Determination of Semi-Empirical Models for Mean Wave Overtopping Using an Evolutionary Polynomial Paradigm
2. Average Wave Overtopping Assessment in Existing Literature
2.1. Goda (2009)
2.2. Mase et al. (2013)
2.3. EurOtop (2018)
2.3.1. Breaking and Non-breaking Wave Conditions
2.3.2. Very (shallow) Foreshores
2.4. Gallach (2018)
- aGallach = 0.0109 − 0.035(1.05 − cot α) and aGallach = 0.109 for cot α > 1.5;
- bGallach = 2 + 0.56(1.5 − cot cot α)1.3 and bGallach = 2 for cot α > 1.5; and
- cGallach = 1.1
3. Scaling Laws in Wave Overtopping
3.1. Volume Flux and Deficit in Freeboard
3.2. Dimensional Analysis
- Q* = 4π2q/gHm0,oTm−1,0,o, based on , see Equation (14): deep-water wave characteristics are employed, after having verified that there is almost no difference (R2 = 99%) between the volume flux calculated employing the deep-water wave height and period with the one calculated based on local wave conditions at the toe.
- Dimensionless freeboard, Rc/Hm0,t.
- In  and , the overtopping is proved to be a function of the deficit of the freeboard, namely 1-Rc/Rumax, rather than the dimensionless freeboard. Notice that we use here the maximum run-up according to . Considering the maximum run-up, the effects of wave periods, foreshore, and dike slope on overtopping discharge are included, as also in .
- The slope parameter proposed in , β = θTm−1,0,o√(g/Hm0,o), is employed to consider the influence of long waves generated by the presence of mild slopes on wave overtopping.
- Relative toe depth employed as shallowness criteria, ht/Hm0,o, is employed as proposed by .
- The ratio between the local water depth, the dike toe, and the local wave height, ht/Hm0,t, is considered as it was proven to have an influence on overtopping discharge , see Equations (2)–(3).
- The combined effect of the foreshore slope and dike slope is taken into account by considering the equivalent slope defined in  and expressed as cotθeq. As an alternative, the imaginary slope described in  and expressed as cotθim will be considered. By means of Evolutionary Polynomial Regression (EPR) application, the influence of the two conceptual slopes on the assessment of mean overtopping discharge will be evaluated and discussed.
- The ratio ht2/Hm0,tLm−1,0,t, which is defined as an impulsiveness ratio in .
4.1. The Evolutionary Polynomial Regression Paradigm
4.2. Model Performance
5. Experimental Datasets for Fitting Overtopping Formula
- EurOtop  database: the new extended database comprises about 13,500 tests on wave overtopping and extends the original CLASH database . The tests include different dike geometries and kinds of structures, 14% of which represent smooth dikes. By excluding cases with zero overtopping, oblique waves, berms, or storm walls and considering only core data, finally 1128 data have been extracted.
- Data described in  and corresponding to tests on sea dikes with shallow foreshores. The tests correspond to five different experimental campaigns carried out at Flanders Hydraulics Research, in Antwerp, and at the Department of Civil Engineering of Ghent University, in Ghent (both in Belgium). These experimental campaigns were conducted to study the influence of mild foreshore slopes and very or extremely shallow water conditions on wave overtopping and loading on smooth sea dikes. The geometry of the dikes resembled typical layouts from the Belgian coast.
- Data from  corresponding to coastal defenses in Japan built in extremely shallow foreshores or on land (emergent toe), namely seawalls with slopes of 1:3, 1:5, and 1:7 with uniform foreshore slope of either 1:10 or 1:30. A Bretschneider–Mitsuyasu spectrum was used to generate random wave trains, differently from other datasets here employed, where a JONSWAP spectrum was employed.
- Data from  corresponding to the experimental campaign carried out in the multi-directional wave basin, at Flanders Hydraulics Research. A 1:2 sloping dike after a 1:35 foreshore slope was employed. All data correspond to cases with mild foreshore slope (β < 0.62) in very and extremely shallow water conditions (ht/Hm0,o < 1). The experimental campaign was conceived to characterize the overtopping discharge of long- and short-crested waves proceeding very obliquely with respect to the dike crest. For the present study, only perpendicular long-crested wave cases are considered.
- Data from Altomare et al. (under review) , where steep foreshore (β > 0.62) is used in combination with a rather steep dike (1:1) and very or extremely shallow waters (ht/Hm0,o < 1). The datasets include cases with crests larger than 0, which have been excluded from the present analysis. The experiments were carried out at the Maritime Engineering Laboratory of Universitat Politècnica de Catalunya – BarcelonaTech (LIM/UPC), in Barcelona, Spain. The scope of the experimental campaign was to measure the overtopping flow properties and analyze the related safety of pedestrian standing on the promenade of a sea dike.
6. A Set of Unified Formulas for the Preliminary Assessment of Wave Overtopping on Smooth Sea Dikes and Vertical Walls
EPR Setting and Results
- Mean overtopping discharge is scaled by the volume flux, including both wave height and period (): dimensionless discharge can be defined as Q* = 4π2q/gHm0,oTm−1,0,o. In this way, the wave period is always considered as an explanatory variable for wave overtopping assessment also for those cases where it is usually not taken into account; see Equation (9) for non-breaking waves.
- Employing deep-water wave characteristics rather than local wave characteristics at the dike toe to define the volume flux and, consequently, the scaling law for overtopping, leads to differences that are negligible. The reader can employ either the former ones or the latter ones, depending on the availability and accuracy of the employed information.
- EPR analysis confirms that an exponential structure, the most employed in literature (see ), leads to the best model representation.
- Both the dimensionless freeboard, Rc/Hm0,t, and the deficit in freeboard, 1-Rc/Rumax, are key explanatory variables for wave overtopping assessment. The main difference is that the definition of the deficit in freeboard requires the assessment of wave run-up, which is not only depending on the wave height but also on the wave period, local water depth, foreshore slope, and dike slope.
- Therefore, the study confirms previous findings from [13,22,23], and partly from , where other variables such as dike and foreshore slope and local water depth play an important role for the calculation of mean discharge rates. By combination these variables in one equivalent or imaginary slope concept, the model accuracy increases.
- EPR results confirm that the imaginary or equivalent slopes provided improve the estimate of mean discharge values, confirming somehow the physical meaning of such concepts. Overtopping reduces when the slope (defined as a cotangent of the angle with the horizontal) increases (heavier wave braking and dissipation over the foreshore slope).
Conflicts of Interest
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|Dataset ID||Dataset Source||# of Data||ht/Hm0,o||cot α||cotθ||Rc/Hm0,t||β|
|1||Altomare et al. (2016)||170||(−0.14)–0.86||2–6||35–50||0.41–3.50||0.29–0.62|
|3||Tamada et al. (2009)||230||(−0.27)–0.50||3, 5, 7||10–30||0.93–8.04||0.44–1.92|
|4||Altomare et al. (under review)||26||0.10–0.36||1||15||2.95–6.19||1.13–1.42|
|5||Altomare et al. (2020)||125||0.12–0.58||1.97||35||0.53–3.24||0.37–0.58|
|Model #||Model Expression||Training||Test|
|μ − 1.64σ||μ + 1.64σ|
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Altomare, C.; Laucelli, D.B.; Mase, H.; Gironella, X. Determination of Semi-Empirical Models for Mean Wave Overtopping Using an Evolutionary Polynomial Paradigm. J. Mar. Sci. Eng. 2020, 8, 570. https://doi.org/10.3390/jmse8080570
Altomare C, Laucelli DB, Mase H, Gironella X. Determination of Semi-Empirical Models for Mean Wave Overtopping Using an Evolutionary Polynomial Paradigm. Journal of Marine Science and Engineering. 2020; 8(8):570. https://doi.org/10.3390/jmse8080570Chicago/Turabian Style
Altomare, Corrado, Daniele B. Laucelli, Hajime Mase, and Xavi Gironella. 2020. "Determination of Semi-Empirical Models for Mean Wave Overtopping Using an Evolutionary Polynomial Paradigm" Journal of Marine Science and Engineering 8, no. 8: 570. https://doi.org/10.3390/jmse8080570