Abstract
In this study, the long-term behavior of cooling an initially quiescent isothermal Newtonian fluid in a rectangular container with an infinite length by unsteady natural convection due to a fixed wall temperature has been investigated by scaling analysis and direct numerical simulation. Two specific cases are considered. Case 1 assumes that the cooling of the fluid is caused by the imposed fixed temperature on the vertical sidewall while the top and bottom boundaries are adiabatic. Case 2 assumes that the cooling is caused by the imposed fixed temperature on both the vertical sidewall and the bottom boundary while the top boundary is adiabatic. The appropriate parameters to represent the long-term behavior of the fluid cooling in the container are the transient average fluid temperature over the whole volume of the container per unit length (i.e., the transient area average fluid temperature, as used in the subsequent numerical simulations) at time and the average Nusselt number on the cooling boundary. A scaling analysis has been carried out which shows that for both cases scales as , where is the dimensionless form of , is the dimensionless time, is the aspect ratio of the container, Ra is the Rayleigh number, and is a proportionality constant. A series of direct numerical simulations with the selected values of , , and Pr (Pr is the Prandtl number) in the ranges of , , and have been carried out for both cases to validate the developed scaling relations. It is found that these numerical results agree well with the scaling relations. The numerical results have also been used to quantify the scaling relations and it is found that and 0.705 respectively for Cases 1 and 2 with Ra, and Pr in the above-mentioned ranges.
2 More- Received 23 December 2003
DOI:https://doi.org/10.1103/PhysRevE.69.056315
©2004 American Physical Society