Natural convection in rectangular enclosures
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Contents 
Heated from the Side
The scale analysis for natural convection in a rectangular enclosure heated from the left side suggested that there are multiple regimes shown in the following table:
Regimes  I: Conduction  II: Tall Systems  III: Boundary layer  IV: Shallow systems 

Condition of occurrence  Ra_{H} < 1  
Flow pattern  Clockwise circulation  Distinct boundary layer on top and bottom walls  Boundary layer on all four walls. Core remains stagnant  Two horizontal wall jets flow in opposite directions. 
Effect of flow on heat transfer  Insignificant  Insignificant  Significant  Significant 
Heat transfer mechanism  Conduction in horizontal direction  Conduction in horizontal direction  Boundary layer convection  Conduction in vertical direction 
Heat transfer  q'˜kHΔT / L  q'˜kHΔT / L  q'˜(k / δ_{T,f})HΔT  q'˜(k / δ_{T,f})HΔT 
Heat transfer in regimes I and II is dominated by conduction and the fluid circulation plays an insignificant role. Therefore, heat transfer in these two regimes is practically equal to heat transfer rate estimated using Fourier’s law. The flow pattern in regime III is of the boundary layer type and the core of the fluid is stagnant and stratified. The fluid flow and heat transfer in regime III can be obtained via boundary layer analysis. Figure 1 shows contours of dimensionless temperature, pressure, horizontal, and vertical velocities for natural convection in a square enclosure (H / L = 1) heated from its left side and cooled from its right side while insulated from the top and bottom. The behavior of boundary layer type flow and heat transfer is apparent because both temperature and velocity gradient peak near the heated and cooled wall. The low gradients of temperature and velocity in the middle of the enclosure further indicate a stagnant and stratified core. For high Rayleigh number, the average Nusselt number for regime III, as obtained from boundary layer analysis, is (Bejan, 2004):

which indicates that natural convection heat transfer in a rectangular enclosure is dominated by both the Rayleigh number and aspect ratio. For laminar natural convection in a rectangular enclosure with aspect ratio H/L between 1 and 10, the following correlations were suggested by Berkovsky and Polevikov (1977):

which is valid for , and

which is valid for . Catton (1978) found that the above correlations agreed well with the experimental data and the results from numerical solutions. For a rectangular enclosure with a larger aspect ratio, MacGregor and Emery (1969) recommended the following correlations:

which is valid for , and

which is valid for. It should be pointed out that the characteristic length for the Rayleigh and Nusselt numbers used in eqs. (4) and (5) is the width of the enclosure, L.
For the shallow enclosure represented by regime IV the horizontal wall jet flows from the left to right on the top wall and from right to the left on the bottom wall. The entire enclosure can be divided into two end regions near the heated and cooled vertical walls and a core region between the two end regions. There are large temperature gradients along the horizontal direction in the end regions, but the core region is essentially stratified (Tichy and Gadgil, 1982). Under fixed aspect ratio, increasing Rayleigh number makes the horizontal wall jets thinner. The core region extends almost the entire length of the shallow enclosure for high Rayleigh number.
Bejan and Tien (1978) performed a scale analysis and obtained an analytical solution via analysis of fluid flow and heat transfer in the core region. Their results compared favorably with experimental (Imberger, 1973) and numerical (Cormack et al., 1974) results. Bejan (2004) pointed out that the result from Bejan and Tien (1978) is not only useful for regime IV (shallow enclosure), but also for regime III (boundary layer regime). The success of Bejan and Tien’s analysis was attributed to the fact that the thermal resistances associated with the vertical end walls were taken into account. For a limited case, the asymptotic heat transfer results can be expressed as:

which is valid when .
Heated from the Bottom
For a rectangular enclosure filled with fluid and heated from the side, natural convection will be initiated as soon as the temperature difference between the two vertical walls is established. For a rectangular enclosure heated from below, natural convection may or may not occur depending on whether the temperature difference between the top and bottom walls exceeds a critical value. If the temperature difference is not sufficient to initiate natural convection, the mechanism of heat transfer is by conduction of the fluid (and radiation for the case where the fluid is a gas). The condition for the onset of natural convection can be expressed in terms of a critical Rayleigh number. For the case that the horizontal dimension is much larger than the height of the enclosure, the criterion for the onset of natural convection is:

When the Rayleigh number just exceeds the above critical Rayleigh number, the flow pattern is twodimensional counter rotating rolls – referred to as Bénard cells (Oosthuizen and Naylor, 1999). As the Rayleigh number further increases to one or two orders of magnitude higher than the above critical Rayleigh number, the twodimensional cells breakup to three dimensional cells whose top view is hexagonal. The function of the twodimensional rolls and threedimensional hexagonal cells is to promote heat transfer from the heated bottom wall to the cooled top wall. Globe and Dropkin (1959) suggested the following empirical correlation

where all thermophysical properties are evaluated at (T_{h} + T_{c}) / 2. Equation (8) is valid for . In addition, H/L must be sufficiently small so that the effect of the sidewalls can be negligible.
Inclined Rectangular Enclosure
When the rectangular enclosure heated from the side is tilted relative to the direction of gravity, additional unstable stratification and thermal instability, which are usually associated with natural convection in an enclosure heated from below, will affect the fluid flow and heat transfer. The tilt angle γ [see Fig. 6.17 (c)] becomes an additional parameter that plays a significant role. When γ is increased from 0° to 90°, the heat transfer mechanism changes from pure conduction at (heated from the top) to singlecell convection at (heated from the side), where both buoyancy effect and Nusselt number are maximized. As the inclination angle γ is further increased beyond 90°, the Nusselt number starts to decrease until a local minimum at γ = γ_{c} is reached. The value of γ_{c} depends on the aspect ratio of the enclosure (see Table below). Beyond γ_{c}, the stable twodimensional singlecell convection becomes unstable and transits to threedimensional rolls. The mechanism of Bénard convection sets in, which results in an increase of the Nusselt number. The variation of Nusselt number as a function of tilt angle is qualitatively shown in Fig. 3.
Table: Critical inclination angle for different aspect ratio (Arnold et al., 1976)
Aspect ratio, H/L  1  3  6  12  >12 

Critical tilt angle, γ_{c}  155°  127°  120°  113°  110° 
Zhong et al. (1983) numerically solved the natural convection in an inclined square enclosure. The transition from a conduction dominated regime at to singlecell convection at is apparent from the isotherms. At , which, according to Table 6.3, is less than the critical inclination angle, the isotherms start to exhibit some behaviors of thermally unstable conditions. They proposed the following correlation for natural convection of air in a square enclosure (H / L = 1) in the region :

where is for pure conduction. While eq. (9) is good for air in a squared enclosure, the following correlation can be applied to other situations:

which indicates that the Nusselt number for an inclined enclosure is related to the Nusselt number at and the actual inclination angle.
References
Arnold, J.N., Catton, I., and Edwards, D.K., 1976, “Experimental Investigation of Natural Convection in Inclined Rectangular Regions of Differing Aspect Ratios,” ASME J. Heat Transfer, Vol. 98, pp. 6771.
Bejan, A., 2004, Convection Heat Transfer, 3rd ed., John Wiley & Sons, New York.
Bejan, A., and Tien, C.L., 1978, “Laminar Natural Convection Heat Transfer in a Horizontal Cavity with Different End Temperatures,” ASME J. Heat Transfer, Vol. 100, pp. 641¬647.
Berkovsky, B.M. and Polevikov, V.K., 1977, “Numerical Study of Problems on HighIntensive Free Convection,” Heat Transfer and Turbulent Buoyant Convection, Eds. Spalding, D.B., and Afgan, N., Vol. II, pp. 443455, Hemisphere Publishing Corporation, Washington, DC.
Caton, I., 1978, “Natural Convection in Enclosures,” Proceedings of the 6th International Heat Transfer Conference, Vol. 6, pp. 1343, Toronto, Canada.
Cormack, D. E., Leal, L. G., and Seinfeld, J. H., 1974, “Natural Convection in a Shallow Cavity with Differentially Heated End Walls: Part 2. Numerical Solutions,” J. Fluid Mech., Vol. 65, pp. 231246.
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
MacGregor, R.K., and Emery, A.P., 1969, “Free Convection through Vertical Plane Layers  Moderate and High Prandtl Number Fluids,” ASME J. Heat Transfer, Vol. 91, pp. 391403
Tichy, J., and Gadgil, A., 1982, “High Rayleigh Number Laminar Convection in Low Aspect Ratio Enclosures with Adiabatic Horizontal Walls and Differentially Heated Vertical Walls,” ASME J. Heat Transfer, Vol. 104, pp. 103110.
Zhong, Z.Y., Lloyd, J.R., Yang, K.T., 1983, “VariableProperty Natural Convection in Tilted Square Cavities,” Numerical Method in Thermal Problems, Vol. III, Eds. Lewis, R.W., Johnson J.A, and Smith W.R., pp. 968979, Pineridge Press, Swansea, UK.