Numerical Calculation of Three-Dimensional MHD Natural Convection Based on Spectral Collocation Method and Artificial Compressibility Method

Abstract

The flow and heat transfer characteristics in a three-dimensional cavity filled with a conducting fluid are investigated in this study in the presence of a constant magnetic field. The three-dimensional Navier–Stokes equations and energy equation are solved directly using a self-developed method, SCM–ACM, which combines the spectral collocation method (SCM) with high-precision and exponential convergence, and the artificial compressibility method (ACM) with easy implementation and good numerical stability. In this paper, we examine the effects of Hartmann numbers (Ha) ranging from 0 to 100, magnetic field directions, and Grashof numbers (Gr) ranging from [Formula: see text] to [Formula: see text] on the structure of the flow and temperature fields, with a Prandtl number (Pr) of 0.71. The results show the Grashof and Hartmann numbers have a significant impact on the flow and temperature distribution in the middle of the cavity, but little effect on that near the walls. As the Grashof number increases, a stable thermal stratification is formed at the center of the cube, and thermal boundary layers are formed near the horizontal walls. The increase in Grashof number enhances the heat transfer rate and increases the temperature difference between the upper hot fluid and the lower cold fluid in the cube. Furthermore, the increase in Grashof number enhances the convective intensity between the isothermal walls, leading to the formation of more vortices, which move toward the corners due to the combined action of centrifugal force and inertia. The Hartmann number has stabilizing effects on the flow and weakens the heat transfer. With a higher Grashof number the magnetic effects become more notable. When [Formula: see text], the magnetic effects are no longer significant. The magnetic field [Formula: see text] perpendicular to the main circulation plane exhibits the strongest stability for the flow.