A simple phase transition relaxation solver for liquid-vapor flows

International audienceDetermining liquid-vapor phase equilibrium is often required in multiphase flow computations. Existing equilibrium solvers are either accurate but computationally expensive, or cheap but inaccurate. The present paper aims at building a fast and accurate specific phase equilibrium solver, specifically devoted to unsteady multiphase flow computations. Moreover, the solver is efficient at phase diagram bounds, where non-equilibrium pure liquid and pure gas are present. It is systematically validated against solutions based on an accurate (but expensive) solver. Its capability to deal with cavitating, evaporating and condensing two-phase flows is highlighted on severe test problems both 1D and 2D

Quelques contributions à la modélisation et simulation numérique des écoulements diphasiques compressibles

This manuscript addresses the theoretical modeling and numerical simulation of compressible twophase flows. In this context, diffuse interface methods are now well-accepted but progress is still needed at the level of numerical accuracy regarding their capture. A new method is developed in this research work, that allows significant sharpening. This method can be placed in the framework of MUSCL-type schemes, widely used in production codes and on unstructured grids. Phase transition is addressed as well through a relaxation process relying on Gibbs free energies. A new instantaneous relaxation solver is developed and happens to be accurate, fast and robust. Moreover, in view of the intended industrial applications, an extension of the thermodynamics of the phases and of the mixture is necessary. A new equation of state is consequently developed. The formulation is convex and based on the “Noble-Abel-Stiffened-Gas” equation of state. In another context, the dispersion of non-miscible fluids under gravity effects is considered as well. This problematic involves large time and space scales and has motivated the development of a new multi-fluid model for “two-layer shallow water” flows. Its numerical resolution is treated as well.Ce manuscrit porte sur la modélisation et la simulation numérique d’écoulements diphasiques compressibles. Dans ce contexte, les méthodes d’interfaces diffuses sont aujourd’hui bien acceptées. Cependant, un progrès est encore attendu en ce qui concerne la précision de la capture numérique de ces interfaces. Une nouvelle méthode est développée et permet de réduire significativement cette zone de capture. Cette méthode se place dans le contexte des méthodes numériques de type “MUSCL”, très employées dans les codes de production, et sur maillages non-structurés. Ces interfaces pouvant être le lieu où une transition de phase s’opère, celle-ci est considérée au travers d’un processus de relaxation des énergies libres de Gibbs. Un nouveau solveur de relaxation `a thermodynamique rapide est développé et s’avère précis, rapide et robuste y compris lors du passage vers les limites monophasiques. En outre, par rapport aux applications industrielles envisagées, une extension de la thermodynamique des phases et du mélange est nécessaire. Une nouvelle équation d’état est développée en conséquence. La formulation est convexe et est basée sur l’équation d’état “Noble-Abel-Stiffened-Gas”. Enfin, sur un autre plan la dispersion de fluides non-miscibles sous l’effet de la gravité est également abordée. Cette problématique fait apparaître de larges échelles de temps et d’espace et motive le développement d’un nouveau modèle multi-fluide de type “shallow water bi-couche”. Sa résolution numérique est également traitée

Some contributions to the theoretical modeling and numerical simulation of compressible two-phase flows

Ce manuscrit porte sur la modélisation et la simulation numérique d’écoulements diphasiques compressibles. Dans ce contexte, les méthodes d’interfaces diffuses sont aujourd’hui bien acceptées. Cependant, un progrès est encore attendu en ce qui concerne la précision de la capture numérique de ces interfaces. Une nouvelle méthode est développée et permet de réduire significativement cette zone de capture. Cette méthode se place dans le contexte des méthodes numériques de type “MUSCL”, très employées dans les codes de production, et sur maillages non-structurés. Ces interfaces pouvant être le lieu où une transition de phase s’opère, celle-ci est considérée au travers d’un processus de relaxation des énergies libres de Gibbs. Un nouveau solveur de relaxation à thermodynamique rapide est développé et s’avère précis, rapide et robuste y compris lors du passage vers les limites monophasiques. En outre, par rapport aux applications industrielles envisagées, une extension de la thermodynamique des phases et du mélange est nécessaire. Une nouvelle équation d’état est développée en conséquence. La formulation est convexe et est basée sur l’équation d’état “Noble-Abel-Stiffened-Gas”. Enfin, sur un autre plan la dispersion de fluides non-miscibles sous l’effet de la gravité est également abordée. Cette problématique fait apparaître de larges échelles de temps et d’espace et motive le développement d’un nouveau modèle multi-fluide de type “shallow water bi-couche”. Sa résolution numérique est également traitéeThis manuscript addresses the theoretical modeling and numerical simulation of compressible two-phase flows. In this context, diffuse interface methods are now well-accepted but progress is still needed at the level of numerical accuracy regarding their capture. A new method is developed in this research work, that allows significant sharpening. This method can be placed in the framework of MUSCL-type schemes, widely used in production codes and on unstructured grids. Phase transition is addressed as well through a relaxation process relying on Gibbs free energies. A new instantaneous relaxation solver is developed and happens to be accurate, fast and robust. Moreover, in view of the intended industrial applications, an extension of the thermodynamics of the phases and of the mixture is necessary. A new equation of state is consequently developed. The formulation is convex and based on the “Noble-Abel-Stiffened-Gas” equation of state. In another context, the dispersion of non-miscible fluids under gravity effects is considered as well. This problematic involves large time and space scales and has motivated the development of a new multi-fluid model for “two-layer shallow water” flows. Its numerical resolution is treated as wel

Modeling Heavy-Gas Dispersion in Air with Two-Layer Shallow Water Equations

Computation of gas dispersal in urban places or hilly grounds requires a large amount of computer time when addressed with conventional multidimensional models. Those are usually based on two-phase flow or Navier-Stokes equations. Different classes of simplified models exist. Among them, two-layer shallow water models are interesting to address large-scale dispersion. Indeed, compared to conventional multidimensional approaches, 2D simulations are carried out to mimic 3D effects. The computational gain in CPU time is consequently expected to be tremendous. However, such models involve at least three fundamental difficulties. The first one is related to the lack of hyperbolicity of most existing formulations, yielding serious consequences regarding wave propagation. The second is related to the non-conservative terms in the momentum equations. Those terms account for interactions between fluid layers. Recently, these two difficulties have been addressed in Chiapolino and Saurel (2018) and an unconditional hyperbolic model has been proposed along with a Harten-Lax-van Leer (HLL) type Riemann solver dealing with the non-conservative terms. In the same reference, numerical experiments showed robustness and accuracy of the formulation. In the present paper, a third difficulty is addressed. It consists of the determination of appropriate drag effect formulation. Such effects also account for interactions between fluid layers and become of particular importance when dealing with heavy-gas dispersion. With this aim, the model is compared to laboratory experiments in the context of heavy gas dispersal in quiescent air. It is shown that the model accurately reproduces experimental results thanks to an appropriate drag force correlation. This function expresses drag effects between the heavy and light gas layers. It is determined thanks to various experimental configurations of dam-break test problems

Numerical investigations of two-phase finger-like instabilities

International audienceThe aim of the present work is to progress in the identification of the effects responsible for the formation of jets in heterogeneous gas-particle cylindrical and spherical explosions. In this direction three two-phase flow models are considered, namely Baer and Nunziato's (BN) (1986) model, Marble's (1963) model and the dense-dilute model of Saurel et al. (2017). The first and third ones involve both non-conservative terms and viscous drag effects while the second one involves viscous drag only as interaction force. Computed results show that viscous drag alone is unable to reproduce finger-like instabilities. The BN model and the dense-dilute one differ significantly by their acoustic properties. It is shown that the only model able to reproduce qualitatively finger-like jets is the dense-dilute model. Mesh dependence of the results is studied as well as presence or absence of viscous drag. It appears that the non-conservative terms seem responsible for jetting effects

Models and methods for two-layer shallow water flows

International audienceTwo-layer shallow water models present at least two fundamental difficulties that are addressed in the present contribution. The first one is related to the lack of hyperbolicity of most existing models. By considering weak compressibility of the phases, a strictly hyperbolic formulation with pressure relaxation is obtained. It is shown to tend to the conventional two-layer model in the stiff pressure relaxation limit. The second issue is related to the non-conservative terms in the momentum equations. Analyzing the Riemann problem structure, local constants appear precisely at locations where the non-conservative products need definition. Thanks to these local constants, a locally conservative formulation of the equations is obtained, simplifying the Riemann problem resolution through a HLL-type Riemann solver. The method is compared to literature data, showing accurate and oscillation free solutions. Additional numerical experiments show robustness and accuracy of the method

Extended Noble–Abel Stiffened-Gas Equation of State for Sub-and-Supercritical Liquid-Gas Systems Far from the Critical Point

International audienceThe Noble-Abel-Stiffened-Gas (NASG) equation of state (Le Métayer and Saurel, 2016) is extended to variable attractive and repulsive effects to improve the liquid phase accuracy when large temperature and pressure variation ranges are under consideration. The transition from pure phase to supercritical state is of interest as well. The gas phase is considered through the ideal gas assumption with variable specific heat rendering the formulation valid for high temperatures. The liquid equation-of-state constants are determined through the saturation curves making the formulation suitable for two-phase mixtures at thermodynamic equilibrium. The overall formulation is compared to experimental characteristic curves of the phase diagram showing good agreement for various fluids (water, oxygen). Compared to existing cubic equations of state the present one is convex, a key feature for computations with hyperbolic flow models

A method to solve Hamilton-Jacobi type equation on unstructured meshes

International audienceA new method is developed to approximate a first-order Hamilton-Jacobi equation in the context of an interface moving along its normal vector field. The interface is tracked with the help of a "Level-Set" function approximated through a finite-volume Godunov-type scheme. Contrarily to most computational approaches that consider smooth Level-Set functions, the present one considers sharp "Level-Set", the numerical diffusion being controlled with the help of the Overbee limiter (Chiapolino et al., 2017). The method requires gradient computation that is addressed through the least squares approximation. Multidimensional results on unstructured meshes are provided and checked against analytical solutions. Geometrical properties such as interfacial area and volume computation are addressed as well. Results show excellent agreement with the exact solutions

Investigations of the Riemann solver with internal reconstruction (RSIR) for the Euler equations

International audienceThe Riemann solver with internal reconstruction (RSIR) of Carmouze et al. (2019) is investigated , revisited and improved for the Euler equations. In this reference, the RSIR approach has been developed to address the numerical resolution of the non-equilibrium two-phase flow model of Saurel et al. (2017). The main idea is to reconstruct two intermediate states from the knowledge of a simple and robust intercell state such as HLL, regardless of the number of waves present in the Riemann problem. Such reconstruction improves significantly the accuracy of the HLL solution, preserves robustness and maintains stationary discontinuities. Consequently, when dealing with complex flow models, such as the aforementioned one, RSIR-type solvers are quite easy to derive compared to HLLC-type ones that may require a tedious analysis of the governing equations across the different waves. In the present contribution, the RSIR solver of Carmouze et al. (2019) is investigated, revisited and improved with the help of thermodynamic considerations, making a simple, accurate, robust and positive Riemann solver. It is also demonstrated that the RSIR solver is strictly equivalent to the HLLC solver of of Toro et al. (1994) for the Euler equations when the Rankine-Hugoniot relations are used. In that sense, the RSIR approach recovers the HLLC solver in some limit as well as the HLL one in another limit and can be simplified at different levels when complex systems of equations are addressed. For the sake of clarity and simplicity, the derivations are performed in the context of the one-dimensional Euler equations. Comparisons and validations against the conventional HLLC solver and exact solutions are presented

Fast 3D computations of compressible flow discharge in buildings and complex networks

Flow computation in complex geometries, such as buildings for example, is challenging at several levels including for instance mesh generation and presence of small elements (doors, windows) that need spatial resolution. When compressible effects are present, such as those resulting from explosions, choking conditions may appear in geometric restrictions. Pressure relaxation to atmospheric condition with existing CFD solvers consequently requires resolved 3D computations, that may be tremendously expensive. In the present work under-resolved computations are addressed, meaning that large computational cells and large time steps are used. Typically, cells’ size is of the order of rooms’ size. Consequently, special care must be taken at geometric discontinuities. This issue is reminiscent of flow computation with discontinuous area change (Le Floch and Thanh, 2003, Andrianov and Warnecke, 2004, Kroner and Thanh, 2005, Thanh, 2009, Han et al., 2012, to cite a few). However, in the present context, both simplifications and complexities appear. On one hand, full Riemann problem solution is not needed, only flux computation is required resulting in significant simplifications. On the other hand, the flow is 3D, meaning that related complexities must be considered, in the frame of unstructured grids. A fast, efficient, and accurate method is developed for such flows. It consists of two steps. The first step deals with a simple and fast mesh generation where no geometric detail, such as a door for example, is needed. Only “footprints” of the 3D geometry are required. The second step deals with a simple, fast and specific Riemann solver that addresses the previously omitted geometric restrictions directly in the solution states and through the flux distribution as well. The proposed method then appears very convenient when hazardous and pressing situations are involved and require knowledge of the pressure fields. It is validated against resolved computations. Examples are shown with 3D computations of a realistic building