Long-term behavior of cooling fluid in a rectangular container

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 T_a(t) 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 t and the average Nusselt number on the cooling boundary. A scaling analysis has been carried out which shows that for both cases θ_a(τ) scales as e^{−C(ARa)−1∕4τ}, where θ_a(τ) is the dimensionless form of T_a(t), τ is the dimensionless time, A is the aspect ratio of the container, Ra is the Rayleigh number, and C is a proportionality constant. A series of direct numerical simulations with the selected values of A, Ra, and Pr (Pr is the Prandtl number) in the ranges of 1∕3⩽A⩽3, 6×10⁶⩽Ra⩽6×10¹⁰, and 1⩽Pr⩽1000 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 C=0.645 and 0.705 respectively for Cases 1 and 2 with Ra, A and Pr in the above-mentioned ranges.