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

Abstract

Purpose -The 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/approach -Governing 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. Findings -It 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/value -The 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.

Time dependent magnetic field effects on the MHD flow and heat transfer in a rectangular duct

“…Thus, the associated momentum equations can be written as (see e.g., in ref. [10] and the therein references):…”

“…Thus, the associated momentum equations can be written as (see e.g., in ref. [10] and the therein references): $$$$\begin{equation} \def\eqcellsep{&}\begin{array}{l}\displaystyle \frac{\partial \bar{u}}{\partial \bar{t}} = \frac{\mu }{\rho } \Delta \bar{u} - \bar{u}\frac{\partial \bar{u}}{\partial \bar{x}} -\bar{v} \frac{\partial \bar{u}}{\partial \bar{y}} - \frac{1}{\rho } \frac{\partial \bar{p}}{\partial \bar{x}}, \\[3pt] \displaystyle \frac{\partial \bar{v}}{\partial \bar{t}} = \frac{\mu }{\rho } \Delta \bar{v} - \bar{u}\frac{\partial \bar{v}}{\partial \bar{x}} -\bar{v} \frac{\partial \bar{v}}{\partial \bar{y}} - \frac{1}{\rho } \frac{\partial \bar{p}}{\partial \bar{y}} - \frac{\sigma }{\rho } B_0^2(\bar{t}) \bar{v}, \\[3pt] \displaystyle \frac{\partial \bar{u}}{\partial \bar{x}} + \frac{\partial \bar{v}}{\partial \bar{y}} = 0, \end{array} \end{equation}$$$$with $$\Delta$$ being the Laplace operator.…”

“…A finite element method has been employed in ref. [10] to approximate a steady natural convection flow in inclined enclosures under the presence of an oblique magnetic field. A more recent finite element method study analysing the natural convection MHD flow and heat transfer characteristics in a corrugated rectangular cavity has been presented in ref.…”

Simulation of coupled elasticity problem with pressure equation: hydroelastic equation

“…The finite element method (FEM) is a popular method for numerically solving differential equations that appear in mathematical and engineering modeling (Lewis et al, 1989(Lewis et al, , 1990Sheshachala and Codina, 2018;Sudheesh and Siva Prasad, 2011;T€ urk and Tezer-Sezgin, 2013). Lewis et al (2007) developed a three-dimensional finite element model for the numerical simulation of metal displacement and heat transfer in the squeeze casting process.…”

Heatlines visualization of convective heat flow during differential heating of porous enclosures with concave/convex side walls

“…In the past few decades, an enormous expansion in the research on convection heat transfer is observed due to versatile natural, industrial and engineering applications such as solidification (Damronglerd et al , 2012; Stapor, 2016), chemical reactions (Chamkha et al , 2012), melting (Wu and Lacroix, 1992, 1993; Lacroix and Benmadda, 1998; Kousksou et al , 2014), cooling (Shuja et al , 2010), thermo-bioconvection (Kuznetsov, 2013) and pool boiling (Elrais et al , 1992). The knowledge of heat and flow characteristics during natural convection within various enclosed cavities is important, and a number of works on natural convection within enclosed cavities for the fluid media via different computational techniques are found in the literature (Turk and Tezer-Sezgin, 2013; Rahman et al , 2014; Salama et al , 2014; Aghighi et al , 2015; Canovas et al , 2015; Miroshnichenko and Sheremet, 2015; Niknami et al , 2015; Yapici and Obut, 2015; Noghrehabadi et al , 2015). In addition to the clear fluid media, the analysis of convection within porous enclosures has been a self-sustained area of research in the heat transfer community and the extensive discussions on natural convection in porous media with the mathematical aspects are documented in various books (Holzbecher, 1998; Bejan, 2013; Ingham and Pop, 1998; Vafai and Hadim, 2000; Pop and Ingham, 2001; Nield and Bejan, 2016).…”