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Convection in superposed fluid and porous layers

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Abstract

This paper reports an analytical study of the stability and natural convection in a system consisting of a horizontal fluid layer over a layer of saturated porous medium. Neumann thermal boundary conditions are applied to the horizontal walls of the enclosure while the vertical walls are impermeable and adiabatic. At the interface between the fluid and the porous layers the empirical slip condition, suggested by Beavers and Joseph, is employed. An analytical solution is obtained using a parallel flow approximation, for constant-flux thermal boundary conditions, for which the onset of supercritical cellular convection occurs at a vanishingly small wavenumber and can thus be predicted by the present theory. The critical Rayleigh number, Ra c , and Nusselt number, Nu, are found to depend on the depth ratio, η, the Darcy number, Da, the thermal conductivity ratio, γ and the slip parameter α. Results are presented for a wide range of each of the governing parameters. The results are compared with limiting cases of the problem and are found to be in agreement.

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Abbreviations

A :

Aspect ratio of the cavity, L′/H′

C f :

Specific heat of the fluid

C :

Dimensionless temperature gradient in the x-direction

Da :

Darcy number, K/H '2

g :

Gravitational acceleration, m/s2

H′:

Height of the enclosure, m

\({{h}'_f }\) :

Thickness of fluid layer, m

\({{h}'_p }\) :

Thickness of porous layer, m

K :

Permeability of porous layer, m2

k f :

Thermal conductivity of the fluid, W/(m K)

k p :

Thermal conductivity of the porous medium, W/(m K)

L′:

Width of enclosure, m

Nu :

Nusselt number, \({Nu=(1/\Delta T)/[(\eta +\gamma \bar \eta)/\gamma ]}\)

P′:

Pressure in the fluid layer, N/m2

\({{P}'_p }\) :

Pressure in the porous medium, N/m2

Pr :

Prandtl number, ν /α f

q′:

Constant heat flux per unit area, W/m2

R :

Darcy–Rayleigh number, \({gK\beta'_T \Delta {T}'{H}'/\alpha_f v}\)

R c :

Critical Darcy–Rayleigh number

Ra :

Rayleigh number, \({g{\beta}'_T \Delta T'H^{'3}/\alpha_f \nu }\)

Ra c :

Critical Rayleigh number

T′:

Temperature in the fluid layer, K

\({{T}'_p }\) :

Temperature in the porous layer, K

\({{T}'_r }\) :

Reference temperature at y′ = 0, K

u′:

Horizontal velocity component in the fluid layer, m/s

\({{u}'_p }\) :

Horizontal velocity component in the porous layer, m/s

v′:

Vertical velocity component in the fluid layer, m/s

\({{v}'_p }\) :

Vertical velocity component in the porous layer, m/s

x :

Dimensionless coordinate axis (x′/H′)

y :

Dimensionless coordinate axis (y′/H′)

α :

Dimensionless slip parameter

α f :

Thermal diffusivity of the fluid, m2/s

\({{\beta}'_T }\) :

Thermal expansion coefficient, K−1

γ :

Dimensionless parameter (k p /k f )

η :

Dimensionless thickness of the porous layer \({({h}'_p /{H}')}\)

\({\bar \eta}\) :

Dimensionless thickness of the fluid layer (1−η)

θ :

Dimensionless temperature variation with y in the fluid layer

θ p :

Dimensionless temperature variation with y in the porous layer

μ :

Dynamic viscosity of the fluid, Ns/m2

ν :

Kinematic viscosity of the fluid, m2/s

ρ :

Density of the fluid, kg/m3

Ψ:

Dimensionless stream function in the fluid layer

Ψ p :

Dimensionless stream function in the porous layer

c :

Critical condition

f :

Refers to the fluid layer

p :

Refers to the porous medium

r :

Reference state

′:

Refers to a dimensional variable

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Authors and Affiliations

  1. Mechanical Engineering Department, Ecole Polytechnique, University of Montréal, C.P. 6079, Succ. “Down-Town”, Montreal, QC, H3C 3A7, Canada

    Zineddine Alloui & Patrick Vasseur

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  1. Zineddine Alloui
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  2. Patrick Vasseur
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Correspondence to Zineddine Alloui.

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Alloui, Z., Vasseur, P. Convection in superposed fluid and porous layers. Acta Mech 214, 245–260 (2010). https://doi.org/10.1007/s00707-010-0284-y

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