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Effects of Porous Fins on Mixed Convection and Heat Transfer Mechanics in Lid-Driven Cavities: Full Numerical Modeling and Parametric Simulations

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Abstract

The present study presents a comprehensive analysis of effects of porous fins on mixed convection heat transfer in lid-driven square cavities, where the top lid has the two-way movement. Porous medium with varying permeability, instead of the solid one, could widely modify its baffling effect on fluid flow and could also be utilized to control fluid flow and heat transfer. Fluid flow within the cavity was solved through the Navier–Stokes equations, while that within a saturated porous medium was governed by the Darcy–Forchheimer model. Numerical results indicate that the adding porous fins with excellent permeability could enhance heat transfer dramatically, especially for more fins. The rate of heat transfer enhancement due to an increase in Darcy number decreases, and similar trends were observed for the quantitative variations of porous fins. Besides, positions of porous fins have significant effects on fluid flow and heat transfer. The correlations of average Nusselt numbers, as functions of various governing parameters, have been proposed. The present investigations could be beneficial to the design of the microelectronic cooling through the installation of porous-alike materials or modules.

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Abbreviations

a :

Coefficients of correlations of the average Nusselt number

A :

Coefficient in the discretized equations

C F :

Inertia coefficient

Da:

Darcy number

g :

Gravity acceleration (m s−2)

Gr:

Grashof number

k :

Thermal conductivity (W m−1 K−1)

K :

Permeability (m2)

H :

Heat function

I, J :

Nodal index along X- and Y-axes, respectively

l fin :

Length of the fin (m)

L :

Length of enclosure (m)

L fin :

Dimensionless length of the fin

n :

Unit normal vector

N :

Number of the fins

Nu:

Nusselt number

p :

Dimensional pressure (Pa)

P :

Dimensionless pressure

Pr:

Prandtl number

Q Φ :

Source term of the discretized governing equation

Ra:

Rayleigh number (Ra = GrPr)

Ri:

Richardson number

R 2 :

Square value of regression coefficient

T :

Dimensional temperature (K)

T c :

Cooling temperature (K)

T h :

High temperature (K)

u 0 :

Dimensional velocity of the lid (m s−1)

U 0 :

Dimensionless velocity of the lid

(u, v):

Dimensional velocities (m s−1)

(U, V):

Dimensionless velocities

(x, y):

Dimensional Cartesian coordinates

(X, Y):

Dimensionless Cartesian coordinates

α :

Thermal diffusivity (m2 s)

β T :

Thermal expansion coefficient (K−1)

δ :

Stop criterion

ε :

Porosity

θ :

Dimensionless temperature

λ :

Thermal conductivity ratio

μ :

Dynamic viscosity (N m−2 s)

υ :

Kinematic viscosity (m2 s−1)

ρ :

Density (kg m−3)

Φ :

General variable (U, V, P or T)

χ :

Parameter for the unification of governing equations

Ψ :

Stream function

Ω :

Control volume for variables

c:

Cooling temperature

e, w, n, s:

Values at the control-volume faces

E, W, N, S:

Values at the nodal points

eff:

Effective

fin:

Porous fin

nb:

Neighbour nodal points

P:

Main nodal points

s:

Porous matrix

m :

Internal iteration number

n :

External iteration number

* :

Uncorrected value

\('\) :

Corrected value

–:

Average value of volume

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Acknowledgements

This research was financially supported by the Natural Science Foundation of China (NSFC Grant No. 51778504; Grant No. U1867221), National Key Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2018YFC0705201, Grant No. 2018YFB0904200), Joint Zhuzhou-Hunan Provincial Natural Science Foundation (Grant No. 2018JJ4064), Zhuzhou Public Welfare Program (Grant No. 50756), National Defense Research Funds for the Central Universities (Grant No. 2042018gf0031, Wuhan University), Beijing Institute of Satellite Environmental Engineering (CAST-BISEE2019-025), Teaching Research Program (Grant No. 2019JG030, Wuhan University) and Shandong Provincial Natural Science Foundation (Grant No. ZR2018MEE035).

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Authors and Affiliations

  1. Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, Wuhan, Hubei Province, People’s Republic of China

    Lei Wang, Wei-Wei Wang, Yang Cai & Fu-Yun Zhao

  2. Hubei Key Laboratory of Waterjet Theory and New Technology (Wuhan University), Wuhan, Hubei Province, People’s Republic of China

    Lei Wang, Wei-Wei Wang, Yang Cai & Fu-Yun Zhao

  3. School of Power and Mechanical Engineering, Wuhan University, Dong-Hu South Road, Wuhan, 430072, Hubei Province, People’s Republic of China

    Lei Wang, Wei-Wei Wang, Yang Cai & Fu-Yun Zhao

  4. College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao, Shandong Province, People’s Republic of China

    Di Liu

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  1. Lei Wang
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  2. Wei-Wei Wang
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  3. Yang Cai
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Correspondence to Fu-Yun Zhao.

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Wang, L., Wang, WW., Cai, Y. et al. Effects of Porous Fins on Mixed Convection and Heat Transfer Mechanics in Lid-Driven Cavities: Full Numerical Modeling and Parametric Simulations. Transp Porous Med 132, 495–534 (2020). https://doi.org/10.1007/s11242-020-01402-3

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