Numerical Study of Lid-Driven Square Cavity with Heat Generation Using LBM

Transcription

1 Amercan Journal of Flud Dynamcs 23, 3(2): 4-47 DOI:.5923/j.ajfd Numercal Study of Ld-Drven Square Cavty wth Heat Generaton Usng LBM M. A. Tahe r, S. C. Saha 2, Y. W. Lee 3, H. D. Km,* Dept of Mechancal Engneerng, Andong Natonal Unversty, Andong, Korea 2 School of Chemstry, Physcs & Mechancal Engneerng, Queensland Unversty of Technology GPO Box 2434, Brsbane, QLD 4, Australa 3 Dept of Mechancal Engneerng, Pukyong Natonal Unversty, Busan, Korea Abstract In ths study, the mxed convecton heat transfer and flud flow behavors n a ld-drven square cavty flled wth hgh Prandtl number flud (Pr = 54, ν =.2-4 m 2 /s) at low Reynolds number s studed usng thermal Lattce Boltzmann method (TLBM) where ν s the vscosty of the flud. The LBM has bult up on the D2Q9 model and the sngle relaxaton tme method called the Lattce-BGK (Bhatnagar-Gross-Krook) model. The effects of the varatons of non dmensonal mxed convecton parameter called Rchardson number(r) wth and wthout heat generatng source on the thermal and flow behavor of the flud nsde the cavty are nvestgated. The results are presented as velocty and temperature profles as well as stream functon and temperature contours for R rangng from. to 5. wth other controllng parameters that present n ths study. It s found that LBM has good potental to smulate mxed convecton heat transfer and flud flow problem. Fnally the smulaton results have been compared wth the prevous numercal and expermental results and t s found to be n good agreement. Keywords Lattce-Boltzmann, Rchardson number, Ld-drven, Heat generaton. Introducton In the recent years, the flud flow and heat transfer n a drven cavty have receved ncreasng attenton because of ts wde applcatons n engneerng and physcal scences. Some of these applcatons nclude ol extracton, coolng of electronc devces and heat transfer mprovement n heat exchanger devces. The Lattce Boltzmann Equaton (LBE) s one of the methods avalable to deal wth such problems effectvely and effcently. It s commonly recognzed that the Lattce Boltzmann Method (LBM) can confdently be used to smulate the ncompressble Naver-Stokes(N-S) equatons wth hgh accuracy. In ths lattce BGK (LBGK) model, the local equlbrum dstrbuton has been chosen to recover the N-S macroscopc equatons by[,2]. In addton, sometmes t s mportant to nvestgate thermal effects smultaneously wth the flud flows. Obvously the temperature dstrbuton n a flow feld s of central nterest n heat transfer problems. In most geophyscal flows, the temperature dfference s the drvng mechansm of the moton of the flud. From a practcal pont of vew, the research on natural and mxed convecton n a cavty has been wdely nvestgated[3, * Correspondng author: kmhd@andong.ac.kr (H. D. Km) Publshed onlne at Copyrght 23 Scentfc & Academc Publshng. All Rghts Reserved 5]. The effect of nclnaton angle on heat transfer rate and flow pattern nsde a ld -drven square cavty wth dfferent mxed convecton parameter has been studed by Darz et al.[6]. In ther study they have shown that, the verty of Nusselt number and Rchardshon number are opposte because the natural convecton changes to mxed convecton or forced convecton when Rchardson number decreases. Consequently, the Nusselt number ncreases. Hussen et al.[7] have nvestgated mxed convecton through a ld-drven ar flled square cavty wth a hot wavy wall. The applcaton of the thermal LBM to the natural convecton flow n a square cavty wth dfferentally heated wall wth a unform heat flux have been dscussed by D'Orazo et al.[8]. Mohammad et al.[9], Dxt et al. [] dscussed a thermal lattce Boltzmann method based on the BGK model. They used ths model to smulate hgh Raylegh number natural convecton n a square cavty. Patl et al.[] showed that the LBM can be successfully appled n deep cavtes wth aspect ratos.5 to 4 wth hgh Reynolds number. Prandtl number effects on MHD flow at a ld -drven cavty consderng porous meda have been numercal nvestgated by Hasanpour et al.[2]. Varous numercal smulatons have been performed usng dfferent thermal LB models or Boltzmann based scheme to nvestgate the mxed convecton flow problems as well as heat transfer related ssues wth dfferent condtons[3-7]. He et al.[8] have dscussed more detals about thermal model for the LBM n ncompressble lmt. In ths model,

2 Amercan Journal of Flud Dynamcs 23, 3(2): they have calculated temperature usng an nternal energy dstrbuton functon, where the macroscopc densty and velocty are stll smulated usng densty dstrbuton functon In addton, the thermal Lattce Bhatnagar-Gross -Krook (TLBGK) model wth some adjustable parameters wth Boussnesq approxmaton has been dscussed by Whte et al.[9]. The more detals TLBM wth some examples are dscussed by many authors[2-26]. In ths study, the problem of mxed convecton n a ld-drven cavty flled wth hgher Prandtl number flud (Pr = 54, ν =.2-4 m 2 /s) and consderng heat source placed at the mddle poston of bottom wall s nvestgated by usng Thermal Lattce Boltzmann Method (TLBM). A few nvestgatons of mxed convecton n a cavty wth upper movng ld, usng TLBM, are found n the lterature, but to the author s knowledge, the above mentoned problem has not been studed yet by ths method. The am of the present study s to examne the effects of mxed convecton parameter, called Rchardson number (R), n presence of heat generatng source at the bottom wall to nvestgate ts effect to the flud flows and temperature felds. 2. Problem Formulaton 2.. Mathematcal Analyss To ensure the model satsfes the N-S equatons for a flud under the nfluence of body force, the most common form of Lattce Boltzmann Equaton (LBE) wth BGK approxmaton can be wrtten as f ( x + t e, t + f ( x, eq 2τ D () = ( f ( x, f ( x, ) + e F τ 2τ 2 A c The functon f ( x, s the partcle dstrbuton functon eq and f s the dscrete equlbrum dstrbuton functon at lattce poston x and tme t wth lattce velocty vector e dscussed by[2]. Here A s the adjustable coeffcent, D s the dmenson, F s appled force, and ω =/τ s the relaxaton parameter that depends on the local macroscopc varables ρ and ρ u.these varables should satsfy the followng laws of conservaton: ρ = f and ρ u = e f (2) In presence of any nteracton or any external forces, the velocty should be modfed by the force term n calculatng the equlbrum dstrbuton functons as τ F u + ρ u eq = (3) The above equatons are the workng horse of the lattce Boltzmann method and replaces N-S equatons n CFD by approprate choce of lattces and f eq ( x,. The form of ths equlbrum dstrbuton functon must be chosen so that the flud mass and momentum are conserved. For two dmensonal D2Q9 model, the equlbrum dstrbuton functon s defned by[2] eq 3 eq f = ρw[ + eu. 2 c 9 eq 2 3 eq2 + ( eu. ) ] 4 u 2 2c 2c Here c s called the Courant-Fredrchs-Le wy (CFL) number and the lattce weghtng factors w depend only on the lattce model. For D2Q9 model, w = 4/ 9, w = /9, =,2,3,4 and w = /36, = 5,6,7,8. Smultaneously, the lattce Boltzmann energy equaton wthout vscous dsspaton[5] can be wrtten as τ (4) g( x+ t e, t+ g(,) x t ( (,) eq = g xt g (,)) xt (5) τ θ Here θ s the relaxaton tme constant for energy equaton. The energy dstrbuton functon g at the equlbrum state can be wrtten as eq 3 eq g = ε w[ + eu. 2 c 9 eq 2 3 eq2 + ( eu. ) ] 4 u 2 2c 2c Where ε s the nternal energy varable of the flud components and s defned as ε ( x, = g( x, (7) It s noted that the nternal energy ε s related (proportonal) to the temperature by the thermodynamc relaton ε = ρ c p T. To mprove the numercal stablty, space-tme ndependent average value of T s used to calculate the lattce speed c = 3RT. Under ths assumpton, the flud corresponds to the deal gas equaton of state, PV = RT, wth specfc heats c v =., c p =2., and wth the adabatc exponent γ = c p / c v = 2.. So the R = c p c v = n lattce unts. It s known that the above equatons recover the N-S equatons both for velocty and temperature felds[2, 5], f the vscosty coeffcent µ and thermal conductvty κ can be dentfed as µ = τ ρrt and κ = τθ RT. ρc p 2 2 These can be wrte n lattce unt as Cs 2 ν = τ 2 and α = τθ Cs 2 2 respectvely. Here υ =µ/ρ s the knematc vscosty of the flu d, α = κ /ρc p s the thermal dffusvty, and Cs = RT s the speed of sound n lattce unt. Therefore, the mean (6)

3 42 M. A. Taher et al.: Numercal Study of Ld-Drven Square Cavty wth Heat Generaton Usng LBM temperature of the flud n ths model can be wrtten as g ( x, ε ( x, T = = (8) ρ c p ρ c p 2.2. Numercal Analyss The physcal confguraton consdered here as shown n Fg.. s a closed square cavty of length H and flled wth hgh Prandtl number flud. The upper horzontal wall moves from left to rght wth a unform velocty U, whle the other three walls are fxed. The horzontal walls are assumed to be nsulated whereas the vertcal walls are mantaned at constant but dfferent temperatures T h (ho and T c (cold). A sold block or thn heater s placed at the bottom wall. gβ TH Ra = Pr Gr =, να where Gr = gβ TH 3 /ν 2 s the Grashof number, Pr = ν/α s the Prandtl number, The rato Gr/Re 2 s the mxed convecton parameter and t s called the Rchardson number (R). Actually, t s a measure of the relatve strength of the natural convecton to forced convecton for a partcular problem. Velocty, v/u X 8 X 2X2 5X x/h 3 Fgure. The confguraton of the problem under consderaton The grd ndependence test has been performed n the LBM smulatons for four dfferent lattce grd resolutons, namely, 8 8,, 2 2 and 5 5 as shown n Fg. 2. The velocty and temperature profles along the horzontal centerlne are llustrated. The velocty profles are vrtually dentcal for grd szes and above wth mperceptble dfferences for low resolutons. It means that the soluton s not very grd senstve. Therefore, any sets of grd can be chosen and hence the unform grd sze of s adopted for numercal smulatons for ths study. The flud propertes consdered here as the Prandtl number, Pr = 54, and vscosty, ν =.2-4 m 2 /s, wth low Reynolds number (Re=). For natural convecton, the momentum and energy equatons are coupled and the flow s drven by temperature or mass gradent,.e. buoyancy force. Hence, there s an extra force term that needs to be consdered n solvng LB equatons. Under Boussnesq approxmaton, the force term per unt mass can be wrtten as F( x, = ρ ( x, g β ( T ( x, Tref ) where T ref s the reference temperature of the flud, g s the gravtaton acceleraton, β s the thermal expanson coeffcent. For s mall temperature dfference, the buoyancy force s balanced by vscous drag and heat dsspaton. The rato of the buoyancy force to the product of vscous force and heat dffuson rates defnes the Raylegh number Temperature, T Fgure 2. Grd refnement test: V-velocty and temperature profles along horzontal centrelne of ld drven cavty for dfferent grd resolutons namely, 8 8,, 2 2 and 5 5 We ntroduce the followng dmensonless varables to normalze the governng equatons, u X = x / H, Y = y / H, U =, U v V = U x/h T T, θ = T T 8 X 8 X 2X2 5X5 c q H t U, q =, t = k( T T ) H h c s c The boundary condtons for the present study are as follows: θ U = Uld, V =, = at Y= Y θ θ U = V = = and = qθs ( for heater) at Y= Y Y θ =, U = V = at X =, Y

4 Amercan Journal of Flud Dynamcs 23, 3(2): and θ = U = V = at X =, Y, where t * s the lattce tme steps, q s the non dmensonal heat generaton parameter or heat source, T s s surface temperature of the heater and θ s s the non dmensonal temperature of the heat source. The characterstc speed U = gβ TH must be chosen carefully(less than about.), so that the low Mach number approxmaton holds n order to nsure the problem s n the ncompressble regme. 3. Results and Dscussons By solvng Eqs.() and (5), we wll get all nformaton that we are nterested n our study. These equatons are solved usng two-dmensonal unform grd system along wth boundary condtons and other equatons descrbed as above. Each numercal tme steps conssts of three stages: () collson, () streamng, and () boundary condtons followed by the LBM approaches. In order to valdate our model, frst, we verfed our computatonal results wth the conventonal benchmark and expermental results as shown n the Fg.3. The present smulaton has been shown aganst the numercal results of Khanafer et al[3] and the expermental results of Krane and Jessee[4] for natural convecton n an enclosure flled wth ar as shown n Fg.3. It can be seen from these fgures that our present numercal smulaton s n very good agreement wth both of numercal and expermental solutons. The velocty and temperature profles have been depcted for Rchardson number, R and heat generatng source, q as shown n Fgs The vertcal velocty component, V s calculated along the horzontal centre lne (see Fg 4) and the horzontal component, U s calculated along the vertcal centre lne of the cavty (see Fg 5). It s found that the flud adjacent to the hot wall receves heat from the hot surface and rses due to buoyancy along the hot wall and flu d adjacent to the cold wall becomes colder and falls along the cold wall due to densty varaton, as expected n Fg.4(a). In more detals, the velocty along the vertcal walls of the cavty shows a hgher level of actvty as predcted by thn layer of hydrodynamc vscous boundary layers. The locatons of the local maxmum and mnmum veloctes for all cases tend to near the walls wth ncreasng R. However, n presence of heat source(q =.5), local maxmum velocty ncreases near the hot wall wth ncreasng R. Therefore, the effect of buoyancy forces as well as heat generaton wth movng ld causes the strong flud moton from left to rght and consequently near the cold wall the flud moton seems not lamnar. Fro m Fg.5, t s seen that the horzontal velocty, U, s zero at Y= and t decreases as Y ncreases, so that t becomes negatve. For hgher values of Y, U ncreases and approaches to ts maxmum value at upper part of the cavty where the top surface s movng horzontally. The effect of dmensonless temperature profles wth R at the centrelne of the cavty as shown n Fg. 6. The temperature s calculated along the horzontal centre lne. Fg. 6(a) represents the temperature profles wthout heat generaton and Fg. 6(b) represents temperature profles wth effect of mposed heat generaton. It s revealed that the varaton of temperature near hot and cold walls versus dmensonless horzontal length s lnear for all R, whch s the characterstc of heat transfer by convecton, whereas, a sgnfcant temperature gradent s observed n the vcnty of the heated surface approxmately, X, and unheated surface.85 X. Moreover, t s seen that thermal stratfcaton and ncrease n temperature n the drecton of heat flow n the core regon X.85. Its physcal meanng s that the temperature should decrease n the ncrease of R. It means that the natural convecton s domnant compare to forced convecton. Ths s because the upper wall s movng wth constant velocty from left to rght and therefore, the flud moton enhances the rate of heat transfer. Wthout heat generaton effect, the thermal boundary layers are seen both left and rght walls (Fg.6a) but n Fg.6(b) t s seen just on left wall. Temperature contours and streamlnes are shown n Fg. 7 and Fg. 8 respectvely for dfferent Rchardson number, R and heat generaton parameters, q. In Fg. 7, when R =.5, the heat s transferred manly by forced convecton between the hot and cold walls, but n presence of heat generaton wth movng ld, the temperatures contours nsde the cavty tend to concentrate at the rght vertcal cold wall. In ths case, thermal boundary layer s observed near both the hot and cold walls. It s also observed that heat transfer reduces when the forced convecton domnated due to presence of heat generaton and movng ld. Smlar phenomenon s seen for R =.. When R = 5., we see that the natural convecton domnants the heat transfer. In ths case both n absence or presence of heat generaton, buoyancy force sgnfcantly enhances the heat transfer due to movng ld as well as heat generaton source effects. The streamlnes for dfferent values of non dmensonal mxed convecton parameter wth heat generaton and wthout heat generaton are shown n Fg. 8. In the absence of heat generaton wth ncreasng R, Fgs.8()-()(a), two prmary vortces are seen. Snce the natural convecton becomes domnant wth the ncreasng of R, near the rght wall vortex approaches larger due to the combned effects of movng ld. Wth ncreasng the values of heat generaton parameter, q>., two prmary vortces become a larger central vortex as heat generaton asssts buoyancy forces by acceleratng the flud flow and the vortex centre moves wth dfferent values q. In addton, the secondary vortces are observed, one s near the top rght corner and another one s bottom left corner for R=.5 due to effect of movng ld as well as heat generatng source at the bottom wall.

5 44 M. A. Taher et al.: Numercal Study of Ld-Drven Square Cavty wth Heat Generaton Usng LBM V-velocty Present-LBM Krane & Jessee [4]-Exp Khanafer et al.[3]-num Temperature Present-LBM Krane & Jessee [4]-Exp Khanafer et al.[3]-num x/h x/h Fgure 3. Comparson of velocty and temperature profles for Pr =.7, Ra =.89 5 (a). q=. R=.5 R=. R=5. (b).3.5 q=.5 R=.5 R=. R=5. V- velocty V-velocty X X Fgure 4. V-velocty profles along horzontal centrelne of the cavty (a) q=. (b) q=.5.8 R=.5 R=. R=5..8 R=.5 R=. R=5. Y.6 Y U- velocty U- velocty Fgure 5. U-velocty profles along vertcal centrelne of the cavty

6 Amercan Journal of Flud Dynamcs 23, 3(2): (a).8 q=. R=.5 R=. R=5. (b).8 (θ) q=.5 R=.5 R=. R=5. (θ) Temperature.6.4 Temperature X X Fgure 6. Temperature profles along horzontal centrelne for R (a) wth no generaton (b) wth heat generaton () R=.5 () R=. () R=5. Fgure 7. The temperature contours for () R=.5, () R=. and () R=5., q=.,.,.5

7 46 M. A. Taher et al.: Numercal Study of Ld-Drven Square Cavty wth Heat Generaton Usng LBM () R=.5 () R=. () R=5. 4. Conclusons Fgure 8. The streamlnes contours for () R=.5, () R=. and () R=5. wth q=.,.,.5 The convecton heat transfer characterstcs and flow performance of hgh Prandtl number flud n a ld drven square cavty have been numercally nvestgated usng TLBGK model. In ths study, the mxed convecton parameter (R) provdes a measurement for the mportance of the thermal natural convecton forces relatve to the mechancally nduced ld-drven forced convecton as well as heat generaton (q) effect. We may summarze the fndngs from the above numercal nvestgaton as follo ws: The nfluenced of heat generaton on flud flow and heat transfer are more sgnfcant for natural convecton domnatng (R = 5) case compare to mxed and forced convecton domnatng cases (R=.5, ). For R =.5 and R =., the overall heat transfer decreases as the buoyancy force and the moton of cavty are not n the same drecton, however t s ncreased wth ncreasng the values of heat generaton parameter. The overall heat transfer ncreases due to movng ld n presence of heat generaton for natural convecton domnatng case, R = 5. For R =.5 and R =, the two prmary vortces me rge and form a larger central vortex f the values of heat generaton (q) ncrease snce heat generaton asssts buoyancy forces by acceleratng the flud flow and the vortex moves wth ncreasng heat generaton. Moreover, for natural convecton domnatng case (R = 5.), the secondary vortces are observed due to effects of movng ld as well as heat generatng source at the bottom wall. REFERENCES

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