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Pré-Publication, Document De Travail Année : 2016

A 2 × 2 HYPERBOLIC SYSTEM MODELLING INCOMPRESSIBLE TWO PHASE FLOWS -THEORY AND NUMERICS

Résumé

We propose a 2 × 2 hyperbolic system of conservation laws to model the dynamics of two incompressible fluids in mechanical disequilibrium. In the theoretical part of the paper we show that this 1D system is not strictly hyperbolic, that the characteristic speed can not a priori be ordered and that the characteristic fields are neither genuinely nonlinear, nor linearly degenerate. We nevertheless prove the existence and uniqueness of an admissible solution to the Riemann problem. This solution remains bounded with positive volume fractions even when one the phases vanishes. We conclude that the multiphase/single phase transition does not imply mechanical equilibrium but displays a non classical wave structure. In the numerical part of the paper we propose some approximate Riemann solvers to simulate the model, especially the multiphase/single phase transition. The classical Riemann solvers have been considered as Godunov scheme, Roe scheme with or without entropy fix. We also propose an in-cell discontinuous reconstruction method which proves to be successful, whereas the other schemes may show some spurious oscillations in some Riemann problem. Finally, as an application we study and simulate the problem of phase separation by gravity. 1. Introduction The flow regime involved in a nuclear reactor core is purely liquid in normal operating conditions but may become a liquid-gas mixture in incidental conditions or even purely gaseous in the case of a serious accident involving a core dewatering. The simulation of the single phase/multiphase transition is numerically challenging and has been a major difficulty in the design of new simulation platforms based on advanced two-fluid models, see for instance [11, 21] and references therein. An important issue is to guarantee the positivity of the volume fraction of each phase. There is an open debate as to whether this positivity is intrinsic to the conservation laws or requires some adequate source terms such as inter-phase friction. The thermal hydraulics platform CATHARE [12] assumes that when a phase disappears, its velocity is equal to the velocity of the other phase. In order to strongly couple the two phase velocities, they use a very high interfacial friction term to deal numerically with these transitions. This paper intends to prove in the case of incompressible phases that the Riemann problem admits a positive solution without any frictional term and that the two velocities are not necessarily equal (section 2), to propose some numerical methods able to deal with vanishing phases (section 3) and then presents some numerical results (section 4). Allowing phase disappearance with mechanical disequilibrium enables for example the modelling of bubbles ascending in a liquid as a consequence of Archimedes' principle.
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Dates et versions

hal-01303133 , version 1 (15-04-2016)

Identifiants

  • HAL Id : hal-01303133 , version 1

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Michael Ndjinga, Thi-Phuong-Kieu Nguyen, Christophe Chalons. A 2 × 2 HYPERBOLIC SYSTEM MODELLING INCOMPRESSIBLE TWO PHASE FLOWS -THEORY AND NUMERICS . 2016. ⟨hal-01303133⟩
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