Long-term behavior of cooling fluid in a vertical cylinder

The long-term behavior of cooling an initially quiescent isothermal Newtonian fluid in a vertical cylinder by unsteady natural convection with a fixed wall temperature has been investigated in this study by scaling analysis and direct numerical simulation. Two specific cases are considered. Case 1 assumes that the fluid cooling is due to the imposed fixed temperature on the vertical sidewall whereas the top and bottom boundaries are adiabatic. Case 2 assumes that the cooling is due to that on both the vertical sidewall and the bottom boundary whereas the top boundary is adiabatic. The long-term behavior of the fluid cooling in the cylinder is well represented by Ta(t), the average fluid temperature in the cylinder at time t, and the average Nusselt number on the cooling boundary. The scaling analysis shows that for both cases θa(τ) scales as e-C(ARa)-1/4τ-1, where θa(τ) is the dimensionless form of Ta(t), τ the dimensionless time, A the aspect ratio of the vertical cylinder, Ra the Rayleigh number, and C 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, and it is found that these numerical results agree well with the scaling relations. These numerical results have also been used to quantify the scaling relations and it is found that C = 1.287 and 1.357 respectively for Case 1 and 2 with Ra, A and Pr in the ranges of 1/3 ⩽ A ⩽ 3, 6 × 106 ⩽ Ra ⩽ 6 × 1010, and 1 ⩽ Pr ⩽ 1000.