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 exchanger, 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]. They also establish the mentioned parameters that have meaningful effect on average Nusselt number at the hot wall and average temperature of the fluid in the enclosure. Saika et al. [9] investigated effect of Hartmann Number on Free Convective Flow of MHD Fluid in a Square Cavity with a Heated Cone of Different Orientation. They reported that a sufficiently large magnetic field can potentially stop fluid movement altogether and heat transfer would be fully by conduction. Hakan and Khaled et al. [10] studied MHD mixed convection in a lid-driven cavity with corner heater. Finite Element Analysis of Magnetohydrodynamic Mixed Convection in a Lid-Driven Trapezoidal Enclosure Having Heated Triangular Block was researched by Sajjad and Alim et al. [11].

The above literature review motivates the author’s, to give attention to the 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.

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 B₀ is applied parallel to the x-axis.

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

∂ u ∂ x + ∂ u ∂ y = 0 (1)

Momentum Equations

ρ ( ∂ u ∂ x + ∂ u ∂ y ) = − ∂ u ∂ x + μ ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) (2)

ρ ( ∂ u ∂ x + ∂ u ∂ y ) = − ∂ u ∂ y + μ ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) + ρ g β ( T − T c ) − σ B 0 2 v (3)

Energy Equation

u ∂ T ∂ x + v ∂ T ∂ y = α ( ∂ 2 T ∂ x 2 + ∂ 2 T ∂ y 2 ) (4)

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, B₀ 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:

X = x L , Y = y L , U = u L α , V = v L α

P = p L 2 ρ α 2 , θ = T − T c T h − T c , σ = ρ 2 α L 2 , α = k ρ C p

Introducing the previous dimensionless variables, the following dimensionless forms of the governing equation are obtained as follow:

Continuity Equation

∂ U ∂ X + ∂ V ∂ Y = 0 (5)

Momentum Equations

U ∂ U ∂ X + V ∂ V ∂ Y = − ∂ P ∂ X + P r ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) (6)

U ( ∂ V ∂ X + ∂ V ∂ Y ) = − ∂ P ∂ Y + P r ( ∂ 2 V ∂ X 2 + ∂ 2 V ∂ Y 2 ) + R a P r θ − H a 2 P r V (7)

Energy Equation

U ∂ θ ∂ X + V ∂ θ ∂ Y = ( ∂ 2 θ ∂ X 2 + ∂ 2 θ ∂ Y 2 ) (8)

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 conductivity of fluid.

The dimensionless parameters are the Prandtl number Pr, Hartmann number Ha and Rayleigh number Ra. which are defined as:

P r = v α , H a 2 = σ B 0 2 L 2 μ , R a = g β L 3 ( T h − T c ) P r v 2

The simulation domain with boundary conditions is shown in Figure 1. Which are No-slip conditions are applied at all cavity boundaries, i.e. u = v = 0. The left and right sidewalls are considered adiabatic. The lower wall is subjected to isothermal heat temperature (T_h). The upper wall is subjected to isothermal cold temperature (T_c). The internal triangular obstacle is taken thermally insulated everywhere, i.e. d T d y = 0 .

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

differential equations which are transferred into a system of integral equations. The quadratic triangular element is used to develop the finite element equations. Different types of grid densities have been selected to assess the accuracy of the numerical simulation procedure. These modified nonlinear equations are transferred in linear algebraic equations with the aid of Newton’s method. Finally, the triangular factorization method is applied for solving those linear equations.

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.

Following portion the estimates are completed for Prandtl number Pr = 0.71, the Rayleigh number Ra = 10³, 10⁵, 5 × 10⁵, 10⁶ and for Hartmann numberHa = 0, 20, 50, 100. The outcomes are pictorial with isotherms, streamlines, velocity profiles, dimensionless temperature and the local Nusselt number.

Streamlines and Isotherms are shown in Figure 3. From the streamlines figure, many cells can be observed inside the cavity. For Ha = 0 where there is no magnetic field are presented in Figure 3 to comprehend the effects of Rayleigh number on the flow field and temperature distribution. At Ra = 10³ and in the non-existence of the magnetic field (Ha = 0) are created four elliptic-shaped cells appear on the top-right, top-left, bottom-right and bottom-left corner of the triangular heated block of the cavity shown in the left side of Figure 3(a). The top-left cells rotate anticlockwise and the top-right cells rotate clockwise. For

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 3(b) atRa = 10³ and Ra = 10⁵. With increasing Rayleigh number, isotherms are many submission which means increasing heat transfer through convection. The differences are clean between Ra = 5 × 10⁵ and Ra = 10⁶ are shown on the right side of Figure 3(c) and Figure 3(d).

Streamlines and Isotherms are shown in Figure 4 where Rayleigh number (Ra = 10⁵) is fixed. Ha is increased to 0, 20, 50 and 100. At Ra = 10⁵ and increasing Ha = 0 to 100 are created four elliptic-shaped cells be present on the top-right, top-left, bottom-right and bottom-left corner of the triangular heated block of the cavity shown in the left side of Figure 4(a). The top-left cells rotate anticlockwise

and the top-right cells rotate clockwise. For higher Hartmann numbers almost the same result as Figures 4(a)-(c) but flow power increases and closer to one another are shown in Figure 4(d). The right side of Figure 4 shows the isotherms for various values of Hartmann number (Ha) with Pr = 0.71 and Ra = 10⁵. Stream function has a symmetrical value about the vertical centerline as the triangular heated obstacle is symmetrical. It is observed that Ha does not have more effect on heat transfer with high Ra. The differences are clean between Ha = 50 and Ha = 100 are shown on the right side of Figure 4(c) and Figure 4(d).

The y-component of velocity along the line parallel to the x axis (y = 0.15) is presented. Figure 5(a) and Figure 5(b) display the effect of different Rayleigh number (Ra) and Hartmann number (Ha) with Pr = 0.71 on the y-component of velocity at y = 0.15. Figure 5(a) display the four profiles corresponding to the four values of Ha which are 0, 20, 50 and 100. It can be visible from Figure 5(a) that the velocity has a larger change for the lower Hartmann number value. Velocity

grows maximum at two locations which is at x = 0.06 and x = 0.14 when Ha = 0. Figure 5(b) displays the four profiles corresponding to the four values of Rayleigh number (Ra) which are 10³, 10⁵, 5 × 10⁵ and 10⁶. It can be visible from Figure 5(b) that velocity has a big change for a higher Rayleigh number value. As seen from Figure 5(a) and Figure 5(b) there is no value of velocity on the triangular heated block. The local Nusselt number at different Hartmann number (Ha) and Rayleigh number (Ra) with Pr = 0.71 on y-component at y = 0.15 is shown in Figure 6(a) and Figure 6(b). At Ha minimum, we get maximum values and atHa maximum, we get minimum shape curve which is shown in Figure 6(a). For different Hartmann numbers, we get different curves. Again the local Nusselt number at a various Rayleigh number with Pr = 0.71 and Ha = 0 fixed on the y-component at y = 0.15 is shown in Figure 6(b). AtRa maximum we get maximum shape curve and if Ra minimum we get the minimum value.

A numerical study investigates Magneto-hydrodynamic natural convection fluid flow and heat transfer in a semi-circular top square cavity filled with heated triangular obstacles. The fluid considered is air. The governing equation of mass, energy equation and momentum were solved using the Galerkin weighted residual method of finite element method. For all cases considered four counter-rotating eddies were shaped inside the cavity. From the present investigation the following conclusions may be drawn as: Rate of heat transfer enhanced for the increase in the buoyancy force via an increase in Rayleigh number with Ha = 0 and reduced for the increasing value of Hartmann number. And also magnetic field strength plays a significant role to control parameters for heat transfer and fluid flow.

The authors declare no conflicts of interest regarding the publication of this paper.

Runa, A.A., Alim, Md.A., Alam, Md.S. and Kabir, K.H. (2022) Finite Element Analysis of Magnetohydrodynamic Natural Convection within Semi-Circular Top Enclosure with Triangular Obstacles. American Journal of Computational Mathematics, 12, 33-43. https://doi.org/10.4236/ajcm.2022.121004