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This paper develops a regularized lattice Boltzmann method for efficiently simulating the flow of N-phase immiscible incompressible fluids based on the phase field model that satisfies conservation and compatibility. By designing auxiliary moments, this method can accurately recover the second-order Allen-Cahn equation and the modified momentum equation. The correctness and effectiveness of the developed N-phase regularized lattice Boltzmann method are validated through numerical simulations of three-phase liquid lens spreading and Kelvin-Helmholtz instability phenomena. Finally, numerical simulations and analyses of three-phase Rayleigh-Taylor instabilities (RTI) were conducted, focusing on the evolution of the phase interface within the Reynolds number range of $ 500 \leqslant Re \leqslant 20000 $ (particularly under high Reynolds number condition of $ Re = 20000 $). Quantitative analyses were performed on the amplitude variations of bubbles and spikes at the two interfaces, as well as the changes in dimensionless velocity. We found that as the Reynolds number increases, the phase interface curls up at multiple locations due to Kelvin-Helmholtz instability, making the fluid more prone to dispersion and fragmentation. This study also simulated the evolution of RTI under different interface perturbations. The results indicate that RTI first develops at the perturbed interface, and its evolution gradually triggers instability at another interface.
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Keywords:
- phase field model /
- N-phase incompressible fluid /
- lattice Boltzmann method /
- Rayleigh-Taylor instability
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图 1 三相液体透镜示意图
Figure 1. The schematic of the spreading of a liquid lens.
图 2 不同界面张力比下的三相液体透镜平衡态 (a) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:1:1 $, (b) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:\sqrt{2}:1 $, (c) $ \sigma_{12}:\sigma_{13}: $$ \sigma_{23}=1:\sqrt{3}:1 $
Figure 2. The equilibrium shapes of liquid by ternary fluids: (a) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:1:1 $, (b) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:\sqrt{2}:1 $, (c) $ \sigma_{12}:\sigma_{13}:\sigma_{23}=1:\sqrt{3}:1 $.
图 3 Kelvin-Helmholtz不稳定性示意图
Figure 3. The schematic of the spreading of Kelvin-Helmholtz instability.
图 4 不同时刻下的密度分布 (a) $ t=2000 $, (b) $ t=3000 $, (c) $ t=5000 $, (d) $ t=9000 $
Figure 4. The density distribution at (a) $ t=2000 $, (b) $ t=3000 $, (c) $ t=5000 $, (d) $ t=9000 $.
图 5 不同时刻下的涡场图 (a) $ t=2000 $, (b) $ t=3000 $, (c) $ t=5000 $, (d) $ t=9000 $
Figure 5. The vorticity field at (a) $ t=2000 $, (b) $ t=3000 $, (c) $ t=5000 $, (d) $ t=9000 $.
图 6 Rayleigh-Taylor不稳定性示意图
Figure 6. The schematic of the Rayleigh-Taylor instability.
图 7 两相情况相界面扰动演化过程 (a) $ Re=30 $, (b) $ Re=150 $, (c) $ Re=3000 $, (d) $ Re=30 $[7], (e) $ Re=150 $[7], (f)$ Re=3000 $[7]
Figure 7. Phase interface disturbance evolution process in two-phase situation: (a) $ Re=30 $, (b)$ Re=150 $, (c) $ Re=3000 $, (d) $ Re= $$ 30 $[7], (e) $ Re=150 $[7], (f) $ Re=3000 $[7].
图 8 两相情况气泡与尖钉振幅随时间的变化 (a) 尖钉振幅, (b) 气泡振幅
Figure 8. Variation of bubble and spike amplitudes with time in two-phase situation: (a)$ H_{{\rm{s}}} $, (b)$ H_{{\rm{b}}} $.
图 9 较低雷诺数对RT不稳定性中相界面演化过程的影响 (a) $ Re=500 $, (b) $ Re=1000 $, (c) $ Re=2000 $
Figure 9. The effect of lower Reynolds numbers on the evolution of the phase interface in RTI: (a) $ Re=500 $, (b) $ Re=1000 $, (c) $ Re=2000 $.
图 10 较高雷诺数对RT不稳定性中相界面演化过程的影响 (a) $ Re=5000 $, (b)$ Re=10000 $, (c) $ Re=20000 $
Figure 10. The effect of higher Reynolds numbers on the evolution of the phase interface in RTI: (a) $ Re=5000 $, (b)$ Re=10000 $, (c) $ Re=20000 $.
图 11 雷诺数对气泡与尖钉随时间演化振幅的影响 (a) 界面一尖钉振幅, (b) 界面二尖钉振幅, (c) 界面一气泡振幅, (d) 界面二气泡振幅
Figure 11. Effect of Reynolds number on the temporal evolution of bubble and spike amplitudes: (a)$ H_{{\rm{s1}}} $, (b)$ H_{{\rm{s2}}} $, (c)$ H_{{\rm{b1}}} $, (d)$ H_{{\rm{b2}}} $.
图 12 雷诺数对无量纲化的气泡和尖钉演化速度的影响 (a) 界面一尖钉, (b) 界面二尖钉, (c) 界面一气泡, (d)界面二气泡
Figure 12. Effect of Reynolds number on the normalized growth rate of bubbles and spikes: (a)$ Fr_{{\rm{s1}}} $, (b)$ Fr_{{\rm{s2}}} $, (c)$ Fr_{{\rm{b1}}} $, (d)$ Fr_{{\rm{b2}}} $.
图 13 上层相界面扰动相界面演化过程 (a) $ Re=500 $, (b) $ Re=1000 $, (c) $ Re=5000 $, (d) $ Re=20000 $
Figure 13. Upper interfacial perturbation and evolution: (a) $ Re=500 $, (b) $ Re=1000 $, (c) $ Re=5000 $, (d) $ Re=20000 $.
图 14 上层相界面扰动时气泡与尖钉振幅随时间的变化 (a) 界面一尖钉振幅, (b) 界面二尖钉振幅, (c) 界面一气泡振幅, (d) 界面二气泡振幅
Figure 14. Bubble and spike amplitude evolution during upper interface perturbation: (a) $ H_{{\rm{s1}}} $, (b) $ H_{{\rm{s2}}} $, (c) $ H_{{\rm{b1}}} $, (d) $ H_{{\rm{b2}}} $.
图 15 下层相界面扰动相界面演化过程 (a) $ Re=500 $, (b) $ Re=1000 $, (c) $ Re=5000 $, (d) $ Re=20000 $
Figure 15. Lower interfacial perturbation and evolution: (a) $ Re=500 $, (b)$ Re=1000 $, (c) $ Re=5000 $, (d) $ Re=20000 $.
图 16 下层相界面扰动时气泡与尖钉振幅随时间的变化 (a) 界面一尖钉振幅, (b) 界面二尖钉振幅, (c) 界面一气泡振幅, (d) 界面二气泡振幅
Figure 16. Bubble and spike amplitude evolution during lower interface perturbation: (a) $ H_{{\rm{s1}}} $, (b) $ H_{{\rm{s2}}} $, (c) $ H_{{\rm{b1}}} $, (d) $ H_{{\rm{b2}}} $.
表 1 不同表面张力比下液体透镜的长度d和高度$ h_1 $, $ h_2 $
Table 1. The length d, $ h_1 $ and $ h_2 $ at equilibrium state with different surface tension ratios.
$ \sigma_{12}:\sigma_{13}:\sigma_{23} $ 解析解 数值解 相对误差 d $ h_1 $ $ h_2 $ d $ h_1 $ $ h_2 $ d $ h_1 $ $ h_2 $ $ 1:1:1 $ 83.10 23.99 23.99 84.26 24.45 24.42 1.40% 1.92% 1.80% $ 1:\sqrt{2}:1 $ 72.67 36.34 15.05 74.02 37.03 15.34 1.86% 1.90% 1.93% $ 1:\sqrt{3}:1 $ 55.05 47.67 7.38 55.83 48.42 7.48 1.42% 1.57% 1.36% 
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