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

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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 convection Square cavity Porous medium Sudden 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

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## References

- Aldabbagh L.B.Y., Manesh H.F., Mohamad A.A.: Unsteady natural convection inside a porous enclosure heated from the side. J. Porous Media.
**11**, 73–83 (2007)CrossRefGoogle Scholar - Banu, N., Rees, D.A., Pop, I.: Steady and unsteady free convection in porous cavities with internal heat generation. In: Heat Transfer 1998, Proceedings of 11th IHTC, vol. 4, pp. 375–380, Kyongju (1998)Google Scholar
- Baytas A.C., Pop I.: Free convection in a square porous cavity using a thermal nonequilibrium model. Int. J. Therm. Sci.
**41**, 861–870 (2002)CrossRefGoogle Scholar - Bejan A.: On the boundary layer regime in a vertical enclosure filled with a porous medium. Lett. Heat Mass Transf.
**6**, 93–102 (1979)CrossRefGoogle Scholar - Bejan, A., Kraus, A.D. (eds.): Heat Transfer Handbook. Wiley, New York (2003)Google Scholar
- Gross, R.J., Bear, M.R., Hickox, C.E.: The application of flux-corrected transport (FCT) on high Rayleigh number natural convection in a porous medium. In: Proceedings of the 8th International Heat Transfer Conference, San Francisco (1986)Google Scholar
- Ingham, D.B., Pop, I. (eds): Transport Phenomena in Porous Media, vol III. Pergamon, Oxford (2005)Google Scholar
- Khashan S.A., Al-Amiri A.M., Pop I.: Numerical simulation of natural convection heat transfer in a porous cavity heated from below using a non-Darcian and thermal non-equilibrium model. Int. J. Heat Mass Transf.
**49**, 1039–1049 (2006)CrossRefGoogle Scholar - Kumar B.V.R., Singh P., Murthy P.V.S.N.: Effect of surface undulations on natural convection in a porous square cavity. ASME J. Heat Transf.
**119**, 848–851 (1997)CrossRefGoogle Scholar - Manole, D.M., Lage, J.L.: Numerical benchmark results for natural convection in a porous medium cavity. In: HTD—vol. 216, Heat and Mass Transfer in Porous Media, ASME Conference, pp. 55–60 (1992)Google Scholar
- Misirlioglu A., Baytas A.C., Pop I.: Free convection in a wavy cavity filled with a porous medium. Int. J. Heat Mass Transf.
**48**, 1840–1850 (2005)CrossRefGoogle Scholar - Nield D.A., Bejan A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)Google Scholar
- Patankar S.V.: Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation, Washington (1980)Google Scholar
- Prasad V., Kulacki F.A.: Convective heat transfer in a rectangular porous cavity—effect of aspect ratio on flow structure and heat transfer. ASME J. Heat Transf.
**106**, 158–165 (1984)CrossRefGoogle Scholar - Pop I., Ingham D.B.: Convective Heat Transfer, Mathematical and Computational Modelling of Viscous Fluids and Porous Media. Pergamon Press, Oxford (2001)Google Scholar
- Saeid N.H.: Natural convection in porous cavity with sinusoidal bottom wall temperature variation. Int. Commun. Heat Mass Transf.
**32**, 454–463 (2005)CrossRefGoogle Scholar - Saeid N.H., Pop I.: Transient free convection in a porous cavity filled with a porous medium. Int. J. Heat Mass Transf.
**47**, 1917–1924 (2004)CrossRefGoogle Scholar - Vafai, K. (eds): Handbook of Porous Media, 2nd edn. Taylor and Francis, Boca Raton (2005)Google Scholar
- Walker, K.L., Homsy, G.M.: Convection in a porous cavity. J. Fluid Mech.
**87**, 449–474 (1978)CrossRefGoogle Scholar