## Section: New Results

### Modelling

#### Accurate steam-water equation of state for two-phase flow LMNC model with phase transition

Member: Y. Penel,

*Coll.: Stéphane Dellacherie, Bérénice Grec, Gloria Faccanoni*

The paper [9] is dedicated to 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. This innovative strategy relies on tabulated values and is proven to satisfy crucial thermodynamic requirements such as positivity, monotonicity, 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.

#### Numerical simulations of Serre - Green-Naghdi type models for dispersive free surface flows

Members: Y. Penel, J. Sainte-Marie

*Coll.: Enrique D. Fernandez-Nieto, Tomas Morales de Luna, Cipriano Escalante Sanchez*

The Serre - Green-Nagdhi equations are simulated under their non-hydrostatic formulation by means of a projection-correction method. This is then extended to the layerwise discretisation of the Euler equations with a special care to the computational cost. An original alternating direction method is used and relies on the tools designed for the monolayer case.

#### Entropy-satisfying scheme for a hierarchy of dispersive reduced models of free surface flow

Member: M. Parisot

The work [12] is devoted to the numerical resolution in the multidimensional framework of a hierarchy of reduced models of the water wave equations, such as the Serre-Green-Naghdi model. A particular attention is paid to the dissipation of mechanical energy at the discrete level, that act as a stability argument of the scheme, even with source terms such space and time variation of the bathymetry. In addition, the analysis leads to a natural way to deal with dry areas without leakage of energy. To illustrate the accuracy and the robustness of the strategy, several numerical experiments are carried out. In particular, the strategy is capable of treating dry areas without special treatment.

#### Congested shallow water model: on floating body

Members: E. Godlewski, M. Parisot, J. Sainte-Marie, F. Wahl

In [22], we are interested in the numerical modeling of body floating freely on the water such as icebergs or wave energy converters. The fluid-solid interaction is formulated using a congested shallow water model for the fluid and Newton's second law of motion for the solid. We make a particular focus on the energy transfer between the solid and the water since it is of major interest for energy production. A numerical approximation based on the coupling of a nite volume scheme for the fluid and a Newmark scheme for the solid is presented. An entropy correction based on an adapted choice of discretization for the coupling terms is made in order to ensure a dissipation law at the discrete level. Simulations are presented to verify the method and to show the feasibility of extending it to more complex cases.

#### Pseudo-compressibility, dispersive model and acoustic waves in shallow water flows

Members: E. Godlewski, M-O. Bristeau, J. Sainte-Marie

In this paper we study a dispersive shallow water type model derived from the compressible Navier-Stokes system. The compressible effects allow to capture the acoustic waves propagation and can be seen as a relaxation of an underlying incompressible model. Hence, the pseudo-compressibility terms circumvent the resolution of an elliptic equation for the non-hydrostatic part of the pressure. For the numerical approximation of shallow water type models, the hy- perbolic part, often approximated using explicit time schemes, is constrained by a CFL condition. Since the approximation of the dispersive terms – im- plicit in time – generally requires the numerical resolution of an elliptic equation, it is very costly. In this paper, we show that when considering the pseudo-compressibility terms a fully explicit in time scheme can be derived. This drastically reduces the cost of the numerical resolution of dispersive models especially in 2d and 3d.

#### Some quasi-analytical solutions for propagative waves in free surface Euler equations

Members: B. Di Martino, M-O. Bristeau, J. Sainte-Marie, A. Mangeney, F. Souillé

This note describes some quasi-analytical solutions for wave propagation in free surface Euler equations and linearized Euler equations. The obtained solutions vary from a sinusoidal form to a form with singularities. They allow a numerical validation of the free-surface Euler codes.

#### Challenges and prospects for dynamical cores of oceanic models across all scales

Members: E. Audusse, J. Sainte-Marie

*Review paper, more than 30 co-authors*

The paper [11] provides an overview of the recent evolution and future challenges of oceanic models dynamical cores used for applications ranging from global paleoclimate scales to short-term prediction in estuaries and shallow coastal areas. The dynamical core is responsible for the discrete approximation in space and time of the resolved processes, as opposed to the physical parameterizations which represent unresolved or under-resolved processes. The paper reviews the challenges and prospects outlined by the modeling groups that participated to the Community for the Numerical Modeling of the Global, Regional, and Coastal Ocean (COMMODORE) workshop. The topics discussed in the paper originate from the experience acquired during the development of 16 dynamical cores representative of the variety of numerical methods implemented in models used for realistic ocean simulations. The topics of interest include the choice of model grid and variables arrangement, vertical coordinate, temporal discretization, and more practical aspects about the evolution of code architecture and development practices.

#### The Navier-Stokes system with temperature and salinity for free surface flows Part I: Low-Mach approximation & layer-averaged formulation

Members: M-O. Bristeau, L. Boittin, A. Mangeney, J. Sainte-Marie, F. Bouchut

In this paper, we are interested in free surface flows where density variations coming e.g. from temperature or salinity differences play a significant role in the hydrodynamic regime. In water, acoustic waves travel much faster than gravity and internal waves, hence the study of models arising in compressible fluid mechanics often requires a decoupling between these waves. Starting from the compressible Navier-Stokes system, we derive the so-called Navier-Stokes-Fourier system in an incompressible context (the density does not depend on the fluid pressure) using the low-Mach scaling. Notice that a modified low-Mach scaling is necessary to obtain a model with a thermo-mechanical compatibility. The case where the density depends only on the temperature is studied first. Then the variations of the fluid density with respect to the temperature and the salinity are considered. We give a layer-averaged formulation of the obtained models in an hydrostatic context. Allowing to derive numerical schemes endowed with strong stability properties – that are presented in a companion paper – the layer-averaged formulation is very useful for the numerical analysis and the numerical simulations of the models. Several stability properties of the layer-averaged Navier-Stokes-Fourier system are proved.

#### The Navier-Stokes system with temperature and salinity for free surface flows - Part II: Numerical scheme and validation

Members: M-O. Bristeau, L. Boittin, A. Mangeney, J. Sainte-Marie, F. Bouchut

In this paper, we propose a numerical scheme for the layer-averaged Euler with variable density and the Navier-Stokes-Fourier systems presented in part I.These systems model hydrostatic free surface flows with density variations. We show that the finite volume scheme presented is well balanced with regards to the steady state of the lake at rest and preserves the positivity of the water height. A maximum principle on the density is also proved as well as a discrete entropy inequality in the case of the Euler system with variable density. Some numerical validations are finally shown with comparisons to 3D analytical solutions and experiments.