Numerical Calculation of Three-Dimensional Turbulent Natural Convection in a Cubical Enclosure Using a Two-Equation Model for Turbulence

This paper presents a model and numerical results for turbulent natural convection in a cubical enclosure heated from below, cooled on a portion of one vertical side wall and insulated on all other surfaces. Three-dimensional balances were derived for material, energy, and the three components of momentum, as well as for the turbulent kinetic energy k and the rate of dissipation of turbulent kinetic energy ε. The constants used in the model were the same as those used by Fraikin et al. for two-dimensional convection in a channel. Illustrative transient calculations were carried out for Ra = 10⁶ and 10⁷ and Pr = 0.7. Both the dominant component of the vector potential and the Nusselt number were found to converge to a steady state. Isothermal lines and velocity vectors for vertical cross sections normal to the cooled wall indicated three-dimensional effects near the side walls. A top view of the velocity vectors revealed a downward spiral flow near the side walls along the cooled vertical wall. A weak spiral flow was also found along the side walls near the wall opposing the partially cooled one. The highest values of the eddy diffusivity were 2.6 and 5.8 times the molecular kinematic viscosity for Ra = 10⁶ and 10⁷, respectively. A coaxial double spiral movement, similar to that previously reported for laminar natural convection, was found for the time-averaged flow field. This computing scheme is expected to be applicable to other thermal boundary conditions.