Numerical study of magneto-convection inside an enclosure using enhanced stream function-vorticity formulation

The buoyancy-driven magneto-hydrodynamic ow is investigated using a twodimensional numerical simulation in a lled square enclosure, which is heated by a uniform volumetric heat density. The top and bottom walls of the enclosure are adiabatic and the side walls have constant temperature. A xed magnetic eld is applied in the direction normal to the left side wall of the enclosure. The stationary dimensionless governing equations are solved numerically for the stream function, vorticity and temperature, with nite dierence method, using MATLAB software. The Prandtl (Pr) number of the uid is assumed to be 0.733; the Rayleigh (Ra) number is made to vary from 104 to 107 and the Hartmann (Ha) number is between 0 and 103. The stream function equation is solved using fast Poisson's equation solver on a rectangular grid (POICALC function in MATLAB) vorticity, and temperature equations are solved using the red-black Gauss-Seidel and biconjugate gradient stabilized (BiCGSTAB) methods, respectively. The proposed method is fast, and there is no need for the variables under-relaxation. It is interesting to know that convergence of this method is superior to other segregated algorithms. The ratio of the Lorentz force to the buoyancy force (Ha2/Ra) is introduced as an index to compare the contribution of natural convection with the magnetic eld strength on heat transfer.