Скачать или смотреть Geoffrey Beck and David Lannes - Wave-structure interactions 1

This course is devoted to the derivation and mathematical analysis of several models describing the interaction of waves with partially immersed objects. A key point of the analysis is to understand initial boundary value problems for hyperbolic systems (e.g., the nonlinear shallow water equations) as well as for dispersive perturbations of such systems (e.g. the Boussinesq systems).


Chapter 1: A general method to model wave-structure interactions.

We describe here a strategy to obtain wave-structure interaction models that take the form of a coupled compressible-incompressible system. Even though the method works in any dimension, we consider in this course (except in Chapter 4) the case of horizontal dimension d=1.

1) Two examples of wave-structure interactions. This talk is focused on two situations exhibiting specific difficulties: a) a fixed object with non vertical sidewalls and b) an object with vertical sidewalls allowed to float freely vertically.

2) Choice of a wave-model. We present here two models commonly used to describe waves in shallow water: the nonlinear shallow water equations (hyperbolic) and a Boussinesq system (nonlinear dispersive).

3) Mixed constraints. We show that the pressure and the surface elevation satisfy different constraints under the object (interior region) and outside (exterior region).

4) Coupling conditions. We comment on the coupling conditions at the contact line.

5) Equation for the solid. The solid is driven by the hydrodynamic force in Newton's equations

6) Links with congested flows.


Chapter 2: Reduction to a transmission problem and mathematical analysis (nonlinear shallow water equations).

We show here in the case of the NSW equations in 1d, and in the case of an object with non vertical walls, that the model derived above can be reduced to atransmission problem on the two connected components of the exterior region. In this formulation, there is no need to compute the pressure exerted at the bottom of the object (it can be recovered once the transmission problem is solved). We then present some important steps in the analysis of (possibly free boundary) hyperbolic systems in 1D.

1) Resolution of the equations in the interior region.

2) Transmission problem in the exterior region

3) Comments in the case of an object with vertical walls

4) Mathematical analysis of 1d initial boundary value hyperbolic systems (Kreiss-Lopatinski condition, compatibility conditions, Kreiss symmetrizers)

5) The case of a free boundary


Chapter 3: Freely floating object with the Boussinesq equations.

The Boussinesq equations are a dispersive perturbationof the hyperbolic nonlinear shallow water equations. This dispersive perturbation induces drastic changes on the behavior and analysis of the associated initial boundary value problem.

1) Augmented formulation. We reformulate the equations as a system of
conservative laws with nonlocal flux and a localized source term; this
system is augmented with an equation on the trace of the surface elevation at the boundary.

2) Mathematical analysis. We insist here on the differences with the hyperbolic case, in particular for the regularity of the solution and the control of its trace at the boundary.

3) Numerical aspects. We propose a numerical scheme based on this augmented formulation.
4) The return to equilibrium case. In the linear regime we comment on the effect of dispersion on a particular setting.


Chapter 4: The two-dimensional case

If time permits, we will show how the handle the case of horizontal dimension d=2for the nonlinear shallow water equations.

https://if-summer2023.sciencesconf.org