# Unsteady natural convection in an enclosure with vertical wavy walls

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s00231-007-0349-1

- Cite this article as:
- Rostami, J. Heat Mass Transfer (2008) 44: 1079. doi:10.1007/s00231-007-0349-1

- 11 Citations
- 207 Downloads

## Abstract

In this paper, unsteady heat transfer and fluid flow characteristics in an enclosure are investigated. The enclosure consists of two vertical wavy and two horizontal straight walls. The top and the bottom walls are considered adiabatic. Two wavy walls are kept isothermal and their boundaries are approximated by a cosine function. Governing equations including continuity, momentum and energy were discretized using the finite-volume method and solved by SIMPLE method in curvilinear coordinate. Simulation was carried out for a range of Grashof number *Gr* = 10^{3}–10^{6}, Prandtl number *Pr* = 0.5–4.0, wave ratio *A* (defined by amplitude/wavelength) 0.0–0.35 and aspect ratio *W* (defined by average width/wavelength) 0.5–1.0. Streamlines and isothermal lines are presented to corresponding flow and thermal fields. Local and average Nusselt number distributions are presented. The obtained results are in good agreement with available numerical and experimental data.

### List of symbols

*a*amplitude of the wave (m)

*A*amplitude–wavelength ratio (2

*a*/λ)*g*gravity vector (m/s

^{2})*Gr*Grashof number

*J*jacobean of the coordinate transformation

*L*total wall curve length–wavelength ratio (

*S*/λ)*n*normal direction to the wavy wall (m)

*Nu*average Nusselt number

*P*pressure (Pa)

*p*dimensionless pressure

*Pr*Prandtl number

*q*_{11},*q*_{12},*q*_{22}the parameters defined at (6)

*Ra*Rayleigh number

*S*total curve length of the wavy wall (m)

*t*time (s)

*T*temperature (K)

*u*dimensionless velocity component in

*x*-direction*U*velocity component in

*x*-direction (m/s)*v*dimensionless velocity component in

*y*-direction*V*velocity component in

*y*-direction (m/s)*w*average width of the cavity (m)

*W*average width–wavelength ratio (

*w*/λ)*x*dimensionless horizontal coordinate

*X*horizontal coordinate (m)

*y*dimensionless vertical coordinate

*Y*vertical coordinate (m)

### Greek symbols

- α
thermal diffusivity (m

^{2}/s)- β
thermal expansion coefficient (1/K)

- λ
surface wavelength (m)

- ρ
density of fluid (kg/m

^{3})- υ
kinematics viscosity (m

^{2}/s)- τ
dimensionless time

- θ
dimensionless temperature

- ξ
curvilinear horizontal coordinate

- η
curvilinear vertical coordinate

### Subscripts

- c
average value based on cold wall

- h
average value based on hot wall

- s
value at steady state

- y
local value

### Superscript

- c
contravariant velocities