Maren Hantke

Modeling phase transition for compressible two-phase flows applied to metastable liquids

The seven-equation model for two-phase flows is a full non-equilibrium model, each phase has its own pressure, velocity, temperature, etc. A single value for each property, an equilibrium value, can be achieved by relaxation methods. This model has better features than other reduced models of equilibrium pressure for the numerical approximations in the presence of non-conservative terms. In this paper we modify this model to include the heat and mass transfer. We insert the heat and mass transfer through temperature and Gibbs free energy relaxation effects. New relaxation terms are modeled and new procedures for the instantaneous temperature and Gibbs free energy relaxation toward equilibrium is proposed. For modeling such relaxation terms, our idea is to make use of the assumptions that the mechanical properties, the pressure and the velocity, relax much faster than the thermal properties, the temperature and the Gibbs free energy, and the ratio of the Gibbs free energy relaxation time to the temperature relaxation time is extremely high. All relaxation processes are assumed to be instantaneous, i.e. the relaxation times are very close to zero. The temperature and the Gibbs free energy relaxation are used only at the interfaces. By these modifications we get a new model which is able to deal with transition fronts, evaporation fronts, where heat and mass transfer occur. These fronts appear as extra waves in the system. We use the same test problems on metastable liquids as in Saurel et al. [R. Saurel, F. Petitpas, R. Abgrall, Modeling phase transition in metastable liquids: application to cavitating and flashing flows, J. Fluid Mech. 607 (2008) 313-350]. We have almost similar results. Computed results are compared to the experimental ones of Simoes-Moreira and Shepherd [J.R. Simoes-Moreira, J.E. Shepherd, Evaporation waves in superheated dodecane, J. Fluid Mech. 382 (1999) 63-86]. A reasonable agreement is achieved. In addition we consider the six-equation model with a single velocity which is obtained from the seven-equation model in the asymptotic limit of zero velocity relaxation time. The same procedure for the heat and mass transfer is used with the six-equation model and a comparison is made between the results of this model with the results of the seven-equation model.

Exact solutions to the Riemann problem for compressible isothermal Euler equations for two-phase flows with and without phase transition

We consider the isothermal Euler equations with phase transition between a liquid and a vapor phase. The mass transfer is modeled by a kinetic relation. We prove existence and uniqueness results. Further, we construct the exact solution for Riemann problems. We derive analogous results for the cases of initially one phase with resulting condensation by compression or evaporation by expansion. Further we present numerical results for these cases. We compare the results to similar problems without phase transition.

EXACT RIEMANN SOLUTIONS TO COMPRESSIBLE EULER EQUATIONS IN DUCTS WITH DISCONTINUOUS CROSS-SECTION

We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross-section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L-M and R-M curves in the velocity-pressure phase plane. The behaviors of the L-M and R-M curves for six basic cases are fully analyzed. Furthermore, we observe that in certain cases the L-M and R-M curves contain the bifurcation which leads to the nonuniqueness of the Riemann solutions. Nevertheless, all possible Riemann solutions including classical as well as resonant solutions are solved in a uniform framework for any given initial data.

THE ROBIN FUNCTION AND ITS EIGENVALUES

Abstract The paper deals with the Robin function and eigenvalue problems generated by the Robin operator. First we show that Greens function of an 𝑛-fold connected domain is the Robin function of an appropriate simply connected domain. The main part of the paper deals with eigenvalue problems for the Robin operator: the mixed Stekloff eigenvalue problem and the membrane problem with mixed boundary conditions. Isoperimetric inequalities are proved for the sum of reciprocal eigenvalues.

Efficient and robust relaxation procedures for multi-component mixtures including phase transition

Abstract We consider a thermodynamic consistent multi-component model in multi-dimensions that is a generalization of the classical two-phase flow model of Baer and Nunziato. The exchange of mass, momentum and energy between the phases is described by additional source terms. Typically these terms are handled by relaxation procedures. Available relaxation procedures suffer from efficiency and robustness resulting in very costly computations that in general only allow for one-dimensional computations. Therefore we focus on the development of new efficient and robust numerical methods for relaxation processes. We derive exact procedures to determine mechanical and thermal equilibrium states. Further we introduce a novel iterative method to treat the mass transfer for a three component mixture. All new procedures can be extended to an arbitrary number of inert ideal gases. We prove existence, uniqueness and physical admissibility of the resulting states and convergence of our new procedures. Efficiency and robustness of the procedures are verified by means of numerical computations in one and two space dimensions.

Numerical Solutions for a Weakly Hyperbolic Dispersed Two-Phase Flow Model

We construct numerical solutions for a dispersed isothermal two-phase flow model. The system is a weakly hyperbolic, isothermal system describing the evolution of mass, momentum as well as volume fraction for the dispersed particles as well as the carrier fluid. The dispersed phase is modeled pressureless. We construct a new HLL-type Riemann solver and perform numerical simulations on the homogeneous part of the model. In each time step, an approximate MUSCL–Hancock finite volume scheme is used in which intercell Riemann problems are solved using the new GHLL solver.