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Research Papers

A Study on Natural Convection in a Cold Square Enclosure With Two Vertical Eccentric Square Heat Sources Using the IB–LBM Scheme

[+] Author and Article Information
S. M. Dash

Aerospace Engineering,
Indian Institute of Technology,
Kharagpur, India
e-mail: smdash@aero.iitkgp.ac.in

S. Sahoo

Mechanical Engineering,
Indian Institute of Technology (ISM),
Dhanbad, India
e-mail: satya@iitism.ac.in

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Thermal Science and Engineering Applications. Manuscript received October 15, 2018; final manuscript received February 3, 2019; published online April 3, 2019. Assoc. Editor: Aaron P. Wemhoff.

J. Thermal Sci. Eng. Appl 11(5), 051013 (Apr 03, 2019) (23 pages) Paper No: TSEA-18-1515; doi: 10.1115/1.4042858 History: Received October 15, 2018; Accepted February 04, 2019

In this article, the natural convection process in a two-dimensional cold square enclosure is numerically investigated in the presence of two inline square heat sources. Two different heat source boundary conditions are analyzed, namely, case 1 (when one heat source is hot) and case 2 (when two heat sources are hot), using the in-house developed flexible forcing immersed boundary–thermal lattice Boltzmann model. The isotherms, streamlines, local, and surface-averaged Nusselt number distributions are analyzed at ten different vertical eccentric locations of the heat sources for Rayleigh number between 103 and 106. Distinct flow regimes including primary, secondary, tertiary, quaternary, and Rayleigh–Benard cells are observed when the mode of heat transfer is changed from conduction to convection and heat sources eccentricity is varied. For Rayleigh number up to 104, the heat transfer from the enclosure is symmetric for the upward and downward eccentricity of the heat sources. At Rayleigh number greater than 104, the heat transfer from the enclosure is better for downward eccentricity cases that attain a maximum when the heat sources are near the bottom enclosure wall. Moreover, the heat transfer rate from the enclosure in case 2 is nearly twice that of case 1 at all Rayleigh numbers and eccentric locations. The correlations for heat transfer are developed by relating Nusselt number, Rayleigh number, and eccentricity of the heat sources.

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References

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Figures

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Fig. 1

Schematic drawing of the computational domains considered in this study is shown. Here the subfigure (a) represents case 1, where two cylinders are at different temperatures (TH and TC); (b) represents case 2, where two cylinders are at the same temperature (TH), and (c) represents the arrangement of the cylinders to investigate vertical eccentricity effects. “S”, “SL”, and “SR” are the directions used to compute the local Nusselt number distribution on the enclosure and square heat source, respectively.

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Fig. 2

Schematic drawing of the computational domains to study the natural convection process from a hot circular cylinder in an enclosure is shown

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Fig. 3

(a) The streamlines and isotherms obtained using the flexible forcing IB-thermal LBM solver are compared with the results of Kim et al. [6] shown in (b).

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Fig. 4

Isotherms and streamlines at χ = 0.0L are shown for Ra varying in the range of 103 to 106. (Contour levels of 1–10 and 1–12 are plotted for isotherms and streamlines, respectively.) In the streamline plots, the counterclockwise and clockwise circulations are presented using solid and dashed lines, respectively. (a) Case 1 and (b) case 2.

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Fig. 5

Isotherms (case 1 in column [i] and case 2 in column [iii]) and streamlines (case 1 in column [ii] and case 1 in column [iv]) are shown for different downward displacements (χs) of the inner cylinders at Ra = 103. (Contour levels of 1–10 and 1–12 are shown for isotherms and streamlines, respectively.) (a) χ = 0.00L, (b) χ = −0.05L, (c) χ = −0.10L, (d) χ = −0.15L, (e) χ = −0.20L, and (f) χ = −0.25L.

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Fig. 6

Isotherms (case 1 in column [i] and case 2 in column [iii]) and streamlines (case 1 in column [ii] and case 1 in column [iv]) are shown for different upward displacements (χs) of the inner cylinders at Ra = 103. (Contour levels of 1–10 and 1–12 are shown for isotherms and streamlines, respectively.) (a) χ = 0.25L, (b) χ = 0.20L, (c) χ = 0.15L, (d) χ = 0.10L, (e) χ = 0.05L, and (f) χ = 0.00L.

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Fig. 7

Isotherms (case 1 in column [i] and case 2 in column [iii]) and streamlines (case 1 in column [ii] and case 1 in column [iv]) are shown for different downward displacements (χs) of the inner cylinders at Ra = 104. (Contour levels of 1–10 and 1–12 are shown for isotherms and streamlines, respectively.) (a) χ = 0.00L, (b) χ = −0.05L, (c) χ = −0.10L, (d) χ = −0.15L, (e) χ = −0.20L, and (f) χ = −0.25L.

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Fig. 8

Isotherms (case 1 in column [i] and case 2 in column [iii]) and streamlines (case 1 in column [ii] and case 1 in column [iv]) are shown for different upward displacements (χs) of the inner cylinders, at Ra = 104. (Contour levels of 1–10 and 1–12 are shown for isotherms and streamlines, respectively.) (a) χ = 0.25L, (b) χ = 0.20L, (c) χ = 0.15L, (d) χ = 0.10L, (e) χ = 0.05L, and (f) χ = 0.00L.

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Fig. 9

Isotherms (case 1 in column [i] and case 2 in column [iii]) and streamlines (case 1 in column [ii] and case 1 in column [iv]) are shown for different downward displacements (χs) of the inner cylinders at Ra = 105. (Contour levels of 1–10 and 1–12 are shown for isotherms and streamlines, respectively.) (a) χ = 0.00L, (b) χ = −0.05L, (c) χ = −0.10L, (d) χ = −0.15L, (e) χ = −0.20L, and (f) χ = −0.25L.

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Fig. 10

Isotherms (case 1 in column [i] and case 2 in column [iii]) and streamlines (case 1 in column [ii] and case 1 in column [iv]) are shown for different upward displacements (χs) of the inner cylinders, at Ra = 105. (Contour levels of 1–10 and 1–12 are shown for isotherms and streamlines respectively.) (a) χ = 0.25L, (b) χ = 0.20L, (c) χ = 0.15L, (d) χ = 0.10L, (e) χ = 0.05L, and (f) χ = 0.00L.

Grahic Jump Location
Fig. 11

Isotherms (case 1 in column [i] and case 2 in column [iii]) and streamlines (case 1 in column [ii] and case 1 in column [iv]) are shown for different downward displacements (χs) of the inner cylinders, at Ra = 106. (Contour levels of 1–10 and 1–12 are shown for isotherms and streamlines, respectively.) (a) χ = 0.00L, (b) χ = −0.05L, (c) χ = −0.10L, (d) χ = −0.15L, (e) χ = −0.20L, and (f) χ = −0.25L.

Grahic Jump Location
Fig. 12

Isotherms (case 1 in column [i] and case 2 in column [iii]) and streamlines (case 1 in column [ii] and case 1 in column [iv]) are shown for different upward displacements (χs) of the inner cylinders, at Ra = 106. (Contour levels of 1–10 and 1–12 are shown for isotherms and streamlines, respectively.) (a) χ = 0.25L, (b) χ = 0.20L, (c) χ = 0.15L, (d) χ = 0.10L, (e) χ = 0.05L, and (f) χ = 0.00L.

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Fig. 13

Distribution of the local Nusselt number (NuL) along the walls of the enclosure for different vertical χs at Ra = 103 is shown. Here, the direction of “S” is followed from Fig. 1. Note that the same line style and color is followed for both case 1 and case 2 for χ < 0.0L and χ > 0.0L, respectively. (a) Case 1 and (a) case 2.

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Fig. 14

Distribution of the local Nusselt number (NuL) along the sides of the square cylinder of case 1 for different vertical χs at Ra = 103 is shown. Here, the arc length direction “SL” and “SR” is followed from Fig. 1. Note that the same line style and color is followed for both (a) and (b) plots for χ < 0.0L and χ > 0.0L, respectively. (a) Case 1: left square heat source and (b) case 1: right square heat source.

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Fig. 15

Distribution of the local Nusselt number (NuL) along the sides of the square cylinder of case 2 for different vertical χs, at Ra = 103 is shown. Here, the arc length direction “SL” is followed from Fig. 1.

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Fig. 16

Distribution of the local Nusselt number (NuL) along the walls of the enclosure for different vertical χs at Ra = 105 is shown. Here, the direction of “S” is followed from Fig. 1. Note that same line style and color are followed for both case 1 and case 2 for χ < 0.0L and χ > 0.0L, respectively. (a) Case 1 and (b) case 2.

Grahic Jump Location
Fig. 17

Distribution of the local Nusselt number (NuL) along the sides of the square cylinder of case 1 for different vertical χs at Ra = 105 is shown. Here, the arc length direction “SL” and “SR” is followed from Fig. 1. Note that same line style and color are followed for both (a) and (b) plots for χ < 0.0L and χ > 0.0L, respectively. (a) Case 1: left square heat source and (b) case 1: right square heat source.

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Fig. 18

Distribution of the local Nusselt number (NuL) along the sides of the square cylinder of case 2 for different vertical χs at Ra = 105 is shown. Here, the arc length direction “SL” is followed from Fig. 1.

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Fig. 19

Distribution of the local Nusselt number (NuL) along the walls of the enclosure for different vertical χs at Ra = 106 is shown. Here, the direction of “S” is followed from Fig. 1. Note that the same line style and color are followed for both case 1 and case 2 for χ < 0.0L and χ > 0.0L, respectively. (a) Case 1 and (b) case 2.

Grahic Jump Location
Fig. 20

Distribution of the local Nusselt number (NuL) along the sides of the square cylinder of case 1 for different vertical χs at Ra = 106 is shown. Here, the arc length direction “SL” and “SR” is followed from Fig. 1. Note that the same line style and color are followed for both (a) and (b) plots for χ < 0.0L and χ > 0.0L, respectively. (a) Case 1: left square heat source and (b) case 1: right square heat source.

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Fig. 21

Distribution of the local Nusselt number (NuL) along the sides of the square cylinder of case 2 for different vertical χs at Ra = 106 is shown. Here, the arc length direction “SL” is followed from Fig. 1.

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Fig. 22

Surface-averaged Nusselt number of the cold enclosure (NuAen) as a function of Ra and eccentricity (χ) is shown. (a) Case 1 and (b) case 2.

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Fig. 23

Surface-averaged Nusselt number of the square cylinder (NuAsqr) as a function of Ra and eccentricity (χ) is shown for both case 1 and case 2. Here, the suffixes “-h” and “-c” represent cold and hot squares, respectively. (a) Case 1 and (b) case 2.

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Fig. 24

Comparison of surface-averaged Nusselt number of the cold enclosure (NuAen) obtained from numerical simulations (solid lines) and prediction from correlation equations (markers) for (a) case 1 and (b) case 2, respectively.

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Fig. 25

Comparison of surface-averaged Nusselt number (NuAsqr) of the square cylinders obtained from numerical simulations (solid lines) and prediction from correlation equations (markers). Here, the suffixes “-h” and “-c” represent cold and hot squares, respectively. (a) Case 1 and (b) case 2.

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