A Heterogeneous Multi-scale Approach to Liquid-Vapour Flow with Phase Transition

描述可压缩液-汽两相流的相变特征是在体相和相界面附近具有完全不同的空间尺度。直接数值模拟的大多数经典数学模型涉及复杂的尺度耦合，使得即使先进的数值技术也不足以在相关的连续机械长度尺度（例如厘米范围）上给出气泡动力学的必要分辨率。为了克服这些困难的尺度分离，我们提出了一类新的非均质多尺度模型：体相的宏观尺度域上的动力学可压缩欧拉方程作为一个宏观尺度模型。对于微尺度模型，我们研究尖锐的接口（SI）以及扩散接口（DI）模型的全维和低维微尺度域接近接口。SI方法用于研究广义Riemann问题，而DI方法通过Navier-Stokes-Korteweg系统和相场模型来实现。求解宏观模型的基本数值工具是非结构网格上的hp自适应间断Galerkin方法。微尺度信息的压缩和重建将通过Ghostfluid的想法来处理。将通过与整个宏观尺度域的扩散界面模型的结果进行比较来验证完整的数值方法，例如在上一个资助期内在项目A1中获得的结果。然后将其应用于涉及单个气泡/液滴或少量气泡/液滴的典型相变问题。特别地，我们将集中讨论蒸发容器中气泡的上升。分析上，我们希望在模型问题的层次上严格地研究异质多尺度模型。一个主要项目是继续我们的研究尖锐的界面限制。

The description of compressible liquid-vapour flow with phase transition is characterized by completely different spatial scales in the bulk phases and in the vicinity of the phase interfaces. Most of the classical mathematical models for direct numerical simulation involve a complex coupling of scales such that even advanced numerical techniques do not suffice to give the necessary resolution for bubble dynamics on a relevant continuum-mechanical length-scale, say the centimeter range. To overcome these difficulties by scale separation we propose a class of new heterogeneous multi-scale models: the dynamics of the bulk phases on the macro-scale domain is modeled by the compressible Euler equations as a macro-scale model. For the micro-scale model we study sharp-interface (SI) as well as diffuse-interface (DI) models on full- and lower-dimensional microscale domains close to the interface. The SI-approach leads to the study of generalized Riemann problems while the DI-approach is realized by Navier- Stokes-Korteweg systems and phase field models. The basic numerical tool to solve the macro-scale model is the hp-adaptive Discontinuous-Galerkin method on unstructured grids. Compression and reconstruction of microscale information will be handled via the Ghostfluid idea. The complete numerical method will be validated by comparison with results for diffuse-interface models on the whole macro-scale domain, e.g. those which have been obtained in the Project A1 in the previous funding period. It will then be applied to typical phase transition problems involving a single bubble/droplet or a small number of bubbles/droplets. In particular we shall focus on the rise of bubbles in an evaporating vessel. Analytically we want to study the heterogeneous multi-scale model rigorously on the level of a model problem. A principal item is the continuation of our studies on sharp interface limits.