Abstract. Nomenclature. A^ = dimensionless half amplitude ' = half amplitude of heat flux variation * - thermal conductivity

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1 Unsteady natural connective flow in an enclosure with a periodically varying side wall heat flux PH. Oosthuizen, J.T. Paul Heat Transfer Laboratory, Department ofmechanical Engineering, Queen's University, Kingston, Ontario, Canada Abstract Unsteady natural convective flow in a square enclosure with one vertical wall heated, with the opposite wall cooled to a umfom lower temperature and with the remaining walls adiabahc has bin numerically studied. The heat flux of the heated wal is spatially uniform but varying sinusoidally with time. The flow has been " expressed dt in terms of stream, function and vorticity and * written in TZZoTh ' ^ '"" *>"* ^ the finite-element hod The solution has as parameters the heat flux Rayleigh number based on the enclosure width, the Prandtl number and the frequency and magnitude of the dimensionless heat flux variation. Results have been ^1 """"7' *" ***** numbers between and 1,000,000. These results have been used to study the effects ^' ^ wall lemperature Nomenclature A^ = dimensionless half amplitude ' = half amplitude of heat flux variation * - thermal conductivity ' " " ""' " " Oux at any instant of time - time_averaged heat flux ''

2 54 Advanced Computational Methods in Heat Transfer Rcf = heat flux Rayleigh number basfed on W T = dimensionless temperature T' = temperature 7"c = temperature of cold wall T = mean dimensionless temperature of hot wall t = dimensionless time t' = time u = dimensionless velocity component in x direction u' = velocity component in x' direction v = dimensionless velocity component in y direction v' = velocity component in y' direction W = width of enclosure x = dimensionless x' coordinate x' = horizontal coordinate position y = dimensionless y' coordinate y' = vertical coordinate position a = thermal diffusivity 0 = bulk coefficient v = kinematic viscosity $ = dimensionless stream function ^' = stream function w = dimensionless vorticity w' = vorticity 1 Introduction Most available studies of free convective flow in an enclosure are concerned with steady flow situations and most studies have considered the case where the wall temperatures are specified. In many real situations, however, the flow in the enclosure is not steady and the heat flux rather than the temperature on at least one of the walls is known and is varying periodically with time. In order to provide some results that indicate the effect of this variation on the flow in the enclosure, a numerical study of laminar free convective flow in a square enclosure with a heat flux on one wall that is varying sinusoidally with time and with the opposite wall cooled to a uniform lower temperature T'c, has been undertaken. The remaining walls are adiabatic. The flow situation is, thus, as shown in Fig. 1. The situation considered is an approximate model of that occurring in the cooling of electronic and electrical equipment. The flow and heat transfer in enclosures that results when the temperature of one vertical wall is suddenly increased has been quite widely studied, e.g. see references [1] to [8]. The case where the hot wall temperature is varying with time has received relatively little

3 Advanced Computational Methods in Heat Transfer 55 w Constant Temperature v Adiabatic Figure 1: Situation considered in present study. attention typical of the available studies m this area being those described m references [9] and [10]. These past studies of unsteady ththt" Z " * ""*"* been concerned with the case wherl the ho wall temperature is varying with time. The present study differs form the past work in that it deals with the practically importam case where the wall heat flux is varying with time portant 2 Governing Equations It has been assumed that the flow is laminar and two- dimensional and hat the fluid properties are constant except for the density change with temperature which gives rise to the buoyancy forces,this having been treated by using the Boussinesq approach. The solution has been obtained in terms of the stream function and vorticity defined, as usual, by: u = dx' dx' dy' (1) The prime (' ) denotes a dimensional quantity. The following dimensionless variables have then been defined:

4 56 Advanced Computational Methods in Heat Transfer T = (7" - T'c)/(q*W'lk), t = t'a I W* (2) where T'c is the temperature of the cold wall and q is the mean heat flux rate from the heated wall section. In terms of these dimensionless variables, the governing equations are: = w (3) du dw ^ f ( W vy du _ # vy dw \ ^ _ p /' d*u> ** + ~JT V^y" "^7 "at ~dy - «f (4) dt, / ^ 97 9^ 97 dt \ dy dx dx dy = 0 (5) Here Rcf is the heat flux Rayleigh number based on the cavity width, W, i.e.: JW = 0 g~q* W* / ki/a (6) The boundary conditions on the solution are as follows: on all walls: $ = 0, d^ldn =0, at x = 0: dt/dx = q, at x = 1: T = 0, on the top and bottom walls, i.e. at y = 0 and at y = 1: dtldy =_0. Here q = q' I q' is the time varying dimensionless heat flux on the heated wall section. It has been here assumed that it varies sinusoidal with time, i.e., it has been assumed, since the time-averaged value of q is 1, that the dimensionless hot wall heat flux is given by: q = I + A sin(27r tl P) (7) As indicated in Fig. 1, A is the dimensionless half amplitude and P is the dimensionless period of the variation. It has been assumed that the fluid is initially at rest and at the cold wall temperature, i.e. it has been assumed that the initial conditions are: t = 0: 4> = 0, T = 0 The above dimensionless equations, subject to the initial and

5 Advanced Computational Methods in Heat Transfer T I 0.40 A= A=0.5 A= Dimensionless Time 0.48 Figure 2: Variation of mean dimensionless heated wall section temperature with dimensionless time for P= 002 S = 1 Rtf = 10* for A = 0.25, 0.5 and 1. boundary conditions, have been solved using a finite element procedure This solution gives, at any instant of dimensionless time the local dimensionless temperature distribution on the heated wall' This variation has then be used to find the instantaneous spatially averaged mean temperature on the heated wall section i e,.1 = / Tdy */ n (8) The adequacy of the present numerical results has been assessed by comparing the results obtained at long times with a step change in wall temperature with standard results for steady state natural convection in a square enclosure and by studying the effects of wide changes dimensionless time step and of the number of nodes and their distribution on the results. These checks indicate that the present results have an uncertainty of significantly less than 2%. 3 Results The solution, in general, has the following parameters: The Rayleigh number, Rtf The Prandtl number, Pr The dimensionless half amplitude of the hot wall heat flux variation, A The dimensionless period of the hot wall heat flux variation, P

6 58 Advanced Computational Methods in Heat Transfer Minimum o.o o.o Figure 3: Effect of A on the maximum and minimum values of mean dimensionless heated wall section temperature for P = 0.02, S = 1 and Rcf = 10*. Because of the possible applications that motivated the study, results have only been obtained for a Prandtl number of 0,7. Results have been obtained for Rayleigh numbers between 100 and 1,000,000 for dimensionless periods between and 0.15 for dimensionless half amplitudes of between 0 and 1. A value of A equal to zero means that there is a constant heat flux at the heated wall section. Because the fluid in the enclosure is initially at rest, there is an initial transient phase at small times but at larger dimensionless times the flow becomes periodic. The dimensionless time taken for the periodic state to be reached was found to be mainly dependent on the Rayleigh number. Except for the very smallest Rayleigh numbers considered, the periodic state was found to have been reached by a dimensionless time of less than 0.4. The initial transient phase will not be considered here, attention only being given to the results in the periodic region i.e. for dimensionless times of greater than 0.4. In all cases it was also found that the ratio of the maximum to the minimum value of the dimensionless wall heat flux was far greater than the ratio of the maximum to the minimum mean wall temperature indicating that a pseudo-steady state flow does not exist in the enclosure under the conditions considered. Figure 2 presents results illustrating the effect of the dimensionless amplitude, A, on the mean hot wall dimensionless temperature in the periodic state. It will be seen that the amplitude of the variation in mean dimensionless wall temperature increases with increasing A This is further illustrated by the results given in Fig. 6 which shows the variations of the maximum and minimum values of

7 Advanced Computational Methods in Heat Transfer P=0.02 P= Dimensionless Time Figure 4: Variation of mean dimensionless heated wall section temperature with dimensionless time for A = = 1 and R* = 10* for f =0.02 and Maximum Mean Minimum p Figure 5: Effect of P on the maximum and minimum values of mean dimensionless heated wall section temperature for A = 0 5 S = 1 and Rcf = 10*. the dimensionless hot wall temperature in the periodic state with dimensionless amplitude A. It will be seen that for the conditions considered, the maximum and minimum values of the dimensionless wall temperature are approximately proportional to A. Figure 3 also shows the variation of the time-averaged mean dimensionless hot wall temperature with A. It will be seen that this average value is essentially constant for the conditions considered and it is equal, essentially, to the value that would exist with steady state heat transfer under the same conditions.

8 60 Advanced Computational Methods in Heat Transfer \^ Minimum \\KXaximum 0.2 o.o E4 1E5 1E6 1E7 Ra + Figure 6: Effect of Re? on the maximum and minimum values of mean dimensionless heated wall section temperature for A = 0.5, 5=1 and P = Heat transfer from a hot moving surface passing through boundary conditions has been obtained using the finite element approach. Figures 4 present results illustrating the effect of the of the dimensionless period P on the mean dimensionless hot wall temperature variations. It will be seen that the variation in the mean dimensionless temperature decreases as the dimensionless period decreases i.e. as the frequency of the wall heat flux variation increases. This is further illustrated by the results given in fig. 5 which shows the variations of the maximum and minimum values of the dimensionless hot wall temperature in the periodic state with dimensionless period P. It will be seen that for dimensionless periods above about 0.02 these maximum values are approximately constant but that at smaller values of P the difference between the maximum and minimum values decreases with decreasing P. Figure 5 also shows the variation of the time-averaged mean dimensionless hot wall temperature with P. It will be seen that this average value is essentially constant for the conditions considered and it is equal, essentially, to the value that would exist with steady state heat transfer under the same conditions. Figures 6 presents results illustrating the effect of the of Re? on the mean dimensionless hot wall temperature variations. It shows the variations of the maximum and minimum values of the dimensionless hot wall temperature in the periodic state with modified Rayleigh number Re?. At small Rayleigh numbers, when the convective motion has a negligible effect on the heat transfer, Re? has, of course, no effect. However, for modified Rayleigh numbers above about 1000, the maximum and minimum values both decrease with increasing

9 Advanced Computational Methods in Heat Transfer 61 R(f. Figure 6 also shows the variation of the time-averaged mean dimensionless hot wall temperature with R<?. It, too, is constant for R# values below about 1000 and then decreases with increasing Rdf at higher values of Rtf. Calculations have also been carried out with A = 0 to determine the final steady state values of the mean dimensionless hot wall temperature for various values of the modified Rayleigh number These final steady state values were found to be essentially equal to the time averaged mean dimensionless hot wall temperatures existing at the same modified Rayleigh number with a fluctuating wall heat flux. 4 Conclusions The results obtained in the present study indicate that, for the range of parameters covered in the present study: 1. Unless the Rayleigh number is small (< approximately 1000), the the flow being periodic for dimensionless times greater than At the smaller values of the dimensionless period, P here considered the range over which this dimensionless mean wall temperature varies decreasing with decreasing P. However at the larger values of P considered the range over which this dimensionless mean wall temperature varies is relatively independent of P. 3. The range over which this dimensionless mean wall temperature vanes is approximately proportional to the dimensionless amplitude, A. 4. The time-averaged mean dimensionless temperature of the heated wall section is essentially equal to the mean wall temperature that would exist at the same Rayleigh number with steady flow in the enclosure. 5 Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada. References 1. Ivey, G.N. Experiments on Transient Natural Convection in a Cavity, Journal of Fluid Mechanics, 1984, 144,

10 62 Advanced Computational Methods in Heat Transfer 2. Kuhn, D.C.S. & Oosthuizen, P.H. Unsteady Natural Convection in a Partially Heated Rectangular Cavity, Journal of Heat Transfer, 1987, 109, No. 3, Hall, J.D., Bejan, A. & Chaddock, J.B. Transient Natural Convection in a Rectangular Enclosure with One Heated Side Wall, Int. J. Heat and Fluid Flow, 1988, 9, Kuhn, D.C.S. & Oosthuizen, P.H. Transient Three-Dimensional Flow in an Enclosure with a Hot Spot on a Vertical Wall, Int. J. Numerical Methods in Fluids, 1988, 8, Schladow, S.G., Patterson, J.C. & Street, R.L. Transient Flow in a Side-Heated Cavity At High Rayleigh Number: a Numerical Study, Journal of Fluid Mechanics, 1989, 200, Hyun, J.M. & Lee, J.W. Numerical Solutions for Transient Natural Convection in a Square Cavity with Different Sidewall Temperatures, Int. J. Heat and Fluid Flow, 1989, 10, Patterson, J.C. & Armfield, S.W. Transient Features of Natural Convection in a Cavity J. Fluid Mechanics, 1990, 219, Jeevaraj, C.G. & Patterson, J.C. Experimental Study of Transient Natural Convection of Glycerol-Water Mixtures in a Side Heated Cavity Int. J. Heat and Mass Transfer, 1992, 35, Kazmierczak, M. & Chinoda, Z. Buoyancy Driven Flow in an Enclosure with Time Periodic Boundary Conditions, Int. J. Heat and Mass Transfer, 1992, 35, Oosthuizen, P.H. & Paul, J.T. Unsteady Natural Convective Flow in an Enclosure with a Partially Heated Wall with a Varying Temperature in Numerical Methods in Thermal Problems, Vol. VIII, Part 1 (ed R.W. Lewis), pp , Proc. 8th Int. Conf., Swansea, Wales, 1993, Pineridge Press, Swansea, U.K., 1993.