Onset of thermal convection in a vertical porous cylinder with conducting wall
TL;DR: In this article, the onset of Rayleigh-Benard convection in a finite circular porous cylinder with vertical axis is investigated analytically, where the cylinder is heated from below, and its top and bottom are assumed impermeable and perfectly heat-conducting.
Abstract: The onset of Rayleigh–Benard convection in a finite circular porous cylinder with vertical axis is investigated analytically. The cylinder is heated from below, and its top and bottom are assumed impermeable and perfectly heat-conducting. The impermeable sidewall of the cylinder is assumed to be heat-conducting so that the temperature perturbation there is zero. The eigenvalue problem is split into two different Helmholtz equations. The resulting eigenmode is expressed in terms of Bessel functions. The preferred mode at the onset of convection is found to be axisymmetric. The critical Rayleigh number is a smooth function of the aspect ratio of the cylinder, in contrast to the standard case of an insulating sidewall.
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TL;DR: In this article, the authors considered the onset of convection in anisotropic porous media subject to a rapid change in boundary conditions and developed new analytical results for sedimentary formations where the average vertical permeability is some fraction of the average horizontal permeability.
Abstract: Previous studies of fluid convection in porous media have considered the onset of convection in isotropic systems and the steady convection in anisotropic systems. This paper bridges between these and develops new results for the onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. These results are relevant to sedimentary formations where the average vertical permeability is some fraction γ of the average horizontal permeability. Linear and global stability analyses are used to define the critical time tc at which the instability occurs as a function of γ and the dimensionless Rayleigh-Darcy number Ra* for both thermal and solute-driven convection in an infinite horizontal slab. Numerical results and approximate analytical solutions are obtained for both a slab of finite thickness and the limit of large slab thickness. For a thick slab, the increase in tc as γ decreases is approximately given by (1+γ)4∕(16γ2). One important application is to the geological storage of carbon dioxide where it is shown that the use of an effective vertical permeability in estimating the critical time is only valid for low permeabilities. The time scale for the onset of convection in geological storage can range from less than a year (for high-permeability formations) to decades or centuries (for low-permeability ones).
255 citations
TL;DR: In this article, an analytical study of linear and nonlinear Darcy-Benard convection of Newtonian liquids and Newtonian nanoliquids confined in a cylindrical porous enclosure is made.
Abstract: An analytical study of linear and nonlinear Darcy-Benard convection of Newtonian liquids and Newtonian nanoliquids confined in a cylindrical porous enclosure is made. The effect of concentric insertion of a solid cylinder into the hollow circular cylinder on onset and heat transport is also investigated. An axisymmetric mode is considered, and the Bessel functions are the eigenfunctions for the problem. The two-phase model is used in the case of nanoliquids. Weakly nonlinear stability analysis is performed by considering the double Fourier-Bessel series expansion for velocity, temperature, and nanoparticle concentration fields. Water well-dispersed with copper nanoparticles of very high thermal conductivity, and one of the five different shapes is chosen as the working medium. The thermophysical properties of nanoliquids are calculated using the phenomenological laws and the mixture theory. It is found that the effect of concentric insertion of a solid cylinder into the hollow cylinder is to enhance the heat transport. The results of rectangular enclosures are obtained as limiting cases of the present study. In general, curvature enhances the heat transport and hence the heat transport is maximum in the case of a cylindrical annulus followed by that in cylindrical and rectangular enclosures. Among the five different shapes of nanoparticles, blade-shaped nanoparticles help transport maximum heat. An analytical expression is obtained for the Hopf bifurcation point in the cases of the fifth-order and the third-order Lorenz models. Regular, chaotic, mildly chaotic, and periodic behaviors of the Lorenz system are discussed using plots of the maximum Lyapunov exponent and the bifurcation diagram.
30 citations
TL;DR: In this article, the onset of thermal convection in a vertical porous cylinder in three dimensions is investigated analytically, and the convection problem is solved for a cylinder wall that is partly conducting and partly penetrative.
Abstract: The onset of thermal convection in a vertical porous cylinder in three dimensions is investigated analytically. Top and bottom of the cylinder are set to be perfectly heat conducting and impermeable, and is uniformly heated from below. The convection problem is solved for a cylinder wall that is partly conducting and partly penetrative. The expressions for semi-conduction and semi-penetration are based on a porous medium separated from its surroundings by a thin wall. The eigenvalue problem is split into two Helmholtz equations, and the results are expressed by Bessel functions in the radial direction. Comparisons are made with existing solutions for the limit cases of a closed cylinder wall that is either conducting or insulating. Two different models are compared for the kinematic limit condition of an open boundary.
19 citations
TL;DR: In this article, the onset of thermal convection in a 2D porous box is investigated analytically, where the lateral walls are partly heat conducting and partly penetrative, and the linear stability problem is solved only for the symmetric configuration of equal conditions at each sidewall.
Abstract: The onset of thermal convection in a 2D porous box is investigated analytically. The lateral walls are partly heat conducting and partly penetrative. The top and bottom are impermeable and perfect heat conductors. The linear stability problem is solved only for the symmetric configuration of equal conditions at each sidewall. The problem is degenerate when the parameters of semi-conduction and semi-penetration coincide. The degenerate problem has one symmetric and one antisymmetric eigenfunctions, and the cell width varies with minimum cell width in the middle. Our primary model for the partly penetrative wall is a thin and highly permeable layer near a closed wall. We also study a secondary model of a partly penetrative wall, with a thin layer of small permeability near a hydrostatic reservoir.
18 citations
Cites methods from "Onset of thermal convection in a ve..."
...Haugen and Tyvand (2003) followed up the 2D analysis by Nilsen and Storesletten (1990) and solved the onset problem for a circular cylinder with conducting and impermeable cylinder walls....
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TL;DR: In this article, the authors investigated the stability of three-dimensional natural convection in an annular porous cavity contained between two vertical coaxial cylinders, using a linear stability analysis and high-order numerical simulations using pseudospectral methods to model the nonlinear regime.
Abstract: We uncover novel features of three-dimensional natural convection in porous media by investigating convection in an annular porous cavity contained between two vertical coaxial cylinders. The investigations are made using a linear stability analysis, together with high-order numerical simulations using pseudospectral methods to model the nonlinear regime. The onset of convection cells and their preferred planform are studied, and the stability of the modes with respect to different types of perturbation is investigated. We also examine how variations in the Rayleigh number affect the convection modes and their stability regimes. Compared with previously published data, we show how the problem inherits an increased complexity regarding which modes will be obtained. Some stable secondary modes or mixed modes have been identified and some overlapping stability regions for different convective modes are determined.
18 citations
Cites background or result from "Onset of thermal convection in a ve..."
...7: Variation of Rac with R for (i) circular cylinder with an insulated sidewall ([33], lowest curve), (ii) circular cylinder with a perfectly conducting sidewall ([13], middle curve), (iii) annular cylinder with perfectly conducting sidewalls and with Rw = 10−4....
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...Haugen and Tyvand [13], who considered a circular cylinder of porous medium with a perfectly heat conducting sidewall, showed that the critical Rayleigh number is a monotonically decreasing function of R and it decreases towards 4π when R → ∞....
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...(The figure is adapted from Figure 1 in [13])....
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...Both [13] and [25] showed that a conducting sidewall does not correspond to a natural cell boundary, and that cells near such a boundary are wider than their insulating sidewall counterparts....
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...of a circular cylinder with perfectly conducting boundaries was considered by Haugen and Tyvand [13], who also compared their findings with Zebib’s results....
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References
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Book•
01 Jan 1992TL;DR: In this paper, an introduction to convection in porous media assumes the reader is familiar with basic fluid mechanics and heat transfer, going on to cover insulation of buildings, energy storage and recovery, geothermal reservoirs, nuclear waste disposal, chemical reactor engineering and the storage of heat-generating materials like grain and coal.
Abstract: This introduction to convection in porous media assumes the reader is familiar with basic fluid mechanics and heat transfer, going on to cover insulation of buildings, energy storage and recovery, geothermal reservoirs, nuclear waste disposal, chemical reactor engineering and the storage of heat-generating materials like grain and coal. Geophysical applications range from the flow of groundwater around hot intrusions to the stability of snow against avalanches. The book is intended to be used as a reference, a tutorial work or a textbook for graduates.
5,570 citations
01 Oct 1948
TL;DR: In this article, it was shown that under certain conditions convective flow may occur in fluid which permeates a porous stratum and is subject to a vertical temperature gradient, on the assumption that the flow obeys Darcy's law.
Abstract: It is shown that under certain conditions convective flow may occur in fluid which permeates a porous stratum and is subject to a vertical temperature gradient, on the assumption that the flow obeys Darcy's law. The criterion for marginal stability is obtained for three sets of boundary conditions, and the motion described. If such convection occurs in a stratum through which a bore-hole passes, the usual method of calculation of the heat flow must be modified, but in general the correction will not be large.
1,234 citations
TL;DR: In this paper, it was shown that the minimum temperature gradient for which convection can occur is approximately 4π2h2μ/kgρ0α D2, where h2 is the thermal diffusivity, g is the acceleration of gravity, μ is the viscosity, k is the permeability, α is the coefficient of cubical expansion, ρ 0 is the density at zero temperature, and D is the thickness of the layer; this exceeds the limiting gradient found by Rayleigh for a simple fluid by a factor of 16D2/27π2
Abstract: The problem is considered of the convection of a fluid through a permeable medium as the result of a vertical temperature‐gradient, the medium being in the shape of a flat layer bounded above and below by perfectly conducting media. It appears that the minimum temperature‐gradient for which convection can occur is approximately 4π2h2μ/kgρ0α D2, where h2 is the thermal diffusivity, g is the acceleration of gravity, μ is the viscosity, k is the permeability, α is the coefficient of cubical expansion, ρ0 is the density at zero temperature, and D is the thickness of the layer; this exceeds the limiting gradient found by Rayleigh for a simple fluid by a factor of 16D2/27π2kρ0. A numerical computation of this gradient, based upon the data now available, indicates that convection currents should not occur in such a geological formation as the Woodbine sand of East Texas (west of the Mexia Fault zone); in view of the fact, however, that the distribution of NaCl in this formation seems to require the existence of ...
796 citations
TL;DR: In this article, the problem of the onset of convection, induced by buoyancy effects resulting from vertical thermal and solute concentration gradients, in a horizontal layer of a saturated porous medium, is treated by linear perturbation analysis.
Abstract: The problem of the onset of convection, induced by buoyancy effects resulting from vertical thermal and solute concentration gradients, in a horizontal layer of a saturated porous medium, is treated by linear perturbation analysis. It is shown that oscillatory instability may be possible when a strongly stabilizing solute gradient is opposed by a destabilizing thermal gradient, but attention is concentrated on monotonic instability. The eigenvalue equation, which involves a thermal Rayleigh number R and an analogous solute Rayleigh number S, is obtained, by a Fourier series method, for a general set of boundary conditions. Numerical solutions are found for some special limiting cases, extending existing results for the thermal problem. When the thermal and solute boundary conditions are formally identical, the net destabilizing effect is expressed by the sum of R and S.
588 citations