Flow of a fluid near its density maximum in a differentially rotating cylinder

Abstract

A numerical study is made of the basic-state flow field of a fluid with a density maximum in a differentially rotating cylinder. The fluid density reaches a maximum ρm at temperature Tm, and a quadratic (ρ-T) relationship is used to model the fluid behavior near Tm. The temperature at the bottom (top) endwall disk is TB (Tτ), with ΔT ≡ TΤ - TB > 0, and Tm lies between TB and Tτ. The rotation rate of the bottom (top) endwall disk is ωB(ωτ), with ε ≡ (ωτ-ωB)/ωB ≪ 1. Numerical solutions were obtained of the Navier-Stokes equations for large rotational Reynolds number and large Rayleigh number. Detailed flow and density fields are portrayed to be strongly dependent on the density inversion factor γ ≡ (Tm - TB)/(Tτ - TB). When γ ≡ 0, the results are qualitatively similar to those of a usual Boussinesq fluid with a linear (ρ-T) relationship. It is shown that a modified thermal wind relation prevails in the interior, in which the vertical shear of azimuthal velocity is balanced by the radial gradient of density. As γ increases, the overall strength of meridional circulation grows. The vertical profiles of azimuthal velocity are plotted as γ varies. The Ekman layer suction is intensified as γ increases. The behavior of average Nusselt number Nu at the bottom disk with varying γ is discussed and physical rationalizations are given. © 2001 Elsevier Science Inc. All rights reserved.