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Hydrodynamic mixed convection in a lid-driven hexagonal cavity with corner heater is numerically simulated in this paper by employing finite element method. The working fluid is assigned as air with a Prandtl num-ber of 0.71 throughout the simulation. The left and right walls of the hex-agonal cavity are kept thermally insulated and the lid moves top to bottom at a constant speed
U_{0}. The top left and right walls of the enclosure are maintained at cold temperature
T_{c}. The bottom right wall is considered with a corner heater whereas the bottom remaining part is adiabatic and inside the cavity a square shape heated block
T_{h}. The focus of the work is to investigate the effect of Hartmann number, Richardson number, Grashof number and Reynolds number on the fluid flow and heat transfer characteristics inside the enclosure. A set of graphical results is presented in terms of streamlines, isotherms, local Nusselt number, velocity profiles, temperature profiles and average Nusselt numbers. The results reveal that heat transfer rate increases with increasing Richardson number and Hartmann number. It is also observed that, Hartmann number is a good control parameter for heat transfer in fluid flow in hexagonal cavity.

Mixed convection in enclosures is encountered in many engineering systems such as cooling of electronic components, ventilation in buildings and fluid movement in solar energy collectors, astrophysics, geology, biology and chemical processes, as well as in many engineering applications. Also mixed convection involving the combined effect of forced and natural convection has been the focus of research due to its occurrence in numerous technological, engineering and natural applications such as: cooling of electronic devices, lubrication technologies, drying technologies, and food processing. Al-Amiri et al. [

To the best of author’s understanding, little attention is given to the problem of MHD mixed convection in a lid-driven cavity with corner heater. There is no previous study on hydrodynamic mixed convection in a lid-driven hexagonal cavity with a corner heater and inside the square heated block. Therefore, this problem could be occurred in many engineering applications such as conveyer belt, escalator and lift as elevator, heating and cooling flows in buildings. The objective of the present study is to investigate numerically hydrodynamic mixed convection in a lid-driven hexagonal cavity with a corner heater, also increase of the lid-driven constant velocity to highlight the applicability of the approach. Numerical results are presented via streamlines, isotherms, velocity profiles, dimensionless temperature, local Nusselt number and average Nusselt number to analyze the effect of Richardson number and Hartmann number of the fluid flow and heat transfer.

The paper is arranged in the following manner: in Section 2, we consider a physical configuration of hexagonal cavity; Section 3 presents the mathematical formulation of this model, corresponding boundary conditions and grid generation for that cavity; in Section 4, detailed analysis of different results and corresponding discussions is added; finally, a brief conclusion is given in Section 5.

The considered two-dimensional model in the present study of mixed convection in a hexagonal cavity with internal heated square block is shown in

The fluid is considered as incompressible, Newtonian and the flow is assumed to

be laminar. Two dimensional, steady equations are written by considering a uniform applied magnetic field. It is assumed that Boussinesq approximation is valid and radiation mode of heat transfer, Joule heating and Hall effects are neglected according to other modes of heat transfer. Thus, using the coordinate system shown in

∂ u ∂ x + ∂ v ∂ y = 0

u ∂ u ∂ x + v ∂ u ∂ y = − 1 ρ ∂ p ∂ x + v ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 )

u ∂ v ∂ x + v ∂ v ∂ y = − 1 ρ ∂ p ∂ y + v ( ∂ 2 v ∂ x 2 + ∂ 2 v ∂ y 2 ) − σ B 0 2 ρ υ + g β ( T − T c )

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

The boundary conditions for the present problem are specified as follows:

On the top left and right walls: u = 0 , v = 0 , T = T c _{ }

On the left and right walls: u = 0 , v = − v 0 , ∂ T ∂ x = 0

On the lower left wall: u = 0 , v = 0 , ∂ T ∂ x = 0

On the lower right wall and square block: u = 0 , v = 0 , T = T h

where x and y are the distance measured along the horizontal and the vertical directions respectively, u and v are the velocity components in x and y direction respectively, T denotes the temperature, υ denotes the kinematic viscosity, α denotes the thermal diffusivity respectively, p is the pressure and ρ is the density.

The governing equations are non-dimensionalized by using the following dimensionless quantities:

X = x L , Y = y L , U = u U 0 , V = v U 0 , P = p ρ U 0 2 , θ = T − T c T h − T c

After substitution of dimensionless variable we get the non-dimensional governing equations which are:

∂ U ∂ X + ∂ V ∂ Y = 0

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

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

U ∂ θ ∂ X + V ∂ θ ∂ Y = 1 R e P r ( ∂ 2 θ ∂ X 2 + ∂ 2 θ ∂ Y 2 )

where U and V are the velocity components in X and Y directions respectively, P is the pressure and θ is the non-dimensional temperature. As we know R i = G r / R e 2 , when R i ≈ 1 both free and forced convection are equally dominant and the flow regime is designated as mixed convection. If R i > 1 then free convection is dominant whereas forced convection is dominant when R i < 1 .

The transformed boundary conditions are:

On the top left and right walls: θ = 0 , U = 0 , V = 0

On the left and right walls: U = 0 , V = − 1 , ∂ θ ∂ X = 0

On the lower left wall: U = 0 , V = 0 , ∂ θ ∂ X = 0

On the lower right wall and square block: θ = 1 , U = 0 , V = 0

We examined five different non-uniform grid systems with the following number of elements within the resolution field. It is observed from

Nodes | 542 | 1230 | 1836 | 5602 | 11908 |
---|---|---|---|---|---|

Elements | 814 | 1127 | 2145 | 6554 | 25482 |

Nu | 13.56730 | 15.39566 | 16.26567 | 20.71909 | 20.80172 |

θ α υ | 0.82835 | 0.81251 | 0.810452 | 0.80167 | 0.80167 |

grid refinement tests. From these values, 11,908 nodes, 25,482 elements can be chosen throughout the simulation to optimize the relation between the accuracy required.

In this section, some representative results are presented to illustrate the effects of various controlling parameters on the fluid flow. These controlling parameters include Richardson number ranging from 0.01 to 10 and the Hartmann number varying from 0 to 50 with fixed Reynolds number R e = 100 and Prandtl number P r = 0.71 . The results are presented in terms of streamlines, isotherms, local Nusselt number, velocity profiles and dimensionless temperature profile along the vertical wall.

Streamlines for different values of Richardson number R i = 0.01 - 10 while H a = 0 and P r = 0.71 are presented in

As Ri increases up to 1, the effect of mixed convection increases and both the primary and secondary circulation do increase. It is observed that moving lids become very strong. For higher Richardson number one circulation cell is formed at upper right corner of the cavity and also the increase of flow strength is shown in

Conduction dominant heat transfer is observed from the isotherms in Figures 3(e)-(h) at H a = 0 and P r = 0.71 respectively. As seen from the figures, isotherms concentrate near the bottom wall and isotherm lines are more bending which means increasing heat transfer throw convection. For higher Richardson number, the isotherm lines are bending near right side wall that means increasing heat transfer throw convection are shown in

Variation of streamlines and isotherms for different values of Richardson number R i = 0.01 - 10 while H a = 50 and P r = 0.71 are shown in

The local Nusselt Number along the vertical wall for different Richardson numbers R i = 0.01 - 10 with H a = 0 and P r = 0.71 of the cavity is shown in

is increased with the increase of Richardson number.

Variation of the vertical velocity component along the horizontal line of the cavity with different Richardson numbers and H a = 0 is shown in

The local Nusselt numbers along vertical wall for different values of Richardson number with H a = 50 and P r = 0.71 of the cavity are shown in

Variations of the vertical velocity component along the horizontal line of the cavity with the Richardson number and for H a = 50 are shown in

Streamlines for G r = 10 4 , R e = 100 and P r = 0.71 are presented in Figures 7(a)-(d) respectively. These figures show the effects of Hartmann number on flow field and temperature distribution. From the streamlines it is found that with the increase of Hartmann number (increase of the strength of the magnetic field), flow strength decreases. At H a = 0 two circular shapes are formed at the below of the heated block of the cavity shown in

Conduction dominant heat transfer is observed from the isotherms in Figures 7(e)-(h) at G r = 10 4 , R e = 100 and P r = 0.71 . At H a = 0 the isotherm lines near the bottom wall and isotherm lines are more bending which means increasing heat transfer throw convection. Formation of the thermal boundary layers can be found that it increases from the isotherms for H a = 0 - 30 . The isotherms are almost parallel to lid-driven wall and they show a linear increase along the corner heater.

The local Nusselt Number along vertical wall for different values of Hartmann number with G r = 10 4 , R e = 100 , P r = 0.71 and R i = 1 of the cavity are shown in

Variation of the vertical velocity component along the horizontal line of the cavity with G r = 10 4 , R e = 100 , P r = 0.71 , R i = 1 and different values of the Hartmann number are shown in

The influence of Richardson number on average Nusselt number versus different Hartmann numbers for vertical wall is displayed in

Numerical study on mixed convection in a lid-driven hexagonal cavity with corner heater has been performed. Results have been presented in terms of streamlines, isotherms, local Nusselt number, velocity profile and dimensionless temperature. The results are obtained for a wide range of pertinent dimensionless groups such as Richardson number and Hartmann numbers. In view of the obtained results, the following findings are précised:

1) The flow characteristics and heat transfer mechanism inside the hexagonal cavity are strongly dependent on the Richardson number.

2) The significant suppression of the convective current in the enclosure is due to increase of Hartmann numbers.

3) For all cases considered, two counter rotating eddies were formed inside the cavity regardless the Richardson number and the Hartmann numbers. The temperature distribution and the flow characteristics inside the cavity strongly depend on both the strength of the magnetic field and the Richardson number.

4) The large values of Richardson number lead to increase the lid-driven effect whereas the small values of Richardson number lead to increase effect of presence of the heat source on the flow and heat characteristics.

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

Munshi, M.J.H., Mostafa, G., Munsi, A.B.S.M. and Waliullah, Md. (2018) Hydrodynamic Mixed Convection in a Lid-Driven Hexagonal Cavity with Corner Heater. American Journal of Computational Mathematics, 8, 245-258. https://doi.org/10.4236/ajcm.2018.83020