Wave Overtopping of Stepped Revetments

Wave overtopping—i.e., excess of water over the crest of a coastal protection infrastructure due to wave run-up—of a smooth slope can be reduced by introducing slope roughness. A stepped revetment ideally constitutes a slope with uniform roughness and can reduce overtopping volumes of breaking waves up to 60% compared to a smooth slope. The effectiveness of the overtopping reduction decreases with increasing Iribarren number. However, to date a unique approach applicable for a wide range of boundary conditions is still missing. The present paper: (i) critically reviews and analyzes previous findings; (ii) contributes new results from extensive model tests addressing present knowledge gaps; and (iii) proposes a novel empirical formulation for robust prediction of wave overtopping of stepped revetments for breaking and non-breaking waves. The developed approach contrasts a critical assessment based on parameter ranges disclosed beforehand between a smooth slope on the one hand and a plain vertical wall on the other. The derived roughness reduction coefficient is developed and adjusted for a direct incorporation into the present design guidelines. Underlying uncertainties due to scatter of the results are addressed and quantified. Scale effects are highlighted.


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Traditionally the main purpose of a coastal structure is to provide coastal safety, taking into account (1) According to [ Wave overtopping is caused by waves running up the face of a dike or seawall [13]. If the wave run-78 up is higher than the freeboard height of the structure water will reach and pass over the crest. Also with a = 0.09 and = 1.5 For very steep slopes ( < 2) and non-breaking waves [13] is referring to the work by [17] 89 who provides the coefficients 90 = 0.09 − 0.01(2 − ) . for < 2 and = 0.09 for ≥ 2 = 1.5 + 0.42(2 − ) . , with a maximum of = 2.35 and = 1.5 for ≥ 2 (6) for Equation (5). But, this formula is valid only for smooth slopes, hence, the friction reduction 91 coefficient has to be set to = 1. For breaking waves, Equation (4) shall be used.

92
The slope roughness is represented in the empirical formulae by the roughness influence factor 93 , representing the permeability and roughness of or on the slope. The mean value approach 94 provided in [13] to predict the mean wave overtopping discharge on coastal dikes and embankments 95 is derived originally by [18].

96
The influence of roughness

97
The influence factor for the permeability or roughness of or on the slope is defined for the wave run- As mentioned by [9], the roughness reduction coefficient is derived for a method described in [19] and is valid only for breaking waves ( , < 1.8). For Iribarren numbers larger than 104 • , = 1.8 (with = 1 in the present study) the roughness reduction coefficient has to be 105 corrected by linear extrapolation between its value at 1.8 along 1.8 < , < 10 to = 1 106 , = + , − 1.8 1 − (8.2) for , > 1.8 Although this approach is based on findings from tests with impermeable rock slopes, [19] 107 advises to also apply Equation (

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(1.6 < ⁄ < 6.7) and plunging and surging wave breaking is covered 1.7 < , < 3.7 . The [10] discusses overtopping data for stepped revetments for steep slopes ( 0.58; 1 ) and the 154 impact of surging waves 3.9 < , < 14.7 . He provides a formula to predict the roughness   For each test, standard JONSWAP spectra were generated with a peak enhancement factor of

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The overtopping reduction coefficient for the stepped revetments which is interpreted as a 221 roughness coefficient is calculated in relation to the wave overtopping on a smooth slope following

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For milder slopes ( ≥ 2) and non-breaking wave conditions (Figure 2 (b)) the mean 239 overtopping discharge for smooth slope conditions is also larger than the prediction with Equation

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(5). The mean overtopping discharge decreases for an increased surface roughness but is mostly 241 larger than the predicted overtopping for a plain vertical wall.

242
For breaking wave conditions and mild slopes ( ≥ 2) (Figure 2 (c)) only a single tests for 243 smooth slopes is available. Nevertheless, also this mean overtopping discharge is larger than 244 predicted according to Equation (4).

245
The values a and b in Equation (4) and (5) as derived from the regression of the smooth slope 246 data given in Figure 2, are given in Table 2. For a 1:6 slope no data for a smooth slope is available, 247 but the tests were conducted in the same wave flume as the present study. A comparable larger mean 248 overtopping rate relative to the prediction is assumed and therefore applied in further calculations.  The regression coefficients listed in Table 2 are empirically derived and applied to the prediction from the stepped surface tests. The influence factor for roughness is calculated by applying Eq.
overtopping volume , the calculated ( ) and corrected ( , ) friction reduction coefficients for 265 stepped revetments are also given.

297
The quality of the derived prediction according to Equation (14) is given for the individual test 298 series in Table 4. Data from the present study show an appropriate goodness of fit for milder slopes ( > 0.9). The goodness of fit is lower for the steep 1:1 slope ( = 0.75) which is mainly caused by 308 and provides an extension of its coverage.
309 Table 4. Goodness of fit ( ), root mean square error ( ) and standard deviation (σ) for data 310 given in Figure 3 according to Equation (14).     converted to potential energy as some energy is dissipated by increasing turbulence.

386
As already discussed it is found that the step geometry has little influence on the energy 387 dissipation for very small step heights ( ≪ ). The flow direction of the wave run-up is still slope-388 parallel. Consequently, these small steps can be considered as micro roughness and are therefore 389 comparable in terms of run-up reduction effectiveness to any other impermeable micro-rough 390 surface.

391
With increasing step height ( < ) the slope-parallel wave run-up is disturbed by means of 392 the step edges effectively. The flow processes on a single step become more important and induced 393 vorticity to diminish kinetic energy. Likewise, vortex shedding occurs at the step edges in 394 dependence of the relative step to wave height and period of the run-up process. In relation to smooth slopes smaller amounts of kinetic energy of the incident wave are converted into potential energy as leads to the lowest overtopping volumes at stepped revetments that is identified for step ratios in the 400 range of 0.5 < / < 2.
Step ratios / > 2 appear to mimic conditions for vertical walls.

401
For a low step ratio, the influence of the offshore located steps decreases due to less energy dissipation 402 in increasing water depth. With increasing step ratio the revetment becomes more reflective and the 403 wave breaking is effected by the step geometry itself. Therefore, also the water depth over the step 404 ( ) influences the wave overtopping. This system performance is comparable to composite walls 405 ( / > 1). If the step height is much larger than the wave height ( ≫ ) the wave overtopping 406 tends to mimic the physical conditions of vertical walls. The kinetic energy of the incident wave is 407 converted to potential energy. Most of the incoming wave energy is reflected at the wall.

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The surface roughness of the tested revetments can be described as very smooth as it is

431
It is expected that the inability to scale the air entrainment will have a more significant effect.

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The importance of the air entrainment for the energy dissipation is discussed by [36].   increasing Iribarren numbers as the step niches are filled water from the previous wave run-up and the macro roughness of the revetment is thereby reduced. In analogy to grass slopes it is shown that the wave overtopping reduction for step ratios smaller than / > 0.067 is negligible. An