An accurate numerical solution study of three-dimensional natural convection in a box

This paper describes the application of the finite difference method to the simulation of three-dimensional natural convection in a box. The velocity–vorticity formulation is employed to represent the mass, momentum, and energy conservations of the fluid medium. We employ a fractional time marching technique for solving seven field variables involving three velocity, three vorticity and one temperature components. By using the fast Fourier transform (FFT) and a tridiagonal matrix algorithm (TDMA), the velocity Poisson equations are advanced in space along with the continuity equation, thus solving efficiently and easily the diagonally dominant tridiagonal matrix equations. Both vorticity and energy equations are discretized through an explicit method (Adams–Bashforth central difference scheme) as a simplified numerical scheme for solving 3D problems, which otherwise requires enormous computational effort. A natural convection in a box for the Rayleigh number equal to 10⁴, 10⁵, 10⁶ and 10⁷ as well as As = L_x / L_z aspect ratios varying from 0.25 to 4 is investigated. It is shown that the benchmark results for temperature and flow fields could be obtained using the present algorithm.