Freely floating objects on a fluid governed by the Boussinesq equations

Abstract

We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave field. Understanding the dispersive contribution to the added mass phenomenon allows us to solve these equations, and a new dispersive hidden regularity effect is used to derive uniform estimates with respect to the dispersive parameter. We then derive an abstract general Cummins equation describing the motion of the solid in the return to equilibrium problem and show that it takes an explicit simple form in two cases, namely, the nonlinear nondispersive and the linear dispersive cases; we show in particular that the decay rate towards equilibrium is much smaller in the presence of dispersion. The latter situation also involves an initial boundary value problem for a nonlocal scalar equation that has interest of its own and for which we consequently provide a general analysis.

Free-Boundary Problems for Wave–Structure Interactions in Shallow-Water: DG-ALE Description and Local Subcell Correction

“…and (iv) η lid and its weak derivatives up to order s are uniformly bounded, then there exists a maximum time T max and a unique solution of the coupled problem (4)-( 9)- (10) 2 , where χ is a diffeomorphism, applying from the initial domain E 0 to the current one E(t), defined in [19] and recalled in (47).…”

“…We follow the lines of [17] while highlighting the differences due to the frame's motion. Let introduce the following projections onto P k (T e h (t)) 2 :…”

“…On the numerical side, taking apart the study of one-way interactions of surface-waves with fixed surface-piercing topography (including for instance wave reflexion, run-up, overtopping and submersion), which can be accurately modeled without considering moving or free-boundary problems, see for instance [12,11], we may refer to [13] for the study of congested flows, to the simulation of surface-piercing structures in heaving motions in [5] or to [4] for the study of a toy-model for wave-energy conversion. On the theoretical side, the recent works [6,2] focus on the interactions between waves in the Boussinesq regime and partly immersed objects are investigated. A general theory for a class of quasi-linear hyperbolic IBVPs with a moving and/or free-boundary is introduced in [19] for the horizontal surface dimension d = 1, and applied to the NSW equations, inspiring the numerical study proposed in the present work.…”

A semi‐implicit finite volume scheme for a simplified hydrostatic model for fluid‐structure interaction

“…Examples of application of the Boussinesq-type approximation can be found in the works of Bingham, 24 Karambas and Loukogeorgaki 25 and Beck and Lannes. 26 A high order discontinuous Galerkin finite element method for the simulation of the coupling of dispersive water waves with structures can be found in Reference 27. For recent and very efficient hyperbolic reformulations of nonlinear dispersive shallow water flows, see for example, References 28-32 and references therein.…”

“…An example of the former model can be found in the work of Rijnsdorp and Zijlema: 22 in their extension of the nonhydrostatic SWASH model, 23 they integrate a body with a fixed position into the domain. Examples of application of the Boussinesq‐type approximation can be found in the works of Bingham, 24 Karambas and Loukogeorgaki 25 and Beck and Lannes 26 . A high order discontinuous Galerkin finite element method for the simulation of the coupling of dispersive water waves with structures can be found in Reference 27.…”