Materials & Engineering Research Institute, Sheffield Hallam University, Howard Street, S1 1WB, United Kingdom.
Department of Engineering and Mathematics, Sheffield Hallam University, Howard Street, S1 1WB, United Kingdom.
Phys Rev E. 2019 Oct;100(4-1):043310. doi: 10.1103/PhysRevE.100.043310.
The utility of an enhanced chromodynamic, color gradient or phase-field multicomponent lattice Boltzmann (MCLB) equation for immiscible fluids with a density difference was demonstrated by Wen et al. [Phys. Rev. E 100, 023301 (2019)2470-004510.1103/PhysRevE.100.023301] and Ba et al. [Phys. Rev. E 94, 023310 (2016)2470-004510.1103/PhysRevE.94.023310], who advanced earlier work by Liu et al. [Phys. Rev. E 85, 046309 (2012)PLEEE81539-375510.1103/PhysRevE.85.046309] by removing certain error terms in the momentum equations. But while these models' collision scheme has been carefully enhanced by degrees, there is, currently, no quantitative consideration in the macroscopic dynamics of the segregation scheme which is common to all. Here, by analysis of the kinetic-scale segregation rule (previously neglected when considering the continuum behavior) we derive, bound, and test the emergent kinematics of the continuum fluids' interface for this class of MCLB, concurrently demonstrating the circular relationship with-and competition between-the models' dynamics and kinematics. The analytical and numerical results we present in Sec. V confirm that, at the kinetic scale, for a range of density contrast, color is a material invariant. That is, within numerical error, the emergent interface structure is isotropic (i.e., without orientation dependence) and Galilean-invariant (i.e., without dependence on direction of motion). Numerical data further suggest that reported restrictions on the achievable density contrast in rapid flow, using chromodynamic MCLB, originate in the effect on the model's kinematics of the terms deriving from our term F_{1i} in the evolution equation, which correct its dynamics for large density differences. Taken with Ba's applications and validations, this result significantly enhances the theoretical foundation of this MCLB variant, bringing it somewhat belatedly further into line with the schemes of Inamuro et al. [J. Comput. Phys. 198, 628 (2004)JCTPAH0021-999110.1016/j.jcp.2004.01.019] and the free-energy scheme [see, e.g., Phys. Rev. E. 76, 045702(R) (2007)10.1103/PhysRevE.76.045702, and references therein] which, in contradistinction to the present scheme and perhaps wisely, postulate appropriate kinematics a priori.
Wen 等人[Phys. Rev. E 100, 023301 (2019)2470-004510.1103/PhysRevE.100.023301]和 Ba 等人[Phys. Rev. E 94, 023310 (2016)2470-004510.1103/PhysRevE.94.023310]展示了增强的动力颜色梯度或相场多组分格子 Boltzmann（MCLB）方程在具有密度差的不可混溶流体中的实用性，Wen 等人的工作改进自 Liu 等人[Phys. Rev. E 85, 046309 (2012)PLEEE81539-375510.1103/PhysRevE.85.046309]，去除了动量方程中的某些误差项。但是，虽然这些模型的碰撞方案已经得到了精心的改进，但在所有模型都共有的分相方案的宏观动力学方面，目前还没有定量的考虑。在这里，通过分析动力学尺度的分相规则（在考虑连续体行为时被忽略），我们推导出了一类 MCLB 的连续流体界面的运动学边界，并对其进行了测试，同时证明了模型动力学和运动学之间的循环关系和竞争关系。我们在第五节中提出的分析和数值结果证实，在动力学尺度上，对于一定的密度对比，颜色是一种材料不变量。也就是说，在数值误差范围内，界面结构是各向同性的（即没有方向依赖性）和伽利略不变的（即没有运动方向的依赖性）。进一步的数值数据表明，在快速流动中使用动力颜色 MCLB 时，对可实现的密度对比度的限制来源于模型运动学的影响，这是由于我们在演化方程中引入的项 F_{1i}，这会纠正模型对大密度差的动力学影响。与 Ba 的应用和验证相结合，这一结果显著增强了这一变种 MCLB 的理论基础，使其在某种程度上滞后于 Inamuro 等人的方案[J. Comput. Phys. 198, 628 (2004)JCTPAH0021-999110.1016/j.jcp.2004.01.019]和自由能方案[例如，参见 Phys. Rev. E. 76, 045702(R) (2007)10.1103/PhysRevE.76.045702，以及其中的参考文献]，与目前的方案形成对比，后两者或许更明智地预先假定了适当的运动学。
