FORCED AND NATURAL CONVECTION HEAT TRANSFER IN A LID-DRIVEN CAVITY

FORCED AND NATURAL CONVECTION HEAT TRANSFER IN A LID-DRIVEN CAVITY Guillermo E. Ovando Cacón UDIM, Instituto Tecnológico de Veracruz, Calzada Miguel Angel de Quevedo 2779, CP. 9860, Veracruz, Veracruz, México. Teléfono (229) 934 500 ext 237. Fax: (229) 934 2279. geocitver@gmail.com Sandy L. Ovando Cacón DACA, Universidad Juárez Autónoma de Tabasco, Avenida Universidad S/N zona de la cultura, CP. 86000, Villaermosa, Tabasco, México. Teléfono (993) 358 585. Fax: (993) 4 2950. ovansandy@gmail.com ABSTRACT Mixed convection eat transfer in a to-dimensional, Cartesian square cavity is numerically studied. A code based on te finite element metod combined it an operator splitting sceme is ritten to solve te governing equations in order to analyze te beavior occurring inside a cavity. Te steady state results are obtained for a Ricardson numbers range of 0. Ri 0 it a Prandtl number of 0.7. Te streamlines, te temperature fields, te velocity profiles and te kinetic energy fields are obtained. For te range of Ricardson numbers studied, it is observed te vortices formation developed by forced convection due to sear motion of te vertical alls and by buoyancy forces due to temperature gradient near te corners of te cavity. For Ri=0., te axial symmetry persist because te vortex formation is dominated by te motion of te moving boundaries generating one vortex near eac vertical all. Wen e increase te Ricardson number, te axial symmetry is lost because te vortex formation is dominated by te temperature gradient generating one vortex near eac corner ere te ot region is located, tis interaction beteen te natural convection and te position of te ot boundaries produces an unbalance in te fluid dynamic generating te symmetry loss and te presence of vortices inard te cavity, e also found tat eat transfer process depends on te position of ot corners observing different beaviors beteen te termal plumes developed at top and bottom ot corners. NOMENCLATURE u ρ Velocity vector u Transversal velocity component u 2 p ρ ν Ω t t f T T T c Axial velocity component Nabla operator Laplacian operator Pressure Density Kinematic viscosity To dimensional region Time Final time Temperature Cool temperature Hot temperature 234

α f ρ g β j x y L V L Ri Re Pe Termal diffusivity Body forces vector Gravity Compressibility coefficient Vertical unitary vector Transversal coordinate Axial coordinate Cavity lengt Velocity of te alls Heated lengt Ricardson number Reynolds number Peclet number INTRODUCTION Force and natural convection eat transfer in a cavity is of considerable importance due to its ide application areas (see [], [2] and [3]). Not only because its importance in fundamental science, but also because it is present in many practical situations, suc as cooling of electronic devices, furnaces, design of building, oscillating flo (Ovando, 2006, see[4]). Oztop et al., [5] studied te relation beteen te natural and forced convection modes on te eat transfer in to-sided lid-driven differentially eated square cavity, tey presented a numerical treatment of te mixed convection eat transfer depending on te direction of te moving alls. Nasr et al., [6] numerically investigated convection in a cavity cooled from te ceiling and eated from loer corner using a square cavity, tey analyzed te effects of eating intensity, te lengt of te eated portion and its position on te resulting flo structure and eat transfer. Most of studies in te literature ere performed bot for orizontal constant moving alls cavities and for side driven differentially eated cavities. Te main intent of tis ork is to investigate te beavior of eat transfer inside a square cavity it ot corners, cool central-part vertical alls and adiabatic central-part orizontal alls, being te aim of tis study to analyze te ot corners effects in te distribution of vortices inside te cavity it moving vertical alls. Tis configuration is inspired by te fact tat can be found in several experimental arrangements to investigate te vortex motion inside a cavity, to tis end, a piston-cylinder device is cosen ere te piston is fixed and te cylinder all moves it a certain velocity. Te ot corners of te cavity can simulate te eating generated by te friction beteen te piston and te cylinder all. Tis researc can be useful to many engineering applications suc as te automotive industry [7], ere te principal component is a piston-cylinder system, te coating industry [8], ere te process can be simulated it a translating lids cavity, and te mixing industry [9], ere te fluid mixing enancement can be investigated it a lid-driven cavity. Te flo inside te cavity presented in tis ork is described by tree non-dimensional parameter: Te Ricardson number (Ri), te Reynolds number (Re) and te Peclet number (Pe). PROBLEM SPECIFICATIONS Te problem considered in tis ork is a to dimensional cavity it a ratio of.0 beteen its vertical and orizontal dimensions as son in figure (). Te vertical alls move simultaneously it an upard constant velocity ile te orizontal alls are fixed. For air as te orking fluid, te sides of te cavity are L=0.6 m. Te four corners of te cavity are maintained at ot temperature ile te central part of te vertical alls is maintained at cold temperature and te central part of te orizontal alls is adiabatic. In all cases te gravitational force direction is parallel to te moving alls and te customary Boussinesq approximation is employed. Te flo region Ω cosen for te numerical simulations is a to dimensional cavity, ere te flo and eat transfer is analyzed. Te conservation equations tat describe tis problem for an incompressible fluid in tis region are te continuity, momentum and energy equations: 235

ρ ere u = ( u, ) u 2 [ 0,t ], ρ u = 0 in Ω f ρ u ρ ρ ρ ρ ν u + u u + P = f in Ω f t T ρ α T + u T = 0 in Ω [ 0,tf ], t [ 0,t ], is te velocity vector, being u and u2 te transversal and axial velocity components P = p/ ρ being p te pressure and ρ te density; t is te time, t f is respectively; ν is te kinematic viscosity, te final time, T is te temperature and α is te termal diffusivity. Te Boussinesq approximation is given troug: () ρ g LT f β = V ( T ) 2 c T j (2) ere g is te gravity, β is te compressibility coefficient, L is te caracteristic lengt, T is te ot temperature, T c is te cold temperature, V is te velocity of te vertical all and j is te vertical unitary vector. Te velocity boundary conditions of te cavity are: u u u u ( x = 0, y,t) = 0, u2 ( x = 0, y,t) = V ( x = L, y,t) = 0, u2 ( x = L, y,t) = V ( x, y = 0,t) = 0, u2 ( x, y = 0,t) = 0, ( x, y = L,t) = 0, u ( x, y = L,t) = 0. 2,, (3) Te temperature boundary conditions are given by: Vertical left all( x = 0,t) T on 0 y L y L L y L T =, Tc on L < y < L L Vertical rigt all ( x = L,t) T on 0 y L y L L y L T =, Tc on L < y < L L Horizontal bottom all (y = 0,t) T on 0 x L y L L x L T =, T y = 0 on L < x < L L Horizontal top all (y = L,t) T on 0 x L y L L x L T =. T y = 0 on L < x < L L (4) 236

L is te lengt of te boundary ere ot temperature is applied at eac corner and in bot direction. L. L is 5% of Te governing equations are cast in dimensionless form using several caracteristic quantities, suc as: te caracteristic velocity V, te caracteristic lengt L, and te caracteristic temperatures T c and T, obtaining te folloing dimensionless variables: * x x =, L * u u =, V * T Tc T =, T T c * y y =, L * u2 u2 =, V * V t = t, L * P p =, 2 V (5) it ic are obtained te folloing non-dimensional numbers: Ricardson number Ri: Reynolds number Re: Peclect number Pe: Ri g ( T ) LT c = β 2 V. (6) VL Re = ν. (7) Pe = RePr. (8) Fig Geometry of te considered square cavity it moving vertical alls and boundary conditions. 237

Te dimensionless form of governing equations can be ritten as: ρ u ρ ρ ρ u + u u + P = RiT j in Ω t Re ρ u = 0 in Ω [ 0,tf ], T ρ T + u T = 0 in Ω [ 0,tf ]. t Pe [ 0,t ], f (9) Governing equations are numerically solved by te finite element metod and te operator splitting sceme, detailed information can be found in te literature (Corin, 973; Gloinski and Pironneau, 992; Turek, 999; Marecuk, 990; Gloinski, 2003; Gloinski and Le Tallec, 989; Dean and Gloinski, 997; Dean, Gloinski and Pan, 998). Te validation of te computer code as been carried out for te mixed convection in a lid-driven cavity by Oztop et al. [5]. Te average Nusselt number obtained it tis code as.32 ic coincides it te value reported (.33) by tese autors. Fig 2 Transversal velocity profiles for tree different meses. GRID INDEPENDENCE STUDY In order to guarantee grid independence of te solution, te simulations ere performed it tree different meses for all cases. For pressure, it as used mes sizes of 2x2, 4x4 and 6x6 it a constant spatial step x in te transversal direction, and a constant spatial step y in te axial direction. Te velocity mes as tice tinner tan te pressure mes. Analyzing te convergence of te tree meses for stationary state e could observe important differences beteen te results obtained it te meses of 2x2 and 4x4, oever te values obtained it te 4x4 and 6x6 meses are practically te same, tis beavior as observed for all cases. In figure (2), is son te transversal velocity profiles for te tree different meses tested it Ri=5. 238

RESULTS Figures (3), (4) and (5) so te streamlines for Ri=0.,.0 and 0.0, respectively. For te loest Ri number, it is observed axial symmetric beavior given by to vortices associated to eac moving vertical boundary. In te middle central vertical line, it can be observed tat te fluid as a strong donard vertical velocity component ile te transversal velocity component of te fluid is zero. Tis beavior is due to te fact tat forced convection is more important tan free convection and te motion of te flo is generated by sear forces. Near te vertical alls te fluid moves upard folloing te motion of tese alls, ten te fluid meets top corners to produce a vortex flo tat launces te fluid troug te central superior part of te cavity ere te fluid comes back to te inferior part of te cavity, in te loer central part of te cavity fluid meets te orizontal inferior all canging its vertical motion to a orizontal motion to produce a sink flo in te bottom corners of te cavity. For Ri=, te flo is still organized in to vortices elongated in te vertical direction and set in motion by te moving boundaries, a counterclockise circulating cell on te rigt and a clockise circulating one on te left. Hoever, in tis case e can observe axial quasi-symmetrical beavior because te buoyancy and forced convection are equally important, and te effect of te termal boundary condition starts to generate tis symmetry loss. Te left cell is larger tan te rigt one. For Ri=0, te vortex distribution inside te cavity is more complex, te vortical structures are stronger tan previous cases and te axial symmetry of te flo is lost. In tis case, it can be observed one vortex near eac corner of te cavity formed by te energy injection at te ot corners. Clockise circulating cells are formed near te left corners and counter-clockise circulating cells are formed near te rigt corners. In te left central part of te cavity a counter-clockise vortex is formed, ile in te rigt central part of te cavity a clockise eak vortex start to form, all tis beavior is due to te buoyancy effect. Te center of te left central vortex moved toard te orizontal symmetry line. Fig 3 Streamlines as a function of position for Ri=0.. 239

Fig 4 Streamlines as a function of position for Ri=.0. Fig 5 Streamlines as a function of position for Ri=0. Figures (6) and (7) so te kinetic energy fields for Ri=0. and 0, respectively. It can be observed tat for te loest Ricardson number, te kinetic energy is ig near te top corners of te cavity and decrease toard te inferior part of te cavity, clearly energy is not uniformly distributed troug te ole cavity for tis case. For Ri=0, te kinetic energy peaks are distributed in te four corners of te cavity, being te inferiors corners kinetic energy iger tan te superiors ones. Tis beavior is due to te fact tat inferior orizontal all does not ave any effect on te convection process of te fluid in tis region ile in te superior region te orizontal all provokes perturbation of te buoyancy forces. Te energy distribution is more uniform en Ricardson number increases, ic is associated it te complex vortex distribution tat can improved engineering process suc as mixing. 240

Fig 6 Distribution of kinetic energy inside of te cavity for Ri=0.. Fig 7 Distribution of kinetic energy inside of te cavity for Ri=0. Figures (8) and (9) so te temperature fields for Ri=0.,.0 and 0.0, respectively. For Ri=0., e can observe from te temperature field tat te eat of te corners fluxes from te corners to te interior part of te cavity, ile te cold region are confined near te cold alls at te central part of te vertical alls. Te central part of te cavity is maintained to a middle temperature and tis is associated it lo vorticity intensity. Wen te Ricardson number increases, e can observe tat te ot temperature region gros toard te interior of te cavity at te upper orizontal all, ile te ot temperature regions are confined to te corner at te loer orizontal all. Te ole corner vortices are located in te region ere tere are ig temperature gradients, tat is in a region it a transition beteen ot fluid and cold fluid. On te oter ands, te left central vortex generates a cold region in te central part of te cavity ic extends toard te loer central part of te cavity. Te rigt cold region tends to extend toard te inferior central part of te cavity driving by te velocity field it a donard large vertical component in tis area, ile te left cold region tends to extend toard te central part of te cavity driving by te left central vortex. Tis asymmetrical beavior is due to te instabilities generates for te different ays it ic te eat fluxes at te upper and loer corner. Near eac ot corner is formed an upard termal boundary layer originating a termal plume it instabilities travelling upards. In te case of loer corners te termal plume moves upard mixing it te cold fluid, but in te case of upper corners te termal plume meets te fixed all being forced to cange its vertical natural direction toard a orizontal direction, see figure (9). Tis complex interaction beteen te eat transfer and fluid dynamic generates axial symmetric-breaking. Te temperature field sos tat te central part of te upper orizontal all is eated ile te central part of te loer orizontal all is cooled. Te strong intensity of te left central vortex generates tat te left cold region extends almost until te central part of te cavity ile te velocity field of te flo around te rigt bottom vortex produces tat te rigt cold region extends toard te inferior lo all. 24

Fig 8 Temperature field inside of te cavity for Ri=0.. Fig 9 Temperature field inside of te cavity for Ri=0. 242

Axial velocity profiles for y=0.25, 0.5 and 0.75 are son in figures (0), () and (2), respectively. For Ri=0., te axial velocity (u 2 ) profile is symmetric at different eigt of te cavity and becomes maximum negative in x=0.5. For Ri=.0, te axial velocity (u 2 ) profile becomes sligtly asymmetric and negative in te central part of te cavity. Increasing Ricardson numbers to Ri=5 or Ri=0, te axial velocity (u 2 ) profiles become more complex tan te previous cases and as to maximum negative values and one maximum positive value. For tese to cases, te velocity of te fluid decreases from te vertical left all and becomes negative, ten sifts its direction upard in te center of te cavity to ten sifts again its direction donard, finally te velocity of te fluid increases folloing te upard movement of te vertical rigt all. Tis complex beavior is driven by te presence of central vortices and its strengt becomes loer for y=0.75 due to te presence of te top orizontal all. In order to analyze te loss of axial symmetry, e calculate te integral of kinetic energy (l 2 ) along te middle vertical line of te cavity, given by: 0 l u dx (0) 2 = 2 Fig 0 Axial velocity profiles as a function of te transversal coordinate for different Ricardson numbers at y=0.25. 243

Fig Axial velocity profiles as a function of te transversal coordinate for different Ricardson numbers at y=0.5. Figure (3) sos te integral of te kinetic energy as a function of Ricardson number. For Ri=, tis integral as a value of zero, indicating a symmetry beavior of te fluid inside te cavity, oever for Ri, te value of te integral is not zero indicating an asymmetric beavior. Te more te Ricardson number, te more te loss of symmetry. Fig 2 Axial velocity profiles as a function of te transversal coordinate for different Ricardson numbers at y=0.75. 244

Fig 3 Integral of te kinetic energy along te axial symmetry line as a function of Ricardson number. DISCUSSION AND CONCLUSIONS Mixed convection in a square cavity it ot temperature at te corners, cold temperature at te central part of te moving up vertical alls and adiabatic condition at te central part of te fixed orizontal alls ave been carried out by solving te Navier-Stokes equations coupled it te energy equation, to tis end, it as developed a numerical code based on finite element metod and an operator splitting sceme. Simulations ere performed for different Ricardson numbers of 0.-0 and ere analyzed te vortex formation associated it te eat transfer regime in mixed convection. For Ri=0., it is observed axial symmetry beavior because te forced convection is more important tan buoyancy and te beavior of te fluid is given by te motion of te vertical alls. For Ri=, ere te natural convection and force convection ave similar effect on te flo, it is observed quasi-symmetric beavior it an axial vortex near eac vertical all. Te temperature field is perturbed by te evolution of te vortices structure and by te temperature boundary condition at te top and bottom corners ic as different termal dynamical beavior. As e increase te Ri number te symmetric beavior is lost and te intensity of te vortices increases, observing vortices near eac corner and also te development of vortices in te central part of te cavity ic generates tat te cold region extends to te central part of te cavity and toard te central part of te inferior all. Te eat of te superior corners tends to flux toard te top interior part of te cavity ile te eat of te inferior corners is confined to tis region due to mixing of ot and cold fluid. Te numerical codes captures te differences of te dynamics of te termal plume at te top and bottom corners ic explains te unbalance of te fluid dynamic and te asymmetrical vortex distribution en Ri number is increased due to te iger injection of energy. We found tat for Ricardson numbers greater tan appear a central vortex ic moves toards te symmetry lines, tis agrees it te result reported by Oztop et al. [5]. For lo Ricardson number, te kinetic energy depends on te impinges of te fluid against te top orizontal all ile for ig Ricardson number, te kinetic energy depends on te injection of energy in te four corners of te cavity. Using te integral of kinetic energy along te middle vertical line is possible to evaluate te axial symmetry loss of flo inside te cavity. REFERENCES [] K. Torrance, R. Davis, K. Eike, D. Gill, D. Gutman, A. Hsui, S. Lyons, H. Zien. 972 Cavity flos driven by buoyancy and sear. J. Fluid Mec. Vol. 5, pp 22-23. [2] R. Iatsu, J. M. Hyun, K. Kuaara. 993 Mixed convection in a driven cavity it a stable vertical temperatura gradient. Int. J. Heat and Mass Transfer, vol. 36, pp 60-608. [3] C. H. Blom, H. C. Kulmann. 2002 Te to-sided lid-driven cavity: experiments on stationary and timedependent flos. J. Fluid Mec. 450, pp 67-95. 245

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