This article provides numerical results for a laminar gas flow at small velocities in the "looped thermosyphon", or "natural circulation loop" : a closed configuration composed of two horizontal adiabatic pipes and two vertical pipes with different fixed wall temperature. To this extent, following Paolucci (1982; 1994) we construct a low-Mach number model capable of taking into account the periodicity and the discontinuities intrinsic to this configuration. This compressible model is richer than the Boussinesq model since it describes the pressure variation and is adapted to the description of flows driven by large temperature gradients. We settle averaged equations through the pipes of small radius compared to the length, this gives a one dimensional system of equations of mass, momentum and energy with two pressures, a dynamical one and a thermodynamical one only function of time. We construct a quasi-exact solution in a laminar and steadystate regime. We approach the low-Mach averaged 1D Model with a coupled numerical method based on the characteristics method considering the presence of the periodic conditions and the discontinuous gravity term with Dirac distributions as derivatives at the corners. The numerical results are confronted and validated by the aforementioned reference solution to determine their accuracy.

The main goal of this paper is to extend the numerical scheme for the transport equation described in previous works [Penel, 2012; Bernard et al., 2014] from one to two dimensional problems. It is based on the method of characteristics, which consists in solving two ordinary differential equations rather than a partial differential equation. Our scheme uses an adaptive 6-point stencil in order to reach second-order accuracy whenever it is possible, and preserves some essential physical properties of the equation, such as the maximum principle. The resulting scheme is proved to be unconditionally stable and to reach second-order accuracy. We show numerical examples with comparisons to the well known Essentially Non-Oscillatory (ENO) scheme [Shu, 1998], in order to illustrate the good properties of our scheme (order of convergence, unconditional stability, accuracy). Using a Gaussian initial condition, several test cases are considered, using a constant or a rotating velocity field, taking into account or not variable source terms. Also, a test is given that shows the possibility of applying the scheme in more realistic fluid mechanics case.

We analyze a class of energy-stable and linearly well-balanced numerical schemes dedicated to the nonlinear shallow water equations with the Coriolis force. The proposed algorithms rely on colocated finite-difference approximations formulated on Cartesian geometries. They involve appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic equilibrium. We show that the resulting methods ensure semidiscrete energy estimates. Among the proposed algorithms, a colocated finite-volume scheme is described. Numerical results show a very clear improvement around the nonlinear geostrophic equilibrium when compared to those of classic Godunov-type schemes.

In the present paper, we investigate a new homogeneous relaxation model describing the behaviour of a two-phase fluid flow in a low Mach number regime, which can be obtained as a low Mach number approximation of the well-known HRM. For this specific model, we derive an equation of state to describe the thermodynamics of the two-phase fluid. We prove some theoretical properties satisfied by the solutions of the model, and provide a well-balanced scheme. To go further, we investigate the instantaneous relaxation regime, and prove the formal convergence of this model towards the low Mach number approximation of the well-known HEM. An asymptotic-preserving scheme is introduced to allow numerical simulations of the coupling between spatial regions with different relaxation characteristic times.

We study the Serre-Green-Naghdi system under a non-hydrostatic formulation, modelling incompressible free surface flows in shallow water regimes. This system, unlike the well-known (nonlinear) Saint-Venant equations, takes into account the effects of the non-hydrostatic pressure term as well as dispersive phenomena. Two numerical schemes are designed, based on a finite volume - finite difference type splitting scheme and iterative correction algorithms. The methods are compared by means of simulations concerning the propagation of solitary wave solutions. The model is also assessed with experimental data concerning the Favre secondary wave experiments [12].

We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.

In geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model. In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to approximate the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity.

This paper is concerned with the design of incomplete equations of state for a two-phase flow with phase transition that are specific to the low Mach number regime. It makes use of the fact that the thermodynamic pressure has small variations in this regime. These equations of state supplement the 2D LMNC model introduced in previous works. Our innovative strategy relies on tabulated values and is proven to satisfy crucial thermodynamic requirements such as positivity, monotonicity and continuity. In particular, saturation values are exact. This procedure is assessed by means of analytical steady solutions and comparisons with standard analytical equations of state, and shows a great improvement in accuracy. (C) 2018 Elsevier Inc. All rights reserved.

The study deals with collocated Godunov type finite volume schemes applied to the two-dimensional linear wave equation with Coriolis source term. The purpose is to explain the wrong behaviour of the classic scheme and to modify it in order to avoid accuracy issues around the geostrophic equilibrium and in geostrophic adjustment processes. To do so, a Hodge-like decomposition is introduced. Then three different well-balanced strategies are introduced. Some properties of the associated modified equations are proven and then extended to the semi-discrete case. Stability of fully discrete schemes under a suitable CFL condition is established thanks to a Von Neumann analysis. Some numerical results reinforce the purpose and exhibit the concrete improvements achieved by the application of these new techniques in both linear and nonlinear cases.

This paper concerns the global stability of weak solutions for the multilayer system introduced by Audusse et al., which models incompressible free surface flows. To do this, it is proven that this model admits the so called BD-entropy and a gain of integrability on the velocity. It allows to obtain enough compactness estimates in order to show the stability of global weak solutions.

The shallowAudusse, Emmanuel water equationsDO, Minh Hieu can be used to modelOmnes, Pascal many phenomena inPenel, Yohan geophysical fluid mechanics. For large scales, the Coriolis force plays an important role and the geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis force is an important feature. In this communication, we investigate the stability condition and the behavior of the so-called Apparent Topography scheme which is capable of capturing a discrete version of the geostrophic equilibrium.

This paper deals with the numerical treatment of two additional terms in the Lmnc-system modelling the coolant in a nuclear reactor core. The latter model was derived and studied by the authors in previous publications. On the one hand, we investigate the influence of the thermal conduction upon steady analytical solutions and upon numerical strategies designed in dimensions 1 and 2. On the other hand, we consider a time-varying thermodynamic pressure that enables to simulate a larger variety of physical situations. Taking into account the resulting terms in the equations lead us to adapt numerical methods to ensure accuracy. In particular, we use an adaptative mesh refinement procedure whose level of refinement depends on the phase.

In order to model the water flow in a nuclear reactor core, the authors carried out several studies coupling a low Mach model - named Low Mach Nuclear Core (LMNC) model - to the stiffened gas law for the equation of state. The LMNC model is derived from the compressible Navier-Stokes equations through an asymptotic expansion with respect to the Mach number commonly assumed to be small in this domain of application. This simplified system of equations provides qualitative results worth of interest under the stiffened gas hypothesis such as analytical solutions in dimension 1 and enables an easier numerical treatment in any dimension compared to the parent compressible model solved in the low Mach regime. Moreover, in the temperature and pressure regime of interest (namely high temperature and pressure situations), the stiffened gas law turns out to be inaccurate, which requires a new modelling of the equation of state. This is why this paper is devoted to the coupling of the LMNC model to an equation of state tuned by means of experimental values (NIST) for thermodynamic variables. The very point in this study consists in presenting an easy-to-implement procedure to fit tabulated values and derivatives satisfying positivity and monotonicity constraints for pure liquid and vapour phases. Modifications of previously published numerical schemes designed for a stiffened gas law are detailed in dimensions 1 and 2 to allow the use of a general equation of state. In the regime of interest and when the coolant is water, numerical results highlight the difference of tabulated equation of state with the stiffened gas law and also show that thermal conduction effects can be ignored.

In the aftermath of the CEMRACS'11 session, a research group named Cdmath was founded to pursue studies upon the modelling of the coolant uid at low Mach number in a nuclear core. This initial work was dedicated to the analysis of the Lmnc model coupled to the Sti ened Gas law. This equation of state was replaced in subsequent works by interpolation formulae to t experimental data. Some theoretical results were provided (explicit solutions, locations of vaporisation, asymptotic behaviour) which were con rmed by numerical studies in 1D. The group then focused on numerical algorithms in dimensions 2 and 3. These works lead to several publications.

In this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a monodimensional low Mach number model supplemented with the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce two variants of a numerical scheme based on the method of characteristics to simulate this model. We study and verify numerically the properties of these schemes. We finally present numerical simulations of a loss of flow accident (LOFA) induced by a coolant pump trip event.

We investigate a simplified model describing the evolution of the coolant within a nuclear reactor core (e.g. of PWR type). This model is named LMNC (for Low Mach Nuclear Core) and consists of the coupling between three equations of different types together with boundary conditions specific to the nuclear framework. After several articles dedicated to dimension 1, we present in this paper some monophasic two-dimensional numerical results when the fluid is modelled by the stiffened gas law describing the pure liquid phase. The underlying numerical strategy is based on the Finite-Element software FreeFem++.

In order to enrich the modeling of fluid flows, we investigate in this paper a coupling between two models dedicated to distinct regimes. More precisely, we focus on the influence of the Mach number as the low Mach case is known to induce theoretical and numerical issues in a compressible framework. A moving interface is introduced to separate a compressible model (Euler with source term) and its low Mach counterpart through relevant transmission conditions. A global steady state for the coupled problem is exhibited. Numerical simulations are then performed to highlight the influence of the coupling by means of a robust numerical strategy.

The Abstract Bubble Vibration model (Abv) is a set of two PDEs consisting of a transport equation and a Poisson equation. It has been derived in order to provide a better understanding of hyperbolic-elliptic couplings which are involved in a more general low Mach number system modelling bubbles. While a local existence theorem has already been proven in any dimension for the Abv model, we get interested in this paper in the one-dimensional case, where we prove the existence of global solutions no matter how smooth the data. We also provide explicit expressions of the solutions by means of the method of characteristics that we apply to the transport equation despite the coupling with the Poisson equation. We then compute these solutions by means of a numerical scheme based on a discrete version of the method of characteristics.