Color-gradient lattice Boltzmann model with nonorthogonal central moments: Hydrodynamic melt-jet breakup simulations

We develop a lattice Boltzmann (LB) model for immiscible two-phase flow simulations with central moments (CMs). This successfully combines a three-dimensional nonorthogonal CM-based LB scheme [De Rosis, Phys. Rev. E 95, 013310 (2017)] with our previous color-gradient LB model [Saito, Abe, and Koyama, Phys. Rev. E 96, 013317 (2017)]. Hydrodynamic melt-jet breakup simulations show that the proposed model is significantly more stable, even for flow with extremely high Reynolds numbers, up to $O\left({10}^6\right)$. This enables us to investigate the phenomena expected under actual reactor conditions.

Figure 1

(a) Snapshots of typical jet breakup regimes in liquid-liquid systems: dripping (0), varicose (I), sinuous without entrainment (IIa), sinuous with entrainment (IIb), and atomization (III) [13]. (b) Map of jet breakup regimes in immiscible liquid-liquid systems [13, 14]

Figure 2

Fluid flow properties revealed at different scales by different simulation methods. The LB method is a mesoscopic simulation approach that lies between microscopic particle-based (e.g., molecular dynamics) and macroscopic Navier-Stokes–based methods.

Figure 3

Boundary conditions for the melt-jet breakup simulations. (a) The boundaries consist of an inflow boundary, wall boundaries, and an outflow boundary. (b) There is a circular inflow boundary of diameter $D_{j0}$ at the top, where the velocity $u_{j0}$ is uniform. Free-slip [67] and convective [91] boundary conditions are imposed at the wall and outflow boundaries, respectively.

Figure 4

Comparison of jet breakup behavior for Case 1 in Table 2 ($D_{j0}=7$ mm) for the (a) UT experiment and (b) simulation. The simulation parameters used were as follows (in lattice units): $\sigma =6.6\times {10}^{-3}$, ${\nu }_j={\nu }_r=2.5\times {10}^{-5}$, ${\nu }_c={\nu }_b=4.5\times {10}^{-5}$, $g=1.3\times {10}^{-6}$, ${\rho }_j={\rho }_r^0=1$, and ${\rho }_c={\rho }_b^0=0.12$. The minimum spatial resolution in this case was $\mathrm{\Delta }x=0.23$ mm. The experimental results clearly show trapped gas, which caused differences in the interface shape compared with the simulation.

Figure 5

Comparison of jet breakup behavior for Case 2 in Table 2 ($D_{j0}=10$ mm) for the (a) UT experiment and (b) simulation. The simulation parameters used were as follows (in lattice units): $\sigma =8.4\times {10}^{-3}$, ${\nu }_j={\nu }_r=2.3\times {10}^{-5}$, ${\nu }_c={\nu }_b=4.3\times {10}^{-5}$, $g=3.4\times {10}^{-6}$, ${\rho }_j={\rho }_r^0=1$, and ${\rho }_c={\rho }_b^0=0.12$. The minimum spatial resolution in this case was $\mathrm{\Delta }x=0.33$ mm. Between $t=0.2$ and $t=0.25$ s, the fragments in the simulation spread out significantly less than in the experiments.

Figure 6

Comparison of jet breakup behavior for Case 3 in Table 2 ($D_{j0}=15$ mm) for the (a) UT experiment and (b) simulation. The simulation parameters used were as follows (in lattice units): $\sigma =4.5\times {10}^{-3}$, ${\nu }_j={\nu }_r=1.4\times {10}^{-5}$, ${\nu }_c={\nu }_b=2.6\times {10}^{-5}$, $g=4.1\times {10}^{-6}$, ${\rho }_j={\rho }_r^0=1$, and ${\rho }_c={\rho }_b^0=0.12$. The minimum spatial resolution in this case was $\mathrm{\Delta }x=0.5$ mm. Although the results spread out more in the lateral direction than the experimental results do between $t=0.16$ and 0.25 s, their final shape is close to that observed experimentally.

Figure 7

Comparison of jet breakup behavior for Case 4 in Table 2 ($D_{j0}=20$ mm) for the (a) UT experiment and (b) simulation. The simulation parameters used were as follows (in lattice units): $\sigma =3.3\times {10}^{-3}$, ${\nu }_j={\nu }_r=1.0\times {10}^{-5}$, ${\nu }_c={\nu }_b=1.9\times {10}^{-5}$, $g=5.3\times {10}^{-6}$, ${\rho }_j={\rho }_r^0=1$, and ${\rho }_c={\rho }_b^0=0.12$. The minimum spatial resolution in this case was $\mathrm{\Delta }x=0.67$ mm. In the simulation, the fragment groups were generated around the jet's leading edge and the side of the jet column, showing similar behavior to that observed in the experiment.

Figure 8

(a) Snapshot of detailed interface structure for Case 4 in Table 2 ($D_{j0}=20$ mm). The color bar indicates the velocity magnitude normalized by the inlet velocity. The liquid column has large velocity, while the generated fragments has small velocity. The regions surrounded by gray dashed lines are shown below. (b) Magnified snapshot for upstream region. The fragments generate from the unstable liquid jet interface. Most of the fragments in this region are stretched, which appear not to be spherical shapes. The velocity magnitude of stretched fragments are large, while that of spherical ones are small. (c) Magnified snapshot for downstream region. Most of the fragments in this region are spherical shapes with low velocity magnitude.

Figure 9

Evolution of the jet's leading edge over time for (a) Case 1 ($D_{j0}=7$ mm), (b) Case 2 ($D_{j0}=10$ mm), (c) Case 3 ($D_{j0}=15$ mm), and (d) Case 4 ($D_{j0}=20$ mm). In all cases, the experimental observations are ahead of the simulation early, but the simulation passes the experiment in the latter stages. Overall, the simulations reproduce the experimental trends well.

Figure 10

Fragment-diameter distribution histograms for (a) Case 1 ($D_{j0}=7$ mm), (b) Case 2 ($D_{j0}=10$ mm), (c) Case 3 ($D_{j0}=15$ mm), and (d) Case 4 ($D_{j0}=20$ mm). Here $d_m$ denotes the mass-median diameter. Larger inlet diameter $D_{j0}$ lead to large fragments appearing more frequently.

Figure 11

Comparison of the experimental and simulated fragment sizes. Here the critical Weber number ${\mathrm{We}}_{\mathrm{cr}}$ in Eq. (52) was assumed to be 18. Both the experimental [23] and simulation results are in accordance with the Kelvin-Helmholtz [Eq. (51)] and critical Weber number [Eq. (52)] theories. However, the Rayleigh-Taylor instability [Eq. (50)] does not correlate with either the numerical or experimental results.

Figure 12

Flow regime region covered by the FT experiment together with the conditions our previous simulations (“o”) [14]. Our current simulations (“$\times$”) involve substantially higher Re values compared with the previous ones. The present simulation condition is located extremely higher $\mathrm{Re}$ condition compared with our previous simulation. The hatched gray region shows the reactor conditions estimated in Ref. [13]; the FT experiment fell within this region. This diagram predicts that the breakup will be in the atomization regime (III). The conditions of UT simulations presented in Sec. 4b are also shown in the same map for reference.

Figure 13

FT simulation results for the coarse (low-resolution) case, showing the (a) interfacial behavior and (b) fragment-diameter distribution. The inlet diameter was set to $D_{j0}=30$ (in lattice units), and the computational domain was discretized into a $240\times 240\times 600$ lattice $(=34560000\mathrm{grid}\mathrm{points})$. The simulation parameters were as follows (in lattice units): $\sigma =1.6\times {10}^{-5}$, ${\nu }_j={\nu }_r=1.4\times {10}^{-6}$, ${\nu }_c={\nu }_b=9.9\times {10}^{-7}$, $g=4.1\times {10}^{-7}$, ${\rho }_j={\rho }_r^0=1$, and ${\rho }_c={\rho }_b^0=0.099$. The minimum spatial resolution in this case was $\mathrm{\Delta }x=1.67$ mm. Although the computation is stable, some unphysical behavior can be seen: The fragments do not appear to be spherical, and the fragments that should appear around the jet's leading edge are not present. In addition, the fragment-diameter distribution appears to be log-normal but is interrupted near the minimum resolution $\mathrm{\Delta }x$.

Figure 14

FT simulation results for the fine (high-resolution) case, showing the (a) interfacial behavior and (b) fragment diameter distribution. The inlet diameter was set to $D_{j0}=60$ (in lattice units), and the computational domain is discretized into a $480\times 480\times 1200$ lattice $(=276480000\mathrm{grid}\mathrm{points})$. The simulation parameters used were as follows (in lattice units): $\sigma =3.2\times {10}^{-5}$, ${\nu }_j={\nu }_r=2.8\times {10}^{-6}$, ${\nu }_c={\nu }_b=2.0\times {10}^{-6}$, $g=2.0\times {10}^{-7}$, ${\rho }_j={\rho }_r^0=1$, and ${\rho }_c={\rho }_b^0=0.099$. The minimum spatial resolution in this case was $\mathrm{\Delta }x=0.83$ mm. Compared with the results in Fig. 13, the interfacial behavior seen in Fig. 14 is clearly different in terms of the fragment sizes and shapes: The fragments are smaller and nearly spherical. In addition, the fragment-diameter distribution in Fig. 14 is smoother than that in Fig. 13, and the higher-resolution domain has also improved the numerical dissolution.