Experimental study on mean overtopping of sloping seawall under oblique irregular waves

In this paper, domestic and abroad research progresses and related calculation formulae of the mean overtopping discharge are summarized. Through integral physical model experiments, the relation between the wave direction and the overtopping discharge on the top of the sloping dike is focused on and put into analysis and discussion; and a modified formula for mean overtopping discharges under oblique irregular waves is proposed. The study shows that the mean overtopping discharge generally goes down as the relative wave obliquity β increases for a fixed measurement point and the mean overtopping discharge generally increases as the wave steepness H/L decreases (the cycle increases) for a fixed relative wave obliquity.


Introduction
The slope-type breakwater dike is an important engineering facility protecting coastal regions from tide and wave attacks. Overtopping discharge is an important hydraulic resistance to estimate the crown elevation of seawall. Besides, it must take the security and engineering cost in consideration when designing the sloping dike. The study of the mean overtopping discharge is meaningful.
There are many factors affecting the overtopping discharge, mainly including structural dimensions of buildings, wave parameters, water flow effects, etc. Integral model experiments are hereinafter conducted in this paper. It emphasizes on the relation between the incident wave directions and the mean overtopping discharge at the crest of sloping dike. After comparing the test results with existent formulae, a formula suitable to estimate the overtopping discharge of sloping dikes with breastwork under multidirectional irregular waves is presented.

Calculation method for mean overtopping discharges on the top of the sloping dike
Since the 20th century, many scholars at home and abroad have been working on researches about the overtopping discharge. Based on experimental data, Saville et al. (1958) found the effect curve of water depth and wave height against the overtopping discharge; Lu (1985) obtained a research result for a calculation method of mean overtopping discharge of regular waves on the top of rubble bedding vertical dike; Wang et al. (1996) presented a calculation method of the mean overtopping discharge on the top of the single-slope dike through a physical model experiment of irregular waves; De Rouck et al. (1998) measured the overtopping discharge on a prototype and compared the results obtained from prototype measurements with those from scale model tests; Van der Meer et al. (2000) analyzed the effects of wave transmission on wave run-up and overtopping. Then, van der Meer's (2002) technical report is used as a guideline for safety assessment and design of dikes in the Netherlands. Li et al. (2007) conducted a three-dimensional model experiment against a sloping dike in respect of oblique waves and multi-directional waves, based on irregular waves, which gave a mean overtopping discharge formula taking into consideration of many factors such as the relative wave obliquity, direction distribution width, relative crest super elevation, relative water depth and wave steepness deep in water. Lorke et al. (2010) studied the effect of current on wave run-up and wave overtopping and the distinction between the wave attracting with and against the current. Etemad-Shahidi and Jafari (2014) (2007) proposed a mean overtopping discharge estimation formula which is suitable for oblique waves and multi-directional irregular waves on the top of the sloping dike featuring concrete pavement and accropode pavement: where, Q stands for the mean overtopping discharge; H s is the significant wave height; T p stands for the spectrum peak wave period; R c is the crest freeboard; m stands for the slope ratio of the dike surface at the seaside; d stands for the water depth in front of the dike; S op is the wave steepness deep in water obtained by the linear wave theory, and ; A and B are equal to coefficients as the function of the wave ratio; γ β stands for the factor of the relative wave obliquity; γ b stands for the factor of pavement structures.

Formula of Hebsgaard et al.
The calculation formula of the mean overtopping discharge presented by Hebsgaard et al. (1998) is: where, β stands for the relative wave obliquity, i.e. the angle between the wave propagation direction and the dike vertical line. Thus, when a wave incidence is vertical to the dike axis, the wave incident angle β is 0; γ f stands for the factor of the pavement layer roughness; k 1 and k 2 are coefficients. When no breastwork exists, k 1 =-0.3 and k 2 =-1.6; when breastwork exists, k 1 =-0.01 and k 2 =-1.0.

Research method proposed by Owen
Owen (1980) presented a mean overtopping discharge of the crest as below in the case of the normal incidence of waves: where, T m is the average wave period of the wave incidence at the dike foot of sea wall; A and B are empirical coefficients that differ between a single-slope dike and a sloping dike with shoulders. This formula is applicable in the range of .
Shown above is the formula of the sloping dike mean overtopping discharge under an unidirectional wave normal incidence effect, which is required to take into consideration of the influence of the oblique wave, with the coefficients A and B being multiplied by relative wave obliquity modification coefficients K A and K B (Yu, 2000), respectively.

Wave overtopping formula
The calculation formula of mean overtopping discharge presented by van der Meer and Janssen (1995) is fully explained as: where, H mo is the significant wave height from spectral moment; ξ 0 is the crushing parameter, and ; tan α is the mean slope; γ b , γ f , γ β , and γ ν are factors of shoulders, roughness, wave incident angles, and breast walls, respectively; K 1 and K 2 are coefficients from the mean value of experimental results, K 1 =4.75 and K 2 =2.6. For safety consideration, K 1 =4.3 and K 2 =2.3.

Influence of wave incident angle
Influence coefficients caused by incident angles of unidirectional and multidirectional waves against wave run-up are calculated by different ways. For the unidirectional wave, the wave incident angle's influence to unidirectional waves is much bigger than that of multidirectional waves. As a safety precaution, it is suggested to adopt the influence coefficient γ b of multidirectional waves which is as follows: 2.5 Methods in Code for Sea Port Hydrology Based on model experiments and researches, scholars in China proposed mean overtopping discharge formulae that are subject to the irregular forward wave, in the case of the sloping dike with or without the breastwork. Among these formulas, the mean overtopping discharge of the sloping dike with the breastwork is calculated as follows: Details about coefficients in the formulas can be found in Code for Sea Port Hydrology (Ministry of Transport of the People's Republic of China, 2013).

Neural network method
Neural network method (http://nn-overtopping. deltares. nl/overtopping.aspx) is a prediction method to estimate overtopping discharge occurring at different types of coastal engineering structures on the basis of a neural network model. It was established under the EU CLASH projects. This method can forecast not only the mean overtopping discharge but also overtopping discharges at different confidence intervals.

Comparison of overtopping discharge influence factors used in various formulas
(1) Among formulas described above, the overtopping discharges of the sloping dike, subject to the wave incident directions to the extent, is determined by oblique-wave-influenced factors in all formulas. In Li et al.'s formula, the range of the relative wave obliquity β is from 0° to 45°; in Owen's formula, the range of the relative wave obliquity β is from 0° to 60°.
(2) The dimensionless overtopping discharges in specifications overseas differ from domestic ones, leading to different computational accuracies. Therefore, the van der Meer and Janssen's formula performs more accurately when being used to calculate overtopping discharges with the wave period of 10 s. This formula is suitable to calculate the mean overtopping discharge of the sea dike inside the British territory.
(3) Li et al.'s formula and the Owen's formula are just suitable for the dikes without breastworks. In regard to armor block, the experiment of this paper, van der Meer and Janssen's formula and Hebsgaard et al.'s formula used accropode blocks. Li et al.'s formula is suitable for concrete slab and dolosse. The formula of Code for Sea Port Hydrology is suitable for concrete slab, rock riprap, dolosse and hollow square. The factor of pavement structures refers to dolosse in this paper.

Experiment conditions
The model experiment was implemented in Coastal Engineering Experiment Hall of the Nanjing Hydraulic Research Institute. The pool is 70 m in length, 52 m in width, 1.2 m in depth, and is equipped with multi-directional irregular wave generators on both sides of the pool. The wave generators generate the desired wave parameter according to the computer automation. Energy dissipaters could be installed in the pool or both sides of the pool to absorb the wave energy in order to avoid wave reflection. This experiment, performed on the basis of Wave Model Test Regulation (Ministry of Transport of the People's Republic of China, 2001), adopted the JONSWAP spectrum. During this experiment, a water butt used for measuring the mean overtopping discharge is 0.2 m in width; the active wave height H s is 10 cm; the accropode block is about 180 g ; the mean cycles T are 1.0 s, 2.0 s and 3.0 s, respectively; and the wave incident angles β are of 9 groups in total, which are respectively 0°, 10°, 20°, 30°, 40°, 50°, 60°, 70° and 80°. To reduce experiment errors, at least 3 times of experiments are required for the same wave element. If differences between experiment results are not significant, the average could be taken as the final experiment result. Otherwise, the number of experiments is to be increased until 3 groups of experiments share a relatively close result. There are 81 groups of experiments in total. See Fig. 1 for the experiment sectional view.  steepness H/L on the mean overtopping discharge, wave obliquity β remains unchanged. Analyzing of the figure can lead to:

Experiment result analysis
(1) The dimensionless overtopping discharge on the top of the sloping dike generally decreases as the relative wave obliquity β increases. Owen proposed that, on a smooth sloping dike, when the relative wave obliquity β is smaller than 30°, the overtopping discharge of the unidirectional wave basically keeps the same, or may exhibit a larger value than that in the case of the normal incidence, and may cause the phenomena so-called "increasing at small degrees of obliquity". During this experiment, when H/L=0.1, the phenomena "increasing at small degrees of obliquity" happens to the wave, i.e. when the relative wave obliquity β=20°, the mean overtopping discharge goes larger than the mean overtopping discharge arising from the forward wave by an increase ≤ 20%. When the relative wave obliquity β=20°-80°, the mean overtopping discharge decreases as the relative wave obliquity increases.
(2) In 1957, Perroud (1957) found stem wave when conducting research on the interaction between an oblique incident solitary wave and a vertical buildings, i.e. when the wave incidence is smaller than 45° relative to the vertical building, there is also wave extending along the vertical building in addition to the incident wave and reflected wave. The stem wave arises from the reflection and diffraction of waves. The wave height and width of which are increased as extending along the vertical building. In the model experiment, the breastwork affects the oblique incident wave in height. When H/L=0.05, the mean overtopping discharge generally goes down as the relative wave obliquity β increases. When β=20°-30°, the mean overtopping discharge increases, but not over the mean overtopping discharge arising from the forward wave. When H/L=0.0333, the mean overtopping discharge generally goes down as the relative wave obliquity β increases. When the relative wave obliquity β=0°-20°, the mean overtopping discharge goes down at a slow speed. When the relative wave obliquity β=20°-60°, the mean overtopping discharge goes down at a relatively fast speed.
(3) When the relative wave obliquities of waves increase at the same angle, the overtopping discharge goes smaller as the wave H/L increases (the cycle gets shorter); in contrast, the overtopping discharge significantly increases as the wave H/L decreases (the cycle gets longer).

Comparison and analysis of calculated and experimental results given by various formulae
The calculated results of the forward dimensionless overtopping discharge are shown in Table 1.
From Calculation of the mean overtopping discharge of the sloping dike by all methods above, and the comparison and analysis against the calculated results may result in a varying curve of dimensionless overtopping discharge and relative wave obliquity β. The comparison plot of dimensionless overtopping discharges is shown in Fig. 3.
When H/L=0.05 and H/L=0.0333, Owen's formula and van der Meer and Janssen's formula present relatively large results, therefore the two formulas are left aside.
Through comparison and analysis from Fig. 5, it can be concluded that: (1) In the case of the irregular wave normal incidence, when H/L=0.1, the practical result is the closest to the Owen's formula result; when H/L=0.05, the practical result is the closest to the result given by Code for Sea Port Hydrology; when H/L=0.0333, the practical result is the closest to the result given by the neural network calculation. Van der Meer and Janssen's formula results are in a relatively large range, while Hebsgaard et al.'s formula results are in a relatively small one. These two methods roughly take into consideration of how the breastwork affects the overtopping discharge, which accordingly affects the calculation result to different extents.
(2) The oblique wave affects the overtopping discharge of the sloping dike to an extent generally shrinking as the increase of the relative wave obliquity β. As the increase of the incident wave steepness H/L, Li et al.'s formula gives a decreasing mean overtopping discharge. Besides, the difference between the calculation result and the practical result increases, therefore, Li et al.'s formula fits more in calculation of the mean overtopping discharge of the small-cycle irregular wave.
(3) As the increase of the incident wave steepness H/L, Owen's formula and van der Meer and Janssen's formula give the increased mean overtopping discharge. Besides, the difference between the calculation result and the practical result increases, therefore, Owen's formula fits more in calculation of the mean overtopping discharge of irregular waves with an average-cycle smaller than 10 s.
(4) When H/L=0.1 and H/L=0.05, the neural network method gives a result varying substantially the same as the practical value. As the relative wave obliquity β becomes larger, the calculated value differs more from the practical value. Therefore, the neural network method fits more in calculation of the mean overtopping discharge of the sloping dike subject to waves.
(5) Six formulae described above are all empirical ones obtained by physical model experiments. Code for Sea Port Hydrology only provides the calculation of the mean overtopping discharge of the sloping dike under the forward wave, and the rest five formulas are all limited in suitable applications and utility. This paper discusses the relation between the relative wave obliquity β and the mean overtopping discharge through performing overtopping discharge model experiments and comparing, analyzing these six present overtopping discharge formulae. Consequently, the proposed formula is Hebsgaard et al.'s formula improved as: where, in the case of breastwork, k 1 =-0.037, k 2 =-1.0, and the rest coefficients are the same as in Section 1.2.
The dimensionless overtopping discharge calculated by Eq. (9) vs. the experiment values can be seen in Fig. 4. Seen from Fig. 4, for mean overtopping discharges, the calculated results match well with the experimental results. And the calculated results are slightly larger than the experimental results, therefore the formula is relatively safe in calculations for various working conditions.

A specific engineering case
To comprehend the newly proposed calculation formula, a specific engineering case is required for its verification. This paper verifies the applicability of the formula with the overall physical model experiment of the First Level Fishing Port in Tiancuo Village.
The First Level Fishing Port in Tiancuo Village has a special geographic location which is located at Meiling Town, Zhao'an County, Zhangzhou City and on the southernmost tip of Fujian Province adjacent to Guangdong Province, and in the southeast of the Gongkou Peninsula. Facing the Dongshan Island across the sea, Tiancuo Fishing Port, adjacent to the Zhao'an Bay on the east, is about 1 km away from the east seafront of Tiancuo Village, which is  shielded by the Shentong Mountain on the south and surrounded by beaches and shelter-belts on the west. The structure of slope-type riprap dike is used for the breakwater dyke constructed in this engineering, as shown in Fig. 5. The 1:47 geometric scale of the model is used in the overall physical model experiment of waves which are irregular waves in this experiment. The wave elements and conditions at the breakwater dyke are shown in Table 2.
The comparison between the mean overtopping discharge results calculated from the formula proposed in this paper and the measured results of the model experiment is made, as shown in Fig. 6. From Fig. 6, for slope-type dike dimensionless overtopping discharges, the experimental results from the overall physical model experiment match well with the calculated results from the formula proposed in this paper. Therefore, the calculation formula of mean overtopping discharges proposed in this paper is extensively applicable and can be applied to engineering case studies.

Conclusions
(1) For a fixed measurement point, the mean overtopping discharge generally decreases as the relative wave ob-liquity β increases. When the relative wave obliquity is relatively small, "oblique increment at a small angle" proposed by Owen will occur as well.
(2) For a fixed relative wave obliquity, the mean overtopping discharge generally increases as the wave steepness H/L decreases (the cycle increases).
(3) Based on the integral physical model experiment, it is suggested to adopt Eq. (9) to calculate the mean overtopping discharge of the sloping dike with breastworks, subject to the oblique wave effect. Formulae proposed in this paper are used only for a certain cross section. Therefore, it is suggested to adopt the integral physical model experiment for analysis and consideration in the case of different cross sections.