具有可证明有界性的两相流守恒扩散界面方法
A conservative diffuse interface method for two-phase flows with provable boundedness properties
摘要：
由于数值过冲和下冲与其色散误差相关，中心有限差分格式在两相流的相位指示函数平流的上下文中一直被避免。然而，我们将展示，对于不可压缩流，在特定的扩散界面模型的背景下，可以保持相场的有界性，同时还利用低成本和易于实施的中心差异来构建对流项的非耗散离散化方案。这是可能的，通过结合平流和重新初始化步骤的保守水平集方案所介绍的奥尔森和Kreiss [J.计算。物理、210，225（2005）]以形成类似于Chiu和Lin [J. Comput.物理、230，185（2011）]。而不是诉诸专门的迎风方法，在这些文章中，我们证明了相场的有界性是保证某些选择的自由参数（$ε$和$γ$），我们提出了一个特定的中心差分方案。建议的离散保守和有界相场方程，这是免费的任何重新初始化或质量重新分配，具有理想的属性，可以利用在耦合有限差分离散的两相动量方程。此外，与现有的保守和有界两相流方法相比，所提出的方法具有竞争力的精度与成本的权衡，小的内存需求，易于实现和良好的并行性，为现实的两相流计算提供了一个可行的替代方案。
英文摘要：
Central finite difference schemes have long been avoided in the context of two-phase flows for the advection of the phase indicator function due to numerical overshoots and undershoots associated with their dispersion errors. We will show however, for an incompressible flow, in the context of a specific diffuse interface model, one can maintain the boundedness of the phase field while also taking advantage of the low cost and ease of implementation of central differences to construct a non-dissipative discretization scheme for the advective terms. This is made possible by combining the advection and reinitialization steps of a conservative level set scheme introduced by Olsson and Kreiss [J. Comput. Phys., 210, 225 (2005)] to form a phase field equation similar to that of Chiu and Lin [J. Comput. Phys., 230, 185 (2011)]. Instead of resorting to specialized upwind methods as in these articles, we prove that the boundedness of the phase field is guaranteed for certain choices of the free parameters ($ε$ and $γ$) for a specific central difference scheme that we propose. The proposed discretely conservative and bounded phase field equation, which is free of any reinitialization or mass redistribution, possesses desirable properties that can be leveraged in the coupled finite difference discretization of the two-phase momentum equation. Additionally, as compared to the state-of-the-art conservative and bounded two-phase flow methods, the proposed method boasts competitive accuracy-vs-cost trade-off, small memory requirements, ease of implementation and excellent parallelizability, providing a viable alternative for realistic two-phase flow calculations.
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