Discrete effects on boundary conditions of the lattice Boltzmann method for fluid flows with curved no-slip walls

The lattice Boltzmann method (LBM) has been formulated as a powerful numerical tool to simulate incompressible fluid flows. However, it is still a critical issue for the LBM to overcome the discrete effects on boundary conditions successfully for curved no-slip walls. In this paper, we focus on the discrete effects of curved boundary conditions within the framework of the multiple-relaxation-time (MRT) model. We analyze in detail a single-node curved boundary condition [Zhao et al., Multiscale Model. Simul. 17, 854 (2019)] for predicting the Poiseuille flow and derive the numerical slip at the boundary dependent on a free parameter as well as the distance ratio and the relaxation times. An approach by virtue of the free parameter is then proposed to eliminate the slip velocity while with uniform relaxation parameters. The theoretical analysis also indicates that for previous curved boundary schemes only with the distance ratio and the halfway bounce-back (HBB) boundary scheme, the numerical slip cannot be removed with uniform relaxation times virtually. We further carried out some simulations to validate our theoretical derivations, and the numerical results for the case of straight and curved boundaries confirm our theoretical analysis. Finally, for fluid flows with curved boundary geometries, resorting to more degrees of freedom from the boundary scheme may have more potential to eliminate the discrete effect at the boundary with uniform relaxation times.

Figure 1

Schematic of a curved-wall boundary along one single lattice direction. The thin solid line is the grid line, and the thick curved one represents the boundary surface. White circles ($\circ$): the fluid nodes; black circle ($•$): the intersection point of the boundary with the grid line; square box ($\square$): the solid node outside the computational domain.

Figure 2

Schematic of the flow and lattice arrangement with a distance ratio $\gamma$. The wall boundary in the HBB boundary condition is placed with $\gamma =1/2$.

Figure 3

Schematic of the Poiseuille flow between two aligned straight walls.

Figure 4

Slip velocity as a function of $\gamma$ under different $l$ at $M=20$ and $\text{Re}=10$. Solid lines denote the slip velocity derived theoretically [Eq. (33)], and symbols denote the predicted slip velocity from numerical simulations.

Figure 5

Velocity profiles predicted from the present analysis with different lattice sizes and (a) $\gamma =0.1$, (b) $\gamma =0.9$. Filled shapes denote the results obtained under $U_s=0$, i.e., Eqs. (36) and (37) with $C=-0.55$. Empty shapes denote the results obtained under $U_s\ne 0$ with $C=-0.55$. Concretely, Eq. (36) and ${\tau }_q=100$ are employed for the case of (a) while $l=0.6$ and Eq. (37) for the case of (b).

Figure 6

Dependence of $l_0$ on $\gamma$ and ${\tau }_s$. The variable $l_0$ is defined as $l_0=\frac{{\gamma }^2+\gamma (2{\tau }_s-1)}{2{\tau }_s-1}$.

Figure 7

Work status of code computations against different $\gamma$ and ${\tau }_s$ at $\text{Re}=10$ and $M=20$ for (a) $C=-0.55$ and (b) $C=-0.6$. The filled circles mean that the computations can converge to the steady state, and the asterisks mean that the computations collapse.

Figure 8

Schematic illustration of the inclined Poiseuille flow with an inclination angle $\theta$.

Figure 9

Velocity profiles of the inclined channel flow under different lattice sizes at $\text{Re}=20$. Filled shapes denote the results predicted by the boundary scheme (15) with Eqs. (36) and (37). Empty shapes denote the results predicted by the HBB scheme with Eq. (40). (a) $\tan \theta =0.2$, (b) $\tan \theta =0.5$, and (c) $\tan \theta =1.5$.

Figure 10

Velocity profiles of the inclined channel flow predicted under different cases of $l$ at $M_{AB}=10$ and $\text{Re}=20$. (a) $\tan \theta =0.5$ and (b) $\tan \theta =1.5$.

Figure 11

Schematic illustration of the circular Couette flow.

Figure 12

Velocity profiles of the cylindrical Couette flow for three cases: (a) $\beta =0.2,{\omega }_1=0.02,{\omega }_2=0$; (b) $\beta =0.5,{\omega }_1=0,{\omega }_2=0.02$; and (c) $\beta =0.8,{\omega }_1=0.02,{\omega }_2=-0.01$. Filled shapes denote the results from the scheme (15) with Eqs. (36) and (37) ($C=-0.6$). Empty shapes denote the results predicted by the HBB scheme with Eq. (40).

Figure 13

Velocity profiles of the cylindrical Couette flow under (a) $\beta =0.2$, (b) $\beta =0.5$, and (c) $\beta =0.8$ at $M=18$.