On the Dynamics of Floating Structures

Abstract

Abstract. This paper addresses the floating body problem which consists in studying the interaction of surface water waves with a floating body. We propose a new formulation of the water waves problem that can easily be generalized in order to take into account the presence of a floating body. The resulting equations have a compressible-incompressible structure in which the interior pressure exerted by the fluid on the floating body is a Lagrange multiplier that can be determined through the resolution of a d-dimensional elliptic equation, where d is the horizontal dimension. In the case where the object is freely floating, we decompose the hydrodynamic force and torque exerted by the fluid on the solid in order to exhibit an added mass effect; in the one dimensional case d = 1, the computations can be carried out explicitly. We also show that this approach in which the interior pressure appears as a Lagrange multiplier can be implemented on reduced asymptotic models such as the nonlinear shallow water equations and the Boussinesq equations; we also show that it can be transposed to the discrete version of these reduced models and propose simple numerical schemes in the one dimensional case. We finally present several numerical computations based on these numerical schemes; in order to validate these computations we exhibit explicit solutions in some particular configurations such as the return to equilibrium problem in which an object is dropped from a non-equilibrium position in a fluid which is initially at rest.

Recent advances on the global regularity for irrotational water waves

“…This includes, of course, well-posedness results in the presence of vorticity analogous to Theorem 2.3. We also refer to recent work of Lannes on the interaction with floating structures [102] and de Poyferré [54] on emerging bottom.…”

Modeling shallow water waves

“…−h0+b ∇P = 0 (see for instance the proof of Proposition 3 in [118] for details of the computations).…”

Quantitative Estimates for Advective Equation With Degenerate Anelastic Constraint

“…It is straightforward to check thatā = 0 out of Ω at the limit. Thus to prove that φ is a renormalized solution to (1) with (2) on the limiting set Ω, it is now enough to pass to the limit in (26).…”

“…Note that quantitative regularity estimates for nonlinear continuity equations at the Eulerian level have also been introduced in [3], [4] using a nonlocal characterization of compactness in the spirit of [7]. PDE's with anelastic constraints are found in many different settings and we briefly refer for instance to [24], [29], [19], [35], [30], [22] in meteorology, to [8], [25], and to [27] for lakes and [33], to [26] for the dynamics of congestion or floating structures, to [17] for astrophysics and to [14] for asymptotic regime of strong electric fields to understand the importance to study PDEs with anelastic constraints especially the advective equation. As an application, we derive a new existence result for the so-called lake equation with possibly vanishing bathymetry which could vanish.…”