Anisotropy and the Onset of the Thermoconvective Instability in a Vertical Porous Layer

2021 ◽  
Author(s):  
Antonio Barletta ◽  
Michele Celli
Keyword(s):  
Porous Layer ◽  
Critical Rayleigh Number ◽  
Normal Modes ◽  
Neutral Stability ◽  
Anisotropic Permeability ◽  
Open Boundaries ◽  
Principal Axes ◽  
Saturated Porous Layer ◽  
Thermoconvective Instability ◽  
The Stability

Abstract The thermoconvective instability of the parallel vertical flow in a fluid saturated porous layer bounded by parallel open boundaries is studied. The open boundaries are assumed to be kept at constant uniform pressure while their temperatures are uniform and different, thus forcing a horizontal temperature gradient across the layer. The anisotropic permeability of the porous layer is accounted for by assuming the principal axes to be oriented along the directions perpendicular and parallel to the layer boundaries. A linear stability analysis based on the Fourier normal modes of perturbation is carried out by testing the effect of the inclination of the normal mode wave vector to the vertical. The neutral stability curves and the critical Rayleigh number for the onset of the instability are evaluated by solving numerically the stability eigenvalue problem.

scholarly journals The Effect of Open Boundaries on the Buoyant Viscoelastic Flow in a Vertical Porous Layer

2021 ◽  
Author(s):  
Stefano Lazzari ◽  
Michele Celli ◽  
Antonio Barletta
Keyword(s):  
Porous Layer ◽  
Viscoelastic Fluid ◽  
Critical Rayleigh Number ◽  
Stationary Flow ◽  
Vertical Axis ◽  
Seepage Flow ◽  
Neutral Stability ◽  
Saturated Porous Layer ◽  
Linear Viscoelastic ◽  
Vertical Height

The Oldroyd–B model for a linear viscoelastic fluid is employed to investigate the buoyant flow in a vertical porous layer with permeable boundaries kept at different uniform temperatures. Seepage flow in the viscoelastic fluid saturated porous layer is modelled through an extended version of Darcy’s law taking into account the Oldroyd–B rheology. The basic stationary flow is parallel to the vertical axis and describes a single–cell vertical pattern where the cell has an infinite vertical height. A linear stability analysis of such a basic flow is carried out to determine the onset conditions for a multicellular pattern. The neutral stability curves and the values of the critical Rayleigh number are evaluated numerically for different retardation time and relaxation time characteristics of the fluid.

scholarly journals On the stability of carbon sequestration in an anisotropic horizontal porous layer with a first-order chemical reaction

2019 ◽  
Vol 475 (2226) ◽  
pp. 20180365 ◽  
Author(s):  
K. Gautam ◽  
P. A. L. Narayana
Keyword(s):  
Chemical Reaction ◽  
Porous Layer ◽  
Nonlinear Stability ◽  
Anisotropy Ratio ◽  
Neutral Stability ◽  
First Order ◽  
Order Chemical Reaction ◽  
Linear And Nonlinear ◽  
First Order Chemical Reaction ◽  
The Stability

Carbon dioxide (CO 2 ) sequestration in deep saline aquifers is considered to be one of the most promising solutions to reduce the amount of greenhouse gases in the atmosphere. As the concentration of dissolved CO 2 increases in unsaturated brine, the density increases and the system may ultimately become unstable, and it may initiate convection. In this article, we study the stability of convection in an anisotropic horizontal porous layer, where the solute is assumed to decay via a first-order chemical reaction. We perform linear and nonlinear stability analyses based on the steady-state concentration field to assess neutral stability curves as a function of the anisotropy ratio, Damköhler number and Rayleigh number. We show that anisotropy in permeability and solutal diffusivity play an important role in convective instability. It is shown that when solutal horizontal diffusivity is larger than the vertical diffusivity, varying the ratio of vertical to horizontal permeabilities does not significantly affect the behaviour of instability. It is also noted that, when horizontal permeability is higher than the vertical permeability, varying the ratio of vertical to horizontal solutal diffusivity does have a substantial effect on the instability of the system when the reaction rate is dominated by the diffusion rate. We used the Chebyshev-tau method coupled with the QZ algorithm to solve the eigenvalue problem obtained from both the linear and nonlinear stability theories.

scholarly journals Stability of a Buoyant Oldroyd-B Flow Saturating a Vertical Porous Layer with Open Boundaries

Fluids ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 375
Author(s):  
Stefano Lazzari ◽  
Michele Celli ◽  
Antonio Barletta
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Critical Rayleigh Number ◽  
Stationary Flow ◽  
Vertical Axis ◽  
Seepage Flow ◽  
Neutral Stability ◽  
Linear Viscoelastic ◽  
Infinite Height ◽  
Limiting Case

The performance of several engineering applications are strictly connected to the rheology of the working fluids and the Oldroyd-B model is widely employed to describe a linear viscoelastic behaviour. In the present paper, a buoyant Oldroyd-B flow in a vertical porous layer with permeable and isothermal boundaries is investigated. Seepage flow is modelled through an extended version of Darcy’s law which accounts for the Oldroyd-B rheology. The basic stationary flow is parallel to the vertical axis and describes a single-cell pattern where the cell has an infinite height. A linear stability analysis of such a basic flow is carried out to determine the onset conditions for a multicellular pattern. This analysis is performed numerically by employing the shooting method. The neutral stability curves and the values of the critical Rayleigh number are evaluated for different retardation time and relaxation time characteristics of the fluid. The study highlights the extent to which the viscoelasticity has a destabilising effect on the buoyant flow. For the limiting case of a Newtonian fluid, the known results available in the literature are recovered, namely a critical value of the Darcy–Rayleigh number equal to 197.081 and a corresponding critical wavenumber of 1.05950.

The effects of boundary imperfections on convection in a saturated porous layer: near-resonant wavelength excitation

1989 ◽  
Vol 199 ◽  
pp. 133-154 ◽  
Author(s):  
D. A. S. Rees ◽  
D. S. Riley
Keyword(s):  
Porous Layer ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Resonant Forcing ◽  
Saturated Porous Layer ◽  
The Stability ◽  
Spatially Varying ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Complementary Study

Weakly nonlinear theory is used to study the porous-medium analogue of the classical Rayleigh-Bénard problem, i.e. Lapwood convection in a saturated porous layer heated from below. Two particular aspects of the problem are focused upon: (i) the effect of thermal imperfections on the stability characteristics of steady rolls near onset; and (ii) the evolution of unstable rolls.For Rayleigh-Bénard convection it is well known (see Busse and co-workers 1974, 1979, 1986) that the stability of steady two-dimensional rolls near onset is limited by the presence of cross-roll, zigzag and sideband disturbances; this is shown to be true also in Lapwood convection. We further determine the modifications to the stability boundaries when small-amplitude imperfections in the boundary temperatures are present. In practice imperfections would usually consist of broadband thermal noise, but it is the Fourier component with wavenumber close to the critical wavenumber for the perfect problem (i.e. in the absence of imperfections) which, when present, has the greatest effect due to resonant forcing. This particular case is the sole concern of the present paper; other resonances are considered in a complementary study (Rees & Riley 1989).For the case when the modulations on the upper and lower boundaries are in phase, asymptotic analysis and a spectral method are used to determine the stability of roll solutions and to calculate the evolution of the unstable flows. It is shown that steady rolls with spatially deformed axes or spatially varying wavenumbers evolve. The evolution of the flow that is unstable to sideband disturbances is also calculated when the modulations are π out of phase. Again rolls with a spatially varying wavenumber result.

Stability of Thermal Convection in a Vertical Porous Layer

1987 ◽  
Vol 109 (4) ◽  
pp. 889-893 ◽  
Author(s):  
L. P. Kwok ◽  
C. F. Chen
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Porous Layer ◽  
Thermal Convection ◽  
Variable Viscosity ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Lateral Heating ◽  
The Stability ◽  
Allowable Temperature

Experiments were carried out to study the stability of thermal convection generated in a vertical porous layer by lateral heating in a tall, narrow tank. The porous medium, consisting of glass beads, was saturated with distilled water. It was found that the flow became unstable at a critical ΔT of 29.2°C (critical Rayleigh number of 66.2). Linear stability analysis was applied to study the effects of the Brinkman term and of variable viscosity separately using a quadratic relationship between the density and temperature. It was found that with the Brinkman term, no instability could occur within the allowable temperature difference across the tank. With the effect of variable viscosity included, linear stability theory predicts a critical ΔT of 43.4°C (Rayleigh number of 98.3).

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.

On Hadley flow in a porous layer with vertical heterogeneity

2012 ◽  
Vol 710 ◽  
pp. 304-323 ◽  
Author(s):  
A. Barletta ◽  
D. A. Nield
Keyword(s):  
Thermal Conductivity ◽  
Temperature Gradient ◽  
Porous Layer ◽  
Plane Waves ◽  
Neutral Stability ◽  
Horizontal Temperature Gradient ◽  
Weighted Residuals ◽  
Thermal Boundary Conditions ◽  
Vertical Heterogeneity ◽  
Thermoconvective Instability

AbstractThe onset of thermoconvective instability in a horizontal porous layer with a basic Hadley flow is studied, under the assumption of weak vertical heterogeneity. Hadley flow is a single-cell convective circulation induced by horizontal linear changes of the layer boundary temperatures. When combined with heating from below, these thermal boundary conditions yield a temperature gradient inclined to the vertical, in the basic state. The linear stability of the basic state is studied by considering small-amplitude disturbances of the velocity field and the temperature field. The linearized governing equations for the disturbances are then solved both by Galerkin’s method of weighted residuals and by a combined use of the Runge–Kutta method and the shooting method. The effect of weak heterogeneity of the permeability and the effective thermal conductivity of the porous medium is studied with respect to neutral stability conditions. It is shown that, among the normal mode disturbances, the most unstable are longitudinal rolls, that is, plane waves with a wave vector perpendicular to the imposed horizontal temperature gradient. The effect of heterogeneity becomes important only for high values of the horizontal Rayleigh number, associated with the horizontal temperature gradient, approximately greater than 60. In this regime, the effect of heterogeneity is destabilizing. It is shown that heterogeneity with respect to thermal conductivity is of major importance in the onset of instability.

Onset of Natural Convection in a Porous Layer Confined by an Elliptic or Semi-Elliptic Box Using the Ritz Method

2020 ◽  
Vol 142 (9) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Natural Convection ◽  
Rayleigh Number ◽  
Porous Layer ◽  
Ritz Method ◽  
Critical Rayleigh Number ◽  
Aspect Ratios ◽  
Side Walls ◽  
The Stability

Abstract Using an efficient Ritz method, the thermoconvective stability of a bottom-heated porous layer in vertical elliptic and semi-elliptic enclosures with adiabatic side walls is studied. The stability mosaics for the critical Rayleigh number and the preferred modes are determined for various aspect ratios.

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.

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