[Paper Review] Implementation and Evaluation of Breaking Detection Criteria for a Hybrid Boussinesq Model

Paola Bacigaluppi, Mario Ricchiuto+1 more|arXiv (Cornell University)|Feb 8, 2019

Coastal and Marine Dynamics52 references21 citations

TL;DR

This paper proposes a novel critical free surface Froude number criterion for wave breaking detection in a hybrid Boussinesq model that switches between enhanced Boussinesq and nonlinear shallow water equations. Using a finite element upwind Petrov-Galerkin method with a custom discontinuity-capturing limiter, the approach enables robust, monotonic simulations of wave shoaling, breaking, and run-up. The physical Froude criterion (Frscr = 1) outperforms alternatives in predicting breaking onset and wave run-up, especially in benchmark cases with experimental data.

ABSTRACT

The aim of the present work is to develop a model able to represent the propagation and transformation of waves in nearshore areas. The focus is on the phenomena of wave breaking, shoaling and run-up. These different phenomena are represented through a hybrid approach obtained by the coupling of non-linear Shallow Water equations with the extended Boussinesq equations of Madsen and Sorensen. The novelty is the switch tool between the two modelling equations: a critical free surface Froude criterion. This is based on a physically meaningful new approach to detect wave breaking, which corresponds to the steepening of the wave's crest which turns into a roller. To allow for an appropriate discretization of both types of equations, we consider a finite element Upwind Petrov Galerkin method with a novel limiting strategy, that guarantees the preservation of smooth waves as well as the monotonicity of the results in presence of discontinuities. We provide a detailed discussion of the implementation of the newly proposed detection method, as well as of two other well known criteria which are used for comparison. An extensive benchmarking on several problems involving different wave phenomena and breaking conditions allows to show the robustness of the numerical method proposed, as well as to assess the advantages and limitations of the different detection methods.

Motivation & Objective

• To develop a physically meaningful wave breaking detection criterion for hybrid Boussinesq models.
• To implement and evaluate three distinct breaking detection criteria within a single finite element numerical framework.
• To assess the robustness and accuracy of the proposed method across diverse wave and bathymetric conditions.
• To investigate the impact of detection criteria on wave run-up, set-up, and energy dissipation in nearshore flows.

Proposed method

• Proposes a critical free surface Froude number criterion (Frscr = 1) as a physically based detection method for wave breaking onset.
• Uses a hybrid model that switches from enhanced Boussinesq equations to nonlinear shallow water equations in breaking regions.
• Employs a finite element upwind Petrov-Galerkin method with a novel discontinuity-capturing limiter to preserve monotonicity and smooth wave resolution.
• Implements and compares three detection criteria: local slope/vertical velocity, convective Froude-based, and a new physical Froude criterion.
• Applies a switch flag (fbreak) to activate the shallow water regime only where wave breaking is detected.
• Performs extensive benchmarking against experimental data, including Hansen and Svendsen tests with varying wave periods and amplitudes.

Experimental results

Research questions

• RQ1 How does the proposed physical Froude-based detection criterion compare to established criteria in predicting wave breaking onset and run-up?
• RQ2 What is the impact of different breaking detection criteria on wave set-up and energy dissipation in nearshore simulations?
• RQ3 How does the choice of detection criterion affect numerical stability and convergence as mesh resolution increases?
• RQ4 To what extent does the implementation of the Froude criterion improve accuracy over local or hybrid criteria in complex wave-bathymetry interactions?

Key findings

• The physical Froude criterion with Frscr = 1 provides the most accurate prediction of breaking onset and post-breaking wave slope, closely matching experimental data from Hansen and Svendsen tests.
• The local criterion consistently overpredicts breaking onset and produces unrealistically high wave set-up due to flagging regions with h < 0 as breaking.
• The hybrid slope/vertical velocity criterion performs well but is outperformed by the physical Froude criterion in cases with moderate to high Ursell numbers.
• The physical Froude criterion (Frscr = 1) yields the best agreement with experimental wave heights and mean water levels, particularly in the 031041 test case with T = 3.33 s and A = 0.0215 m.
• Grid convergence studies reveal that the local criterion converges to the non-breaking Boussinesq solution, indicating a fundamental flaw in its closure behavior as ∆x → 0.
• The study highlights that the classical asymptotic vertical velocity profile may be less suitable for the Madsen and Sørensen model than for other Boussinesq formulations, suggesting a need for model-specific refinement.