Natural convection in shallow and tall cavities with four walls differentially heated

Steady laminar natural convection heat transfer inside airfilled rectangular cavities with differentially heated horizontal and vertical opposite walls, is studied numerically. Both cases of bottom endwall either at higher or lower temperature than the top endwall are considered. A computer code based on the SIMPLE-C algorithm is employed for the solution of the mass, momentum, and energy conservation equations. Simulations are performed for different values of the aspect ratio of the cavity from 0.25 to 4, and the Rayleigh number based on the cavity height in the range between 10 and 10, whose effects on the flow and temperature fields, and on the heat transfer rate across the enclosure, are analyzed in details. INTRODUCTION Most of the scientific work conducted in the past on natural convection inside rectangular enclosures has been substantially oriented to study the unidirectional heat flows which originate when the imposed heat flux or temperature difference is either simply horizontal or vertical. Actually, much more complex boundary conditions may be encountered in practical cases, which has motivated a number of researchers to carry out studies on situations wherein a multi-directional heat flow is established across the enclosure. This is, e.g., the case of the studies performed by: Kimura and Bejan [1], on enclosures heated from one side and cooled from below; November and Nansteel [2], and Aydin et al. [3], on enclosures heated from below and cooled at a single side, or heated at one side and cooled from the top; Ganzarolli and Milanez [4], and Basak et al. [5], on enclosures heated from below and cooled at both sidewalls; Ostrach and Raghavan [6], Shiralkar and Tien [7], and Cianfrini et al. [8], on enclosures with differentially heated horizontal and vertical opposite walls. With specific reference to the last configuration cited above, in which a horizontal and a vertical temperature gradients are simultaneously imposed across the cavity, most of the interest has been addressed to the square geometry, mainly with the aim of evaluating the effects of either stabilizing or destabilizing vertical temperature differences on the horizontal heat transfer. In contrast, despite the importance in a number of applications, a much smaller degree of attention has been paid to the role played by the aspect ratio of the enclosure. In this context, the aim of the present paper is to study the behavior of air-filled shallow and tall cavities, whose opposite walls are differentially heated, under the assumption that the horizontal and vertical temperature differences are the same, and that the mean temperature level of the horizontal walls is the same as that of the vertical sidewalls, which means that two adjacent walls of the enclosure are heated at same temperature TH, while the other two walls are cooled at same temperature TC. Both thermal configurations of bottom wall either heated or cooled with respect to the opposite top wall are examined. The study is carried out numerically under the assumption of two-dimensional laminar flow. A computer code based on the SIMPLE-C algorithm is used for the solution of the mass, momentum and energy conservation equations. Simulations are executed for different values of the height-to-width aspect ratio of the cavity in the range between 0.25 and 4, and the Rayleigh number based on the cavity height in the range between 10 and 10, whose effects on the flow and temperature fields, as well as on the heat transfer rates, are analyzed in full detail and discussed. Dimensionless heat transfer correlating equations are also proposed. MATHEMATICAL FORMULATION An air-filled rectangular enclosure of height H and width L, is considered. The cavity is heated at one side, and cooled at the other side, maintained at temperatures TH and TC, respectively. The same temperature differential (TH − TC) is imposed at the horizontal walls. Both cases of enclosure with bottom endwall either heated or cooled are analyzed, the former case being denoted as configuration BH, the latter as configuration BC, as sketched in Fig. 1, where the (x,y) coordinate system adopted is also represented.