Scaling analysis and simulation of natural convection cooling of fluid with Pr < 1 in a rectangular container

Abstract

The unsteady natural convection cooling of fluid with Pr < 1 in a rectangular container with an imposed lower temperature on the vertical sidewalls is dominated by the boundary-layer development stage adjacent to the sidewall, the stratification stage, and the cooling-down stage, respectively. Various scaling laws to describe the unsteady flow behavior at these respective stages are developed with scaling analysis and are verified and quantified by direct numerical simulation with selected values of the aspect ratio of the cylinder A, the Rayleigh number Ra, and the Prandtl number Pr in the ranges of 1/3< A < 3, 10^6 < Ra < 10^10, and 0.01 < Pr < 0.5. Three-dimensional direct numerical simulation results have also been obtained and shown that the two-dimensional assumption made in the scaling analysis is correct, at least for the boundary-layer development and initial stratification stages of the flow.