Transport in Porous Media

, Volume 77, Issue 3, pp 463–474 | Cite as

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

Article

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 Th and the right-hand vertical wall is maintained at temperature Tc (Th > Tc) 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

cp

Specific heat at constant pressure (J kg−1 K−1)

g

Acceleration due to gravity (m s−2)

K

Permeability of the porous medium (m2)

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)

Th

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)

Tc

Temperature of the right-hand vertical wall at t≥ 0 (K)

T0

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

αe

Effective thermal diffusivity (m2 s−1)

β

Coefficient of thermal expansion (K−1)

\({\epsilon}\)

Dimensionless constant

θ

Dimensionless temperature

υ

Kinematic viscosity (m2 s−1)

ρf, ρm

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 (m2 s−1)

Subscripts

f

Fluid

i

Initial condition

m

Porous medium

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.SultanpurIndia

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