Finite Element Analysis of Magnetohydrodynamic Natural Convection within Semi-Circular Top Enclosure with Triangular Obstacles

The phenomena of magneto-hydrodynamic natural convection in a two-dimensional semicircular top enclosure with triangular obstacle in the rectangular cavity were studied numerically. The governing differential equations are solved by using the most important method which is finite element method (weighted-residual method). The top wall is placed at cold Tc and bottom wall is heated Th. Here the sidewalls of the cavity assumed adiabatic. Also all the wall are occupied to be no-slip condition. A heated triangular obstacle is located at the center of the cavity. The study accomplished for Prandtl number Pr = 0.71; the Rayleigh number Ra = 103, 105, 5 × 105, 106 and for Hartmann number Ha = 0, 20, 50, 100. The results represent the streamlines, isotherms, velocity and temperature fields as well as local Nusselt number.


Introduction
Convection is a mode of heat transfer which takes place through the movement of collective masses of heated atoms and molecules within fluids such as gases and liquids, including molten rock. Application of natural convection heat transfer is very important in science, engineering researcher and fields such as thermal insulation, heating and cooling buildings, solar collector, heat exchang-er, crystal growing, food processing, energy drying processes, lakes and geothermal reservoirs, nuclear energy, underground water flow, etc. Several numerical and experimental systems have been advanced to investigate flow characteristics inside the cavities with and without obstacles. Most of the cavities repeatedly used in industries are rectangular, square, trapezoidal, cylindrical, elliptic and triangular, etc.
Earlier studies were mainly developed on physics of the various flow systems in different cavity. Reddy [1] introduced finite element analysis to develop the energy and momentum equations subject to the boundary conditions simultaneously and the finite element solutions of differential equations with constant coefficients are exact at the nodes. Hussein et al. [2] developed entropy generation analysis of a natural convection inside a sinusoidal surrounding with various shapes of cylinders. Their outcomes showed that the entropy generations due to heat transfer, fluid friction and total entropy generation enhance with increasing values of Rayleigh number, while the local Bejan number decreases.
Arun et al. [3] investigate on natural convection heat transfer problems by Lattice Boltzmann Method. They found the importance of the natural convection problem and the ability of lattice Boltzmann method to apply in the computational field. Seyyedi et al. [4] studied natural convection heat transfer under constant heat flux wall in a nanofluid filled annulus enclosure. Sheikholeslami et al. [5] investigated natural convection heat transfer in a nanofluid filled inclined L-shaped enclosure. It can be terminated that the turning angle of the enclosure can be a control parameter for heat and fluid flow. The outcomes publish that average Nusselt number is an increasing function of nanoparticle volume fraction and Rayleigh number. Natural convection in a triangular top wall enclosure with a solid strip was extensively experimentally and numerically researched by Hussain et al. [6]. Bhuiyan et al. [7] studied magneto hydrodynamic natural convection in a square cavity with semicircular heated obstacle. They investigated the effect of Rayleigh and Hartmann numbers on the flow field in a square cavity with semicircular heated block along uniform magnetic field. Numerical simulation of natural convection in a rectangular lacuna with triangles of different orientation in presence of magnetic field was researched by Alam et al. [8]. problem of natural convection in enclosure with obstacles of different shapes like triangles, circles, solid strips and so on. To the best of author's knowledge no investigation has been done yet on finite element analysis of MHD natural convection within semi-circular top enclosure with triangular obstacle. So the proposed study is to address the issue.

Physical Configuration
The physical model deliberated in the present study of a two-dimensional rectangular cavity and semi-circular top enclosure with heated triangular obstacles is shown in FIGURE1 is considered for simulation purposes. The height and the width of the cavity are denoted by L. The lower wall is kept at heated (T h ) and the upper wall is kept atcold (T c ) under all situations T h > T c condition is maintained. The right and left wall are adiabatic. The gravitational force g, acts vertically downward. The magnetic field of strength B 0 is applied parallel to the x-axis.

Mathematical Formulation
The flow is considered steady, laminar, incompressible and two-dimensional. The field of governing equations solved during the simulation for the free convection flow inside the domain are, conservation of mass, momentum and energy can be written as: Continuity Equation Energy Equation where x and y are the distances measured along the horizontal and vertical directions respectively; u and v are the velocity components in the x and y directions respectively; T denote the fluid temperature, T c denotes the reference temperature for which buoyant force vanishes, ρ is the fluid density, g is the acceleration due to gravity, β is the volumetric coefficient of thermal expansion, σ is the electrical conductivity, B 0 is the magnetic induction, α is the thermal diffusivity and ν kinematic viscosity of the fluid.
The governing equations are non-dimensionalized using the following dimensionless variables: Introducing the previous dimensionless variables, the following dimensionless forms of the governing equation are obtained as follow: Continuity Equation Momentum Equations Energy Equation where X and Y are the coordinates varying along horizontal and vertical directions, respectively, U and V are the velocity components in the X and Y directions, respectively, θ is the dimensionless temperature and P is the dimensionless pressure. C p is the fluid specific heat at constant pressure; k is the thermal con-

Numerical Procedure
The dimensionless governing equations are solved with the required boundary conditions using the finite element method, the Galerkin weighted residual technique. Mass, momentum and energy are two dimensional non-linear partial

Validation of Numerical Procedure
Validation of the numerical procedure was made by comparing streamlines and isotherms with results shown in Figure 2(a) and Figure 2(b) by Bhuiyan et al.
(2014). They investigate the effect of magnetic field in a square cavity with semicircular heated block. From this figures as seen the obtained outcomes show excellent agreement.  higher Rayleigh numbers almost the similar outcomes as Figure 3(a) but flow power increases are shown in Figures 3(b)-(d). The right side of Figure 3 shows the isotherms for different values of Rayleigh number (Ra) with Pr = 0.71 and Ha = 0. The dimensionless temperature whose value range is 0 -1. Stream function has a symmetrical value about the vertical center line as the triangular heated obstacle is symmetrical. It is observed that isothermal lines slightly move from the heated surfaces to cold surfaces on the right side of Figure 3(a) and     Figure 6(b). At Ra maximum we get maximum shape curve and if Ra minimum we get the minimum value.