The Rayleigh—Jeffreys problem with boundary slab of finite conductivity

1968 ◽  
Vol 32 (2) ◽  
pp. 393-398 ◽  
Author(s):  
D. A. Nield
Keyword(s):  
Finite Thickness ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Wave Number ◽  
Linear Perturbation ◽  
Finite Conductivity ◽  
Solid Layer ◽  
Onset Of Convection ◽  
Infinite Conductivity ◽  
Bounded Below

Linear perturbation analysis is applied to the problem of the onset of convection in a horizontal layer of fluid heated uniformly from below, when the fluid is bounded below by a rigid plate of inlinite conductivity and above by a solid layer of finite conductivity and finite thickness. The critical Rayleigh number and wave-number are found for various thickness ratios and thermal conductivity ratios. Both numbers are reduced by the presence of a boundary of finite (rather than infinite) conductivity in qualitative agreement with the observation of Koschmieder (1966).

The Onset of Convection in a Layer of Cellular Porous Material: Effect of Temperature-Dependent Conductivity Arising From Radiative Transfer

2010 ◽  
Vol 132 (7) ◽  
Author(s):  
D. A. Nield ◽  
A. V. Kuznetsov
Keyword(s):  
Rayleigh Number ◽  
Porous Material ◽  
Effective Conductivity ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Wave Number ◽  
Effect Of Temperature ◽  
Onset Of Convection ◽  
Cellular Foams ◽  
Variation Parameter

The onset of convection in a horizontal layer of a cellular porous material heated from below is investigated. The problem is formulated as a combined conductive-convective-radiative problem in which radiative heat transfer is treated as a diffusion process. The problem is relevant to cellular foams formed from plastics, ceramics, and metals. It is shown that the variation of conductivity with temperature above that of the cold boundary leads to an increase in the critical Rayleigh number (based on the conductivity of the fluid at that boundary temperature) and an increase in the critical wave number. On the other hand, the critical Rayleigh number based on the conductivity at the mean temperature decreases with increase in the thermal variation parameter if the radiative contribution to the effective conductivity is sufficiently large compared with the nonradiative component.

Theory on Thermal Instability of Binary Gas Mixtures in Porous Media

1976 ◽  
Vol 98 (1) ◽  
pp. 35-41 ◽  
Author(s):  
M. L. Lawson ◽  
Wen-Jei Yang ◽  
S. Bunditkul
Keyword(s):  
Fluid Layer ◽  
Perturbation Technique ◽  
Critical Rayleigh Number ◽  
Lower Boundary ◽  
Horizontal Layer ◽  
Linear Perturbation ◽  
Onset Of Convection ◽  
Hurwitz Stability ◽  
Binary Gas Mixture ◽  
The Galerkin Method

A theory is developed which predicts the instability of a horizontal layer of porous medium saturated with a binary gas mixture. The lower boundary of the system is maintained at a higher temperature and the upper one at low temperature. The transport equations and coefficients are developed on the basis of kinetic theory. A linear perturbation technique is employed to reduce the governing equations for momentum, heat, and mass transfer to eigenvalue differential equations which are solved by the Finlayson method, the combination of the Galerkin method and the Routh-Hurwitz stability criterion. Only neutral stationary stability is found to occur in the system. Its criterion can be predicted by a simple algebraic equation. Both the critical Rayleigh and wave numbers for the onset of convection are governed by five independent dimensionless parameters, two of which are most influential. The critical Rayleigh number may be lower or greater than that for pure fluid layer depending upon whether thermal diffusion induces the heavier component of the mixture to move toward the cold or hot boundary, respectively. The theory compares well with the experimental results.

Onset of Convection in a Fluid Saturated Porous Layer Overlying a Solid Layer Which is Heated by Constant Flux

1999 ◽  
Vol 121 (4) ◽  
pp. 1094-1097 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Critical Rayleigh Number ◽  
Hydrothermal Systems ◽  
Solid Layer ◽  
Onset Of Convection ◽  
Constant Flux ◽  
Saturated Porous Layer ◽  
Bottom Heating ◽  
Temperature Heating

The thermoconvective stability of a porous layer overlying a solid layer is important in seafloor hydrothermal systems and thermal insulation problems. The case for constant flux bottom heating is considered. The critical Rayleigh number for the porous layer is found to increase with the thickness of the solid layer, a result opposite to constant temperature heating.

The thermohaline Rayleigh-Jeffreys problem

1967 ◽  
Vol 29 (3) ◽  
pp. 545-558 ◽  
Author(s):  
D. A. Nield
Keyword(s):  
Rayleigh Number ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Stability Boundary ◽  
Boundary Curve ◽  
Solute Concentration ◽  
Horizontal Layer ◽  
Wave Number ◽  
Convection Cell ◽  
Onset Of Convection

The onset of convection induced by thermal and solute concentration gradients, in a horizontal layer of a viscous fluid, is studied by means of linear stability analysis. A Fourier series method is used to obtain the eigenvalue equation, which involves a thermal Rayleigh numberRand an analogous solute Rayleigh numberS, for a general set of boundary conditions. Numerical solutions are obtained for selected cases. Both oscillatory and monotonic instability are considered, but only the latter is treated in detail. The former can occur when a strongly stabilizing solvent gradient is opposed by a destablizing thermal gradient. When the same boundary equations are required to be satisfied by the temperature and concentration perturbations, the monotonic stability boundary curve in the (R, S)-plane is a straight line. Otherwise this curve is concave towards the origin. For certain combinations of boundary conditions the critical value ofRdoes not depend onS(for some range ofS) or vice versa. This situation pertains when the critical horizontal wave-number is zero.A general discussion of the possibility and significance of convection at ‘zero’ wave-number (single convection cell) is presented in an appendix.

scholarly journals The effect of controlled vibrations on Rayleigh-Benard convection

2021 ◽  
Vol 2057 (1) ◽  
pp. 012012
Author(s):  
A I Fedyushkin
Keyword(s):  
Rayleigh Number ◽  
Convective Flow ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Wave Number ◽  
Lower Wall ◽  
Vibration Effect ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Abstract The paper presents the results of a numerical study of convective heat transfer in a long horizontal layer heated from below with and without the vibration effect of the lower wall. The simulation was carried out on the basis of solving the Navier-Stokes 2D equations for an incompressible fluid in the Boussinesq approximation. It is shown that the influence of vibrations of the lower heated wall on the wave number of the convective flow roll structure, on the time and on the critical Rayleigh number of convection. The influence of controlled harmonic vibrations of wall on the structure of convective flow in the Rayleigh-Benard problem has been investigated. It is shown that the wave number of the periodic convective structure, the critical Rayleigh number, and the time of occurrence of Rayleigh-Benard convection under the vertical vibration effect on the horizontal layer from the lower wall are reduced.

Finite Amplitude Convection in a Rotating Porous Medium

1987 ◽  
Vol 42 (1) ◽  
pp. 13-20
Author(s):  
B. S. Dandapat
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Interstitial Fluid ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Linear Stability Theory ◽  
Onset Of Convection

The onset of convection in a horizontal layer of a saturated porous medium heated from below and rotating about a vertical axis with uniform angular velocity is investigated. It is shown that when S ∈ σ >1, overstability cannot occur, where ε is the porosity, σ the Prandtl number and S is related to the heat capacities of the solid and the interstitial fluid. It is also shown that for small values of the rotation parameter T1, finite amplitude motion with subcritical values of Rayleigh number R (i.e. R < Re, where Re is the critical Rayleigh number according to linear stability theory) is possible. For large values of T1, overstability is the preferred mode.

Natural Convection in an Inclined Fluid Layer With a Transverse Magnetic Field: Analogy With a Porous Medium

1995 ◽  
Vol 117 (1) ◽  
pp. 121-129 ◽  
Author(s):  
P. Vasseur ◽  
M. Hasnaoui ◽  
E. Bilgen ◽  
L. Robillard
Keyword(s):  
Magnetic Field ◽  
Transverse Magnetic Field ◽  
Hartmann Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Integral Form ◽  
Transverse Magnetic ◽  
Horizontal Layer ◽  
Governing Equations ◽  
Onset Of Convection

In this paper the effect of a transverse magnetic field on buoyancy-driven convection in an inclined two-dimensional cavity is studied analytically and numerically. A constant heat flux is applied for heating and cooling the two opposing walls while the other two walls are insulated. The governing equations are solved analytically, in the limit of a thin layer, using a parallel flow approximation and an integral form of the energy equation. Solutions for the flow fields, temperature distributions, and Nusselt numbers are obtained explicitly in terms of the Rayleigh and Hartmann numbers and the angle of inclination of the cavity. In the high Hartmann number limit it is demonstrated that the resulting solution is equivalent to that obtained for a porous layer on the basis of Darcy’s model. In the low Hartmann number limit the solution for a fluid layer in the absence of a magnetic force is recovered. In the case of a horizontal layer heated from below the critical Rayleigh number for the onset of convection is derived in term of the Hartmann number. A good agreement is found between the analytical predictions and the numerical simulation of the full governing equations.

Onset of convection in a variable-viscosity fluid

1982 ◽  
Vol 120 ◽  
pp. 411-431 ◽  
Author(s):  
Karl C. Stengel ◽  
Dean S. Oliver ◽  
John R. Booker
Keyword(s):  
Rayleigh Number ◽  
Viscosity Ratio ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Onset Of Convection ◽  
Numerical Predictions ◽  
Dependent Viscosity

The Rayleigh number R, in a horizontal layer with temperature-dependent viscosity can be based on the viscosity at T0, the mean of the boundary temperatures. The critical Rayleigh number Roc for fluids with exponential and super-exponential viscosity variation is nearly constant at low values of the ratio of the viscosities at the top and bottom boundaries; increases at moderate values of the viscosity ratio, reaching a maximum at a ratio of about 3000, and then decreases. This behaviour is explained by a simple physical argument based on the idea that convection begins first in the sublayer with maximum Rayleigh number. The prediction of Palm (1960) that certain types of temperature-dependent viscosity always decrease Roc is confirmed by numerical results but is not relevant to the viscosity variations typical of real liquids. The infinitesimal-amplitude state assumed by linear theory in calculating Roc does not exist because the convection jumps immediately to a finite amplitude at R0c. We observe a heat-flux jump at R0c exceeding 10% when the viscosity ratio exceeds 150. However, experimental measurements of R0c for glycerol up to a viscosity ratio of 3400 are in good agreement with the numerical predictions when the effects of a temperature-dependent expansion coefficient and thermal diffusivity are included.

Onset and Development of Natural Convection Above a Suddenly Heated Horizontal Surface

1995 ◽  
Vol 117 (4) ◽  
pp. 808-821 ◽  
Author(s):  
R. J. Goldstein ◽  
R. J. Volino
Keyword(s):  
Rayleigh Number ◽  
Heated Surface ◽  
Critical Rayleigh Number ◽  
Fluid Motion ◽  
Horizontal Layer ◽  
Onset Of Convection ◽  
Surface Convection ◽  
Heating From Below ◽  
Conduction Layer ◽  
Moving Pattern

The onset and development of flow in a thick horizontal layer subject to a near-constant flux heating from below has been studied experimentally. The overall heat-flux-based Rayleigh number, Ra*, ranges from 2 × 108 to 7 × 1010. Flow visualization shows the growth and breakdown of a conduction layer adjacent to the heated surface. Convection is characterized by the release of warm meandering plumes and thermals from a boundary layer. The planform of convection at the heated surface begins with a pattern of small spots suggestive of Be´nard cells. Some of these cells expand, forming a larger cell pattern. This continues until a quasi-steady state is reached in which the former cell boundaries form a slowly moving pattern of warm lines on the heated surface. The lines are believed to be the source of the plumes and thermals. Quantitatively, the onset of convection occurs at a constant (critical) Rayleigh number based on the conduction layer thickness, Raδ. Based on the first observation of fluid motion, this critical Rayleigh number is approximately 1300. Based on the heated surface temperature the critical Rayleigh number is 2700. The nondimensional wavenumber associated with the observed instabilities at the onset of convection is about 2.2.

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