Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions

In this work, we propose an interfacial scheme accompanying the lattice Boltzmann method for convection-diffusion equations with general interfacial conditions, including conjugate conditions with or without jumps in heat and mass transfer, continuity of macroscopic variables and normal fluxes in ion diffusion in porous media with different porosity, and the Kapitza resistance in heat transfer. The construction of this scheme is based on our boundary schemes [Huang and Yong, J. Comput. Phys. 300, 70 (2015)] for Robin boundary conditions on straight or curved boundaries. It gives second-order accuracy for straight interfaces and first-order accuracy for curved ones. In addition, the new scheme inherits the advantage of the boundary schemes in which only the current lattice nodes are involved. Such an interfacial scheme is highly desirable for problems with complex geometries or in porous media. The interfacial scheme is numerically validated with several examples. The results show the utility of the constructed scheme and very well support our theoretical predications.

Figure 1

Schematic depiction of the lattice and a straight boundary. Solid circles, interior nodes; open circles, exterior nodes; gray squares, boundary nodes.

Figure 2

Schematic map of the lattice and the curved boundary. Solid circles, interior nodes; open circles, exterior nodes; gray squares, boundary nodes.

Figure 3

Schematic map of the lattice and curved interface. Solid circles, interior nodes in ${\mathrm{\Omega }}^{\text{B}}$; open circles, interior nodes in ${\mathrm{\Omega }}^{\text{A}}$; gray squares, interface points.

Figure 4

Schematic map of the mesh grid for steady and unsteady convection-diffusions in a rectangular channel containing two fluids in domain A ($0\le y\le l$) and domain B ($l\le y\le H$).

Figure 5

Contour map of the numerical solution ${\varphi }^{\text{num}}$ for the steady-state convection-diffusion in the channel. Parameters: $h=H/128=1/128, {\tau }^{\text{A}}=0.75, {\tau }^{\text{B}}=1.25$, Pe = 20, and $\gamma =0.5$.

Figure 6

Profiles of $\varphi$ and $\frac{\partial \varphi }{\partial y}$ at the interface $y=l$ in domain A for the steady convection-diffusion in a rectangular channel. Parameters: $h=H/256=1/256, {\tau }^{\text{A}}=0.75, {\tau }^{\text{B}}=1.25$, Pe = 20, and $\gamma =0.3$.

Figure 7

Relative ${\ell }^2$-errors $E, E_{\text{grad}}$, and $E_{\text{interface}}$ and absolute ${\ell }^2$ error of interfacial fluxes versus the mesh size $h$ for steady convection-diffusion in a rectangular channel with different $\gamma$. Parameters: ${\tau }^{\text{A}}=0.75, {\tau }^{\text{B}}=1.25$, and Pe = 0.

Figure 8

Relative ${\ell }^2$ errors $E, E_{\text{grad}}, E_{\text{interface}}$, and $E_{\text{interfacial}\text{flux}}$ versus the mesh size $h$ for steady convection-diffusion in a rectangular channel with different $\gamma$. Parameters: ${\tau }^{\text{A}}=0.75, {\tau }^{\text{B}}=1.25$, and Pe = 20.

Figure 9

Relative ${\ell }^2$ errors $E, E_{\text{grad}}, E_{\text{interface}}$, and $E_{\text{interfacial}\text{flux}}$ versus the mesh size $h$ for steady convection-diffusion in a rectangular channel with different ${\tau }^{\text{A}}$. Parameters: $\gamma =0.3$ and Pe = 20.

Figure 10

Profiles of $\varphi$ and $\frac{\partial \varphi }{\partial y}$ at the interface $y=l$ for unsteady convection-diffusion in a rectangular channel at time $T=1$. Parameters: ${\tau }^{\text{A}}=0.75, {\tau }^{\text{B}}=1.25$, Pe = 20, $h=1/256$, and $\gamma =0.3$.

Figure 11

Relative ${\ell }^2$ errors $E, E_{\text{grad}}, E_{\text{interface}}$, and $E_{\text{interfacial}\text{flux}}$ versus the mesh size $h$ for unsteady convection-diffusion in a rectangular channel with different $\gamma$ at time $T=1$. Parameters: ${\tau }^{\text{A}}=0.75, {\tau }^{\text{B}}=1.25$, and Pe = 0.

Figure 12

Relative ${\ell }^2$ errors $E, E_{\text{grad}}, E_{\text{interface}}$, and $E_{\text{interfacial}\text{flux}}$ versus the mesh size $h$ for unsteady convection-diffusion in a rectangular channel with different $\gamma$ at time $T=1$. Parameters: ${\tau }^{\text{A}}=0.75, {\tau }^{\text{B}}=1.25$, and Pe = 20.

Figure 13

Relative ${\ell }^2$ errors $E, E_{\text{grad}}, E_{\text{interface}}$, and $E_{\text{interfacial}\text{flux}}$ versus the mesh size $h$ for unsteady convection-diffusion in a rectangular channel with different ${\tau }^{\text{A}}$ at time $T=1$. Parameters: $\gamma =0.3$ and Pe = 20.

Figure 14

Profiles of $\varphi$ and $\frac{\partial \varphi }{\partial r}$ at the interface $r=R_1$ for steady-state heat conduction in a circular domain. Parameters: ${\tau }^{\text{A}}=3, R_c=1$, and $h=1/256$.

Figure 15

Relative ${\ell }^2$ errors $E, E_{\text{grad}}, E_{\text{interface}}$, and $E_{\text{interfacial}\text{flux}}$ versus the mesh size $h$ for steady-state heat conduction in a circular domain. Parameter: $R_c=0$.

Figure 16

Relative ${\ell }^2$ errors $E, E_{\text{grad}}, E_{\text{interface}}$, and $E_{\text{interfacial}\text{flux}}$ versus the mesh size $h$ for steady-state heat conduction in a circular domain. Parameter: $R_c=1$.

Figure 17

Relative ${\ell }^2$ errors $E, E_{\text{grad}}, E_{\text{interface}}$, and $E_{\text{interfacial}\text{flux}}$ versus the mesh size $h$ for unsteady-state heat conduction in a circular domain at time $T=1$.