Understanding overtopping flow thickness and velocity over coastal defences! (part 1)

From today, I’m diving fully into my PhD thesis work — and revisiting some of the papers I’ve read along the way. Writing them up here helps me reflect and internalise key concepts. And this one? Absolutely blew my mind. It’s a 2019 study by Mares-Nasarre et al., titled "A new method to estimate overtopping layer thickness and flow velocity on mound breakwaters". It was carried out at the wave flume of the Laboratorio de Puertos y Costas at the Universitat Politècnica de València (LPC-UPV), using a physical model with a gentle beach slope and a structure slope of 1/50.

The experimental setup: tons of tests

They ran 123 physical tests on scaled rubble mound breakwaters (1:40 scale) using three different armour layers: single-layer Cubipod®, double-layer rock, and double-layer randomly placed cubes. Of these, 66 tests measured both overtopping layer thickness (OLT) and overtopping flow velocity (OFV), while the remaining 57 measured OLT only.

OLT was recorded at the centre of the breakwater crest (B/2) using a capacitive gauge inside a stilling cylinder. Flow velocity was measured using micro-propellers that captured the peak of each overtopping event.

The wave conditions varied between regular and JONSWAP-type irregular waves, with \( H_s \) and \( T_p \) ranging from 1.2 to 2.8 s (model scale). The freeboard \( R_c \) was adjusted for each setup to maintain low to moderate overtopping rates, avoiding full wave breaking.

To ensure this, they controlled the Iribarren number (also known as surf similarity parameter):

\[ \xi = \frac{\tan(\alpha)}{\sqrt{H_s / L_0}} \quad \text{with} \quad L_0 = \frac{g T_p^2}{2\pi} \]

And yes: \( H_s \) was taken at the paddle (offshore), not at the toe of the breakwater — a critical detail for scaling and interpretation.

Previous models: not quite there

Before jumping into their contributions, let’s talk about the methods they tested and ultimately improved upon. They looked at:

  • EuroTop (2018) – assuming smooth surfaces and applying generic roughness factors \( \gamma_f = 1.0 \)
  • Van Gent (2001, 2002) – included empirical coefficients tailored for some breakwater configurations
  • Schüttrumpf & Van Gent (2003) – used other empirical coefficients based on earlier datasets

What did they find? That none of these approaches captured OLT for mound breakwaters accurately. Some overestimated, some underestimated. Worse still, most were developed for smooth dikes and relied on roughness corrections that just don’t translate well to coarse, porous rubble structures.

Fig. 8 in the paper shows these discrepancies beautifully. Depending on the coefficients used and the assumption about \( \gamma_f \), results ranged from realistic to wildly off-mark. It set the stage for something better.

Part 1: Modelling overtopping thickness (OLT)

They proposed this formula for estimating the 2% exceedance thickness:

\[ \frac{h_{c,2\%}(B/2)}{H_s} = c_{A,h}^* \left( \frac{Ru_{2\%} - z_A}{H_s} \right) \]

Important: the position \( B/2 \) is the centre of the crest — not the edge.

To move from the armour layer (at \( R_c \)) to the crest centre, they applied an exponential decay:

\[ \frac{h_{c,2\%}(x_c)}{h_{A,2\%}(R_c)} = \exp\left(-c_{c,h}^* \frac{x_c}{B} \right) \]

They calibrated both coefficients using data from all tests. The final distribution for OLT followed an exponential law:

\[ F\left( \frac{h_c(B/2)}{h_{c,2\%}(B/2)} \right) = 1 - \exp\left(-K_1 \cdot \frac{h_c(B/2)}{h_{c,2\%}(B/2)} \right) \]

Where \( K_1 = 4.2 \) and the rMSE = 0.162 — a fantastic fit, especially for tail events.

Part 2: Modelling overtopping velocity (OFV)

Now for the tricky part. They proposed:

\[ u_{c,2\%}(B/2) = K_2 \sqrt{g \cdot h_{c,2\%}(B/2)} \]

with calibrated values:

  • Cubipod®: \( K_2 = 0.57 \)
  • Rock (2L): \( K_2 = 0.47 \)
  • Cubes (2L): \( K_2 = 0.60 \)

The velocity distribution followed a Rayleigh distirbution:

\[ F\left( \frac{u_c(B/2)}{u_{c,2\%}(B/2)} \right) = 1 - \exp\left( -K_3 \left[ \frac{u_c(B/2)}{u_{c,2\%}(B/2)} \right]^2 \right) \]

with \( K_3 = 3.6 \) and an rMSE = 0.271. Not bad, considering the flow velocity is subject to higher variability.

What normalisation works best?

They tested:

  • \( u_c(B/2)/(H_s/T_{m-1,0}) \) – best fit
  • \( u_c(B/2)/\sqrt{g H_s} \) – poor fit
  • \( u_c(B/2)/\sqrt{g h_{c,2\%}(B/2)} \) – poor fit

So, overtopping velocity scales best with wave height and period (not gravity alone).

Part 3: Are thickness and speed related?

Short answer: no. Surprisingly, thickness and velocity from the same overtopping event didn’t correlate. This is a big deal, because you might expect more water to also mean faster flow — but the data disagrees. To rigorously test this, the authors used two types of non-parametric statistical tests.

The first one was the Wald–Wolfowitz runs test, which checks whether the sequence of OLT and OFV values show signs of statistical dependence. The result? Only 5 out of 47 test series showed a possible correlation, which wasn’t enough to reject the null hypothesis at a 10% significance level. So far, no evidence of a link.

But the most elegant and intuitive part was their randomisation test (Fig. 18 of the paper). Here's how it worked:

  • They took the 20 highest overtopping events from each test based on OLT values.
  • For each of those events, they matched it with its measured velocity, creating 66 sets of 20 pairs: \( h_{c,i}(B/2), u_{c,i}(B/2) \).
  • They then multiplied each pair to get a synthetic overtopping discharge: \( q_i = h_{c,i}(B/2) \cdot u_{c,i}(B/2) \), and averaged them into \( \bar{q} \).
  • Next, they shuffled the velocities randomly within each set 100 times — destroying any existing order — and recalculated new average discharges \( \bar{q}_k \).

If thickness and velocity were truly dependent, then the real average \( \bar{q} \) should be significantly higher than those from the shuffled versions. But it wasn’t. Out of 6,600 fake averages, only 3.72% were higher than the real one — not enough to reject independence at a 10% level.

In other words: OLT and OFV are statistically independent.

My interpretation? They’re driven by different wave mechanisms:

  • OLT responds to low-frequency oscillations (infragravity)
  • OFV responds to short, high-frequency wave fronts

This decoupling is important — if you’re modelling overtopping loads or risks, you can’t assume that one explains the other.

Final (and personal) thought

This paper brings much-needed clarity to how we should model overtopping properties for mound structures. The experiments are rigorous, the calibrations are thoughtful, and the proposed formulas are both simple and powerful.

And the most elegant part? Realising that overtopping thickness and flow speed aren’t really friends. They’re cousins raised by different wave regimes. In fact, my personal hypothesis is that flow thickness is governed by low-frequency oscillations, while flow velocity is driven by high-frequency oscillations that might explain the lack of correlation, and it’s a working hypothesis I’ll carry forward in my own simulations.

Until next time — this is Que onda con los papers.

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