scholarly journals Subcritical convection of liquid metals in a rotating sphere using a quasi-geostrophic model

2016 ◽  
Vol 808 ◽  
pp. 61-89 ◽  
Author(s):  
Céline Guervilly ◽  
Philippe Cardin
Keyword(s):  
Prandtl Number ◽  
Nonlinear Oscillations ◽  
Liquid Metals ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Zonal Flow ◽  
Onset Of Convection ◽  
Nonlinear Convection ◽  
Rotating Sphere ◽  
Weak Branch

We study nonlinear convection in a rapidly rotating sphere with internal heating for values of the Prandtl number relevant for liquid metals ($Pr\in [10^{-2},10^{-1}]$). We use a numerical model based on the quasi-geostrophic approximation, in which variations of the axial vorticity along the rotation axis are neglected, whereas the temperature field is fully three-dimensional. We identify two separate branches of convection close to onset: (i) a well-known weak branch for Ekman numbers greater than $10^{-6}$, which is continuous at the onset (supercritical bifurcation) and consists of thermal Rossby waves and (ii) a novel strong branch at lower Ekman numbers, which is discontinuous at the onset. The strong branch becomes subcritical for Ekman numbers of the order of $10^{-8}$. On the strong branch, the Reynolds number of the flow is greater than $10^{3}$, and a strong zonal flow with multiple jets develops, even close to the nonlinear onset of convection. We find that the subcriticality is amplified by decreasing the Prandtl number. The two branches can co-exist for intermediate Ekman numbers, leading to hysteresis ($Ek=10^{-6}$, $Pr=10^{-2}$). Nonlinear oscillations are observed near the onset of convection for $Ek=10^{-7}$ and $Pr=10^{-1}$.

scholarly journals Bifurcations from steady to quasi-periodic flows in a laterally heated cavity filled with low Prandtl number fluids

2018 ◽  
Vol 861 ◽  
pp. 223-252 ◽  
Author(s):  
A. Medelfef ◽  
D. Henry ◽  
A. Bouabdallah ◽  
S. Kaddeche
Keyword(s):  
Prandtl Number ◽  
Liquid Metals ◽  
Three Dimensional ◽  
Numerical Continuation ◽  
Arnoldi Method ◽  
Periodic Flows ◽  
Bifurcation Points ◽  
Prandtl Numbers ◽  
Stable Flow ◽  
Flow States

This study deals with the transition toward quasi-periodicity of buoyant convection generated by a horizontal temperature gradient in a three-dimensional parallelepipedic cavity with dimensions $4\times 2\times 1$ (length $\times$ width $\times$ height). Numerical continuation techniques, coupled with an Arnoldi method, are used to locate the steady and Hopf bifurcation points as well as the different steady and periodic flow branches emerging from them for Prandtl numbers ranging from 0 to 0.025 (liquid metals). Our results highlight the existence of two steady states along with many periodic cycles, all with different symmetries. The bifurcation scenarios consist of complex paths between these different solutions, giving a succession of stable flow states as the Grashof number is increased, from steady to periodic and quasi-periodic. The change of these scenarios with the Prandtl number, in connection with the crossing of bifurcation points, was carefully analysed.

scholarly journals Density Stratification Effects on the Thermal Convection in a Rotating Spherical Shell

2001 ◽  
Vol 203 ◽  
pp. 195-197
Author(s):  
N. Nishikawa ◽  
K. Kusano
Keyword(s):  
Spherical Shell ◽  
Thermal Convection ◽  
Structural Transition ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Density Stratification ◽  
Nonlinear Convection ◽  
Solar Convection Zone ◽  
Solar Convection ◽  
The Stability

The density stratification effects on the thermal convection in a rotating spherical shell, which is the representative of the solar convection zone, are investigated by three dimensional numerical simulations. It is found that, the convection structure in the strongly stratified system is switched from parallel cells aligned to the rotation axis to zonal rolles dominated by the longitudinally averaged mode, as the Rayleigh number increases much larger than the stability threshold. Corresponding to this structural transition, the averaged kinetic helicity reverses the sign in each hemisphere (from negative to positive in the northern hemisphere). The results indicate that the density stratification is much important for the nonlinear convection process in the rotating spherical shell.

Nonlinear convection in a layer with nearly insulating boundaries

1980 ◽  
Vol 96 (2) ◽  
pp. 243-256 ◽  
Author(s):  
F. H. Busse ◽  
N. Riahi
Keyword(s):  
Thermal Conductivity ◽  
Prandtl Number ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Thermal Conductance ◽  
Convection Flow ◽  
Surprising Result ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Steady Convection

A general class of solutions is studied describing three-dimensional steady convection flows in a fluid layer heated from below with boundaries of low thermal conductivity. Non-linear properties of the solutions are analysed and the physically realizable convection flow is determined by a stability analysis with respect to arbitrary three-dimensional disturbances. The most surprising result is that square-pattern convection is preferred in contrast to two-dimensional rolls that represent the only form of stable convection in a symmetric layer with highly conducting boundaries. The analysis is carried out in the limit of infinite Prandtl number and for a particular boundary configuration. But it is shown that the results hold for arbitrary Prandtl number to the order to which they have been derived and that other assumptions about the boundaries require only minor modifications as long as their thermal conductance remains low.

Spiralling columnar convection in rapidly rotating spherical fluid shells

1992 ◽  
Vol 236 ◽  
pp. 535-556 ◽  
Author(s):  
K. Zhang
Keyword(s):  
Temperature Field ◽  
Kinetic Energy ◽  
Prandtl Number ◽  
Taylor Number ◽  
Total Kinetic Energy ◽  
Significant Finding ◽  
Mean Flow ◽  
Onset Of Convection ◽  
Nonlinear Convection ◽  
Cylindrical Annulus

It is shown that the fundamental features of both thermal instabilities and the corresponding nonlinear convection in rapidly rotating spherical systems (in the range of the Taylor number 109 < T < 1012) are determined by the fluid properties characterized by the size of the Prandtl number. Coefficients of the asymptotic power law for the onset of convection at large Taylor number are estimated in the range of the Prandtl number 0.1 ≤ Pr ≤ 100. For fluids of moderately small Prandtl number, a new type of convective instability in the form of prograde spiralling drifting columnar rolls is discovered. The linear columnar rolls extend spirally from near latitude 60° to the equatorial region, and each spans azimuthally approximately five wavelengths with the inclination angle between a spirally elongated roll and the radial direction exceeding 45°. As a consequence, the radial lengthscale of the linear roll becomes comparable with the azimuthal lengthscale. A particularly significant finding is the connection between the new instability and the predominantly axisymmetric convection. Though non-axisymmetric motions are preferred at the onset of convection, the nonlinear convection (at the Rayleigh number of the order of (R—Rc)/Rc = O(0.1)) bifurcating supercritically from the spiralling mode is primarily dominated by the component of the axisymmetric zonal flow, which contains nearly 90% of the total kinetic energy. For fluids of moderately large Prandtl numbers, thermal instabilities at the onset of convection are concentrated in a cylindrical annulus coaxial with the axis of rotation; the position of the convection cylinder is strongly dependent on the size of the Prandtl number. The associated nonlinear convection consists of predominantly non-axisymmetric columnar rolls together with a superimposed weak mean flow that contains less than 10% of the total kinetic energy at (R—Rc)/Rc = O(0.1). A double-layer structure of the temperature field (with respect to the basic state) forms as a result of strong nonlinear interactions between the nonlinear flow and the temperature field. It is also demonstrated that the aspect ratio of the spherical shell does not substantially influence the fundamental properties of convection.

scholarly journals 2-Amino-5-chloropyridinium cis-diaquadioxalatochromate(III) sesquihydrate

2012 ◽  
Vol 68 (6) ◽  
pp. m824-m825 ◽  
Author(s):  
Ichraf Chérif ◽  
Jawher Abdelhak ◽  
Mohamed Faouzi Zid ◽  
Ahmed Driss
Keyword(s):  
Rotation Axis ◽  
Three Dimensional ◽  
Octahedral Geometry ◽  
Distorted Octahedral Geometry ◽  
Water Molecules ◽  
Lattice Water ◽  
Compound C ◽  
Hydrogen Bonding Interactions ◽  
Distorted Octahedral ◽  
Independent Lattice

In the crystal structure of the title compound, (C5H6ClN2)[Cr(C2O4)2(H2O)2]·1.5H2O, the CrIII atom adopts a distorted octahedral geometry being coordinated by two O atoms of two cis water molecules and four O atoms from two chelating oxalate dianions. The cis-diaquadioxalatochromate(III) anions, 2-amino-5-chloropyridinium cations and uncoordinated water molecules are linked into a three-dimensional supramolecular array by O—H...O and N—H...O hydrogen-bonding interactions. One of the two independent lattice water molecules is situated on a twofold rotation axis.

scholarly journals Prandtl Number in Classical Hard-Sphere and One-Component Plasma Fluids

Molecules ◽  
2021 ◽  
Vol 26 (4) ◽  
pp. 821
Author(s):  
Sergey Khrapak ◽  
Alexey Khrapak
Keyword(s):  
Prandtl Number ◽  
Hard Sphere ◽  
Solid Phase ◽  
Freezing Point ◽  
Three Dimensional ◽  
Order Unity ◽  
Strongly Coupled ◽  
Dielectric Fluids ◽  
Weakly Coupled ◽  
Component Plasma

The Prandtl number is evaluated for the three-dimensional hard-sphere and one-component plasma fluids, from the dilute weakly coupled regime up to a dense strongly coupled regime near the fluid-solid phase transition. In both cases, numerical values of order unity are obtained. The Prandtl number increases on approaching the freezing point, where it reaches a quasi-universal value for simple dielectric fluids of about ≃1.7. Relations to two-dimensional fluids are briefly discussed.

scholarly journals Crystal structure ofcatena-poly[[diaquabis(4-formylbenzoato-κO1)cobalt(II)]-μ-pyrazine-κ2N:N′]

2015 ◽  
Vol 71 (4) ◽  
pp. 339-341 ◽  
Author(s):  
Gülçin Şefiye Aşkın ◽  
Fatih Çelik ◽  
Nefise Dilek ◽  
Hacali Necefoğlu ◽  
Tuncer Hökelek
Keyword(s):  
Hydrogen Bonds ◽  
Rotation Axis ◽  
Three Dimensional ◽  
Water Molecules ◽  
Carboxylate Group ◽  
Polymeric Compound ◽  
Distorted Octahedral Coordination ◽  
Linear Chains ◽  
Dimensional Network ◽  
Distorted Octahedral

In the title polymeric compound, [Co(C8H5O3)2(C4H4N2)(H2O)2]n, the CoIIatom is located on a twofold rotation axis and has a slightly distorted octahedral coordination sphere. In the equatorial plane, it is coordinated by two carboxylate O atoms of two symmetry-related monodentate formylbenzoate anions and by two N atoms of two bridging pyrazine ligands. The latter are bisected by the twofold rotation axis. The axial positions are occupied by two O atoms of the coordinating water molecules. In the formylbenzoate anion, the carboxylate group is twisted away from the attached benzene ring by 7.50 (8)°, while the benzene and pyrazine rings are oriented at a dihedral angle of 64.90 (4)°. The pyrazine ligands bridge the CoIIcations, forming linear chains running along theb-axis direction. Strong intramolecular O—H...O hydrogen bonds link the water molecules to the carboxylate O atoms. In the crystal, weak O—Hwater...Owaterhydrogen bonds link adjacent chains into layers parallel to thebcplane. The layers are linkedviaC—Hpyrazine...Oformylhydrogen bonds, forming a three-dimensional network. There are also weak C—H...π interactions present.

A Numerical Study of Three-Dimensional Natural Convective Heat Transfer From a Plate With a “Wavy” Surface

2001 ◽  
Author(s):  
Patrick H. Oosthuizen ◽  
Matt Garrett
Keyword(s):  
Heat Transfer ◽  
Prandtl Number ◽  
Surface Waves ◽  
Heat Transfer Rate ◽  
Transfer Rate ◽  
Three Dimensional ◽  
Wavy Surface ◽  
Surface Waviness ◽  
Dimensionless Amplitude ◽  
The Mean

Abstract Natural convective heat transfer from a wide isothermal plate which has a “wavy” surface, i.e., has a surface which periodically rises and falls, has been numerically studied. The surface waves run parallel to the direction of flow over the surface and have a relatively small amplitude. Two types of wavy surface have been considered here — saw-tooth and sinusoidal. Surfaces of the type considered are approximate models of situations that occur in certain window covering applications, for example, and are also sometimes used to try to enhance the heat transfer rate from the surface. The flow has been assumed to be laminar. Because the surface waves are parallel to the direction of flow, the flow over the surface will be three-dimensional. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy forces, this being treated by means of the Boussinesq type approximation. The governing equations have been written in dimensionless form, the height of the surface being used as the characteristic length scale and the temperature difference between the surface temperature and the temperature of the fluid far from the plate being used as the characteristic temperature. The dimensionless equations have been solved using a finite-element method. Although the flow is three-dimensional because the surface waves are all assumed to have the same shape, the flow over each surface thus being the same, and it was only necessary to solve for the flow over one of the surface waves. The solution has the following parameters: the Grashof number based on the height, the Prandtl number, the dimensionless amplitude of the surface waviness, the dimensionless pitch of the surface waviness, and the form of the surface waviness (saw-tooth or sinusoidal). Results have been obtained for a Prandtl number of 0.7 for Grashof numbers up to 106. The effects of Grashof number, dimensionless amplitude and dimensionless pitch on the mean heat transfer rate have been studied. It is convenient to introduce two mean heat transfer rates, one based on the total surface area and the other based on the projected frontal area of the surface. A comparison of the values of these quantities gives a measure of the effectiveness of the surface waviness in increasing the mean heat transfer rate. The results show that while surface waviness increases the heat transfer rate based on the frontal area, the modifications of the flow produced by the surface waves are such that the increase in heat transfer rate is less than the increase in surface area.

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