Analysis of a Simplified Model of Rigid Structure Floating in a Viscous Fluid

Abstract

We study the interaction of surface water waves with a floating solid constraint to move only in the vertical direction. The first novelty we bring in is that we propose a new model for this interaction, taking into consideration the viscosity of the fluid. This is done supposing that the flow obeys a shallow water regime (modeled by the viscous Saint-Venant equations in one space dimension) and using a Hamiltonian formalism. Another contribution of this work is establishing the well-posedness of the obtained PDEs/ODEs system in function spaces similar to the standard ones for strong solutions of viscous shallow water equations. Our wellposedness results are local in time for every initial data and global in time if the initial data are close (in appropriate norms) to an equilibrium state. Moreover, we show that the linearization of our system around an equilibrium state can be described, at least for some initial data, by an integro-fractional differential equation related to the classical Cummins equation and which reduces to the Cummins equation when the viscosity vanishes and the fluid is supposed to fill the whole space. Finally, we describe some numerical tests, performed on the original nonlinear system, which illustrate the return to equilibrium and the influence of the viscosity coefficient.

Modelling and simulation of a wave energy converter

“…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]. We consider an incompressible, irrotational, inviscid and homogeneous fluid in a shallow water regime, which occurs in the region where the OWC is installed.…”

Shallow water waves generated by a floating object: a control theoretical perspective

“…There is another interesting formulation for the governing equations (2.13). As in [27], we can define the Langrangian L and the action functional S as…”

“…only partially immersed in the fluid, is setup studied in John [18,19] under simplified assumptions. Recently, Lannes gave in [23] a new formulation of the governing equations and proposed a formulation of the problem as a coupling between a standard wave model (in which the surface elevation is free and the pressure is constrained) and a congested model containing an object (where the pressure is free and the surface elevation is constrained); this method can be implemented with various asytmptotic models: non-viscous 1D shallow water model in Iguchi and Lannes [17], viscous 1D shallow water model in Maity et al [27], 2D radial symmetric shallow water equations in Bocchi [5], Boussinesq equations in Bresch et al [6] and also in Beck and Lannes [3]. We also refer to Godlewski et al [14] where the constraint for the equations with the object is released, using a typical "low Mach" technique.…”

Local and global well-posedness of one-dimensional free-congested equations

“…This result has been then extended to the dispersive Boussinesq case in [1,6], and to the two-dimensional axisymmetric case in [3]. Finally, the study [18] includes viscosity effects, still in an axisymmetric configuration. The congestion problem (1.1) is similar to the viscous WSI problem [18] in the sense that it can be formulated as a mixed initial-boundary value problem with a implicit coupling of the (parabolic) PDEs describing the dynamics in the free/exterior zone with a nonlinear ODE.…”

“…Finally, the study [18] includes viscosity effects, still in an axisymmetric configuration. The congestion problem (1.1) is similar to the viscous WSI problem [18] in the sense that it can be formulated as a mixed initial-boundary value problem with a implicit coupling of the (parabolic) PDEs describing the dynamics in the free/exterior zone with a nonlinear ODE. This ODE (see (2.15) below) represents the evolution of the free-congested interface in (1.1), while in [18] the ODE models the vertical motion of the structure (the contact line between the fluid and the structure is there constant due to the axisymmetric hypothesis).…”