Buoyancy driven instabilities in rotating layers with parallel axis of rotation

Busse, Friedrich H.; Kropp, M.

doi:10.1007/bf00944739

Cited by 8 publications

(11 citation statements)

References 8 publications

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“…We have considered the onset of convection in a rotating fluid layer confined between isothermal, rigid boundaries which are in differential motion. In a limited number of particular cases, analytical progress has been possible, building on the work of Busse and co-workers (Busse 1970;Kropp & Busse 1991;Busse & Kropp 1992;Auer et al 1995), by reducing the problem to an equivalent eigenvalue problem resembling that of standard Rayleigh-Bénard convection, but with a modified Rayleigh number. We confirm the unexpected result of Busse & Kropp (1992) that when the shear flow and rotation vector are parallel, oblique rolls are preferred and the critical Rayleigh number is reduced.…”

Section: Discussionmentioning

confidence: 99%

“…At first sight it may seem that since both rotation and shear flow individually favour aligned rolls, the combination of the two will have the same effect. Busse & Kropp (1992) discovered the remarkable fact that this is not the case, and in fact the preferred mode of convection consists of rolls with their axes at a slight angle to the direction of Ω and U 0 . A similar phenomenon also occurs for Poiseuille flow in a rotating pipe (Pedley 1969;Joseph & Carmi 1969).…”

Section: Aligned Rotation and Shear Flowmentioning

confidence: 99%

“…We allow the angle ψ between the rotation axis and the shear flow to be arbitrary. The special cases ψ = 0 and ψ = ±π/2 have been analysed by others (Hathaway, Toomre & Gilman 1980;Kropp & Busse 1991;Busse & Kropp 1992).…”

Section: Introductionmentioning

confidence: 99%

“…The combined effects of rotation and imposed shear have been analysed in special cases by Kropp & Busse (1991) and Busse & Kropp (1992), for thermal convection between differentially heated coaxial cylinders that rotate about their common axis. By assuming a narrow gap between the cylinders, they reduced the problem to an equivalent problem in a plane layer.…”

Section: Introductionmentioning

confidence: 99%

“…They identified longitudinal and transverse rolls (with axes aligned parallel and perpendicular to the direction of the shear flow, respectively) as being of particular significance, but discovered that other, oblique, rolls could be preferred in certain parameter ranges. Busse & Kropp (1992) examined a system similar to our plane layer but with the shear flow and rotation vector aligned. A counter-intuitive, but nonetheless correct, result of their analysis is that although the rotation and shear flow would individually impose a preference for longitudinal rolls, in combination they select oblique rolls.…”

Section: Introductionmentioning

confidence: 99%

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Linear stability of rotating convection in an imposed shear flow

Matthews

Cox

1997

J. Fluid Mech.

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In many geophysical and astrophysical contexts, thermal convection is influenced by both rotation and an underlying shear flow. The linear theory for thermal convection is presented, with attention restricted to a layer of fluid rotating about a horizontal axis, and plane Couette flow driven by differential motion of the horizontal boundaries.The eigenvalue problem to determine the critical Rayleigh number is solved numerically assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection rolls is found, for different orientations of the rotation vector with respect to the shear flow. For moderate rates of shear and rotation, the preferred roll orientation depends only on their ratio, the Rossby number.It is well known that rotation alone acts to favour rolls aligned with the rotation vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls parallel to the shear flow are preferred. However, it is found that when the rotation vector and shear flow are parallel, the two effects lead counter-intuitively (as in other, analogous convection problems) to a preference for oblique rolls, and a critical Rayleigh number below that for Rayleigh–Bénard convection.When the boundaries are poorly conducting, the eigenvalue problem is solved analytically by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the stability problem is found to be qualitatively similar to that for fixed-temperature boundaries.Fully nonlinear numerical simulations of the convection are also carried out. These are generally consistent with the linear stability theory, showing convection in the form of rolls near the onset of motion, with the appropriate orientation. More complicated states are found further from critical.

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Section: Discussionmentioning

confidence: 99%

Section: Aligned Rotation and Shear Flowmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linear stability of rotating convection in an imposed shear flow

Matthews

Cox

1997

J. Fluid Mech.

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The Onset and Development of Thermal Convection in Fully Developed Shear Flows

Kelly

1994

Advances in Applied Mechanics

100

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Convection in directionally solidifying alloys under inclined rotation

Chung

Chen

2000

J. Fluid Mech.

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In an experiment on binary alloys directionally solidifying from below, Sample & Hellawell (1984) showed that the plume convection can be successfully prohibited by rotating the cooling tank around an inclined axis. In the present paper we interpret their experimental observation by an analytical approach. Results show that there is a flow induced by the inclination. The induced flow in the fluid layer is a parallel shear flow consisting of three parts: the thermal boundary-layer flow, the solute boundary- layer flow, and the Ekman-layer flow. In the mush, the induced flow is also a parallel flow but of much smaller velocity, consisting of two flows of opposite directions. The induced velocity in the fluid layer increases with inclination angle and decreases with the effective Taylor number Te. The induced velocity in the mush also increases with inclination angle but remains virtually the same on varying the speed of rotation. The linear stability analysis of the mushy layer shows that, due mostly to the reduction of buoyancy, the mush becomes more stable as the inclination angle increases. In the precession-only case, the most-unstable mode of instability is the longitudinal mode, which propagates in a direction perpendicular to the induced flow. In the spin (with or without precession) case, the instability modes propagating in different directions are of equal stability. Because the induced flow changes direction with a frequency equal to the spin angular velocity, the flow scans over all the directions of the system and stabilizes equally the modes in different directions. We conclude on the basis of the present results and from the practical point of view that spin-only rotation is more effective than the precession-only rotation in stabilizing the convection during solidification.

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Buoyancy driven instabilities in rotating layers with parallel axis of rotation

Cited by 8 publications

References 8 publications

Linear stability of rotating convection in an imposed shear flow

Linear stability of rotating convection in an imposed shear flow

The Onset and Development of Thermal Convection in Fully Developed Shear Flows

Convection in directionally solidifying alloys under inclined rotation

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