Workshop on Liquid-Vapor Flows. March 2 to 4, 2016

Nina Aguillon (Paris) 

Christophe Berthon (Nantes)

Christophe Chalons (Versailles) 

Frédéric Coquel (Palaiseau)

Johannes Daube (Freiburg)

Bruno Després (Paris)

Christian Dickopp (Aachen)

Florence Drui (Châtenay-Malabry) 

Robert Eymard  (Marne-La-Vallée)

Jan Giesselman (Stuttgart)

Nina Aguillon (Paris)  Numerical approximation of hyperbolic systems containing an interface 

Abstract. We present an approach to approximate numerically the solution of coupled hyperbolic conservation laws. The coupling is achieved through a fixed interface, in which interface conditions are linking the traces on both sides. The main interest of the scheme is that it does not use the knowledge of the solution to the Riemann problem, and hence it is quite flexible and easy to implement. The idea of the scheme is to balance the effects ot the waves that enter the interface, in order to numerically mimic the structure of the Riemann problem. The scheme is well balanced with respect to all the piecewise equilibria associated with the interface conditions. If one of the interface condition is the conservation of a conserved variable, the scheme maintains it exactly. We will present a detailed analysis in the classical case, and a variety of test cases assessing the quality of the method. This is a joint work with Raul Borsche (Technische Universität Kaiserslautern)

Abstract. It is the purpose of this contribution to provide an overview on recent advances in the development of all-regime Lagrange-Remap numerical schemes for compressible fluids systems with source terms. We will consider in particular the case of large friction coefficients and the case of low-Mach numbers. More precisely, we will present a discretization strategy for gas dynamics equations for unstructured grids based on a Lagrange-Remap approach that does not involve any moving mesh. A natural semi-implicit extension of the method that allows to remain stable under a CFL condition involving only the material velocity will be given, together with an extremely simple modification that allows to provide an accurate and stable solver for simulations involving low-Mach regions in the flow. The stability properties of the proposed schemes and several numerical experiments will be presented. This contribution is based on a series of joint works with Mathieu Girardin and Samuel Kokh. These works were performed during M. Girardin’s PhD thesis.

Bruno Després (Paris) Modeling uncertainties with  kinetic equations

Abstract: The modeling of uncertainties is fundamental in industry and in CFD. For  nonlinear equations, it   questions the compatibility of L1-BV techniques (for conservation laws)  with L2 approaches (for the uncertainties). I will review recent progresses on the modeling at the kinetic level (with B. Perthame), and present recent ideas  which show connection of the so-called kinetic polynomials with optimal control (with E. Treat).

Christian Dickopp (Aachen) Coupling of (elastic)-plastic solids with compressible two-phase fllows for cavitation damaging

Abstract. As a model problem to investigate cavitation damaging the collapse of a single gas bubble collapsing near to an elastic or elastic-plastic solid wall is simulated numerically. This transient three-phase system is  modeled by the compressible Euler equations completed by a stiffened gas law for both fluids, where the liquid and the gas phase are distinguished  by a level set approach, and either the pure elastodynamical equations for a linear-elastic solid or an extension to describe plastic effects. A weak coupling strategy connects the alternating calculations of the fluid solver and  the solid solver using transient boundary conditions that are updated by the other solver.

Abstract. Compressible two-fluid models offer a potential solution for simulating separated two-phase flows configurations. On the other hand, a specific family of such models has been developed for the regime of dispersed gas bubbles and show good agreement with experiments in the case of small acoustic perturbations. On the way to connect both types of flows, we propose here a hierarchy of homogeneous two-fluid models. Starting with Hamilton’s variational principle and adding thermodynamically consistent dissipative structures, we built a new connected hierarchy, each level of which being mathematically well-posed. Every new relaxation small parameter is physically identified through acoustic linearization and analysis of the systems dispersion relations. Furthermore, numerical methods based on finite volume schemes are developed so as to preserve the properties of the models at the continuous level and to asymptotically handle the transition from each subsystem to another. Finally, simulations of simple academic configurations are performed and show the expected properties of the first models of the hierarchy, the numerical methods and dynamically adaptive mesh techniques with the potential for massively parallel simulations.

 

Jan Giesselman (Stuttgart) A priori error analysis of DG approximations of two-phase flows

Abstract. In this talk we consider a one dimensional  model for isothermal two-phase flows using Lagrangian coordinates. The model is of diffuse interface type with a non-monotone pressure law. We will present a priori error analysis of a semi-discrete discontinuous Galerkin method, which satisfies a discrete version of the energy inequality which is valid on the continuous level. It also satisfies a relative energy type stability theory. Combining this stability framework with suitable projection operators for the exact solution allows us to derive optimal order error estimates. We will also present numerical results obtained using a fully-discrete version of the scheme, which validate our theoretical results.

 

 

Mirko Kraenkel (Freiburg) Discontinuous Galerkin schemes for the Navier-Stokes-Allen-Cahn system

Dietmar Kroener (Freiburg) Conservation laws on surfaces

Rüdiger Müller (Berlin)  The Lippmann equation for liquid metal electrodes

Abstract. The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Recently we have derived a general thermodynamically consistent model for electrochemical interfaces, which shows a remarkable agreement to single crystal experimental data. In this talk, we apply the model to a curved liquid metal electrode in contact with an electrolyte. By matched asymptotic analysis we obtain the Lippmann equation whenever the Debye length is small compared to electrode curvature radius. The interfacial tension of the Lippmann equation is composed of the surface tension of our general model, and additional contributions arising from the adjacent space charge layers. Comparison with experimental data of several mercury-electrolyte interfaces confirms the theoretical results. This is a joint work with Wolfgang Dreyer, Clemens Guhlke, and Manuel Landstorfer.

 

Carlos Pares (Malaga) Nonconservative products and Shallow Water models: an overview

Abstract. Many hyperbolic nonlinear PDE systems that include source terms and nonconservative products arise in the simulation of geophysical flows by means of depth-averaged models.  In this talk, a review of the numerical techniques developed in last years by my group of research and collaborators to solve this type of systems will be presented together with a discussion of the main difficulties  and challenges in this field. Some applications to the simulation of real flows will be shown to illustrate this topic.

 

Arnold Reusken (Aachen) Space-time unfitted FEM for problems with moving discontinuities

Abstract. In this talk we will discuss unfitted finite element methods (or CutFEM) in  a space-time setting. The motivation for the development of these methods comes from two-phase incompressible flows. We explain how these techniques can be applied for the accurate discretization of a mass transport equation and a two-phase flow (Navier-)Stokes equation. The idea of the method, rigorous error bounds for certain problem classes and results of numerical experiments will be presented.

---