The onset of convection in fluid layers heated rapidly in a time-dependent manner

Jhaveri, Bharat S.; Homsy, G. M.

doi:10.1017/s0022112082000135

Cited by 68 publications

(37 citation statements)

References 17 publications

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“…The problem of hydrodynamic stability in porous media has been studied extensively (Horton and Rogers 1945;Lapwood 1948; Morrison and Rogers 1949;Katto and Masuoka 1967;Beck 1972;Caltagirone 1980;Nield and Bejan 1999). The stability analysis for plain fluids under a variety of boundary conditions has also been thoroughly investigated (Morton 1957, Lick 1965, Foster 1965a,b, 1968, Mahler et al 1968, Jhavery and Homsy 1982, Tan and Thorpe 1996. For example, Kaviany (1984) studied the onset of thermal convection in a porous medium both theoretically and experimentally.…”

Section: Introductionmentioning

confidence: 99%

Stability of a fluid in a horizontal saturated porous layer: effect of non-linear concentration profile, initial, and boundary conditions

2006

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Carbon dioxide injected into saline aquifers dissolves in the resident brines increasing their density, which might lead to convective mixing. Understanding the factors that drive convection in aquifers is important for assessing geological CO 2 storage sites. A hydrodynamic stability analysis is performed for non-linear, transient concentration fields in a saturated, homogenous, porous medium under various boundary conditions. The onset of convection is predicted using linear stability analysis based on the amplification of the initial perturbations. The difficulty with such stability analysis is the choice of the initial conditions used to define the imposed perturbations. We use different noises to find the fastest growing noise as initial conditions for the stability analysis. The stability equations are solved using a Galerkin technique. The resulting coupled ordinary differential equations are integrated numerically using a fourth-order Runge-Kutta method. The upper and lower bounds of convection instabilities are obtained. We find that at high Rayleigh numbers, based on the fastest growing noise for all boundary conditions, both the instability time and the initial wavelength of the convective instabilities are independent of the porous layer thickness. The current analysis provides approximations that help in screening suitable candidates for homogenous geological CO 2 sequestration sites.

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Section: Introductionmentioning

confidence: 99%

Stability of a fluid in a horizontal saturated porous layer: effect of non-linear concentration profile, initial, and boundary conditions

2006

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“…Another issue is the criteria for determining the onset of instability, which is somewhat arbitrary due to the imprecise definition of critical time. Foster (1965aFoster ( , 1968, and Jhavery and Homsy (1982) used this method for a thermal-diffusing boundary layer in an incompressible fluid. Nield (1968) studied the onset of thermohaline convection in a porous medium for various boundary conditions.…”

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confidence: 99%

Linear Stability Analysis of Double-Diffusive Convection in Porous Media, with Application to Geological Storage of CO2

2009

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Onset of double-diffusive buoyancy-driven flow resulted from vertical temperature and concentration gradients in a horizontal layer of a saturated and homogenous porous medium is investigated using amplification factor theory. After injection of CO 2 into a deep saline aquifer, the density of the brine saturated with CO 2 increases slightly. This increase in density induces natural convection. The effect of geothermal gradient is also considered in this work as a second incentive for convection and the double-diffusion convection was studied. Linear stability analysis is used to predict the inception of instabilities and initial wavelength of the convective instabilities. The analysis presented is applied to acid gas injection (as an analogue for CO 2 storage) into saline aquifers in the Alberta basin. It is found that the geothermal gradient does not have significant effect on the onset of convection for these aquifers. It is shown that the geothermal effects on the onset of natural convection are negligible as compared to the solutal effects induced by dissolution and diffusion of CO 2 in deep saline aquifers. Therefore, the linear stability analysis and the long-term numerical simulation of CO 2 sequestration into such saline aquifers may be assumed to be isothermal in terms of natural convection occurrence. List of SymbolsA Time components of the perturbed velocity amplitude function a Dimensionless wave number B Time components of the perturbed concentration amplitude function C Concentration (kg/m 3 ) 123 442 M. Javaheri et al.D Molecular diffusion (m 2 /s) D Derivative operator E Time components of the perturbed temperature amplitude function d Derivative g Gravitational acceleration (m/s 2 ) H Porous layer thickness (m) i Imaginary number K Thermal conductivity (W/m K) k Permeability (m 2 ) Le Lewis number (dimensionless) p Pressure (kg/m s 2 ) Ra Rayleigh number (dimensionless) T Temperature (K) T 1 Temperature of the bottom layer (K) t Time (s) u Velocity in x-direction (m/s) V Volume (m 3 ) v Velocity in y-direction (m/s) v Vector of Darcy velocity (m/s) w Velocity in z-direction (m/s) w Amplification factor (dimensionless) x Coordinate direction (m) y Coordinate direction (m) z Vertical coordinate direction (m) α Thermal diffusivity (m 2 /s) β C Coefficient of density increase by concentration (m 3 /kg) β T Coefficient of thermal expansion (K −1 ) φ Porosity (dimensionless) ρ Density (kg/m 3 ) μ Viscosity (kg/m s) ρ Density difference (kg/m 3 ) σ Matrix to fluid specific heat capacity (dimensionless) ∇ 2 Laplacian operator ∇ 2 H ∂ 2 /∂ x 2 D + ∂ 2 /∂ y 2 D

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“…Indeed the problems discussed here are very similar to the ones arising in convective ramping experiments, where it is known that the sensitivity of the evolution to the initial amplitude of the fluctuations is great (16,17). Moreover, the onset of manifest spatial organization cannot be obtained from the linear theory alone, and the derivation of amplitude equations with the full nonlinearity of the problem is necessary to obtain a fair description of the concentration or convective patterns.…”

Section: Iaxy2'mentioning

confidence: 82%

Spatial patterns and double diffusion in chemical reactions

Dewel

Borckmans

Walgraef

1983

Proc. Natl. Acad. Sci. U.S.A.

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Spatial ordering has been observed recently during various photochemical reactions. Convoluted concentration bands first appear near the surface of shallow irradiated solutions. They thereafter extend into the bulk, and finger-like structures spontaneously develop. We discuss here the possible role of double-diffusion effects in the onset of this phenomenon. Indeed, chemical reactions occurring near the surface or evaporation of the solvent, or both, induce in the bulk adverse gradients of a pair of properties (concentrations of solute or concentration and tem- (10)(11)(12).For the sake of clarity, we consider the following model. IaXy2'The main problem is concerned with the determination of the time of onset for the instability to occur in the presence of the time developing gradients aAA(z,t) and adBO(z,t). A simple method to estimate this time is provided by the quasistationary approximation. It consists in freezing the concentration profile at a given time and determining its stability. Although this approximation is known to yield inaccurate values of the growth rate of the disturbances, it seems in fact to predict the time at which the structuration first develops in the layer accurately (13).As the concentration gradients are slowly developing in time (DAB 10-5 cm2s-1), their effects can only be felt in a small layer at the top of the solution, the thickness of which is of the order of (DAt)'12 (10-2 after 10 s). An The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

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The onset of convection in fluid layers heated rapidly in a time-dependent manner

Cited by 68 publications

References 17 publications

Stability of a fluid in a horizontal saturated porous layer: effect of non-linear concentration profile, initial, and boundary conditions

Stability of a fluid in a horizontal saturated porous layer: effect of non-linear concentration profile, initial, and boundary conditions

Linear Stability Analysis of Double-Diffusive Convection in Porous Media, with Application to Geological Storage of CO2

Spatial patterns and double diffusion in chemical reactions

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