Forcing term in single-phase and Shan-Chen-type multiphase lattice Boltzmann models

Numerous schemes have been proposed to incorporate a bulk forcing term into the lattice Boltzmann equation. In this paper we present a simple and straightforward comparative analysis of five popular schemes [Shan and Chen, Phys. Rev. E 47, 1815 (1993); Phys Rev Lett. 81, 1618 (1998); He et al., Phys. Rev. E 57, R13 (1998); Guo et al., Phys. Rev. E 65, 046308 (2002); Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)] in which their differences and similarities are identified. From the analysis we classify the schemes into two groups; the behaviors of the schemes in each group are proven to be identical up to second order. Numerical test simulating the two-dimensional unsteady Taylor-Green vortex flow problem demonstrate that all five schemes are of comparable accuracy for single-phase flow. However, for two-phase flow the situation is different, which is demonstrated by incorporating these schemes into different Shan-Chen-type multiphase models. The forcing scheme in the original Shan-Chen (SC) multiphase model turns out to be inaccurate in terms of the resulting surface tension for different density ratios and relaxation times. In the numerical tests, a typical equation of state and interparticle interactions including next-nearest neighbors were incorporated into the SC model. Our results confirm that the surface-tension values obtained from the original SC lattice Boltzmann method (LBM) simulation depend on the value of the relaxation time $\tau$. For $\tau <0.7\Delta t$, the surface tension agree well with the analytical solutions. However, when $\tau >0.7\Delta t$, the surface tension turns out to be systematically larger than the analytical one, exceeding it by more than a factor of 2 for $\tau =2\Delta t$. In contrast, with the application of the scheme proposed by He et al., the SC LBM produces very accurate surface tensions independent of the value of $\tau$. We also found that the densities of the coexisting liquid and gas can be adjusted to match those at thermodynamic equilibrium if the particle interaction term includes next-nearest-neighbor contributions. The obtained results will be useful for further studies of two-phase flow with high density ratios using the SC LBM approach.

Figure 1

Error of the velocity field (a) when there is no body force applied and (b)–(d) as a function of the nondimensional time $t$ for (b) $\tau =1.0\Delta t$, (c) $\tau =1.5\Delta t$, and (d) $\tau =0.7\Delta t$, when body force is applied.

Figure 2

Error of the densities of liquid and gas as a function of mesh size. The slope of the top thick line is $-$2. The line represents the exact second-order spatial accuracy, which is used to guide the eyes. The other solid, dashed, and dotted lines correspond to schemes I, III, and V, respectively.

Figure 3

Results obtained by the SC model with the CS EOS and forcing scheme III for different values of $\tau$: (a) reduced density of liquid, (b) reduced density of gas, (c) surface tension, and (d) liquid-to-gas density ratio as a function of $T/T_c$.

Figure 4

Results obtained by the original SC model (scheme I) for different values of $\tau$: (a) reduced density of liquid, (b) reduced density of gas, (c) surface tension, and (d) liquid-to-gas density ratio as a function of $T/T_c$ for the CS EOS.

Figure 5

Results obtained by the SC model in combination with scheme V for different values of $\tau$: (a) reduced density of liquid, (b) reduced density of gas, (c) surface tension, and (d) density ratio as a function of $T/T_c$ for the CS EOS.

Figure 6

Surface tension $\sigma$ as a function of $T/T_c$ for the vdW EOS.

Figure 7

Stability of forcing schemes I, III, and V as a function of $\tau$. The upper right region of each line is the corresponding stable region of each scheme. The CS EOS is used with $a=1$, $R=1$, and $b=4$.

Figure 8

Surface-tension comparison between the SC model with forcing scheme III and the analytical solution for different $A_1$ and $A_2$. In the simulation, the interparticle force including next-nearest neighbors for the CS EOS is used and $\tau =\Delta t$.

Figure 9

Coexistence curve for the CS EOS: The solid line is the analytical solution obtained from the Maxwell construction (corresponding to thermodynamic consistency). The numerical results were obtained by the original SC model with $\tau =\Delta t$.