Study of phase-field lattice Boltzmann models based on the conservative Allen-Cahn equation

Conservative phase-field (CPF) equations based on the Allen-Cahn model for interface tracking in multiphase flows have become more popular in recent years, especially in the lattice-Boltzmann (LB) community. This is largely due to their simplicity and improved efficiency and accuracy over their Cahn-Hilliard–based counterparts. Additionally, the improved locality of the resulting LB equation (LBE) for the CPF models makes them more ideal candidates for LB simulation of multiphase flows on nonuniform grids, particularly within an adaptive-mesh refinement framework and massively parallel implementation. In this regard, some modifications—intended as improvements—have been made to the original CPF-LBE proposed by Geier et al. [Phys. Rev. E 91, 063309 (2015)] which require further examination. The goal of the present study is to conduct a comparative investigation into the differences between the original CPF model proposed by Geier et al. [Phys. Rev. E 91, 063309 (2015)] and the so-called improvements proposed by Ren et al. [Phys. Rev. E 94, 023311 (2016)] and Wang et al. [Phys. Rev. E 94, 033304 (2016)]. Using the Chapman-Enskog analysis, we provide a detailed derivation of the governing equations in each model and then examine the efficacy of the above-mentioned models for some benchmark problems. Several test cases have been designed to study different configurations ranging from basic yet informative flows to more complex flow fields, and the results are compared with finite-difference simulations. Furthermore, as a development of the previously proposed CPF-LBE model, axisymmetric formulations for the proposed model by Geier et al. [Phys. Rev. E 91, 063309 (2015)] are derived and presented. Finally, two benchmark problems are designed to compare the proposed axisymmetric model with the analytical solution and previous work. We find that the accuracy of the model for interface tracking is roughly similar for different models at high viscosity ratios, high density ratios, and relatively high Reynolds numbers, while the original CFP-LBE without the additional time-dependent terms outperforms the so-called improved models in terms of efficiency, particularly on distributed parallel machines.

Figure 1

Moving bubble in a uniform vertical flow. Model A is shown with a black solid line, model B is shown with a red dashed line, model C is shown with a blue dash-dot-dot line, and model D is shown with a green long-dashed line (all lines are on top of each others). $\mathrm{Pe}=1$. Left to right is $t=5T_f,t=10T_f,t=15T_f,t=20T_f$, and $t=25T_f$.

Figure 2

Moving bubble in a uniform vertical flow. Model A is shown with a black solid line (circular shape in the center), and model B is shown with a red dashed line (circular shape in the center). Models A and B are on top of each other. Model C is shown with a blue dash-dot-dot line (horizontal elliptical shape), and model D is shown with a green long-dashed line (vertical elliptical shape). $\mathrm{Pe}=5$. Left to right is $t=5T_f,t=10T_f,t=15T_f,t=20T_f$, and $t=25T_f$.

Figure 3

Moving bubble in a uniform vertical flow. Model A is shown with a black solid line (circular shape in the center), and model B is shown with a red dashed line (circular shape in the center). Models A and B are on top of each other. Model C is shown with a blue dash-dot-dot line (horizontal elliptical shape), and model D is shown with a green long-dashed line (vertical elliptical shape). $\mathrm{Pe}=20$. Left to right is $t=5T_f,t=10T_f,t=15T_f,t=20T_f$, and $t=25T_f$.

Figure 4

Moving bubble in a uniform vertical flow. Model A is shown with a black solid line (circular shape in the center), and model B is shown with a red dashed line (circular shape in the center). Models A and B are on top of each other. Model C is shown with a blue dash-dot-dot line (irregular shape expanded in the $x$ direction), and model D is shown with a green long-dashed line (irregular shape expanded in the $y$ direction). $\mathrm{Pe}=50$. Left to right is $t=5T_f,t=10T_f,t=15T_f,t=20T_f$, and $t=25T_f$.

Figure 5

Moving bubble in a uniform diagonal flow at $t=10T_f$. (a) $\mathrm{Pe}=5$ and (b) $\mathrm{Pe}=50$. Model A is shown with a black solid line (circular shape in the center), and model B is shown with a red dashed line (circular shape in the center). Models A and B are on top of each other. Model C is shown with a blue dash-dot-dot line, and model D is shown with a green long-dashed line (figure is a zoom-in). (a) All the lines are on top of each other. (b) Models A and B are on top of each other. Model C is a horizontal elliptical shape when the $x$ axis rotates ${135}^{\circ }$ counterclockwise. Model D is a horizontal elliptical shape when the $x$ axis rotates ${45}^{\circ }$ counterclockwise.

Figure 6

Zalesak's disk at $\mathrm{Pe}=75$. (a) Model A. (b) Model B. Black solid line is initial condition and red line is $t=T_f$. Lines are on top of each other.

Figure 7

Rotation of disk with time. Left to right is $t=\frac{1}{4}{T}_f$, $t=\frac{1}{2}{T}_f$, $t=\frac{3}{4}{T}_f$, and $t=T_f$. The black line is model A and the red line is model B. Lines are on top of each other.

Figure 8

Rotation of disk with time. Left to right is $t=\frac{1}{4}{T}_f$, $t=\frac{1}{2}{T}_f$, $t=\frac{3}{4}{T}_f$, and $t=T_f$. The black line is model A and the red line is model B. Lines are on top of each other.

Figure 9

Time evolution of the vortex drop. Left to right is $t=0$, $t=\frac{1}{4}{T}_f$, $t=\frac{1}{2}{T}_f$, $t=T_f$. Top to bottom $L_0=100$ ($\frac{W}{L_0}=0.05$), $L_0=200$ ($\frac{W}{L_0}=0.025$), $L_0=400$ ($\frac{W}{L_0}=0.0125$), $L_0=800$ ($\frac{W}{L_0}=0.00625$). The black line is model A and the red dashed line is model B. Lines are on top of each other.

Figure 10

The phase-field contour after $5000000$ iterations is shown. Model A is shown with a black solid line, model B is shown with a red dashed line, and the initial condition is denoted with a black dot. Lines are on top of each other.

Figure 11

Evolution of bubble rising in a continuous phase. Left to right is (a) case A1, (b) case A2, (c) case A3, and (d) case B4. Nondimensional times for (a), (b), and (c) are $t^{*}=0$, 1.5, 3, 4.5, 6, and 7.5 and for (d) are $t^{*}=0$, 1.5, 3, 5.5, and 10. Solid black line is model A and dashed red line is model B. Lines are on top of each other.

Figure 12

Time history of the normalized mass for both models.

Figure 13

(a) Profile of the phase field and additional term as a function of $r$ for two different interface locations of $r_0=10$ and $r_0=20$. (b) Highest absolute value of the additional term for different interface locations.

Figure 14

Evolution of droplet oscillation in the density ratio $\frac{{\rho }_h}{{\rho }_l}=100$ at (a) $t=0$, (b) $t=1700$, (c) $t=2700$, and (d) $t=4000$. Model A is shown with a black line and model B is shown with a red dashed line. Lines are on top of each other.

Figure 15

Time evolution of the half-axis length $R_r$ at different density ratios.