scholarly journals Thermal convection in a magnetized conducting fluid with the Cattaneo–Christov heat-flow model

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160649 ◽  
Author(s):  
J. J. Bissell
Keyword(s):  
Rayleigh Number ◽  
Heat Flow ◽  
Prandtl Number ◽  
Thermal Convection ◽  
Flow Model ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Conducting Fluid ◽  
Heat Flow Model ◽  
The Impact

By substituting the Cattaneo–Christov heat-flow model for the more usual parabolic Fourier law, we consider the impact of hyperbolic heat-flow effects on thermal convection in the classic problem of a magnetized conducting fluid layer heated from below. For stationary convection, the system is equivalent to that studied by Chandrasekhar ( Hydrodynamic and Hydromagnetic Stability, 1961), and with free boundary conditions we recover the classical critical Rayleigh number R c ( c ) ( Q ) which exhibits inhibition of convection by the field according to R c ( c ) → π 2 Q as Q → ∞ , where Q is the Chandrasekhar number. However, for oscillatory convection we find that the critical Rayleigh number R c ( o ) ( Q , P 1 , P 2 , C ) is given by a more complicated function of the thermal Prandtl number P 1 , magnetic Prandtl number P 2 and Cattaneo number C . To elucidate features of this dependence, we neglect P 2 (in which case overstability would be classically forbidden), and thereby obtain an expression for the Rayleigh number that is far less strongly inhibited by the field, with limiting behaviour R c ( o ) → π Q / C , as Q → ∞ . One consequence of this weaker dependence is that onset of instability occurs as overstability provided C exceeds a threshold value C T ( Q ); indeed, crucially we show that when Q is large, C T ∝ 1 / Q , meaning that oscillatory modes are preferred even when C itself is small. Similar behaviour is demonstrated in the case of fixed boundaries by means of a novel numerical solution.

scholarly journals On the sensitivity of 3-D thermal convection codes to numerical discretization: a model intercomparison

2014 ◽  
Vol 7 (5) ◽  
pp. 2065-2076 ◽  
Author(s):  
P.-A Arrial ◽  
N. Flyer ◽  
G. B. Wright ◽  
L. H. Kellogg
Keyword(s):  
Rayleigh Number ◽  
Numerical Simulations ◽  
Spherical Harmonics ◽  
Thermal Convection ◽  
Mantle Convection ◽  
Numerical Study ◽  
Critical Rayleigh Number ◽  
Initial Condition ◽  
Numerical Discretization ◽  
The Impact

Abstract. Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how the discretization of the governing equations affects the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh–Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in developing mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes – an established, community-developed, and benchmarked finite-element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBFs) – are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ = 4). How this pattern varies to perturbations in the initial condition and Rayleigh number is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ = 6). Although both methods first converge to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady-state pattern is presented as a combination of third- and fourth-order spherical harmonics leading to a five-cell or hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.

scholarly journals On the sensitivity of 3-D thermal convection codes to numerical discretization: a model intercomparison

2014 ◽  
Vol 7 (2) ◽  
pp. 2033-2064
Author(s):  
P.-A. Arrial ◽  
N. Flyer ◽  
G. B. Wright ◽  
L. H. Kellogg
Keyword(s):  
Rayleigh Number ◽  
Numerical Simulations ◽  
Spherical Harmonics ◽  
Thermal Convection ◽  
Mantle Convection ◽  
Numerical Study ◽  
Critical Rayleigh Number ◽  
Initial Condition ◽  
Numerical Discretization ◽  
The Impact

Abstract. Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how does the discretization of the governing equations affect the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh-Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in development mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes: an established, community-developed, and benchmarked finite element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBF) are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ =4). How variations in the behavior of the cubic pattern to perturbations in the initial condition and Rayleigh number between the two numerical discrezations is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ = 6). Although both methods converge first to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady state pattern is presented as a combination of order 3 and 4 spherical harmonics leading to a five cell or a hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.

Linear stability of rotating convection in an imposed shear flow

1997 ◽  
Vol 350 ◽  
pp. 271-293 ◽  
Author(s):  
PAUL MATTHEWS ◽  
STEPHEN COX
Keyword(s):  
Rayleigh Number ◽  
Shear Flow ◽  
Eigenvalue Problem ◽  
Linear Stability ◽  
Thermal Convection ◽  
Critical Rayleigh Number ◽  
Rotation Vector ◽  
Linear Stability Theory ◽  
Fixed Temperature ◽  
Differential Motion

In many geophysical and astrophysical contexts, thermal convection is influenced by both rotation and an underlying shear flow. The linear theory for thermal convection is presented, with attention restricted to a layer of fluid rotating about a horizontal axis, and plane Couette flow driven by differential motion of the horizontal boundaries.The eigenvalue problem to determine the critical Rayleigh number is solved numerically assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection rolls is found, for different orientations of the rotation vector with respect to the shear flow. For moderate rates of shear and rotation, the preferred roll orientation depends only on their ratio, the Rossby number.It is well known that rotation alone acts to favour rolls aligned with the rotation vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls parallel to the shear flow are preferred. However, it is found that when the rotation vector and shear flow are parallel, the two effects lead counter-intuitively (as in other, analogous convection problems) to a preference for oblique rolls, and a critical Rayleigh number below that for Rayleigh–Bénard convection.When the boundaries are poorly conducting, the eigenvalue problem is solved analytically by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the stability problem is found to be qualitatively similar to that for fixed-temperature boundaries.Fully nonlinear numerical simulations of the convection are also carried out. These are generally consistent with the linear stability theory, showing convection in the form of rolls near the onset of motion, with the appropriate orientation. More complicated states are found further from critical.

scholarly journals Nonlinear feedback in a six-dimensional Lorenz Model: impact of an additional heating term

2015 ◽  
Vol 2 (2) ◽  
pp. 475-512
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the steamfunction is referred to as a secondary streamfunction mode, while the two additional modes, that appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. The implications of this and previous work, as well as future work, are also discussed.

scholarly journals Heat flow model for pulsed laser melting and rapid solidification of ion implanted GaAs

2010 ◽  
Vol 108 (1) ◽  
pp. 013508 ◽  
Author(s):  
Taeseok Kim ◽  
Manoj R. Pillai ◽  
Michael J. Aziz ◽  
Michael A. Scarpulla ◽  
Oscar D. Dubon ◽  
...  
Keyword(s):  
Heat Flow ◽  
Rapid Solidification ◽  
Flow Model ◽  
Pulsed Laser ◽  
Laser Melting ◽  
Heat Flow Model ◽  
Ion Implanted

Experiments on the inhibition of thermal convection by a magnetic field

1957 ◽  
Vol 240 (1220) ◽  
pp. 108-113 ◽  
Keyword(s):  
Magnetic Field ◽  
Electrical Conductivity ◽  
Rayleigh Number ◽  
Magnetic Fields ◽  
Thermal Convection ◽  
Kinematic Viscosity ◽  
Critical Rayleigh Number ◽  
Theoretical Relation ◽  
Horizontal Layers

Experiments on the magnetic inhibition of thermal convection in horizontal layers of mercury heated from below are described. A large 36½ in. cyclotron magnet reconditioned for hydromagnetic studies was used in these experiments. By using layers of mercury of depth 3 to 6 cm and magnetic fields of strength 500 to 8000 gauss, it has been possible to determine the dependence of the critical Rayleigh number for the onset of instability on the parameter Q 1 ( = σH 2 d 2 / π 2 ρν , where H denotes the strength of the field, σ the electrical conductivity, ν the coefficient of kinematic viscosity, ρ the density and d the depth of the layer) for Q 1 varying between 40 and 1·6 × 10 6 . The experiments fully confirm the theoretical relation derived by Chandrasekhar.

Cold-specific feeding response of rats to cold exposure and energy density of body weight change

1981 ◽  
Vol 51 (2) ◽  
pp. 327-334 ◽  
Author(s):  
S. D. Morrison
Keyword(s):  
Body Weight ◽  
Food Intake ◽  
Energy Density ◽  
Heat Flow ◽  
Cold Exposure ◽  
Weight Change ◽  
Flow Model ◽  
Energy Yield ◽  
Body Weight Change ◽  
Heat Flow Model

The increased food intake of rats exposed to cold is the result of increased intake due to cold (cold-specific compartment; A) and decreased intake due to simultaneously decreased body weight (weight-specific compartment; B). The two compartments are evaluated at 5, 13, and 17 degrees C. B is evaluated as the food intake of theoretical, isogravimetric control (identical to cold-exposed rats with respect to body weight and rate of change of body weight and identical to nonexposed rats in all other respects) that takes into account both the change in energy expenditure due to decreased body weight and the energy yield from tissue catabolism represented by change of body weight. A is the observed food intake minus B. A theoretical heat-flow model, in which expected changes in heat flow during cold exposure drive food intake to maintain or restore preexposure body weight status, corroborated the partition derived from experimental data. However, both the experimental results and the heat-flow model imply that the energy density of body weight change is negatively correlated with rate of body weight change. The energy density of weight change is high with high rates of weight loss and low with high rats of weight gain.

Effect of a stabilizing gradient of solute on thermal convection

1968 ◽  
Vol 34 (2) ◽  
pp. 315-336 ◽  
Author(s):  
George Veronis
Keyword(s):  
Rayleigh Number ◽  
Temperature Gradient ◽  
Prandtl Number ◽  
Critical Rayleigh Number ◽  
Effective Diffusion ◽  
Stratified Fluid ◽  
Finite Amplitude ◽  
Oscillatory Motion ◽  
Linear Stability Theory ◽  
Onset Of Convection

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

Non-linear effects in a Rayleigh–Bénard experiment

1970 ◽  
Vol 42 (1) ◽  
pp. 161-175 ◽  
Author(s):  
D. R. Caldwell
Keyword(s):  
Rayleigh Number ◽  
Heat Flow ◽  
Density Profile ◽  
Critical Rayleigh Number ◽  
Temperature Drop ◽  
Density Profiles ◽  
Heating Curve ◽  
Linear Effects ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Observations of temperature drop as a function of heat flow in Rayleigh–Bénard convection with curved density profiles show: (1) reversal of slope in the heating curve, (2) oscillations with time, (3) history dependence, and (4) an increase in critical Rayleigh number as the curvature of the density profile is increased. Some of the results are quite similar to the predictions of Busse.

scholarly journals Effect of Vertical Vibrations on the Onset of Binary Convection

2013 ◽  
Vol 18 (3) ◽  
pp. 899-910 ◽  
Author(s):  
M.S. Swamy
Keyword(s):  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Lewis Number ◽  
Wave Number ◽  
Closed Form Expression ◽  
Effective Control ◽  
Regular Perturbation ◽  
Binary Fluid ◽  
Vertical Vibrations

Abstract In the present work the linear stability analysis of double diffusive convection in a binary fluid layer is performed. The major intention of this study is to investigate the influence of time-periodic vertical vibrations on the onset threshold. A regular perturbation method is used to compute the critical Rayleigh number and wave number. A closed form expression for the shift in the critical Rayleigh number is calculated as a function of frequency of modulation, the solute Rayleigh number, Lewis number, and Prandtl number. These parameters are found to have a significant influence on the onset criterion; therefore the effective control of convection is achieved by proper tuning of these parameters. Vertical vibrations are found to enhance the stability of a binary fluid layer heated and salted from below. The results of this study are useful in the areas of crystal growth in micro-gravity conditions and also in material processing industries where vertical vibrations are involved

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