Implementation of contact angles in pseudopotential lattice Boltzmann simulations with curved boundaries

The pseudopotential multiphase lattice Boltzmann (LB) model is a very popular model in the LB community for simulating multiphase flows. When the multiphase modeling involves a solid boundary, a numerical scheme is required to simulate the contact angle at the solid boundary. In this work, we aim at investigating the implementation of contact angles in the pseudopotential LB simulations with curved boundaries. In the pseudopotential LB model, the contact angle is usually realized by employing a solid-fluid interaction or specifying a constant virtual wall density. However, it is shown that the solid-fluid interaction scheme yields very large spurious currents in the simulations involving curved boundaries, while the virtual-density scheme produces an unphysical thick mass-transfer layer near the solid boundary although it gives much smaller spurious currents. We also extend the geometric-formulation scheme in the phase-field method to the pseudopotential LB model. Nevertheless, in comparison with the solid-fluid interaction scheme and the virtual-density scheme, the geometric-formulation scheme is relatively difficult to implement for curved boundaries and cannot be directly applied to three-dimensional space. By analyzing the features of these three schemes, we propose an improved virtual-density scheme to implement contact angles in the pseudopotential LB simulations with curved boundaries, which does not suffer from a thick mass-transfer layer near the solid boundary and retains the advantages of the original virtual-density scheme, i.e., simplicity, easiness for implementation, and low spurious currents.

Figure 1

Sketch of the characteristic lines of a point in the ghost contact-line region.

Figure 2

Illustration of the intersection point $D_2$ for different contact angles.

Figure 3

Illustration of the halfway bounce-back boundary scheme.

Figure 4

Static contact angles obtained by the virtual-density scheme. (a) ${\rho }_w=4.5$, (b) ${\rho }_w=3.25$, and (c) ${\rho }_w=1.5$. From left to right $\theta \approx {31}^{\circ }$, 65°, and 121°, respectively.

Figure 5

Static contact angles obtained by the improved virtual-density scheme. (a) $\phi =1.4$, (b) $\phi =1$, and (c) $\mathrm{\Delta }\rho =0.5$. From left to right $\theta \approx {34}^{\circ }$, 88°, and 125°, respectively.

Figure 6

Static contact angles obtained by the solid-fluid interaction scheme. (a) $G_w=-0.6$, (b) $G_w=0.3$, and (c) $G_w=1.2$. From left to right $\theta \approx {38}^{\circ }$, 59°, and 119°, respectively.

Figure 7

The fluid density profiles along the central vertical line, i.e., $x=N_x/2$. (Left) The density profiles near the bottom of the cylinder for the results shown in Figs. 4, 5, and 6. (Right) The density profiles near the top of the cylinder for the results shown in Figs. 4, 5, and 6. The solid circular cylinder is located at $y\in \left[60,200\right]$.

Figure 8

Static contact angles obtained by the geometric-formulation scheme. (a) ${\theta }_a={60}^{\circ }$, (b) ${\theta }_a={90}^{\circ }$, and (c) ${\theta }_a={120}^{\circ }$. From left to right $\theta \approx {58}^{\circ }$, 88°, and 121°, respectively.

Figure 9

The spurious currents produced by (a) the solid-fluid interaction scheme at $G_w=1.2$, (b) the virtual-density scheme at ${\rho }_w=1.5$, and (c) the improved virtual-density scheme at $\mathrm{\Delta }\rho =0.5$.

Figure 10

Comparison of the maximum spurious currents yielded by different contact angle schemes.

Figure 11

Comparison of the (a) maximum and (b) minimum fluid densities obtained by the simulations using different contact angle schemes.

Figure 12

Simulations of Poiseuille flow between two parallel solid plates. The channel confined by the two solid plates is fully filled with the liquid phase. Comparison of the velocity profiles obtained by (a) the virtual-density scheme and (b) the improved virtual-density scheme.

Figure 13

Droplet impact on a cylindrical surface at $\mathrm{Re}=600$ and $\theta \approx {60}^{\circ }$. A comparison of the results obtained by (a) the virtual-density scheme, (b) the geometric-formulation scheme, and (c) the improved virtual-density scheme. From left to right $t=100{\delta }_t$, $300{\delta }_t$, $4000{\delta }_t$, and $10000{\delta }_t$, respectively.

Figure 14

Validation of the improved virtual-density scheme for simulating 3D contact angles on a curved surface. A 3D view is shown in the upper row, while in the lower row the density contours of the x-z cross-section at $y=100$ are presented. (a) $\phi =1.2$ with $\theta \approx {53}^{\circ }$, (b) $\phi =1$ with $\theta \approx {88}^{\circ }$, and (c) $\mathrm{\Delta }\rho =0.55$ with $\theta \approx {145}^{\circ }$.