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/s112420089285x
 Cite this article as:
 Kumari, M. & Nath, G. Transp Porous Med (2009) 77: 463. doi:10.1007/s112420089285x
Abstract
Unsteady natural convection flow in a twodimensional square cavity filled with a porous material has been studied. The flow is initially steady where the lefthand vertical wall has temperature T _{h} and the righthand vertical wall is maintained at temperature T _{c} (T _{h} > T _{c}) and the horizontal walls are insulated. At time t > 0, the lefthand 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 temperatureNomenclatures
 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 lefthand vertical wall at t = 0 (K)
 \({\bar{T}_{\rm h}}\)

Temperature of the lefthand vertical wall at t > 0 (K)
 T _{c}

Temperature of the righthand 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