MHD Natural Convective Flow in an Isosceles Triangular Cavity Filled with Porous Medium due to Uniform/Non-Uniform Heated Side Walls

1 Introduction

Study of natural convection in fluid flow confined in the enclosure of any shape has attracted many researchers due to its numerous applications in engineering and industrial processes. Examples include energy efficient building, cooling system of nuclear reactor, fibrous insulation, heat transfer systems associated with boilers and condensers, energy efficient dryers, cooling systems of automobile engines and many others [1–6]. A new regime in this field started when convective heat transfer filled with porous media gained motivation due to a multitude of applications in thermal and geothermal energy, porous catalysts and soil pollution, etc. For this purpose, a number of models start from simplest to the complex, including the Darcy model [7], the Forchhiemer-extended Darcy model [8], the Brinkmann-extended Darcy model [9], and general model [10], which have been proposed in this article. Later on, we show the comprehensive physical and mathematical aspects of convection in porous media, a monograph written by Nield and Bejan [11] and several the books written by Pop and Ingham [12], Ingham and Pop [13] and Vafai [14, 15] among others., which serve as an excellent review on what has been achieved in this field up to now.

Conversely, several investigations in the form of research papers have been published in different science journals which are as follows: Tong and Subramanian [16] studied natural convection in a porous vertical enclosure using Brinkman-extended Darcy model. Considering no slip condition, they applied the modified Oseen technique to calculate the solution. Vafai and Tien [17] carried out a study on boundary and inertia effects on heat transfer and fluid flow in a porous medium. They used local volume averaging technique to establish governing equations and numerically investigated temperature and velocity profiles near impermeable boundary in a porous medium. Poulikakos and Bejan [18] also studied natural convection in a porous layer which is heated and cooled along one vertical side wall. Poulikakos et al. [19] again presented a study on convection in fluid overlying a porous bed with a high Rayleigh number. Numerical results were presented for various values of parameters, Rayleigh number Ra = 10²–10⁶, Darcy number Da = 10⁻⁷–10⁻⁴ and aspect ratio A = 0.2–1. Blake and Bejan [20] investigated the natural convection in water at temperature 4 °C saturated in a porous layer heated from below. The ranges of different parameters in this article are 200 ≤ Ra_p  ≤ 10⁴, 0.167 ≤ H/L ≤ 1, and 4 °C ≤ T_H  ≤ 8 °C. Baytas and Pop [21] have studied the free convection in a porous triangular cavity using the local thermal non-equilibrium (LTNE) model. Varol et al. [22] have analysed the natural convection in non-isothermally heated triangular cavity filled with porous media. They reported that heat transfer enhances when vertical and inclined walls were isothermal, while the bottom wall was kept at non-uniform temperature. Further, Varol et al. [23] investigated the steady-state free convection heat transfer in a right-angled triangular enclosure filled with porous media and concluded that an augmentation in heat transfer takes place with the decrease in aspect ratio. Recently Sun and Pop [24, 25], studied the steady-state free convection heat transfer and fluid flow in a porous triangular cavity filled with a nanofluid. Bhardwaj et al. [26] numerically analysed the heat transfer and entropy generation characteristics in a two-dimensional porous right-angled triangular enclosure with undulations on the left wall. The enclosure is heated sinusoidally from the bottom wall, while the left wall is maintained at a uniform temperature lower than the bottom one, and the right inclined wall is kept adiabatic. This study is very important in view of efficient utilization of energy resources that the process of natural convection should be efficient while transferring heat along with minimum degradation of energy, i.e. minimum entropy generation. Basak et al. [27] investigated natural convection in an isosceles triangular cavity filled with porous medium under various thermal boundary conditions. Bejan and Poulikakos [28] discussed non-Darcy regime for vertical boundary layer natural convection in porous medium. In another paper, Basak et al. [29] presented a study on convection in a square cavity filled with porous medium. They calculated numerical results for uniformly and non-uniformly heated bottom wall, where the upper wall is insulted and vertical side walls are maintained cold. Streamlines and isotherms are shown for wide range of Rayleigh numbers Ra (10³ ≤ Ra ≤ 10⁶), Prandtl numbers Pr (0.71 ≤ Pr ≤ 10), and the Darcy number Da (10⁻⁵ ≤ Da ≤ 10⁻³).

In the light of the aforementioned literature review, it appears that the magneto-hydrodynamic effects on the natural convective flow in an isosceles triangle filled with isotropic porous medium subjected to heated inclined walls has not been studied yet. It is, therefore, the aim of the present study to investigate the effects of uniformly and non-uniformly heated inclined walls of cavity in the presence of uniform magnetic field on the fluid flow inside the triangle. The governing equations for conservation of mass, momentum, and energy are normalised using the non-dimensional variables and are solved numerically. The Galerkin finite element method with penalty parameter is used to solve the obtained nonlinear partial differential equations. For the computation purpose, non-orthogonal grid generation has been done with iso-parametric mappings inside the triangular enclosure. Results are presented for different physical parameters in terms of streamlines, isotherms, and heat transfer rate in terms of local Nusselt number and average Nusselt numbers, which are discussed in detail.

2 Mathematical Model

The configuration consideration in the present study consists of an isosceles triangular enclosure enclosing porous medium of an isotropic nature. The electrically conducting viscous incompressible fluid is flowing through the porous medium inside the enclosure, as shown in Figure 1. In the porous region, both solid matrix and the voids through which fluid is flowing are assumed to be at the same temperature and under the local thermal equilibrium state [11]. The thermo physical properties of the fluid in flow field are assumed constant, except density of the fluid is taken as temperature dependent. After using the Boussinesq approximation [30], the density variation causes a body force term in the momentum equation.

Figure 1:

Triangular cavity flow with uniformly/non-uniformly heated side walls.

Moreover, it is assumed that a horizontal uniform magnetic field B with constant magnitude B₀ is applied to the enclosure in the absence of induced magnetic field being small as compare to B₀ (low-R_m approximation [31]), letting the walls of the cavity be electrically insulated with no Hall effects. The electromagnetic force is reduced to the damping factor -B₀²v [32], where v is the vertical component of velocity. Thus, the Lorentz force depends upon the velocity component only, which is perpendicular to the magnetic field. Under these assumptions in the absence of Forchheimer's inertia term and following the earlier work [17, 33], the governing equations for conservation of mass, momentum, and energy in the absence of viscous dissipation can be written as

and

The boundary conditions can be defined as

where x, y are the components in Cartesian coordinate system; u, v are velocity components along horizontal and vertical directions, respectively; P is pressure; ρ is density; K is specific permeability of the medium; α is thermal diffusivity; L is the length of base of cavity; and υ is kinematic viscosity. The non-dimensional variables are introduced as follows:

After substituting the transformation (6) in (1)–(4), we get

and the boundary conditions (5) are reduced to

Here, U and V are the non-dimensional velocity components, θ is the non-dimensional temperature, Pr is the Prandtl number, Ra is the Rayleigh number, Ha is the Hartman number, and Da is the Darcy number. The heat transfer coefficient h appearing in Newton's law of cooling may be determined in the dimensionless form from the local Nusselt numbers Nu. The local Nusselt number for bottom, left, and right walls are defined as follows:

The average Nusselt number Nu¯ for the bottom, left and right walls are defined as follows:

3 Method of Solution

The penalty method is introduced to eliminate the pressure term from momentum equations (8, 9) with the help of the continuity equation [34]. The incompressibility condition is defined by introducing penalty parameter γ as follows:

In order to satisfy the continuity equation, we have to take a large value for γ, generally equal to 10⁷, which returns consistent solutions. Upon using (14) in (8) and (9), we get the following form:

and

We approximate velocity and temperature profiles by using bi-quadratic basis set {ϕk}k = 1N with 6-nodal triangular elements as follows:

In order to solve these partial differential equations, the Galerkin weighted residual technique of finite element method is used, in which the residual equations are obtained for internal domain Ω as follows:

The obtained nonlinear residual (18)–(20) are solved by using the Newton-Raphson technique. The motion of fluid in terms of stream functions can be obtained from velocity components by using the following relation:

This may be reduced into the following single equation:

where stream functions are also approximated by the basis set {ϕ}k = 1N as

and after applying again the Galerkin weighted residual technique, the following residual equation for stream function is obtained:

and its solution is obtained using no-slip conditions along all the boundaries.

4 Validation

In order to develop the grid independent solution of the discussed problem, the numerical values of the computed average Nusselt number at the bottom wall is demonstrated in Table 1 against different refinement levels of non-uniform initial mesh. It is noted that with increase in the number of elements or by increasing the refinement level, the percentage error of the solution with the solution at previous refinement level is decreased. It is as minimum as 0.1 % at the fourth refinement level; therefore, throughout the study, the third refinement level is used for the solution with 1776 number of 6-nodal triangular elements. Once the grid independence is achieved, the code is further validated against the results of Basak et al. [27] as a limiting case, as shown in Figure 2. The left column contains the results of Basak et al. [27] and the right column contains results obtained by present investigation in case of a uniformly heated side wall with Pr = 0.7, Ra = 10⁶ and Da = 10⁻⁵. The results are evidently accurate and in good agreement with the results of Basak et al. [27].

Figure 2:

Contours for stream functions and isotherms with Pr = 0.7, Ra = 10⁶ and Da = 10⁻⁵.

5 Results and Discussion

In this section, results obtained by numerical simulations for two-dimensional laminar convective flow through a porous medium inside the triangular cavity under the effects of MHD are discussed. Discussion is divided into two cases: (i) uniformly and (ii) non-uniformly heated inclined side walls, where the bottom wall is considered as cold. Heat transfer rate in terms of local Nusselt number and average Nusselt number have also been computed and shown in figures. The graphs are plotted for a wide range of parameters, Rayleigh number Ra (10³ ≤ Ra ≤ 10⁶), Prandtl number Pr (0.026 ≤ Pr ≤ 10), Da (10⁻⁵ ≤ Da ≤ 10⁻³), and Hartman number Ha (50 ≤ Ha ≤ 10³).

5.1 Uniformly Heated Side Walls

In this case of study, side walls are subjected to a constant temperature (θ = 1), and the bottom wall is taken at cold temperature. Therefore, there appears jump type finite discontinuity at the lower left and right corners of the cavity, as these corners are at the intersection of walls at different temperatures. This discontinuity needs special attention and has been addressed according to the procedure given by Ganzarolli and Milanez [35]. Temperature at these corner nodes is taken to be the average temperature of bottom and corresponding side walls. However, the adjacent nodes are taken at corresponding boundary wall temperature to avoid singularity. As the bottom wall is kept on cold temperature and inclined side walls are heated uniformly, fluid present adjacent to the side walls is at a higher temperature than the bottom wall. Hence, the fluid near the side walls is less dense than that near the bottom cold wall due to the fact that the hot fluid is less dense than that of cold fluid. In consequence, the variation of density of fluid near the walls produces circulation of fluid in the enclosure in clockwise/anti-clockwise directions. The hot fluid expands, becomes more buoyant, and transfers the energy, and again descends down to the cold wall through the central vertical line of the cavity, resulting in two rolls of symmetric circulations as shown in the figures. Streamlines with positive values are shown as anti-clockwise circulation, and streamlines with negative values are shown as clockwise circulations according to the definition of stream function.

Figures 3 and 4 illustrate contour plots for streamlines and isotherms at Da = 10⁻³, Pr = 0.7, Ha = 50 for Ra = 4 × 10⁵ and 10⁶, respectively. It is observed that heat flow in the cavity is purely due to conduction and isotherms appear to be smooth and monotonic in this case. Also two symmetric rolls of clockwise and anti-clockwise circulations of streamlines are observed. The maximum value of stream function (|ψ|_max) is noted to be 1.5, as shown in Figure 3. It is observed through Figure 4 that with the increase in Rayleigh number, the strength of the circulation also increased. The upper corner of the cavity is observed to be empty due to weak effects of circulation and isotherms. It may be seen from Figure 4 that isotherms are pushed towards the bottom wall, and contour lines are concentrated in lower half of the cavity, where magnitude of stream function is increased to |ψ|_max = 2.5. It is due to the reason that the left and right corners are the places where the difference in the temperature is maximum.

Figure 3:

Streamlines and isotherms for uniformly heated inclined walls with Ra = 4 × 10⁵, Pr = 0.7, Da = 10⁻³ and Ha = 50.

Figure 4:

Streamlines and isotherms for uniformly heated inclined walls with Ra = 10⁶, Pr = 0.7, Da = 10⁻³ and Ha = 50.

Figures 5 and 6 contain plots for fluids for which thermal diffusivity dominates, i.e. Pr = 0.026 and momentum diffusivity dominates, i.e. Pr = 10, respectively, when other parameters are fixed at Ra = 10⁶, Da = 10⁻³ and Ha = 50. It is observed that a low Prandtl number corresponds to weaker clockwise and anti-clockwise circulation of streamlines, which is clearly due to conduction dominant effects. When the Prandtl number is 0.026, magnitude of circulation is noted to be 2, and when the Prandtl number is increased to 10, magnitude of circulation is also increased to 2.5. The isotherms for smaller Prandtl number are more pressed towards bottom wall, as shown in Figure 5.

Figure 5:

Streamlines and isotherms for uniformly heated inclined walls with Ra = 10⁶, Pr = 0.026, Da = 10⁻³ and Ha = 50.

Figure 6:

Streamlines and isotherms for uniformly heated inclined walls with Ra = 10⁶, Pr = 10, Da = 10⁻³ and Ha = 50.

Figures 7 and 8 show numerical outputs at for Pr = 0.026 and Pr = 10 respectively with large Hartman number Ha = 10³ and Ra = 10⁶, Da = 10⁻³. It is seen that higher value of Hartman number results into very weak circulation of streamlines, but the isotherms which were clustered in lower portion of the cavity at Ha = 50 started expanding up in the cavity. Again isotherms are noted to be smooth, monotonic symmetric about vertical line passing through centre of horizontal wall. Magnitudes of stream function are observed to be |ψ|_max = 0.06 at Pr = 0.026, and |ψ|_max = 0.07 at Pr = 10 in Figures 7 and 8 respectively.

Figure 7:

Streamlines and isotherms for uniformly heated inclined walls with Ra = 10⁶, Pr = 0.026, Da = 10⁻³ and Ha = 10³.

Figure 8:

Streamlines and isotherms for uniformly heated inclined walls with Ra = 10⁶, Pr = 10, Da = 10⁻³ and Ha = 10³.

5.2 Non-Uniformly Heated Side Walls

In this case, both left and right inclined walls are subject to sinusoidal heat wave θ = Sin(πy). This type of heating is taken due to reason that it removes singularity from the bottom left and right corners.

Figures 9–11 show contour plots for stream function and isotherms for Ra = 10⁶, Pr = 0.026, 0.7, Da = 10⁻³ and Ha = 50, 10³ with non-uniformly heated inclined side walls. Strong circulation is observed in the case of non-uniformly heated side walls and it is shifted to the centre of the enclosure from bottom corners (as observed in previous case). It is further noted that, by increasing the value of Prandtl and Hartmann numbers, streamlines and isotherms become smoother. It is pointed out that, two rolls of symmetric circulations are observed in each case, and streamlines are seen to be pushed towards the inclined side walls. It is seen through Figure 9 that when sinusoidal heat wave is applied to side walls of cavity, isotherms are observed to be compressed towards side walls and distributed throughout the triangular cavity. It is important to note that the high temperature gradient is seen near the upper vertex of the enclosure in this figure. However, for a uniform heating case, it was observed that isotherms are compressed towards the bottom wall, and temperature gradient is observed to be concentrated in the lower half of the cavity and especially near the lower two corners of the cavity due to the maximum temperature difference there. It is further noted that isotherms for θ ≤ 0.6 are pushed towards the bottom wall, and for θ ≤ 0.7, isotherms are pressed to the side walls. As we increase the Prandtl number from 0.026 to 0.7, isotherms near the bottom wall get concaved up from concaved down in a small interval about the centre of the bottom wall, as shown in Figure 10.

Figure 9:

Streamlines and isotherms for non-uniformly heated inclined walls with Ra = 10⁶, Pr = 0.026, Da = 10⁻³ and Ha = 50.

Figure 10:

Streamlines and isotherms for non-uniformly heated inclined walls with Ra = 10⁶, Pr = 0.7, Da = 10⁻³ and Ha = 50.

Figure 11:

Streamlines and isotherms for non-uniformly heated inclined walls Ra = 10⁶, Pr = 0.026, Da = 10⁻³ and Ha = 10³.

When the Prandtl number is increased from 0.026 to 0.7, magnitude of stream function is noted to increase from |ψ|_max = 3 (see Fig. 9) to |ψ|_max = 3.6 (see Fig. 10); it is due to the phenomenon that the convection helps the fluid flow through buoyancy. Conversely, when the Hartman number is increased from 50 to 10³, magnitude of stream function is reduced from |ψ|_max = 3 (see Fig. 9) to |ψ|_max = 0.04 (see Fig. 11), as the Hartman number is a ratio of electromagnetic force to the viscous forces and an increase in the Hartman number is due to the dominance of electromagnetic force, which in consequence produces resistance to the flow, and as a result |ψ|_max is reduced to 0.04.

Figure 12a and b is drawn to show the heat transfer rate in terms of the local Nusselt number along the bottom wall (a) and along the inclined side walls (b). Because the cavity under consideration is symmetric about the vertical line passing through the centre of the bottom wall, and both of the side walls are subjected to the same temperature, consequently, the heat transfer rate at both side walls is observed as the same. Therefore, we have shown a graph of the Nusselt number for the left side wall only. In this figure, the solid lines represent the case of uniformly heated side wall and dashed lines represent the case of non-uniformly heated side walls. Curves are plotted for different values of the Darcy number Da and the Prandtl number Pr, where the Rayleigh number Ra and the Hartmann number Ha are 10⁶ and 50, respectively. It is observed that when side walls are heated uniformly (solid curves), the heat transfer rate is noted as very large at the left and right edges of the bottom wall, as shown in Figure 12a. This is due to the fact that both boundaries meeting at these edges are at different temperatures and causes maximum temperature difference to occur. It is further seen that the heat transfer rate is minimum at the centre of the bottom wall for all values of Da and Pr. Similarly, heat transfer rate along side walls is maximum at the bottom edge, where distance is taken to be zero, as shown in Figure 12b, which is also due to having a maximum temperature difference at this point. Further, the heat transfer rate is almost zero at the upper vertex due to the fact that no temperature difference at this vertex causes no transfer of energies shown in right-hand side of Figure 12b; the increase in the Nusselt number due to Pr and Da is also observed through Figure 12a and b.

Figure 12:

(a, b) Local Nusselt numbers for cold bottom wall and inclined side walls heated uniformly (solid lines) and non-uniformly (dashed lines) with different values of Pr and Da, where Ra = 10⁶ and Ha = 50 are fixed.

For the case of non-uniformly heated side walls (dashed curves), heat transfer rate Nu along the bottom wall is also maximum at vertices and minimum at the centre as compared to the uniformly heated side walls, and heat transfer for this case is considerably minimum. Whereas along the side walls (Fig. 12b), heat transfer rate shows a sinusoidal nature due to non-uniformly heated side walls. The heat transfer rate at the upper vertex of the cavity for the case of non-uniformly heated side walls is also minimum due to minimum temperature differences at this point. The variation in heat transfer rate Nu for different values of the Darcy number Da and the Prandtl number Pr is also noted from the figure as a sinusoidal wave.

The average Nusselt number at the bottom and side walls against the Hatmann number Ha for different values of the Darcy number Da is shown in Figure 13a and b. It is seen that the average Nusselt number decreases with an increased Hartmann number and attains constant values against Ha > 500. Similarly, by reducing the values of the Darcy number, the average Nusselt number decreases up to fixed values. The same observation is noted at the bottom and side walls and can be further proved from Table 2.

Figure 13:

(a, b) Average Nusselt number for uniform and non-uniform heating case against different values of Da and Ha, where Ra = 10⁶ and Pr = 10 are fixed.

Table 2

Average Nusselt numbers along different walls of cavity against various values of flow parameters, Ha, Pr, for fixed Da.

Ha	Da	Pr	Uniform heating case		Non-uniform heating case
			Nu-avg. bottom wall	Nu-avg. side wall	Nu-avg. bottom wall	Nu-avg. side wall
0	10−3	0.026	7.4549	4.9936	5.1623	2.4169
50	–	–	7.4093	4.9625	4.9550	2.3206
100	–	–	7.3113	4.8950	4.8646	2.2807
200	–	–	7.1637	4.7919	4.7495	2.2285
500	–	–	7.0962	4.7382	4.6656	2.1876
1000	–	–	7.0922	4.7323	4.6541	2.1816
0	–	0.7	7.6294	5.1212	5.0725	2.3580
50	–	–	7.5108	5.0397	4.9990	2.3437
100	–	–	7.3350	4.9175	4.8995	2.3065
200	–	–	7.1070	4.7730	4.7418	2.2340
500	–	–	7.0822	4.6817	4.6552	2.1911
1000	–	–	7.0456	4.3435	4.6443	2.1850

6 Conclusions

A computational study is performed to investigate the two-dimensional, laminar, steady-state MHD natural convection flow within the isosceles triangular enclosure filled with isotropic porous medium. The side walls of the triangular enclosure are subjected to uniform and non-uniform heat. The finite element method is used to obtain the solution, governing conservation of mass and momentum, and energy equations are nonlinear requiring an iterative technique solver to solve these equations by considering Prandtl number Pr = 0.026–10, Hartman number Ha = 50 – 10³ and Rayleigh number Ra = 10³–10⁷. For this purpose, we applied the Galerkin weighted residual method with a penalty parameter.

For the case of uniform heating, it is observed that the strength of circulations of streamlines increases when the Rayleigh number is increased above critical value (Ra = 4 × 10⁵), but an increase in the Hartman number results in a decrease in the strength of streamline circulation. The effects on circulations of streamlines due to the Prandtl number are similar to that of the Rayleigh number. Isotherm contours gets closer towards the bottom of the cavity as the Rayleigh number or Prandtl number are increased. Whereas with an increase in the Hartman number, isotherms move towards the upper portion of the cavity. Conversely, it is found that heat transfer rate is maximum at corners of the bottom wall due to a maximum temperature difference at the corner nodes, and it appears to be constant near centre of the bottom wall. Heat transfer is higher near the lower end of side walls due to the same reasons; it is almost zero near upper corners, as two side walls are at the same temperature. When a side wall is subjected to a sinusoidal heat wave, streamlines are pushed towards side walls of cavity, and strength of circulation of streamline is increased with an increase in the Prandtl number. Conversely, isotherms >0.7 are pushed towards side walls while other isotherms are pushed near the bottom wall when Hartman and Prandtl numbers are increased. It is further seen that the circulation strength of streamlines is decreased with increases in Hartman numbers in a non-uniform heating case as well. Furthermore, the heat transfer rate at the bottom wall for a non-uniformly heating case is considerably less than that of a uniform heating case, and it is a farther minimum at the centre of the bottom wall. However, it is observed that the sinusoidal nature is present along the side walls and minimum at the upper vertex of the cavity.