Conservative phase-field lattice Boltzmann model for interface tracking equation

Based on the phase-field theory, we propose a conservative lattice Boltzmann method to track the interface between two different fluids. The presented model recovers the conservative phase-field equation and conserves mass locally and globally. Two entirely different approaches are used to calculate the gradient of the phase field, which is needed in computation of the normal to the interface. One approach uses finite-difference stencils similar to many existing lattice Boltzmann models for tracking the two-phase interface, while the other one invokes central moments to calculate the gradient of the phase field without any finite differences involved. The former approach suffers from the nonlocality of the collision operator while the latter is entirely local making it highly suitable for massive parallel implementation. Several benchmark problems are carried out to assess the accuracy and stability of the proposed model.

Figure 1

Diagonal translation of a circular interface at $\mathrm{Pe}=60$. The phase-field contour ($\varphi =0$) is shown: Black dashed lines represent the initial profile ($T=0$) and red solid lines show the interface after 10 cycles ($T=10T_f$). (a) Results of the proposed SRT model of Refs. [4, 5] and conservative LBM using (b) finite differences and (c) moments to compute the interface normal.

Figure 2

Rotation of the Zalesak's disk with time.

Figure 3

Zalesak's disk at $\mathrm{Pe}=60$. The phase-field contour ($\varphi =0$) is shown at $T=0$ (black dashed line) and $T=2T_f$ (red solid line). (a) Results of the SRT model of Refs. [4, 5] and conservative LBM using (b) finite differences and (c) moments to compute the interface normal.

Figure 4

Evolution of a circular interface under shear flow.

Figure 5

A circular interface under shear flow at $\mathrm{Pe}=60$. The phase-field contour ($\varphi =0$) is shown at $T=T_f$ (upper row) and $T=2T_f$ (lower row). (a, d) Results of the SRT model of Refs. [4, 5]; (b, e) conservative LBM using finite differences; (c, f) conservative LBM using moments to compute the interface normal.

Figure 6

Evolution of a circular interface subject to a deformation flow field.

Figure 7

Deformation of a circular interface at $\mathrm{Pe}=60$. The phase-field contour ($\varphi =0$) is shown at $T=T_f/2$ (upper row) and $T=T_f$ (lower row). (a, d) Results of the SRT model of Refs. [4, 5]; (b, e) conservative LBM using finite differences; (c, f) conservative LBM using moments to compute the interface normal.

Figure 8

Deformation of a circular interface at $\mathrm{Pe}=60$ subjected to a smoothed velocity field. The phase-field contour ($\varphi =0$) is shown at $T=T_f/2$ (upper row) and $T=T_f$ (lower row). (a, d) Results of the SRT model of Refs. [4, 5]; (b, e) conservative LBM using finite differences; (c, f) conservative LBM using moments to compute the interface normal.

Figure 9

Convergence study: $L_2$ norm versus number of grid points for the diagonal translation of a circular interface at $\mathrm{Pe}=60$ and $\mathrm{Cn}=0.015$.