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Potential flow reconstruction of plunging breaking wave measured with PIV

This study combines experimental Particle Image Velocimetry (PIV) measurements with fully nonlinear potential flow (FNPF) modeling to investigate plunging breaking waves, revealing that the quasi-universal breaking onset criterion of $B=0.85$ is satisfied only after the wave has already begun to overturn.

Original authors: Jeffrey Harris, Lili Kimmoun, Sunil Mohanlal, Jensen McTighe, Jason Dahl, Stephan Grilli

Published 2026-06-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jeffrey Harris, Lili Kimmoun, Sunil Mohanlal, Jensen McTighe, Jason Dahl, Stephan Grilli

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Catching a Wave Before It Crashes

Imagine you are trying to predict exactly when a giant ocean wave will crash. For engineers building offshore oil rigs or coastal defenses, knowing when and how hard a wave hits is crucial. If you get it wrong, the structure could be damaged.

Scientists have long tried to create computer models to simulate these waves. However, once a wave starts to break and turn into white foam, the physics get messy and chaotic. Most computer models struggle with this "messy" part.

This paper is about a team of researchers who wanted to test if their super-accurate computer model could predict the exact moment a wave decides to crash, and if that prediction matches what they see in a real-life wave tank.

The Experiment: A "Wave Machine" in a Lab

The researchers built a long, narrow pool of water (a wave flume) at a university in France. Think of it as a giant, high-tech bathtub.

The Wave Maker: On one end, they used a giant paddle that swings back and forth. By moving the paddle in a very specific, complex rhythm (like a DJ mixing tracks), they could force the water to create a single, massive "focused" wave.
The Goal: They wanted this wave to grow taller and taller until it reached a specific spot in the middle of the pool and then crash (plunge) forward, just like a surfer's wave.
The Measurement: They didn't just watch; they used two tools:
1. Wave Gauges: Like rulers sticking out of the water to measure height.
2. PIV (Particle Image Velocimetry): This is the cool part. They added tiny, harmless particles to the water and shone a laser sheet through it. A high-speed camera took thousands of photos per second. By tracking how the particles moved, they could map the speed and direction of the water inside the wave, right up to the very top of the crest.

The Computer Model: The "Perfect" Simulator

On the computer side, they used a "Fully Nonlinear Potential Flow" (FNPF) model.

The Analogy: Imagine the water as a perfectly smooth, frictionless sheet of silk. In the real world, water is sticky and messy (viscous). In this computer model, they pretend the water is perfect silk.
The Challenge: Real waves break because of that stickiness and chaos. Since the computer model ignores stickiness, it can't naturally simulate the "crash" and the foam. If they just let the model run, the wave would just keep curling over forever without ever splashing.
The Fix: To make the model act like a real breaking wave, the researchers added a "brake." When the computer detected the wave was about to break, it applied a special "absorbing pressure" (like a sponge) to the top of the wave to drain its energy, mimicking the energy loss of a real crash.

The Big Discovery: The "Magic Number"

For years, scientists have debated a specific rule to predict when a wave will break. The rule is based on a ratio called B.

The Ratio: It compares how fast the water at the very top of the wave is moving ( $u_c$ ) versus how fast the wave shape itself is traveling ( $c_c$ ).
The Theory: If the water at the top moves faster than 85% of the wave's speed ( $B = 0.85$ ), the wave is doomed to break.

What the researchers found:

The Timing is Tight: They found that the wave actually starts to tip over (overturn) almost at the exact same moment this 0.85 ratio is reached. It's like a domino falling; the moment the domino hits the 0.85 mark, it's already tipping.
The Catch: Because the wave starts tipping so quickly after hitting 0.85, it is extremely hard for computer models to catch this exact moment. If the model is even a tiny bit slow, the wave has already crashed, and the "perfect silk" model breaks down.
The Data Match: Despite the difficulty, their computer model matched the real-life camera and laser measurements incredibly well. The speed of the water inside the wave, measured by the laser, matched the computer's prediction almost perfectly, even right near the top of the wave.

The Energy Loss: How Hard Does It Hit?

The paper also looked at how much energy is lost when the wave breaks.

The Analogy: Think of a car crashing. The "breaking strength" is like how hard the car hits the wall.
The Finding: They tried to use a formula that other scientists had created to predict this crash strength. However, because their wave was so intense and focused (like a high-speed car crash), the old formula didn't quite work. They had to tweak the "braking" in their computer model to match the real-world energy loss. They found that for these specific, intense waves, the energy dissipation followed a different pattern than what was seen in gentler waves.

The "Magic Trick" of Simplification

One of the most clever parts of the paper is how they analyzed the data.

The Problem: The computer model generates millions of data points, which is hard to compare to the laser measurements.
The Solution: The researchers realized that the complex movement of the water near the crest could be described by just three invisible "poles" (mathematical points) floating in the air above the water.
The Analogy: Imagine trying to describe the shape of a cloud. Instead of mapping every single water droplet, you could just say, "There are three invisible magnets pulling the cloud into this shape." They found that using just three of these mathematical "magnets" was enough to perfectly recreate the speed and shape of the crashing wave. This is a huge simplification that makes understanding the wave much easier.

Summary

The researchers successfully proved that:

Their computer model is accurate enough to predict the exact moment a wave starts to crash.
The "0.85 rule" (where the water speed hits 85% of the wave speed) is a very reliable indicator that a wave is about to break.
Even though the water gets messy when it breaks, the flow just before the crash follows simple, predictable rules that can be described with very few mathematical "poles."

They have made all their data and measurements available for other scientists to use as a "gold standard" benchmark to test their own wave models against.

Technical Summary: Potential Flow Reconstruction of Plunging Breaking Wave Measured with PIV

Problem Statement
Accurate modeling of breaking water waves is critical for offshore structure design and coastal engineering, yet it remains challenging due to the complex physics of energy dissipation and overturning. While Computational Fluid Dynamics (CFD) models can simulate breaking, they are computationally expensive and often restricted to small scales. Consequently, reduced-order models, such as Fully Nonlinear Potential Flow (FNPF) models, are widely used but rely on approximate parameterizations for energy dissipation and breaking onset. A specific kinematic criterion, $B = u_c/c_c \approx 0.85$ (where $u_c$ is the crest fluid velocity and $c_c$ is the crest celerity), has been proposed as a universal precursor to breaking. However, there is a lack of high-resolution experimental evidence validating this criterion and the associated energy dissipation parameterization, particularly regarding the precise kinematics near the crest just before and during overturning. Furthermore, measuring these velocities experimentally is difficult due to the complexity of the free surface and the limitations of Particle Image Velocimetry (PIV) near the crest.

Methodology
The study employs a dual approach combining precision laboratory experiments with high-fidelity numerical simulations:

Experimental Setup: Experiments were conducted in the precision wave flume of École Centrale de Marseille (ECM). A flap-type wavemaker generated focused plunging breaking waves using a modified Ricker amplitude density spectrum to achieve frequency focusing. The water depth was fixed at $h = 0.667$ m.
- Measurements: Surface elevation was recorded by nine wave gauges. Flow kinematics in the crest region were measured using a high-speed video camera and Particle Image Velocimetry (PIV) seeded with tracer particles.
- Repeatability: Multiple tests were performed to ensure repeatability, with wave gauge measurements showing differences of less than 0.5 mm at the breaking point.
Numerical Model: A 2D Fully Nonlinear Potential Flow (FNPF) model based on the Boundary Element Method (BEM) was used to simulate the experiments.
- Governing Equations: The model solves Laplace's equation for the velocity potential with fully nonlinear kinematic and dynamic boundary conditions on the free surface, using a mixed Eulerian-Lagrangian framework.
- Breaking Onset and Dissipation: The model incorporates a breaking detection mechanism based on the $B = u_c/c_c$ criterion. Once the threshold $B_t = 0.85$ is reached, an "absorbing" surface pressure is applied to simulate energy dissipation. The dissipation strength was parameterized using a constant non-dimensional breaking strength $b$ , calibrated against experimental data.
- Velocity Reconstruction: To compare numerical results with PIV data (which often has gaps or noise near the crest), the authors developed a reduced-order analytic representation of the near-crest velocity field. This method approximates the flow using a superposition of discrete complex poles (singularities) located outside the fluid domain, fitted to the BEM surface velocities.

Key Contributions

High-Resolution Validation: The study provides a direct comparison between FNPF model results and precision PIV measurements of a plunging breaker, specifically focusing on the kinematics near the crest at the onset of breaking.
Validation of the $B$ -Criterion: The research investigates the timing of the $B = 0.85$ criterion relative to the physical overturning of the wave.
Reduced-Order Velocity Modeling: A novel analytic approach using complex poles is introduced to reconstruct the internal velocity field near the overturning crest. This method allows for accurate comparison with experimental data without the computational cost of evaluating singular integrals at every point and handles the complex geometry of the overturning surface.
Dissipation Parameterization: The study evaluates different formulations for the breaking strength parameter ( $\gamma$ and $b$ ), finding that for these specific focused plunging waves, a parameterization based on Romero et al. (2022) provided better agreement with experimental energy loss than the formulation by Derakhti et al. (2019).

Results

Model Accuracy: The FNPF-BEM model accurately reproduced the free-surface evolution and phase of the waves up to the point of overturning. The numerical results showed excellent agreement with wave gauge time series and PIV-derived velocity fields, both before and after the breaking event (when dissipation was active).
Breaking Onset Timing: The analysis revealed that the criterion $B = 0.85$ is satisfied almost simultaneously with the wave front becoming vertical and the onset of overturning. The time difference between reaching $B=0.85$ and the front becoming vertical was found to be negligible (approx. 0.02 s), suggesting that for rapidly breaking plunging waves, the criterion may be reached too late to serve as a predictive precursor in some numerical schemes without immediate dissipation.
Velocity Comparison: The analytic pole model successfully reconstructed the velocity field. Comparisons between the pole model, the full BEM solution, and PIV measurements showed strong agreement for horizontal and vertical velocities at locations slightly below the free surface. Discrepancies were noted very close to the crest, consistent with PIV measurement limitations (noise, reflection).
Dissipation Calibration: Sensitivity analysis indicated that a constant breaking strength parameter $b \approx 0.0431$ (derived from Romero et al.'s empirical relationship) yielded the best match to experimental wave heights downstream of the breaker. The study noted that the high intensity of the focused wave resulted in a breaking strength parameter $\gamma$ that was outside the range of previous datasets used to derive alternative dissipation laws.

Significance
The paper concludes that the dataset generated from these experiments, characterized by high repeatability and detailed kinematic measurements, serves as a robust experimental benchmark for validating breaking onset criteria and energy dissipation models in reduced-order wave simulations. The study confirms that FNPF models, when coupled with appropriate dissipation mechanisms and the $B$ -criterion, can accurately simulate the pre-breaking and early overturning stages of plunging waves. The proposed analytic pole representation offers a promising tool for reduced-order modeling, capable of capturing the localized, near-singular behavior of breaking waves with a small number of parameters. The authors modestly note that while the simplified representation captures the kinematics well, it does not yet provide a detailed assessment of breaking criteria itself, leaving that for future work. The dataset is made available as supplementary material to facilitate further research in this domain.