Multiphase flows are very important in industrial application. In present study, the force schemes in the pseudopotential LBM for two-phase flows have been compared in detail and the force schemes include Shan-Chen, EDM, MED, and Guo’s schemes. Numerical simulations confirm that all four schemes are consistent with the Laplace law. For Shan-Chen scheme, the smaller
Multiphase flows are very important in industrial application [
In pseudopotential LBM, the interactions between fluids are simulated by an artificial interparticle potential. So, the force scheme is very important to simulate two-phase flows accurately. There are four main kinds of force schemes in the pseudopotential LBM: the first is Shan and Chen’s force scheme [
Based on above analysis, although some studies have been carried out on the force schemes in pseudopotential LBM, but the detailed comparison of different force schemes is scarce in available literature. In present study, a detailed comparison of force schemes including Shan-Chen, EDM, MED, and Guo’s schemes will be carried out.
The first pseudopotential lattice Boltzmann model for two-phase flows is proposed by Shan and Chen [
The equilibrium distribution functions can be calculated by
The density and flow velocity can be obtained by the following:
The pseudopotential lattice Boltzmann model with EDM [
The real fluid velocity can be obtained from Ginzburg and Adler [
The pseudopotential lattice Boltzmann model with Guo’s force scheme [
The density and flow velocity can be obtained by the following [
The pseudopotential lattice Boltzmann model with MED force scheme [
The density and flow velocity can be obtained by
In order to get relatively large density ratio, the Carnahan-Starling Equation of State (C-S EOS) is used in the present study [
In this section, two-phase separation will be used to test four schemes of Shan-Chen, EDM, MED, and Guo. The force term can be calculated by the following:
In the present work, the interaction potential is defined according to the method by Chen et al. [
In computation,
Predicted coexistence densities of two-phase separation by of Shan-Chen, EDM, MED, and Guo’s schemes
It can be known from Figure
Surface tension is of great importance in the two-phase flows and its relationship with bubble radius is consistent with the Laplace law. In this section, Laplace’s law will be used to verify four schemes and the effect of
The effect of
|
|
|
---|---|---|
EDM scheme | ||
|
||
0.75 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.8 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.85 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.95 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
MED scheme | ||
|
||
0.8 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.85 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.95 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
Guo’s scheme | ||
|
||
0.85 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.9 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
0.95 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Relationship between bubble radius and pressure difference at different temperatures for
Shan-Chen scheme
EDM scheme
Guo’s scheme
MED scheme
Relationship between bubble radius and pressure difference at different
EDM scheme
Guo’s scheme
MED scheme
The density distribution of stable bubble with EDM force scheme.
It can be seen from Figures
The study shows that the simulation will become unstable when
The minimum
It can be seen from Figure
The spurious current is one of the important criteria to evaluate pseudopotential models because it will lead to the computation instability. In order to study the effect of four force schemes on spurious current, a series of tests have been carried out. The maximum spurious currents for different
Maximum spurious currents for Shan-Chen, EDM, Guo’s, and MED schemes.
|
|
| |||
---|---|---|---|---|---|
Shan-Chen scheme | EDM scheme | Guo’s scheme | MED scheme | ||
0.95 | 0.90 |
|
|
|
|
0.85 |
|
|
|
|
|
0.80 |
|
|
|
|
|
0.75 |
|
|
|
|
|
0.70 |
|
|
|
|
|
|
|||||
0.90 | 0.90 |
|
|
|
|
0.85 |
|
|
|
|
|
0.80 |
|
|
|
|
|
0.75 |
|
|
|
|
|
0.70 |
|
|
|
|
|
|
|||||
0.85 | 0.90 |
|
|
|
|
0.85 |
|
|
|
|
|
0.80 |
|
|
|
|
|
0.75 |
|
|
|
|
|
0.70 |
|
|
|
|
|
0.65 |
|
|
— |
|
The maximum spurious currents of Shan-Chen, EDM, MED, and Guo’s schemes for different
Moreover, it can be seen from Figure
Comparison of maximum spurious currents for EDM scheme between Kupershtokh et al. [
In present study, the force schemes in the pseudopotential LBM for two-phase flows have been compared in detail and the force schemes include Shan-Chen, EDM, MED, and Guo’s schemes. The LBM with four schemes have been used to study the two-phase separation and surface tension. Besides, the maximum two-phase density ratio and spurious currents also are discussed in detail. Based on the above study, the following conclusions can be drawn:
(1) Numerical simulations confirm that all four schemes are consistent with the Laplace law. For the EDM, MED, and Guo’s schemes,
(2) When
(3) Except for the EDM scheme, the curves of the maximum spurious currents
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The first author would like to acknowledge the financial support of the National Natural Science Foundation of China (Grants nos. 51409183, 51579166, and 51611130203). Besides, this work was also supported by the National Key Technologies R&D Program of China (no. 2015BAD24B01).