Investigation of Magnetic Field Effect on Natural Convection Flow using Fast Method

ABSTRACT: The laminar steady magneto-hydrodynamics natural convection in a filled square enclosure with internal heat generation is investigated by two-dimensional numerical simulation. The enclosure is heated by a uniform volumetric heat density and walls have constant temperature. A fixed magnetic field is applied to the enclosure. The dimensionless governing equations are solved numerically for the stream function, vorticity and temperature using finite difference method for various Rayleigh (Ra) and Hartmann (Ha) numbers in MATLAB software. The stream function equation is solved using fast Poissonʹs equation solver on a rectangular grid (POICALC function in MATLAB), voricity and temperature equations are solved using red-black Gauss-Seidel and bi-conjugate gradient stabilized (BiCGSTAB) methods respectively. The proposed method is fast and there is no need to the under-relaxation factors for variables compared to commercial methods (e.g. SIMPLE) and is applicable for high Rayleigh numbers in steady state. The ratio of the Lorentz force to the buoyancy force (Ha2/Ra) is introduced as an index to compare the contribution of natural convection and magnetic field strength on heat transfer.