An Analytical Study on Natural Convection in Isotropic and Anisotropic Porous Channels

1990 ◽  
Vol 112 (2) ◽  
pp. 396-401 ◽  
Author(s):  
T. Nilsen ◽  
L. Storesletten
Keyword(s):  
Analytical Study ◽  
Vertical Direction ◽  
Flow Patterns ◽  
Heat Conducting ◽  
Two Dimensional ◽  
Anisotropic Porous Media ◽  
Onset Of Convection ◽  
Linear Temperature ◽  
Weakly Nonlinear ◽  
Rayleigh Numbers

This paper is an analytical study on natural two-dimensional convection in horizontal rectangular channels filled by isotropic and anisotropic porous media. The channel walls, assumed to be impermeable and perfectly heat conducting, are nonuniformly heated to establish a linear temperature distribution in the vertical direction. We derive the critical Rayleigh numbers for the onset of convection and examine the steady flow patterns at moderately supercritical Rayleigh numbers. The stability properties of these flow patterns are examined against two-dimensional perturbations using a weakly nonlinear theory.

Numerical simulation of thermal convection in a two-dimensional finite box

1989 ◽  
Vol 199 ◽  
pp. 1-28 ◽  
Author(s):  
Isaac Goldhirsch ◽  
Richard B. Pelz ◽  
Steven A. Orszag
Keyword(s):  
Prandtl Number ◽  
Flow Patterns ◽  
Chaotic Regime ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Chaotic Flows ◽  
Chaotic Flow ◽  
Rayleigh Numbers ◽  
Pseudo Spectral ◽  
Rayleigh Benard

The problems of dynamical onset of convection, textural transitions and chaotic dynamics in a two-dimensional, rectangular Rayleigh-Bénard system have been investigated using well-resolved, pseudo-spectral simulations. All boundary conditions are taken to be no-slip. It is shown that the process of creating the temperature gradient in the system, is responsible for roll creation at the side boundaries. These rolls either induce new rolls or move into the interior of the cell, depending on the rate of heating. Complicated flow patterns and textural transitions are observed in both non-chaotic and chaotic flow regimes. Multistability is frequently observed. Intermediate-Prandtl-number fluids (e.g. 0.71) have a quasiperiodic time dependence up to Rayleigh numbers of order 106. When the Prandtl number is raised to 6.8, one observes aperiodic (chaotic) flows of non-integer dimension. In this case roll merging and separation is observed to be an important feature of the dynamics. In some cases corner rolls are observed to migrate into the interior of the cell and to grow into regular rolls; the large rolls may shrink and retreat into corners. The basic flow patterns observed do not change qualitatively when the chaotic regime is entered.

Numerical Study of Thermal Convection in Boussinesq-Stokes Suspension Occupying a Rectangular Box

2011 ◽  
Author(s):  
S. Manjunath ◽  
N. P. Chandrashekara
Keyword(s):  
Aspect Ratio ◽  
Thermal Convection ◽  
Couple Stress ◽  
Numerical Study ◽  
Critical Rayleigh Number ◽  
Vertical Direction ◽  
Heat Conducting ◽  
Linear Temperature ◽  
Vertical Walls ◽  
Vertical Cavity

This paper is a Fourier–series assisted numerical study of two-dimensional thermal convection in Boussinesq–Stokes suspensions occupying a cavity. The suspension is modeled as a couple stress liquid. The horizontal walls of the cavity are assumed to be perfectly heat conducting and the vertical walls are non-uniformly heated to establish a linear temperature in the vertical direction. The critical Rayleigh number is obtained numerically as a function of couple stress parameter and aspect ratio, and the same is plotted graphically. The results of slender vertical cavity, classical Rayleigh-Be´nard convection, rectangular and square cavities of finite aspect-ratio heated from below are obtained as limiting cases of the study.

Characteristics of high Rayleigh number two-dimensional convection in an open-top porous layer heated from below

1999 ◽  
Vol 394 ◽  
pp. 241-260 ◽  
Author(s):  
ABDELLAH S. M. CHERKAOUI ◽  
WILLIAM S. D. WILCOCK
Keyword(s):  
Rayleigh Number ◽  
Control Volume ◽  
Flow Patterns ◽  
Convective System ◽  
Two Dimensional ◽  
Simpler Algorithm ◽  
Convective Pattern ◽  
Volume Method ◽  
Rayleigh Numbers ◽  
Type Of Transition

Using a control-volume method and the simpler algorithm, we computed steady-state and time-dependent solutions for two-dimensional convection in an open-top porous box, up to a Rayleigh number of 1100. The evolution of the convective system from onset to high Rayleigh numbers is characterized by two types of transitions in the flow patterns. The first type is a decrease in the horizontal aspect ratio of the cells. We observe two such bifurcations. The first occurs at Ra = 107.8 when the convective pattern switches from a steady one-cell roll to a steady two-cell roll. The second occurs at Ra ≈ 510 when an unsteady two-cell roll evolves to a steady four-cell roll. The second type of transition is from a steady to an unsteady pattern and we also observe two of these bifurcations. The first occurs at Ra ≈ 425 in the two-cell convective pattern; the second at Ra ≈ 970 in the four-cell pattern. Both types of bifurcations are associated with an increase in the average vertical convective heat transport. In the bi-cellular solutions, the appearance of non-periodic unsteady convection corresponds to the onset of the expected theoretical scaling Nu ∝ Ra and also to the onset of plume formation. Although our highest quadri-cellular solutions show signs of non-periodic convection, they do not reach the onset of plume formation. An important hysterisis loop exists for Rayleigh numbers in the range 425–505. Unsteady convection appears only in the direction of increasing Rayleigh numbers. In the decreasing direction, steady quadri-cellular flow patterns prevail.

Asymptotic theory of convection in a rotating, cylindrical annulus

1986 ◽  
Vol 173 ◽  
pp. 545-556 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Equatorial Plane ◽  
Asymptotic Theory ◽  
Rossby Waves ◽  
Numerical Results ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Rotation Rates ◽  
Cylindrical Annulus ◽  
Simple Equations ◽  
Rayleigh Numbers

The problem of convection in a rotating cylindrical annulus heated from the outside and cooled from the inside is considered in the limit of high rotation rates. The constraint of rotation enforces the two-dimensional character of the motion when the angle of inclination of the axisymmetric end surfaces with respect to the equatorial plane is small. Even when the angle of inclination is large only the dependences on the radial and the azimuthal coordinates need to be considered. The dependence on time at the onset of convection is similar to that of Rossby waves. But at higher Rayleigh numbers a transition to vacillating solutions occurs. In the limit of high rotation rates simple equations can be derived which permit the reproduction and extension of previous numerical results.

The Onset of Convection in a Porous Medium Occupying an Enclosure of Variable Width or Height

2007 ◽  
Vol 129 (12) ◽  
pp. 1714-1718 ◽  
Author(s):  
D. A. Nield ◽  
A. V. Kuznetsov
Keyword(s):  
Porous Medium ◽  
Analytical Study ◽  
Saturated Porous Medium ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Stabilizing Effect

An analytical study is made of the onset of convection in a saturated porous medium occupying a two-dimensional enclosure of uniform height, but whose width is slowly varying in an arbitrary manner, or one of uniform width, but whose height is slowly varying in an arbitrary manner. It is found that the variation of width generally has a stabilizing effect, whereas variation of height generally has a destabilizing effect.

Finite amplitude two-dimensional convection in a finite rotating system

1978 ◽  
Vol 363 (1713) ◽  
pp. 195-215 ◽  
Keyword(s):  
Multiple Scales ◽  
Horizontal Stress ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Rotating System ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Aspect Ratios ◽  
Rayleigh Numbers

This paper considers the effect of rotation, measured by a Taylor number T , on two-dimensional Bénard convection between horizontal stress-free boundaries which are maintained at different constant temperatures. The fluid is confined laterally by rigid sidewalls which are assumed only approximately insulating, the possibility of small lateral heat losses, which are observed experimentally, being incorporated in the theory. The distance between the sidewalls is 2 L times the height of the layer. A weakly nonlinear theory based on the method of multiple scales is de­veloped to describe the motion for slightly supercritical Rayleigh numbers R , and large aspect ratios ( L ≫ 1), although the results are also valid for finite values of L if the speed of rotation is large ( T ≫ 1). In the exchange case a steady finite amplitude solution evolves if the Prandtl number of the fluid σ is greater than 0.577, but subcritical instability and bursting can occur for a certain range of Taylor numbers if σ < 0.577. In the over­stable case disturbances propagate between the sidewalls, and ultimately either decay or, for Rayleigh numbers greater than a critical value depending on both σ and T , attain an equilibrium state controlled by reflexion at the sidewalls.

An analytical study on three-dimensional versus two-dimensional water table-induced flow patterns in a Tóthian basin

2017 ◽  
Vol 31 (22) ◽  
pp. 4006-4018 ◽  
Author(s):  
Jun-Zhi Wang ◽  
Xiao-Wei Jiang ◽  
Zhi-Yuan Zhang ◽  
Li Wan ◽  
Xu-Sheng Wang ◽  
...  
Keyword(s):  
Water Table ◽  
Analytical Study ◽  
Three Dimensional ◽  
Flow Patterns ◽  
Two Dimensional ◽  
Induced Flow

Nonhomogeneous Porosity and Thermal Diffusivity Effects on a Double-Diffusive Convection in Anisotropic Porous Media

2016 ◽  
Vol 17 (5) ◽  
pp. 205-220 ◽  
Author(s):  
Akil J. Harfash
Keyword(s):  
Porous Media ◽  
Thermal Diffusivity ◽  
Three Dimensional ◽  
Vertical Direction ◽  
Double Diffusive Convection ◽  
Anisotropic Porous Media ◽  
Onset Of Convection ◽  
Diffusive Convection ◽  
Dimensional Simulation ◽  
Double Diffusive

AbstractA model for double-diffusive convection in anisotropic and inhomogeneous porous media has been analysed. In particular, the effects of variable permeability, thermal diffusivity and variable gravity with respect to the vertical direction, have been studied. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using three dimensional simulation. Our results show that the linear theory produce a good predicts on the onset of instability in the basic steady state. It is known that as${R_c}$increases the onset of convection is more likely to be via oscillatory convection as opposed to steady convection, and the three dimensional simulation results show that as$Rc$increases, the actual threshold moving toward the nonlinear stability threshold and the behaviour of the perturbation of the solutions becomes more oscillated.

A simplified numerical and analytical study for assessing the seismic response of a gravity concrete lock

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Neander Berto Mendes ◽  
Lineu José Pedroso ◽  
Paulo Marcelo Vieira Ribeiro
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Analytical Study ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Fluid Structure ◽  
Analytical Formulation ◽  
Acoustic Cavity ◽  
Water Chamber ◽  
Structure Problem

ABSTRACT: This work presents the dynamic response of a lock subjected to the horizontal S0E component of the El Centro earthquake for empty and completely filled water chamber cases, by coupled fluid-structure analysis. Initially, the lock was studied by approximation, considering it similar to the case of a double piston coupled to a two-dimensional acoustic cavity (tank), representing a simplified analytical model of the fluid-structure problem. This analytical formulation can be compared with numerical results, in order to qualify the responses of the ultimate problem to be investigated. In all the analyses performed, modeling and numerical simulations were done using the finite element method (FEM), supported by the commercial software ANSYS.

scholarly journals Onset of Convection in Two-Dimensional Porous Cavities with Open and Conducting Boundaries

2021 ◽  
Vol 136 (3) ◽  
pp. 791-812
Author(s):  
Peder A. Tyvand ◽  
Jonas Kristiansen Nøland
Keyword(s):  
Critical Rayleigh Number ◽  
Numerical Results ◽  
Temperature Perturbation ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Order Problem ◽  
Fourth Order Problem ◽  
Thermally Conducting ◽  
Perturbation Pressure ◽  
Special Case

AbstractThe onset of thermal convection in two-dimensional porous cavities heated from below is studied theoretically. An open (constant-pressure) boundary is assumed, with zero perturbation temperature (thermally conducting). The resulting eigenvalue problem is a full fourth-order problem without degeneracies. Numerical results are presented for rectangular and elliptical cavities, with the circle as a special case. The analytical solution for an upright rectangle confirms the numerical results. Streamlines penetrating the open cavities are plotted, together with the isotherms for the associated closed thermal cells. Isobars forming pressure cells are depicted for the perturbation pressure. The critical Rayleigh number is calculated as a function of geometric parameters, including the tilt angle of the rectangle and ellipse. An improved physical scaling of the Darcy–Bénard problem is suggested. Its significance is indicated by the ratio of maximal vertical velocity to maximal temperature perturbation.

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