Abstract. Introduction

Transcription

1 Combined forced and natural convection in a square cavity - numerical solution and scale analysis A.T. Franco/ M.M. Ganzarolli'' "DAMEC, CEFET, PR , Curitiba, PR Brasil >>DE, FEM, UNICAMP , Campinas, SP Brasil Abstract Combined forced and natural convection in a square cavity is analyzed Vertical walls are maintained at different temperatures and the horizontal walls are adiabatic. The mass flow is induced by a shear force resulting from the motion of the upper wall combined with buoyancy force due to lateral wall heating. Laminar and two-dimensional flow is assumed and the problem is solved numerically by using the Finite Volume-SOLA method Numerical results are obtained for the Grashof number ranging from 10* to 10 and the Reynolds number values of 100, 400 and The effect of the Prandtl number, varying from 0.01 to 7.0, is also investigated. The average Nusselt number is reported for different values of the above parameters in order to ascertain the relative importance of natural and forced convection The proper scaling of the heat transfer (Nusselt number) in the different asymptotic regions leads to the criterion for the transition from one convection mechanism to another, represented by the Richardson number (Ri) for Pr < 1 and Ri/Pr^ for Pr > 1. Introduction Mixed convection heat transfer in lid-driven cavities have been the subject of many studies stimulated by different reasons. This phenomenon combines, on the one hand, the hydrodynamic cavity where the flow is induced by the upper moving wall; and, on the other hand, the heat transfer by

2 96 Advanced Computational Methods in Heat Transfer natural convection inside a thermal cavity with differentially heated side walls. Most of these studies are concentrated on cavities subjected to vertical temperature gradients. Iwatsu et al.* and Morzynski & PopieP analyzed a cavity cooled from below and heated from the upper moving wall. A cavity subjected to a instable vertical temperature gradient, that is, heated from below and cooled from the top wall, was studied by Mohamad & Viskanta^. The Prandtl (Pr) number effect for an also gravitationally unstable configuration was described by Moallemi & Jang'*. In the present work, combined forced and natural convection in a square cavity with the vertical walls maintained at different temperatures and the horizontal walls kept adiabatic is analyzed. The flow is induced by a shear force resulting from the upper wall motion and the buoyancy force due to the lateral wall heating. This work combines two classic benchmark problems: the lid-driven cavity and the thermal cavity heated from the sides, commonly employed to validate numerical algorithms. The problem is solved numerically by using the Finite Volume-SOLA method. Numerical results are presented for the Grashof number ranging from 10^ to 10^ and the Reynolds number values of 100, 400 and The effect of the Prandtl number, varying from 0.01 to 7, is also investigated. The average Nusselt number is reported as a function of the above parameters and a scale analysis is presented in order to determine the proper scalings of the heat transfer and the criterion for the transition from one convection mechanism to another. Problem formulation The cavity studied and the boundary conditions are schematically showed in Figure 1. The cavity is considered two-dimensional of equal length and height H, completely filled with a newtonian fluid. The vertical walls are maintained at constant temperatures T/,, (hot) and TC (cold) while the top surface moves from left to right at a constant speed Up. Laminar flow is assumed and the customary Boussinesq approximation for the governing equations is employed. The set of governing equations can be written in dimensionless form by using the following group of variables (X, Y) = (x, y)/h; (U, V) = (u, «)/[/ ; r = at/h*; 0 = (T - T,)/(T, - TJ (1) and the conservation equations for mass, momentum and energy become _!_%/ du_ du _dp_ l_ PC Or * dx dy ~~ OX * Re * ~*~ * ^ >

3 Advanced Computational Methods in Heat Transfer 97 H H Figure 1: The coordinate system of the cavity. 9P _l_^ _J^ ^ ^ " where the quantities Re = UpH/v, Pe = Re x Pr and Gr(= the Reynolds, Peclet and Grashof numbers. The corresponding boundary conditions are g/3h*/i>*) are = Q,y = 0 and oy = V = 0 and 8 = (6) The average Nusselt number is defined as M-/'fV (7) J 0 O y\. being determined from the numerical results either at the hot or at the cold wall.

4 98 Advanced Computational Methods in Heat Transfer Numerical procedure The system of equations above is solved by using the Finite Volume - SOLA method. This procedure is an amended version of the MAC-method and is described in detail by Hirt et ala The process consists of advancing in time (level n -f 1) the velocity and pressure fields from the previous values of velocities (time level n) and the steady state is reached when max < 10-5 and 1+1 < 10-5 (8) where (/> = w, v ou T. In order to verify the grid dependence of parameters like as the Nusselt number, several tests were performed for different values of Reynolds, Prandtl and Grashof numbers. Due to the large temperature and velocity gradients near the solid walls, a non-uniform grid in both x and?/ directions is adopted and the hyperbolic sine function is used to diminish the control volume size near the walls. The grid dependence of the average Nusselt number, for Gr = 10 and Pr = 0.71, is showed in Table 1. Table 1: Effect of mesh size. Reynolds Mesh 15 x x 30 GO x GO 80 x V% 8.G V% V% It may be noticed that when the number of grid points is increased from GO x GO to 80 x 80 the Nusselt number variation is within 0.2%. In view of these results, a non-uniform grid with 60 x GO grid points is adopted in the present work. The numerical method was also validated by comparing the results with the calculations of Hortmann et al. for a thermal cavity with the side walls at different temperatures. The comparison revealed a very good agreement with deviation of about 0.5% for the average Nusselt number when the Grashof number is 10.

5 Results and discussion Advanced Computational Methods in Heat Transfer 99 Numerical results are obtained for values of the Grashof number ranging from 10* to 10* and the Reynolds number values of 100, 400 and The Prandtl number values of 0.01, 0.1, 0.71 and 7.0 are also considered and its effect on heat transfer is investigated. The flow pattern consisting of a main clockwise cell of size of the cavity is found. Secondary eddies can occur, but this structure remanis the same over the range of variation of the parameters covered in this study. Contour maps showing isotherms are plotted for nine equally spaced values between the maximum (0 = 1) and the minimum (6 = 0) values of the dimensionless temperature 0. Figure 2 shows isotherms for the values of Gr and Re indicated in the figure caption and the Prandtl number fixed at 0.71, a typical value for air. Due to the flow induced by the upper moving wall, the plots do not exhibit the centersymmetry that characterizes the pure natural convection regime inside a cavity heated from the sides. The Figures 2a and 2b shows, for a low Grashof number (Gr = 10*), the effect of the increase of the Reynolds number. Since the contribution of buoyancy is small for low Grashof numbers, these isotherms are very close to that corresponding to a pure forced convection regime. Otherwise, following the sequence 2a-c-d, for Re=100, it can be seen how the increase of the Grashof number changes the isotherms toward a typical pure natural convection configuration. By comparing Figures 2d and 2f, for a large Grashof number (Gr = 10*), the isotherms are very similar in spite of the increase of the Reynolds number. It can be noticed that the core cavity is thermally stratified and the isotherms are compressed near the vertical walls, indicating intense temperature gradients in this region. These features characterizes the boundary layer regime for thermal cavities ( Gill*). On the other extreme, when the Reynolds number is large (#e=1000) and the Grashof number is moderate, the Figures 2b and 2e shows the insignificant effect of the Grashof number on the temperature field and the heat transfer is pure forced convection dominated. These plots show, in a qualitative way, that in mixed convection there are limiting cases where the heat transfer is dominated either by natural or by forced convection heat transfer mechanisms. From the boundary layer regime scale analysis for laminar natural convection on a vertical surface in an infinite medium, and for laminar forced flow over aflatplate parallel to a uniform stream ( Bejan*), the Nusselt number variation with respect to the Rayleigh number (Ra = Gr x Pr) and the Reynolds number is expressed by the proportionality Pr^Ra*/* natural convection /2 forced convection ^ where a and b are functions of the Prandtl number (Pr > I or Pr < I). In order to verify these scales, the Nusselt number is plotted for different

6 100 Advanced Computational Methods in Heat Transfer a-) V V c-) Figure 2: Isotherms a-) #c = 100 and Ea = 10^, b-) T^e = 1000 and /?.% = 10^, c-) Ec = 100 and #o = 10^, d-) ^e = 100 and ^a = 10\ e-) 7f.e = 1000 and Ea = 10\ f-) #e = 1000 and 7?a = 10^.

7 Advanced Computational Methods in Heat Transfer 101 Re= 100 Re = 400 Re= moo Nu - Rot(1/4) 10 10" 10" 10* 10* Ra a-) 10 Nu Gr = 10t2-0 Gr = 10t4 G_r_= 10t7 Re b-) 1Cr" Figure 3: The average Nusselt number as a function of (a) the Rayleigh number (b) the Reynolds number. 10^

8 102 Advanced Computational Methods in Heat Transfer values of the Prandtl number as a function of the Rayleigh and the Reynolds numbers. Figures 3a and 3b illustrates the Nusselt number variation as a function of Rayleigh and Reynolds numbers, for Pr=0.01, 0.71 and 7.0, and the respective power law from equation (9). For low to moderate values of Rayleigh (or Grashof) number, the Nusselt number is less sensitive to the Rayleigh number increase as the Reynolds number varies from 100 to The Nusselt number increases very close to the power law Re*/'*, as showed in Figure 3b, approaching the pure forced convection dominated heat transfer mode. On the opposite situation, that is, for large values of the Rayleigh (or Grashof) number, the Reynolds number pratically does not affect the Nusselt number (Figure 3b) and the phenomenon obeys, in a aproximate way, the power law Ra*/* which indicates the enhancement of natural convection as a dominant heat transfer mechanism. The transition from forced to natural convection dominance is represented by the knee in each Reynolds number curve, which shifts to the right as the Prandtl number increase. For low Prandtl number (Pr 0.01) and moderate values of Reynolds and Rayleigh (Re < 400 and Ra < 10^), as ilustrated in Figure 3a, the Nusselt number remains constant (Nu = 1) regardless the increase of these parameters. In this case, the temperature varies almost linearly across the cavity and the heat transfer corresponds to the pure conduction limit. Since the temperature profile is quasi-linear, the overall heat transfer is of order of k AT and, consequently, the order of magnitude of the Nusselt number becomes Nu = constant = 0(1) (10) for the pure conduction heat transfer limit. The key question now is to determine (approximately) the criterion for the transition from one convection mechanism to another. To find out this criterion, a scale analysis similar to that presented by Bejan^ is performed. This analysis compares, in an order of magnitude sense, the thermal boundary layer thickness for natural convection (<5/v) on a vertical surface with that of forced convection (6p) over a flat plate. The type of convection mechanism is decided by the smaller of the two boundary layer thickness, that is > (9(1) natural convection,, < (9(1) forced convection ^ ' This ratio leads to the criterion for the transition from one convection mechanism to another, represented by J*L\'"I " = 1A for Pr>l Pr"J n = 0, for Pr < 1 ^ '

9 Advanced Computational Methods in Heat Transfer 103 where Ri is the Richardson number (Ri = Gr/Re^), This criterion and the proper scaling of the Nusselt number for the distinct asymptotic regions (natural or forced convection domains) can be verified in Figure 4. The dashed lines represent the pure natural and forced convection Nusselt number scales for the vertical wall and for the flat plate and could explain, aproximately, the Nusselt number results obtained from the numerical simulations. It can be seen that, excluding the conduction limit, all the numerical results are close to the dashed lines, demonstrating the correct scaling of both the abscissa and ordinate. For the extreme values of the parameter Ri/Pr, that is, the natural and the forced convections domains, the Nusselt number results approaches the respective limit. Figure 4 suggests that the abscissa must be raised to less than the I/4th power in the natural convection limit, as it was seemingly found by Moallemi & Jang^. The knee of the numerical results occurs by about Ri/Pr = 1, regardless the Prandtl number, confirming this parameter as the appropriate criteria for the transition from a forced convection to a natural convection dominant heat transfer mode and vice-versa. QJ = 10' 1 Q0. Pr < 1, m = 1/2 and n=0 Pr>1,m = n=1/3 a Cond. limit a a a a a 1...-L.-I forced convection */' natural convection io ' Ri/Prtn Figure 4: Nusselt number variation as a function of Re, Ri and Pr. Concluding remarks The phenomenon of combined natural and forced convection heat transfer inside a lid-driven cavity diferentially heated from the sides is studied numerically. The results show the existence of limiting cases where the heat transfer may be ruled either by natural or forced convection mechanisms. Nevertheless, for low values of the Prandtl number and moderate values of

10 104 Advanced Computational Methods in Heat Transfer the Rayleigh and Reynolds number, the Nusselt number remains constant and the heat transfer is dominated by pure conduction mechanisms. A scale analysis similar to that conducted for the boundary layer regime on a vertical surface and over a flat plate leads to the criterion for the transition from one convection mechanism to another, represented by the Richardson number (Ri) for Pr < I and Ri/Pr^ for Pr > 1fluids.Eventually, the results confirm this parameter as the most adequate for the combined natural and forced convection heat transfer analysis. Acknowledgments We are very grateful by the utilization of the computational support of the CENAPAD and NUPES-CEFET/PR. References 1. Iwatsu, R., Hyun, J. M. & Kuwahara, K., Mixed convection in a driven cavity with a stable vertical temperature gradient, Int. J. Heat Mass Tnma/er, 1993, Vol. 36, No. 6, p Morzynski, M. & Popiel, Cz. O., Laminar heat transfer in a twodimensional cavity covered by moving wall, Numerical Heat Transfer, 1988, Vol. 13, p Mohamad, A. A. & Viskanta, R., Effects of the upper lid shear on the stability of flow in a cavity heated from below, Int. J. Heat Mass Tmng/er, 1989, Vol. 32, p Moallemi, M. K. & Jang, K. S., Prandtl number effects on laminar mixed convection heat transfer in a lid-driven cavity, Int. J. Heat Mass 7hmg/er, 1992, Vol. 35, No. 8, p Hirt, C. W., Nichols, B. D. & Romero, N. C., SOLA- Numerical Solution Algorithm for Transient Fluid Flow, Los Alamos Laboratory, Report LA-5852, Hortmann, M., Peric, M.& Scheurer, G., Finite volume multigrid prediction of laminar natural convection: bench-mark solutions. Int. J. Numerical Methods in Fluids, 1990, Vol. 11, Gill, A. E., The boundary layer regime for convection in a rectangular cavity", J. FMd Mec/aamcs, 1966, Vol. 26, p Bejan, A.,"Con%ec%(m #W Trans/e/', Wiley, New York, 1984.