Hydrodynamic behavior of the pseudopotential lattice Boltzmann method for interfacial flows

The lattice Boltzmann method (LBM) is routinely employed in the simulation of complex multiphase flows comprising bulk phases separated by nonideal interfaces. The LBM is intrinsically mesoscale with a hydrodynamic equivalence popularly set by the Chapman-Enskog analysis, requiring that fields slowly vary in space and time. The latter assumptions become questionable close to interfaces where the method is also known to be affected by spurious nonhydrodynamical contributions. This calls for quantitative hydrodynamical checks. In this paper, we analyze the hydrodynamic behavior of the LBM pseudopotential models for the problem of the breakup of a liquid ligament triggered by the Plateau-Rayleigh instability. Simulations are performed at fixed interface thickness, while increasing the ligament radius, i.e., in the “sharp interface” limit. The influence of different LBM collision operators is also assessed. We find that different distributions of spurious currents along the interface may change the outcome of the pseudopotential model simulations quite sensibly, which suggests that a proper fine-tuning of pseudopotential models in time-dependent problems is needed before the utilization in concrete applications. Taken all together, we argue that the results of the proposed paper provide a valuable insight for engineering pseudopotential model applications involving the hydrodynamics of liquid jets.

Figure 1

Initial configuration for the numerical simulations. A cylindrical ligament of radius $R_0$ is perturbed with a sinusoidal wave along the axial ($x$) coordinate. The perturbation wavelength corresponds to the fastest growing mode of the Plateau-Rayleigh instability [6].

Figure 2

Panel (a): We report the LBM density evolution as a function of the nondimensional time ($t^{★}$) for the SRT collision operator and for different resolutions. Panel (b): We report the dimensionless breakup times as a function of the ligament radius for both the SRT and the MRT. The filled symbols refer to the presence of satellite droplets in postbreakup conditions. Panel (c): postbreakup conditions for both the SRT and the MRT.

Figure 3

Influence of the initial perturbation $\epsilon =\frac{\delta }{R_0}$ for the two collision operators on the nondimensional breakup time and comparison with literature data [52] and linear stability analysis prediction [49].

Figure 4

Panel (a): Normalized perturbation with respect to the initial perturbation value at $x\approx L_x/2$ (and $x=L_x/8$ in the inset) as a function of nondimensional time. Panel (b): Three snapshots of interface evolution at $t^{★}=7–9$ both for the SRT and for the MRT.

Figure 5

Normalized minimum ligament height near the breakup time.

Figure 6

Comparison of the time evolution of the normalized velocity magnitude according to the SRT and the MRT collision operators.