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Research Papers: Micro/Nanoscale Heat Transfer

Heat Transfer Evaluation on Curved Boundaries in Thermal Lattice Boltzmann Equation Method

[+] Author and Article Information
Like Li

e-mail: likelichina@ufl.edu

Renwei Mei

Professor
Member ASME
e-mail: rwmei@ufl.edu

James F. Klausner

Professor
Fellow ASME
e-mail: klaus@ufl.edu
Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611-6250

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Heat Transfer. Manuscript received February 18, 2013; final manuscript received July 10, 2013; published online October 25, 2013. Assoc. Editor: Zhuomin Zhang.

J. Heat Transfer 136(1), 012403 (Oct 25, 2013) (14 pages) Paper No: HT-13-1087; doi: 10.1115/1.4025046 History: Received February 18, 2013; Revised July 10, 2013

An efficient and accurate approach for heat transfer evaluation on curved boundaries is proposed in the thermal lattice Boltzmann equation (TLBE) method. The boundary heat fluxes in the discrete velocity directions of the TLBE model are obtained using the given thermal boundary condition and the temperature distribution functions at the lattice nodes close to the boundary. Integration of the discrete boundary heat fluxes with effective surface areas gives the heat flow rate across the boundary. For lattice models with square or cubic structures and uniform lattice spacing the effective surface area is constant for each discrete heat flux, thus the heat flux integration becomes a summation of all the discrete heat fluxes with constant effective surface area. The proposed heat transfer evaluation scheme does not require a determination of the normal heat flux component or a surface area approximation on the boundary; thus, it is very efficient in curved-boundary simulations. Several numerical tests are conducted to validate the applicability and accuracy of the proposed heat transfer evaluation scheme, including: (i) two-dimensional (2D) steady-state thermal flow in a channel, (ii) one-dimensional (1D) transient heat conduction in an inclined semi-infinite solid, (iii) 2D transient heat conduction inside a circle, (iv) three-dimensional (3D) steady-state thermal flow in a circular pipe, and (v) 2D steady-state natural convection in a square enclosure with a circular cylinder at the center. Comparison between numerical results and analytical solutions in tests (i)–(iv) shows that the heat transfer is second-order accurate for straight boundaries perpendicular to one of the discrete lattice velocity vectors, and first-order accurate for curved boundaries due to the irregularly distributed lattice fractions intersected by the curved boundary. For test (v), the computed surface-averaged Nusselt numbers agree well with published results.

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Figures

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Fig. 1

Discrete velocity set {eα} for the D3Q7 TLBE model. eα = (0, 0, 0), (±1, 0, 0), (0, ±1, 0), and (0, 0, ±1).

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Fig. 2

llustration of the heat transfer evaluation on a curved boundary (closed circles: field nodes; closed squares: boundary nodes; open circles: exterior nodes)

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Fig. 3

Schematic representation of the lattice covering a 2D channel of height H=(Ny-3+2Δ)δy

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Fig. 4

Wall heat flux error E2_qw versus the grid resolution, 1/H, for the channel flow Dirichlet problem at (a) Δ = 0.50, 0.25, and 0.75, and (b) Δ = 0.50, 0.01, and 0.99

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Fig. 5

Wall heat flux error E2_qw versus the lattice link fraction Δ at Ny = 34 for the channel flow Dirichlet problem

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Fig. 6

Wall heat flux error E2_qw versus (τD − 0.5) at Ny = 66 for the channel flow Dirichlet problem

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Fig. 7

Interior gradient error E2_gx in x-direction versus the grid resolution, 1/H, for the channel flow Dirichlet problem at (a) Δ = 0.50, 0.25, and 0.75, and (b) Δ = 0.50, 0.01, and 0.99

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Fig. 8

Interior gradient error E2_gy in y-direction versus the grid resolution, 1/H, for the channel flow Dirichlet problem at (a) Δ = 0.50, 0.25, and 0.75, and (b) Δ = 0.50, 0.01, and 0.99

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Fig. 9

Interior gradient errors of (a) E2_gx, and (b) E2_gy in x- and y-directions, respectively, versus the grid resolution, 1/H, for the channel flow Neumann problem

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Fig. 10

Layout of the lattice around an inclined semi-infinite solid. The blue lines depict the boundary wall at different inclination angles θ = tan -1(1.0), tan -1(1.2), tan -1(1.5), and tan -1(2.0).

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Fig. 11

Variations of the heat transfer rate with time for the Dirichlet condition T(l = 0) = t1/2 at the end of an inclined semi-infinite solid (symbols: LBE results; lines: exact solutions).

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Fig. 12

Variations of the heat transfer rate with time for the Dirichlet condition T(l = 0) = 1-exp(-0.1t) at the end of an inclined semi-infinite solid (symbols: LBE results; lines: exact solutions)

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Fig. 13

Schematic layout of the lattice around a circle

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Fig. 14

Variations of the error E(t*) defined in Eq. (35) with time at different grid resolution for the transient heat conduction inside a circle

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Fig. 15

Heat transfer error, E2, defined in Eq. (36) versus the grid resolution, 1/r0, for the transient heat conduction inside a circle

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Fig. 16

Heat transfer error E2-Q·w versus the grid resolution, 1/r0, for the circular pipe flow with a Dirichlet boundary condition

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Fig. 17

Schematic depiction of the computational domain (r0 = 0.2L) and the Dirichlet thermal and velocity boundary conditions on the inner and outer walls

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Fig. 18

Isotherms in the field between the circular cylinder and the cavity walls at (a) Ra = 103, (b) Ra = 104, (c) Ra = 105, and (d) Ra = 106

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Fig. 19

Streamlines in the field between the circular cylinder and the cavity walls at (a) Ra = 103, (b) Ra = 104, (c) Ra = 105, and (d) Ra = 106

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