Abstract
In the present paper, a three-dimensional natural convection induced by a heat generating block placed in the lower adiabatic surface of a cubical cavity is studied numerically. The upper surface of the cavity is maintained cold at a constant temperature, while the other surfaces are considered adiabatic. The cooling fluid is air \(\left( {{\text{Pr}}\, = \,0.72} \right)\) and the solid block to the fluid thermal conductivity ratio, \(\left( {k^{*} \, = \,{{k_{s} } \mathord{\left/ {\vphantom {{k_{s} } {k_{a} }}} \right. \kern-0pt} {k_{a} }}} \right),\) covers the range 0.1 to 200. The generated power and considered domain dimension correspond to values of the Rayleigh number ranging between \(10^{4}\) and \(10^{6} .\) The block’s dimension (height and width) is one of the important controlling parameters of the present problem. The three-dimensional flow structures and the corresponding heat transfer rates are illustrated for different combinations of the governing parameters. The study focuses on the analysis of the effect of three-dimensionalities with the aim of ruling on the conditions of validity of the two-dimensional approach in terms of Rayleigh number. This goal is achieved for various sizes of the block and thermal conductivity ratios.
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Abbreviations
- \(B\):
-
Block’s dimensionless height, \(\left( {{{h^{\prime}} \mathord{\left/ {\vphantom {{h^{\prime}} {H^{\prime}}}} \right. \kern-0pt} {H^{\prime}}}} \right)\)
- \(k^{ * }\):
-
Solid block to fluid thermal conductivities ratio, \(\left( {{{k_{s} } \mathord{\left/ {\vphantom {{k_{s} } {k_{a} }}} \right. \kern-0pt} {k_{a} }}} \right)\)
- \(q_{vol}{\prime}\):
-
Block’s heat generation per unit volume, \(\left( {{\text{Wm}}^{ - \,3} } \right)\)
- \({\text{Ra}}\):
-
Rayleigh number, \({{\left( {g\beta q^{\prime}_{{{\text{vol}}}} H^{{\prime}{5}} } \right)} \mathord{\left/ {\vphantom {{\left( {g\beta q^{\prime}_{{{\text{vol}}}} H^{{\prime}{5}} } \right)} {\left( {k_{a} \nu \alpha_{a} } \right)}}} \right. \kern-0pt} {\left( {k_{a} \nu \alpha_{a} } \right)}}\)
- \(T_{a\left( s \right)}\):
-
Dimensionless temperature, \((T^{\prime}_{{a(s)}} - T^{\prime}_{c} )/q_{{{\text{vol}}}} ^{\prime } H^{{\prime 2}} /k_{a}\)
- \(a\):
-
Air
- \(s\):
-
Solid block
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Mouhtadi, D., Raji, A., Hasnaoui, M., Amahmid, A., Bennacer, R. (2024). Two- and Three-Dimensional Bifurcations in the Case of Natural Convection Induced by a Heat Generating Block. In: Ali-Toudert, F., Draoui, A., Halouani, K., Hasnaoui, M., Jemni, A., Tadrist, L. (eds) Advances in Thermal Science and Energy. JITH 2022. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-031-43934-6_8
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