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Research Papers: Natural and Mixed Convection

Finite Element Simulation on Natural Convection Flow in a Triangular Enclosure Due to Uniform and Nonuniform Bottom Heating

[+] Author and Article Information
S. Roy

Department of Mathematics,  Indian Institute of Technology Madras, Chennai-600036, Indiasjroy@iitm.ac.in

Tanmay Basak1

Department of Chemical Engineering,  Indian Institute of Technology Madras, Chennai-600036, Indiatanmay@iitm.ac.in

Ch. Thirumalesha

Department of Chemical Engineering,  Indian Institute of Technology Madras, Chennai-600036, India

Ch. Murali Krishna

Department of Mathematics,  Indian Institute of Technology Madras, Chennai-600036, India

1

Corresponding author.

J. Heat Transfer 130(3), 032501 (Mar 06, 2008) (10 pages) doi:10.1115/1.2804934 History: Received July 19, 2006; Revised August 30, 2007; Published March 06, 2008

A penalty finite element analysis with biquadratic elements has been carried out to investigate natural convection flows within an isosceles triangular enclosure with an aspect ratio of 0.5. Two cases of thermal boundary conditions are considered with uniform and nonuniform heating of bottom wall. The numerical solution of the problem is illustrated for Rayleigh numbers (Ra), 103Ra105 and Prandtl numbers (Pr), 0.026Pr1000. In general, the intensity of circulation is found to be larger for nonuniform heating at a specific Pr and Ra. Multiple circulation cells are found to occur at the central and corner regimes of the bottom wall for a small Prandtl number regime (Pr=0.026−0.07). As a result, the oscillatory distribution of the local Nusselt number or heat transfer rate is seen. In contrast, the intensity of primary circulation is found to be stronger, and secondary circulation is completely absent for a high Prandtl number regime (Pr=0.7–1000). Based on overall heat transfer rates, it is found that the average Nusselt number for the bottom wall is 2 times that of the inclined wall, which is well, matched in two cases, verifying the thermal equilibrium of the system. The correlations are proposed for the average Nusselt number in terms of the Rayleigh number for a convection dominant region with higher Prandtl numbers (Pr=0.7 and 10).

Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic diagram of the physical system

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Figure 2

(a) The mapping of a triangular domain to a square domain in the ξ‐η coordinate system and (b) the mapping of an individual element to a single element in the ξ‐η coordinate system

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Figure 3

Temperature and stream function contours for a uniformly heated bottom wall, θ(X,0)=1, and cooled inclined walls, θ(X,Y)=0, with Ra=103 and Pr=0.026 (Case I). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 4

Temperature and stream function contours for a uniformly heated bottom wall, θ(X,0)=1, and cooled inclined walls, θ(X,Y)=0, with Ra=5×103 and Pr=0.026 (Case I). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 5

Temperature and stream function contours for a uniformly heated bottom wall, θ(X,0)=1, and cooled inclined walls, θ(X,Y)=0, with Ra=104 and Pr=0.026 (Case I). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 6

Temperature and stream function contours for a uniformly heated bottom wall, θ(X,0)=1, and cooled inclined walls, θ(X,Y)=0, with Ra=4×104 and Pr=0.026 (Case I). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 7

Temperature and stream function contours for a uniformly heated bottom wall, θ(X,0)=1, and cooled inclined walls, θ(X,Y)=0, with Ra=105 and Pr=0.026 (Case I). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 8

Temperature and stream function contours for a uniformly heated bottom wall, θ(X,0)=1, and cooled inclined walls, θ(X,Y)=0, with Ra=105 and Pr=0.7 (Case I). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 9

Temperature and stream function contours for a uniformly heated bottom wall, θ(X,0)=1, and cooled inclined walls, θ(X,Y)=0, with Ra=105 and Pr=1000 (Case I). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 10

Temperature and stream function contours for nonuniformly heated bottom walls, θ(X,0)=sin(πX∕2), and cooled inclined walls, θ(X,Y)=0, with Ra=5×103 and Pr=0.026 (Case II). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 11

Temperature and stream function contours for nonuniformly heated bottom walls, θ(X,0)=sin(πX∕2), and cooled inclined walls, θ(X,Y)=0, with Ra=105 and Pr=0.026 (Case II). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 12

Temperature and stream function contours for nonuniformly heated bottom walls, θ(X,0)=sin(πX∕2), and cooled inclined walls, θ(X,Y)=0, with Ra=105 and Pr=0.7 (Case II). Clockwise and anticlockwise flows are shown with negative and positive signs of stream function, respectively.

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Figure 13

Variation of the local Nusselt number with distance for low Pr at the (a) bottom wall and (b) inclined wall for uniformly (—) and nonuniformly (---) heated bottom walls and cold isothermal inclined walls.

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Figure 14

Variation of the local Nusselt number with distance for high Pr at the (a) bottom wall and (b) inclined wall for uniformly (—) and nonuniformly (---) heated bottom walls and cold isothermal inclined walls.

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Figure 15

Variation of the average Nusselt number with the Rayleigh number for uniformly heated ((a) and (b)) and nonuniformly heated bottom walls ((c) and (d)) with Pr=0.026 (—) and Pr=0.07 (---). The insets show the plot of local Nusselt numbers (Nub and Nus) versus the distance for three sets of Ra.

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Figure 16

Variation of the average Nusselt number with the Rayleigh number for uniformly heated ((a) and (b)) and nonuniformly heated bottom walls ((c) and (d)) with Pr=0.7 (—) and Pr=10.0 (---). The insets show the log-log plot of the average Nusselt number versus the Rayleigh number for convection dominant regimes.

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