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Mixed convection flow is one of the essential criteria of fluid flow and heat transfer. And its application has been increased due to modernization of society. So, to compete with the global world an analysis has been investigated numerically. In this study we have considered 2D double lid driven cavity with two - sided adiabatic wall s . This problem is illustrated mathematically by a collection of governing equations and the developed model has been solved numerically by using Finite Difference Method (FDM). The goal of the present study is to analyze numerically the thermal behaviour and parameters effect on heat transfer inside the 2D chamber. Also this analysis has been observed for the case where the upper wall is moving at positive direction and lower wall is moving at negative direction with constant speed. Furthermore, we have tried to analyze the velocity and temperature profiles for a vast range of dimensionless parameters namely Reynolds number (Re) , Richardson number (Ri) and Prandtl number (Pr) and presented graphically. Moreover, it is found that these flow parameters have significant effects in controlling the flow behavior inside the cavity. A comparison has been done to validate our code and found a good agreement. Finally, average Nusselt number (Nu) ha s been studied for the effects of these parameters and presented in tabular form.

Mixed convection, takes place when natural convection and forced convection mechanisms perform simultaneously to exchange heat due to temperature differences. This is likewise characterized as circumstances where both pressure and buoyancy forces interface. The heat transfer behavior of the fluid as convection is mostly depended on the flow, temperature, geometry and inclination. In recent years, the natural and mixed convection flow has gained interest from their hypothetical and practical perspectives.

“The double lid-driven cavity problem has been gained extensive interest among the scientists and researchers because of its simplest geometrical settings with all fluid mechanical structure and applications such as cooling of electronic gadgets, drying instrument, softening procedures and so forth. Additionally, movement of side walls is also responsible for the fluid convective behavior within the enclosed square cavity” [

In recent past, Pal et al. [

In the present experiment, a steady state, incompressible flow inside a cavity has been studied. The physical model along with boundary conditions is shown in

The following governing equation has been presented for 2D and incompressible flow in non-dimensional form:

∂ U ∂ X + ∂ V ∂ Y = 0 (1)

U ∂ U ∂ X + V ∂ U ∂ Y = − ∂ P ∂ X + 1 Re ( ∂ 2 U ∂ X 2 + ∂ 2 U ∂ Y 2 ) (2)

U ∂ V ∂ X + V ∂ V ∂ Y = − ∂ P ∂ Y + 1 Re ( ∂ 2 V ∂ X 2 + ∂ 2 V ∂ Y 2 ) + R i θ (3)

U ∂ θ ∂ X + V ∂ θ ∂ Y = 1 P r R e ( ∂ 2 θ ∂ X 2 + ∂ 2 θ ∂ Y 2 ) (4)

where U and V are the velocity components along X and Y directions, θ is the non-dimensional temperature and P is the non-dimensional pressure component. Here, the dimensionless parameters in Equations (1)-(4) are: Ri is the Richardson number, Re is the Reynolds number, Gr is the Grashof number, is the Prandtl number. The governing parameters in Equations (1)-(4) and non-dimensional length and velocities are as follow:

X = x L ; Y = y L ; U = u V L ; V = v V L ; P = p ρ V L 2 ; θ = T − T C T H − T C ;

R i = G r R e 2 ; R e = V L L ν ; G r = g β Δ T L 3 ν 3 ; P r = ν α ;

where ρ , g , α , β and ν are the fluid density, gravitational acceleration, thermal diffusivity, the coefficient of thermal expansion and kinematic viscosity, respectively. The velocity and temperature flow fields boundary conditions are shown in

U = V = 0 , ∂ θ ∂ x = 0 at X = 0 and X = 1 U = − 1 , V = 0 , θ = 0 at Y = 0 U = 1 , V = 0 , θ = 1 at Y = 1 } (5)

Also, the definition of local and average Nusselt number is described in Pal et al. [

We have used finite difference method to deal with the governing equations (1)-(4) along with the boundary situation (5). Then an in house code (DGK) has been implemented to represent the flow and thermal behaviors. However, grid independent test has been done to verify the code which is also presented in Pal et al. [

In this research, variation of velocity and temperature profiles with different Re, Ri and Pr inside the square cavity with the given boundary conditions (5) are studied and explained graphically in Figures 2-4. Additionally, variation of average Nusselt number has been studied and presented in tabular form in

The flow and thermal fields within the double lid driven cavity for Ri = 1.5 and Pr = 10.0 are represented in

Re = 100 | Avg Nu | ||
---|---|---|---|

Pr | Ri = 0.1 | Ri = 1.0 | Ri = 1.5 |

0.71 | 4.15 | 1.84 | 1.52 |

2 | 7.58 | 2.69 | 2.09 |

5 | 11.89 | 4.00 | 2.86 |

10 | 15.62 | 6.09 | 3.77 |

Re = 500 | |||

Pr | Ri = 0.1 | Ri = 1.0 | Ri = 1.5 |

0.71 | 9.24 | 2.04 | 1.81 |

2 | 15.47 | 3.12 | 2.63 |

5 | 21.69 | 4.85 | 3.93 |

10 | 33.68 | 7.41 | 5.20 |

Re = 1000 | |||

Pr | Ri = 0.1 | Ri = 1.0 | Ri = 1.5 |

0.71 | 11.98 | 2.12 | 1.82 |

2 | 16.95 | 3.29 | 2.67 |

5 | 20.59 | 5.05 | 4.21 |

10 | 31.25 | 8.11 | 5.64 |

difference region near the moving lid for Re = 500. Additionally, the center thermal region separate into two parts. One part moving towards the hot region and another part moving towards the cold region. These all makes a steep temperature gradient along the adiabatic wall. This is due to the presence of shear stress. As an increase of Reynolds number Re = 1000, shows hot fluid mixes up with the cold fluid very well and the thermal boundary layer diminishes.

The flow phenomena and heat transfer are observed inside a bounded domain where the upper wall is moving in the positive direction and lower wall is moving in the negative direction. The geometrical characteristics have been set to the Equations (1)-(5) and then solved by 4th order FDM. The effects of dimensionless parameters Re, and Pr have been presented graphically and in tabular form. It is noticed that there is no effect of Pr on the velocity profiles but strong effect on temperature profiles. Also, it is found that for increasing of Ri, rate of heat exchange decreases. Besides, for low Richardson number flow is dynamically unstable and for higher values of Ri, the flow is stable. Furthermore, as Re increases, the augmentation of average Nu increases for all cases. Overall, higher intensification of average Nu is pragmatic when flow becomes turbulent and deterioration of average Nu can be realistic when flow becomes laminar.

This work has been fully supported by University Grant Commission (UGC), Bangladesh via grant number Reg./Admin-3/76338 (Year: 2017-2018).

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

Pal, D.R., Saha, G. and Saha, K.C. (2018) Parameters Effect on Heat Transfer Augmentation in a Cavity with Moving Horizontal Walls. Journal of Applied Mathematics and Physics, 6, 1907-1915. https://doi.org/10.4236/jamp.2018.69162