Family of parametric second-order boundary schemes for the vectorial finite-difference-based lattice Boltzmann method

In this paper, we propose a family of parametric second-order boundary schemes for the vectorial finite-difference-based lattice Boltzmann method (FD-LBM), which consist of convex combinations. The FD-LBM unifies several different numerical schemes for the Navier-Stokes equations, and thereby these boundary schemes are naturally applicable for the standard LBM. The accuracy of the boundary schemes is independent of the boundary location, and it is validated by several numerical experiments, two- and three-dimensional flow problems, with straight and curved boundaries.

Figure 1

Construction of the boundary scheme. ${\mathit{x}}_l$ and ${\mathit{x}}_f$ are lattice nodes inside the computational domain $\mathrm{\Omega }$, and ${\mathit{x}}_r$ is a lattice node outside the domain. ${\mathit{x}}_b$ is the intersection of the lattice line and the boundary. ${\mathit{x}}_m$ and ${\mathit{x}}_c$ are two auxiliary points on the lattice line. $\gamma$ and $\lambda$ are two scaled distances.

Figure 2

Convergence order of the vectorial FD-LBM (5) together with the boundary scheme (14) for the Couette flow. From top to bottom: $d=A, d=A^2, d=A^2(2-A), d=A=1$ and from left to right: $\gamma =0.25, \gamma =0.5, \gamma =0.75$ with $\lambda =0.3\gamma , 0.5\gamma , min\{\gamma ,0.5\}$.

Figure 3

Illustration of the irregular domain. The fluid domain is that enclosed with the thick curve.

Figure 4

Convergence order of the vectorial FD-LBM (5) together with the boundary scheme (14) for the Taylor-Green vortex flow in an irregular domain $\mathrm{\Omega }$. From left to right: $\lambda =0.3\gamma , 0.5\gamma , min\{\gamma ,0.5\}$ with $d=A, d=A^2, d=A^2(2-A), d=A=1$.

Figure 5

Convergence order of the vectorial FD-LBM (5) together with the boundary scheme (14) for the Ethier-Steinman exact Navier-Stokes solution in a sphere domain $\mathrm{\Omega }$. From left to right: $\lambda =0.3\gamma , 0.5\gamma , min\{\gamma ,0.5\}$, with $d=A, d=A^2, d=A^2(2-A), d=A=1$.

Figure 6

Convergence order of the vectorial FD-LBM (5) together with the boundary scheme (14) for the Hagen-Poiseuille flow in a cylindrical pipe. From left to right: $\lambda =0.3\gamma , 0.5\gamma , min\{\gamma ,0.5\}$, with $d=A, d=A^2, d=A^2(2-A), d=A=1$.