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.

scholarly journals Vadasz Number Effects on Convection in a Horizontal Porous Layer Subjected to Internal Heat Generation and G-Jitter

Fluids ◽  
2020 ◽  
Vol 5 (3) ◽  
pp. 124
Author(s):  
Saneshan Govender
Keyword(s):  
Heat Transfer ◽  
Rayleigh Number ◽  
Porous Layer ◽  
Heat Generation ◽  
Critical Rayleigh Number ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Onset Of Convection ◽  
Vadasz Number ◽  
The Impact

Flow and heat transfer in a horizontal porous layer subjected to internal heat generation and g-jitter is considered for the Dirichlet thermal boundary condition. A linear stability analysis is used to determine the convection threshold in terms of the critical Rayleigh number. For the low amplitude, high frequency approximation, the results show that vibration has a stabilizing effect on the onset of convection when the porous layer is heated from below. When the porous layer is cooled from below and heated from above, the vibration has a destabilizing effect in the presence of internal heat generation. It is also demonstrated that when the top and bottoms walls are cooled and rigid/impermeable, the critical Rayleigh number is infinitely large and conduction is the only possible mode of heat transfer. The impact of increasing the Vadasz number is to stabilize the convection, in addition to reducing the transition point from synchronous to subharmonic solutions.

Thermal Instability of Convection in a Rotating Nanofluid Saturated Porous Layer Placed at a Finite Distance From the Axis of Rotation

2016 ◽  
Vol 138 (10) ◽  
Author(s):  
S. Govender
Keyword(s):  
Porous Layer ◽  
Critical Rayleigh Number ◽  
Vertical Axis ◽  
Analytical Investigation ◽  
Onset Of Convection ◽  
Axis Of Rotation ◽  
Offset Distance ◽  
Gravity Effects ◽  
Saturated Porous Layer ◽  
Finite Distance

An analytical investigation for the onset of convection in a vertical porous layer saturated by a nanofluid is presented when the porous layer is placed some finite distance from the axis of rotation. A linear stability analysis is used to determine the convection threshold in terms of the key parameters for the nanofluid. This study reconfirms that the Taylor number and gravity effects are passive, and that the most critical mode is roll cells aligned with the vertical axis of rotation. The critical Rayleigh number is presented in terms of the nanofluid parameters and offset distance for stationary convection.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

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.

Throughflow effects on convective instability in superposed fluid and porous layers

1991 ◽  
Vol 231 ◽  
pp. 113-133 ◽  
Author(s):  
Falin Chen
Keyword(s):  
Porous Layer ◽  
Convective Instability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Vertical Direction ◽  
Single Layer ◽  
Layer Depth ◽  
Onset Of Convection ◽  
Precise Control ◽  
Porous Layers

We implement a linear stability analysis of the convective instability in superposed horizontal fluid and porous layers with throughflow in the vertical direction. It is found that in such a physical configuration both stabilizing and destabilizing factors due to vertical throughflow can be enhanced so that a more precise control of the buoyantly driven instability in either a fluid or a porous layer is possible. For ζ = 0.1 (ζ, the depth ratio, defined as the ratio of the fluid-layer depth to the porous-layer depth), the onset of convection occurs in both fluid and porous layers, the relation between the critical Rayleigh number Rcm and the throughflow strength γm is linear and the Prandtl-number (Prm) effect is insignificant. For ζ ≥ 0.2, the onset of convection is largely confined to the fluid layer, and the relation becomes Rcm ∼ γ2m for most of the cases considered except for Prm = 0.1 with large positive γm where the relation Rcm ∼ γ3m holds. The destabilizing mechanisms proposed by Nield (1987 a, b) due to throughflow are confirmed by the numerical results if considered from the viewpoint of the whole system. Nevertheless, from the viewpoint of each single layer, a different explanation can be obtained.

Numerical simulation of two-dimensional compressible convection

1975 ◽  
Vol 70 (4) ◽  
pp. 689-703 ◽  
Author(s):  
Eric Graham
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Fluid Velocity ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Onset Of Convection ◽  
Local Sound ◽  
Constant Viscosity ◽  
Convection Problem

A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.

Experimental investigation of convective stability in a superposed fluid and porous layer when heated from below

1989 ◽  
Vol 207 ◽  
pp. 311-321 ◽  
Author(s):  
Falin Chen ◽  
C. F. Chen
Keyword(s):  
Porous Layer ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Convective Stability ◽  
Onset Of Convection ◽  
Temperature Distributions ◽  
Critical Wavelength ◽  
Crystal Film ◽  
Flux Curve

Experiments have been carried out in a horizontal superposed fluid and porous layer contained in a test box 24 cm × 12 cm × 4 cm high. The porous layer consisted of 3 mm diameter glass beads, and the fluids used were water, 60% and 90% glycerin-water solutions, and 100% glycerin. The depth ratio ď, which is the ratio of the thickness of the fluid layer to that of the porous layer, varied from 0 to 1.0. Fluids of increasingly higher viscosity were used for cases with larger ď in order to keep the temperature difference across the tank within reasonable limits. The top and bottom walls were kept at different constant temperatures. Onset of convection was detected by a change of slope in the heat flux curve. The size of the convection cells was inferred from temperature measurements made with embedded thermocouples and from temperature distributions at the top of the layer by use of liquid crystal film. The experimental results showed (i) a precipitous decrease in the critical Rayleigh number as the depth of the fluid layer was increased from zero, and (ii) an eightfold decrease in the critical wavelength between ď = 0.1 and 0.2. Both of these results were predicted by the linear stability theory reported earlier (Chen & Chen 1988).

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

1996 ◽  
Vol 326 ◽  
pp. 399-415 ◽  
Author(s):  
M. Wanschura ◽  
H. C. Kuhlmann ◽  
H. J. Rath
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Critical Rayleigh Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Onset Of Convection ◽  
Buoyant Convection ◽  
Thermal Boundary Conditions ◽  
Aspect Ratios ◽  
Prandtl Numbers

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

Thermal Convective Instabilities in a Porous Medium

1984 ◽  
Vol 106 (1) ◽  
pp. 137-142 ◽  
Author(s):  
M. Kaviany
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Temperature Distribution ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Onset Of Convection ◽  
Long Time ◽  
Nonlinear Temperature

The onset of convection due to a nonlinear and time-dependent temperature stratification in a saturated porous medium with upper and lower free surfaces is considered. The initial parabolic temperature distribution is due to uniform internal heating. The medium is then cooled by decreasing the upper surface temperature linearly with time. Linear stability theory is applied to the more formally developed governing equations. In order to obtain an asymptotic solution for transient problems involving very long time scales, the critical Rayleigh number for steady-state, nonlinear temperature distribution is also obtained. The effects of porosity, permeability, and Prandtl number on the time of the onset of convection are examined. The steady-state results show that the critical Rayleigh number depends only on the ratio of porosity to permeability and when this ratio exceeds a value of one thousand, the critical Rayleigh number is directly proportional to this ratio.

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.

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