Three-dimensional lattice Boltzmann flux solver for simulation of fluid-solid conjugate heat transfer problems with curved boundary

A three-dimensional (3D) lattice Boltzmann flux solver is presented in this work for simulation of fluid-solid conjugate heat transfer problems with a curved boundary. In this scheme, the macroscopic governing equations for mass, momentum, and energy conservation are discretized by the finite-volume method, and the numerical fluxes at the cell interface are reconstructed by the local solution of lattice Boltzmann equation. For solving the 3D fluid-solid conjugate heat transfer problems, the density distribution function (D3Q15 model) is utilized to compute the numerical fluxes of continuity and momentum equations, and the total enthalpy distribution function (D3Q7 model) is introduced to calculate the numerical flux of the energy equation. The connections between the macroscopic fluxes and the local solution of the lattice Boltzmann equation are provided by the Chapman-Enskog expansion analysis. As compared with the lattice Boltzmann method, in which the time step and grid spacing are correlated, the local solution of the lattice Boltzmann equation at each cell interface used in the present scheme is independent of each other. As a result, the drawback of the tie-up between the time step and grid spacing can be effectively removed and the developed method applies very well to nonuniform mesh and curved boundaries. To validate the performance of the developed method, the steady and unsteady natural convection in a finned 3D cavity and in a finned 3D annulus are simulated. Numerical results showed that the present scheme can effectively solve the 3D conjugate heat transfer problems with a curved boundary.

Figure 1

Distribution of equilibrium functions at the cell interface and its surrounding points (D3Q15 model for illustration).

Figure 2

Schematic of unsteady heat conduction in an infinite system.

Figure 3

Accuracy analysis of LBFS for unsteady heat conduction in an infinite system.

Figure 4

Schematic of natural convection in a finned 3D cavity.

Figure 5

Temperature contours for steady natural convection in a finned 3D cavity at (a) $\mathrm{Ra}={10}^4$ and (b) $\mathrm{Ra}={10}^6$.

Figure 6

Comparison of temperature contours at the $z=0.5$ plane for steady natural convection in a finned 3D cavity at (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, and (d) $\mathrm{Ra}={10}^6$. (Present: colored background with black solid line; fluent: white dashed line.)

Figure 7

Comparisons of $u$ velocity profile along the vertical centerline and $v$ velocity profile along the horizontal centerline at the $z=0.5$ plane for steady natural convection in a finned 3D cavity at (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, and (d) $\mathrm{Ra}={10}^6$.

Figure 8

Comparison of temperature profile along the horizontal centerline at $z=0.5$ plane for unsteady natural convection in a finned 3D cavity at (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, and (d) $\mathrm{Ra}={10}^6$.

Figure 9

Comparison of $u$ velocity profile along the vertical centerline at the $z=0.5$ plane for unsteady natural convection in a finned 3D cavity at (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, and (d) $\mathrm{Ra}={10}^6$.

Figure 10

Schematic of natural convection in a finned 3D annulus.

Figure 11

Temperature contours for steady natural convection in a finned 3D annulus at (a) $\mathrm{Ra}={10}^4$ and (b) $\mathrm{Ra}={10}^5$.

Figure 12

Comparison of temperature contours at $z=0$ plane for steady natural convection in a finned 3D annulus at (a) $\mathrm{Ra}={10}^4$ and (b) $\mathrm{Ra}={10}^5$. (Present: colored background with black solid line; fluent: white dashed line.)

Figure 13

Comparisons of $u$ velocity profile along the vertical centerline and $v$ velocity profile along the horizontal centerline at $z=0$ plane for steady natural convection in a finned 3D annulus at (a) $\mathrm{Ra}={10}^4$ and (b) $\mathrm{Ra}={10}^5$.

Figure 14

Comparison of temperature profile along the horizontal centerline at $z=0$ plane for unsteady natural convection in a finned 3D annulus at (a) $\mathrm{Ra}={10}^4$ and (b) $\mathrm{Ra}={10}^5$.

Figure 15

Comparison of $u$ velocity profile along the vertical centerline at $z=0$ plane for unsteady natural convection in a finned 3D annulus at (a) $\mathrm{Ra}={10}^4$ and (b) $\mathrm{Ra}={10}^5$.