Turbulent buoyant confined jet with variable source temperature

Transcription

1 Tubulen buoyan confined je wih vaiable souce empeaue M. F. El-Amin 1,, Amgad Salama 1 and Shuyu Sun 1 1 King Abdullah Univesiy of Science and Technology (KAUST), Thuwal , Kingdom of Saudi Aabia Depamen of Mahemaics, Faculy of Science, Aswan Univesiy, Aswan 8158, Egyp Absac In his wok, expeimenal and numeical invesigaions ae consideed fo confined buoyan ubulen je wih vaying inle empeaues. Resuls of he expeimenal wok and numeical simulaions fo he poblem unde consideaion ae pesened. Fou cases of diffeen vaiable inle empeaues and diffeen flow aes ae consideed. The ealizable k ε ubulence model is used o model he ubulen flow. Compaisons show good ageemens beween simulaed and measued esuls. The aveage deviaion of he simulaed empeaue by ealizable k ε ubulen model and he measued empeaue is wihin %. The esuls indicae ha empeaues along he veical axis vay, geneally, in nonlinea fashion as opposed o he appoximaely linea vaiaion ha was obseved fo he consan inle empeaue ha was done in a pevious wok. Fuhemoe, hemal saificaion exis paiculaly close o he enance egion. Fuhe away fom he enance egion he vaiaion in empeaues becomes elaively smalle. The saificaion is obseved since he sa of he expeimen and coninues duing whole ime. Numeical expeimens fo consan, monoone inceasing and monoone deceasing of inle empeaue ae done o show is effec on he buoyancy foce in ems of Richadson numbe. Keywods: ealizable k ε model, ubulen je, saificaion, hea soes, CFD 1

2 Nomenclaue A 0 consan defined in Eq. (3) A s consan defined in Eq. (4) b CFD c local je widh [m] Compuaional Fluid Dynamics laeal spead ae of he je C 1 paamee defined in Eq. (0) C consan equals 1.9 C p specific hea [m /s K] C 1ε consan equals 1.44 C 3ε paamee defined in Eq. (0) C µ coefficien defined in Eq. (1) d inle diamee [m] g acceleaion due o gaviy [m/s ] G b geneaion of ubulence due o buoyancy G k geneaion of ubulen kineic enegy due o he mean velociy gadien I 0 inensiy of ubulence a he nozzle inle k R Re S T ubulen kineic enegy adial axis [m] Richadson numbe Reynolds numbe sain ae empeaue [K]

3 T exp measued empeaue [K] T in inle empeaue [K] T sim simulaed empeaue [K] p P P 0 ime [s] pessue [Pa] dynamic pessue [Pa] inle dynamic pessue [Pa] P c ceneline dynamic pessue [Pa] P u u c ubulen Pandl numbe fo enegy mean axial velociy componen [m/s] ceneline velociy [m/s] u 0 inle velociy [m/s] v z mean adial velociy componen [m/s] axial axis [m] Geek symbols Δ ime sep [s] Δ T empeaue diffeence [K] Δ ρ densiy diffeence [kg/m 3 ] β ε λ hemal expansion coefficien [1/K] ubulence dissipaion ae hemal conduciviy W/(m.K) µ dynamic viscosiy [kg/m s] µ ubulen viscosiy 3

4 ν kinemaics viscosiy [m s] ρ densiy [kg/m 3 ] θ consan defined in Eq. (5) σ k σ ε ubulen Pandl numbe fo k ubulen Pandl numbe foε Inoducion Alhough saificaion of fluids due o he exisence of empeaue gadiens is no desiable in many pocesses ha equie homogenizaion, i is, in ohe pocesses (e.g., hea soage anks) desiable because of he low mixing mechanisms involved which help mainaining he equied empeaue disibuion. The poblem, howeve, in hemally saified hea soage anks is is sensiiviy o exenal disubance. Tha is he sae of hemal saificaion could be desoyed once sufficien disubance is inoduced. In paicula, he condiions a he inle ae consideed as one example of such disubance souces. Theefoe, i is impoan o sudy he effec of inle condiions on saificaion behavio of such sysems. Since, in hea soage anks and in many ohe applicaions, fluids ene he ank in he fom of buoyan je, gea deal of woks on je flow have been consideed eihe expeimenally o numeically. Howeve, he poblem of buoyan heaed je wih vaiable souce empeaue which can be found in many indusial and envionmenal applicaions, has eceived elaively lile aenion. Je flow can be divided mainly ino hee ypes, pue je, pue plume and foced plume. In pue jes, fluids ine he domain wih high momenum fluxes which essenially cause highe inensiy of ubulen mixing. In pue plume, on he ohe hand, buoyancy fluxes cause local acceleaion leading o ubulen mixing. In he geneal case of a foced plume a combinaion of iniial momenum and buoyancy fluxes ae esponsible fo ubulen mixing. Seveal echniques wee developed o sudy jes in confined spaces as will be explained lae. Laely, wih he incease of 4

5 compues efficiencies and capaciies, compuaional fluid dynamics (CFD) became one of he essenial ools o exploing on fluids behavio of such fundamenal impoance. In ubulen flows Reynolds Aveage Navie-Sokes echnique (RANS) ae usually adoped in ode o make he sysem amenable o soluion. The poblem of using RANS appoach, howeve, is ha ill now, hee is no unifying se of equaion o model all kinds of ubulen flows and hea ansfe scenaios. Theefoe, i is impoan o choose he model which suies he case unde invesigaion and even o calibae is coefficiens in ode o fi expeimenal esuls. El-Amin e al. [1] invesigaed he D upwad, axisymmeic ubulen confined je and developed seveal models o descibe flow paens using ealizable k ε ubulence model. Fuhemoe, CFD analysis of he flow sucue of a hoizonal wae je eneing a ecangula ank has been done by El-Amin e al. [, 3]. Thei findings wee lae used by Panhalookaan e al. [4] o calibae boh ealizable and RNG k ε ubulence models so ha hey may be used fo simulaing saified ho wae soage anks. Compehensive eviews of je flows wee pesened by Rajaanam [5] and Lis [6]. Fuhemoe seveal expeimenal woks wee conduced o highligh he ineesing paens and he govening paamees peinen o his kind of flows including he wok of Wygnanski and Fiedle [7], Rodi [8], Panchapakesan and Lumley [9-10], Fukushima e al. [11] Agawal e al. [1], O Hen e al. [19] and many ohes. On he ohe hand, seveal phenomena peinen o buoyan jes wee invesigaed by many auhos. Fo example, he poblem of enainmen by a plume o je a densiy ineface was consideed by Baines [13], mechanisms involved in ansiion o ubulence in buoyan plume flow was invesigaed by Kmua and Bejan [14], ound buoyan jes wee also invesigaed by Shabbi and Geoge [15], and Papanicolaou and Lis [16], bifucaion in a buoyan hoizonal lamina je was sudied by Aakei, e al. [17]. Wih espec o he kind of fluids used in buoyan je sudies, seveal eseaches have consideed diffeen fluids fo eihe expeimenal o numeical invesigaions. Fo example O Hen e al. [18] pefomed expeimenal wok on a ubulen buoyan helium plume. El-Amin and Kanayama [19, 0] sudied buoyan je esuling fom hydogen leakage. 5

6 They developed he similaiy fomulaion and soluions of he ceneline quaniies such as velociy and concenaion. Fuhemoe, El-Amin [1] inoduced a numeical invesigaion of a veical axisymmeic non-boussinesq buoyan je esuling fom hydogen leakage in ai as an example of injecing a low-densiy gas ino high-densiy ambien. On he ohe hand, he mechanics of buoyan je flows issuing wih a geneal hee-dimensional geomey ino an unbounded ambien envionmen wih unifom densiy o sable densiy saificaion and unde sagnan o seady sheaed cuen condiions is invesigaed by Jika []. He fomulaed an inegal model fo he consevaion of mass, momenum, buoyancy and scala quaniies in he ubulen je flow. Fuhemoe Jika [3] exended his wok o also encoune plane buoyan je dynamics esuling fom he ineacion of muliple buoyan je effluxes spaced along a diffuse line. In he pevious wok by El-Amin e al. [1], analyses of he componens of D axisymmeic veical unheaed/heaed ubulen confined je using ubulence ealizable k ε model wee conduced. Moeove expeimenal wok was elaboaed fo empeaue measuemens of such sysem o povide veificaion of he models used. In ha wok, seveal models wee consideed o descibe axial velociy, ceneline velociy, adial velociy, dynamic pessue, mass flux, momenum flux and buoyancy flux fo boh unheaed (non-buoyan) and heaed (buoyan) je. In ha wok inle empeaues wee consideed fixed. Howeve, in many applicaions inle empeaues ae no, geneally fixed. An expeimenal sudy of a saified hemal soage unde vaiable inle empeaue fo diffeen inle designs was pefomed by Abo-Hamdan e al. [5]. Fuhemoe, Yoo and Kim [4] inoduced appoximae analyical soluions fo saified hemal soage unde vaiable inle empeaue. In his wok, analysis of veical ho wae je eneing a cylindical ank filled wih cold wae wih vaiable inle empeaue is conduced. The inle empeaue of he buoyan je is allowed o change wihin a small ange and is consideed as a funcion of ime. Numeical invesigaions unde he above menioned condiions ae pefomed in ode o obain fields of pessue, velociy, 6

7 empeaue and ubulence. D axisymmeic simplificaion is assumed o educe he gid size in he soluion domain and he ealizable k ε model is used o model ubulen flow. Measuemens Schemaic diagam of he expeimenal seup is shown in Fig. 1. The cylindical ank, made of m hick galvanized ion shees, has a diamee of 0.36 m and a heigh of m is shown in Fig. a. The inle pipe is locaed a cene of he boom of he ank wih an inne diamee of 0.0 m. he inle pipe is inseed in he ank up o a heigh of 0.06 m above he base of he ank. The oule pipe, locaed a cene of he op of he ank, has an inne diamee of 0.0 m wih a deph in he ank of m fom he op plae. The unceainy in he diamee of he inle and oule pipes is ± m. This geomey suggess ha he ank, he inle, and he oule may be modeled as axisymmeic aound he veical axis. Themal effecs ae measued by hemocouples of K-ype which wee calibaed agains a sandad PT-5 esisance hemomee wih an aveage calibaion eo of ± 0.5 K. Flow ae is measued using a magneic-ype wih a calibaion eo ± 3.5%. The empeaue is ecoded in Kelvin using a daa acquisiion sysem conneced wih a pesonal compue. The daa acquisiion sysem has an eo abou ± 1K. The above esimaed eos ae included in he measued daa. Themal effecs ae measued by inseing a veical od wih 9 sainless seel sheahed K-ype hemocouples. The nine sensos ae disibued a diffeen heighs as, 0.06, 0.1, 0.18, 0.4, 0.30, 0.36, 0.4, 0.48 and 0.54 m measued fom he boom. The disance beween he symmey axis and he hemocouples od is 0.09 m, i.e. in he middle beween he symmey axis and he ank wall. The inle empeaue was measued using anohe hemocouple in which is locaed a he inle pipe. I is impoan o indicae ha all pecauions have been aken o make sue he geomeical symmey is achieved in he sense of he alignmen of he inle and oule ubes, he smoohness of ube maeials, he inle geomey, ec. 7

8 Fo he vaiable inle empeaue, he used paamees ae lised in Table 1. The duaion fo measuemens fo each case was appoximaely 30 min. The inle empeaue and he iniial empeaue ae given in column and 3 of he Table 1, especively, and he flow ae is given in column 4. The inle velociy, Reynolds numbe and ubulence inensiy a he inle nozzle ae calculaed fom he given daa. The bes fiing fo he given cuves of he inle vaiable empeaue, Fig. 3a, can be epesened by a 5 h ode polynomial as a funcion of ime, Eqs. (A.1-A.4) in he Appendix. The coesponding Reynolds numbes, Fig. 3b, and inle ubulence inensiy, Fig. 3c, ae calculaed as funcions of inle empeaue which in un have been epesened by 5 h ode polynomials of ime, Eqs. (A.5-A.8) and (A.9-A.1), especively, wih he aid of Eqs. (13)-(14). These funcions may be epesened by ohe polynomials wih less ode bu he deviaion fom he measued daa will incease. The following empiical elaion (Fluen Use s Guide, Fluen Inc. 003, ch. 6) is used o descibe he ubulen inensiy a he inle nozzle as a funcion of he Reynolds numbe: 0.15 I = 0.16 (13) 0 (Re) The Reynolds numbe wih he inle diamee as a lengh scale is defined by he elaion: u 0 d Re = (14) ν The measued empeaue pofiles fo vaiable inle empeaue ae ploed as a funcion of ime, a diffeen sensos posiions (cases 1-4) in Fig. 4 (a-d). I is appaen ha he empeaues along he veical axis vay in nonlinea fashion wih ime, especially, a lage heighs z 0.18 m (plume 8

9 9 egion). Also, he figues indicae ha he empeaue inceases as ime inceases. Themal saificaion is obseved looking a he diffeence in empeaue op (highe) o boom (lowe). The degee of saificaion, howeve, seems o be moe ponounced in he lowe half of he ank han in he op half. The saificaion is veified fom he beginning of he expeimen and coninues duing whole ime. Mahemaical Fomulaion A compaison sudy was done by El-Amin e al. [1] o es diffeen ubulence models when simulaing confined buoyan je, and hey epoed ha he bes model is he ealizable ε k model. Theefoe, in his wok we conside his model o simulae he poblem unde consideaion. The ealizable ε k model developed by Shih e al. [6] involves a new eddy-viscosiy fomula oiginally poposed by Reynolds [7] and a new model equaion fo dissipaion ε based on he dynamic equaion of he mean-squae voiciy flucuaions. The Reynolds-aveaged Navie-Sokes equaions (RANS) ae given in Eqs. () - (3), and he enegy equaion is epesened by Eq. (4). The govening equaions of mass, momenum and ubulence ake he fom: Coninuiy equaion: 0 ) ( 1 = v z u (1) Axial momenum equaion: z k T T g z v u z u z z p u v z u u u = 3 ) ( ) ( 1 ) ( 1 1 ) ( 0 β µ µ ρ µ µ ρ ρ () Radial momenum equaion:

10 10 k v z v u z p v v z v u v = ) ( 3 ) ( 1 ) ( 1 1 ) ( µ µ ρ µ µ ρ ρ (3) Enegy equaion: = T c z T c z T v z T u T p p ) P ( 1 ) P ( 1 ) ( µ λ ρ µ λ ρ (4) Tubulen kineic enegy (k) equaion: [ ] ρε ρ σ µ µ ρ σ µ µ ρ = b k k k G G k z k z k v z k u k 1 ) ( 1 ) ( 1 ) ( (5) Tubulence dissipaion ae (ε ) equaion: k G C C k C S C z z v z u b ε νε ε ε ε σ µ µ ρ ε σ µ µ ρ ε ε ε ε ε ε ε ) ( 1 ) ( 1 ) ( = (6) In he above equaions, u and v ae he mean axial and adial velociy componens, especively. The ohe quaniies ae ime,, densiy ρ, acceleaion due o gaviy, g, pessue, p, dynamic viscosiy, µ, kinemaics viscosiy, ν, hemal conduciviy, λ, ubulen Pandl numbe fo k, k σ, ubulen Pandl numbe foε, ε σ, T is he empeaue, 0 T is he efeence opeaing empeaue. The ubulence dissipaion ae is denoed byε, while k is he ubulen kineic enegy of he ubulen flucuaions pe uni mass. The ubulen viscosiy µ is defined as: ε ρ µ µ / k C = (7) whee µ C is coefficien, which is a new vaiable defined in he ealizable ε k model and given by he elaion:

11 Cµ = ε /( A 0 ε As ks ) (8) whee A = 4.04, (9) 0 A = 6 1/ cos θ, (10) s 1 θ = (1/ 3)cos (6 1/ S ), (11) ( v / z u ) S = 0.5 / (1) The eddy viscosiy fomulaion is based on he ealizabiliy consains, he posiiviy of he nomal sess and Schwaz inequaliy fo ubulen shea sesses. Fuhemoe, in Eq. (0), C 1 defined by he fom: and C = max[0.43, Sk /( Sk 5 )] (13) 1 ε G k = µ S (14) is he geneaion of ubulen kineic enegy due o he mean velociy gadien. C 3 ε = anh u / v (15) The velociy componen u paallel o he gaviaional veco and v is he componen of he velociy pependicula o he gaviaional veco. In his way, C 1 fo buoyan shea layes fo which he main flow diecion is aligned wih he diecion of gaviy (he case unde sudy). Fo buoyan shea 3 ε = layes ha ae pependicula o he gaviaional veco, C 0. 3 ε = The geneaion of ubulence due o buoyancy is given by he elaion: G b µ T = β g (16) P z 11

12 P is he ubulen Pandl numbe fo enegy and β is he hemal expansion coefficien. The model consans of he k ε model ae esablished o ensue ha he model pefoms well fo ceain canonical flows such as pipe flow, je flow, and bounday laye flow. C 1 ε = 1.44, C =1.9, σ k =1. 0, σ = 1. and P = ε. Shih e al. [6] have compaed hei model (ealizable k ε ubulence model) wih expeimenal daa as well as wih he sandad k ε model fo a ound je flow and ohe flows. The compaison shows a good maching beween hei model and he expeimenal daa han he sandad model. The ealizable k ε model implies ha he model saisfies specific consains on he Reynolds s sesses ha make he model moe consisen wih he physics of ubulen flows and hence moe accuae han he ohe ubulen model. This model conains a new anspo equaion fo he ubulen dissipaion ae. Also, a ciical coefficien of he model, C µ, is expessed as a funcion of mean flow and ubulence popeies, ahe han assumed o be consan as in he sandad model. This allows he model o saisfy ceain mahemaical consains on he nomal sesses consisen wih he physics of ubulence (ealizabiliy). Addiionally, he ealizable k ε model uses diffeen souces and sinks ems in he anspo eddy dissipaion. The modified pedicion of ε along wih he modified calculaion of µ, makes his model supeio o he ohe k ε models. Fo he je flow his model does bee in pedicing he speading ae especially, fo nea egion z<0.35 (see El-Amin e al. [1]). Bounday condiions need o be specified on all sufaces of he compuaional domain. Boundaies pesened in his sudy include inflow (Inle), ouflow (oule), solid wall and axis of symmey as shown in Fig. b. Ω, Ω and Ω wall denoe he bounday of he inle, oule and wall, in ou 1

13 especively. In addiion o he non-ealisic bounday on he axis of symmey Ω axis. The velociyinle bounday condiions imposed a he nozzle ae, u = u on 0, v = 0, and T = T in Ωin (17) T in is defined in Eq. (A.1-A.4) fo he cases of vaiable inle empeaue. Due o he sagnan condiions of wae inside he ank befoe he beginning of he influx, all velociy componens wee iniially se o zeo. Hea ansfe hough he walls of he ank is no aken ino consideaion (adiabaic walls). Densiy of wae, specific hea, hemal conduciviy and viscosiy ae fomulaed as a piecewise-linea pofile of empeaue. The ubulence inensiy and hydaulic diamee chaaceizes he ubulence a he inle bounday. The following equaion of empiical coelaion fo pipe flows is used o descibe he ubulence inensiy a he inle bounday as a funcion of he Reynolds numbe, 0.15 I 0 = 0.16(Re) on Ω in (18) Alenaively, one can use he following k and ε on he inle bounday as (see Kadem e al. [8]), k ( 0.3( / ) ) 3 / = kin = 0.005u0, ε = ε in = kin / d on (19) Ωin whee d is nozzle diamee. 13

14 The bounday condiion on he axis of symmey is epesened by fee-slip condiion which is a non-ealisic wall wih no-ficion when velociy and ohe componens nea he wall ae no eaded. Unlike he no-slip bounday condiion fo which flow has zeo velociy in he wall, fee-slip flow is angen o he suface. On he axis of symmey, he adial velociy componen v, and he gadien of he ohe dependen vaiables (u, ε, k, T) wee equal o zeo. So, one may wie hem as, u = 0, v = 0, T = 0, k = 0, ε = 0 on Ω axis (0) Solid wall bounday condiions ae epesened along he solid walls; he no-slip bounday condiion fo velociies, zeo value fo ubulen kineic enegy, and zeo gadiens fo empeaue and enegy dissipaion ae wee used. T ε u = 0, v = 0, on Ω wall, = 0, k = 0, = 0, on n n 1 1 Ω wall (1) whee n 1 is he ouwad nomal of he wall. Finally, he oule bounday which wae dischaged ouside i feely, can be fomulaed as, u n = 0, v n = 0, T n = 0, k n = 0, ε = 0 n on Ω ou () whee n is he ouwad nomal of he oule bounday. 14

15 Also, saic pessue can be defined a a known given poin in he domain and Fluen exapolaes all ohe condiions fom he ineio of he domain. Iniial condiions ae descibed as follows, u = 0, v = 0, T = T0, k = 0, ε = 0 a = 0 (3) In fac, vey small values ae given as iniial condiions fo k and ε insead of zeo which only speed up convegence of he soluion. Numeical Invesigaions Fig. b shows he compuaional domain wih dimensions of: adius=0.18 m and heigh=0.605 m. The adius of he inle and oule pipes is 0.01 m, while he inle heigh inside he ank is 0.06 m and he oule deph in he ank is m. The meshes ae buil up of Quadaic Submap cells. The numbe of gid elemens used fo all calculaions is 7,984. Fluen 6.1 and he gid geneaion ool Gambi [9] ae used o model he flow in he ank by solving he coninuiy, momenum, ubulence and enegy equaions. In ode o pove gid independence, numeical expeimen fo case 4, is epeaed on he sysemaically efined gids of sizes 7,984 (gid-1), 9,40 (gid-), 1,880 (gid-3) and 3,560 (gid-4) quadilaeal cells, especively. The minimum disances beween he nodes in he especive gids ae m, 0.00 m, m and m and he maximum disances beween he nodes ae m, m, m and m especively in he ode of efinemen. Figs. 5 (a, b) show he esuls of he gid efinemen sudies fo he axial velociy and Tempeaue, especively. The maximum deviaion caused by gid is abou 3 % fo he velociy, and 0.14 % fo he empeaue which could be negligible. 15

16 In ode o achieve convegence, Unde-Relaxaion is applied on pessue, velociies, enegy, ubulen viscosiy, ubulence kineic enegy and ubulen dissipaion ae calculaions. Body Foce Weighed Disceizaion is used fo pessue and he velociy-pessue coupling is eaed using he SIMPLE algoihm. A Second-Ode Upwind scheme is used in he equaions of momenum, enegy, ubulence kineic enegy and ubulence dissipaion ae. Segegaed Implici Solve wih he Implici Second-Ode scheme is used. In ode o use a suiable ime sep, we pefomed a compaison fo one case wih diffeen ime seps as 0.01, 0.1, 0.5 and 1 s which ae shown in Table. This compaison includes empeaue, axial velociy, ubulen kineic enegy and ubulen dissipaion ae of kineic enegy. Fom his able one can noe ha he diffeences ae negligible values. Then, o educe he ime of calculaion we have o use he ime sep of 1 sec. Compaisons Boh measued and simulaed empeaues as a funcion of he ank heigh fo vaious imes and vaiable inle empeaues (cases 1-4) ae ploed in Figs. 6 (a-d), especively. Good ageemen beween he expeimenal and numeical daa is obseved. The maximum eo obseved is 0.35 K, howeve, fo cases 1,, 3 and 4 he maximum eo is 0., 0., 0.35 and 0.35 K, especively, occus afe 30 min of chaging pocess. Axial and Radial Velociies The mean posiive axial velociy u (excluding he efleced velociy wih he negaive values) is nomalized by he ceneline velociy u c agains /cz ( nomalized by he je widh b=cz, c is he laeal spead ae of he je) wih diffeen heighs, fo he case 4 a 15 min, is ploed in Fig. 7. I can be seen fom his figue ha axial velociy pofile shows self-simila behavio. Theefoe, axial 16

17 velociy may be epesened by a Gaussian disibuion using ceneline velociy, u c, heigh, z, and widh, b, as paamees. The Gaussian funcion akes he fom: u = u exp (4) c b This empiical model is ploed in Fig. 8 wih compaison wih he simulaed axial velociy. In his sudy, he paamee of laeal spead ae of he je c=0.11 which lies in he ange of he sandad values as epoed by Fische e al. [30]. One can noe a elaively lage eo a small velociies a he boh ends of he bell-shape cuve. Using axial velociy definiion, Eq. (4), ceneline axial velociy (velociy a he axis of symmey) can be given as: u c = u A /( z ) 0 (5a) 0 u z such ha u c = u( 0). Alenaively, he ceneline velociy may be wien in he fom: u c = u B d /( z ) 0, (5b) 0 u z o be compaable wih he common fomula of he ceneline velociy given in lieaue. I is noable ha Au = Bud, B u specifies he decay ae of he ime aveaged ceneline velociy. Dimensional agumens ogehe wih expeimenal obsevaions sugges ha he mean flow vaiables, which ae known as similaiy soluions, ae confoming wih Eqs. (5) (Fishe e al. [30], Hussein e al. [31], and Shabbi and Geoge [3]). The coninuiy equaion, Eq. (1), fo he imeaveaged velociies can be solved by subsiuing he axial velociy fom ino Eq. (1) o obain he coss-seam adial velociy in he fom: v uc c 5 5 = exp( η ) η exp( η ) η 6 6 (6) 17

18 whee, η = b( z) = / c( z z ) / 0 Dynamic Pessue The dynamic pessue behaves simila o he axial velociy bu of couse i is scala quaniy such ha we do no see negaive beaks of he cuve. The dynamic pessue can be defined accoding o he equaion: ( u v ) 1 P d = ρ (7) A inle u, v) = ( u,0), heefoe, P ( = ρ u 0 is he je nozzle dynamic pessue. On he ohe hand, one can model he simulaed dynamic pessue by he elaion: P = P exp (8a) d c h z o P = P exp (8b) d c b whee P c is he ceneline dynamic pessue, and h = c/. Figue 9 illusaes a compaison beween he simulaed dynamic pessue and is Gaussian fiing using Eq. (8) as a funcion of fo diffeen posiions of z of he unheaed je a =5 min (case 4). This figue shows a good maching fo his Gaussian disibuion of he dynamic pessue. Seleced Simulaed Resuls In Fig. 10 empeaue pofiles ae ploed agains z a diffeen imes. One may noice elaively highe empeaues close o he inle up o, appoximaely, z 0. 1 m, and hen i 18

19 deceases as z inceases. As he ime poceeds, he empeaue close o he inle deceases as shown in he figue while i inceases fuhe away. The ubulence inensiy as a funcion of he axis of symmey z fo vaious imes is shown in Fig. 11. The ubulence inensiy is defined as he aio of he oo-mean-squae of he ubulen velociy flucuaions and he mean velociy. Appaenly close o he inle velociy flucuaions inceases due o he impingemen of he je in he elaively quiescen fluid in he ank. Howeve, away fom he inle he inensiy of ubulence deceases because of he decease in velociy as he je speads laeally as manifesed in Fig. 1. Fo z m he ubulence inensiy is he same duing all ime duaion, while fo z > m he ubulence inensiy deceases wih ime. I is ineesing o noe ha inside he oule pipe he ubulence inensiy inceases as manifesed by he shap incease in ubulence inensiy owads he oule pipe as a esul of he influence of pipe wall. The velociy magniude as a funcion of adial axis disance,, a diffeen posiions of z a =10 min is ploed in Fig. 1. The velociy magniude in boom pa of he ank is lage close o he axis of symmey while i has smalle values fa fom i (i.e., as inceases). As z inceases velociy magniude deceases close o axis of symmey z while i inceases as inceases. This behavio may be explained by he fac ha he je leaves he inle wih a highe velociy and dispeses laeally as i moves fa fom he souce. Figue 13 shows empeaue as a funcion of fo vaious values of z a =10 min. A he boom of he ank (i.e., small z), he empeaue is highe close o he symmey axis and i is shaply deceases fa fom i (i.e., as inceases). Theefoe, as z inceases and he je dispeses moe laeally, he empeaue close o he axis of symmey deceases while inceasing as inceases. Je Richadson Numbe Richadson numbe is defined as a aio of he buoyancy and he ineia foces. Bu fo moe convenience we will define he Richadson numbe accoding o he local ceneline velociy. 19

20 Richadson numbe is calculaed using buoyancy-elaed ems (densiy diffeence) and he velociy a he same poin. In je flows, Richadson numbe akes he fom, [30]: π R = 1/ gδρ d 4 u c 1/ (9) Richadson numbe is ploed in Fig. 14 agains he heigh z, a diffeen imes fo Case 3. Fom his figue i can be seen ha Richadson numbe is educed in he egion close o he nozzle inle, and hen i inceases wih he heigh. In he boom pa he ineia effec dominaes he buoyancy effec (je-like zone), heefoe Richadson numbe deceases. In he op pa of he ank, on he ohe hand, he buoyancy effec dominaes he ineia (plume-like zone) as manifesed by he incease of Richadson numbe. Also, in his zone Richadson numbe inceases wih ime because empeaue inceases wih ime and enhances he buoyancy while i deceases close o he inle as he inle empeaue is se o decease. In ode o examine he effec of vaying he inle empeaue on Richadson numbe we pefom hee numeical expeimens, one of hem wih consan inle empeaue, and wo wih monoony inceasing and monoony deceasing inle empeaue, especively. The inle empeaues fo hese numeical expeimens ae defined as: T = K, fo he consan inle empeaue, in T in = , fo he monoone inceasing inle empeaue, T in = , fo he monoone deceasing inle empeaue, whee, 1 30 min, fo monoone inceasing inle empeaue, T and fo monoone deceasing inle empeaue, T in Figue 15 shows Richadson numbe fo he case of consan inle empeaue. Fom his figue i can be seen ha Richadson numbe behavio is simila fo all imes close o he inle (i.e., in he je-like egion). In he plume-like egion Richadson numbe inceases wih ime because of he in 0

21 incease in empeaue. Figues 16 and 17 illusae Richadson numbe fo he monoone inceasing and monoone deceasing inle empeaue, especively. One can noice ha Richadson numbe in he plume-like egion in he case of monoone inceasing inle empeaue inceases wih ime as a manifesaion of he inceased buoyancy, Fig. 16. On he ohe hand, fo he monoone deceasing inle empeaue, Fig.17, Richadson numbe deceases in he Je-like egion as a manifesaion of he deceased empeaue. Conclusions This wok is devoed o invesigae he poblem of non-unifom inle empeaue of buoyan je. An analysis fo veical ho wae jes eneing a cylindical ank filled wih cold wae unde he condiion of vaiable inle empeaue is inoduced. The vaiable inle empeaue is consideed as a funcion of ime of chaging pocess. Expeimenal measuemens ae pefomed fo he diffeen cases in sequenial ime seps fo boh consan and vaiable inle empeaue. Numeical invesigaions unde he above menioned condiions ae pefomed. The ealizable k ε ubulence model is used o simulae ubulen flow fo his poblem. Compaisons beween he measued and simulaed empeaue show good ageemens. Seleced empiical Gaussian model wih sandad paamees ae used o epesen he simulaed esuls. Seleced simulaed quaniies such as velociy magniude, empeaue and ubulence inensiy ae invesigaed. The esuls indicae ha empeaue vaies, appoximaely, linealy wih ime fo he consan inle empeaue cases, while, i seems o be, appoximaely, polynomial o logaihms funcions of ime fo he vaiable inle empeaue, especially, fo plume egion. Also, hemal saificaion exis; howeve hemal layes in op pa of he ank hinne han hem in he boom pa. The saificaion is veified fom he beginning of expeimen and coninues duing whole ime. 1

22 Acknowledgemen The fis auho would like o hank he Alexande von Humbold (AvH) Foundaion, Gemany fo funding his fellowship and fo suppoing of his eseach pojec. Refeences: 1. El-Amin MF, Sun S, Heidemann W, Mülle-Seinhagen H (010) Analysis of a ubulen buoyan confined je modeled using ealizable k ε model. Hea Mass Tansfe, 46(8): El-Amin MF, Heidemann W, Mülle-Seinhagen H (005) Tubulen je flow ino a wae soe. Poc. Hea Tansfe in Componens and Sysems fo Susainable Enegy Technologies, 5-7 Apil 005, Genoble, Fance, El-Amin MF, Heidemann W, Mülle-Seinhagen H (004) Unseady buoyancy-induced and ubulen flow fom a ho hoizonal je enance ino a sola wae soage. WSEAS In. Conf. Hea and Mass Tansfe (HMT 004), Cofu Island, Geece, Aug , Panhalookaan V, El-Amin MF, Heidemann W, Mülle-Seinhagen H (008) Calibaed models fo simulaion of saified ho wae hea soes. In. J. Enegy Res. 3: Rajaanam N (1976) Tubulen jes. Elsevie Science, New Yok 6. Lis EJ (198) Tubulen jes and plumes. Annu Rev Fluid Mech 14: Wygnanski I, Fiedle H (1969) Some measuemens in a self-peseving je. J Fluid Mech 38: Rodi W (1975) A new mehod of analyzing ho-wie signals in highly ubulen flow and is evaluaion in a ound je. DISA Infomaion 17, Febuay Panchapakesan NR, Lumley JL (1993) Tubulence measuemens in axisymmeic jes of ai and helium. Pa 1. Ai je. J Fluid Mech 46:197 3

23 10. Panchapakesan NR, Lumley JL (1993) Tubulence measuemens in axisymmeic jes of ai and helium. Pa. Helium je. J Fluid Mech 46: Fukushima C, Aanen L, Weseweel J (000) Invesigaion of he mixing pocess in an axisymmeic ubulen je using PIV and LIF. 10h Inenaional symposium on applicaions of lase echniques o fluid mechanics, July, Lisbon, Pougal 1. Agawal A, Pasad AK (003) Inegal soluion fo he mean flow pofiles of ubulen jes, plumes, and wakes. ASME J Fluids Eng 15: Baines WD (1975) Enainmen by a plume o je a a densiy ineface. J Fluid Mech 68(): Kmua S, Bejan A (1983) Mechanism fo ansiion o ubulence in buoyan plume flow. In J Hea Mass Tansf 6: Shabbi A, Geoge K (1994) Expeimens on a ound ubulen buoyan plume. J Fluid Mech 15: Papanicolaou PN, Lis EJ (1988) Invesigaions of ound veical ubulen buoyan jes. J Fluid Mech 195: Aakei JH, Das D, Sinivasan J (000) Bifucaion in a buoyan hoizonal lamina je. J Fluid Mech 41: O Hen TJ, Weckman EJ, Geha AL, Tieszen SR, Scefe RW (005) Expeimenal sudy of a ubulen buoyan helium plume. J Fluid Mech 544: El-Amin MF, Kanayama H (009) Inegal soluions fo seleced ubulen quaniies of smallscale hydogen leakage: a non-buoyan je o momenum-dominaed buoyan je egime. In J Hydogen Enegy 34: El-Amin MF, Kanayama H (009) Similaiy consideaion of he buoyan je esuling fom hydogen leakage. In J Hydogen Enegy 34:

24 1. El-Amin MF (009) Non-Boussinesq ubulen buoyan je esuling fom hydogen leakage in ai. In J Hydogen Enegy 34: Jika GH (004) Inegal model fo ubulen buoyan jes in unbounded saified flows. Pa 1: single ound je. Envion Fluid Mech 4: Jika GH (006) Inegal model fo ubulen buoyan jes in unbounded saified flows. Pa : plane je dynamics esuling fom mulipo diffuse jes. Envion Fluid Mech 6: Yoo H, Kim CJ, Kim CW (1999) Appoximae analyical soluions fo saified hemal soage unde vaiable inle empeaue. Sola Enegy 66: Abo-Hamdan MG, Zuiga YH, and Ghaja, AJ (199) An expeimenal sudy of a saified hemal soage unde vaiable inle empeaue fo diffeen inle designs. In. J. Hea Mass Tansfe 35: Shih TH, Liou WW, Shabbi A, Yang Z, Zhu J (1995) A new k ε eddy-viscosiy model fo high Reynolds numbe ubulen flows-model developmen and validaion. Compu Fluids 4(3): Reynolds WC (1987) Fundamenals of ubulence fo ubulence modeling and simulaion. Lecue noes fo Von Kaman insiue, Agad Repo No Kadem K, Maaoui A, Salem A, Younsi R (007) Numeical simulaion of hea ansfe in an axisymmeic ubulen je impinging on a fla plae. Adv Model Opim 9(): Fluen 6.1 (003) Use s Guide, Fluen Inc. 30. Fische HB, Lis EJ, Koh RCY, Imbege J, Books NH (1979) Mixing in inland and coasal waes. Academic Pess, San Diego 31. Hussein JH, Capp SP, Geoge WK (1994) Velociy measuemens in a high-reynolds-numbe, momenum-conseving, axisymmeic, ubulen je. J Fluid Mech 58: Shabbi A, Geoge K (1994) Expeimens on a ound ubulen buoyan plume. J Fluid Mech 15:1 3 4

25 Appendix: The inle vaiable empeaue may be given as a funcion of ime fo each case as follows: T6 ( ) = (A.1) T7 ( ) = (A.) T8 ( ) = (A.3) T9 ( ) = (A.4) These polynomials ae ploed in Fig. 3a. The anges of vaiaion of he inle empeaue ae: T6 ( ) T7 ( ) T8 ( ) T9 ( ) K K K K The coesponding Reynolds numbes ae: Re 6 ( ) = (A.5) Re 7 ( ) = (A.6) Re 8 ( ) = (A.7) Re 9 ( ) = (A.8) 5

26 These polynomials ae shown in Fig. 3b. The anges of vaiaion of he Reynolds numbes ae: 144 Re 6 ( ) Re 7 ( ) Re8 ( ) Re9 ( ) 711 Also, he nozzle inle ubulence inensiy can be epesened in a polynomial fom of ime: I ( ) = (A.9) I ( ) = (A.10) I ( ) = (A.11) I ( ) = (A.1) These pofiles ae illusaed in Fig. 3c. The anges of vaiaion of he inle ubulence inensiies ae: I 06( ) I 07 ( ) I 08( ) I 09( )

27 Table Capions Table 1: Summay of he expeimenal daa wih vaiable inle empeaue Table : Time sep compaison of empeaue, axial velociy, ubulen kineic enegy and ubulen dissipaion ae of kineic enegy, fo case 4 7

28 Figue Capions Fig. 1: Schemaic diagam of he expeimenal seup. Fig. (a, b): Schemaic epesenaion of he calculaion domain. Fig. 3 (a, b, c): Vaiable inle (a) empeaue, (b) Reynolds numbe and (c) ubulence inensiy as a funcion of ime fo cases 1-4. Fig. 4 (a, b, c, d): Pofiles of measued empeaue as a funcion of ime, a diffeen sensos posiions, fo vaiable inle empeaue, cases 1-4. Fig. 5 (a, b): Gid independence es by (a) axial velociy, and (b) empeaue. Fig. 6 (a, b, c, d): Compaison beween measued and simulaed empeaue as funcion of ank heigh fo vaiable inle empeaue (cases 1-4). Fig. 7: Nomalized axial velociy as a funcion of /cz a diffeen posiions of z of case 4 a =15 min. Fig. 8: Compaison beween simulaed and empiical Gaussian model of axial velociy as a funcion of fo diffeen posiions of z of he case 4 a =15 min. Fig. 9: Compaison beween he simulaed dynamic pessue and is Gaussian fiing as a funcion of fo diffeen posiions of z of he case 4 a =15 min. Fig. 10: Tempeaue as a funcion of he axis of symmey z(=0) wih vaies imes, case 1. Fig. 11: Tubulence inensiy as a funcion of he axis of symmey z(=0) wih vaies imes, case 1. Fig. 1: Velociy magniude as a funcion of wih vaies values of z a =10 min, case 3. Fig. 13: Tempeaue as a funcion of wih vaies values of z a =10 min, case 3. Fig. 14: Richadson numbe as a funcion of he heigh z, a diffeen imes, fo case 3. Fig. 15: Richadson numbe as a funcion of he heigh z, a diffeen imes, wih consan inle empeaue. Fig. 16: Richadson numbe as a funcion of he heigh z, a diffeen imes, wih monoone inceasing inle empeaue. 8

29 Fig. 17: Richadson numbe as a funcion of he heigh z, a diffeen imes, wih monoone deceasing inle empeaue. 9

30 Table 1: Case Inle Iniial Flow ae Inle Re [-] Tubulence empeaue empeaue [m 3 /s] velociy inensiy [%] [K] [K] [m/s] 1 T ( ) Re 6 ( ) I ( ) 6 T ( ) Re 7 ( ) I ( ) 7 3 T ( ) Re 8 ( ) I ( ) 8 4 T ( ) Re 9 ( ) I ( ) Table : Δ 0.01 s 0.1 s 0.5 s 1 s T u k ε 1.33E E E E-05 30

31 Themocouples Daa acquisiion sysem Oule Pump Tank Heae Inle Flow- mee Fig. 1 31

32 0.18m 0.055m 0.09m Oule Tank Top 0.605m S9 0.56m S8 0.48m S7 0.4m Axis of Symmey S6 S5 S4 0.36m 0.30m 0.4m g Senso Seies S3 S 0.18m 0.1m 0.06m Inle S10 S1 0.06m 0.00m Tank Boom Fig. a 3

33 Wall z-axis Oule Wall z-axisymmey (ceneline) Inle -axis Wall Fig. b 33

34 Tin [K] Time [min] Fig. 3a Case 1 Case Case 3 Case 4 34

35 Re() [-] Case 1 Case Case 3 Case Time [min] Fig. 3b 35

36 I0() [-] Case 1 Case Case 3 Case Time [min] Fig. 3c 36

37 Fig. 4a 37

38 Fig. 4b 38

39 Fig. 4c 39

40 Fig. 4d 40

41 u [m/s] [m] Fig. 5a gid-1 gid- gid-3 gid-4 41

42 Tempeaue [K] gid-1 gid- gid-3 gid [m] Fig. 5b 4

43 Fig. 6a 43

44 Fig. 6b 44

45 Fig. 6c 45

46 Fig. 6d 46

47 Fig. 7 47

48 Fig. 8 48

49 Fig. 9 49

50 Fig

51 Fig

52 Velociy Magniude[m/s] [m] Fig. 1 z=0.0 m z=0.35 m z=0.50 m 5

53 Tempeaue[K] z=0.0 m z=0.35 m z=0.50 m [m] Fig

54 Fig

55 Richadson Numbe_R [-] =5 min =10 min =0 min =30 min z [m] Fig

56 Richadson Numbe_R [-] =5 min =10 min =0 min =30 min z [m] Fig

57 Richadson Numbe_R [-] =5 min =10 min =0 min =30 min z [m] Fig