Mixed Convection of Heat Transfer in a Square Lid-Driven Cavity

. Three dimensional steady state mixed convection in a lid driven cubical cavity heating from below has been investigated numerically. Two sided walls are maintained at a constant ambient temperature Ttop > Tbottom, while the vertical walls are thermally insulated. Governing equations expressing in a dimensionless form are solved by using finite element method. The Reynolds number is fixed at Re=100, while the Richardson number is varied from 0.001 to 10. Parametric studies focusing on the effect of the Richardson number on the fluid flow and heat transfer have been performed. The flow and heat transfer characteristics, expressed in terms of streamlines, isotherms and average wall Nusselt number are presented for the entire range of Richardson number considered. Multiple correlations in terms of the heat transfer rate and Richardson number has been established.


INTRODUCTION
In recent years the mixed convection in rectangular or square cavities has been investigated by many researchers. This attempt is due to the fact that heat transfer in a cavity can be found in many industrial and engineering applications such as electronic component cooling, food drying process, nuclear reactors etc… Flow and heat transfer phenomena caused by buoyancy and shear forces in enclosures have been studied extensively in the literature. For example, Iwatsu and Hyun [1] studied numerically three dimensional flows in cubical containers. The top moving wall is maintained at a higher temperature than the bottom wall. Numerical solutions are obtained over a wide range of physical parameters, 102 ≤ Re ≤ 2×103, 0 ≤ Ri ≤ 10 and Pr = 0.71. Numerical flow visualizations demonstrate the explicit effects of Ri as well as Re. Mohamed and Viskanta [2] investigated the effects of a sliding lid on the fluid flow and thermal structures in a lid-driven cavity. Moallemi and Jang [3] studied numerically mixed convective flows in a bottom heated square lid-driven enclosure. They investigated the effect of Prandtl number on the flow and heat transfer process. They found that the effects of buoyancy are more pronounced for higher values of Prandtl numbers, and they also derived a correlation for the average Nusselt number in terms of the Prandtl number, Reynolds number and Richardson number. Prasad and Koseff [4] performed an experimental investigation of mixed convection flow in a lid-driven cavity for different Richardson numbers, ranging from 0.1 to 1000. Their results indicate that the overall heat transfer rate is a very weak function of the Grashof number for the examined range of Reynolds numbers. They have also analyzed the mean heat flux values over the entire boundary to produce Nusselt number and Stanton number correlations which are very useful for design applications. Sharif [5] performed a numerical investigation with supplementary flow visualization of laminar mixed convective heat transfer in two-dimensional shallow rectangular driven cavities of aspect ratio 10. The top moving plate of the cavity is set at a higher temperature than the bottom stationary plate. Computations are reported for Rayleigh numbers ranging from 105 to 107 while keeping the Reynolds number fixed at 408.21, thus encompassing the wide spectrum of dominating forced convection, mixed convection, and dominating natural convection flow regimes. The fluid Prandtl number is taken as 6, representative of water. The effects of inclination of such a cavity on the flow and thermal fields are also investigated for inclination angles ranging from 0° to 30°. The author observed that the local Nusselt number at the heated moving plate starts with a high value and decreases rapidly to a small value towards the right side. The local Nusselt number at the cold plate reveals an oscillatory behavior near the right side due to the presence of a separation bubble at the cold surface in that location.
In the present work, the effect of temperature gradient orientation on the fluid flow and heat transfer in a lid-driven square cavity is investigated numerically.

Governing equations
For laminar, incompressible and three-dimensional mixed convection, after invoking the Boussinesq approximation and neglecting the viscous dissipation, can be expressed in the dimensionless form as:

Initial and Boundary conditions
No slip condition at and side walls. The upper lid has a constant velocity, u 0 . The upper lid wall has an isothermal condition with temperature, T=T c . The bottom wall is at rest and isotherm, i.e., T=T h (T c < T h ). Finally, the remaining walls are adiabatic (see Fig.1).
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Numerical procedure
Numerical computations were carried out with an in-house computer code written in the FORTRAN programming language.
In the FORTRAN code, the unsteady Navier-Stokes and energy equations are discretized by a second-order time stepping finite difference procedure. The procedure adopted here deserves a detailed explanation. First, the non-linear terms in Eqs. (2) are treated explicitly with a second-order Adams-Bashforth scheme. Second, the convective terms in Eq. (3) are treated semi implicitly. Third, the diffusion terms in Eqs. (2) and (3) are treated implicitly. In order to avoid the difficulty that the strong velocity-pressure coupling brings forward, we selected a projection method described in the work of Peyret and Taylor [6], Achdou and Guermond [7].
A finite-volume method is implemented to discretize the Navier-Stokes and energy equations (Patankar [8], Moukhalled and Darwish [9], Kobayachi and Pereira [10]). In this method, the solution domain is subdivided into small finite control volumes (CV). The grid used is more refined than that in areas in need and bigger in other zones.
In fact, it is necessary to use a fine mesh at the top and bottom wall, capable of accurately modeling the heat transfer and flow. With this approach, we can reduce the number of grid points without losing accuracy of calculation.
The advective terms in Eqs.
(2) are discretized using a QUICK third-order scheme whereas a second-order central differencing (Hayase, Humphrey and Greif [11]) is applied in Eq. (3). The discretized momentum and energy equations are solved employing the red and black successive over relaxation method (RBSOR) [12], while the Poisson pressure correction equation is solved utilizing a full multi-grid method (Hortmann, Peric and Scheuerer [13], M.S. Mesquita and M.J.S. de Lemos [14], E. Nobile [15]). If specific details about the computational methodology are needed, the reader is directed to Ben-Cheikh et al. [16].
Here, represents a dependent variable u, v, w, or θ, the indexes i, j, k indicate a grid point, and the index m is the current iteration at the grid level. The convergence criterion was set to 10−6.

CODE VALIDATION
The present numerical code is validated against a documented numerical study. Namely, the numerical solution reported by Iwatsu and Hyun [1]. The findings of the comparisons are documented in Table 2 for the average Nusselt number. The comparisons illustrate close proximity in the predictions made between the various solutions. These validation cases boost up the confidence in the numerical outcome of the present work.  The effect of grid resolution was also examined in order to select the appropriate grid density. Table 1 presents the results of a grid independency study showing the effects of number of grid points on Nusselt. A 48 3 non uniform grid is found to meet the requirements of both the grid independency study and the computational time costs.

RESULTS AND DISCUSSION
The effect of the downward temperature gradient on the streamline and isotherm patterns for Ri = 0.001, 1, and 10 is shown in Fig. 2. When Ri = 10, a single central primary vortex is observed covering the cavity domain. The vortex is driven by the moving lid. From subfigure 2c. the isotherms are symmetrical and clamped in the lower part of the plane (YOZ) and also we observed the appearance of the plume is believed to be the reason for the more enhancements in heat transfer.   Comparisons of the average Nusselt number between the numerical results and those obtained by the correlation are reported in Figs.3. As a result, the average Nusselt number computed from the above equations together with the numerical results agree well.

CONCLUSION
Numerical simulations of mixed convection in a lid-driven square cavity are made to investigate the effects of temperature gradient on the flow field and heat transfer characteristics for five different values of the Richardson number (Ri = 0.001≤ Ri≤10).From the numerical results, the following conclusions may be drawn:

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When large Ri is united with low Re, a single central primary vortex is observed covering the cavity domain. This vortex decreases slightly when Ri decreased. The rate of heat transfer increases with the Richardson number. A correlation in terms of the heat transfer rate and Richardson number has been established also in this case.
Numerical results demonstrate that the heat transfer rate increase due to the increased buoyancy effects in the lower portion of the cavity.