Analysis of MHD Mixed Convection Flow Within a Square Enclosure Containing a Triangular Obstacle

Finite element method is used to solve two-dimensional governing mass, momentum and energy equations for steady state, mixed convection problem inside a lid driven square enclosure containing a triangular hot obstacle located at the centre of the enclosure. The enclosure top wall is adiabatic while the bottom wall and triangular obstacle are maintained at a uniform temperature higher than the vertical side walls. The left vertical wall is moving with a uniform velocity by unity from bottom to top. All solid boundaries are in no slip condition. The aim of the study is to describe the effect of magnetic field, Prandtl number and the size of triangular obstacle on mixed convection fluid flow, heat transfer and temperature of the fluid. The investigation is conducted for various values of magnetic parameter Ha, obstacle size (area) A, Richardson number Ri, and Prandtl number Pr. Various results such as streamlines, isotherms, heat transfer rates in terms of average Nusselt number Nu, and average temperature θav of the fluid in the enclosure are presented for different parameters. It is observed that the obstacle size and dimensionless parameters Ha and Pr have significant effect on both the flow and thermal fields. The results also indicate that the average Nusselt number at the heated surface is strongly dependent on the configurations of the system studied under different geometrical and physical configurations.


Introduction
The study of mixed convection of electrically-conducting fluid in the magnetohydrodynamic (MHD) devices such as coolers of nuclear reactors, thermal insulators and microelectronic devices should account for the effect of a transverse magnetic field on the fluid flow and the heat transfer mechanism. It is found that the fluid flow experiences a Lorentz force due to the influence of magnetic field.
Combined free and forced convective flow in lid driven cavities occurs as a result of two competing mechanisms. The first one is due to shear flow caused by the movement of one of the walls of the enclosure, while the following one is due to buoyancy flow produced by thermal non-homogeneity of the enclosure boundaries. Analysis of mixed convective flow in a lid-driven enclosure finds applications in material processing, flow and heat transfer in solar ponds, dynamics of lakes, reservoirs and cooling ponds, crystal growing, float glass production, metal casting, food processing, galvanizing, and metal coating.
There are many investigations on mixed convective flow in lid-driven cavities. Many different configurations and combinations of thermal boundary conditions have been considered and analyzed by various investigators.

Literature Review
Alchaar et al [1] studied natural convection heat transfer in a rectangular enclosure with a transverse magnetic field. In this paper the effect of, transverse magnetic field on buoyancydriven convection in a shallow rectangular cavity is numerically investigated. They found that at high Hartmann number, the velocity gradient in the core revealed by both analytical and numericalanalysesisconstant outside the two Hartmann layers at the vicinity of the walls normal tothe magnetic field. Akhter and Alim [2] studied the effects of pressure work on natural convection flow around a sphere with radiation heat loss. They found that increasing Prandtl number leads to decrease the velocity and temperature, increase the skin friction and reduce the rate of heat transfer. Mousa [3] investigated the modeling of laminar buoyancy convection in a square cavity containing an obstacle. He found that in case of low Rayleigh numbers (10 2 -10 4 ), the rate of heat transfer decreases when the aspect ratio of the adiabatic square obstacle increases. Nasrin [4] investigated mixed magnetoconvection in a lid driven cavity with a sinusoidal wavy wall and a central heat conducting body. She considered a heat conducting square body located at the center of cavity. The cavity horizontal walls are perfectly insulated while the corrugated right vertical surface is maintained at a uniform temperature higher than the left lid. She found that the influence of Ha does not affect significantly the thermal current activities. But the flow pattern changes dramatically owing to the hindrance of the imposed magnetic field. Aydin [5] studied the effects of moving wall on aiding and opposing mechanisms of mixed convection in a shear and buoyancy driven cavity. Oztop and Dagtekin [6] extended this idea to two-sided lid driven cavity. They investigated on mixed convection in two-sided lid-driven differentially heated square cavity. Saha et al [7] investigated numerically mixed convection heat transfer in a lid driven cavity with wavy bottom surface. They found that Reynolds number, Grashof number and the number of undulations of the wavy surface have significant effect on the flow fields, temperature distributions and heat transfer in the cavity. Oztop et al [8] studied numerically MHD mixed convection in a liddriven cavity with corner heater. They reported that magnetic field plays an important role to control heat transfer and fluid flow. They showed that heat transfer decreases with increasing of Hartmann number and the rate of reduction is higher for high values of the Grashof number. Rahman and Alim [9] investigated MHD mixed convection flow in a vertical liddriven square enclosure including a heat conducting horizontal circular cylinder with Joule heating. They reported that mixed convection parameter Ri affects significantly on the flow structure and heat transfer inside the enclosure and the overall heat transfer decreases with the increase of joule heating parameter J . Moallemi and Jang [10] studied numerically mixed convective flow in a bottom heated square lid-driven enclosure. They investigated the effect of Prandtl number on the flow and heat transfer process. Prasad and Koseff [11] reported experimental results for mixed convection in deep liddriven cavities heated from below. They observed that the heat transfer is rather insensitive to the Richardson number. YadollahiFarsani and Ghasemi [12] investigated magnetohydrodynamic mixed convective flow in a cavity. They used a cavity of which lower surface is heated from below whereas other walls of the cavity are thermally isolated. They found that as Hartmann number increases the Nusselt number, representing heat transfer from the cavity decreases. In the present paper the main objective is to examine the fluid flow and heat transfer in a lid-driven square enclosure containing a triangular obstacle in presence of magnetic field.

Governing Equations
In the present problem, the flow is considered to be steady, two-dimensional, laminar, and incompressible and there is no viscous dissipation. The gravity force acts in the vertically downward direction, fluid properties are constant and fluid density variations are neglected except in the buoyancy term. Radiation effect is neglected. Under the usual Boussinesq approximation, the governing equations for the present problem can be described in dimensionless form by the following equations.
The dimensionless variables are defined as:

Numerical Validation
To validate the present numerical code, the results for mixed convection flow in an enclosed enclosure with heated upper wall have been compared with those obtained by Saha et al [7].

Numerical Technique
The numerical procedure used in this work is based on the Galerkin weighted residual method of finite element formulation. The application of this technique is well described by Taylor and Hood [13] and Dechaumphai [14]. In this method, the solution domain is discretized into finite element meshes, which are composed of non-uniform triangular elements. Then the nonlinear governing partial differential equations (i.e. mass, momentum and energy equations) are transferred into a system of integral equations by applying Galerkin Residual method. The integration involved in each term of these equations is performed by using Gauss's quadrature method. The nonlinear algebraic equations so obtained are modified by imposition of boundary conditions. These modified nonlinear equations are transferred into linear algebraic equations by Newton-Raphson iteration. Finally, these linear equations are solved by using Triangular Factorization method.

Effect of Hartmann Number
For

Effect of Triangular Obstacle Size
Effect of obstacle size on streamlines and isotherms is presented in fig.6 and fig.7 . The flow structure for the four different values of triangular obstacle area in the forced convection dominated region has been shown in the bottom row of fig.6. In absence of obstacle, it is seen that a uni-cellular vortex appears nearer to the left vertical moving wall, which is owing to the effect of shear force and buoyancy force. This shows that presence and increasing size of triangular obstacle has great effect on fluid temperature. This can be attributed to the fact that a large centered triangular obstacle narrows the regions available for warm fluid flows and enhances fluid temperature.

Effect of Prandtl Number
The influence of Prandtl number on the flow field at three different values of Ri is shown in fig.9  in the forced convection dominated region is shown in the bottom row of fig.9. Here the fluid flow is due to the shear induced force by the moving lid only. Next at 1 = Ri , the balance between the shear and buoyancy effect is manifested in the formation of a new small vortex near the obstacle. As Ri rises further to 10, the heat transfer is mostly by natural convection in the enclosure. Three vertical vortices are formed in two sides of the obstacle. Among these, the larger two are bi-cellular vortices and the smaller one is adjacent to lid driven wall. As Pr increases the smaller vortex becomes larger. Temperature field has been simulated using isotherms for the mentioned parameters. Effect of Prandtl number on isotherms is presented in fig.10  . For Pr=0.03 the isotherms nearer to side walls become almost parallel to the vertical walls resembling the conduction like heat transfer in the enclosure and the isotherms adjacent to obstacle are mostly nonlinear. Thus when the fluid is considered as liquid metal it has almost no effect on convection process. As Pr increases (the fluid is considered as air and water at various temperatures) the bending of isothermal lines increases and they are accumulating near the obstacle in the forced convection dominated region. But in the natural convection dominated region the isotherms depart from the hot wall and begin to crowd near the cold walls forming thermal boundary layers in the enclosure. As Pr

Conclusion
From this investigation, it is found that all the parameters magnetic field, Prandtl number, Richardson number and size of triangular obstacle play significant role on average heat transfer and fluid temperature. The major conclusions have been drawn as followings: Heat transfer rate is higher in absence of magnetic field. Presence of magnetic field decreases Nu drastically, but increasing value of Ha decreases Nu gradually. av θ increases abruptly for lower Ha but it increases linearly for higher Ha.