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Thermal Convection in a Rotating Annulus of Liquid: Effect of Viscosity on the Transition Between Axisymmetric and Non-Axisymmetric Flow Regimes

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  • 1 Massachusetts Institute of Technology, Cambridge
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Abstract

When a vertical rotating annulus of liquid is subject to a horizontal temperature gradient, provided that the coefficient of kinematical viscosity, ν¯, is not too great and the angular velocity of rotation,Ω is sufficiently high, four distinct regimes of hydrodynamical flow are possible, as shown in previous work by Hide. Only one of these regimes is characterized by symmetry about the axis of rotation.

The principal properties of the flow depend largely on the dimensionless parameters Π2d/(ba), Π4gdΔρ/ρ¯Ω2(ba)2, Π5≡4Ω2(ba)5/ν¯2d and Π6≡ν¯/κ¯, where d is the depth of the fluid,b and a are the radi of curvature of the surfaces of the annulus, g is the acceleration of gravity, ρ¯ is the mean density of the fluid, Δρ is the density contrast associated with the impressed horizontal temperature gradient and κ¯ is the thermal diffusivity of the fluid. In a diagram with log10Π5 as abscissa and log10Π4 as ordinate, axisymmetric flow is found outside an anvil-shaped region whose upper boundary lies below the line 11,&=2.0, the flow being symmetric for an 114 when 115 where lis. The lower boundary of the anvil-shaped region is given by the equation.
10461025

Theory suggests that the non-axisymmetric flow is a manifestation of “baroclinic instability,” a process first studied in the investigation of planetary-scale atmospheric motions. The experimental results are discussed in terms of Eady' theoretical baroclinic instability model, as extended by others to include effects due to viscous boundary layers. Qualitatively, agreement between theory and experiment is satisfactory, and at the highest values of II attained in the experiments the quantitative agreement is remarkably good. The poor quantitative agreement found at the lowest values of II5 attained indicates that a major source of frictional dissipation has not yet been properly accounted for in the theory.

Abstract

When a vertical rotating annulus of liquid is subject to a horizontal temperature gradient, provided that the coefficient of kinematical viscosity, ν¯, is not too great and the angular velocity of rotation,Ω is sufficiently high, four distinct regimes of hydrodynamical flow are possible, as shown in previous work by Hide. Only one of these regimes is characterized by symmetry about the axis of rotation.

The principal properties of the flow depend largely on the dimensionless parameters Π2d/(ba), Π4gdΔρ/ρ¯Ω2(ba)2, Π5≡4Ω2(ba)5/ν¯2d and Π6≡ν¯/κ¯, where d is the depth of the fluid,b and a are the radi of curvature of the surfaces of the annulus, g is the acceleration of gravity, ρ¯ is the mean density of the fluid, Δρ is the density contrast associated with the impressed horizontal temperature gradient and κ¯ is the thermal diffusivity of the fluid. In a diagram with log10Π5 as abscissa and log10Π4 as ordinate, axisymmetric flow is found outside an anvil-shaped region whose upper boundary lies below the line 11,&=2.0, the flow being symmetric for an 114 when 115 where lis. The lower boundary of the anvil-shaped region is given by the equation.
10461025

Theory suggests that the non-axisymmetric flow is a manifestation of “baroclinic instability,” a process first studied in the investigation of planetary-scale atmospheric motions. The experimental results are discussed in terms of Eady' theoretical baroclinic instability model, as extended by others to include effects due to viscous boundary layers. Qualitatively, agreement between theory and experiment is satisfactory, and at the highest values of II attained in the experiments the quantitative agreement is remarkably good. The poor quantitative agreement found at the lowest values of II5 attained indicates that a major source of frictional dissipation has not yet been properly accounted for in the theory.

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