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ABSTRACT
Transient natural convection in a differentially heated square cavity that has one vertical wavy wall has been studied numerically. The domain of interest is bounded by vertical isothermal walls and horizontal adiabatic walls. The governing equations formulated in dimensionless stream function, vorticity, and temperature, within the Boussinesq approximation with corresponding initial and boundary conditions, have been solved using an iterative implicit finite-difference method. The main objective is to investigate the effect of the dimensionless time 0 ≤ τ ≤ 0.4, Rayleigh number 104 ≤Ra ≤ 106, surface emissivity 0 ≤ ε < 1, undulations number 1 ≤ κ ≤ 6, and shape parameter 0.6 ≤ a ≤ 1.4 on fluid flow and heat transfer. Results are presented in the form of streamlines, isotherms, and distribution of average total Nusselt number at the wavy wall.
Nomenclature
| a | = | shape parameter |
| A = H/D | = | aspect ratio |
| b | = | wavy contraction ratio |
| D | = | length of cavity (m) |
| Fk−i | = | view factor from k-th to i-th element of cavity |
| g | = | gravitational acceleration (m · s−2) |
| H | = | height of cavity (m) |
| k | = | thermal conductivity (W · m−1 · K−1) |
| = | radiation number | |
| = | average Nusselt number at left wavy wall | |
| NS | = | number of surface elements in cavity |
| p | = | dimensional pressure (Pa) |
| Pr = ν/α | = | Prandtl number |
| Qrad | = | dimensionless net radiative heat flux |
| = | Rayleigh number | |
| Rk | = | dimensionless radiosity of k-th element of cavity |
| t | = | dimensional time (s) |
| T | = | dimensional temperature (K) |
| Tc | = | dimensional cooled wall temperature (K) |
| Th | = | dimensional heated wall temperature (K) |
| T0 = (Th + Tc)/2 | = | dimensional mean temperature of heated and cooled walls (K) |
| u | = | dimensional velocity component in x-direction (m · s−1) |
| v | = | dimensional velocity component in y-direction (m · s−1) |
| U | = | dimensionless velocity component in X-direction |
| V | = | dimensionless velocity component in Y-direction |
| x, y | = | dimensional Cartesian coordinates (m) |
| x1 | = | dimensional location of left vertical wall (m) |
| x2 | = | dimensional location of right vertical wall (m) |
| X, Y | = | dimensionless Cartesian coordinates |
| Greek Symbols | = | |
| α | = | thermal diffusivity (m2 · s−1) |
| β | = | thermal expansion coefficient (K−1) |
| γ = Tc/Th | = | temperature parameter |
| Δ | = | dimensional distance between vertical walls (m) |
| Δτ | = | dimensionless time step |
| ϵ | = | surface emissivity |
| η, ξ | = | Cartesian coordinates of computational domain |
| Θ | = | dimensionless temperature |
| κ | = | number of undulations |
| ν | = | kinematic viscosity (m2 · s−1) |
| ρ | = | fluid density (kg · m−3) |
| σ | = | Stefan–Boltzmann constant |
| τ | = | dimensionless time |
| Ψ | = | dimensionless stream function |
| Ω | = | dimensionless vorticity |
| Subscripts | = | |
| c | = | cooled wall |
| h | = | heated wall |
Nomenclature
| a | = | shape parameter |
| A = H/D | = | aspect ratio |
| b | = | wavy contraction ratio |
| D | = | length of cavity (m) |
| Fk−i | = | view factor from k-th to i-th element of cavity |
| g | = | gravitational acceleration (m · s−2) |
| H | = | height of cavity (m) |
| k | = | thermal conductivity (W · m−1 · K−1) |
| = | radiation number | |
| = | average Nusselt number at left wavy wall | |
| NS | = | number of surface elements in cavity |
| p | = | dimensional pressure (Pa) |
| Pr = ν/α | = | Prandtl number |
| Qrad | = | dimensionless net radiative heat flux |
| = | Rayleigh number | |
| Rk | = | dimensionless radiosity of k-th element of cavity |
| t | = | dimensional time (s) |
| T | = | dimensional temperature (K) |
| Tc | = | dimensional cooled wall temperature (K) |
| Th | = | dimensional heated wall temperature (K) |
| T0 = (Th + Tc)/2 | = | dimensional mean temperature of heated and cooled walls (K) |
| u | = | dimensional velocity component in x-direction (m · s−1) |
| v | = | dimensional velocity component in y-direction (m · s−1) |
| U | = | dimensionless velocity component in X-direction |
| V | = | dimensionless velocity component in Y-direction |
| x, y | = | dimensional Cartesian coordinates (m) |
| x1 | = | dimensional location of left vertical wall (m) |
| x2 | = | dimensional location of right vertical wall (m) |
| X, Y | = | dimensionless Cartesian coordinates |
| Greek Symbols | = | |
| α | = | thermal diffusivity (m2 · s−1) |
| β | = | thermal expansion coefficient (K−1) |
| γ = Tc/Th | = | temperature parameter |
| Δ | = | dimensional distance between vertical walls (m) |
| Δτ | = | dimensionless time step |
| ϵ | = | surface emissivity |
| η, ξ | = | Cartesian coordinates of computational domain |
| Θ | = | dimensionless temperature |
| κ | = | number of undulations |
| ν | = | kinematic viscosity (m2 · s−1) |
| ρ | = | fluid density (kg · m−3) |
| σ | = | Stefan–Boltzmann constant |
| τ | = | dimensionless time |
| Ψ | = | dimensionless stream function |
| Ω | = | dimensionless vorticity |
| Subscripts | = | |
| c | = | cooled wall |
| h | = | heated wall |
Acknowledgements
This work was conducted as a government task for the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K.