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Research Papers: Natural and Mixed Convection

# High Rayleigh Number Natural Convection Inside 2D Porous Enclosures Using the Lattice Boltzmann Method

[+] Author and Article Information
Ramanathan Vishnampet, V. Babu

Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

Arunn Narasimhan1

Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, Indiaarunn@iitm.ac.in

1

Corresponding author.

J. Heat Transfer 133(6), 062501 (Mar 08, 2011) (9 pages) doi:10.1115/1.4003534 History: Received April 19, 2010; Revised January 13, 2011; Published March 08, 2011; Online March 08, 2011

## Abstract

Lattice Boltzmann method (LBM) is employed to investigate natural convection inside porous medium enclosures at high Rayleigh numbers. Volume averaged porous medium model is coupled with the lattice Boltzmann formulation of the momentum and energy equations for fluid flow. A parallel implementation of the single relaxation time LBM is used, which allows the porous medium modified Rayleigh number $Ram$ to be as high as $108$. Heat transfer results in the form of the enclosure averaged Nusselt number Nu are obtained for higher modified Rayleigh numbers $104≤Ram≤108$. The Nu values are compared with values in the absence of the form drag term. The form drag due to the porous medium is found to influence Nu considerably. The effect of the form drag on Nu is studied by using a form drag modified Rayleigh number $RaC$ (extended from $Ram$). Utilizing the results for Nu in the high $Ram$ range, a correlation is proposed between Nu and $RaC$ for Darcy numbers $10−6≤Da≤10−2$, encompassing the non-Darcy flow regime.

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## Figures

Figure 1

Schematic of the square porous medium enclosure

Figure 2

Discrete lattice velocities for the D2Q9 lattice; ξ0=(0,0)

Figure 3

Streamlines (top) and isotherms (bottom) for Da=10−6 and (a) Ram=10, (b) Ram=102, (c) Ram=103, and (d) Ram=104(Pr=1)

Figure 4

Streamlines (top) and isotherms (bottom) for Da=10−4 and (a) Ram=103, (b) Ram=104, (c) Ram=105, and (d) Ram=106(Pr=1)

Figure 5

Streamlines (top) and isotherms (bottom) for Da=10−2 and (a) Ram=105, (b) Ram=106, (c) Ram=107, and (d) Ram=108(Pr=1)

Figure 6

Temperature profiles at y=L/2 for Da=10−6 and various Ram

Figure 7

Temperature profiles at y=L/2 for (a) Da=10−4 and (b) Da=10−2 for the entire enclosure and temperature profiles for (c) Da=10−4 and (d) Da=10−2 in the region near the vertical boundaries

Figure 8

Comparison of generalized (Eqs. 2,4) and Darcy models (Eq. 5) for Da=10−2 and 10−4

Figure 9

Variation of Nu with Ra for 10−6≤Da≤10−2

Figure 10

Variation of Nu with Ram for 10−6≤Da≤10−2 and comparison of present results with Nu=0.577 Ram0.5(4)

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