Mathematical analysis of low Mach number flows for some single and two phase models

• Benoit Desjardins (FMJH)

Room: Amphi Lavoisier A, 3rd floor There has been a lot of progress in the last 10 years for the mathematical analysis of low Mach number flows in the case of single phase flows. Similar questions can be raised in the two phase framework, where one or the two phases are slightly compressible: this talk presents some of the additional difficulties of low Mach number analysis for two phase compressible models, which are widely used in industrial applications. Only one pressure four equations models will be considered (no temperature equations).

Low Mach number flow in plate-type fuel cores

• Anne Charmeau (DEN/DANS/DM2S, CEA Saclay)

Room: Amphi Lavoisier A, 3rd floor

High temperature thermalhydraulics modeling of a Molten salt: application to the molten salt nuclear reactor

• Pablo Rubiolo (IN2P3, CNRS Grenoble)

Room: Amphi Lavoisier A, 3rd floor An overview of the ongoing efforts in the area of the thermal-hydraulics modeling of a Molten Salt Fast Reactor (MSFR) is presented. The MFSR employs a flowing liquid fuel based on a high temperature lithium fluoride salt. A molten fuel salt flow can be considered in many situations as an incompressible flow (low Match). However, several phenomena intrinsic to a molten fuel salt flow posse unique challenges (radiative heat transfer, volumetric heat source, phase change, strong neutronics feedbacks, etc.). To study some of these phenomena and to improve current CFD models an experimental facility called SWATH (Salt at WAll: Thermal ExcHanges) will be built as part of the European project SAMOFAR (2015-2019).

On the stability of IMEX schemes for singular hyperbolic PDE's

• Sebastian Noelle (RWTH Aachen, Germany)

Room: Amphi Lavoisier A, 3rd floor I will discuss systems of conservation laws where some wave speeds become singular. The classic example is the low Mach number limit in gas dynamics. In the singular limit, hyperbolicity gets lost, and near the limit, explicit time discretizations become either inefficient or unstable, both due to the CFL condition. The established concept to design efficient and stable algorithms near the singular limit is a time-Implicit-Explicit splitting, called IMEX. The recent asymptotic preserving (AP) IMEX schemes are consistent with the singular limit. A key question is the asymptotic stability of these schemes. I discuss two examples: a well-known, but unstable, scheme, and an also well-known, but stable scheme. Then I present a new stability analysis for IMEX schemes, which explains the outcomes of these experiments. I also give an outlook to a new concept of splittings, the so called RS-IMEX schemes (Reference Solution IMEX), and first aplications.

CDMATH library and low-Mach models applied to two-phase flows with Adaptive Mesh Refinement

• Anouar Mekkas (DEN/DANS/DM2S, CEA SAclay)
• Arthur Talpaert (DEN/DANS/DM2S, CEA Saclay & CMAP)

Room: Amphi Lavoisier A, 3rd floor This work presents the incremental implementation of the simulation of two-phase flows in low-Mach conditions, particularly for dilating bubbles in a nuclear core. We use CDMATH, a new easy-to-use and open-source library, which relies on the rich MED Coupling library for powerful visualization. We provide simple tools for patch-based Adaptive Mesh Refinement (AMR) designed with parallel and balanced computing in mind. With AMR, we refined the coarse mesh on a set of patches in order to locally improve the precision in regions of interest and such that we finely capture changes located at the interface. The models we present are simplified but necessary steps to obtain efficient simulation of the incompressible Navier-Stokes model and of the low Mach model with interface. This research is a joint work together with Grégoire Allaire, Stéphane Dellacherie and Samuel Kokh. It is sponsored by the CEA (French Atomic Energy Commission) and the DGA (French ministry of defense).

An Asymptotic Preserving scheme in the low-Mach number limit for the Euler system

• Marie-Hélène Vignal (IMT, Université Toulouse 3)

Room: Amphi Lavoisier A, 3rd floor I am interested in the so-called Asymptotic preserving schemes. These schemes are well known to be well adapted for the resolution of multiscale problems in which several regimes are present. I will present the particular case of the low Mach number limit for the Euler system. We recall that when the Mach number tends to zero, the pressure waves are very fast and this yields the fluid incompressible. When a standard explicit finite volume scheme is used, it is well known that its time step is constrained by the C.F.L. (Courant-Friedrichs, Levy) condition. In the low Mach number regime, this leads to time steps invertely proportional to the very large pressure waves velocity. Thus, explicit schemes suffer from a severe numerical constraint in low-Mach regimes. Furthermore, these schemes are not consistent in this regime. This means that they do not capture the incompressible limit even if they are used with constrained meshes. Then, it is necessary to develop new schemes for bypassing these limitations. These new schemes must be stable and consistent in all regimes: from low Mach numbers to order one Mach numbers. I will show how to construct such a scheme for the Euler system and I will present numerical results showing the good behavior of these schemes in all regimes.

A low Mach correction for the Godunov scheme applied to the linear wave equation with porosity

• Jonathan Jung (EFREI)

Room: Amphi Lavoisier A, 3rd floor We study the low Mach number behavior of the Godunov finite volume scheme applied to the linear wave equation with porosity. More precisely, we extend the Hodge decomposition to a weighted L^2 space. We illustrate the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergent-free velocity while this is not the case in the cartesian case. We study the properties of the modified equation associated to this Godunov scheme and we propose some correction that is continuous with respect to the Mach number.

On some models for bifluid flows

• Didier BRESCH (Université de Savoie)

Room: Amphi Lavoisier A, 3rd floor In this talk I will discuss some low Mach number systems for bifluid flows. More precisely I will focus on mathematical structures for systems used for bilayer shallow-water flows in a fixed channel or incompressible bifluid models and mixture systems used for instance for pollutant spreading or aerated avalanches (miscible flows). This corresponds respectively to joint works with M. Renardy and with V. Giovangigli, E. Zatorska.

Session poster Room: Amphi Lavoisier A, 3rd floor

An all-regime Lagrange-Projection like scheme for 2D homogeneous models for two-phase flows on unstructured meshes

• Mathieu Girardin (CMAP, Ecole Polytechnique)

Room: Amphi Lavoisier A, 3rd floor We propose an all regime Lagrange-Projection like numerical scheme for 2D homogeneous models for two-phase flows. By all regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization, i.e. a mesh size and time step much bigger than the Mach number M of the mixture. The key idea is to decouple acoustic, transport and phase transition phenomenon using a Lagrange-Projection decomposition in order to treat implicitly (fast) acoustic and phase transition phenomenon and explicitly the (slow) transport phenomena. Then, extending a strategy developed in the case of the usual gas dynamics equations, we alter the numerical flux in the acoustic approximation to obtain an uniform truncation error in term of M. This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and preserving the mass fraction within the interval (0,1). Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.

Low-velocity scheme for hyperbolic conservation laws with constrains

• Martin Parisot (ANGE, Inria Paris Rocquencourt)

Room: Amphi Lavoisier A, 3rd floor This talk is devoted to the numerical approximation of first order conservation laws under constraints. The proposed strategy is based on a relaxation of the constraints in order to get a system of hyperbolic conservation laws. However, the new system have to be considerate at the limit of the relaxation parameter goes to zero to ensure the (formal) consistency with the initial model. We will explain how this limit is analogous to a low-Mach asymptotic and we will propose several examples and illustrations.

Low Mach Number Modeling of Stratified Astrophysical Flows

• Ann Almgren (Lawrence Berkeley Lab, USA)

Room: Amphi Lavoisier A, 3rd floor Computational astrophysics has traditionally relied on discretizations of either the fully compressible equations for fluid dynamics, or the anelastic approximation, supplemented by equations describing the thermonuclear reactions and heat release. The low Mach number formulation, like the anelastic approximation, analytically removes acoustic wave propagation from the system. However, the more general low Mach number approach retains nonlinear compressibility effects resulting from nuclear burning, compositional changes and changing radial stratification. This model is a generalization of the pseudo-incompressible approximation to systems with a non-ideal gas equation of state and a time-varying base state. I will discuss the derivation of the low Mach number equation set for astrophysics, focusing on the similarities with and differences from numerical models of the Earth's atmosphere.

Turbulent kinetic energy transfers in low-Mach wall-bounded flows

• Adrien Toutant (PROMES, Université de Perpignan)

Room: Amphi Lavoisier A, 3rd floor The context of the study is high temperature solar receivers (SR). The main characteristics of flow inside a SR are turbulence (because flow rates are important) and temperature gradient (because only one face of the SR receives concentrated sunlight). A specific algorithm for the resolution of low-Mach equations is proposed in order to improve mass and energy conservations. Direct numerical simulations (DNS) of a simplified solar receiver in a bi-periodic plane channel are carried out in order to analyze the coupling between turbulence and temperature gradients. The results show that the temperature gradient creates asymmetric profiles of mean and fluctuating velocities. This asymmetry cannot only be explained by the fluid property variations as a function of temperature. It is truly related to the coupling of velocity / temperature, which mainly leads to an increase of the turbulent intensity at the cold side and to its decrease at the hot side. The analysis of the turbulent kinetic energy in the physical and in the spectral domains shows that temperature gradient changes the mechanisms of production, transfer and dissipation.

Asymptotic limits of the Shallow Water equations

• Carine Lucas (MAPMO, Université d'Orléans)

Room: Amphi Lavoisier A, 3rd floor In this talk, I will explain the asymptotic limits that can be obtained from the Shallow Water equations. In the Shallow Water equations, the values of two non-dimensional numbers (the Strouhal and the Froude numbers) can be chosen in order to catch various physical phenomena. We will see that, depending on the links between the Strouhal number, the Froude number, and the scales of the topography, we get other well-known equations such as the "Lake equation" for example. This work has been done in collaboration with Didier Bresch (LAMA, Université de Savoie, France) and Rupert Klein (Free University of Berlin, Germany).

Godunov type schemes for low Froude flows with Coriolis force

• Emmanuel Audusse (LAGA, Université Paris 13)

Room: Amphi Lavoisier A, 3rd floor in this talk, we are interested in the numerical simulation of free surface geophysical flows. Our objective is to study the relation between two well known categories of numerical schemes: the well-balanced schemes that are designed to preserve some stationary states of the model and the low Froude number schemes that are designed to remain accurate in some asymptotic regime. We study two particular cases that are related to the presence of topography and Coriolis source terms. More particularly we analyze the linearized system in order to identify the kernel of the associated operator and to derive stable and accurate numerical schemes.