Numerical Analysis of Natural Convection of Thermoelectrically Conducting Fluids in a Square Cavity under a Uniform Magnetic Field. Estimation of Induced Heating Term.

A mixed‐interpolation finite element method for incompressible thermal flows of electrically conducting fluids

“…Here, the mean Nusselt number N u at the hot or cold wall is defined by It is confirmed for both Hartmann numbers that the values of the mean Nusselt number at the walls draw close to each other with time, which indicates that the time varying MHD convections make the transition to the convections at steady state. These results are in good agreement with the convergence histories of N u for the two cases of a low Eckert number, which are obtained under the same calculation conditions (but with a different numerical scheme), in Reference . Figure shows the vector plots of the velocity and magnetic flux density for the two different Hartmann numbers at t = 10 5 .…”

“…The fixed dimensionless quantities in this numerical analysis are as follows: R e = G r = 6.967 × 10 6 , R m = 10 5 , P e = 1.75 × 10 5 , and E c = 10 − 4 . These parameter values are used in Reference for the cases of a low Eckert number. Notice that the magnetic Reynolds number is increased by a factor of 10 5 compared with the previous channel flow problem.…”

“…where Equations (16) and (17) are derived using Equations (13) and (15), respectively. After the error of the discretized continuity equation becomes smaller than a specified convergence criterion, the iteratively corrected velocity and pressure values are replaced by v nC1 D O v .mC1/ and p nC1 D O p .mC1/ , respectively.…”

“…Magnetohydrodynamic (MHD) fluid flows for liquid metals, which are viscous incompressible flows of conducting fluids, have been extensively studied for decades and many valuable theoretical and experimental achievements have been made in various flow regimes [1][2][3][4][5][6] (see, in particular, Reference [2] for details on engineering applications such as magnetic damping in casting operations and magnetic levitation using a high-frequency induction coil). In order to investigate detailed flow structures and their associated electromagnetic fields in a wide range of MHD problems, many numerical methods have also been proposed in the framework of finite difference, finite volume, finite element, and spectral discretizations (e.g., References [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] in which numerical articles pertaining to finite element methods are mainly selected). Although these numerical methods have been proved to be useful in two-dimensional (2D) and three-dimensional (3D) simulations of fluid flows confined in relatively simple geometries, it is much desired, especially for the calculation of MHD flows in engineering applications, that numerical schemes have the capability to generate accurate solutions to the problems involving arbitrary-shaped boundaries within a reasonable amount of time.…”

Magneto hydrodynamic buoyant convection of an electrically conducting fluid in a tilted square cavity

“…Oki and Tanahashi [6] proposed a version of generalized simplified marked and cell finite element method to solve unsteady magneto hydrodynamic natural convection in a square cavity considering the effects of Joulean heating. It was pointed out that when Joulean heating is small, the effect of convection inhibition by Lorentz force becomes more significant.…”