Stability of unstably stratified shear flow between parallel plates

Fujimura, Kaoru; Kelly, R. E.

doi:10.1016/0169-5983(88)90006-8

Cited by 28 publications

(18 citation statements)

References 9 publications

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“…Provided that |Ω Ω Ω v | is sufficiently large an Ekman layer forms on the lower boundary. We have explored the ensuing mode competition between hydrodynamic and convective instabilities that occurs in the presence of buoyancy forces; in the absence of rotation such competition has been investigated by Fujimura & Kelly (1988) and Mohamad & Viskanta (1989). Our study has been complicated by the fact that the marginal stability surface defined by the neutral Rayleigh number Ra = Ra(k, φ) generally has multiple minima.…”

Section: Discussionmentioning

confidence: 99%

The onset of thermal convection in EkmanCouette shear flow with oblique rotation

2003

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The onset of convection of a Boussinesq fluid in a horizontal plane layer is studied. The system rotates with constant angular velocity Ω Ω Ω, which is inclined at an angle ϑ to the vertical. The layer is sheared by keeping the upper boundary fixed, while the lower boundary moves parallel to itself with constant velocity U 0 normal to the plane containing the rotation vector and gravity g (i.e. U 0 g × Ω Ω Ω). The system is characterized by five dimensionless parameters: the Rayleigh number Ra, the Taylor number τ 2 , the Reynolds number Re (based on U 0 ), the Prandtl number Pr and the angle ϑ. The basic equilibrium state consists of a linear temperature profile and an Ekman-Couette flow, both dependent only on the vertical coordinate z. Our linear stability study involves determining the critical Rayleigh number Ra c as a function of τ and Re for representative values of ϑ and Pr.Our main results relate to the case of large Reynolds number, for which there is the possibility of hydrodynamic instability. When the rotation is vertical ϑ = 0 and τ 1, so-called type I and type II Ekman layer instabilities are possible. With the inclusion of buoyancy Ra = 0 mode competition occurs. On increasing τ from zero, with fixed large Re, we identify four types of mode: a convective mode stabilized by the strong shear for moderate τ , hydrodynamic type I and II modes either assisted (Ra > 0) or suppressed (Ra < 0) by buoyancy forces at numerically large τ , and a convective mode for very large τ that is largely uninfluenced by the thin Ekman shear layer, except in that it provides a selection mechanism for roll orientation which would otherwise be arbitrary. Significantly, in the case of oblique rotation ϑ = 0, the symmetry associated with U 0 ↔ −U 0 for the vertical rotation is broken and so the cases of positive and negative Re exhibit distinct stability characteristics, which we consider separately. Detailed numerical results were obtained for the representative case ϑ = π/4. Though the overall features of the stability results are broadly similar to the case of vertical rotation, their detailed structure possesses a surprising variety of subtle differences.

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Section: Discussionmentioning

confidence: 99%

The onset of thermal convection in EkmanCouette shear flow with oblique rotation

2003

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“…There have been some theoretical studies of 3D Rayleigh-Bénard convection (RBC) in the presence of a plane Couette shear flow [43]. The theory assumed the usual theoretical geometry of a fluid layer confined between infinite, perfectly conducting horizontal planes.…”

Section: Discussionmentioning

confidence: 99%

Bifurcations in annular electroconvection with an imposed shear

2001

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We report an experimental study of the primary bifurcation in electrically-driven convection in a freely suspended film. A weakly conducting, submicron thick smectic liquid crystal film was supported by concentric circular electrodes. It electroconvected when a sufficiently large voltage V was applied between its inner and outer edges. The film could sustain rapid flows and yet remain strictly two-dimensional. By rotation of the inner electrode, a circular Couette shear could be independently imposed. The control parameters were a dimensionless number R, analogous to the Rayleigh number, which is ∝ V 2 and the Reynolds number Re of the azimuthal shear flow. The geometrical and material properties of the film were characterized by the radius ratio α, and a dimensionless number P, analogous to the Prandtl number. Using measurements of current-voltage characteristics of a large number of films, we examined the onset of electroconvection over a broad range of α, P and Re. We compared this data quantitatively to the results of linear stability theory. This could be done with essentially no adjustable parameters. The current-voltage data above onset were then used to infer the amplitude of electroconvection in the weakly nonlinear regime by fitting them to a steady-state amplitude equation of the Landau form. We show how the primary bifurcation can be tuned between supercritical and subcritical by changing α and Re.

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“…One way of systematically studying this dependence is to superpose simple flows or rotations on well-understood instabilities. For example, the standard case of buoyancy-driven Rayleigh-Bénard convection (RBC) has been studied in the presence of rotation [7,8,9,10] and shearing due to an open throughflow [11,12], as well as in the geophysically interesting cases of radial gravitation with rotation [5,6]. The phenomenology of two-dimensional (2D) electroconvection in a rectangular geometry was the subject of our previous experimental [13,14,15,16,17] and theoretical [2,3] work.…”

Section: Introductionmentioning

confidence: 99%

Electrically driven convection in a thin annular film undergoing circular Couette flow

1999

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We investigate the linear stability of a thin, suspended, annular film of conducting fluid with a voltage difference applied between its inner and outer edges. For a sufficiently large voltage, such a film is unstable to radially-driven electroconvection due to charges which develop on its free surfaces. The film can also be subjected to a Couette shear by rotating its inner edge. This combination is experimentally realized using films of smectic A liquid crystals. In the absence of shear, the convective flow consists of a stationary, azimuthally one-dimensional pattern of symmetric, counter-rotating vortex pairs. When Couette flow is applied, an azimuthally traveling pattern results. When viewed in a co-rotating frame, the traveling pattern consists of pairs of asymmetric vortices. We calculate the neutral stability boundary for arbitrary radius ratio α and Reynolds number Re of the shear flow, and obtain the critical control parameter Rc (α, Re) and the critical azimuthal mode number mc (α, Re). The Couette flow suppresses the onset of electroconvection, so that Rc (α, Re) > Rc (α, 0). The calculated suppression is compared with experiments performed at α = 0.56 and 0 ≤ Re ≤ 0.22. Fluids, 11, 3613 (1999). See also http://mobydick.physics.utoronto.ca. Physics of

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Stability of unstably stratified shear flow between parallel plates

Cited by 28 publications

References 9 publications

The onset of thermal convection in EkmanCouette shear flow with oblique rotation

The onset of thermal convection in EkmanCouette shear flow with oblique rotation

Bifurcations in annular electroconvection with an imposed shear

Electrically driven convection in a thin annular film undergoing circular Couette flow

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