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 ρ m at temperature T m, and a quadratic ( ρ– T) relationship is used to model the fluid behavior near T m. The temperature at the bottom (top) endwall disk is T B (T T ) , with ΔT≡ T T− T B>0, and T m lies between T B and T T. The rotation rate of the bottom (top) endwall disk is Ω B (Ω T ) , with ε≡(Ω T −Ω B )/Ω B ≪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≡( T m− T B)/( T T− T B). 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.