Flow patterns and heat transfer in square cavities with perfectly conducting horizontal walls: the case of high Rayleigh numbers ( )

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1 Advances in Fluid Mechanics VII 391 Flow patterns and heat transfer in square cavities with perfectly conducting horizontal walls: the case of high Rayleigh numbers ( ) R. L. Frederick & S. Courtin Universidad de Chile, Departamento de Ingeniería Mecánica, Santiago, Chile Abstract Flow patterns and heat transfer in a square cavity with perfectly conducting horizontal walls are described by direct simulation for Rayleigh numbers of The code uses second order time and space discretisations and non-uniform grids. For Ra = 10 6 or less a steady final state is found. Periodic characteristics are obtained at Ra = 10 7, and non periodic behaviour is found at higher Ra. Time averaged wall and mid plane Nusselt numbers (not previously known for this case) are reported and correlated. Their oscillation frequencies are determined to characterize the regime. The mid plane Nusselt number, Nu mid, always exceeds wall Nusselt numbers. Nu mid represents the total heat moved by the cavity under the perfectly conducting condition. At Rayleigh numbers from 10 8 onwards, Nu mid exceeds the corresponding value for the adiabatic problem. Keywords: natural convection, enclosures, perfectly conducting walls. 1 Introduction Natural convection in differentially heated square enclosures has received considerable attention in the CFD and Heat Transfer literature. Although the condition of perfectly conducting horizontal walls is more realistic when air is the test fluid, most studies use the adiabatic condition. In the present study, numerical simulations of the natural convection in a square cavity with perfectly conducting horizontal walls at high Rayleigh numbers are reported. Nusselt number values for some cases not previously treated in the literature are given. doi: /afm080381

2 392 Advances in Fluid Mechanics VII Attempts to simulate the flow in differentially heated cavities at very high Rayleigh numbers have been relatively scarce in the literature until very recently. A direct numerical simulation of cases with Rayleigh numbers in the turbulent range seems at first sight to be possible because the confined geometry limits the size of the largest turbulent structures. Even so, the need to describe in full detail even the smallest structures in the viscous layers near the walls is a very severe difficulty. Very high Rayleigh numbers, above 10 9, are associated to turbulence or to the succession of chaotic flow patterns that result in turbulence. Successful simulation of flow and temperature fields must account for the extremely high temperature and velocity gradients at the walls, which require small grid steps. Time scales get shorter as Ra grows, requiring small time steps. However, simulation times must be long enough to get meaningful time averages of all quantities. The present study is restricted to a maximum Rayleigh number of Two-dimensional simulations of differentially heated cavities at high Rayleigh numbers, with adiabatic horizontal boundaries have been reported. Paolucci [1, 2] used the finite volume method to give detailed description of the flow dynamics. The first paper [1] dealt with the transition to turbulence, while [2] conducted a direct numerical simulation of turbulent flow for air, taken as a non-boussinesq fluid, at Ra = Le Quéré et al [3] used spectral methods to obtain numerical simulations at Ra = , using the Boussinesq assumption. The main goal in these contributions was to characterize the mechanisms governing the onset of unsteadiness, the transition to chaos and the turbulence. In the chaotic state, internal wave activity was not suppressed, as shown by high amplitude and low frequency oscillations in average Nusselt number at the vertical midline. High frequencies are observed in wall Nusselt number oscillations, which are associated to motion within the boundary layers. Three-dimensional solutions for differentially heated, cubical enclosures with adiabatic walls were given by Fusegi et al [4], Janssen and Henkes [5], Tric et al [6] and others. In [5, 6] 2D results were also reported for Rayleigh numbers up to From the 2D results in [6], the Ra exponent in the Nusselt Rayleigh number relation is 0.3 at low Ra ( ) and in the upper range ( ). Le Quéré and Behnia [6] implies a 0.26 exponent in [4] for Ra = Information on Nusselt number values for the perfectly conducting square cavity is scarce, especially at high Rayleigh numbers ( ). An experimental benchmark for cubical cavities up to Ra = 10 8 is available [7], and numerical simulations for the same situation have been performed [8]. In this paper direct numerical simulations are reported for a square cavity with perfectly conducting horizontal walls for Ra = After a brief description of flow and heat transfer in this situation, the dependences of wall and central Nusselt number on Rayleigh number are described and explained. Then the results are generalized in the form of Nusselt number correlations. 2 Formulation and numerical method Consider a square cavity of side L, filled with an incompressible Boussinesq fluid such as air (Pr = 0.71). From a uniform temperature, motionless state, a

3 Advances in Fluid Mechanics VII 393 difference T between the hot left wall and the cold one is imposed at t = 0. The dimensionless governing equations of continuity, momentum and energy in terms of the Rayleigh number (Ra), based on L and T are: V t U V + = 0 U U U P + U + V = + Pr 2 U t V V P 2 + U + V = + Pr V + Ra Pr Θ Θ Θ Θ + U + V = 2 Θ t Y and V are the vertical coordinate and velocity, respectively. The following boundary conditions apply: X = 0, Θ = 0.5; X = 1, Θ = -0.5, Y = 0 and Y = 1, Θ = X U = V = 0 on all walls Initial conditions for the interior points in the region are: Θ = 0, U = V = 0 at t = 0. A finite volume code based on the SIMPLER method with second order time and space discretisations and non-uniform grids in both directions was used. The coordinates of nodes in the deformed grid are expressed in terms of the coordinates in an equally spaced grid with the same number of nodes (X ), according to the rule of Janssen and Henkes [3]: 1 ' ' X = X sin(2πx ) 2π A grid sensitivity analysis showed that appropriate grids were of nodes for Ra 10 8 and nodes for higher Rayleigh numbers. The latter grid gave 40 temperature nodes at the zone of high wall temperature gradient at Ra = For validation the results of the code for a square cavity with adiabatic horizontal walls and Pr = 0.71 were compared with a benchmark for that problem [6]. For Rayleigh numbers up to 10 6, the maximum differences in horizontal and vertical peak velocities never exceed 0.2%. The Nusselt Numbers did not deviate from the benchmark by more than 0.1%. In addition, left and right average Nusselt numbers agreed to within 0.02% in our runs for the adiabatic case. Finally, our Nusselt number results for a perfectly conducting cavity agreed to within 0.1% with the results available in [8] at Ra = Results and discussion Differences between the adiabatic and perfectly conducting cases will be briefly described. With perfectly conducting horizontal walls, heat transfer is

4 394 Advances in Fluid Mechanics VII characterized by three average horizontal Nusselt numbers: the two wall Nusselt numbers, Nu hot, Nu cold, and the vertical midline average Nusselt number, Nu mid. While Nu hot = Nu cold, Nu mid is higher than the other two. The difference is caused by heat flow across the horizontal walls. The flow in the adiabatic case at Ra = 10 6 is in boundary layer regime, with a stratified core. A similar flow pattern occurs in the perfectly conducting case, but, as the horizontal walls are active, the velocities and circulation rate increase in comparison with the adiabatic case. However, the heat transfer at the vertical walls ( Nu hot ) is lower. Fig. 1 shows the horizontal local Nusselt number profile at the hot (left) wall as a function of Y. In the adiabatic case (not shown) the local Nu in the vicinity of the arriving corner (Y = 0) is the highest. In the perfectly conducting case, Nu is unity at both corners due to the linear temperature on the horizontal walls. Buoyancy forces imposed by the lower wall at 0 < X < 0.5 on the fluid coming from the cold wall, cause it to flow upwards, forming an inclined stream that moves toward the hot wall. These two facts result in a lower Nu hot than in the adiabatic case at each Ra, in spite of the higher circulation rate. Figure 1: Horizontal Nusselt number profile at the hot wall, Ra = The vertical local Nusselt numbers across the perfectly conducting walls are zero at the corners, as vertical wall temperatures are uniform. A net vertical heat flow (fig. 2) enters the cavity in the interval 0 < X < 0.5, and it is compensated by an equal heat output in the region 0.5 < X < 1. As a result, the average horizontal Nusselt number varies with X and has a maximum at X = 0.5, which results from

5 Advances in Fluid Mechanics VII 395 the sum of the heat transferred through the hot wall and the contribution from the horizontal walls from X = 0 to 0.5. Therefore Nu mid represents the total heat moved by the cavity under the perfectly conducting condition. Figure 2: Net vertical heat flow across the horizontal walls of the cavity. 3.1 Cases Ra = 10 7, 10 8 and 10 9 The solutions for the two higher Rayleigh numbers were run from the uniform temperature motionless state, but the case 10 7 was run using the temperature, velocity and pressure fields for Ra = 10 8 as initial condition. In all these cases, the final regime is time dependent, with oscillations of great amplitude in Nu mid. At variance with Ra = 10 6, in which the final regime is permanent, at Ra = 10 7 the two wall Nusselt numbers are unequal at each time instant, and display roughly sinusoidal, out of phase oscillations (fig. 3). From fig. 3, it is possible to identify the main oscillation frequencies. These are, for Nu mid, of and for the wall Nusselt numbers, A time averaged temperature field (fig. 4) shows complete stratification in the nucleus and full symmetry about the vertical axis. The boundary layer thickness is considerably reduced with respect to Ra = Inclined fluid jets discharged by the active walls are visible. The similarity of this time- averaged temperature field with the steady one found at Ra = 10 6 shows that, although at Ra = 10 7 the regime is unsteady, it is fully laminar. Non-periodic oscillations occur at Ra = In a comparison of cold wall Nusselt number with the result of Janssen and Henkes (23.9), the percent differences are very low (0.1 %). Fig. 5 shows the evolution of Nusselt number

6 396 Advances in Fluid Mechanics VII in the final regime at Ra = Nu mid displays complex oscillations in which its value at a given instant cannot be predicted from previous values. High amplitude oscillations are associated to internal wave motion that is not damped at this Ra. By Fourier analysis one main frequency was found for Nu mid (1881.9), probably associated with internal wave motion. Oscillations of wall Nusselt number are of high complexity and show an additional frequency of As this frequency is not found in the central Nusselt number, it must be caused by near- wall motions. These frequencies agree in order of magnitude with the ones reported by Paolucci [1] for the adiabatic case. The behaviour at this Ra can be considered as chaotic. Fig. 6 shows a time- averaged temperature field. It is very complex compared to the one at Ra = Local circulations appear at different heights in the vicinity of the hot wall. Thermal plumes are generated by interaction of hot fluid with the cold part of the upper wall, (also ascending plumes are generated in the hot part of the lower wall). This local behaviour is similar to a Rayleigh-Bénard situation. These plumes and the incidence of cold and hot fluids on the hot and cold walls respectively produce many local circulations (of a spatial scale smaller than the overall motion). This seems to be the cause of the instability of the final regime, which is transient at high Ra. Figure 3: Evolution of mid plane (bold line) and wall Nusselt numbers at Ra = Heat transfer Characteristic values of the solutions are given in Table 1. Maximum velocities at the cavity axes and Nusselt numbers are steady state values for Ra up to 10 6 and time averages for higher Ra.

7 Advances in Fluid Mechanics VII 397 Table 1: Maximum velocities and overall Nusselt numbers. Ra U max V max Nu mid Nu cold All velocities are a 15 20% higher than the corresponding values for the adiabatic problem. Nusselt numbers are in general lower than their adiabatic counterparts are. This holds for wall Nusselt numbers at all Ra, but at Ra = 10 8 and 10 9, Nu mid exceeds the adiabatic value. The results in Table 1 can be expressed by the following equations: Nu mid = 0.16Ra Nu cold = 0.15Ra Figure 4: Time averaged temperature field, Ra = According to these trends the midline Nusselt numbers would progressively exceed the ones for adiabatic cavities (in which the Rayleigh number exponent is lower) as Ra grows.

8 398 Advances in Fluid Mechanics VII Figure 5: Evolution of mid plane (bold line) and wall Nusselt numbers at Ra = Figure 6: Time averaged temperature field, Ra = 10 9.

9 Advances in Fluid Mechanics VII Conclusions The simulation of the high Rayleigh number cases (up to Ra = 10 9 ) has been done successfully. The phenomenon is characterized by the generation of boundary layers on vertical walls with thickness that decrease as Ra grows. These boundary layers are extended to the adjacent horizontal walls. The internal wave regime characteristic of this phenomenon at all Rayleigh numbers is suppressed in the final regime up to Ra = At higher Rayleigh numbers internal wave motion remains, with oscillation of high amplitude. The wall Nusselt numbers, up to Ra = 10 6 are equal at all times, during the transient and final regimes. For Ra = 10 7 or higher, both wall Nusselt numbers oscillate out of phase. At Ra = 10 9, the oscillations of the two wall Nusselt numbers are completely independent. The Nusselt number behaviour in the present situation had not been described previously. New results of wall and mid plane Nusselt number are the main contribution of this work. At Rayleigh numbers from 10 8 on, the total heat moved by the cavity under the perfectly conducting condition (Nu mid ) exceeds the corresponding value for the adiabatic problem. References [1] Paolucci, S. & Chenoweth, D.R., Transition to chaos in a differentially heated vertical cavity, J. Fluid Mechanics, 201, pp , [2] Paolucci, S., Direct numerical simulation of two-dimensional turbulent natural convection in an enclosed cavity. J. Fluid Mechanics, 215, pp [3] Le Quéré, P. & Behnia, M., From onset of unsteadiness to chaos in a differentially heated square cavity, J. Fluid Mechanics, 359, pp , [4] Fusegi, T., Hyun, J.M., Kuwahara, K. & B. Farouk, B., A Numerical Study of Three-dimensional Natural Convection in a Differentially Heated Cubical Enclosure, Int. J. Heat Mass Transfer, 34, pp , [5] Janssen, R.J.A. & Henkes, R.A.W., Instabilities in three-dimensional differentially heated cavities with adiabatic horizontal walls, Phys. Fluids, 8, pp 62 74, [6] Tric, E., Labrosse, G. & Betrouni, M., A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions, Int. J. Heat Mass Transfer, 43, pp , [7] Leong, W.H., Hollands, K.G.T. & Brunger, A.P., Experimental Nusselt numbers for a cubical cavity benchmark problem in natural convection, Int. J. Heat Mass Transfer, 42, pp , [8] Henkes, R.A.W. & Le Quéré, P., Three dimensional transitions of natural convection flows, J. Fluid Mech., 319, pp , 1996.