Phase appearance or disappearance in two-phase flows

This paper is devoted to the treatment of specific numerical problems which appear when phase appearance or disappearance occurs in models of two-phase flows. Such models have crucial importance in many industrial areas such as nuclear power plant safety studies. In this paper, two outstanding problems are identified: first, the loss of hyperbolicity of the system when a phase appears or disappears and second, the lack of positivity of standard shock capturing schemes such as the Roe scheme. After an asymptotic study of the model, this paper proposes accurate and robust numerical methods adapted to the simulation of phase appearance or disappearance. Polynomial solvers are developed to avoid the use of eigenvectors which are needed in usual shock capturing schemes, and a method based on an adaptive numerical diffusion is designed to treat the positivity problems. An alternate method, based on the use of the hyperbolic tangent function instead of a polynomial, is also considered. Numerical results are presented which demonstrate the efficiency of the proposed solutions

An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations

We present an Asymptotic-Preserving 'all-speed' scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and Two-dimensional numerical results provide a validation of the Asymptotic-Preserving 'all-speed' properties

Influence of Interfacial Forces on the Hyperbolicity of the Two-Fluid Model

The Two-Fluid Model, an averaged model widely used in the modeling of two-phase compressible flows, generally fails to be hyperbolic in its basic formulation. However, interfacial forces such as the interfacial pressure term and the virtual mass force, bringing new differential terms to the system can change the previous analysis and make the problem hyperbolic. The case where the two phases are incompressible has been studied by Stuhmiller in 1977, but till now, no proof of their efficiency in rendering the model hyperbolic exists in the compressible case. The aim of this paper is to detail the effects these forces have on the hyperbolicity of the Two-Fluid Model in the compressible case. We characterise the location and topology of the non hyperbolic regions, and propose a closure for the interfacial pressure that makes the system unconditionnally hyperboli

A conservative pressure based solver with collocated variables on unstructured grids for two-fluid flows with phase change

International audienceNumerical simulation of two-phase flows based on the two-fluid six-equation model is the focus of this work. This approach is widely used in thermo-hydraulics for many industrial applications, particularly nuclear power plant safety analysis. A semi-implicit numerical method that has been successfully adopted in several industrial thermo-hydraulic codes, because of its efficiency and robustness, has been extended. In this paper, the governing equations are solved on unstructured grids with collocated variables to accommodate complicated geometries, and a deferred-correction method is used to deal with the non-orthogonality of unstructured grids. In addition, the numerical method is conservative, meaning that it is capable of capturing a shock wave exactly in single-phase flows, and ensuring the conservation of physically conservative quantities of a two-phase flow mixture. Numerical benchmarks and industrial test cases are performed in order to validate the numerical method and to evaluate its behavior with respect to different physical configurations

Asymptotic Preserving Property of a Semi-implicit Method

International audienceThis work focuses on the study of the asymptotic preserving property of a semi-implicit method. The semi-implicit method, which is a pressure-based method, has been successfully used to simulate two-phase flows in numerous industrial applications. This method is used in our studies due to the fact that pressure-based methods generally perform well at low Mach numbers. The semi-implicit method is applied to the homogeneous equilibrium model (HEM) in this work to simulate two-phase flows. We show that the semi-implicit method is asymptotic preserving, i.e. the discretization for a compressible model tends to a consistent discretization for the related incompressible model at the low Mach number limit. Finally, test cases are performed to show that the numerical method is able to deal with low Mach number flows, as well as flows with a wide range of Mach numbers

Notions clés des méthodes numériques dans le projet de NEPTUNE

We will try to present herein the main issues of our investigation in numerical methods for two-phase flow modeling, within the framework of the NEPTUNE project, which benefits from both contributions of CEA and EDF. These may be recast in five work packages. The first two are devoted to the mathematical and numerical modeling of two-phase flows with interfaces and the two-fluid two pressure approach. This in particular includes investigation of relaxation methods in order to establish correct links with standard two-fluid models, which are the core of the third work package. Computations of the interaction of shock waves with bubbles will be presented. Some new results concerning two-fluid and three-field flow modeling will also be briefly presented. Part of the work in the third work package concerns benchmarking, and comparison with several hyperbolic solvers, but also improvement of numerical treatment of source terms, multi-field models and suitable boundary conditions. The fourth one, which deals with the interfacial coupling of codes, is probably the most important one since it requires connecting all models together. Since little attention has been paid to this crucial point, part of the focus will be given in this paper on the coupling of equations of state, one-dimensional and three-dimensional codes, porous approach and free medium approach, but also on ongoing work concerning relaxed and unrelaxed hyperbolic two-phase flow models. The fifth work package gathers all classical contributions in numerical methods, including: recent applications of fictitious domain methods ; preconditioning of so-called “low Mach number” two-phase flows (with applications to the motion of rising bubbles in water) ; parallel and multigrid techniques (with applications to steam generators in nuclear power plants) ; Finite Volume Element methods (with applications to the standard two-fluid models) ; construction and validation of new exact or approximate Riemann solvers (in order to cope with vanishing phases). The latter five obviously aim at improving accuracy, stability and also at reducing CPU time. A few examples will enable to highlight the main advantages and possible drawbacks of these new developments, and the impact of the current and future increasing computational facilities. Main past achievements, and key points of current and future work on all these issues will be discussed. All available references will be given in order to help the reader getting a more accurate insight on these various contributions. The whole has clearly benefited from contributions of several PhD students : Thomas Fortin, Vincent Guillemaud, Olivier Hurisse, Angelo Muronne, Isabelle Ramière, Jean-Michel Rovarch and Nicolas Seguin

Key issues for numerical methods in NEPTUNE project

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Advanced three-dimensional two-phase flow simulation tools for application to reactor safety (ASTAR)

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