Convective Instability in Porous Media

A special issue of Fluids (ISSN 2311-5521).

Deadline for manuscript submissions: closed (31 August 2017) | Viewed by 56273

Special Issue Editors


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Guest Editor
Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK
Interests: convection; porous media; instability; numerical simulation; asymptotic analysis; non-Newtonian fluids
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Guest Editor
Department of Industrial Engineering, Alma Mater Studiorum Università di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy
Interests: stability analysis of convection flows in porous media; convection and instability of dissipative and non-Newtonian fluid flows; new features for the instability of shear flows.
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Many practical problems involving porous media in engineering, geophysics and CO2 sequestration involve the simulation of what might be termed convection in a very wide sense. Such instabilities are always brought about by nonuniform buoyancy forces due to density changes which, in turn, have arisen because of variations in the temperature and/or chemical composition of the fluid, or by miscible or immiscible displacements of heavier fluids by lighter fluids. The aim of this Special Issue is to collect together a wide variety of papers which have, as their unifying theme, the onset and subsequent development of convective instability. We intend there to be a strong emphasis on the methodology of solution, for reasons of pedagogy, for both analytical and numerical methods. Papers which involve those variants of Darcy’s law for which there is no formal support (i.e., REV averaging or experimental validation), will not feature in this issue. Numerical accuracy will be of paramount importance.

Dr. D. Andrew S. Rees
Prof. Dr. Antonio Barletta
Guest Editors

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Keywords

  • porous media
  • instability
  • numerical simulation
  • Darcy-Bénard convection
  • boundary layers
  • fingering
  • bifurcations

Published Papers (14 papers)

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Research

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2620 KiB  
Article
Onset of Convection in the Presence of a Precipitation Reaction in a Porous Medium: A Comparison of Linear Stability and Numerical Approaches
by Parama Ghoshal, Min Chan Kim and Silvana S. S. Cardoso
Fluids 2018, 3(1), 1; https://doi.org/10.3390/fluids3010001 - 22 Dec 2017
Cited by 5 | Viewed by 3736
Abstract
Reactive convection in a porous medium has received recent interest in the context of the geological storage of carbon dioxide in saline formations. We study theoretically and numerically the gravitational instability of a diffusive boundary layer in the presence of a first-order precipitation [...] Read more.
Reactive convection in a porous medium has received recent interest in the context of the geological storage of carbon dioxide in saline formations. We study theoretically and numerically the gravitational instability of a diffusive boundary layer in the presence of a first-order precipitation reaction. We compare the predictions from normal mode, linear stability analysis, and nonlinear numerical simulations, and discuss the relative deviations. The application of our findings to the storage of carbon dioxide in a siliciclastic aquifer shows that while the reactive-diffusive layer can become unstable within a timescale of 1 to 1.5 months after the injection of carbon dioxide, it can take almost 10 months for sufficiently vigorous convection to produce a considerable increase in the dissolution flux of carbon dioxide. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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1308 KiB  
Article
Horizontal Cellular Oscillations Caused by Time-Periodic Resonant Thermal Forcing in Weakly Nonlinear Darcy-Bénard Convection
by Ibrahim M. Jais and D. Andrew S. Rees
Fluids 2017, 2(4), 60; https://doi.org/10.3390/fluids2040060 - 04 Nov 2017
Viewed by 3142
Abstract
The onset of Rayleigh-Bénard convection in a horizontally unbounded saturated porous medium is considered. Particular attention is given to the stability of weakly nonlinear convection between two plane horizontal surfaces heated from below. The primary aim is to study the effects on postcritical [...] Read more.
The onset of Rayleigh-Bénard convection in a horizontally unbounded saturated porous medium is considered. Particular attention is given to the stability of weakly nonlinear convection between two plane horizontal surfaces heated from below. The primary aim is to study the effects on postcritical convection of having small amplitude time-periodic resonant thermal forcing. Amplitude equations are derived using a weakly nonlinear theory and they are solved in order to understand how the flow evolves with changes in the Darcy-Rayleigh number and the forcing frequency. When convection is stationary in space, it is found to consist of one of two different types depending on its location in parameter space: either a convection pattern where each cell rotates in the same way for all time with a periodic variation in amplitude (Type I) or a pattern where each cell changes direction twice within each forcing period (Type II). Asymptotic analyses are also performed (i) to understand the transition between convection of types I and II; (ii) for large oscillation frequencies and (iii) for small oscillation frequencies. In a large part of parameter space the preferred pattern of convection when the layer is unbounded horizontally is then shown to be one where the cells oscillate horizontally—this is a novel form of pattern selection for Darcy-Bénard convection. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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767 KiB  
Article
Convective Flow in an Aquifer Layer
by Dambaru Bhatta and Daniel Riahi
Fluids 2017, 2(4), 52; https://doi.org/10.3390/fluids2040052 - 08 Oct 2017
Cited by 4 | Viewed by 3613
Abstract
Here, we investigate weakly nonlinear hydrothermal two-dimensional convective flow in a horizontal aquifer layer with horizontal isothermal and rigid boundaries. We treat such a layer as a porous medium, where Darcy’s law holds, subjected to the conditions that the porous layer’s permeability and [...] Read more.
Here, we investigate weakly nonlinear hydrothermal two-dimensional convective flow in a horizontal aquifer layer with horizontal isothermal and rigid boundaries. We treat such a layer as a porous medium, where Darcy’s law holds, subjected to the conditions that the porous layer’s permeability and the thermal conductivity are variable in the vertical direction. This analysis is restricted to the case that the subsequent hydraulic resistivity and diffusivity have a small rate of change with respect to the vertical variable. Applying the weakly nonlinear approach, we derive various order systems and express their solutions. The solutions for convective flow quantities such as vertical velocity and the temperature that arise as the Rayleigh number exceeds its critical value are computed and presented in graphical form. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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233 KiB  
Article
Modelling Bidispersive Local Thermal Non-Equilibrium Flow
by Franca Franchi, Roberta Nibbi and Brian Straughan
Fluids 2017, 2(3), 48; https://doi.org/10.3390/fluids2030048 - 18 Sep 2017
Cited by 10 | Viewed by 2924
Abstract
In this work, we present a system of equations which describes non-isothermal flow in a bidispersive porous medium under conditions of local thermal non-equilibrium. The porous medium consists of macro pores, and in the solid skeleton are cracks or fissures which give rise [...] Read more.
In this work, we present a system of equations which describes non-isothermal flow in a bidispersive porous medium under conditions of local thermal non-equilibrium. The porous medium consists of macro pores, and in the solid skeleton are cracks or fissures which give rise to micro pores. The temperatures in the solid skeleton and in the fluids in the macro and micro pores are all allowed to be independent. After presenting the general model, we derive a result of universal stability, which guarantees exponential decay of the solution for all initial data. We further present a concrete example by specializing the model to the problem of thermal convection in a layer heated from below. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
339 KiB  
Article
Thermal Convection in a Rotating Anisotropic Fluid Saturated Darcy Porous Medium
by Shatha Haddad
Fluids 2017, 2(3), 44; https://doi.org/10.3390/fluids2030044 - 21 Aug 2017
Cited by 10 | Viewed by 2829
Abstract
The stability of the thermal convection in a fluid-saturated rotating anisotropic porous material is investigated. We take into account the rotation of a layer of saturated porous medium about an axis orthogonal to the planes bounding the layer. The permeability is allowed to [...] Read more.
The stability of the thermal convection in a fluid-saturated rotating anisotropic porous material is investigated. We take into account the rotation of a layer of saturated porous medium about an axis orthogonal to the planes bounding the layer. The permeability is allowed to be an anisotropic tensor. In particular, we restrict our attention to the case where the permeability in the vertical direction is different to that in the horizontal plane. The linear instability and nonlinear stability analysis, in the case where the inertial term vanishes, are performed. It is shown, by using an energy method, that the nonlinear critical Rayleigh numbers coincide with those of the linear analysis. The results reveal that the system becomes more stable when the rotation is present. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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418 KiB  
Article
Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux
by Abdoulaye Gueye, Mohamed Najib Ouarzazi, Silvia C. Hirata and Haikel Ben Hamed
Fluids 2017, 2(3), 42; https://doi.org/10.3390/fluids2030042 - 22 Jul 2017
Cited by 5 | Viewed by 4037
Abstract
We analyze the thermal convection thresholds and linear characteristics of the primary and secondary instabilities for viscoelastic fluids saturating a porous horizontal layer heated from below by a constant flux. The Galerkin method is used to solve the eigenvalue problem by taking into [...] Read more.
We analyze the thermal convection thresholds and linear characteristics of the primary and secondary instabilities for viscoelastic fluids saturating a porous horizontal layer heated from below by a constant flux. The Galerkin method is used to solve the eigenvalue problem by taking into account the elasticity of the fluid, the ratio between the viscosity of the solvent and the total viscosity of the fluid and the lateral confinement of the medium. For the primary instability, we found out that depending on the rheological parameters, two types of convective structures may appear when the basic conductive solution loses its stability: stationary long wavelength instability as for Newtonian fluids and oscillatory convection. The effect of the lateral confinement of the porous medium by adiabatic walls is to stabilize the oblique and longitudinal rolls and therefore selects transverse rolls at the onset of convection. In the range of the rheological parameters where stationary long wave instability develops first, we use a parallel flow approximation to determine analytically the velocity and temperature fields associated with the monocellular convective flow. The linear stability analysis of the monocellular flow is performed, and the critical conditions above which the flow becomes unstable are determined. The combined influence of the viscoelastic parameters and the lateral confinement on the characteristics of the secondary instability is quantified. The major new findings concerning the secondary instabilities may be summarized as follows: (i) For concentrated viscoelastic fluids, computations showed that the most amplified mode of convection corresponds to oscillatory transverse rolls, which appears via a Hopf bifurcation. This pattern selection is independent of both the fluid elasticity and the lateral confinement of the porous medium. (ii) For diluted viscoelastic fluids, the preferred mode of convection is found to be oscillatory transverse rolls for a very laterally-confined medium. Otherwise, stationary or oscillatory longitudinal rolls may develop depending on the fluid elasticity. Results also showed the destabilizing effect of the relaxation fluid elasticity and the stabilizing effect of the viscosity ratio for the onset of both primary and secondary instabilities. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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1449 KiB  
Article
Interaction of the Longwave and Finite-Wavelength Instability Modes of Convection in a Horizontal Fluid Layer Confined between Two Fluid-Saturated Porous Layers
by Tatyana P. Lyubimova and Igor D. Muratov
Fluids 2017, 2(3), 39; https://doi.org/10.3390/fluids2030039 - 16 Jul 2017
Cited by 14 | Viewed by 3282
Abstract
The onset of convection in a three-layer system consisting of two fluid-saturated porous layers separated by a homogeneous fluid layer is studied. It is shown that both a longwave convective regime developing in the whole system and a finite-wavelength regime of convection concentrated [...] Read more.
The onset of convection in a three-layer system consisting of two fluid-saturated porous layers separated by a homogeneous fluid layer is studied. It is shown that both a longwave convective regime developing in the whole system and a finite-wavelength regime of convection concentrated in the homogeneous fluid layer are possible. Due to the hydraulic resistance of the porous matrix, the flow intensity in the longwave convective regime is much lower than that in the finite-wavelength regime. Moreover, it grows at a much slower pace with the increase of the Grashof number. Because of that, the long-wave convective regime becomes unstable at small supercriticalities and is replaced by a finite-wavelength regime. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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5200 KiB  
Article
Convective to Absolute Instability Transition in a Horizontal Porous Channel with Open Upper Boundary
by Antonio Barletta and Michele Celli
Fluids 2017, 2(2), 33; https://doi.org/10.3390/fluids2020033 - 14 Jun 2017
Cited by 9 | Viewed by 3445
Abstract
A linear stability analysis of the parallel uniform flow in a horizontal channel with open upper boundary is carried out. The lower boundary is considered as an impermeable isothermal wall, while the open upper boundary is subject to a uniform heat flux and [...] Read more.
A linear stability analysis of the parallel uniform flow in a horizontal channel with open upper boundary is carried out. The lower boundary is considered as an impermeable isothermal wall, while the open upper boundary is subject to a uniform heat flux and it is exposed to an external horizontal fluid stream driving the flow. An eigenvalue problem is obtained for the two-dimensional transverse modes of perturbation. The study of the analytical dispersion relation leads to the conditions for the onset of convective instability as well as to the determination of the parametric threshold for the transition to absolute instability. The results are generalised to the case of three-dimensional perturbations. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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2124 KiB  
Article
Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media
by Vandita Sharma, Satyajit Pramanik and Manoranjan Mishra
Fluids 2017, 2(2), 32; https://doi.org/10.3390/fluids2020032 - 11 Jun 2017
Cited by 9 | Viewed by 4698
Abstract
Viscous fingering is ubiquitous in miscible displacements in porous media, in particular, oil recovery, contaminant transport in aquifers, chromatography separation, and geological CO2 sequestration. The viscosity contrasts between heavy oil and water is several orders of magnitude larger than typical viscosity contrasts [...] Read more.
Viscous fingering is ubiquitous in miscible displacements in porous media, in particular, oil recovery, contaminant transport in aquifers, chromatography separation, and geological CO2 sequestration. The viscosity contrasts between heavy oil and water is several orders of magnitude larger than typical viscosity contrasts considered in the majority of the literature. We use the finite element method (FEM)-based COMSOL Multiphysics simulator to simulate miscible displacements in homogeneous porous media with very large viscosity contrasts. Our numerical model is suitable for a wide range of viscosity contrasts covering chromatographic separation as well as heavy oil recovery. We have successfully captured some interesting and previously unexplored dynamics of miscible blobs with very large viscosity contrasts in homogeneous porous media. We study the effect of viscosity contrast on the spreading and the degree of mixing of the blob. Spreading (variance of transversely averaged concentration) follows the power law t 3 . 34 for the blobs with viscosity O ( 10 2 ) and higher, while degree of mixing is found to vary non-monotonically with log-mobility ratio. Moreover, in the limit of very large viscosity contrast, the circular blob behaves like an erodible solid body and the degree of mixing approaches the viscosity-matched case. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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447 KiB  
Article
Hexagonal Cell Formation in Darcy–Bénard Convection with Viscous Dissipation and Form Drag
by D. Andrew S. Rees and Eugen Magyari
Fluids 2017, 2(2), 27; https://doi.org/10.3390/fluids2020027 - 20 May 2017
Cited by 5 | Viewed by 3301
Abstract
Recent interest in the effects of viscous dissipation on convective flows in porous media has centred almost exclusively on forced convection flows. In this paper, we investigate the manner in which it affects the onset and early stages of convection in Darcy–Bénard convection. [...] Read more.
Recent interest in the effects of viscous dissipation on convective flows in porous media has centred almost exclusively on forced convection flows. In this paper, we investigate the manner in which it affects the onset and early stages of convection in Darcy–Bénard convection. A weakly nonlinear theory is described briefly, and it is shown that hexagonal cells are preferred over rolls when the Rayleigh number is sufficiently close to 4 π 2 . At higher Rayleigh numbers, two-dimensional rolls are preferred. When weak form drag is included, then subcritical convection eventually disappears as the Forchheimer parameter increases, yielding a highly novel situation wherein hexagonal convection arises supercritically. The range of stability of hexagons is found to increase. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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659 KiB  
Article
Linear Stability Analysis of Penetrative Convection via Internal Heating in a Ferrofluid Saturated Porous Layer
by Amit Mahajan, Sunil and Mahesh Kumar Sharma
Fluids 2017, 2(2), 22; https://doi.org/10.3390/fluids2020022 - 04 May 2017
Cited by 4 | Viewed by 4162
Abstract
Penetrative convection due to purely internal heating in a horizontal ferrofluid-saturated porous layer is examined by performing linear stability analysis. Four different types of heat supply functions are considered. The Darcy model is used to incorporate the effect of the porous medium. Numerical [...] Read more.
Penetrative convection due to purely internal heating in a horizontal ferrofluid-saturated porous layer is examined by performing linear stability analysis. Four different types of heat supply functions are considered. The Darcy model is used to incorporate the effect of the porous medium. Numerical solutions are obtained by using the Chebyshev pseudospectral method, and the results are discussed for all three boundary conditions: when both boundaries are impermeable and conducting; when both boundaries are conducting with lower boundary impermeable and free upper boundary; and when both boundaries are impermeable with lower boundary conducting and upper with constant heat flux. The effect of the Langevin parameter, width of ferrofluid layer, permeability parameter, and nonlinearity of the fluid magnetization has been observed at the onset of penetrative convection for water- and ester-based ferrofluids. It is seen that the Langevin parameter, width of ferrofluid layer, and permeability parameter have stabilizing effects on the onset of convection, while the nonlinearity of the fluid magnetization advances the onset of convection. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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1698 KiB  
Article
The Onset of Convection in an Unsteady Thermal Boundary Layer in a Porous Medium
by Biliana Bidin and D. Andrew S. Rees
Fluids 2016, 1(4), 41; https://doi.org/10.3390/fluids1040041 - 08 Dec 2016
Cited by 4 | Viewed by 3532
Abstract
In this study, the linear stability of an unsteady thermal boundary layer in a semi-infinite porous medium is considered. This boundary layer is induced by varying the temperature of the horizontal boundary sinusoidally in time about the ambient temperature of the porous medium; [...] Read more.
In this study, the linear stability of an unsteady thermal boundary layer in a semi-infinite porous medium is considered. This boundary layer is induced by varying the temperature of the horizontal boundary sinusoidally in time about the ambient temperature of the porous medium; this mimics diurnal heating and cooling from above in subsurface groundwater. Thus if instability occurs, this will happen in those regions where cold fluid lies above hot fluid, and this is not necessarily a region that includes the bounding surface. A linear stability analysis is performed using small-amplitude disturbances of the form of monochromatic cells with wavenumber, k. This yields a parabolic system describing the time-evolution of small-amplitude disturbances which are solved using the Keller box method. The critical Darcy-Rayleigh number as a function of the wavenumber is found by iterating on the Darcy-Rayleigh number so that no mean growth occurs over one forcing period. It is found that the most dangerous disturbance has a period which is twice that of the underlying basic state. Cells that rotate clockwise at first tend to rise upwards from the surface and weaken, but they induce an anticlockwise cell near the surface at the end of one forcing period, which is otherwise identical to the clockwise cell found at the start of that forcing period. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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Review

Jump to: Research

3580 KiB  
Review
Instability and Route to Chaos in Porous Media Convection
by Peter Vadasz
Fluids 2017, 2(2), 26; https://doi.org/10.3390/fluids2020026 - 18 May 2017
Cited by 12 | Viewed by 3622
Abstract
A review of the research on the instability of steady porous media convection leading to chaos, and the possibility of controlling the transition from steady convection to chaos is presented. The governing equations consisting of the continuity, the extended Darcy, and the energy [...] Read more.
A review of the research on the instability of steady porous media convection leading to chaos, and the possibility of controlling the transition from steady convection to chaos is presented. The governing equations consisting of the continuity, the extended Darcy, and the energy equations subject to the assumption of local thermal equilibrium and the Boussinesq approximation are converted into a set of three nonlinear ordinary differential equations by assuming two-dimensional convection and expansion of the dependent variables into a truncated spectrum of modes. Analytical (weak nonlinear), computational (Adomian decomposition) as well as numerical (Runge-Kutta-Verner) solutions to the resulting set of equations are presented and compared to each other. The analytical solution for the transition point to chaos is identical to the computational and numerical solutions in the neighborhood of a convective fixed point and deviates from the accurate computational and numerical solutions as the initial conditions deviate from the neighborhood of a convective fixed point. The control of this transition is also discussed. Full article
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5804 KiB  
Review
The Elder Problem
by John W. Elder, Craig T. Simmons, Hans-Jörg Diersch, Peter Frolkovič, Ekkehard Holzbecher and Klaus Johannsen
Fluids 2017, 2(1), 11; https://doi.org/10.3390/fluids2010011 - 21 Mar 2017
Cited by 14 | Viewed by 8862
Abstract
This paper presents an autobiographical and biographical historical account of the genesis, evolution and resolution of the Elder Problem. It begins with John W. Elder and his autobiographical story leading to his groundbreaking work on natural convection at Cambridge in the 1960’s. His [...] Read more.
This paper presents an autobiographical and biographical historical account of the genesis, evolution and resolution of the Elder Problem. It begins with John W. Elder and his autobiographical story leading to his groundbreaking work on natural convection at Cambridge in the 1960’s. His seminal work published in the Journal of Fluid Mechanics in 1967 became the basis for the modern benchmark of variable density flow simulators that we know today as “The Elder Problem”. There have been well known and major challenges with the Elder Problem model benchmark—notably the multiple solutions that were ultimately uncovered using different numerical models. Most recently, it has been shown that the multiple solutions are indeed physically realistic bifurcation solutions to the Elder Problem and not numerically spurious artefacts. The quandary of the Elder Problem has now been solved—a major scientific breakthrough for fluid mechanics and for numerical modelling. This paper—records, reflections, reminiscences, stories and anecdotes—is an historical autobiographical and biographical memoir. It is the personal story of the Elder Problem told by some of the key scientists who established and solved the Elder Problem. 2017 marks the 50 year anniversary of the classical work by John W. Elder published in Journal of Fluid Mechanics in 1967. This set the stage for this scientific story over some five decades. This paper is a celebration and commemoration of the life and times of John W. Elder, the problem named in his honour, and some of the key scientists who worked on, and ultimately solved, it. Full article
(This article belongs to the Special Issue Convective Instability in Porous Media)
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