Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number

Dellacherie, Stéphane

doi:10.1016/j.jcp.2009.09.044

Cited by 179 publications

(350 citation statements)

References 41 publications

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“…As a consequence, there are two different time scales in the compressible NavierStokes system used to model the flow in the nuclear core: a first time scale which is the material time scale t mat and a second time scale which is the acoustic time scale t ac . Since t ac t mat and since the usefull time scale is t mat in the study of thermal-hydraulic phenomena in a nuclear core whose the flow is at low Mach number, the discretization of the compressible Navier-Stokes system needs to use an implicit scheme in such a way that the time step is of the order or greater than t mat , implicit scheme which may be expensive from a CPU point of view and which may suffer from a lack of robustness and/or of a lack of accuracy [6,10].…”

Section: Introductionmentioning

confidence: 99%

On a low Mach nuclear core model

Dellacherie

2012

ESAIM: Proc.

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Abstract. We propose to formally derive a low Mach number model adapted to the modeling of a water nuclear core (e.g. of PWR-or BWR-type) in the forced convection regime or in the natural convection regime by filtering out the acoustic waves in the compressible Navier-Stokes system. Then, we propose a monodimensional stationary analytical solution with regular and singular charge loss when the equation of state is a stiffened gas equation. Moreover, we show that this solution may not be admissible from a physical or a mathematical point of view for a particular choice of the mass flux and we study the consistency between this solution and the solution obtained from a Boussinesq approximation. Let us underline that the modeling of the nuclear core is simplified in this paper. For example, the flow is a single-phase flow and we do not model neither the porosity nor the turbulence. Nevertheless, it will be possible to enrich the modeling in the future.Résumé. On se propose de formellement dériver un modèle bas Mach adaptéà la modélisation d'un coeur de réacteur nucléaireà eau (par exemple de type REP ou REB) en régime de convection forcée ou en régime de convection naturelle en filtrant les ondes acoustiques dans un modèle de type NavierStokes compressible. On construit ensuite une solution analytique stationnaire monodimensionnelle avec perte de charge régulière et singulière dans le cas où l'équation d'état est de type gaz raidi. Puis, on montre que cette solution peut ne pasêtre physiquement ou mathématiquement admissible pour un choix particulier du flux de masse et onétudie la cohérence entre cette solution et la solution obtenueà partir d'une approximation de Boussinesq. Soulignons que la modélisation proposée du coeur nucléaire est ici simplifiée. Par exemple, l'écoulement est monophasique et on ne modélise ni la porosité, ni la turbulence. Il sera par contre toutà fait possible d'enrichir la modélisation par la suite.

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Section: Introductionmentioning

confidence: 99%

On a low Mach nuclear core model

Dellacherie

2012

ESAIM: Proc.

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“…The scheme was shown to be L 2 stable under a suitable CFL condition. The proofs extend the ideas introduced in [5] for the study of the homogeneous wave equations in low Mach number regimes. In this paper, our objective is to study in the same context the numerical scheme introduced in [3] as a well-balanced (WB) scheme for the Shallow Water equations with Coriolis source term (1).…”

Section: Introductionmentioning

confidence: 71%

Analysis of Apparent Topography Scheme for the Linear Wave Equation with Coriolis Force

Audusse

Hieu

Penel

2017

Springer Proceedings in Mathematics &Amp; Statistics

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“…It requires adjustments to solve for 2D or 3D flows, at low Mach number, on a cartesian mesh. As Mach number goes to zero, in some extreme 2D or 3D cases, the native method does converge but the solution is not consistent with the analytical or experimental solution [4]. A so-called "pressure correction" is added to the initial solver to solve for the right physical solution.…”

Section: Introductionmentioning

confidence: 99%

“…Le schéma numérique du code FLICA4 est basé sur une technique de volumes finis où les flux numériques convectifs sont calculésà l'aide d'un solveur colocalisé appelé Roe [3]. L'analyse de cette méthode numérique montre qu'à bas nombre de Mach, il est nécessaire d'introduire des modifications spécifiques aux géométries 2D ou 3D sur un maillage cartésien sans quoi la solution ne converge pas vers la bonne solution lorsque le nombre de Mach tend vers zéro [4]. C'est la raison pour laquelle une correction dite "correction de pression" est appliquée.…”

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A New Numerical algorithm for two-phase flows drift-flux model with staggered grid in porous media

et al. 2017

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Abstract. FLICA4 is a 3D compressible code dedicated to reactor core analysis. FLICA4 solves for two-phase flows in porous media using a compressible drift-flux model [2]. To derive convective fluxes, FLICA4 uses a specific finite volume numerical method based on an extension of the Roe's approximate Riemann colocated solver [3]. The method is developped to solve 1D flows. It requires adjustments to solve for 2D or 3D flows, at low Mach number, on a cartesian mesh. As Mach number goes to zero, in some extreme 2D or 3D cases, the native method does converge but the solution is not consistent with the analytical or experimental solution [4]. A so-called "pressure correction" is added to the initial solver to derive an accurate solution. However, this extra term produces checkerboard oscillations in space for most applications of interest, even when solving for 1D flows. These checkerboard oscillations are sometimes critical and may lead to unstable solutions or divergence of the solver. The paper presents the analysis of an alternative numerical algorithm to solve for compressible drift-flux model in the low Mach regime. The current study aims at investigating a new compressible scheme that is both accurate and robust at low Mach number on staggered grids. The actual behavior of the new scheme is tested against simplified nuclear core "boiling channel" study cases in the low Mach regime. The tests are extended to compressible regime for flows exhibiting both shock waves and no shock waves. The paper shows that the alternative numerical algorithm leads to numerical solutions that are very close to analytical solutions. Moreover, no more checkboard oscillations are generated. Therefore, the algorithm is a real improvement to FLICA4 current numerical method. 59Résumé. FLICA4 est un logiciel de simulation 3D dédiéà l'analyse desécoulements dans les coeurs de réacteurs nucléaires et qui résout un modèle compressible diphasiqueà 4équationsà l'échelle poreuse [2]. Le schéma numérique du code FLICA4 est basé sur une technique de volumes finis où les flux numériques convectifs sont calculésà l'aide d'un solveur colocalisé appelé Roe [3]. L'analyse de cette méthode numérique montre qu'à bas nombre de Mach, il est nécessaire d'introduire des modifications spécifiques aux géométries 2D ou 3D sur un maillage cartésien sans quoi la solution ne converge pas vers la bonne solution lorsque le nombre de Mach tend vers zéro [4]. C'est la raison pour laquelle une correction dite "correction de pression" est appliquée. Cette "correction de pression" nécessaireà la précision du schéma numériqueà bas nombre de Mach pour des configurations 2D ou 3D sur un maillage cartésien introduit presque systématiquement des oscillations en espace de type mode enéchiquier dans les configurationsétudiées ici, surtout en 1D. Comme ces oscillations spatiales peuventêtre trés fortes dans certains cas etéventuellement conduireà une divergence de certains calculs. Nousétudions un nouvel algorithme numérique pour résoudre le modèle compressible diphasiqueà...

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Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number

Cited by 179 publications

References 41 publications

On a low Mach nuclear core model

On a low Mach nuclear core model

Analysis of Apparent Topography Scheme for the Linear Wave Equation with Coriolis Force

A New Numerical algorithm for two-phase flows drift-flux model with staggered grid in porous media

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