Low-Prandtl-number convection in a rotating cylindrical annulus

Plaut, Emmanuel; Busse, Friedrich H.

doi:10.1017/s0022112002008923

Cited by 39 publications

(39 citation statements)

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“…The boundary terms are not included, i.e., they are cancelled. Some of them vanish because of the Neumann boundary conditions (21) and (23), the other ones are taken to be zero because the Dirichlet boundary conditions (19), (20) and (22) are imposed by a penalization method; some diagonal coefficients of the matrix A are modified accordingly [18]. In the non-Newtonian case, our temporal scheme demands to update A because of the evolution of the viscosity as time goes on.…”

Section: Numerical Schemementioning

confidence: 99%

Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids

Jenny

Plaut

Briard

2015

Journal of Non-Newtonian Fluid Mechanics

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Section: Numerical Schemementioning

confidence: 99%

Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids

Jenny

Plaut

Briard

2015

Journal of Non-Newtonian Fluid Mechanics

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“…In order to avoid the creation of an unphysical mean pressure gradient in the annulus (see e.g. Plaut & Busse 2002;Plaut 2003), (3.4a) has to be supplemented by the mean component of the azimuthal Navier-Stokes equation,…”

Section: 2mentioning

confidence: 99%

“…A first version of our Reynolds-stress reformulation was given implicitly in Plaut & Busse (2002) ( § 4.2; figure 5) and more explicitly in Plaut & Busse (2005) ( § § 7 and 8; figures 10 and 11). In both cases 'Cartesian' quasi-geostrophic (QG) models of rotating convection in a closed container were studied: a small-gap approximation was used to unfold the natural annular geometry of the systems.…”

Section: Introductionmentioning

confidence: 99%

Reynolds stresses and mean fields generated by pure waves: applications to shear flows and convection in a rotating shell

et al. 2008

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A general reformulation of the Reynolds stresses created by two-dimensional waves breaking a translational or a rotational invariance is described. This reformulation emphasizes the importance of a geometrical factor: the slope of the separatrices of the wave flow. Its physical relevance is illustrated by two model systems: waves destabilizing open shear flows; and thermal Rossby waves in spherical shell convection with rotation. In the case of shear-flow waves, a new expression of the Reynolds-Orr amplification mechanism is obtained, and a good understanding of the form of the mean pressure and velocity fields created by weakly nonlinear waves is gained. In the case of thermal Rossby waves, results of a three-dimensional code using no-slip boundary conditions are presented in the nonlinear regime, and compared with those of a two-dimensional quasi-geostrophic model. A semi-quantitative agreement is obtained on the flow amplitudes, but discrepancies are observed concerning the nonlinear frequency shifts. With the quasi-geostrophic model we also revisit a geometrical formula proposed by Zhang to interpret the form of the zonal flow created by the waves, and explore the very low Ekman-number regime. A change in the nature of the wave bifurcation, from supercritical to subcritical, is found. IntroductionIn hydrodynamic stability theory and turbulence modelling, it is natural and customary to separate the velocity field into a mean flow V and a fluctuating part v. In the Navier-Stokes equation for V , the nonlinear term expressing the feedback of the fluctuating flow onto the mean flow is usually written as the divergence of the Reynolds stress tensorwhere the angle brackets indicate a suitable averaging. A good understanding or modelling of τ is therefore required to explain the form of the mean flow, and other mean properties of the flow, such as the flow rate and head losses, in the case of an open system for instance. The tensor τ is also quite important for energy since in purely hydrodynamical systems its contraction with the mean strain rate tensor is the only possible source of growth of the fluctuating kinetic energy, as shown in a landmark paper by Reynolds (1895) Busse (1983) in the case of a fluctuating field corresponding to a columnar quasi-two-dimensional wave. He noted in his § 3 (see also his figure 3), a link between the variations of the 'phase function' of the wave and the relevant cross-diagonal Reynolds stress, which was revisited by Zhang (1992). The reformulation proposed by Zhang for the Reynolds stress, however, is limited to a special form of the streamfunction. In the somewhat more general case of a two-dimensional, x, y, fluctuating field, Pedlosky (1987) established ( § 7.3, p. 502) a link between the product v x v y that controls the most important Reynolds stress, i.e. the cross-diagonal stress τ xy , and the slope of the streamlines of v. Pedlosky offered no simple formula for the average v x v y , however.The primary aim of this paper is to complement these pioneering works by propo...

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“…For comparable Ra, convection in low Prandtl number fluids is expected to be different from the more conventional Pr T 1 case in that inertia effects are enhanced for small Pr (e.g., refs. [20][21][22]. All else being equal, nonrotating convection experiments and simulations find that low Pr fluids produce more vigorous convection, yet lower Nu values than moderate Pr fluids (23,24).…”

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confidence: 98%

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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Low-Prandtl-number convection in a rotating cylindrical annulus

Cited by 39 publications

References 20 publications

Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids

Numerical study of subcritical Rayleigh–Bénard convection rolls in strongly shear-thinning Carreau fluids

Reynolds stresses and mean fields generated by pure waves: applications to shear flows and convection in a rotating shell

Turbulent convection in liquid metal with and without rotation

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