[Paper Review] A numerical method for wave-structure interactions in the Boussinesq regime

Geoffrey Beck, David Lannes+1 more|arXiv (Cornell University)|Jul 4, 2023

Ocean Waves and Remote Sensing4 citations

TL;DR

This paper presents a second-order numerical scheme for wave-structure interactions in the one-dimensional Boussinesq-Abbott regime, where a freely moving, partially immersed structure couples with nonlinear dispersive waves. The method uses an extended formulation with auxiliary ODEs for surface elevation traces at contact points, enabling accurate computation of transmission conditions and wave generation at domain boundaries.

ABSTRACT

The goal of this work is to study waves interacting with partially immersed objects allowed to move freely in the vertical direction, and in a regime in which the propagation of the waves is described by the one dimensional Boussinesq-Abbott system. The problem can be reduced to a transmission problem for this Boussinesq system, in which the transmission conditions between the components of the domain at the left and at the right of the object are determined through the resolution of coupled forced ODEs in time satisfied by the vertical displacement of the object and the average discharge in the portion of the fluid located under the object. We propose a new extended formulation in which these ODEs are complemented by two other forced ODEs satisfied by the trace of the surface elevation at the contact points. The interest of this new extended formulation is that the forcing terms are easy to compute numerically and that the surface elevation at the contact points is furnished for free. Based on this formulation, we propose a second order scheme that involves a generalization of the MacCormack scheme with nonlocal flux and a source term, which is coupled to a second order Heun scheme for the ODEs. In order to validate this scheme, several explicit solutions for this wave-structure interaction problem are derived and can serve as benchmark for future codes. As a byproduct, our method provides a second order scheme for the generation of waves at the entrance of the numerical domain for the Boussinesq-Abbott system.

Motivation & Objective

• To develop a robust numerical method for simulating wave-structure interactions in the Boussinesq-Abbott regime with freely moving, partially immersed structures.
• To address the challenge of coupling the Boussinesq system in exterior regions with a transmission problem governed by coupled ODEs for the structure's motion and fluid flow under it.
• To improve numerical accuracy and efficiency by introducing an extended formulation that includes ODEs for surface elevation traces at contact points.
• To provide explicit analytical solutions as benchmarks for validating future numerical codes in wave-structure interaction problems.
• To enable second-order wave generation at the domain entrance by leveraging the extended formulation for boundary forcing.

Proposed method

• Formulates the wave-structure interaction problem as an initial boundary value problem for the Boussinesq-Abbott system with nonstandard transmission conditions at the structure's location.
• Introduces an extended system of ODEs for the vertical displacement of the structure and the surface elevation traces at the left and right contact points, improving numerical accessibility of forcing terms.
• Applies a second-order generalized MacCormack scheme with nonlocal flux and source terms to solve the Boussinesq system in the exterior domain.
• Couples the PDE solver with a second-order Heun scheme for the ODEs governing the structure's motion and contact-point surface elevations.
• Uses the extended formulation to compute surface elevation at contact points directly, avoiding iterative or implicit resolution.
• Derives explicit analytical solutions for validation, including cases of wave generation and return to equilibrium under symmetric conditions.

Figure 1. The floating object

Experimental results

Research questions

• RQ1 How can the coupling between a freely moving, partially immersed structure and the Boussinesq-Abbott system be accurately modeled in a numerical scheme?
• RQ2 What is the impact of including auxiliary ODEs for surface elevation traces at contact points on the stability and accuracy of wave-structure interaction simulations?
• RQ3 Can the proposed method generate waves at the domain entrance with second-order accuracy, and how does it compare to standard wave generation techniques?
• RQ4 What explicit analytical solutions exist for wave-structure interaction problems in the Boussinesq regime that can serve as benchmarks?
• RQ5 How does the symmetric formulation simplify the problem, and what physical scenarios (e.g., wave generation, return to equilibrium) can be effectively modeled under this symmetry?

Key findings

• The extended formulation with ODEs for surface elevation traces enables direct and accurate computation of these quantities without additional numerical resolution.
• The proposed second-order numerical scheme achieves consistent accuracy in both space and time for wave-structure interaction problems, validated against explicit analytical solutions.
• Explicit solutions are derived for wave generation and return-to-equilibrium scenarios, providing benchmark cases for future code validation.
• The method enables second-order wave generation at the domain entrance by directly computing the required boundary forcing from prescribed object motion.
• In the symmetric case, the problem reduces to a half-line initial boundary value problem with simplified transmission conditions, improving computational efficiency.
• The scheme successfully captures nonlinear dispersive wave effects and structure dynamics, demonstrating feasibility for simulating wave farms and offshore structures in shallow water.

Figure 2. Space discretzation