Rogerio Fernandes Brito Member, ABCM

Transcription

1 Tubulent Natual Convecton n Enclosues Usng Lage-Eddy Smulaton wth Rogeo Fenandes Bto Membe, ABCM ogbto@unfe.edu.b Genéso José Menon ogbto@unfe.edu.b Fedeal Unvesty of Itauba UNIFEI Depatment of Mechancal Engneeng Itaubá, MG, Bazl Macelo José Pan Membe, ABCM pan@ufba.b Fedeal Unvesty of Baha UFBA Depatment of Mechancal Engneeng Salvado, BA, Bazl Tubulent Natual Convecton n Enclosues Usng Lage-Eddy Smulaton wth Localzed Heatng fom Hozontal Bottom Suface and Coolng fom Vetcal Sufaces Tubulent natual convecton of a that happens nto nne squae cavty wth localzed heatng fom hozontal bottom suface has been numecally nvestgated. Localzed heatng s smulated by a centally located heat souce on the bottom wall, and two values of the dmensonless heat souce length ae consdeed n the pesent wok. Solutons ae obtaned fo seveal Raylegh numbes wth Pandtl numbe P = The hozontal top suface s themally nsulated and the vetcal sufaces ae assumed to be the cold sothemal sufaces wheeas the heat souce on the bottom wall s sothemally heated. In ths study, the Nave-Stokes equatons ae used consdeng a two-dmensonal and tubulent flow n unsteady state. The Fnte Element Method (FEM) wth a Galekn scheme s utlzed fo solvng the consevaton equatons. The fomulaton of consevaton equatons s caed out fo tubulent flow and the mplementaton of tubulent model s made by Lage-Eddy Smulaton (LES). The dstbutons of the steam functon and of the tempeatue ae detemned as functons of themal and geometcal paametes. The aveage Nusselt numbe Num s shown to ncease wth an ncease n the Raylegh numbe Ra as well as n the dmensonless heat souce length. The esults of ths wok can be appled to the desgn of electonc components. Keywods: cavtes, fnte element, tubulence, natual convecton, LES Intoducton 1 Natual convecton n enclosues s an aea of nteest due to ts wde applcaton and geat mpotance n engneeng. Tansent natual convecton flows occu n many technologcal and ndustal applcatons. Theefoe, t s mpotant to undestand the heat tansfe chaactestcs of natual convecton n an enclosue. Along the yeas, eseaches have looked fo moe flows wth the obectve to appoxmate the eal case found n geophyscal o ndustal means. Then, we can defne fou basc types of bounday condtons. They ae: the natual convecton due to a unfomly heated wall (wth a tempeatue o a constant heat flux); the natual convecton nduced by a local heat souce; the natual convecton unde multple heat souces wth the same stength and type; and the natual convecton conugated wth nne heat-geneatng conductve body o conductve walls. The bounday condtons mentoned pevously ae based on a sngle tempeatue dffeence between the dffeentally heated walls. Most of the pevous studes have addessed natual convecton n enclosues due to ethe a hozontally o vetcally mposed tempeatue dffeence. Howeve, depatues fom ths basc stuaton ae often found n felds such as electoncs coolng. The coolng of electonc components s essental fo the elable pefomance. The chaactestcs of flud flow and heat tansfe unde the multple tempeatue dffeences ae moe complcated and have a pactcal mpotance n themal management and desgn. In the pesent wok, a two-dmensonal numecal smulaton n a cavty s caed out fo a tubulent flow. The tubulence study s a complex and challengng assumpton. Thee ae few woks n the lteatue that deal wth natual convecton n closed cavtes usng the tubulence model LES. The motvaton to accomplsh ths wok eles on the fact that thee s a geat numbe of poblems n Pape accepted Decembe, Techncal Edto: Fancsco R. Cunha engneeng that can use ths geomety. One tubulence model s mplemented hee togethe wth the fnte element method. A Lage Eddy Smulaton (LES) seems as a pomsng appoach fo the analyss of the hgh Gashof numbe tubulence that contans thee-dmensonal and unsteady chaactestcs. A dect smulaton of tubulence gves us moe accuate and pecse data than expements; t s essentally unsutable fo hgh Gashof numbe flows because of computatonal lmtatons. It s known that the LES enables an accuate pedcton of tubulence, but spends much less CPU tme than the dect smulaton. In lteatue, a lage numbe of theoetcal and expemental nvestgatons ae epoted on natual convecton n enclosues. Natual convecton of a n a two-dmensonal ectangula enclosue wth localzed heatng fom below and symmetcal coolng fom the sdes was numecally nvestgated by Aydn and Yang (2000). Localzed heatng was smulated by a centally located heat souce on the bottom wall, and fou dffeent values of the dmensonless heat souce length, 1/5, 2/5, 3/5 and 4/5 wee consdeed. Solutons wee obtaned fo Raylegh numbe values fom 10 3 to The aveage Nusselt numbe at the heated pat of the lowe wall, Nu, was shown to ncease wth an ncease of the Raylegh numbe, Ra, o of the dmensonless heat souce length. Peng and Davdson (2001) studed the tubulent natual convecton n a closed enclosue n whch vetcal lateal walls wee mantaned at dffeent tempeatues. Both the Smagonsk and the dynamc models wee appled to the tubulence smulaton. Peng and Davdson (2001) modfed the Smagonsk model by addng the buoyancy tem to the tubulent vscosty calculaton. Ths model would be called the Smagonsk model wth buoyancy tem. The computed esults wee compaed to expemental data and showed a stable themal statfcaton unde a low tubulence level (Ra = 1.58 x 10 9 ). Deng et al. (2002) studed numecally a two-dmensonal lamna natual convecton n a ectangula enclosue wth dscete heat souces on walls n the unsteady egme. A new combned tempeatue scale was suggested to nondmensonalze the J. of the Baz. Soc. of Mech. Sc. & Eng. Copyght 2009 by ABCM July-Septembe 2009, Vol. XXXI, No. 3 / 199

2 Rogeo Fenandes Bto et al. govenng equatons of natual convecton nduced by multple tempeatue dffeences. The Raylegh numbes used wee Ra = 10 3 to It was pefomed n the wok of Olvea and Menon (2002) a numecal study of tubulent natual convecton n squae enclosues. The fnte volume method togethe wth LES was used. The enclosue lateal sufaces wee kept to dffeent sothemal tempeatues, and the uppe and lowe sufaces wee solated. The flow was studed fo low Raylegh numbes Ra = 1.58 x Thee tubulence LES models wee used. Ampofo and Kaayanns (2003) conducted an expemental study of low-level tubulence natual convecton n an a flled vetcal squae cavty. The cavty was 0.75 m hgh x 1.5 m deep gvng se to a 2D flow. The hot and cold walls of the cavty wee sothemal at 50 and 10 ºC espectvely, that s, a Raylegh numbe equals to 1.58 x The expements that wee caed out on Ampofo wok and Kaayanns (2003) wee conducted wth vey hgh accuacy and as such the esults fomed expemental benchmak data and wee useful fo valdaton of computatonal flud dynamcs codes. Matoell et al. (2003) wok dealt wth the natual convecton flow and heat tansfe fom a hozontal plate cooled fom above. Expements wee caed out fo ectangula plates havng aspect atos between φ = and 0.43 and Raylegh numbes n the ange of 290 Ra w These values of Ra w and φ wee selected to the desgn of pnted ccut boads. The esults showed that such a low Ra w effect could be accounted fo n a physcally consstent manne by addng a constant tem to the heat tansfe coelaton. In the pesent wok, tubulent natual convecton of a that happens nto nne squae cavty wth localzed heatng fom hozontal bottom suface has been numecally nvestgated. The obectve of the analyses of heat tansfe s to nvestgate the Nusselt numbe dstbuton on the vetcal walls and heated lowe hozontal suface. Anothe obectve s to vefy the effect of heght vaaton I of the hozontal heated lowe suface on the tubulent flow. Sx cases ae studed numecally. The Raylegh numbe Ra s vaed and so s the dmensonless length the heat souce, whee (1- )/2 x (1 )/2 and x s the coodnate component n the x decton. Fo the cases 1, 2 and 3, the dmenson s fxed n = 0.4 and the Raylegh numbes Ra s vaed, n Ra = 10 7, 10 8 and Fo the cases 1, 2, and 3, t s used a non-stuctued mesh of fnte elements wth 5,617 tangle elements and 2,908 nodal ponts. The othe cases also used a non-stuctued mesh of fnte elements wth lnea tangle elements. In cases 4, 5, and 6, s fxed n = 0.8. The cases 1 and 4, 2 and 5, 3 and 6 ae smulated, espectvely, fo Ra = 10 7, 10 8 and The cases 4, 5, and 6 ae smulated wth one mesh wth 5,828 elements and 3,015 nodes. The tubulence model used n all cases s the Lage-Eddy Smulaton (LES) wth the second-ode stuctue-functon sub-gd scale model (F 2 ). It s adopted a geomety wth an aspect ato A = H/L = 1.0. Compasons ae made wth expemental data and numecal esults found n Tan and Kayanns (2000), Olvea and Menon (2002), Lankhost (1991) and Cesn et al. (1999). Nomenclatue C θ = Cossng tubulent flux d = Dstance d fom the taget pont u T = Flteed vaable poducts that descbe the tubulent heat tanspot u u = Flteed vaable poducts that descbe the tubulent momentum tanspot = Leonad Tenso L L θ R = Leonad tubulent flux = Reynolds sub-gd tenso S " = Defomaton tenso ate A = Dmensonless constant c = Enclosue aspect ato C = Cossng tenso g = Gavty acceleaton, m/s 2 H = Chaactestc dmenson of cavty I = Length of the heated hozontal lowe suface L = Chaactestc dmenson of cavty l = Scale lengths and the velocty N = Numbe of ponts fom the neghbohood n = Unt vecto nomal to the suface o bounday Nu = Nusselt numbe p = Pessue, Pa P = Pandtl numbe q = Velocty, m/s = Dstance between two ponts, m Ra = Raylegh numbe S = Souce tem, Suface T = Tempeatue, ºC t = Tme, s t CPU = CPU pocessng tmes, s u = Velocty n x decton, m/s v = Velocty n y decton, m/s x = Coodnate component n x decton y = Coodnate component n y decton Geek Symbols δ ϕ τ θ ν T ε φ Ω ρ β ϕ 1 2 ν ψ α ω = Konecke delta = Lage eddy component = Reynolds tenso = Sub-gd tubulent flux = Tubulent knematc vscosty = Dsspaton of the tubulent knetc enegy ε = Aspect ato = Dmensonless heat souce length = Studed doman = Flud densty = Flud volumetc expanson coeffcent = Geneal vaable = Geometc mean of dstances d fom neghbong elements to the pont whee v T s calculated = Flte length n x decton = Flte length n y decton = Knematc vscosty = Steam functon = Themal dffusvty = Votcty Subscpts m elatve to mean elatve to dectons elatve to dectons k elatve to k dectons T elatve to tubulent c elatve to cold h elatve to hot w Wall 1,2,3,4,5 elatve to sufaces 1,2,3,4,5 200 / Vol. XXXI, No. 3, July-Septembe 2009 ABCM

3 Tubulent Natual Convecton n Enclosues Usng Lage-Eddy Smulaton wth Poblem Descpton and Hypothess Fgue 1 shows the geomety wth the doman Ω. It wll be consdeed a squae cavty. The uppe hozontal suface S 4 s themally nsulated and the vetcal sufaces S 1 and S 3 ae assumed to be the cold sothemal sufaces. The bottom hozontal sufaces S 5 and S 6 ae also themally nsulated. Localzed heatng s smulated by a centally located heat souce on the bottom wall, S 2. The ntal condton n Ω s: T = 0 wth ψ = ω = 0. All physcal popetes of the flud ae constant except the densty n the buoyancy tem, whee t obeys the Boussnesq appoxmaton. It s assumed that the thd dmenson of the cavtes s lage enough so that the flow and heat tansfe ae two-dmensonal. In the Lage Eddy Smulaton (LES), a vaable decomposton smla to the one n the Reynolds decomposton s pefomed, whee the quantty ϕ s splt as follows: ' = ϕ ϕ (4) ϕ whee ϕ s the lage eddy component and ' ϕ s the small eddy component. Fgue 2 shows one of the meshes used n the numecal smulatons of the pesent wok. Fgue 2. Mesh aangement fo cases 1, 2 and 3. Fgue 1. Cavty geomety Theoy of Sub-Gd Scale Modellng The govenng consevaton equatons ae: u = 0 (1) u 1 p (2) = v gβ ρ T T = T α S ( T T0 ) δ whee x ae the axal coodnates x and y, u ae the velocty components, p s the pessue, T s the tempeatue, ρ s the flud densty, ν s the knematc vscosty, g s the gavty acceleaton, β s the flud volumetc expanson coeffcent, δ s the Konecke delta, α s the themal dffusvty, and S the souce tem. The last tem n Eq. (2) s the Boussnesq buoyancy tem whee T 0 s the efeence tempeatue. (3) The followng hypotheses ae employed n the pesent wok: unsteady tubulent egme; ncompessble two-dmensonal flow; constant flud physcal popetes, except the densty n the buoyancy tems. The followng flteed consevaton equatons ae shown afte applyng the flteng opeaton to Eqs. (1) to (3). It s done by usng the volume flte functon pesented n Kanovc (1998). The densty s constant. u u = 0 (5) ( T T0 ) δ 2 (6) 1 p = v gβ ρ T T = T α S In the Eqs. (5) to (7), u u and u T ae the flteed vaable poducts that descbe the tubulent momentum tanspot and the heat tanspot, espectvely, between the lage and sub-gd scales. (7) J. of the Baz. Soc. of Mech. Sc. & Eng. Copyght 2009 by ABCM July-Septembe 2009, Vol. XXXI, No. 3 / 201

4 Rogeo Fenandes Bto et al. Accodng to Olvea and Menon (2002), the poducts uu and u T ae splt nto othe tems by ncludng the Leonad tenso L, the Cossng tenso C, the Reynolds sub-gd tenso R, the Leonad tubulent flux L θ, the Cossng tubulent flux C θ and the sub-gd tubulent flux θ. The Cossng and Leonad tems, accodng to Padlla (2000), can be neglected. Afte the development shown n Olvea and Menon (2002), the followng consevaton equatons ae obtaned: = 0 (8) ( T T0 ) δ 2 u u 1 p u (9) τ = ν gβ ρ x x T T T θ = α (10) whee P s the Pandtl numbe wth α = ν / P. The tensos τ and θ that appea n Eqs. (9) and (10) ae modeled n the fothcomng topcs. Sub-gd scale model Many sub-gd scale models use the dffuson gadent hypothess smla to the Boussnesq one that expesses the subgd Reynolds tenso n functon of the defomaton ate and knematc enegy. Accodng to Slvea-Neto (1998), the Reynolds tenso s defned as: τ 2 3 = 2ν T S δ S (11) kk whee v T s the tubulent knematc vscosty, δ s the Konecke delta, and S s defomaton tenso ate gven by: S = (12) Substtutng S, fom Eq. (12), n Eq. (11) and manpulatng equatons, we have: u gβ ( T T0 ) δ 2 1 p u u u ν = ν T ρ x x x (13) In a smla way, the enegy equaton s obtaned: The sub-gd models gve the followng expesson fo the tubulent vscosty v T : ν = c l q (16) T whee c s a dmensonless constant, l and q ae the scale lengths and the velocty, espectvely. The paamete l s elated to the flte sze and t s usually used n the two-dmensonal case wth a ectangula element as: _ l = = ( 1 2 ) 1/2 (17) whee 1 and 2 ae the flte lengths n x and y dectons. The second-ode stuctue-functon sub-gd scale model (F 2 ) The tubulent vscosty v T s calculated as follows: v 2 ( x,, t) = 0.104C 3/ F 2( x, t) T k, (18) whee C k = 1.4 s the Kolmogoov constant (Kolmogoov, 1941). The vaable s the geometc mean of dstances d fom neghbong elements to the pont whee v T s calculated and s gven by: and F 2( x,, t) N N Π =1 = d (19) s the stuctue functon of second ode veloctes. Accodng to Kolmogoov (1941) law that establshes that the stuctue functon of second ode veloctes s popotonal to (ε) 2/3, whee s the dstance between two ponts, the stuctue functon can be calculated as: F 2 N 1 = Σ N = 1 {{[ u ( x d e, t) u( x, t) ] 2 [ υ ( x d e, t) υ( x, t) ] } whee u ( x d e t) and ( x d e t ), 2 d 2/ 3 (20) υ, ae the veloctes at the pont of the neghbong centod placed at a dstance d fom the taget pont, u( x, t) and υ ( x, t) ae the veloctes at ths pont of the element, N s the numbe of ponts fom the neghbohood, t s the tme and e the vecto on decton. T T = ( α α ) T T whee the tubulent themal dffusvty α T s calculated as: (14) The tubulent themal dffuson s estmated fom the tubulent knematc vscosty, by assumng: P T = ν T / α T = 0.4 (21) and P T s the tubulent Pandtl numbe. α T = ν T / P T (15) 202 / Vol. XXXI, No. 3, July-Septembe 2009 ABCM

5 Tubulent Natual Convecton n Enclosues Usng Lage-Eddy Smulaton wth Intal and bounday condtons Fom ths secton on, the uppe bas that mean aveage values wll be omtted. Fgue 1 pctues the enclosue on whch the ntal bounday condtons ae as follows: ( x, y,0) ) = 0, v( x, y,0) ) = 0, T( x, y,0) ) = 0 u (22) u = v = 0, T = Tc = 0 (23) u = v = 0, T = Th = 1 (24) u = v = 0, T / y = 0 (25) The flow feld can be descbed by the steam functon ψ and the votcty ω dstbutons gven by: ( / ) ( y) u = ψ / y, υ = ψ /, ω = υ / (26) whee u and ν ae the velocty components n x and y dectons, espectvely. Hence, the contnuty equaton gven by Eq. (1) s exactly satsfed. Wokng wth dmensonless vaables, t s possble to deal wth Raylegh numbe Ra, Pandtl numbe P and the enclosue aspect ato A gven by: [ g ( T T ) H / v ] 10,10 and 10 Ra = P β, P = ν /α = 0.7, h c = A = H/L = 1.0 (27) whee T h and T c ae the tempeatues on sufaces S 2 and S 1 - S 3, espectvely. H s the chaactestc dmenson of cavty. Numecal method Equatons (8) to (10) ae solved though the fnte element method (FEM) wth lnea tangula elements usng the Galekn fomulaton. The system of equatons s solved wth the Gauss Quadatue. The poblem soluton follows the steps below: (1) though Eq. (26), the steam functon feld ψ s solved; (2) the wall votcty s detemned n matcal fom, accodng to Slvea-Neto et al. (2000); (3) the bounday condtons fo votcty ae appled; (4) the votcty n the nteo s calculated accodng to Eq. (26); (5) the tempeatue feld s solved though Eq. (10); (6) the local Nusselt numbe Nu s obtaned usng Eq. (28); (7) the tme s nceased wth the tme step t and the teaton wth unty, and then t tuns to the fst step (1). It stats all ove agan tll t eaches the stop cteon. The local Nusselt numbe Nu s defned as: ( T n) H ( T T ) Nu = (28) w whee n s the unt vecto nomal to the suface o bounday, whee the local Nusselt numbe Nu s calculated. Numecal method valdaton In the pesent wok, a study of the effect of mesh efnement on the aveage Nusselt numbe Nu m calculated on hot lowe suface S 2 s conducted. The themal paametes used ae: Raylegh numbe Ra = 10 6 and Pandtl numbe P = The geometc paametes used ae: cavty aspect ato A = 1.0 and dmensonless length of heated souce = 0.5. Fve mesh types ae used. Table 1 shows the esults obtaned n ths mesh study. Afte ths study, we adopted a computatonal mesh between meshes D and E. In ode to compae the esults wth the ones found n lteatue and then to valdate the computatonal code n FORTRAN, two cases ae taken fom Bto et al. (2002) and Bto et al. (2003). Bto et al. (2002) and Bto et al. (2003) use the same tubulence model LES as the one used n the pesent wok. In the fst compason, the study of the natual tubulent flow n a squae enclosue wth dffeent tempeatues fo vaous Raylegh numbes s caed out n Bto et al. (2002). The second compason s made n Bto et al. (2003) consdeng a lamna flow n a ectangula enclosue wth an ntenal cylnde. h c Table 1. Numec esults obtaned by Nusselt Nu m n the heated lowe suface S 2. Mesh Numbe of elements NE Numbe of nodes NO Nu m S2 Devaton of Nu m S2 t CPU [ s ] A B 1, C 3,022 1, D 5,384 2, , E 5,981 3, , In the fst compason, t s also used the Lage Eddy Smulaton (LES). The esults n Bto et al. (2002) ae compaed not only to the expemental and numecal ones n Peng and Davdson (2001), and Tan and Kaayanns (2000), but also to the numecal ones n Lankhost (1991). In the compasons ealzed n Bto et al. (2002), measues fo the cente of squae cavty fo dmensonless aveage velocty ae made. The esults showed good concodance wth the expemental esults. The second compason s made n Bto et al. (2003) whose esults ae compaed to the ones n Cesn et al. (1999). Cesn et al. (1999) consdeed a two-dmensonal lamna flow. Fo the numecal smulaton made by Cesn et al. (1999), a dmenson z s adopted n such a way that the flow can be consdeed two-dmensonal. Cesn et al. (1999) study a ectangula enclosue whee the hozontal suface has a constant convecton heat tansfe wheeas the hozontal lowe suface s submtted to solaton. The vetcal sufaces ae sothemal havng a low tempeatue T c. On the othe J. of the Baz. Soc. of Mech. Sc. & Eng. Copyght 2009 by ABCM July-Septembe 2009, Vol. XXXI, No. 3 / 203

6 Rogeo Fenandes Bto et al. hand, the cylnde suface has a hgh tempeatue T h. In the second compason, the maxmum devaton s % wth Raylegh numbe equals to 3.4 x 10 3 usng a mesh wth 5,790 elements and 3,011 node ponts. The mno devaton s 7.53 % to Raylegh numbe equals to 3.0 x Results The man obectve of ths study s to analyze the nfluence of Raylegh numbe s vaaton and the length I of the heated hozontal lowe suface on the flow feld. The geomety s chosen n ode to smulate the coolng of the a n cavtes wth electonc components placed on the lowe hozontal suface. A ange of Raylegh numbes n a low tubulence flow s used. The themal paametes used ae: Ra = 1.0 x 10 7, 1.0 x 10 8, and 1.0 x 10 9 wth P = The geomety paametes used n the sx cases mentoned pevously ae: H = 1.0; L = 1.0; T h = 1; T c = 0 and A = H / L = 1.0. In ode to model the tubulence, t s used the Lage-Eddy Smulaton (LES) wth the second-ode stuctue-functon sub-gd scale model (F 2 ). In ths wok, we also make a study of effect of the mesh efnement, amng to obtan the best tme computatonal cost. It s used a pogam n FORTRAN, wth the Compaq Vsual Fotan v6.6 complato, fo the ealzaton of the numec smulaton. The numec esults ae obtaned usng one Intel Pentum III pocesso of 800 MHz wth 128 MB memoy RAM (see Table 1 fo CPU pocessng tmes t CPU n seconds). Fgues 3 and 4 pesent the local Nusselt numbe Nu vesus the coodnate x fo hozontal lowe sufaces S 2, S 5 and S 6. Fgues 5-10 show the aveage Nusselt numbe Nu m vesus tme fo all sx cases. Fgues show the flow felds and the tempeatue n tems of steam functon lnes ψ, sothems T m and velocty vectos u. The tme step t adopted n ths pesent wok s based on Peng and Davdson wok (2001), whee t = t 0, t 0 = H / (g β t H) 1/2. In the pesent wok, due to the lmtaton of the hadwae (pocesso), we adopt one tme step t thee tmes bgge than the value adopted n Peng and Davdson wok (2001). In Fgues 5-10, the aveage tme to obtan the aveage quanttes s fom 400 to 600t 0, t = ( ) t 0. Fgues show the steam functon ψ wth a lne spacng equals to 10 ( ψ = 10). Fo the sothems, we adopt the same lne spacng n all Fgs , t m = The steam functon ψ s shown fo the last nteacton, t = 600t 0. The sothems ae calculated at each nodal pont consdeng an aveage tme, that s, t = ( ) t 0. The same was done to the velocty vectos u. Fgues 3 and 4 show the dstbuton of local Nusselt numbe Nu along all the lowe hozontal suface S 2. Fgues 3 and 4 show the esults fo heated lengths = 0.4 and 0.8, fo the last tme (t = 600t 0 ). We obseve that nceasng, Nu nceases n the hozontal lowe heated suface. Fo a fxed value of, Ra ncease does not esult n a heat exchange on suface S 2. We also obseve a cetan symmety of the heat tansfe n the mddle of the cavty (x = L/2), even fo hghe Ra (Ra = 10 9 ). the values of Nu m. Fgues 6 and 9, fo Ra = 10 8, show that the ncease educes Nu m on S 2. In Fgs. 7 and 10, fo Ra = 10 9, we obseve the same behavo found n Fgs. 6 and 9 wth Ra = Then, we can conclude that the flow become oscllatng fo Ra = 10 8 and = 0.8, and, as t can be seen n Fg. 10, the heat tansfe ates ae lage on all the sufaces, ncludng the uppe hozontal suface S 4. Fgue 3. Local Nusselt numbe Nu on S 2, S 5 and S 6 sufaces fo Ra = 10 7, 10 8 e 10 9 wth t = 600 t 0 fo cases 1, 2 and 3 ( = 0.4). Fgue 4. Local Nusselt numbe Nu on S 2, S 5 and S 6 sufaces fo Ra = 10 7, 10 8 e 10 9 wth t = 600 t 0 fo cases 4, 5 and 6 ( = 0.8). Fgues 5-10 show the aveage Nusselt numbes Nu m calculated on sufaces S 1, S 2, S 3 and S 4, vesus tme t fo a tme ange t = ( ) t 0. Fgues 5, 6, and 7, show that the hghe the Raylegh numbe, the hghe the convecton n all cavty sufaces studed fo fxed values of. Fgues 8, 9 and 10 show that heat tansfe s hghe when the Raylegh numbe s nceased. In Fg. 10, the Nu m values oscllate, due to the effect of the tubulence nsde the cavty. The ates of heat tansfe ae a lttle lage than those pesented n Fgs. 5, 6, and 7. In Fgues 5 to 10, whee Ra = 10 7, the ncease does not consdeably nfluence 204 / Vol. XXXI, No. 3, July-Septembe 2009 ABCM

7 Tubulent Natual Convecton n Enclosues Usng Lage-Eddy Smulaton wth Fgue 5. Nu m vesus t on S 1, S 2, S 3 and S 4 wth P = 0.70, = 0.4 and t = ( ) t 0, fo Ra = Fgue 8. Nu m vesus t on S 1, S 2, S 3 and S 4 wth P = 0.70, = 0.8 and t = ( ) t 0, fo Ra = Fgue 6. Nu m vesus t on S 1, S 2, S 3 and S 4 wth P = 0.70, = 0.4 and t = ( ) t 0, fo Ra = Fgue 9. Nu m vesus t on S 1, S 2, S 3 and S 4 wth P = 0.70, = 0.8 and t = ( ) t 0, fo Ra = Fgue 7. Nu m vesus t on S 1, S 2, S 3 and S 4 wth P = 0.70, = 0.4 and t = ( ) t 0, fo Ra = Fgue 10. Nu m vesus t on S 1, S 2, S 3 and S 4 wth P = 0.70, = 0.8 and t = ( ) t 0, fo Ra = J. of the Baz. Soc. of Mech. Sc. & Eng. Copyght 2009 by ABCM July-Septembe 2009, Vol. XXXI, No. 3 / 205

8 Rogeo Fenandes Bto et al. Fgues show the effect of Raylegh numbe, whee 10 7 Ra 10 9, and the effect of the dmensonless length of heat souce fo = 0.4 and 0.8. Due to the symmetcal bounday condtons along the vetcal walls, the flow and the tempeatue felds have a elatve symmety n the mddle of the cavty. Fo the tempeatue feld, we obseve that ths symmety s bette vsualzed, because the sothems ae obtaned though an aveage n the tme fo t = ( ) t 0. These same symmetcal bounday condtons n the vetcal decton esult n two geat flud aeas that symmetcally ecculate. As the flow tends to the oscllatng egme fo Ra = 10 9, ths symmety s lost. In Fgs. 11, 12 and 13, whee = 0.4, t s obseved that the ntenal flud ecculaton s moe sgnfcant as Ra nceases. Fo Ra 10 7, themal plumes ae fomed ove the hot suface S 2. The hot flud, whch s n the lowe egon of the cavty, moves up due to buoyant foces. Dung ts tavelng to the uppe pat of the cavty, the flud s cooled n vetcal lateal walls. It can be noted n Fg. 12 that wth the Ra ncease to Ra 10 8, a egon wth lowe heat tansfe s bought about gvng se to a smalle themal plume. In Fg. 13, fo Ra 10 9, pactcally all the flud nsde the cavty has a stable aveage tempeatue between the maxmum and mnmum values stated by the bounday condtons. Fom Fgs. 11, 12 and 13, the aveage velocty vectos pctue the flud behavo n the tme ange t = ( ) t 0. Fgues 14, 15, and 16 show the esults fo = 0.8 and Ra between 10 7 Ra Fo Fgs. 14, 15 and 16, whee = 0.8, the ncease of the heated suface length S 2 makes the heat tansfe ncease n all sufaces, as seen n the gaphcs of Nu m vesus the tme t. The suface S 2 has the educton of the Nu m value calculated fo the ange 400 to 600 t 0 wth Ra = Fo the sothems, Fgs. 15 and 16 show few dffeences. The steamlnes n Fgues 14, 15 and 16 show two bg flud egons that eccle n opposte dectons. Dscusson In ths nvestgaton, the esults of a numecal study of buoyancy-nduced flow and heat tansfe n a two-dmensonal squae enclosue wth localzed heatng fom below and symmetcal coolng fom the sdes ae pesented. The man paametes of nteest ae Raylegh numbe Ra and the dmensonless heat souce length. One knd of sub-gd scale model s used: lage-eddy smulaton (LES) wth the second-ode stuctue-functon subgd scale model (F 2 ) (moe detals n Slvea-Neto, 1998). The consevaton equatons ae dscetzed by the Galekn fnte element method wth lnea tangula elements. Two cases ae used fo valdaton of the computatonal doman of the pesent wok. As n Bto et al. (2002) and Bto et al. (2003), the same tubulence model LES togethe wth the fnte element method s used n the pesent wok. It s obseved that nceasng Ra, the ate of heat tansfe also nceases, as expected. Fo a fxed value of Ra, the ncease also nceases the heat tansfe. Fo Ra = 10 9 and = 0.8, although the flow s consdeed two-dmensonal, t s notced that the flow becomes oscllatng n tme, whch s a typcal chaactestc of a flow n tanston to tubulence. The aveage tempeatue T m and velocty vectos u dstbutons ae pesented fo Raylegh numbe 10 7 Ra 10 9 and Pandtl numbe P = 0.70 fo t = ( ) t 0. The esults of steam functon ψ dstbutons ae pesented fo t = 600 t 0. Acknowledgements The authos thank the fnancal suppot fom CNPq and CAPES wthout whch ths wok would be mpossble. Fgue 11. Case 1 Steamfuncton ψ fo t = 600 t 0 ( ψ = 10), aveage tempeatue T m ( T m = 0.01) fo t = ( ) t 0, and velocty vectos fo t = ( ) t 0 Ra = 10 7 P = 0.7 = / Vol. XXXI, No. 3, July-Septembe 2009 ABCM

9 Tubulent Natual Convecton n Enclosues Usng Lage-Eddy Smulaton wth Fgue 12. Case 2 Steamfuncton ψ fo t = 600 t 0 ( ψ = 10), aveage tempeatue T m ( T m = 0.01) fo t = ( ) t 0, and velocty vectos fo t = ( ) t 0 Ra = 10 8 P = 0.7 = 0.4. Fgue 13. Case 3 Steamfuncton ψ fo t = 600 t 0 ( ψ = 10), aveage tempeatue T m ( T m = 0.01) fo t = ( ) t 0, and velocty vectos fo t = ( ) t 0 Ra = 10 9 P = 0.7 = 0.4. Fgue 14. Case 4 Steamfuncton ψ fo t = 600 t 0 ( ψ = 10), aveage tempeatue T m ( T m = 0.01) fo t = ( ) t 0, and velocty vectos fo t = ( ) t 0 Ra = 10 7 P = 0.7 = 0.8. J. of the Baz. Soc. of Mech. Sc. & Eng. Copyght 2009 by ABCM July-Septembe 2009, Vol. XXXI, No. 3 / 207

10 Rogeo Fenandes Bto et al. Fgue 15. Case 5 Steamfuncton ψ fo t = 600 t 0 ( ψ = 10), aveage tempeatue T m ( T m = 0.01) fo t = ( ) t 0, and velocty vectos fo t = ( ) t 0 Ra = 10 8 P = 0.7 = 0.8. Fgue 16. Case 6 Steamfuncton ψ fo t = 600 t 0 ( ψ = 10), aveage tempeatue T m ( T m = 0.01) fo t = ( ) t 0, and velocty vectos fo t = ( ) t 0 Ra = 10 9 P = 0.7 = 0.8. Refeences Ampofo, F. and Kaayanns, T.G., 2003, Expemental benchmak data fo tubulent natual convecton n an a flled squae cavty, Intenatonal Jounal of Heat and Mass Tansfe, Vol. 46, pp Aydn, O. and Yang, W.-J., 2000, Natual convecton n enclosues wth localzed heatng fom below and symmetcal coolng fom sdes, Intenatonal Jounal of Numecal Methods fo Heat & Flud Flow, Vol. 10, No. 5, pp Bto, R. F., Gumaães, P. M., Slvea-Neto, A., Olvea, M. and Menon, G. J., 2003, Tubulent natual convecton n a ectangula enclosue usng lage eddy smulaton, Poceedngs of the COBEM 17 th Intenatonal Congess of Mechancal Engneeng, São Paulo, Bazl, pp Bto, R. F., Slvea-Neto, A., Olvea, M. and Menon, G. J., 2002, Convecção natual tubulenta em cavdade etangula com um clndo nteno, Poceedngs of the MECOM 1 st South-Amecan Congess on Computatonal Mechancs, and III Bazlan Congess on Computatonal Mechancs, and VII Agentne Congess on Computatonal Mechancs, Santa Fe, Agentna, pp (In Potuguese) Cesn, G., Paoncn, M., Cotella G. and Manzan, M., 1999, Natual convecton fom a hozontal cylnde n a ectangula cavty, Intenatonal Jounal of Heat and Mass Tansfe, Vol. 42, pp Deng, Q.-H., Tang, G.-F. and L, Y., 2002, A combned tempeatue scale fo analyzng natual convecton n ectangula enclosues wth dscete wall heat souces, Intenatonal Jounal of Heat and Mass Tansfe, Vol. 45, pp Kolmogoov, A. N., 1941, The local stuctue of tubulence n ncompessble vscous flud fo vey lage Reynolds numbes, Dokl. Akad. Nauk SSSR, Vol. 30, No.4, pp , and also n: Poceedngs of Royal Socety of London, 1991, Vol. 434, pp Kanovc, S., 1998, Lage-Eddy Smulaton of the Flow Aound a Suface Mounted Sngle Cube n a Channel, M.Sc. Thess, Chalmes Unvesty of Technology, Gotebog, Sweden. Lankhost, A. M., 1991, Lamna and Tubulent Natual Convecton n Cavtes Numecal Modellng and Expemental Valdaton, Ph.D. Thess, Technology Unvesty of Delft, The Nethelands. Matoell, I., Heeo, J. and Gau, F. X., 2003, Natual convecton fom naow hozontal plates at modeate Raylegh numbes, Intenatonal Jounal of Heat and Mass Tansfe, Vol. 46, pp / Vol. XXXI, No. 3, July-Septembe 2009 ABCM

11 Tubulent Natual Convecton n Enclosues Usng Lage-Eddy Smulaton wth Olvea, M. and Menon, G. J., 2002, Smulação de gandes escalas utlzada paa convecção natual tubulenta em cavdades, Poceedngs of the ENCIT 9 th Congesso Basleo de Engenhaa e Cêncas Témcas, Caxambu, Bazl, CD ROM, pp (In Potuguese) Padlla, E. L. M., 2000, Smulação Numéca de Gandes Escalas com Modelagem Dnâmca, Aplcada à Convecção Msta, M.Sc. Thess, Fedeal Unvesty of Ubelanda, Ubelânda, Bazl. (In Potuguese) Peng, S. H. and Davdson, L., 2001, Lage-eddy smulaton fo tubulent buoyant flow n a confned cavty, Intenatonal Jounal of Heat and Flud Flow, Vol. 22, pp Slvea-Neto, A., 1998, Smulação de gandes escalas de escoamentos tubulentos, In: I ETT Escola de Pmavea de Tansção e Tubulênca, Vol.1, Ro de Janeo, Bazl, pp (In Potuguese) Slvea-Neto, A., Bto, R. F., Das, J. B. and Menon, G. J., 2000, Aplcação da smulação de gandes escalas no método de elementos fntos paa modela escoamentos tubulentos, In: II ETT Escola Baslea de Pmavea, Tansção e Tubulênca, Ubelânda, Bazl, pp (In Potuguese) Tan, Y. S. and Kaayanns, T. G., 2000a, Low tubulence natual convecton n an a flled squae cavty pat I: the themal and flud flow felds, Intenatonal Jounal of Heat and Mass Tansfe, Vol. 43, pp Tan, Y. S. and Kaayanns, T. G., 2000b, Low tubulence natual convecton n an a flled squae cavty - pat II: the tubulence quanttes, Intenatonal Jounal of Heat and Mass Tansfe, Vol. 43, pp J. of the Baz. Soc. of Mech. Sc. & Eng. Copyght 2009 by ABCM July-Septembe 2009, Vol. XXXI, No. 3 / 209