# Influence of the variable thermophysical properties on the turbulent buoyancy-driven airflow inside open square cavities

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## Abstract

The effects of the air variable properties (density, viscosity and thermal conductivity) on the buoyancy-driven flows established in open square cavities are investigated, as well as the influence of the stated boundary conditions at open edges and the employed differencing scheme. Two-dimensional, laminar, transitional and turbulent simulations are obtained, considering both uniform wall temperature and uniform heat flux heating conditions. In transitional and turbulent cases, the low-Reynolds *k* − ω turbulence model is employed. The average Nusselt number and the dimensionless mass-flow rate have been obtained for a wide and not yet covered range of the Rayleigh number varying from 10^{3} to 10^{16}. The results obtained taking into account variable properties effects are compared with those calculated assuming constant properties and the Boussinesq approximation. For uniform heat flux heating, a correlation for the critical heating parameter above which the burnout phenomenon can be obtained is presented, not reported in previous works. The effects of variable properties on the flow patterns are analyzed.

## Keywords

Mass Flow Rate Rayleigh Number Average Nusselt Number Critical Heat Flux Boussinesq Approximation## List of symbols

*b*Width of the vent, m (Fig. 1)

*C*_{1},*C*_{2}Correlation factors

*c*_{p}Specific heat at constant pressure, J kg

^{−1}K^{−1}*g*Gravitational acceleration, m s

^{−2}- Gr
_{H} Grashof number for isothermal cases,

*g*β(*T*_{ w }−*T*_{∞})*H*^{3}/ν_{∞}^{2}- Gr
_{H} Grashof number for heat flux cases,

*g*β*q**H*^{4}/ν_{∞}^{2}κ_{∞}*h*_{x}Local heat transfer coefficient, −κ(∂

*T*/∂*n*)_{ w }/(*T*_{ w }−*T*_{∞}), W m^{−2}K^{−1}*I*Turbulence intensity, Eq. 18

*k*Turbulent kinetic energy, Eq. 17, m

^{2}s^{−2}*H*Height of the cavity (and the heated wall) (Fig. 1), m

*L*Length of the cavity (Fig. 1), m

*L*_{c}Typical length, m

*M*Dimensionless mass-flow rate,

*m*/ρ_{∞}α_{∞}*m*Two-dimensional mass-flow rate, kg m

^{−1}s^{−1}*n*Coordinate perpendicular to wall, m

- Nu
_{H} Average Nusselt number based on

*H*, isothermal cases, Eq. 6- Nu
_{H} Average Nusselt number based on

*H*, heat flux cases, Eq. 7- Nu
_{x} Local Nusselt number,

*h*_{ x }*H*/κ*P*Average reduced pressure, N m

^{−2}*P*_{T}Total-average reduced pressure, N m

^{−2}*p*Pressure, N m

^{−2}- Pr
Prandtl number, μ

*c*_{ p }/κ*q*Wall heat flux, W m

^{−2}*R*Constant of the gas, J kg

^{−1}K^{−1}- Ra
_{H} Rayleigh number based on

*H*, (Gr_{ H })(Pr)*S*_{ij}Mean-strain tensor, s

^{−1}*T*,*T*′Average and turbulent temperatures, respectively, K

- \(-\overline{T'u_{j}}\)
Average turbulent heat flux, K m s

^{−1}*U*_{j},*u*_{j}Average and turbulent components of velocity, respectively, m s

^{−1}- \(-\overline{u_{i}u_{j}}\)
Turbulent stress, m

^{2}s^{−2}*u*_{τ}Friction velocity,

*u*_{τ}= (τ_{ w }/ρ)^{1/2}, m s^{−1}*x*,*y*Cartesian coordinates in the vertical and horizontal directions, m

*y*^{+}ρ

*y*_{1}*u*_{τ}/μ, with*y*_{1}the distance between the wall and the first grid point

## Greek symbols

- α
Thermal diffusivity, κ/ρ

*c*_{ p }, m^{2}s^{−1}- β
Coefficient of thermal expansion, 1/

*T*_{∞}, K^{−1}- δ
_{ij} Krönecker delta

- δ
_{T} Thickness of the thermal boundary layer, m

- ϕ
Dependent variable

- φ
Correlation factor, Eq. 28

- κ
Thermal conductivity, W m

^{−1}K^{−1}- \(\Uplambda\)
Heating parameter, Eqs. 2 and 4 for UWT and UHF heating, respectively

- μ
Viscosity, kg m

^{−1}s^{−1}- ν
Kinematic viscosity, μ/ρ, m

^{2}s^{−1}- θ
Dimensionless temperature difference, θ = (

*T*−*T*_{∞})/(*T*_{ w }−*T*_{∞})- ρ
Density, kg m

^{−3}- τ
_{w} Wall shear stress, N m

^{−2}- ω
Specific dissipation rate of

*k*(or turbulent frequency), s^{−1}

## Subscripts

*B*Constant properties and Boussinesq approximation

*c*Critical value

*l*Laminar

- max
Maximum value

- ref
Reference mesh

*t*Turbulent

*w*Wall

- ∞
Ambient or reference conditions

## Superscript

- −
Averaged value

## Abbreviations

- DNS
Direct numerical simulation

- UHF
Uniform heat flux

- UWT
Uniform wall temperature

## Notes

### Acknowledgments

We would like to acknowledge to Prof. J. Hernández (UNED, Spain), for his great teachings.

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