# Unsteady Natural Convection Within a Porous Enclosure of Sinusoidal Corrugated Side Walls

## Abstract

Numerically investigation of free convection within a porous cavity with differential heating has been performed using modified corrugated side walls. Sinusoidal hot left and cold right walls are assumed to receive sudden differentially heating where top and bottom walls are insulated. Air is considered as working fluid and is quiescent, initially. Numerical experiments reveal 3 distinct stages of developing pattern including initial stage, oscillatory intermediate, and finally steady-state condition. Implicit Finite Volume Method with TDMA solver is used to solve the governing equations. This study has been performed for the Rayleigh numbers ranging from 100 to 10,000. Outcomes have been reported in terms of isotherms, streamline, velocity and temperature plots and average Nusselt number for various Ra, corrugation frequency, and corrugation amplitude (CA). The effects of sudden differential heating and its resultant transient behavior on fluid flow and heat transfer characteristics have been shown for the range of governing parameters. The present results show that the transient phenomena are enormously influenced by the variation of the Rayleigh Number with CA and frequency.

## Keywords

Natural convection Transient behavior Differentially heating Porous enclosure Sinusoidal corrugation## List of Symbols

- CA
Corrugation amplitude

- CF
Corrugation frequency

- \(g\)
Gravitational acceleration (\(\hbox {m/s}^{2}\))

- \(h\)
Convective heat transfer coefficient (\(\hbox {W/m}^{2}\hbox {K}\))

- \(H\)
Height of the enclosure (m)

- \(k\)
Thermal conductivity (W/mK)

- \(K\)
Permeability of porous media

- Nu
Average Nusselt number, Eq. (7)

- Pr
Prandtl number

- Ra
Rayleigh Number

- \(t\)
Dimensional time (s)

- \(T\)
Temperature (K)

*Tc*Temperature of the cold surface (K)

*Th*Temperature of the hot surface (K)

- \(\varDelta T\)
Temperature difference of cold and hot walls (K)

- \(u\)
Velocity component in x, y-direction (m/s)

- \(U\)
Dimensionless velocity component in X- direction, \(uH/\alpha \)

- \(v\)
Velocity component in y-direction (m/s)

- \(V\)
Dimensionless velocity component in Y- direction, \(vH/\alpha \)

- \(x, y\)
Cartesian coordinates (m)

- \(X, Y\)
dimensionless Cartesian coordinates, \((x,y)/ H\)

## Greek Symbols

- \({\varGamma }\)
dummy variable

- \(\sigma \)
ratio of composite material heat capacity to convective fluid heat capacity

- \(\tau \)
dimensionless time, \(\alpha t/H^{2}\)

- \(k\)
Thermal conductivity of fluid (W/mk)

- \(\alpha \)
Thermal diffusivity (\(\hbox {m}^{2}/\hbox {s}\))

- \(\beta \)
Coefficient of volumetric expansion (1/K)

- \(\theta \hbox {s}\)
local dimensionless surface temperature

- \(\theta \)
Dimensionless temperature

- \(\upsilon \)
kinematic viscosity (\(\hbox {m}^{2}/\hbox {s}\))

- \(\uppsi \)
Dimensionless stream function

- \(\Psi \)
Dimensional stream function (\(\hbox {m}^{2}/\hbox {s}\))

## Subscripts

- h
Cold temperature (K)

- c
Hot temperature (K)

- s
Surface

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