Improved forcing scheme in pseudopotential lattice Boltzmann methods for multiphase flow at arbitrarily high density ratios

The pseudopotential lattice Boltzmann method has been widely used to simulate many multiphase flow applications. However, there still exist problems with reproducing realistic values of density ratio and surface tension. In this study, a higher-order analysis of a general forcing term is derived. A forcing scheme is then constructed for the pseudopotential method that is able to accurately reproduce the full range of coexistence curves. As a result, multiphase flow of arbitrarily high density ratios independent of the surface tension can be simulated. Furthermore, the interface width can be tuned to allow for grid refinement and systematic error reduction. Numerical results confirm that the proposed scheme enables independent control of density ratio, surface tension, and interface width simultaneously.

Figure 1

Simulation results for gas densities in one dimension, for $\psi =exp(-1/\rho )$. The method of Guo (diamonds), the EDM (triangles), and the present method (crosses) are shown compared with the Maxwell construction (black line).

Figure 2

Simulation results for gas densities in one dimension, for the Carnahan-Starling equation of state. The method of Guo with $\alpha =1$ (crosses) and $\alpha =1/2$ (pluses) and the EDM with $\alpha =1$ (triangles) and $\alpha =1/2$ (diamonds) are shown compared with the Maxwell construction (black line). In both cases, increasing the interface width (by decreasing $\alpha$) does not improve the results. (Liquid densities are all in close agreement with the Maxwell construction and are therefore not shown.)

Figure 3

Simulation results for gas densities in one dimension, for the Carnahan-Starling equation of state. The present method with increasing interface widths, using $\alpha =1$ (triangles), $\alpha =1/4$ (crosses), and $\alpha =1/16$ (circles), is shown compared with the Maxwell construction (black line). $\alpha =1$, $1/4$, and $1/16$ give interface widths of, for example, $W=3.0$, $5.8$, and $11.5$, respectively, at $T=0.07$, reducing to $W=3.2$ and $6.1$ for $\alpha =1/4$ and $1/16$ at $T=0.04$ (at which point the interface width at $\alpha =1$ has become too small for a stable solution). Significant improvement over existing methods (shown in Fig. 2) is observed, with results tending to the correct density with increasing interface width (decreasing $\alpha$). (Liquid densities are all in close agreement with the Maxwell construction and are therefore not shown.)

Figure 4

Error in gas density (compared with the predicted gas density from the Maxwell construction) with interface width, $W$, for the present forcing scheme with the Carnahan-Starling equation of state. Three different density ratios are shown: ${\rho }_r\approx 10$ using $T=0.0790$ (diamonds, solid line), ${\rho }_r\approx 100$ using $T=0.0585$ (circles, dashed line), and ${\rho }_r\approx 1000$ using $T=0.0455$ (crosses, dotted line).

Figure 5

Error in gas density (compared with the predicted gas density from the Maxwell construction) with varying surface tension parameter, ${\sigma }_c$, for the present forcing scheme. The result is for the Carnahan-Starling equation of state with $T=0.0455$ (density ratio of approximately 1000). ${\sigma }_0$ is ${\sigma }_c$ with $\kappa =0$. Two different interface widths are shown: $W=9.5$ (triangles, dashed line) and $W=13.3$ (circles, solid line).

Figure 6

Laplace law results for $T=0.0585$ giving a density ratio of approximately 100 and an interface width of $W=6.1$, including linear fits. Theoretical surface tensions (normalized by surface tension at $\kappa =0$, ${\sigma }_0$) are 1 (circles, solid black line), $1/4$ (crosses, dashed line), $1/16$ (diamonds, solid gray line), and $1/64$ (pluses, dotted line).

Figure 7

Measured surface tension, ${\sigma }_m$, against theoretical surface tension ${\sigma }_{th}$ (normalized by surface tension at $\kappa =0$, ${\sigma }_0$). $T=0.0585$ giving ${\rho }_r\approx 100$. Results shown are for measurements made using the Laplace law, for two fixed interface widths: $W=6.1$ (circles) and $W=8.7$ (crosses). Solid and dashed lines show the theoretical linear relationship, with gradient equal to ${\sigma }_0$ for each case.

Figure 8

Measured surface tension, ${\sigma }_m$, against theoretical surface tension ${\sigma }_{th}$ (normalized by surface tension at $\kappa =0$, ${\sigma }_0$). $T=0.0455$ giving ${\rho }_r\approx 1000$. Results shown are for measurements made using the Laplace law, for two fixed interface widths: $W=9.4$ (circles) and $W=13.3$ (crosses). Also shown for the $W=13.3$ case are measurements made using oscillating droplets (diamonds). Solid and dashed lines show the theoretical linear relationship, with gradient equal to ${\sigma }_0$ for each case.

Figure 9

Variation in isotropy, $I$, defined in Eq. (47), and average spurious velocity, $\overline{u}$, with measured surface tension, ${\sigma }_m$. $T=0.0585$ giving ${\rho }_r\approx 100$. Results for isotropy (crosses) and spurious velocity (circles) in black are for an interface width of $W=6.1$. Results for isotropy (pluses) and spurious velocity (diamonds) in grey are for an interface width of $W=8.7$.

Figure 10

Variation in isotropy, $I$, defined in Eq. (47), and average spurious velocity, $\overline{u}$, with measured surface tension, ${\sigma }_m$. $T=0.0455$ giving ${\rho }_r\approx 1000$. Results for isotropy (crosses) and spurious velocity (circles) in black are for an interface width of $W=9.4$. Results for isotropy (pluses) and spurious velocity (diamonds) in gray are for an interface width of $W=13.3$.

Figure 11

Variation in isotropy, $I$, defined in Eq. (47), and average spurious velocity, $\overline{u}$, with measured interface width, $W$. $T=0.0585$ giving ${\rho }_r\approx 100$. Results for isotropy (crosses) and spurious velocity (circles) in black are for a surface tension of $\sigma =0.0077$. Results for isotropy (pluses) and spurious velocity (diamonds) in gray are for a surface tension of $\sigma =0.00091$.

Figure 12

Variation in isotropy, $I$, defined in Eq. (47), and average spurious velocity, $\overline{u}$, with measured interface width, $W$. $T=0.0455$ giving ${\rho }_r\approx 1000$. Results for isotropy (crosses) and spurious velocity (circles) in black are for a surface tension of $\sigma =0.0063$. Results for isotropy (pluses) and spurious velocity (diamonds) in gray are for a surface tension of $\sigma =0.00074$.