An Accurate Phase Field Method for 2-Phase Flow With Soluble Surfactants
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摘要: 基于相场理论，该文提出了一种精确的考虑可溶性表面活性剂作用的两相流相场方法.为保证相界面处动量输运的守恒性，引入一致性的质量通量以考虑相变量扩散对质量守恒的影响，应用有限体积方法离散守恒形式的控制方程；选择五阶WENO格式处理控制方程的对流项，改善界面处理的精度和稳定性.此外，还构造了多组二维差分模板以进一步改善表面张力项中的梯度离散，并证实了对应格子Boltzmann D2Q9模型的模板能够显著降低伪势速度，改善表面活性剂浓度的计算精度.通过对静态液滴、双液滴融合、大密度比气泡上升以及剪切流中的单液滴变形与破裂等问题进行数值研究，充分验证了所提出的两相流相场方法的精度、守恒性与鲁棒性.Abstract: An accurate phase field method for 2-phase flow with soluble surfactants was developed based on the phase field theory. The key point of this method was the utilization of consistent and conservative mass flux to ensure the conservation of momentum transport across the interface. The finite-volume method was used to discretize the governing equations in their conservative form. The 5th-order WENO scheme was chosen to effectively handle the convective terms, aimed to enhance accuracy and robustness in addressing steep variations in the interfacial region. Furthermore, various 2D difference templates were designed to optimize gradient discretization in the surface tension term. Particularly, with the template corresponding to the lattice Boltzmann D2Q9 model, a notable reduction of the spurious velocity and a significant improvement of the accuracy of surfactant concentration prediction were achieved. Various examples such as static droplets, fusion of 2 droplets, bubble rise with a large density ratio, deformation and breakage of individual droplets in shear flow demonstrate the accuracy, conservative properties, and robustness of the proposed method.
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Key words:
- 2-phase flow /
- phase field method /
- soluble surfactant /
- continuous surface force model
edited-byedited-by1) （我刊编委张良奇、曾忠来稿） -
图 1 梯度算子模板点示意图
Figure 1. The schematic diagram of the template points of the gradient operator
图 2 数值计算流程图
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 2. The flowchart for the numerical calculation
图 5 不同半径不同表面张力系数下液滴内外压差的数值解与理论解的对比
Figure 5. Comparison of numerical and theoretical solutions of pressure differences between inside and outside droplets with different radii and different surface tension coefficients
图 6 R=0.25时相界面分布的数值解与理论解的对比
Figure 6. Comparison of numerical and theoretical solutions for the phase interface distribution at R=0.25
图 9 不同网格分辨率情况下液滴体积随时间的变化
Figure 9. The variations of droplet volumes with time for different grid resolutions
图 11 气泡上升过程中某些时刻的界面形态
Figure 11. The interface morphologies at some moments during the rise of the bubble
图 12 大密度比气泡的上升速度以及质量中心
Figure 12. The rising velocity and the centre of mass of the bubble with a large density ratio
图 13 水平中心线上表面活性剂分布
Figure 13. The surfactant distributions on the horizontal centre line
图 16 剪切流中干净液滴变形系数
Figure 16. The deformation coefficient of the clean droplet in shear flow
图 17 不同表面活性剂浓度对液滴变形系数的影响
Figure 17. Effects of different surfactant concentrations on the deformation coefficients of the droplet
图 18 不同表面活性剂浓度对液滴变形的影响(Ca=0.25)
Figure 18. Effects of different surfactant concentrations on droplet deformations (Ca=0.25)
图 19 不同时刻的干净液滴界面形状(Re=10, Ca=0.35)
Figure 19. Interface shapes of the clean droplet at different moments(Re=10, Ca=0.35)
图 20 不同表面活性剂浓度对液滴破裂的影响
Figure 20. Effects of surfactant concentrations on the droplet breakage
表 1 不同网格分辨率下伪势速度的范数误差
Table 1. Errors of the pseudopotential velocity at different grid resolutions
grid resolution L2 L∞ present normal present normal 32×32 6.314×10－5 3.157×10－4 3.680×10－4 1.418×10－3 64×64 2.459×10－5 1.585×10－4 1.354×10－4 5.982×10－4 128×128 3.585×10－5 1.506×10－4 3.732×10－5 2.565×10－4 
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表 2 水平中心线上表面活性剂分布误差
Table 2. Errors of surfactant distributions on the horizontal centreline
ψb L2 L∞ present normal present normal 0.01 1.163×10－4 1.238×10－4 6.067×10－4 6.631×10－4 0.05 5.065×10－4 5.433×10－4 2.721×10－3 3.001×10－3
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