Convection in a rotating cylinder. Part 2. Linear theory for low Prandtl numbers

Goldstein, H. F.; Knobloch, Edgar; Mercader, Isabel; Net, Marta

doi:10.1017/s0022112094000510

Cited by 52 publications

(50 citation statements)

References 19 publications

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“…8,10 However, in our present configuration, due to the broken centrosymmetry of the system, these two steady modes merge into a complex coalescence when they are about to cross, thus producing totally new dynamics: oscillation. That the broken symmetry can play a significant role in determining the system dynamics has also been studied by Ecke et al 24 and Goldstein et al 25,26 for convection in a rotating cylinder. In this prototype problem, these authors …”

Section: Variations Of Leading Eigenvalues With Grmentioning

confidence: 78%

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Chen

Zhan

et al. 2010

Physics of Fluids

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Double diffusive convection in a porous layer saturated with viscoelastic fluid using a thermal non-equilibrium model Phys. Fluids 23, 054101 (2011) Influence of vibrations on thermodiffusion in binary mixture: A benchmark of numerical solutions Phys. Fluids 19, 017111 (2007) Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field Phys. Fluids 18, 073302 (2006) Additional information on Phys. Fluids Double-diffusive buoyancy convection in an open-top rectangular cavity with horizontal temperature and concentration gradients is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude ͑buoyancy ratio R =−1͒. In this case, a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Gr c . Linear stability analysis and direct numerical simulation show that depending on the cavity aspect ratio A, the first primary instability can be oscillatory, while that in a closed cavity is always steady. Near a codimension-two point, the two leading real eigenvalues merge into a complex coalescence that later produces a supercritical Hopf bifurcation. As Gr further increases, this complex coalescence splits into two real eigenvalues again. The oscillatory flow consists of counter-rotating vortices traveling from right to left and there exists a critical aspect ratio below which the onset of convection is always oscillatory. Neutral stability curves showing the influences of A, Lewis number Le, and Prandtl number Pr are obtained. While the number of vortices increases as A decreases, the flow structure of the eigenfunction does not change qualitatively when Le or Pr is varied. The supercritical oscillatory flow later undergoes a period-doubling bifurcation and the new oscillatory flow soon becomes unstable at larger Gr. Random initial fields are used to start simulations and many different subcritical steady states are found. These steady states correspond to much stronger flows when compared to the oscillatory regime. The influence of Le on the onset of steady flows and the corresponding heat and mass transfer properties are also investigated.

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Section: Variations Of Leading Eigenvalues With Grmentioning

confidence: 78%

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Chen

Zhan

et al. 2010

Physics of Fluids

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“…This lack of dependence on rigid rotation may be contrasted with a large class of 3D and quasi-2D rotating Rayleigh-Bénard systems [4,[48][49][50][51][52], where rotation produces added stability but the absence of strictly 2D flow results in a time-dependent (precessing) convection pattern in the co-rotating frame. The precession seen in these systems is analogous to the travelling of the sheared patterns that we observed only to the extent that rotation and shear break the same symmetry, which allows travelling patterns.…”

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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“…Our most weakly supercritical experiment, labeled case "b" in Fig. 3, convects heat by oscillatory motions in the outer half of the cylinder, s > 0:5h, where s is cylindrical radius (48). Fig.…”

Section: Methodsmentioning

confidence: 97%

Turbulent convection in liquid metal with and without rotation

King

Aurnou

2013

Proc. Natl. Acad. Sci. U.S.A.

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The magnetic fields of Earth and other planets are generated by turbulent, rotating convection in liquid metal. Liquid metals are peculiar in that they diffuse heat more readily than momentum, quantified by their small Prandtl numbers, Pr = 1. Most analog models of planetary dynamos, however, use moderate Pr fluids, and the systematic influence of reducing Pr is not well understood. We perform rotating Rayleigh-Bénard convection experiments in the liquid metal gallium (Pr = 0:025) over a range of nondimensional buoyancy forcing (Ra) and rotation periods (E). Our primary diagnostic is the efficiency of convective heat transfer (Nu). In general, we find that the convective behavior of liquid metal differs substantially from that of moderate Pr fluids, such as water. In particular, a transition between rotationally constrained and weakly rotating turbulent states is identified, and this transition differs substantially from that observed in moderate Pr fluids. This difference, we hypothesize, may explain the different classes of magnetic fields observed on the Gas and Ice Giant planets, whose dynamo regions consist of Pr < 1 and Pr > 1 fluids, respectively.T he interiors of Earth and other terrestrial bodies, as well as the Gas Giant planets, contain vast oceans of flowing metals. Mixed by turbulent convection as these planets cool, the flowing conductors produce electric currents that maintain planetary magnetic fields. These bodies also rotate, and it is expected that the resulting Coriolis forces strongly influence convective dynamo processes. Beyond linear theory (1), however, not much is known about the dynamics of rotating convection in liquid metal that underlie planetary magnetic field generation.Theory and experiments often focus on a simplified analog of geophysical and astrophysical systems, Rayleigh-Bénard convection (RBC). The RBC system consists of a fluid layer contained between flat, horizontal plates separated by distance h, the bottom of which is warmer than the top by ΔT, and with downward pointing gravity, g. Near the bottom boundary, the fluid warms and expands by a volumetric factor α (per degree kelvin) and similarly cools and contracts near the top boundary, such that the fluid is unstably stratified.Rotating convection is explored by spinning the RBC system about a vertical axis at an angular rate Ω, which is adopted as the reference frame for the model. The fluid dynamics of the system are characterized by three dimensionless parameters. The Prandtl number, Pr ≡ ν=κ, defines the diffusive transport properties of the fluid, where ν and κ are its viscous and thermal diffusivities. The Rayleigh number, Ra ≡ αgΔTh 3 =ðνκÞ, prescribes the magnitude of the buoyancy force. Finally, the rotation period is specified by the Ekman number, E ≡ ν=ð2Ωh 2 Þ. The primary diagnostic used in many convection studies, including this one, is the Nusselt number, which characterizes the efficiency of heat transport by convection as Nu ≡ qh=ðkΔTÞ, where q is total heat flux and k is the fluid's thermal conductivi...

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Convection in a rotating cylinder. Part 2. Linear theory for low Prandtl numbers

Cited by 52 publications

References 19 publications

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Onset of double-diffusive convection in a rectangular cavity with stress-free upper boundary

Bifurcations in annular electroconvection with an imposed shear

Turbulent convection in liquid metal with and without rotation

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