Consistent second-order boundary implementations for convection-diffusion lattice Boltzmann method

In this study, an alternative second-order boundary scheme is proposed under the framework of the convection-diffusion lattice Boltzmann (LB) method for both straight and curved geometries. With the proposed scheme, boundary implementations are developed for the Dirichlet, Neumann and linear Robin conditions in a consistent way. The Chapman-Enskog analysis and the Hermite polynomial expansion technique are first applied to derive the explicit expression for the general distribution function with second-order accuracy. Then, the macroscopic variables involved in the expression for the distribution function is determined by the prescribed macroscopic constraints and the known distribution functions after streaming [see the paragraph after Eq. (29) for the discussions of the “streaming step” in LB method]. After that, the unknown distribution functions are obtained from the derived macroscopic information at the boundary nodes. For straight boundaries, boundary nodes are directly placed at the physical boundary surface, and the present scheme is applied directly. When extending the present scheme to curved geometries, a local curvilinear coordinate system and first-order Taylor expansion are introduced to relate the macroscopic variables at the boundary nodes to the physical constraints at the curved boundary surface. In essence, the unknown distribution functions at the boundary node are derived from the known distribution functions at the same node in accordance with the macroscopic boundary conditions at the surface. Therefore, the advantages of the present boundary implementations are (i) the locality, i.e., no information from neighboring fluid nodes is required; (ii) the consistency, i.e., the physical boundary constraints are directly applied when determining the macroscopic variables at the boundary nodes, thus the three kinds of conditions are realized in a consistent way. It should be noted that the present focus is on two-dimensional cases, and theoretical derivations as well as the numerical validations are performed in the framework of the two-dimensional five-velocity lattice model.

Figure 1

Schematic description of boundary nodes around a curved boundary.

Figure 2

Concentration distribution at various times obtained from the BGK model based boundary scheme for cases of (a) $\mathrm{Pe}=1$; (b) $\mathrm{Pe}=10$.

Figure 3

Relative error at different Pe for the concentration versus mesh resolution $L=\left(N_x-1\right)\delta x$ for the unsteady 1D tests obtained from the BGK model based boundary scheme. The blue dashed reference line has a slope of −2, which reflects the second-order accuracy of the results. The calculated slopes of the best fit line for the error trends, as well as a blue dashed reference line with a slope of −2, are provided.

Figure 4

Relative error at various τ values for the temperature (a) and the interior temperature gradient (b) vs mesh resolution $H=\left(N_y-1\right)\delta x$ for heat transfer in a 2D channel flow with the Dirichlet condition obtained from the BGK model based boundary scheme. The calculated slopes of the best fit line for the error trends, as well as a pink dashed reference line with a slope of −2, are provided.

Figure 5

Relative error at various τ values for the temperature (a) and the interior temperature gradient (b) vs mesh resolution $H$ for heat transfer in a 2D channel flow with the Neumann condition obtained from the BGK model based boundary scheme. The calculated slopes of the best fit line for the error trends, as well as a pink dashed reference line with a slope of −2, are provided.

Figure 6

Relative error comparisons between the BGK collision operator and the MRT collision operator based on the tests of heat transfer in a 2D channel flow with the Dirichlet condition (a) and Neumann condition (b). The calculated slopes of the best fit line for the error trends, as well as a pink dashed reference line with a slope of −2, are provided.

Figure 7

Relative error for the temperature vs relaxation parameter values (i.e., $\tau$ for the BGK model and $1/\phantom{1s_1}{s}_1$ for the MRT model) for the tests of heat transfer in a 2D channel flow with the Dirichlet condition (a) and Neumann condition (b) obtained from the BGK collision operator and the MRT collision operator, respectively.

Figure 8

Relative error at various τ values for the concentration (a) and the interior concentration gradients (b) vs mesh resolution $L$ for the Helmholtz equation in a square domain obtained from the BGK model based boundary scheme. The calculated slopes of the best fit line for the error trends, as well as a pink dashed reference line with a slope of −2, are provided.

Figure 9

Relative error comparisons between the present scheme for BGK model and the Huang et al. [60] scheme based on the test of the Helmholtz equation in a square domain: (a) concentration; and (b) error interior concentration gradients.

Figure 10

Relative error comparisons between the BGK collision operator and the MRT collision operator based on the test of the Helmholtz equation in a square domain. The calculated slopes of the best fit line for the error trends, as well as a pink dashed reference line with a slope of −2, are provided.

Figure 11

Relative error for the concentration vs relaxation parameter values (i.e., $\tau$ for the BGK model and $1/\phantom{1s_1}{s}_1$ for the MRT model) for test of the Helmholtz equation in a square domain obtained from the BGK collision operator and the MRT collision operator, respectively.

Figure 12

Relative error at various τ values for the temperature (a), interior temperature gradients (b), boundary value (c), and the boundary flux (d) vs mesh resolution $r_0=\left(N_r-1\right)\delta x$ for the heat conduction inside a circle with Dirichlet condition obtained from the BGK model based boundary scheme. The calculated slopes of the best fit line for the error trends are provided.

Figure 13

Relative error at various τ values for the temperature (a), interior temperature gradients (b), and the boundary flux (c) vs mesh resolution $r_0$ for the heat conduction inside a circle with Neumann condition obtained from the BGK model based boundary scheme. The calculated slopes of the best fit line for the error trends are provided. Note: the relative errors for the boundary value are not presented here due to the additional treatments of the boundary nodes needed for the present case.

Figure 14

Relative error at various τ values for the temperature (a), interior temperature gradients (b), boundary value (c), and the boundary flux (d) vs mesh resolution $r_0$ for the heat conduction inside a circle with linear Robin condition obtained from the BGK model based boundary scheme. The calculated slopes of the best fit line for the error trends are provided.

Figure 15

Relative error comparisons at $\tau =0.75$ for the temperature distribution between the present scheme for BGK model and the Huang et al. [60] scheme based on the test of the heat conduction inside a circle with Dirichlet condition. Note: comparisons for the boundary value are not presented here because the reference data are not provided in Huang et al. [60].

Figure 16

Relative error comparisons at $\tau =0.75$ between the present scheme for BGK model and the Huang et al. scheme [60] based on the test of the heat conduction inside a circle with linear Robin condition: (a) error for the temperature distribution; (b) error for the boundary values.

Figure 17

Relative error comparisons between the BGK collision operator and the MRT collision operator based on the tests of heat conduction inside a circle with Dirichlet condition (a), Neumann condition (b), and linear Robin condition (c). The calculated slopes of the best fit line for the error trends, as well as a pink dashed reference line with a slope of −2, are provided.

Figure 18

Isotherms in the region between the circular cylinder and the cavity walls for Ra values of (a) ${10}^3$; (b) ${10}^4$; (c) ${10}^5$; and (d) ${10}^6$ obtained from the BGK model based boundary scheme.

Figure 19

Streamlines in the region between the circular cylinder and the cavity walls for Ra values of (a) ${10}^3$; (b) ${10}^4$; (c) ${10}^5$; and (d) ${10}^6$ obtained from the BGK model based boundary scheme.