一种用于可压缩性不大的Cahn-Hilliard-Navier-Stokes方程的数值方法，用于变量密度流
A numerical method for the quasi-incompressible Cahn-Hilliard-Navier-Stokes equations for variable density flows with a discrete energy law
AI总结：
本文提出一种基于扩散界面模型的数值方法，用于不同密度流体的Navier-Stokes方程，通过设计C^0有限元方法和特殊时间方案，保持离散能量守恒，用于研究液滴合并和上升现象。
摘要：
本文研究了一种扩散界面模型用于Navier-Stokes方程中不同密度流体的界面问题。通过对系统进行小幅度改写，证明了系统下存在连续能量守恒定律，假设所有变量具有合理正则性。文献中指出，保持能量守恒的方法在多相问题中表现更佳。因此，对于改写后的系统，我们设计了C^0有限元方法和特殊时间方案，以在离散层面保持能量守恒。这种离散能量守恒（几乎与连续能量守恒相同）对于变量密度两相流模型，以前从未用C^0有限元方法建立过。引入牛顿方法以线性化高度非线性的离散化方案。通过自适应网格进行一些数值实验，以研究不同密度比下的液滴合并和上升现象。展示了界面演化的快照以及不同时间的自适应网格，以证明我们的数值方案可以平滑地处理包括液滴破裂/捏合在内的演化。检查了系统的离散能量功能，以证明我们的方案在离散层面保持能量守恒。
英文摘要：
In this paper, we investigate numerically a diffuse interface model for the Navier-Stokes equation with fluid-fluid interface when the fluids have different densities \cite{Lowengrub1998}. Under minor reformulation of the system, we show that there is a continuous energy law underlying the system, assuming that all variables have reasonable regularities. It is shown in the literature that an energy law preserving method will perform better for multiphase problems. Thus for the reformulated system, we design a $C^0$ finite element method and a special temporal scheme where the energy law is preserved at the discrete level. Such a discrete energy law (almost the same as the continuous energy law) for this variable density two-phase flow model has never been established before with $C^0$ finite element. A Newton's method is introduced to linearise the highly non-linear system of our discretization scheme. Some numerical experiments are carried out using the adaptive mesh to investigate the scenario of coalescing and rising drops with differing density ratio. The snapshots for the evolution of the interface together with the adaptive mesh at different times are presented to show that the evolution, including the break-up/pinch-off of the drop, can be handled smoothly by our numerical scheme. The discrete energy functional for the system is examined to show that the energy law at the discrete level is preserved by our scheme.
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