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Onset of internal convection in superposed air-porous layer with heat source depending on solid volume fraction: influence of different modeling

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Abstract

A linear stability analysis for the onset of internal convection in a composite air-porous layer with the heat source strength depending on the solid volume fraction is carried out through the shooting method. The method is based on the algorithm for constructing a fundamental system of partial solution vectors. We compare the onset values produced by the following three approaches: (1) Darcy approach with the interfacial condition of vanishing tangential velocity (D), (2) Darcy modeling with the Beavers–Joseph condition (\({D_\mathrm{BJ}}\)), and (3) Darcy–Brinkman formulation (DB). The relative difference between the values produced by the \({D_\mathrm{BJ}}\) and DB approaches versus the results obtained by the simplest D model is presented at various solid volume fractions. This difference is the smallest one in the limiting case, when the solid volume fraction tends to unity and the local convection dominates. On the other hand, it can also significantly decrease at some intermediate value of the solid volume fraction that corresponds to an abrupt jump-like transition from local to large-scale convective flows. It has been found that the solid volume fraction and relative depth of the air sublayer speed up the internal convection onset. The onset internal Darcy–Rayleigh number for large-scale convection cannot be reduced by more than ten times, whereas its value for local convection may vary over a wide range covering several orders of the initial magnitude obtained in the absence of air sublayer.

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Abbreviations

b :

Ratio of volumetric heat capacities

C :

Specific heat capacity

d :

Depth ratio

\({{\mathrm{Da}}}\) :

Darcy number

g :

Acceleration of gravity

\({h_\mathrm{a}}\) :

Depth of air sublayer

\({h_\mathrm{p}}\) :

Depth of porous sublayer

K :

Permeability

k :

Wave number

m :

Porosity, \(\left( {1 - \phi } \right) \)

P :

Pressure excluding the hydrostatic additive

\({\Pr _\mathrm{p}}\) :

Prandtl number in the porous sublayer

\({Q_\mathrm{s}}\) :

Volumetric heat source strength in the solid phase

\({R_I}\) :

Internal Darcy–Rayleigh number normalized by the solid volume fraction

\(R{a_I}\) :

Internal Darcy–Rayleigh number, \(\phi {R_I}\)

T :

Temperature

t :

Time

\({\mathbf{{V}}_\mathrm{a}}\) :

Velocity in the air sublayer

\({\mathbf{{V}}_\mathrm{p}}\) :

Seepage velocity in the porous sublayer

x :

Coordinate of longitudinal axis

z :

Coordinate of transversal axis

\({\alpha _\mathrm{BJ}}\) :

Beavers–Joseph coefficient

\(\beta \) :

Thermal expansion coefficient

\({{\gamma }}\) :

Unit vector of the z-axis

\({\delta _{Ra}}\) :

Relative deviation of the Darcy–Rayleigh number from the reference value

\(\kappa \) :

Thermal conductivity ratio

\({\kappa _\mathrm{a}}\) :

Thermal conductivity of air sublayer

\({\kappa _\mathrm{p}}\) :

Thermal conductivity of porous sublayer

\(\lambda \) :

Growth rate of the disturbance

\(\nu \) :

Air kinematic viscosity

\({\rho _0}\) :

Air density at \(T = {T_0}\)

\({\chi _\mathrm{eff}}\) :

Effective thermal diffusivity of porous sublayer, \({{{\kappa _p}} / {{{\left( {{\rho _0}C} \right) }_a}}}\)

\({\chi _\mathrm{a}}\) :

Thermal diffusivity of air sublayer, \({{{\kappa _a}} / {{{\left( {{\rho _0}C} \right) }_a}}}\)

\(\phi \) :

Solid volume fraction

\(\nabla \) :

Nabla operator

\(\Theta \) :

Dimensionless temperature deviation from the reference value \({T_0}\)

a :

Air sublayer

l :

The lower boundary of porous sublayer

p :

Porous sublayer

s :

Solid

u :

The upper boundary of air sublayer

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Acknowledgements

The work was supported by the Russian Science Foundation (Grant No. 21-71-10045), https://rscf.ru/en/project/21-71-10045/. The authors also would like to thank the very competent Reviewers for their valuable comments and suggestions.

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  1. Department of General Physics, Perm State University, Bukireva, 15, Perm, Russian Federation, 614990

    Ekaterina Kolchanova & Nikolay Kolchanov

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  1. Ekaterina Kolchanova
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Correspondence to Ekaterina Kolchanova.

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Kolchanova, E., Kolchanov, N. Onset of internal convection in superposed air-porous layer with heat source depending on solid volume fraction: influence of different modeling. Acta Mech 233, 1769–1788 (2022). https://doi.org/10.1007/s00707-022-03204-8

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  • DOI: https://doi.org/10.1007/s00707-022-03204-8

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