Convection in a rapidly rotating spherical shell with an imposed laterally varying thermal boundary condition

2009 ◽  
Vol 641 ◽  
pp. 335-358 ◽  
Author(s):  
CHRISTOPHER J. DAVIES ◽  
DAVID GUBBINS ◽  
PETER K. JIMACK
Keyword(s):  
Spherical Shell ◽  
Outer Boundary ◽  
Homogeneous Problem ◽  
Steady Flows ◽  
Rapid Rotation ◽  
Rotation Rates ◽  
Thermally Driven ◽  
Layer Flow ◽  
Steady Solutions ◽  
Convection Rolls

We investigate thermally driven convection in a rotating spherical shell subject to inhomogeneous heating on the outer boundary, extending previous results to more rapid rotation rates and larger amplitudes of the boundary heating. The analysis explores the conditions under which steady flows can be obtained, and the stability of these solutions, for two boundary heating modes: first, when the scale of the boundary heating corresponds to the most unstable mode of the homogeneous problem; second, when the scale is larger. In the former case stable steady solutions exhibit a two-layer flow pattern at moderate rotation rates, but at very rapid rotation rates no steady solutions exist. In the latter case, stable steady solutions are always possible, and unstable solutions show convection rolls that cluster into nests that are out of phase with the boundary anomalies and remain trapped for many thermal diffusion times.

scholarly journals The most massive white dwarfs in the solar neighbourhood

2021 ◽  
Vol 503 (4) ◽  
pp. 5397-5408
Author(s):  
Mukremin Kilic ◽  
P Bergeron ◽  
Simon Blouin ◽  
A Bédard
Keyword(s):  
White Dwarf ◽  
White Dwarfs ◽  
Transverse Velocity ◽  
Maximum Mass ◽  
Model Calculations ◽  
Rapid Rotation ◽  
Rotation Rates ◽  
Binary Mergers ◽  
Chandrasekhar Limit

ABSTRACT We present an analysis of the most massive white dwarf candidates in the Montreal White Dwarf Database 100 pc sample. We identify 25 objects that would be more massive than $1.3\, {\rm M}_{\odot }$ if they had pure H atmospheres and CO cores, including two outliers with unusually high photometric mass estimates near the Chandrasekhar limit. We provide follow-up spectroscopy of these two white dwarfs and show that they are indeed significantly below this limit. We expand our model calculations for CO core white dwarfs up to M = 1.334 M⊙, which corresponds to the high-density limit of our equation-of-state tables, ρ = 109 g cm−3. We find many objects close to this maximum mass of our CO core models. A significant fraction of ultramassive white dwarfs are predicted to form through binary mergers. Merger populations can reveal themselves through their kinematics, magnetism, or rapid rotation rates. We identify four outliers in transverse velocity, four likely magnetic white dwarfs (one of which is also an outlier in transverse velocity), and one with rapid rotation, indicating that at least 8 of the 25 ultramassive white dwarfs in our sample are likely merger products.

scholarly journals Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

2001 ◽  
Vol 435 ◽  
pp. 103-144 ◽  
Author(s):  
M. RIEUTORD ◽  
B. GEORGEOT ◽  
L. VALDETTARO
Keyword(s):  
Lyapunov Exponent ◽  
Velocity Field ◽  
Spherical Shell ◽  
Asymptotic Properties ◽  
Analytical Form ◽  
Shear Layers ◽  
Three Dimensions ◽  
Geometrical Approach ◽  
Steady Flows ◽  
Whole Space

We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincaré's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor (the associated Lyapunov exponent is always negative or zero). We show that these attractors exist in bands of frequencies the size of which decreases with the number of reflection points of the attractor. At the bounding frequencies the associated Lyapunov exponent is generically either zero or minus infinity. We further show that for a given frequency the number of coexisting attractors is finite.We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. Then, using a sphere immersed in a fluid filling the whole space, we study the critical latitude singularity and show that the velocity field diverges as 1/√d, d being the distance to the characteristic grazing the inner sphere.We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general reveal an attractor expected at the eigenfrequency of the mode. Investigating the structure of these shear layers, we find that they are nested layers, the thinnest and most internal layer scaling with E1/3, E being the Ekman number; for this latter layer, we give its analytical form and show its similarity to vertical 1/3-shear layers of steady flows. Using an inertial wave packet travelling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in [0, 2Ω], contrary to the case of the full sphere (Ω is the angular velocity of the system).Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers (10−10–10−20), which are out of reach numerically, and this for a wide class of containers.

Linear Stability of Momentum Boundary Layer Flow and Heat Transfer Over a Moving Wedge

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Ramesh B. Kudenatti ◽  
Noor E. Misbah ◽  
M. C. Bharathi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Linear Stability ◽  
Boundary Layer Flow ◽  
Shooting Methods ◽  
Flow And Heat Transfer ◽  
Moving Wedge ◽  
Layer Flow ◽  
Steady Solutions ◽  
Similar Solutions

Abstract This paper studies the linear stability of the unsteady boundary-layer flow and heat transfer over a moving wedge. Both mainstream flow outside the boundary layer and the wedge velocities are approximated by the power of the distance along the wedge wall. In a similar manner, the temperature of the wedge is approximated by the power of the distance that leads to a wall exponent temperature parameter. The governing boundary layer equations admit a class of self-similar solutions under these approximations. The Chebyshev collocation and shooting methods are utilized to predict the upper and lower branch solutions for various parameters. For these two solutions, the velocity, temperature profiles, wall shear-stress, and temperature gradient are entirely different and need to be assessed for their stability as to which of these solutions is practically realizable. It is shown that algebraically growing steady solutions do exist and their effects are significant in the unsteady context. The resulting eigenvalue problem determines whether or not the steady solutions are stable. There are interesting results that are linked to bypass an important class of boundary layer flow and heat transfer. The hydrodynamics behind these results are discussed in some detail.

Instability of rotating convection

2000 ◽  
Vol 403 ◽  
pp. 153-172 ◽  
Author(s):  
S. M. COX ◽  
P. C. MATTHEWS
Keyword(s):  
Small Angle ◽  
Large Scale ◽  
Resonant Mode ◽  
Heteroclinic Cycle ◽  
Mean Flow ◽  
Full Account ◽  
Rapid Rotation ◽  
Critical Wavelength ◽  
Mode Interactions ◽  
Convection Rolls

Convection rolls in a rotating layer can become unstable to the Küppers–Lortz instability. When the horizontal boundaries are stress free and the Prandtl number is finite, this instability diverges in the limit where the perturbation rolls make a small angle with the original rolls. This divergence is resolved by taking full account of the resonant mode interactions that occur in this limit: it is necessary to include two roll modes and a large-scale mean flow in the perturbation. It is found that rolls of critical wavelength whose amplitude is of order ε are always unstable to rolls oriented at an angle of order ε2/5. However, these rolls are unstable to perturbations at an infinitesimal angle if the Taylor number is greater than 4π4. Unlike the Küppers–Lortz instability, this new instability at infinitesimal angles does not depend on the direction of rotation; it is driven by the flow along the axes of the rolls. It is this instability that dominates in the limit of rapid rotation. Numerical simulations confirm the analytical results and indicate that the instability is subcritical, leading to an attracting heteroclinic cycle. We show that the small-angle instability grows more rapidly than the skew-varicose instability.

scholarly journals On the multiple solutions of coating and rimming flows on rotating cylinders

2017 ◽  
Vol 835 ◽  
pp. 540-574 ◽  
Author(s):  
André v. B. Lopes ◽  
Uwe Thiele ◽  
Andrew L. Hazel
Keyword(s):  
Stokes Equations ◽  
Bond Number ◽  
Lubrication Approximation ◽  
Dynamics Model ◽  
Rotation Rates ◽  
Gravitational Forces ◽  
Number Ratio ◽  
Gradient Dynamics ◽  
Steady Solutions ◽  
Lubrication Models

We consider steady solutions of the Stokes equations for the flow of a film of fluid on the outer or inner surface of a cylinder that rotates with its axis perpendicular to the direction of gravity. We find that previously unobserved stable and unstable steady solutions coexist over an intermediate range of rotation rates for sufficiently high values of the Bond number (ratio of gravitational forces relative to surface tension). Furthermore, we compare the results of the Stokes calculations to the classic lubrication models of Pukhnachev (J. Appl. Mech. Tech. Phys., vol 18, 1977, pp. 344–351) and Reisfeld & Bankoff (J. Fluid Mech., vol. 236, 1992, pp. 167–196); an extended lubrication model of Benilov & O’Brien (Phys. Fluids, vol. 17, 2005, 052106) and Evans et al. (Phys. Fluids, vol. 16, 2004, pp. 2742–2756); and a new lubrication approximation formulated using gradient dynamics. We quantify the range of validity of each model and confirm that the gradient-dynamics model is most accurate over the widest range of parameters, but that the new steady solutions are not captured using any of the simplified models because they contain features that can only be described by the full Stokes equations.

Transfer of Heat in Rotating Systems

1976 ◽  
Author(s):  
A. D. Gosman ◽  
M. L. Koosinlin ◽  
F. C. Lockwood ◽  
D. B. Spalding
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Fluid Flow ◽  
Rotating Disk ◽  
Experimental Measurements ◽  
Steady Flows ◽  
Disk System ◽  
Rotating Systems ◽  
Flow And Heat Transfer ◽  
Layer Flow

A calculation procedure has been developed for predicting fluid-flow and heat-transfer phenomena in axisymmetrical, rotating, turbulent, steady flows, with special reference to those mainly confined within cavities. The procedure has been used for predicting boundary-layer flow between a rotating disk and a stationary one, and flow and heat transfer in a shrouded-disk system. Agreement with experimental measurements is satisfactory.

Spherical shell rotating convection in the presence of a toroidal magnetic field

1995 ◽  
Vol 448 (1933) ◽  
pp. 245-268 ◽  
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Spherical Shell ◽  
Lorentz Force ◽  
Thermal Convection ◽  
Toroidal Magnetic Field ◽  
Wave Number ◽  
Convective Motions ◽  
Magnetic Convection ◽  
Convection Rolls

Convective instabilities of a self-gravitating, rapidly rotating fluid spherical shell are investigated in the presence of an imposed azimuthal axisymmetric magnetic field in the form of the toroidal decay mode that satisfies electrically insulating boundary conditions and has dipole symmetry. Concentration is on two major questions: how purely thermal convection of the different forms (Zhang 1992, 1994) is affected by the Lorentz force, the strength of which is measured by the Elsasser number ∧, and in what manner purely magnetic instabilities in a spherical shell (Zhang & Fearn 1993, 1994) are associated with magnetic convection. It is found that the two-dimensionality of purely thermal convection (Busse 1970) survives under the influence of a strong Lorentz force. Convective motions always attempt to satisfy the Proudman–Taylor constraint and remain predominantly two-dimensional in the whole range of ∧, 0 ≤ ∧ ≤ ∧ c , where ∧ c ═ O (10) is the critical Elsasser number for purely magnetic instabilities. Though the optimum azimuthal wave number m of convection rolls decreases drastically, from m ~ O ( T 1/6 ) at ∧ ═ 0 to m ═ O (5) at ∧ ═ O (1). We show that there exist no optimum values of ∧ that can give rise to an overall minimum of the (modified) Rayleigh number R *; the optimum value of R * is a monotonically, smoothly decreasing function of ∧, from R * ═ O ( T 1/6 ) at ∧ < O ( T -1/6 ) to R * ═ O (–10) at ∧ ═ 20. We also show that the influence of the magnetic field on thermal convection is crucially dependent on the size of the Prandtl number. At sufficiently small Prandtl number, the Poincaré convection mode (Zhang 1994) is preferred in the region 0 ≤ ∧ < ∧ c , and is only slightly affected by the presence of the toroidal magnetic field. Analytical solutions of the magnetic convection problem are then obtained based on a perturbation analysis, showing a good agreement with the numerical solution.

scholarly journals Determining Rotation Rates from Light Curves: RW Mon and RW Tau

1989 ◽  
Vol 107 ◽  
pp. 373-373
Author(s):  
W. Van Hamme
Keyword(s):  
Light Curve ◽  
Emission Line ◽  
Rotation Rate ◽  
Firm Value ◽  
Primary Component ◽  
Light Curves ◽  
Line Broadening ◽  
Rapid Rotation ◽  
Rotation Rates ◽  
Algol System

AbstractWe present light curve solutions for the non-synchronously rotating Algols RW Mon and RW Tau, and we illustrate how rotation rates are determined from light curves. We find RW Mon’s primary component to spin at about 5 times the synchronous rate, which confirms the indication of fast rotation from reported emission line activity. RW Tau turns out to be only a mildly rapidly rotating Algol system, and our light curve solutions do not yield any firm value for the rotation rate of the primary component. It is suggested that continued efforts should be made to do good quality line broadening studies in order to find rotation rates for systems with only modest degrees of rapid rotation, and in order to further test photometric rotation rates against those of line broadening studies.

Comments on “heat transfer in a square porous cavity in presence of square solid block”

2019 ◽  
Vol 30 (2) ◽  
pp. 704-723 ◽  
Author(s):  
D.A.S. Rees
Keyword(s):  
Systematic Approach ◽  
Outer Boundary ◽  
Finite Difference Methods ◽  
Content Type ◽  
Porous Cavity ◽  
Richardson’S Extrapolation ◽  
Steady Solutions ◽  
Internal Block ◽  
Error Testing ◽  
Solid Block

Purpose The purpose of this paper is to discuss the need to attend correctly to the accuracy and the manner in which the value of the streamfunction is determined when two or more impermeable boundaries are present. This is discussed within the context of the paper by Nandalur et al. (2019), which concerns the effect of a centrally located conducting square block on convection in a square sidewall-heated porous cavity. Detailed solutions are also presented which allow the streamfunction to take the natural value on the surface of the internal block. Design/methodology/approach Steady solutions are obtained using finite difference methods. Three different ways in which insulating boundary conditions are implemented are compared. Detailed attention is paid to the iterative convergence of the numerical scheme and to its overall accuracy. Error testing and Richardson’s extrapolation have been used to obtain very precise values of the Nusselt number. Findings The assumption that the streamfunction takes a zero value on the boundaries of both the cavity and the embedded block is shown to be incorrect. Application of the continuity-of-pressure requirement shows that the block and the outer boundary take different constant values. Research limitations/implications The Darcy–Rayleigh number is restricted to values at or below 200; larger values require a finer grid. Originality/value This paper serves as a warning that one cannot assume that the streamfunction will always take a zero value on all impermeable surfaces when two or more are present. A systematic approach to accuracy is described and recommended.

Flow instabilities in the wide-gap spherical Couette system

2013 ◽  
Vol 738 ◽  
pp. 184-221 ◽  
Author(s):  
Johannes Wicht
Keyword(s):  
Shear Layer ◽  
Differential Rotation ◽  
Outer Boundary ◽  
Fluid Flows ◽  
Zonal Flow ◽  
Comprehensive Overview ◽  
Jet Instability ◽  
Rotation Rates ◽  
Boundary Rotation ◽  
Spherical Couette

AbstractThe spherical Couette system is a spherical shell filled with a viscous fluid. Flows are driven by the differential rotation between the inner and the outer boundary that rotate with $\Omega $ and $\Omega + \mathrm{\Delta} \Omega $ about a common axis. This setup has been proposed for second-generation dynamo experiments. We numerically explore the different instabilities emerging for rotation rates up to $\Omega = (1/ 3)\times 1{0}^{7} $, venturing also into the nonlinear regime where oscillatory and chaotic solutions are found. The results provide a comprehensive overview of the possible flow regimes. For low values of $\Omega $ viscosity dominates and an equatorial jet in meridional circulation and zonal flow develops that becomes unstable as the differential rotation is increased beyond a critical value. For intermediate $\Omega $ and an inner boundary rotating slower than the outer one, new double-roll and helical instabilities are found. For large $\Omega $ values Coriolis effects enforce a nearly two-dimensional fundamental flow where a Stewartson shear layer develops at the tangent cylinder. This shear layer is the source of nearly geostrophic non-axisymmetric instabilities that resemble columnar Rossby modes. At first, the instabilities differ significantly depending on whether the inner boundary rotates faster $( \mathrm{\Delta} \Omega \gt 0)$ or slower $( \mathrm{\Delta} \Omega \lt 0)$ than the outer one. For very large outer boundary rotation rates, however, both instabilities once more become comparable. Fast inertial waves similar to those observed in recent spherical Couette experiments prevail for larger $\Omega $ values and $ \mathrm{\Delta} \Omega \lt 0$ in when $ \mathrm{\Delta} \Omega $ and $\Omega $ are of comparable magnitude. For larger differential rotations $ \mathrm{\Delta} \Omega \gg \Omega $, however, the equatorial jet instability always takes over.

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