Floating Structures in Shallow Water: Local Well-posedness in the Axisymmetric Case

Abstract

The floating structure problem describes the interaction between surface water waves and a floating body, generally a boat or a wave energy converter. As shown by Lannes in [18], the equations for the fluid motion can be reduced to a set of two evolution equations on the surface elevation and the horizontal discharge. The presence of the object is accounted for by a constraint on the discharge under the object; the pressure exerted by the fluid on this object is then the Lagrange multiplier associated with this constraint. Our goal in this paper is to prove the well-posedness of this fluid-structure interaction problem in the shallow water approximation under the assumption that the flow is axisymmetric without swirl. We write the fluid equations as a quasilinear hyperbolic mixed initial boundary value problem and the solid equation as a second order ODE coupled to the fluid equations. Finally we prove the local in time well-posedness for this coupled problem, provided some compatibility conditions on the initial data are satisfied.

Modelling and simulation of a wave energy converter

“…The present research is essentially motivated by a series of works by Rezanejad and collaborators on the experimental and numerical study of nearshore OWCs, in particular, we refer to Rezanejad and Soares [14], where the authors used a linear potential theory to do simulations and showed the improvement of the efficiency when a step is added. Our goal is to numerically study this type of WEC considering as the governing equations for this wave-structure interaction the nonlinear shallow water equations derived by Lannes in [8], whose local well-posedness was obtained by Iguchi and Lannes in [7] the one-dimensional case and by Bocchi in [1] in the two-dimensional axisymmetric case. In the Boussinesq regime and for a fixed partially immersed solid similar equations were studied by Bresch, Lannes and Métivier in [2] and in the shallow water viscous case by Maity, San Martín, Takahashi and Tucsnak in [11] and by Vergara-Hermosilla, Matignon, and Tucsnak in [15].…”

“…where u(t, x, z) is the horizontal component of the fluid velocity vector field. Let us first give the boundary conditions related to (1). The relevance of these boundary conditions will be explained in Section 2.2.…”

“…In Section 2, we derive the model used in the numerical simulations following [1,2,7]. In particular, we show that the equations (1) can be reformulated as two transmission problems, one related to the step in the bottom topography and one related to the wave-structure interaction at the entrance of the chamber.…”

An overview on congestion phenomena in fluid equations

“…Among the possible general extensions, let us mention the case of heterogeneous maximal density constraints ρ ≤ ρ * (t, x) (see for instance [24], [38], [60] or [7] for applications to traffic flows). Depending on the applications, other reference fluid systems can be preferred to the classical Navier-Stokes equations: non-linear shallow water or Boussinesq equations for wave-structure interactions (see [38], [9], [36]), gradient flow formulations in the modeling of crowds [50], porous media equations and Hele-Shaw free boundary problems for tissue growth modeling (see for instance [35], [61]), Bingham equations for complex geophysical flows [21], etc. For the clarity of the presentation, we shall stick in this paper to the two "toy" systems (1.1) and (1.4) which already raise important and difficult analysis problems.…”

Congested shallow water model: on floating body

“…The surface pressure is then obtained by an elliptic equation. An analysis of this shallow water type model in the two dimensional case with radial symmetry is done in [7]. In [11], a phenomena of dispersive boundary layer is highlighted.…”