Improved three-dimensional color-gradient lattice Boltzmann model for immiscible two-phase flows

In this paper, an improved three-dimensional color-gradient lattice Boltzmann (LB) model is proposed for simulating immiscible two-phase flows. Compared with the previous three-dimensional color-gradient LB models, which suffer from the lack of Galilean invariance and considerable numerical errors in many cases owing to the error terms in the recovered macroscopic equations, the present model eliminates the error terms and therefore improves the numerical accuracy and enhances the Galilean invariance. To validate the proposed model, numerical simulations are performed. First, the test of a moving droplet in a uniform flow field is employed to verify the Galilean invariance of the improved model. Subsequently, numerical simulations are carried out for the layered two-phase flow and three-dimensional Rayleigh-Taylor instability. It is shown that, using the improved model, the numerical accuracy can be significantly improved in comparison with the color-gradient LB model without the improvements. Finally, the capability of the improved color-gradient LB model for simulating dynamic two-phase flows at a relatively large density ratio is demonstrated via the simulation of droplet impact on a solid surface.

Figure 1

Density contours of a moving droplet simulated by (a) the original model and (b) the improved model. From left to right: $t=60000{\delta }_t$, $80000{\delta }_t$, and $95000{\delta }_t$.

Figure 2

Schematic of the layered two-phase flow between two parallel plates.

Figure 3

Simulation of layered two-phase flow in a channel. Comparison of the velocity profiles obtained by the original and improved color-gradient LB models for (a) case A and (b) case B.

Figure 4

Simulation of layered two-phase flow in a channel. Comparison of the velocity profiles obtained by the original and improved color-gradient LB models for case C.

Figure 5

Convergence order of the present color-gradient LB model.

Figure 6

Simulation of three-dimensional Rayleigh-Taylor instability. Snapshots of the fluid interface obtained by (a) the original model and (b) the improved model at $t^{*}=1$, 2, and 3 (from left to right).

Figure 7

Comparison of the fluid interface at $t^{*}=4$ obtained by (a) the original color-gradient model, (b) the improved color-gradient model, and (c) a multiphase flux solver in Ref. [51].

Figure 8

The positions of the bubble front, the spike tip, and the saddle point versus time. The Atwood number is 0.5 and the Reynolds number is 1024.

Figure 9

Snapshots of droplet impact on a flat surface at $\mathrm{Re}=1000$ with ${\rho }_R/\phantom{{\rho }_R{\rho }_B}{\rho }_B=100$. (a) $t=2000{\delta }_t$, (b) $t=4000{\delta }_t$, (c) $t=10000{\delta }_t$, (d) $t=30000{\delta }_t$, (e) $t=60000{\delta }_t$, and (f) $t=90000{\delta }_t$.

Figure 10

Comparison of the maximum spreading factor between the present numerical results, the experimental correlation in Ref. [53], and the experimental data in Ref. [54].