Stability of Thermal Convection in a Vertical Porous Layer

1987 ◽  
Vol 109 (4) ◽  
pp. 889-893 ◽  
Author(s):  
L. P. Kwok ◽  
C. F. Chen
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Porous Layer ◽  
Thermal Convection ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Lateral Heating ◽  
The Stability ◽  
Allowable Temperature

Experiments were carried out to study the stability of thermal convection generated in a vertical porous layer by lateral heating in a tall, narrow tank. The porous medium, consisting of glass beads, was saturated with distilled water. It was found that the flow became unstable at a critical ΔT of 29.2°C (critical Rayleigh number of 66.2). Linear stability analysis was applied to study the effects of the Brinkman term and of variable viscosity separately using a quadratic relationship between the density and temperature. It was found that with the Brinkman term, no instability could occur within the allowable temperature difference across the tank. With the effect of variable viscosity included, linear stability theory predicts a critical ΔT of 43.4°C (Rayleigh number of 98.3).

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.

Stability of Gravity Driven Convection in a Cylindrical Porous Layer Subjected to Vibration

2006 ◽  
Author(s):  
Saneshan Govender
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Density Gradient ◽  
Porous Layer ◽  
Binary Alloy ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Direct Result ◽  
Porous Cylinder

In both pure fluids and porous media, the density gradient becomes unstable and fluid motion (convection) occurs when the critical Rayleigh number is exceeded. The classical stability analysis no longer applies if the Rayleigh number is time dependant, as found in systems where the density gradient is subjected to vibration. The influence of vibrations on thermal convection depends on the orientation of the time dependant acceleration with respect to the thermal stratification. The problem of a vibrating porous cylinder has numerous important engineering applications, the most important one being in the field of binary alloy solidification. In particular we may extend the above results to understanding the dynamics in the mushy layer (essentially a reactive porous medium) that is sandwiched between the underlying solid and overlying melt regions. Alloyed components are widely used in demanding and critical applications, such as turbine blades, and a consistent internal structure is paramount to the performance and integrity of the component. Alloys are susceptible to the formation of vertical channels which are a direct result of the presence convection, so any technique that suppresses convection/the formation of channels would be welcomed by the plant metallurgical engineer. In the current study, the linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogeneous cylindrical porous layer heated from below. The linear stability results show that increasing the frequency of vibration stabilizes the convection. In addition the aspect ratio of the porous cylinder is shown to influence the stability of convection for all frequencies analysed. It was also observed that only synchronous solutions are possible in cylindrical porous layers, with no transition to sub harmonic solutions as was the case in Govender (2005a) for rectangular layers or cavities. The results of the current analysis will be used in the formulation of a model for binary alloy systems that includes the reactive porous medium model.

Stability of Heat Pipes in Vapor-Dominated Systems

1999 ◽  
Author(s):  
Pouya Amili ◽  
Yanis C. Yortsos
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Oil Recovery ◽  
Critical Rayleigh Number ◽  
Stability Limit ◽  
Wave Number ◽  
Nuclear Waste Repository ◽  
Geothermal Systems ◽  
Dimensionless Numbers ◽  
The Stability

Abstract We study the linear stability of a two-phase heat pipe zone (vapor-liquid counterflow) in a porous medium, overlying a superheated vapor zone. The competing effects of gravity, condensation and heat transfer on the stability of a planar base state are analyzed in the linear stability limit. The rate of growth of unstable disturbances is expressed in terms of the wave number of the disturbance, and dimensionless numbers, such as the Rayleigh number, a dimensionless heat flux and other parameters. A critical Rayleigh number is identified and shown to be different than in natural convection under single phase conditions. The results find applications to geothermal systems, to enhanced oil recovery using steam injection, as well as to the conditions of the proposed Yucca Mountain nuclear waste repository. This study complements recent work of the stability of boiling by Ramesh and Torrance (1993).

On the Effect of Mechanical and Thermal Anisotropy on the Stability of Gravity Driven Convection in Rotating Porous Media

2005 ◽  
Author(s):  
Saneshan Govender ◽  
Peter Vadasz
Keyword(s):  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Linear Stability Theory ◽  
Mechanical Anisotropy ◽  
Onset Of Convection ◽  
Thermal Anisotropy ◽  
Gravity Driven ◽  
The Stability ◽  
Rayleigh Benard

We investigate Rayleigh-Benard convection in a porous layer subjected to gravitational and Coriolis body forces, when the fluid and solid phases are not in local thermodynamic equilibrium. The Darcy model (extended to include Coriolis effects and anisotropic permeability) is used to describe the flow whilst the two-equation model is used for the energy equation (for the solid and fluid phases separately). The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection and the effect of both thermal and mechanical anisotropy on the critical Rayleigh number is discussed.

The Effect of Suction on the Stability of Fluid between Horizontal Plates at Differing Temperatures

1985 ◽  
Vol 199 (2) ◽  
pp. 145-151 ◽  
Author(s):  
M M Sorour ◽  
M A Hassab ◽  
F A Elewa
Keyword(s):  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Stability Criteria ◽  
Thermal Boundary Conditions ◽  
Wide Range ◽  
Fluid Layers ◽  
The Stability ◽  
Horizontal Fluid Layer

The linear stability theory is applied to study the effect of suction on the stability criteria of a horizontal fluid layer confined between two thin porous surfaces heated from below. This investigation covers a wide range of Reynolds number 0 ≥ Re ≥ 30, and Prandtl number 0.72 ≥ Pr ≥ 100. The results show that the critical Rayleigh number increases with Peclet number, and is independent of Pr as far as Re < 3. However, for Re > 3 the critical Rayleigh number is function of both Pr and Pe. In addition, the analysis is extended to study the effect of suction on the stability of two special superimposed fluid layers. The results in the latter case indicate a more stabilizing effect. Furthermore, the effect of thermal boundary conditions is also investigated.

Onset of Natural Convection in a Porous Layer Confined by an Elliptic or Semi-Elliptic Box Using the Ritz Method

2020 ◽  
Vol 142 (9) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Natural Convection ◽  
Rayleigh Number ◽  
Porous Layer ◽  
Ritz Method ◽  
Critical Rayleigh Number ◽  
Aspect Ratios ◽  
Side Walls ◽  
The Stability

Abstract Using an efficient Ritz method, the thermoconvective stability of a bottom-heated porous layer in vertical elliptic and semi-elliptic enclosures with adiabatic side walls is studied. The stability mosaics for the critical Rayleigh number and the preferred modes are determined for various aspect ratios.

scholarly journals Two-Dimensional Convection Induced by Gravity and Centrifugal Forces in a Rotating Porous Layer Far Away from the Axis of Rotation

1998 ◽  
Vol 4 (2) ◽  
pp. 73-90 ◽  
Author(s):  
Peter Vadasz ◽  
Saneshan Govender
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Temperature Fields ◽  
Wave Number ◽  
Marginal Stability ◽  
Two Dimensional ◽  
Wide Range ◽  
Gravity Driven ◽  
Gravity Driven Flow ◽  
The Stability

The stability and onset of two-dimensional convection in a rotating fluid saturated porous layer subject to gravity and centrifugal body forces is investigated analytically. The problem corresponding to a layer placed far away from the centre of rotation was identified as a distinct case and therefore justifying special attention. The stability of a basic gravity driven convection is analysed. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number for different values of the gravity related Rayleigh number. For any given value of the gravity related Rayleigh number there is a transitional value of the wave number, beyond which the basic gravity driven flow is stable. The results provide the stability map for a wide range of values of the gravity related Rayleigh number, as well as the corresponding flow and temperature fields.

scholarly journals Effect of Vadasz Number on Magnetoconvection in a Darcy Porous Layer with Concentration Based Internal Heating

2019 ◽  
pp. 1-13
Author(s):  
C. Israel-Cookey ◽  
L. Ebiwareme ◽  
E. Amos
Keyword(s):  
Magnetic Field ◽  
Rayleigh Number ◽  
Porous Layer ◽  
Internal Heat ◽  
Lewis Number ◽  
Mode Analysis ◽  
Internal Heating ◽  
Vadasz Number ◽  
The Stability ◽  
Solutal Rayleigh Number

In this research article, the effect of Vadasz number on magnetoconvection in a Darcy Porous Layer with concentration based internal heating is studied. The flow is governed by the Oberbeck-Boussineq model for Newtonian fluid. The stability analysis method based on the perturbation of infinitesimal amplitude is carried out using the normal mode analysis. The onset criterion for both the stationary and oscillatory convection on the stability of system is obtained. The analysis examines the effects of pertinent parameters on the stability of the system: magnetic field parameter, solutal Rayleigh number, Lewis number and Vadasz number. The result show that, internal heat parameter,  and Lewis number, , hastens the onset of instability in the system, whereas magnetic field, , Vadasz number,  and solutal Rayleigh number,  delay the onset of instability.

Stability and Convection in Impulsively Heated Porous Layers

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
M. J. Kohl ◽  
M. Kristoffersen ◽  
F. A. Kulacki
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Heated Surface ◽  
Critical Rayleigh Number ◽  
Axisymmetric Flow ◽  
Temperature Fluctuations ◽  
Lower Surface ◽  
Onset Of Convection ◽  
Upward Moving ◽  
The Stability

Experiments are reported on initial instability, turbulence, and overall heat transfer in a porous medium heated from below. The porous medium comprises either water or a water-glycerin solution and randomly stacked glass spheres in an insulated cylinder of height:diameter ratio of 1.9. Heating is with a constant flux lower surface and a constant temperature upper surface, and the stability criterion is determined for a step heat input. The critical Rayleigh number for the onset of convection is obtained in terms of a length scale normalized to the thermal penetration depth as Rac=83/(1.08η−0.08η2) for 0.02<η<0.18. Steady convection in terms of the Nusselt and Rayleigh numbers is Nu=0.047Ra0.91Pr0.11(μ/μ0)0.72 for 100<Ra<5000. Time-averaged temperatures suggest the existence of a unicellular axisymmetric flow dominated by upflow over the central region of the heated surface. When turbulence is present, the magnitude and frequency of temperature fluctuations increase weakly with increasing Rayleigh number. Analysis of temperature fluctuations in the fluid provides an estimate of the speed of the upward moving thermals, which decreases with distance from the heated surface.

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.

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