Benchmark solutions for natural convection in a cubic cavity using the high-order time-space method

Benchmark numerical solutions for a three-dimensional natural convection heat transfer problem in a cubical cavity are presented in this paper. The 3-D cavity has two differentially heated and isothermal vertical walls and also four adiabatic walls. The computations are conducted for three Rayleigh numbers of 10⁴, 10⁵ and 10⁶. The filled fluid is with air and the Prandtl number is fixed at 0.71. The computed results are efficiently obtained by using the time-space method, which was proposed by Saitoh (1991) as a highly efficient and fast solver for general heat transfer and fluid flow problems. In our computations, the high-accuracy finite differences of a fourth-order were employed for the spatial discretization of governing equations and boundary conditions. In addition the third-order backward finite difference was used in timewise discretization. The resultant converged flow and temperature characteristics are also presented. The spatial grid dependency of the solutions was examined on a uniform grid. In addition, the grid-independent benchmark solutions were obtained by Richardson extrapolation for three cases. The present benchmark solutions will be useful for checking the performance and accuracy of any numerical methodologies.