Auteur/autrice : davidlannes

Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids

7 janvier 2021 / / 0 Comments

B. Desjardins, D. Lannes, J.-C. Saut, Water Waves (2020), 1-40 Download

Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical stratification of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm–Liouville problem associated with the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so-called Brunt–Vaisälä frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approxima-tion (i.e., neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a nonlinear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluid systems.

MSRI semester: Mathematical problems in fluid dynamics

1 janvier 2021 / / 0 Comments

Website: https://www.msri.org/programs/327

Organizers

Thomas Alazard (Ecole Normale Supérieure Paris-Saclay; Centre National de la Recherche Scientifique (CNRS)), Hajer Bahouri (Laboratoire Jacques-Louis Lions; Centre National de la Recherche Scientifique (CNRS)), Mihaela Ifrim (University of Wisconsin-Madison), Igor Kukavica (University of Southern California), David Lannes (Institut de Mathématiques de Bordeaux; Centre National de la Recherche Scientifique (CNRS)), LEAD Daniel Tataru (University of California, Berkeley)

Program description

Fluid dynamics is one of the classical areas of partial differential equations, and has been the subject of extensive research over hundreds of years. It is perhaps one of the most challenging and exciting fields of scientific pursuit simply because of the complexity of the subject and the endless breadth of applications.

The focus of the program is on incompressible fluids, where water is a primary example. The fundamental equations in this area are the well-known Euler equations for inviscid fluids, and the Navier-Stokes equations for the viscous fluids. Relating the two is the problem of the zero viscosity limit, and its connection to the phenomena of turbulence. Water waves, or more generally interface problems in fluids, represent another target area for the program. Both theoretical and numerical aspects will be considered.


The Hydrodynamics of Wave Energy Convertors 2

3 juin 2019 / / 0 Comments

(Bordeaux, salle 2 de l’IMB)

This conference addresses the modelling, mathematical analysis and numerical simulation of wave energy convertors and more broadly to wave-structure interactions: conference website

It is organized in the framework of the work packages 3 and 5 of the Excellence Cluster Sysnum as one of two twin events devoted to marine renewable energies organized in Bilbao (the VI Marine Energy Conference, June 25th) and Bordeaux. A Transborder Laboratory of Mathematics and Applications (Transmath) was created in december 2018 to foster collaborations between the Idex of Bordeaux and Euskampus.

As for the first edition of this conference, also supported by the Oceanera-net Midwest and the Fondation Del Duca,  which has held in 2017 at the BCAM in Bilbao, the goal of this conference is to:

1- Focus on PDE and numerical modelling techniques. We will have talks on advanced and recent approaches including
– Depth averaged/Boussinesq type approximation and models
– Potential and Euler models
– Full CFD simulations

2- Give an overview of examples of industrial techniques and applications with several European industrial actors

Field data-based evaluation of methods for recovering surface wave elevation from pressure measurements

20 mai 2019 / / 0 Comments

A. Mouragues, P. Bonneton, D. Lannes, B. Castelle, and V. Marieu, Coastal Engineering, 150:147 – 159, 2019 Download

We compare different methods to reconstruct the surface elevation of irregular waves propagating outside the surf zone from pressure measurements at the bottom. The traditional transfer function method (TFM), based on the linear wave theory, predicts reasonably well the significant wave height but cannot describe the highest frequencies of the wave spectrum. This is why the TFM cannot reproduce the skewed shape of nonlinear waves and strongly underestimates their crest elevation. The surface elevation reconstructed from the TFM is very sensitive to the value of the cutoff frequency. At the individual wave scale, high-frequency tail correction strategies associated with this method do not significantly improve the prediction of the highest waves. Unlike the TFM, the recently developed weakly-dispersive nonlinear reconstruction method correctly reproduces the wave energy over a large number of harmonics leading to an accurate estimation of the peaked and skewed shape of the highest waves. This method is able to recover the most nonlinear waves within wave groups which some can be characterized as extreme waves. It is anticipated that using relevant reconstruction method will improve the description of individual wave transformation close to breaking.

Location map with the field site of La Salie indicated by the black circle. (b) Unmanned aerial vehicle photo of the field site at mid-tide during the experiment. A video system was installed on the pier shown in the lefthand side of the image. The yellow star and the red star show the location of the video system and the instrument, respectively, during the experiment.


Waves interacting with a partially immersed obstacle in the Boussinesq regime

13 février 2019 /

D. Bresch, D. Lannes, G. Métivier, to appear in Analysis & PDE, Analysis & PDE, 14 (2021), 1085-1124 Download

This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= 1 for 2 x 2 hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a conguration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the hyperbolic nonlinear shallow water equations), and in the presence of a fixed partially immersed obstacle. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. In order to obtain existence and uniform bounds on the solutions over the relevant time scale, a control of this dispersive boundary layer and of the oscillations in time it generates is necessary. This analysis leads to a new notion of compatibility condition that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. To the authors’ knowledge, this is the rst time that these phenomena (likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems) are exhibited.


MathOcean du 25/03/2019

13 février 2019 /

(Bordeaux, salle 2 de l’IMB)

  • 14h00-15h00 Jean-François Filipot (France Energies Marines) : Le projet DiMe: vers une meilleure caractérisation du déferlement dans les états de mer de tempête pour le dimensionnement des machine EMR
  • 15h00-16h00 Damien Sous (Institut Méditerranéen d’Océanologie): Hydrodynamique d’un récif barrière corallien

Jean-François Filipot: Le projet DiMe: vers une meilleure caractérisation du déferlement dans les états de mer de tempête pour le dimensionnement des machine EMR. Dans le cadre du projet DiMe (Dimensionnement et Météocéan : caractérisation des états de mer extrêmes déferlants pour le dimensionnement des systèmes EMR), un consortium de 17 partenaires mènent des recherches dans l’objectif d’affiner la connaissance du déferlement et de ses statistiques en conditions extrêmes. Les travaux s’organisent autour de la définition de nouvelles paramétrisations du déferlement en eau profonde et peu profonde pour les modèles de vagues à phases moyennées et résolues. Ces efforts se nourrissent d’observations collectées depuis le phare de La Jument, un ouvrage mythique exposé à des états de mer exceptionnellement énergétiques. Cette expérience fait intervenir de manière couplée des observations des vagues et de leur déferlement par stéréo vidéo, radar bande X, houlographe, ADCP. Elles devraient permettre d’interpréter les mesures d’accélération et de pression s’exerçant sur le phare.

Damien Sous: Hydrodynamique d’un récif barrière corallien. Les vagues jouent un rôle déterminant dans les processus côtiers: érosion, submersion, circulation et transport des masses d’eaux, sédiments, contaminants et nutriments. Un point clé de la transformation de la houle à l’approche du rivage est la prise en compte des terrains complexes, type rocheux ou récifaux qui s’écartent souvent des hypothèses posées pour les plages sableuses ouvertes. En particulier, les fortes pentes et la complexité géométrique des fonds ont une forte influence sur les processus de réflexion, déferlement et dissipation des vagues et ondes longues associées. Le travail de recherche présenté ici vise à comprendre le fonctionnement hydrodynamique des récifs barrières, où les interactions entre vagues, niveaux moyens et courants sont largement controlés par la bathymétrie. La compréhension de ces processus complexes nécessite la combinaison de mesures sur le terrain et de simulations numériques. Le site atelier choisi est le récif barrière de Ouano en Nouvelle Calédonie. L’exposé présentera les méthodes utilisées et leur limitations, les résultats obtenus ainsi que les travaux encore en cours sur les questions en suspens.

Generating boundary conditions for a Boussinesq system

3 février 2019 /

D. Lannes, L. Weynans, Nonlinearity 33 (2020), 6868 Download

We present a new method for the numerical implementation of generating boundary conditions for a one dimensional Boussinesq system. This method is based on a reformulation of the equations and a resolution of the dispersive boundary layer that is created at the boundary when the boundary conditions are non homogeneous. This method is implemented for a simple first order finite volume scheme and validated by several numerical simulations. Contrary to the other techniques that can be found in the literature, our approach does not cause any increase in computational time with respect to periodic boundary conditions


Hyperbolic free boundary problems and applications to wave-structure interactions

31 janvier 2019 /

T. Iguchi, D. Lannes, to appear in Indiana University Journal of Mathematics, Download

Motivated by a new kind of initial boundary value problem (IBVP) with a free boundary arising in wave-structure interaction, we propose here a general approach to one dimensional IBVP as well as transmission problems. For general strictly hyperbolic N xN
quasilinear hyperbolic systems, we derive new sharp linear estimates with refi ned dependence on the source term and control on the traces of the solution at the boundary. These new estimates are used to obtain sharp results for quasilinear IBVP and transmission problems, and we also use them to propose a general approach to 2 x2 quasilinear IBVP and transmission problems with a moving or possibly free boundary. In the latter case, two kinds of evolution equations for the boundary are considered. The fi rst one is of « kinematic type » in the sense that the velocity of the interface has the same regularity as the trace of the solution. Several applications that fall into this category are considered: the interaction of waves with a lateral piston, and a new version of the well-known stability of shocks (classical and undercompressive) that improves the results of the general theory by taking advantage of the specifi cities of the one-dimensional case. We also consider « fully nonlinear » evolution equations characterized by the fact that the velocity of the interface is one derivative more singular than the trace of the solution. This confi guration is the most challenging; it is motivated by a free boundary problem arising in wave-structure interaction, namely, the evolution of the contact line between a floating object and the water. This problem is solved as an application of the general theory developed here.

An example of free boundary with kinematic boundary condition: the piston problem
An example of free boundary with fully nonlinear boundary condition : the floating structure


The shoreline problem for the one-dimensional shallow water and Green- Naghdi equations

16 janvier 2019 /

D. Lannes, G. Métivier, J. Ec. Polytech. Math. 5 (2018), 455–518. Download

The Green-Naghdi equations are a nonlinear dispersive perturbation of the nonlinear shallow water equations, more precise by one order of approximation. These equations are commonly used for the simulation of coastal flows, and in particular in regions where the water depth vanishes (the shoreline). The local well-posedness of the Green-Naghdi equations (and their justification as an asymptotic model for the water waves equations) has been extensively studied, but always under the assumption that the water depth is bounded from below by a positive constant. The aim of this article is to remove this assumption. The problem then becomes a free-boundary problem since the position of the shoreline is unknown and driven by the solution itself. For the (hyperbolic) nonlinear shallow water equation, this problem is very related to the vacuum problem for a compressible gas. The Green-Naghdi equation include additional nonlinear, dispersive and topography terms with a complex degenerate structure at the boundary. In particular, the degeneracy of the topography terms makes the problem loose its quasilinear structure and become fully nonlinear. Dispersive smoothing also degenerates and its behavior at the boundary can be described by an ODE with regular singularity. These issues require the development of new tools, some of which of independent interest such as the study of the mixed initial boundary value problem for dispersive perturbations of characteristic hyperbolic systems, elliptic regularization with respect to conormal derivatives, or general Hardy-type inequalities.