWS. NWS cases he diffeence ( v ) in he adial diecion and acceleaes flid paicles in he cenal coe. In he enance egion he velociy pofile changes fo a nifo disibion a he pipe inle o a paabolic one in he flly developed egion. The flid paicles in he cenal coe ae acceleaed. The agnide of he eqied nondiensional acceleaion poe is KE = 0.7 hich is deived fo he diffeence in kineic enegy beeen he flo a he inle and ha in he flly developed egion. The occence of he ansiion depends on he acceleaion poe RW given by NWS: (a) hen RW 0.7 ansiion akes place; (b) hen RW Z 0.7 ansiion neve akes place. deailed sdy of he physical echanis behind NWS and he occence of ansiion ill be a fe ok. cknoledgen. We ish o epess o sincee appeciaion o Eeis Pofesso. Senge of he Univesiy of Uah Pofesso T. K. DeLillo of Wichia Sae Univesiy and D. K. Shiokai of SGI Japan fo sefl advice and encoageen and o he Infoaion Synegy ene Tohok Univesiy fo is osanding copaional sevices. REERENES [1] R.-Y. HEN lo in he enance egion a lo ynolds nbes J. lids Eng. 9 (1973) pp [2] P. G. DRZIN ND W. H. REID Hydodynaic Sabiliy abidge Univesiy Pess 191 p [3] S. GOLDSTEIN Moden Developens in lid Dynaics ol. 1 Dove 196 pp [4] R.. GRNGER lid Mechanics Dove 199 pp [] L. M. HUNG ND T. S. HEN Sabiliy of he developing laina pipe flo Phys. lids 17 (1974) pp [6] L. M. HUNG ND T. S. HEN Sabiliy of developing pipe flo sbjeced o non-aisyeic disbances J. lid Mech. 63 (1974) pp [7] H. KND Neical sdy of he enance flo and is ansiion in a cicla pipe ISS (Insie of Space and sonaical Science) Tokyo po No [] H. KND opeized odel of ansiion in cicla pipe flos. Pa 2. alclaion of he ini ciical ynolds nbe SME (eican Sociey of Mechanical Enginees) ED 20 (1999) pp [9] H. KND Laina-blen ansiion: alclaion of ini ciical ynolds nbe in channel flo RIMS (seach Insie fo Maheaical Science Kyoo Univ.) Kokyok Bessas B pp [10] H. KND ND T. YNGIY Hyseesis cve in epodcion of ynolds s colo-band epeiens J. lids Eng. 130 (200) (10 pages). [11] R. R. KERSWELL cen pogess in ndesanding he ansiion o blenve in a pipe Nonlineaiy 1 (200) pp. R17 R44. [12] D. KONDEPUDI ND I. PRIGOGINE Moden Theodynaics John Wiley & Sons 199 pp [13] M. J. LIGHTHILL Laina Bonday Layes L. Rosenhead ed. Dove 19 p. 4. [14] R. L. PNTON Incopessible lo Wiley-Inescience 194 p [1] P. J. ROHE ndaenals of opaional lid Dynaics Heosa 199 pp [16] O. REYNOLDS n epeienal invesigaion of he cicsances hich deeine hehe he oion of ae shall be diec o sinos and of he La of esisance in paallel channels Tans. Royal Soc. London 174 (13) pp [17]. H. SHPIRO R. SIEGEL ND S. J. KLINE icion faco in he laina eny egion of a sooh be Poc. 2nd Nal. ong ppl. Mech. SME 194 pp [1] K. SHIMOMUKI ND H. KND Neical sdy of noal pesse disibion in enance flo beeen paallel plaes. I. inie diffeence calclaions Elecon. Tans. Ne. nal. 23 (2006) pp hp://ena.ah.ken.ed/vol /pp di/pp hl.

18 + ) ) ETN Ken Sae Univesiy hp://ena.ah.ken.ed LULTION O MINIMUM RITIL REYNOLDS NUMBER 1 [19] K. SHIMOMUKI ND H. KND Neical sdy of noal pesse disibion in enance pipe flo Elecon. Tans. Ne. nal. 30 (200) pp hp://ena.ah.ken.ed/vol /pp10-2.di/pp10-2.hl. [20] S. TNEKOD lid Dynaics by Leaning fo lo Iages (in Japanese) saka Pblishing o. Tokyo 1993 p. 16. [21] T. TTSUMI Sabiliy of he laina inle-flo pio o he foaion of Poisseile egie J. Phy. Soc. of Japan 7 (192) pp ppendi..1. Noenclae. (*) + = pipe diaee = 2 = conacion aio = + = belloh diaee õ = enhalpy = l + 7 hee is vole = aial poin of gid syse = nbe of aial gid poins = adial poin of gid syse = nbe of adial gid poins KE = incease in kineic enegy (ni is poe); see (3.6) NWS = noal all sengh; see (3.3) = pesse = 6 nn B pogl = adial coodinae = 6 = pipe adis = 6 + = 0. = ynolds nbe = l + *) ) RW = poe done by NWS o ae of change of ok; see (4.10) = ie = l +þ = epeae = aial velociy l = ean aial velociy a pipe inle l = inenal enegy = adial velociy = velociy veco o vole WK = ok done by NWS = aial coodinae = 6 6 = acal aial coodinae 4 = aial coodinae = RR = 6 o y = flid densiy y = seafncion = 6 2l + = voiciy = + l = angle in cylindical coodinaes = kineaic viscosiy P' = pesse dop P = adial gid size P! = aial gid size Spescip: 6 = diensional qaniy + RRX

19 ª ETN Ken Sae Univesiy hp://ena.ah.ken.ed 16 H. KND.2. locha fo eplici ieaion ehod. Sa 1) Se iniial and bonday condiions 2) alclae SBž fo Y Eq. (2.1) eplici y 3) alclae Ÿ Bž ž Bž y fo and Ÿ Bž Eq. (2.2) Gass-Seidel Bž 4) alclae Ÿ ž y fo Ÿ Ež ž Eq. (2.2) ) heck ÇŸ Bž ž YBž - D 6) heck ynbž - ys D - 7) Se iniial vales fo pesse Eq. (2.) ) alclae bee pesse Eq. (2.7) heck + ž - + E - Gass-Seidel End IG..1. locha fo eplici ieaion ehod.