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Numerical Analysis of Free Convection flow in Square Enclosure Partially Heated from below using the Multigrid Method

— The present work investigates the efficiency of the Multigrid method when applied to solve two-dimensional laminar steady free convection flow in a square enclosure partially heated from below. The numerical method includes finite volume discretization with upwind scheme on structure orthogonal regular meshes. The performance of the correction storage (CS) Multigrid algorithm is compared for different numbers of sweeps in each grid level. Up to two grids, for both Multigrid V- and W- cycles, are presented. The results are mainly analyzed in terms of the average heat transfer at the walls of the enclosure and Multigrid performance on the rate of convergence. It is also shown that convective heat transfer has a characteristic behavior for each boundary conditions adopted in given ranges of the governing parameters.


I. INTRODUCTION
This study promotes a discussion surrounding the efficiency of the Multigrid method applied in a specific configuration. Multigrid methods have been used in many different calculations as a result of its facilitated converge capability.In single-grids, convergence rates solutions are greater in the beginning of calculations, reducing this sensibility while the iterative processes goes on. This hardto-converge behavioris due to the iterative methods which smooth out only those Fourier error components of wavelengths smaller than or equal to the grid size. Naturally, this effect becomes more significant as the mesh becomes refined. At another level, Multigrid methods cover a broader range of wavelengths trough relaxation on more than one grid, making it easier to converge.
A Multigrid cycle is a repetitive procedure practiced at each grid level according to the grid hierarchy. The V-and W-cycles are types of Multigrid that determines the convergence criterion and the number of iterations in each step along consecutive grid levels visited by the algorithm. Within each cycle, the intermediate solution is relaxed before (pre-) and after (post-smoothing) the transportation of values to coarser (restriction) or to finer (prolongation) grids [[1]- [3]].
The Multigrid method can be roughly classified into two major categories. The CS formulation, where algebraic equations are solver for the corrections of the variables and the Full Approximation Storage (FAS) scheme, where variables themselves are handled in all grid levels. Since much work has been done on both major classifications, specific recommendations are admitted. The application of  [3]].Therefore, an exception has been disclosed in the work of [ [4]], who reported predictions for the Navier-Stokes equations successfully applying the Multigrid CS formulation. In the literature, however, not too many attempts in solving non-linear problems with Multigrid linear operators are found.
Acknowledging the advantages of using multiple grids, [ [5]] presented numerical computations applying this technique to recirculating flows in several geometries of engineering interest. There, the correction storage (CS) formulation was applied to non-linear problems. Later [ [6]], analyzed the effect of Peclet number and the use of different solution cycles when solving the temperature field within flows with a given velocity distribution. In all those cases, the advantages in using more than one grid in iterative solution was confirmed, furthermore, [ [7]], introduced the solution of the energy equation in their Multigrid algorithm. Temperature distribution was calculated solving the whole equation set together with the flow field as well as uncoupling the momentum and energy equations. A study on optimal relaxation parameters was there reported. More recently, [[8]] analyzed the influence of the increase of points of the mesh and optimal values of the parameters of the Multigrid cycle for different geometries. Also, [[8]- [11]], presented a study on optimal convergence characteristics in solution of conductive-convective problems.
Much work has been done with enclosure geometries, studying heat and mass transfer, (simultaneously or not) because of its engineering response value. The reason for the attention behind the physical nature of buoyance-induced flows is well represented in [ [12]]. Cooling of electronic devices before its excessive heating, recovery of remnant oil in petroleum reservoirs, dispersion of atmospheric pollution and its implications in adjacent cities,spreading chemical and nuclear waste in soil, are just some examples of its importance.
The current work considers that free convection conditions can be imposed inside a square cavity and aims to study the interactions between buoyancy forces and heat elements inside. This application involves the work showed in the literature that has been discussed by many authors. The interaction between buoyancy forces and the heated elements and the numerical analysis of Multigrid solution applied into momentum and heat transfer forms the main objective of current work.

II. GOVERNING EQUATIONS
The following equations emerge from the mathematical descriptions of fluid flow and convective heat transfer in the enclosure. These governing equations are based on twodimensional, incompressible, laminar flow in Cartesian coordinate system.
Where U and V are the velocity components in x and y directions respectively,  is the density of the fluid, p is the total pressure and  is the kinematic viscosity of the fluid. The gravity acceleration is defined by g and T  is the thermal expansion coefficient. T and ref T are the temperature and the reference temperature, respectively, and  is the thermal diffusivity.
The transport dimensionless parameters, Grashoff ( Gr ), which provides the relationship between fluid buoyancy and viscosity, Prandtl ( Pr ), that provides the relationship of momentum diffusivity and thermal diffusivity and the Rayleigh number ( Ra ), which is an associated number of buoyancy-driven flow (natural convection) are given by:

III. NUMERICAL MODEL
The solution domain consist on a number of rectangular control volumes (CV), resulting in a structure orthogonal non-uniform mesh. Grid points are located according to a After integrating it over the CV of A set of algebraic equations are resulted from the integration of terms in (8), as one can name it, convection, diffusion, and source. Since these procedures are described somewhere else (e.g. Error! Reference source not found.) they are not repeated in this paper. To summarize, convective terms are discretized using the upwind differencing scheme (UDS) and diffusive fluxes make use of the central differencing scheme (CDS).
The final discretization for grid node P is done by using the integrated transport equation (8) with the substitution of all approximate expressions for interface values and With the east face coefficient, for example, being defined as: In (10)

IV. MULTIGRID TECHNIQUE
The wanted effect of the Multigrid approach in this work is for the convergence rate to become independent on the grid spacing and the numerical solution faster. The task behind the Multigrid method is to involve a hierarchy of successively coarsened grids into the iterative solution process. When done, an adequate strategy for the movement through the different grid levels should follow, along with consistently transferring data with the discretization scheme between the grids. This process allows an efficient error reduction over a wide spectrum of frequencies.
If an iterative scheme as the one described below is applied to the system of equations on a given grid, it turns out only those frequencies of the solution error can be reduced efficiently, which corresponds to the grid spacing. The high frequencies of the error are reduced a few iterations, while the low frequencies nearly remain unchanged. At another level, the steps usually taken in Multigrid algorithm are the reduction of high frequency errors(smoothing), computation of residual error(residual computation), decimation of the residual error to a coarser grid(restriction), and the interpolation into a finer grid. The present development about the Multigrid technique is also presented in [ [5], [6], [10], [11]] and for this reason the development is not repeated here.

V. RESULTS AND DISCUSSION
The computer code was run on an IBM PC machine with an INTEL CORE 2 DUO 2.0 GHz processor. Grid independence studies were conducted such that the solutions presented herein are essentially grid independent. For both cycles, pre-and post-smoothing iterations were accomplished via the Gauss-Seidel algorithm while, at the coarsest-grid, the TDMA method has been applied [ [13]]. The

Fig. 2 -a) Geometries and boundary conditions and b) computational grid.
The main difference between the present work and the work from [ [14]] is that here it is used the prescribed values (temperature -H T ) while in the reference work is used heat fluxes. This particularity reinforces a permanent regime with constant physical properties of the flux.
In order to understand better the implications of this new configuration and construct a simpler idea of the square cavity, for comparison meaning, Fig. 3 shows the streamlines and isotherms, respectively, of a clear square cavity heated on the left and cooled from the opposing side for Rayleigh numbers ranging from 3 10 1 to 6 10 1 In Fig. (3a), 3 10 1  Ra , the streamlines indicates an existence ofone centered vortex while corresponding isotherms, Fig. (3b), indicates a conductive heat transfer, expressed by the almost parallel pattern with the heated wall. The vortex is generated by the horizontal temperature gradient across the section. This gradient, y T   is negative everywhere, inducing a clockwise oriented vorticity.
When the Rayleigh number is increased to 4 10 1  Ra , Fig. (3c), the vortex in the middle of the  cavity starts to shape differently, slightly into a more elliptic configuration. The isotherms, Fig (3d), therefore, have a considerable convection advance, and the parallel configuration is undone, especially in the middle of the cavity.Temperature gradients are stronger near the vertical walls, but decrease in the center region.  /dx.doi.org/10.22161/ijaers.73.51  ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.com Page | 359 Fig. (3e), the elliptical behavior continuous and the centered vortex is stretched so that two secondary vortices can be shown inside of it. Heat transfer by convection in the viscous boundary layer alters the temperature distribution to such an extent that temperature gradients in the center of the domain are close to zero. Fig.  (3e) shows that, with this change in the sign of the source term negative, vorticity is induced within the domain. This also causes the development of secondary vortices in the core. Fig. (3f) continuous its convection advance in the square cavity indicating a faster movement of the flux closer to the walls.

Ra
, one can see an initiative attempt for a three vortices configuration inside the main vortex. Corresponding Fig. (3h) shows that heat transfer is mostly convective, again due to the faster movement on boarders.

Reference [[12]] Present Results
In Fig. 5, it is easy to see the similarities in the results of bothworks. Studying these results, one can note the plume shaped structure starting at the bottom of the cavity moving towards the top. Circulatory motion brings the bottom hot  /dx.doi.org/10.22161/ijaers.73.50  ISSN: 2349-6495(P) | 2456-1908(O) www.ijaers.com Page | 360 temperature stream up to the top wall, substantially penetrating into the flow core.
From these results, is expected a better understanding and a certain familiarization with the final configuration, Fig.  2a, due to the heated bottom in the square cavity. Additionally, Fig. 5 representsthe only possible solutions for that Ra number, reinforcing the final results and the geometry qualitatively.
Final configuration, square cavity partially heated from above, is separated into two different forms, denoted as 1 C and 2 C . The 1 C structure hasboth lateral walls cooled at temperature c T and top wall adiabatic, while 2 C hasleft lateral wall and top wall cooled at temperature c T and right lateral wall adiabatic. Fig. 6 is a representative scheme of 1 C and 2 C . Notice that for allconfigurations the total surfaces of the cooled walls are identical.