Effects of thermal boundary conditions on natural convection flows within a square cavity

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Abstract

A numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 103  Ra  105 and Prandtl number Pr, 0.7  Pr  10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rates at the center of the bottom wall than the uniform heating case for all Rayleigh numbers; however, average Nusselt numbers show overall lower heat transfer rates for the non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented.

Introduction

Natural convection in a closed square cavity has occupied the center stage in many fundamental heat transfer analysis which is of prime importance in certain technological applications. In fact, buoyancy-driven convection in a sealed cavity with differentially heated isothermal walls is a prototype of many industrial application such as energy efficient design of buildings and rooms, operation and safety of nuclear reactors and convective heat transfer associated with boilers. Buoyancy driven flows are complex because of essential coupling between the transport properties of flow and thermal fields. In particular, internal flow problems are considerably more complex than external ones. This is because at large Rayleigh number (product of Prandtl and Grashof numbers) classical boundary layer theory can assume the simplifications for external flow problems, namely, the region outside the boundary layer is unaffected by the boundary layer. For confined natural convection, in contrast, boundary layers form near the walls but the region external to them is enclosed by the boundary layers and forms a core region. Since the core is partially or fully encircled by the boundary layers, the core flow is not readily determined from the boundary conditions but depend on the boundary layer, which, in turn, is influenced by the core. The interactions between the boundary layer and core constitute a major complexity in the problem. In fact, the situation is even more intricate because it often appears that more than one global core flow is possible and flow subregions, such as, cells and layers, may be embedded in the core. A literature survey shows that the comprehensive review of these problems was made by Ostrach [1], [2], [3], Gebhart [4] and Hoogendoorn [5] in which each emphasizes essentially various aspects of the subject.

Perusal of prior numerical investigations by Patterson and Imberger [6], Nicolette et al. [7], Hall et al. [8], Hyun and Lee [9], Fusegi et al. [10], Lage and Bejan [11], [12], and Xia and Murthy [13] reveal that several attempts have been made to acquire a basic understanding of natural convection flows and heat transfer characteristics in an enclosure. However, in most of these studies, one vertical wall of the enclosure is cooled and another one heated while the remaining top and bottom walls are well insulated. November and Nansteel [14] and Valencia and Frederick [15] have shown a specific interest to focus on a natural convection within a rectangular enclosure wherein a bottom heating and/or a top cooling are involved. Studies on natural convection in rectangular enclosures heated from below and cooled along a single side or both sides have been carried out by Ganzarolli and Milanez [16]. Later, the case of heating from one side and cooling from the top has been analyzed by Aydin et al. [17] who investigated the influence of aspect ratio for air-filled rectangular enclosures. Also, Kirkpatrick and Bohn [18] examined experimentally the case of high Rayleigh number natural convection in a water-filled cubical enclosure heated simultaneously from below and from the side. Recently, Corcione [19] has studied natural convection in a air-filled rectangular enclosure heated from below and cooled from above for a variety of thermal boundary conditions at the side walls. Numerical results were reported for several values of both width-to-height aspect ratio of the enclosure and Rayleigh number.

The aim of the present study is to investigate natural convection in a square cavity when bottom wall is heated (uniformly and non-uniformly) and top wall is well insulated while two vertical walls are cooled by means of two constant temperature baths (see Fig. 1). In case of uniformly heated bottom wall, the finite discontinuities in temperature distribution appear at the edges of the bottom wall. The discontinuities can be avoided by choosing a non-uniform temperature distribution along the bottom wall (i.e., non-uniformly heated bottom wall) as discussed by Minkowycz et al. [20] where an investigation is made for a mixed convection flow on a vertical plate, which is either heated or cooled. In the current study, we have used Galerkin finite element method with penalty parameter to solve the non-linear coupled partial differential equations of flow and temperature fields for both uniform and non-uniform temperature distribution prescribed at the bottom wall. The results will be illustrated for Ra = 103–105 with Pr = 0.7–10 to represent influence of natural convection on heat transfer rates in terms of local and average Nusselt numbers at the bottom and side walls.

Section snippets

Mathematical formulation

The flow model is based on the assumptions that the fluid is Newtonian and that the properties are constant with the exception of the density in the body force term of the momentum equation. The Boussinesq approximation is invoked for the fluid properties to relate density changes to temperature changes, and so to couple in this way the temperature field to the flow field. The governing equations for natural convection flow using conservation of mass, momentum and energy can be written as:ux+

Solution procedure

The momentum and energy balance equations (8), (9), (10) are solved using the Galerkin finite element method. The continuity equation (7) will be used as a constraint due to mass conservation and this constraint may be used to obtain the pressure distribution (Basak and Ayappa [21]; Reddy [22]). In order to solve equations (8), (9), (10), we use the penalty finite element method where the pressure P is eliminated by a penalty parameter γ and the incompressibility criteria given by Eq. (7) (see

Stream function

The fluid motion is displayed using the stream function ψ obtained from velocity components U and V. The relationships between stream function, ψ (Batchelor [23]) and velocity components for two dimensional flows are:U=ψYandV=-ψX,which yield a single equation2ψX2+2ψY2=UY-VX.Using the above definition of the stream function, the positive sign of ψ denotes anti-clockwise circulation and the clockwise circulation is represented by the negative sign of ψ. Expanding the stream function (ψ

Numerical tests

The computational domain consists of 20 × 20 bi-quadratic elements which correspond to 41 × 41 grid points. The bi-quadratic elements with lesser number of nodes smoothly capture the non-linear variations of the field variables which are in contrast with finite difference/finite volume solutions available in the literature [10], [11], [12]. In order to assess the accuracy of the numerical procedure, the algorithm based on the grid size (41 × 41) for a square enclosure with a side wall heated were

Conclusions

The prime objective of the current investigation is to study the effect of continuous and discontinuous Dirichlet boundary conditions on the flow and heat transfer characteristics due to natural convection within a square enclosure. The penalty finite element method helps to obtain smooth solutions in terms of stream functions and isotherm contours for wide ranges of Pr and Ra with uniform and non-uniform heating of the bottom wall. It has been demonstrated that the formation of boundary layers

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