Conservative phase-field method with a parallel and adaptive-mesh-refinement technique for interface tracking

Based on Fick's second law and Cahn-Hilliard theory, a conservative phase-field model is developed to track interface. The phase-field variable changes in a hyperbolic tangent behavior across the diffuse interface over which the interface curvature can be easily calculated. Different from the frequently used lattice-Boltzmann–based discrete method, the phase-field equation is discretized using a fourth-order Runge-Kutta method. Accordingly, the present numerical scheme alleviates the programming burden, reduces the memory usage, but maintains a high numerical accuracy. To achieve large-scale interface tracking, a parallel and adaptive-mesh-refinement algorithm is developed to reduce the computing overhead. Various cases of the interface evolutions under steady flow fields indicate that the proposed numerical scheme can capture the interface with high accuracy. Furthermore, the robustness of the numerical scheme is validated by simulating the Rayleigh-Taylor instability, and good agreement with previous work is achieved.

Figure 1

Schematic diagram of the Cartesian uniform grid. The shadow box denotes the volume element associated with the grid point $(i,j)$.

Figure 2

Five-level (see $I\text{−}V$) hierarchical data structure of Zalesak's slotted disk. The mesh and patchbox are depicted by the black and red lines, respectively. The patchbox includes a collection of meshes with the same size. (a) The configuration of the data structure, including the meshes and patchboxes. (b)–(f) The layout of patchboxes on each level, and (g) shows the overall superimposed configuration.

Figure 3

$\varphi \text{−}\varepsilon$ profile along the normal direction of the circle interface with different interface parameters $\delta$, i.e., 2, 5, and 10, respectively.

Figure 4

Simulation results of Zalesak's slotted disk using Euler forward (see the first row) and RK4 (see the second row) discretization methods. Each column corresponds to the results using the time step $dt=0.2$, 0.5, and 1 from left to right, respectively. The final one (see thin solid line, especially in the zoomed-in image) after one period $T_f$ is superimposed on the initial one (see thick solid line). The width ratio of the two kinds of line is 6.

Figure 5

Simulation results under different configurations regarding the number of grid levels, i.e., from 1 to 6 levels for (a)–(f), respectively.

Figure 6

Relative deviation $E_{\varphi }$ vs (a) time step $dt$ and (b) spatial step $dx$ at a constant Péclet number, respectively.

Figure 7

Relative deviation $E_{\varphi }$ vs (a) initial velocity $U_0$, (b) interface thickness $\delta$, and (c) mobility $M$ under the same time step and spatial step, respectively.

Figure 8

Rotation of Zalesak's slotted disk with a four-level hierarchical architecture for the 2D case (see the first row) and 3D case (see the second row), respectively. Each column corresponds to the time at $t=0$, $0.25T_f$, $0.5T_f$, $0.75T_f$, and $T_f$, respectively. The boxes shown are the patchboxes rather than the meshes.

Figure 9

Evolution of the circular or spherical interface with four-level hierarchical architecture under the shear flow field for the 2D case (see the first row) and 3D case (see the second row), respectively. Each column corresponds to the time at $t=0$, $0.25T_f$, $0.5T_f$, $0.75T_f$, and $T_f$, respectively. The boxes shown are the patchboxes rather than the meshes.

Figure 10

Distribution of the patchboxes superimposed on the meshes at $t=0.5T_f$. The number of the grid level is 4, i.e., from I to IV in (b). The patchboxes and meshes are depicted by the red and black lines, respectively.

Figure 11

Evolution of the circular or spherical interface with four-level hierarchical architecture after reversing the flow field at $t=0.5T_f$ for the 2D case (see the first row) and 3D case (see the second row), respectively. Each column corresponds to the time at $t=0.75T_f$ and $T_f$, respectively. The boxes shown are the patchboxes rather than the meshes.

Figure 12

Evolution of the circular interface with four-level hierarchical architecture under the deformation flow field. The first and the second rows correspond to the cases with and without $cos(\pi t/T_f)$ in the velocity field, respectively. Each column corresponds to the time at $t=0$, $0.25T_f$, $0.5T_f$, $0.75T_f$, and $T_f$, respectively. The boxes shown are the patchboxes rather than the meshes.

Figure 13

Evolution of the spherical interface with four-level hierarchical architecture under the deformation flow field at $t=0$, $0.25T_f$, $0.5T_f$, $0.75T_f$, and $T_f$, respectively. The boxes shown are the patchboxes rather than the meshes.

Figure 14

Evolution of the two-phase interface of the 2D Rayleigh-Taylor instability at dimensionless times $t^{\prime }=1,2,3$, 4, and 5, respectively.

Figure 15

Evolution of the two-phase interface of the 3D Rayleigh-Taylor instability at dimensionless times $t^{\prime }=1,2,3$, and 4, respectively.

Figure 16

Variation with time of the positions of the bubble front, saddle point (only in 3D), and spike tip in (a) 2D and (b) 3D, respectively.