# Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

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## Abstract

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number \(10^{-5}\le \hbox {Da}\le 10^{-3}\), porous layer height ratio \(0\le d/L\le 1\), thermal conductivity ratio \(1\le k_{1,3}\le 20\), and dimensionless time \(0\le \tau \le 1000\) on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.

### Keywords

Conjugate natural convection Darcy model Brinkman-extended Darcy model Boussinesq approximation Stream function—vorticity formulation### List of Symbols

### Variables

- Bi
Biot number, \({hL}/{k_{1}}\)

- d
Height of the horizontal porous layer

- \(D\)
Height of the vertical cylindrical cavity

- Da
Darcy number, \(K/{L^{2}}\)

- \(g\)
Gravitational acceleration

- \(h\)
Heat-transfer coefficient

- \(k_{1}\)
Thermal conductivity of the solid walls

- \(k_{2}\)
Thermal conductivity of the fluid

- \(k_{3}\)
Effective thermal conductivity of the porous medium

- \(k_{i,j}\)
Thermal conductivity ratio, \({k_i}/{k_j}\)

- \(K\)
Permeability of the porous medium

- \(l\)
Thickness of the solid walls

- \(L\)
Radius of the vertical cylindrical cavity

- \(\hbox {Nu}_{\mathrm{avg}}\)
Average Nusselt number at \(Z=l/L, \int \limits _{0}^{1}{\left| {\frac{\partial \Theta }{\partial Z}}\right| _{Z=l/L} dR}\)

- \(p\)
Pressure

- Pr
Prandtl number, \({\nu _2}/{{\alpha }_{2}}\)

- \(r, z\)
Dimensional cylindrical coordinates

- \(R, Z\)
Dimensionless cylindrical coordinates

- Ra
Rayleigh number, \(\frac{g\beta \left( {T_0 -T^\mathrm{e}} \right) L^{3}}{\nu _{2}{\alpha }_{2}}\)

- \(t\)
Dimensional time

- \(T\)
Dimensional temperature

- \(T^{\mathrm{e}}\)
Dimensional ambient temperature

- \(T_0\)
Dimensional initial temperature

- \(U\)
Dimensionless velocity component in \(R\)-direction,

- \(V\)
Dimensionless velocity component in \(Z\)-direction,

- \(V_{r}\)
Dimensional velocity component in \(r\)-direction

- \(V_{z}\)
Dimensional velocity component in \(z\)-direction

### Greek Symbols

- \({\alpha }_{1}\)
Thermal diffusivity of the solid walls

- \({\alpha }_{2}\)
Thermal diffusivity of the fluid

- \({\alpha }_{3}\)
Effective thermal diffusivity of the porous medium

- \({\alpha }_{i,j}\)
Thermal diffusivity ratio, \({{\alpha }_{i}}/{{\alpha }_{j}}\)

- \(\beta \)
Thermal expansion coefficient

- \(\Delta \tau \)
Dimensionless time step

- \(\Theta \)
Dimensionless temperature

- \(\Theta _{\mathrm{avg}}\)
Dimensionless average temperature in the cylindrical cavity

- \(\kappa \)
Slip coefficient in the Beavers–Joseph condition

- \(\tilde{\kappa }\)
Stress jump coefficient

- \(\mu _{2}\)
Dynamic viscosity of the fluid

- \(\mu _{3}\)
Effective dynamic viscosity of the porous medium

- \(\nu _{2}\)
Kinematic viscosity of the fluid

- \(\rho \)
Density of the fluid

- \(\sigma \)
Heat capacity ratio

- \(\tau \)
Dimensionless time

- \(\psi \)
Dimensional stream function

- \(\Psi \)
Dimensionless stream function

- \(\left| \Psi \right| _{\mathrm{max}}\)
Maximum absolute value of the dimensionless stream function

- \(\omega \)
Dimensional vorticity

- \(\Omega \)
Dimensionless vorticity

### Subscripts

- 1
Solid walls

- 2
Fluid

- 3
Porous medium

- avg
Average value

- e
Environment

- max
Maximum value

## Notes

### Acknowledgments

This work was supported by the Grants Council (under the President of the Russian Federation), Grant No. MD-1506.2014.8 and by the Federal Grant-in-Aid Program “Human Capital for Science and Education in Innovative Russia” (Governmental Contract No. 8345).

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