Freely floating cylinder on a 3D fluid governed by the Boussinesq equations in the axisymmetric without swirl case

TL;DR

This work analyzes the interaction between dispersive surface waves and a freely floating cylinder in a 3D fluid under axisymmetric without swirl assumptions using the Boussinesq framework. By deriving an Augmented formulation that couples nonlocal, dispersive PDEs in the exterior domain with a boundary SHODE for the cylinder, the authors obtain a well-posed, energy-consistent model that captures added-mass effects and dispersive boundary layers. They establish two complementary well-posedness approaches (ODE perspective and semi-linear fixed-point) and derive a Cummins-type equation in the linear regime to study long-time decay, proving that in 2D the center of mass cannot decay faster than $\mathcal{O}(t^{-1/2})$ without viscosity and $\mathcal{O}(t^{-3/2})$ with viscosity, with no exponential decay possible in 2D. The analysis further characterizes the dispersive boundary layer via inverse dispersive operators and transfer functions, linking the exterior wave-field to boundary dynamics. Overall, the paper provides rigorous foundational results for the PDE-ODE coupling in dispersive wave-structure interactions and offers a pathway toward numerical schemes based on the Augmented formulation.

Abstract

This paper deals with the interactions of waves governed by a non-linear dispersive Boussinesq type system with the vertical displacement of a cylindrical floating structure in an axisymmetric without swirl situation. The Boussinesq regime is a good approximation of free surface Euler's equations when the non-linear parameter and the shallowness parameter are small. The vertical motion of the floating body is governed by the Newton equation. The full coupled wave-structure interaction problem under consideration is reduced to a boundary problem. The boundary condition satisfied by the discharge is given in terms of the vertical displacement of the floating cylinder. The latter is calculated using an ODE, which requires knowledge of the trace of the surface elevation and its second-time derivative. We use the dispersion in order to exhibit a hidden second order ODE on the trace of the surface elevation. This finally allows us to rewrite the waves-structure interaction problem as a system of non-local conservative PDEs plus bounded radial terms with a dispersive boundary layer, combined with an ODE at the boundary. This is what we call the Augmented formulation. Afterwards we showed that this formulation is well-posed with two different methods. Finally, we study the return to equilibrium situation in the linear regime. In particular, we improved previous results on the explicit time decay. We showed that the center mass of the floating body cannot converge to its equilibrium faster than $\mathcal{O}(t^{-1/2})$ in 2D without viscosity and faster than $\mathcal{O}(t^{-3/2})$ with viscosity.

Figures (5)

• Figure 1: Waves propagation with a floating cylinder.
• Figure 2: Illustration of a slice of the problem. The function $f$ refers to any spatial function.
• Figure 3: Illustration of $f$, $g$, $k$ (defined in the proof of Lemma \ref{['lem:ineg']} and above ), with $R = 1$ and $\kappa = 0.1$.
• Figure 4: Branching points of $\sqrt{1+\nu s + \kappa^2 s^2}$ depending on $\kappa$ and $\nu$.
• Figure 5: Real part and imaginary part of $\mathcal{P}$ and points where they both cancel, with $\nu = 0.3$, $\kappa = 0.5$.

Theorems & Definitions (96)

• Theorem 1.1: Augmented formula
• Definition 1
• Theorem 1.2: Wellposedness of the Augmented formula
• Theorem 1.3: Return to the equilibrium
• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3: Energy balance of the waves-structure system
• proof
• Proposition 2.4