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

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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.