Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows

In this paper, we present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models. In addition, the proposed model can achieve superior numerical accuracy compared with previous Allen-Cahn type of LB models. Several benchmark two-phase problems, including static droplet, layered Poiseuille flow, and spinodal decomposition are simulated to validate the present LB model. It is found that the present model can achieve relatively small spurious velocity in the LB community, and the obtained numerical results also show good agreement with the analytical solutions or some available results. Lastly, we use the present model to investigate the droplet impact on a thin liquid film with a large density ratio of 1000 and the Reynolds number ranging from 20 to 500. The fascinating phenomena of droplet splashing is successfully reproduced by the present model and the numerically predicted spreading radius exhibits to obey the power law reported in the literature.

Figure 1

Static droplet tests with density ratio ${\rho }_l/{\rho }_g=1000:1$. (a) The velocity distribution of the whole domain at the equilibrium state. The solid and dashed lines represent the equilibrium shape of the droplet and its initial shape, respectively; (b) the density distributions along the horizontal center line $(y=N_y/2)$ at different values of the mobility.

Figure 2

Comparisons of the horizontal velocity profiles obtained by the present model and the existing Allen-Cahn based LB models [27, 28] with various density ratios: (a) ${\rho }_l:{\rho }_g=10:1$; (b) ${\rho }_l:{\rho }_g=100:1$; (c) ${\rho }_l:{\rho }_g=150:1$; (d) ${\rho }_l:{\rho }_g=1000:1$. The solid lines represent the corresponding analytical solutions.

Figure 3

Time evolution of the density distribution during the phase separating processes, (a) $t=0$; (b) $t=0.05$; (c) $t=0.25$; (d) $t=0.5$; (e) $t=5$; (f) t = 25; (g) $t=50$; (h) $t=125$.

Figure 4

Schematic of the initial setup for the droplet impact on a thin liquid film.

Figure 5

Snapshots of droplet impact on a thin liquid film with $\text{Re}=500,\text{We}=8000$, and ${\rho }_l/{\rho }_g=1000$. The time instants $t^{*}$ have been normalized by the characteristic time $D/U$.

Figure 6

Snapshots of droplet impact on a thin liquid film with $\text{Re}=100,\text{We}=8000$, and ${\rho }_l/{\rho }_g=1000$. The time instants $t^{*}$ have been normalized by the characteristic time $D/U$.

Figure 7

Snapshots of droplet impact on a thin liquid film with $\text{Re}=20,\text{We}=8000$, and ${\rho }_l/{\rho }_g=1000$. The time instants $t^{*}$ have been normalized by the characteristic time $D/U$.

Figure 8

The numerically predicted spreading radius versus the dimensionless time. The solid line represents the theoretical power law.