Gas-kinetic Scheme Based on Cahn-Hilliard Phase-field Equation

doi:10.19596/j.cnki.1001-246x.8862

Abstract

Abstract:

The gas kinetic scheme based on Cahn-Hilliard phase field equation with interfacial fluxes determined by the Chapman-Enskog first-order approximation is constructed in the framework of finite volume. It is further demonstrated that the proposed model can be accurately recovered to Cahn-Hilliard equation. The method is tested through several cases and compared with the corresponding lattice Boltzmann method. The results show that the method in this paper has excellent accuracy and numerical stability for interface trapping. The study extends application of the gas-kinetic scheme to phase-field theory and provides a new solution for the simulation of two-phase fluid systems.

Key words: two-phase system, phase field model, Gas-kinetic scheme, Chapman-Enskog expansion

Hao ZHONG, Lianhua ZHU, Jin BAO, Zhaoli GUO. Gas-kinetic Scheme Based on Cahn-Hilliard Phase-field Equation[J]. Chinese Journal of Computational Physics, 2025, 42(2): 171-181.

Figures/Tables 19

Fig.1 Interface diagram of GKS

Fig.2 Shape of phase interface of diagonally translating droplet (Pe=5) (a) LSG-LBM; (b) GKS (solid line t=0, dashed line t=1T)

Fig.3 Shape of phase interface of diagonally translating droplet (Pe=5) (a) LSG-LBM; (b) GKS (solid line t=0, dashed line t=10T)

Fig.4 Shape of phase interface of diagonally translating droplet (Pe=2 000) (a) LSG-LBM; (b) GKS (solid line t=0, dashed line t=1T)

Fig.5 Shape of phase interface of diagonally translating droplet (Pe=2 000) (a) LSG-LBM; (b) GKS (solid line t=0, dashed line t=10T)

Table 1 Global error of translating droplets along the diagonal after 1 period with different Pe numbers

Pe	$ \\|E(\phi)\\|_2$		$\\|E(\phi)\\|_{\max } $
Pe	LSG-LBM	GKS	LSG-LBM	GKS
5	0.112 3	0.036 4	0.611 4	0.223 7
50	0.160 3	0.083 7	0.518 0	0.382 5
500	0.240 2	0.103 5	0.479 6	0.328 0
2 000	0.290 4	0.132 9	0.481 6	0.356 4

Fig.6 L2 relative error with number of periods for GKS and LBM methods

Table 2 Error of center-of-mass distance along diagonal translation of droplet after 1 and 10 periods at different Pe numbers

Pe	t=1T		t=10T
Pe	LSG-LBM	GKS	LSG-LBM	GKS
5	0.75	0.20	4.16	0.94
50	1.01	0.51	3.41	1.54
500	0.83	0.53	2.29	1.31
2 000	0.81	0.51	1.76	1.25

Fig.7 Computational time between GKS and LBM methods for along diagonal translation of droplet (Pe=500, n=10)

Fig.8 Rotation of Zalesak disc with time

Fig.9 Shape of the phase interface of rotating disk (Pe=5) (a) LSG-LBM; (b) GKS (solid line t=0, dashed line t=1T)

Fig.10 Shape of the phase interface of rotating disk(Pe=50) (a) LSG-LBM; (b) GKS (solid line t=0, dashed line t=1T)

Fig.11 Shape of the phase interface of rotating disk (Pe=2 000) (a) LSG-LBM; (b) GKS (solid line t=0, dashed line t=1T)

Table 3 Global errors of rotating discs at different Pe numbers

Pe	$ \\|E(\phi)\\|_2$		$ \\|E(\phi)\\|_{\max }$
Pe	LSG-LBM	GKS	LSG-LBM	GKS
5	0.072 6	0.055 7	0.978 3	0.972 5
50	0.069 6	0.042 4	0.915 0	0.900 9
500	0.088 0	0.055 8	0.947 5	0.810 5
2 000	1.017 0	0.073 2	4.859 2	0.781 6

Fig.12 Interface reconfiguration at different moments of single vortex shear flow(n=2) (a) t=0; (b) t=0.25T; (c) t=0.5T; (d) t=0.75T; (e) t=T

Fig.13 Interface reconfiguration at different moments of single vortex shear flow (n=4) (a) t=0; (b) t=0.25T; (c) t=0.5T; (d) t=0.75T; (e)t=T

Fig.14 (a) Order parameter conservation errors over time in one period; (b) L2 relative error with number of periods

Table 4 Relative error of single vortex shear flow at different Pe numbers after one period

Pe	$\\|E(\phi)\\|_2 $
Pe	n=2	n=4
5	0.038 2	0.112 9
50	0.048 3	0.089 6
500	0.065 5	0.144 3
2 000	0.052 9	0.140 2

Fig.15 Interfacial reconfiguration of 3D single-vortex shear flow at different moments of flow (a) t=0; (b) t=0.25T; (c) t=0.5T; (d) t=0.75T; (e) t=T

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