FEM solution of natural convection flow in square enclosures under magnetic field

Abstract

PurposeThe purpose of the paper is to obtain finite element method (FEM) solution of steady, laminar, natural convection flow in inclined enclosures in the presence of an oblique magnetic field. The momentum equations include the magnetic effect, and the induced magnetic field due to the motion of the electrically conducting fluid is neglected. Quadratic triangular elements are used to ensure accurate approximation for second order derivatives of stream function appearing in the vorticity equation.Design/methodology/approachGoverning equations in terms of stream function and vorticity are solved by FEM using quadratic triangular elements. Vorticity boundary conditions are obtained through Taylor series expansion of stream function equation by using more interior stream function values to improve the accuracy. Isothermally heated or cooled and/or adiabatic conditions for the temperature are imposed. Results are obtained for Rayleigh number values and Hartmann number values up to 1000000 and 100, respectively.FindingsIt is observed that streamlines form a thin boundary layer close to the heated walls as Ha increases. The same effect is seen in the vorticity contours, and isotherms are not affected much. As Ra increases streamlines are deformed moving from the heated walls through cooled walls. Vorticity starts to develop boundary layers close to heated and adjacent walls. Isotherms are pushed towards the sinusoidally heated wall whereas in the case of linearly heated left and bottom walls they expand towards cooled part of the cavity as Ra increases.Originality/valueThe application of FEM with quadratic elements for solving natural convection flow problem under the effect of a magnetic field is new in the sense that the results are obtained for large values of Rayleigh and Hartmann numbers.