# Unsteady Natural Convection Flow in a Square Cavity Filled with a Porous Medium Due to Impulsive Change in Wall Temperature

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11242-008-9285-x

- Cite this article as:
- Kumari, M. & Nath, G. Transp Porous Med (2009) 77: 463. doi:10.1007/s11242-008-9285-x

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## Abstract

Unsteady natural convection flow in a two-dimensional square cavity filled with a porous material has been studied. The flow is initially steady where the left-hand vertical wall has temperature *T*_{h} and the right-hand vertical wall is maintained at temperature *T*_{c} (*T*_{h} > *T*_{c}) and the horizontal walls are insulated. At time *t* > 0, the left-hand vertical wall temperature is suddenly raised to \({{\bar{T}}_{\rm h}\,({\bar{T}}_{\rm h} > T_{\rm h})}\) which introduces unsteadiness in the flow field. The partial differential equations governing the unsteady natural convection flow have been solved numerically using a finite control volume method. The computation has been carried out until the final steady state is reached. It is found that the average Nusselt number attains a minimum during the transient period and that the time required to reach the final steady state is longer for low Rayleigh number and shorter for high Rayleigh number.

### Keywords

Unsteady natural convectionSquare cavityPorous mediumSudden change in wall temperature### Nomenclatures

*c*_{p}Specific heat at constant pressure (J kg

^{−1}K^{−1})*g*Acceleration due to gravity (m s

^{−2})*K*Permeability of the porous medium (m

^{2})*k*Thermal conductivity (W m

^{−1}K^{−1})*L*Height/length of the cavity (m)

*Nu*Local Nusselt number

- \({\overline{Nu}}\)
Average Nusselt number

*Ra*Rayleigh number

*t*Time (s)

*t*^{*}Dimensionless time

*T*Fluid temperature (K)

*T*_{h}Temperature of the left-hand vertical wall at

*t*= 0 (K)- \({\bar{T}_{\rm h}}\)
Temperature of the left-hand vertical wall at

*t*> 0 (K)*T*_{c}Temperature of the right-hand vertical wall at

*t*≥ 0 (K)*T*_{0}Average temperature at

*t*= 0 (K)*u*,*v*Velocity components along

*x*and*y*directions, respectively (m s^{−1})*U*,*V*Dimensionless velocity components along

*x*and*y*directions, respectively*x*,*y*Cartesian coordinates (m)

*X*,*Y*Dimensionless Cartesian coordinates

### Greek symbols

*α*_{e}Effective thermal diffusivity (m

^{2}s^{−1})*β*Coefficient of thermal expansion (K

^{−1})- \({\epsilon}\)
Dimensionless constant

*θ*Dimensionless temperature

*υ*Kinematic viscosity (m

^{2}s^{−1})*ρ*_{f},*ρ*_{m}Density of the fluid and porous medium, respectively (kg m

^{−3})*σ*Ratio of composite material heat capacity to convective fluid heat capacity

*ψ*Dimensionless stream function

*ψ*^{*}Stream function (m

^{2}s^{−1})

### Subscripts

- f
Fluid

- i
Initial condition

- m
Porous medium