Crest modifications to reduce wave overtopping of non-breaking waves over a smooth dike slope

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Coastal Engineering 101 (2015) 69–88

Contents lists available at ScienceDirect

Coastal Engineering journal homepage: www.elsevier.com/locate/coastaleng

Crest modiﬁcations to reduce wave overtopping of non-breaking waves over a smooth dike slope Koen Van Doorslaer ⁎, Julien De Rouck, Sarah Audenaert 1, Valerie Duquet 2 Dept. Of Civil Engineering, Ghent University, Technologiepark 904, 9052 Zwijnaarde, Belgium

a r t i c l e

i n f o

Article history: Received 16 January 2014 Received in revised form 6 February 2015 Accepted 9 February 2015 Available online 27 April 2015 Keywords: Wave overtopping Reduction factors Storm wall Parapet Promenade Stilling wave basin

a b s t r a c t The formula to quantify the average wave overtopping discharge of non-breaking waves over a dike according to the TAW-report (2002) and included in the EurOtop Manual (2007), only contains the inﬂuence of the roughness of the dike slope and the obliqueness of the waves on wave overtopping. Unlike the formula for breaking waves, the reductive effect of a storm wall or a berm is not included in this formula. Over 1000 scale model tests with non-breaking waves on a wide variety of structures have been carried out for this paper. The crest of the dike slope has been modiﬁed with a storm wall, a storm wall and parapet, a stilling wave basin, a promenade, and a combination of a promenade and storm wall with or without parapet. For all these modiﬁed crests, a reduction factor has been deducted to include in the wave overtopping formula of non-breaking waves. This paper presents the results of all investigated measures to reduce wave overtopping over a smooth dike slope in an easy and logical way to serve as a guidance for use by designers. An example is given where all presented measures are compared to each other, and to other well-known structures. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Although the Belgian coast has only a length of 67 km, it is a densely populated area of which every meter is intensively used. A similar trend exists worldwide: coastal zones belong to the areas with high population densities, even though these regions are the most vulnerable to the risk of storm surges and overtopping. In addition, due to climate change and sea level rise, an increase in the number and the intensity of violent storms is expected. As a result, large amounts of water can overtop which may lead to dike instability and ﬂooding, a disaster for safety of mankind and economy. From this perspective, extensive research has been carried out in order to improve the safety level of the coastal areas. In Belgium, the Integrated Masterplan for Coastal Safety (IMCS) was approved by the government in 2011. This masterplan states that the Belgian coastal zones have to be protected against a storm with a return period of 1000 years and that measures have to be taken in vulnerable areas (Mertens et al. (2009)).

⁎ Corresponding author at: Dredging, Environmental and Marine Engineering (DEME), Scheldedijk 30, 2070 Zwijndrecht, Belgium. Tel.: +32 32505211. E-mail addresses: [email protected], [email protected] (K. Van Doorslaer), [email protected] (J. De Rouck), [email protected] (S. Audenaert), [email protected] (V. Duquet). 1 Present address: International Marine and Dredging Consultants (IMDC), Coveliersstraat 15, 2600 Antwerp, Belgium. Tel.: +32 32709295. 2 Present address: SBE Engineering Consultants, Slachthuisstraat 71, 9100 Sint-Niklaas, Belgium. Tel.: +32 37779519.

http://dx.doi.org/10.1016/j.coastaleng.2015.02.004 0378-3839/© 2015 Elsevier B.V. All rights reserved.

The typical geometry of a large part of the Belgian coastline consists of a sandy beach under a mild slope (1:100–1:50) followed by a smooth dike slope (1.5 ≤ cot(α) ≤ 3) and a quasi-horizontal part, further called “promenade”. Just next to this promenade, apartment buildings are present at several locations. Before 2004,3 during nearly every winter storm, the water reached the sea dike in Ostend (Fig. 1), leading to a high possible risk of wave overtopping and resulting damage. The IMCS recommends both ‘soft’ and ‘hard’ measures to reduce wave overtopping. The soft measures are beach nourishments, while the hard measures can be new constructions on top of the smooth dike (Fig. 2) slope or the modiﬁcation of the crest of an existing sea dike. Whatever measures are designed, they have to take the strict spatial restrictions into account: increasing the height of the sea dike should be limited due to visual implications for people living close behind the coastline, while a landward expansion of the promenade is often impossible due to the presence of apartments and buildings. The construction of permanent or mobile storm walls with or without parapet, or the integration of a so called stilling wave basin in the crest of the sea dike are such measures. The inﬂuence of these hard measures on the overtopping discharges in non-breaking wave conditions forms the subject of this paper.

3 After 2004, the Flemish government decided not to wait for the IMCS to be published and works to start, since the risk in Ostend was too high. A beach nourishment was carried out as “emergency measure” before the improvements as suggested in the IMCS will be implemented. This prevented waves overtopping the dike slope during normal winter storms.

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Fig. 1. Sea dike in Ostend, at the Belgian coastline, during a winter storm before 2004.

1970s datasets using irregular waves were produced leading to a more realistic simulation of sea states. A large number of data gathered from different laboratories with many variation in hydraulic and geometrical boundary conditions, were brought together in Van der Meer and Janssen (1994). A dimensionless plot of all data led to a best ﬁtting trend line which allows calculating wave overtopping over a sloping structure. Results for a wider variety of structures and hydraulic parameters are published in TAW (2002) and EurOtop manual (2007). The British guidelines for wave overtopping propose a slightly different formula, based on a (smaller) data set by Owen (1980). Besides the study of wave overtopping in an empirical way, also numerical models and prototype measurements were used to investigate wave overtopping (CLASH project, De Rouck et al., 2009). In this way, complex structures and scale effects could be studied. Another method to calculate wave overtopping is by using a neural network (NN), based on the CLASH database (Fig. 3). Over 10,000 tests on many different geometries were collected and identiﬁed by 31 (geometrical and hydraulic) parameters. By using such a neural network (Van Gent et al., 2007; Verhaeghe et al., 2008) overtopping can be calculated for a wider range of geometries and sea states. In this paper a comparison with the existing empirical formulae of overtopping over sloping dikes will be made, based on the new data set of about 1000 tests with non-breaking waves. 2.1. Smooth dike slopes According to the TAW-report, and included as such in the EurOtop manual (2007), dimensionless average wave overtopping discharge q (in m3/s/m) is expressed as a function of the dimensionless freeboard of the construction. The formula giving the average overtopping discharge for breaking waves on a sloping dike becomes: q 0:067 R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ γb ξm−1;0 exp −4:75 C Hm0 ξm−1;0 γb γ f γβ γ v tanα g H 3m0

!

ð1Þ Fig. 2. Artist impression of a coastal dike, after implementation of reductive measures (promenade and storm wall) to reduce overtopping. Impression by ‘Coastal Division, Flemish Government’ for IMCS.

Over 1000 tests have been carried out with and without overtopping reducing measures. This happened during several test campaigns at Ghent University between 2005 and 2012. The test matrix was based on hydraulic boundary conditions present along the Belgian Coast, but opened up to a broader spectrum of relevant parameters to have a good base for analysis (e.g. from small to large wave periods, from small to large dimensionless freeboards, etc.). Each year, the knowledge of the authors increased, and new details and geometries were investigated. All separate measures have been published on international conferences and/or in local master theses at Ghent University (to be downloaded from www.lib.ugent.be; see reference list for the independent titles). This paper serves as the summary of all these different reductive measures. The results are presented in the (standard) nondimensional log-linear graph, from which reduction factors ‘γ’ for each individual measure are deducted. The existing formulae according to EurOtop Manual 2007 are updated, and the data are again plotted in the log-linear graph to visualize the increased performance due to the reduction factor γ. 2. Literature study Several methods can be found in literature to quantify the wave overtopping discharge over a sea defense structure. The oldest method is by means of experimental formulae, which have been set up by testing a scale model of a structure in a wave ﬂume or wave tank. In the beginning only regular waves were investigated, while since the

with a maximum for non-breaking waves: ! q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:6 C : Hm0 γ f γ β g H 3m0

ð2Þ

Overtopping over a smooth impermeable dike with a simple slope and perpendicular wave attack can be calculated by setting all γ-factors equal to one. The reductive effects that certain geometrical (γb: berm in the dike, γf: roughness of the slope, γv: vertical wall on top of the dike) and hydraulic (γβ: wave obliqueness) variations bring along, can be accounted for by using the individual values of γ b 1 as deﬁned in the TAW-report. The transition from the formula for breaking waves to the one for non-breaking waves occurs for a wave breaker parameter ξm−1,0 (calculated with spectral wave period Tm−1,0) with a value of about 1.82. In the current dataset, only non-breaking waves (formula (2)) are considered, since the boundary conditions of the model tests are based on the geometry of the Belgian sea dikes before the improvements according to the IMCS-report were implemented, leading to non-breaking wave conditions. Note that new research by van der Meer and Bruce (2013) has improved the Eqs. (1) and (2) mainly for small freeboards Rc/Hm0 b 0.5– 1. For larger freeboards, the updated formulae by Van der Meer and Bruce lead to similar results as formulae (1) and (2), so they can be maintained. In our research, mainly the range Rc/Hm0 N 0.5–1 is present. Consequently our results will be compared with the approach by EurOtop (2007). The formula for non-breaking waves (2) does not contain a γv or γb reduction factor. The fact that the overtopping reducing effect of a

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Fig. 3. Parameters from the NN and CLASH database to quantify wave overtopping discharges over coastal structures.

vertical wall or a promenade cannot be taken into account in formula (2) forms the starting point for the research carried out for this paper. EurOtop (2007) mentions an alternative overtopping formula developed by Owen in 1980, based on a smaller dataset than formulae (1) and (2) (and therefore a smaller range of validity). The average overtopping according to Owen can be calculated with formula (3): ! q Q0 RC qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ exp −b pﬃﬃﬃﬃﬃﬃﬃﬃ : HS s0;m s0;m g H3m0

ð3Þ

The coefﬁcients b and Q0 are listed in Owen's paper (1980) and depend on the slope of the dike. This formula (3) shows three remarkable differences compared to formula (2) for the non-breaking waves: - There is no distinction between breaking waves and non-breaking waves in the Owen formula (3). This formula overestimates overtopping for steep slopes. - Wave overtopping discharge is inﬂuenced by the wave period by means of the wave steepness s0,m according to Owen, while formula (2) for non-breaking waves shows no dependency of the wave period. - The dike slope angle is another active parameter according to Owen (different coefﬁcients b and Q0 for different slopes) while formula (2) shows no dependency for non-breaking waves. Apart from Owen, also the ﬁnal report of the Dike-3D project (Kortenhaus et al. (2006)) mentions a possible inﬂuence of the wave period, and they include a correction factor γs0 to account for the wave steepness in the non-breaking formula. In (2) and conﬁrmed by many international researchers, the wave period does not inﬂuence the wave overtopping discharge for non-breaking waves on slopes cot(α) = 2 to 3. Also a recent work by Victor (2012) conﬁrms this; he states that there is a small inﬂuence of slope angle and wave period for non-breaking waves, but this inﬂuence is insigniﬁcant within the range tested in this study (see further, Table 1 and others). Not the presentation by Owen, but the international standardized presentation as in formula (2) will be used throughout the paper. Nevertheless, the inﬂuence of the wave period and slope angle on wave overtopping for non-breaking waves will be checked in the current data set, and conclusions will be given for each geometry. 2.2. Storm wall and parapet EurOtop (2007) includes a section on the effect of wave walls on sloping structures. It is stated that the knowledge on this structure is limited and only a few model studies were available. Therefore the

use of the neural networkis recommended for more reliable calculations. Nevertheless, a method to calculate the reduction factor γv is proposed in the manual, to include in formula (1), not in (2). A ﬁrst step is to calculate the average slope by changing the vertical wall by a 1:1 slope. This allows calculating the wave breaker parameter ξm−1,0 in order to use the breaking or non-breaking overtopping formula. Next, EurOtop (2007) mentions a reduction factor γv = 0.65 when a wall is present, and γv = 1 when no wall is present. This leads to the following interpolated formula: γv ¼ 1:35−0:0078 αwall

ð4Þ

where αwall is the angle of the steep slope in degrees (between 45° for a 1:1 slope and 90° for a vertical wall). Even though no factor γv is included in formula (2) (in contrast to formula (1)), it will be veriﬁed in our data if formula (4) is a good method to predict overtopping of non-breaking waves over a smooth dike slope with storm wall. Kortenhaus et al. (2001) discusses Storm Surge Protection (SPP) wall, which is similar to our dike slope with promenade and storm wall. In the current study, the promenade is above the SWL, while in the SPP-project there is a considerable water depth on the promenade, which is a different hydraulic situation. Instead of an overtopping bore (like in the current study), waves can face the SPP wall. The focus is mainly on impact forces on the wall, but the paper by Kortenhaus et al. also deals with the overtopping reduction shortly. It is mentioned that a so-called overtopping reducer, which is comparable to what we call parapet, is an efﬁcient method to further reduce wave overtopping, especially for dimensionless freeboards Rc/Hs N 1.2. Another reductive measure is the implementation of “underwater barriers” on the promenade. This looks a bit like the stilling wave basin (SWB), but again with the difference that the SWB in our study is located above the SWL. One of the conclusions by Kortenhaus et al. (2001) is that a certain horizontal space should be available between the underwater barrier and the SPP wall, to allow wave energy dissipation. A last important ﬁnding in the SPP project is that the wave period has a signiﬁcant inﬂuence on the overtopping rates for the tested geometries (dike slope with promenade and storm wall). Also Pearson et al. (2004), which studies the effectiveness of parapets on vertical walls, claim to see a clear dependency of the wave period on the overtopping discharges. However, both papers don't work out a generic method to account for this dependency. As mentioned in the previous section, this will be veriﬁed in the current data set. Recently, new work by Tuan (2013) was published which gives an approach to calculate the reduction due to crown-walls on low sea dikes. Tuan states that a reduction factor for the inﬂuence of a vertical wall on top on the slope has to be included in the non-breaking formula (2) just as in formula (1). He also introduces a reduction factor for a promenade, but the length of his promenade is based on typical Vietnamese coastal dikes and is much smaller (1 m–2 m prototype)

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and therefore out of the range of this study. He proposes a reduction factor for a vertical wall as follows: γ wall ¼

1 1 þ 1:60

ð5Þ

hwall 1 Rc ξ0m

where hwall is the wall height, and Rc⁎ the difference between the base of the storm wall and the still water level, which is not in agreement with the international standard.4 In the current paper, Rc will respect the international interpretation by deﬁning Rc as the difference between the highest point of the structure and the still water level. Tuan (2013) also changes the Rc-value in formula (2) by his Rc⁎. Furthermore, his study advises to use the actual slope angle α instead of the equivalent slope angle as is done in the EurOtop manual. The data in the current paper are also analyzed by always using the actual slope angle. Tuan (2013) states that the total reduction is always achieved by multiplying all contributing reduction factors. That is because he only has data of the combined geometry promenade with storm wall and no independent data for the promenade alone; the Vietnamese promenades are simply too small for that. However, making a general statement that reduction factors can always be multiplied to account for the combined effect is not correct. The current research shows that it can be an overestimation of the reductive effect: the ﬁnal γ by multiplication of the independent values can be lower than the real value of γ, leading to an underestimation of the wave overtopping discharge. For the dike slope with storm wall, both the EurOtop approach as Tuan's approach will be compared with the current data set. When a parapet is available, EurOtop (2007), Cornett et al. (1999), Kortenhaus et al. (2003), Pearson et al. (2004) only give a reduction related to vertical walls (caisson, quay wall, …) and not to sloping structures. Cornett et al. (1999) investigated a caisson with 3 angles of overhanging wall geometry: 30°, 45° and 60°. They found a γ-factor starting from 0.9 up to 0.7. A value of γv = 0.7 means a reduction in wave overtopping up to 95% for Rc/Hm0 = 1.67. Even a relatively modest overhang inclined 30° with respect to vertical can reduce overtopping ﬂows by a factor of 10 or more. The extent of the decrease in overtopping discharges at the overhanging wall was found to be highly variable, depending on the water level and wave conditions. No generic method is proposed in Cornett et al. (1999) to calculate the reduction factor. The other references (Kortenhaus et al., 2003; Pearson et al., 2004) did develop a generic method to determine a k-factor deﬁned as the ratio of overtopping discharge with parapet to the overtopping without parapet. This is unfortunately not applicable here since our study always has a sloping dike which does not ﬁt the parameters used by Kortenhaus et al. (2003). Some studies exist with a storm wall with or without parapet on top of a rubble mound breakwater, which is a sloping structure. Coeveld et al. (2006) deﬁne a parameter Q' which is similar to the k-factor by Kortenhaus et al. (2003): the ratio of overtopping discharge of a breakwater with crest element to one without crest element. This ratio Q' seems to be dependent on a number of parameters: - An exponential decreasing trend for increasing Rc/Hm0 was found. - No relationship between Q' and the wave period was found. - A decrease of the ratio Q' was observed for an increase in nose length. - A decrease of the ratio Q' was observed for an increase in crest width. The ratio Q' is deﬁned as 0

−4

Q ¼ 1:55 e

4

hwall H m0

− 0:4

B H m0

NL m0

− 2H

Symbols used in Tuan's paper have been changed to avoid misunderstandings.

ð6Þ

where hwall is the wall height, B the crest width and NL the length of the parapet nose. The authors believe that the ratio ﬁlters out the roughness of the rubble mound breakwater and is therefore also applicable on smooth slopes with and without storm wall with parapet. Nevertheless, Coeveld et al. (2006) state that their data show bad comparison with the TAW prediction. A better comparison was found to compute the overtopping over the rubble mound breakwater without storm wall with the Neural Network approach, and then using formula (6) to include the effect of the storm wall and parapet. This approach will be tested on our data. By adding a storm wall or a parapet on the slope, the slope virtually increases. The extreme boundary of a virtually increased slope would be a vertical wall. New analysis of vertical wall data by van der Meer and Bruce (2013) shows that for small freeboards (Rc/Hm0 b 0.91) the formula by Allsop et al. (1995) and for large freeboards (Rc/Hm0 N 0.91) the formula by Franco et al. (1994) should be used. This line, with bending point at Rc/Hm0 = 0.91, will be plotted along with the data of a sloping dike with storm wall and parapet (Fig. 10). Also the line of a steep slope cot(a) = 1, according to the data analysis by Victor (2012) and formulae mentioned in van der Meer and Bruce (2013), will be added to this graph for reasons of comparison.

2.3. Promenade In this paper, the (quasi) horizontal part at crest level is called promenade. Keep in mind that it contains a gentle slope of 1% to 2% to stimulate draining from rainfall and overtopped water towards the sea. A promenade is different than what is meant by the term “berm” in the EurOtop manual. A berm is a (quasi) horizontal part in the dike slope and often located around the design water level, to reduce the average slope and thereby reduce wave overtopping discharges. The promenade in this study is at crest level, clearly above SWL. Thereby, the reduction coefﬁcient as presented in EurOtop (2007) might not be the best prediction tool. Nevertheless, it will be checked with our data. EurOtop (2007) also acknowledges that if the overtopping is not measured at the end of the slope, but a few meters backwards the hazard effect of overtopping will reduce. As a rule of thumb, a reduction qeffective = qcrest/x where x is the backward distance between 5 and 25, is mentioned. This is however not based on measurements, but just an indication that the promenade can signiﬁcantly reduce wave overtopping effects. It will be further studied in the present data set.

2.4. Stilling wave basin This paper introduces a number of other crest modiﬁcations, such as a storm wall at the end of a (fairly long) promenade, or a stilling wave basin. Some literature is available on these topics, like Geeraerts et al. in the Dike-3D project, but it all belongs to the research project of Ghent University on the reduction of wave overtopping, and is thereby included in this paper. On the stilling wave basin, some independent literature was found on overspill basins in Italian marina breakwaters (van der Meer and Bruce (2013)). It shows that a storm wall and an overspill basin can have similar behavior in terms of reducing average overtopping discharges, but the overspill basin acts twice as good as the storm wall regarding individual wave overtopping events. Both measures can decrease the average discharge up to a factor of 2 under the tested conditions. Further, it is stated that the volume of the overspill basin plays a role, but the freeboard remains the most important parameter. A last important statement is that the water in the basin should be allowed rapid drainage in order to maintain the efﬁciency of the basin. This latter is conﬁrmed by Burchart and Andersen (2006).

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

73

Fig. 4. Position of wave gauges in the 2D ﬂume of Ghent University (distances in mm).

3. Test set-up The objective to predict the reduction on average overtopping discharge has been accomplished by means of experimental research carried out in the wave ﬂume (L = 30.00 m, W = 1.00 m, H = 1.20 m) of the Coastal Engineering Department of Ghent University. Waves are generated using a piston type wave paddle, and the steering of this paddle features active wave absorption. Each tested time series contained approximately 1000 waves, in order to obtain reliable average overtopping discharges. Waves are measured using resistance type wave gauges, positioned as shown in Fig. 4: two in front of the wave paddle (on behalf of the active wave absorption), three at deeper water, and three in front of the structure (at a distance of about 0.4 L from the structure). By means of these groups, incident and reﬂected wave conditions can be separated from each other and the incoming wave height can be determined, using the method by Mansard and Funke (1980). The height of the foreshore is 27 cm, its length is 2 wavelengths. The water depth on the foreshore is large enough to avoid wave breaking, see details for each geometry. The wave spectra at the deeper water and on the foreshore are very comparable, no loss of energy takes place. Wave overtopping is captured by a tray on top of the smooth dike slope, and lead to a 30 liter basin that is constantly weighed on a balance. When the basin is full, water is pumped back to the wave ﬂume in order to maintain the correct water level in the ﬂume during the test. Total overtopping volume can be deducted from the balance's weight registration in time. A part of the tests also had individual overtopping measurements, but this is not further treated in this paper. Non-breaking wave conditions were tested on a smooth dike slope (γf = 1) with perpendicular wave attack (γβ = 1). Both a dike slope 1 V:2 H and 1 V:3 H were tested. A few of the test results with ξm−1,0 just above 1.82 gave results that could not be identiﬁed as “non-breaking waves” with full conﬁdent. Therefore, in the current data set, the limit was set on ξm−1,0 N 2.1 to clearly deﬁne a test as non-breaking. The majority of the tests were performed with a JONSWAP 3.3 spectrum while some were performed with a Pierson–Moskowitz spectrum, both single peak spectra. No inﬂuence of the spectrum on the overtopping volumes could be noticed for the range of dimensionless freeboards and tested spectra in the current data set.

combinations of a promenade with a wall with or without parapet, are related to this reference situation. A sketch of this reference geometry is given in Fig. 5, the arrow with indication ‘q’ shows where the overtopping is measured. To determine the reduction factors for all different geometries, the measured data are not compared to formula (2), but to a reference formula which is determined out of 80 new tests on a smooth dike slope as shown in Fig. 5. In this way, the test set-up, measuring devices and -techniques are the same for the reference case as for cases with reductive measures, which guarantee good comparison and exclude model effects. The range of parameters of these 80 tests is summarized in model values in Table 1. All test results are plotted in a semi-logarithmic diagram with the dimensionless freeboard (Rc/Hm0) on the horizontal axis and the qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dimensionless overtopping discharge q= gH3m0 on the vertical axis. An exponential trend line is ﬁtted through the data and gives the reference formula to calculate the average overtopping discharge on a smooth dike slope under non-breaking wave conditions.

Fig. 5. Smooth dike slope.

Rc/Hmo

1.0E+00 0

0.5

1

1.5

2

2.5

3

3.5

1.0E-01

4.1. Smooth dike slopes The smooth dike slope serves as a reference situation. The reduction factors which take into account the inﬂuence of overtopping reducing measures such as a wall, a parapet, an SWB, a promenade and

q/(g ·Hmo³)1/2

4. Results 1.0E-02

y = 0.2e-2.28x R² = 0.94

1.0E-03

1.0E-04 EurOtop average trendline 5% lower boundary

Table 1 Summary of the characteristics of the tests (non-breaking waves) on a smooth dike slope (scale model values). Slope angle of the smooth dike slope Mean spectral wave period Dimensionless freeboard Freeboard (top of structure to SWL) Spectral wave height Water depth at toe of the structure Wave breaker parameter

cot(α) Tm−1,0 Rc/Hm0 Rc Hm0 d ξm−1,0

2 and 3 0.91–2.45 s 0.83–3.15 0.12–0.27 m 0.07–0.21 m 0.35–0.49 m 2.10–4.90

1.0E-05

y = 0.2e-2.6x

5% upper boundary Smooth dike slope: measured data

1.0E-06

Exponential (smooth dike slope: measured data)

Fig. 6. Reference data set of non-breaking waves on a smooth dike slope.

q R qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:28 c : Hm0 g H3m0

ð7Þ

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

It can be noticed that the scatter around this trend line is small. When taking the exponential coefﬁcient as normally distributed stochastic variable, the mean value is 2.28 with a standard deviation of 0.15. Additionally, almost all results are located above the average trend line in EurOtop, 2007 (formula (2)) resulting in a slightly higher trend line and thus a smaller coefﬁcient in the exponent: 2.28 instead of 2.6. If the CLASH dataset is used, but limited to the data with similar geometry β = 0° for perpendicular wave attack, γf = 1 for smooth slopes, 1.5 ≤ cot(α) ≤ 3.0 for the dike slope, Rc N 0 for positive crest freeboards, B = 0 for the absence of berms, and Gc = 0 for the absence of promenades at crest level; overtopping is measured direct at the end of the dike slope - ξm−1,0 N 2.1 like in our own database, in order to only work with non-breaking waves.

-

only 472 of the 10,532 tests remain. When also reliability factor RF is set to one, 255 data points remain, which are shown in Fig. 7. The average trend line through these 255 tests has a coefﬁcient 2.29, very close to the reference line of the data from Fig. 6. The data from Fig. 6 are plotted together with the considered CLASH data in Fig. 7, and they are in line with each other, which means that the new data are comparable to what was found in other laboratories with similar boundary conditions. The CLASH data have larger scatter, probably due to the fact that they are found in different laboratories all over the world. The coefﬁcient 2.6 in Eq. (2) is the average coefﬁcient (normally distributed with a standard deviation of 0.35) of all different geometries and hydraulic conditions for non-breaking waves over sloping structures. For reasons of comparison, all reductive measures further in the paper will be referred to formula (7) since all experiments are run in the same wave ﬂume. The internationally accepted exponential coefﬁcient 2.6 in Eq. (2) is not being called into question to predict wave overtopping of non-breaking waves over smooth dike slopes. Hence we propose to use the reduction coefﬁcients as proposed later in this paper in combination with formula (2) with coefﬁcient 2.6 to calculate average overtopping discharges. For deterministic design, EurOtop (2007) manual advises to use coefﬁcient 2.3. This will be further explained in Section 5. As mentioned in Section 2, the new dataset was subjected to a detailed analysis on the inﬂuences of slope angle and wave period on the mean overtopping discharge. Fig. 8 contains 29/80 data points on the smooth dike slope where the focus was put on the slope angle and

Rc/Hmo

1.0E+00

0

0.5

1

1.5

2

2.5

3

3.5

4

q/(g ·Hmo³)1/2

1.0E-02

1.0E-03

1.0E-04

1.0E-06

0

0.5

1

1.5

2

2.5

3

3.5

1.0E-01

1.0E-02

1.0E-03

1.0E-04

1.0E-05

1.0E-06

EurOtop average trendline 5% lower boundary 5% upper boundary Smooth dike slope 1:2 Tm-1,0 = 1.64 Smooth dike slope 1:2 Tm-1,0 = 2.36 Smooth dike slope 1:3 Tm-1,0 = 1.64 Smooth dike slope 1:3 Tm-1,0 = 2.36 Exponential (smooth dike slope: all measured data)

y = 0.2e-2.28x R² = 0.94 y = 0.2e-2.6x

Fig. 8. 29 data points on a smooth dike slope split up by wave period and slope angle α.

wave period, where the red trend line is the same one as in Fig. 6 based on all 80 points. - Slope angle: when comparing the slope angles cot(α) = 2 versus cot(α) = 3 the mildest slope is slightly more overtopped since this slope reﬂects less energy and allows more overtopping. Another explanation is that the layer thickness of the “tongue” of the overtopping wave is somewhat larger for milder slopes, giving a larger overtopping discharge (Bosman et al., 2008). The difference between both slopes is however almost negligible in the obtained data, from which it can be concluded that the slope angle only has very limited inﬂuence within the used range of tested parameters. This ﬁnding is conﬁrmed by Victor (2012); wave overtopping has a weak dependency on the slope angle for mild dike slopes (1.5 ≤ cot α ≤ 3). - Wave period: on a slope 1:2 there is no inﬂuence of the wave period on the average overtopping discharge, while on slope 1:3 a minor difference exists: an increase of wave overtopping occurs for increasing wave period. This is in line with the ﬁndings by Victor (2012); an (minor) inﬂuence of the wave period on wave overtopping exists, but its inﬂuence is limited compared to the effect of the relative crest freeboard.

Due to the minor differences in overtopping discharge for tests in the current data set with different wave periods or different slope angles, and due to conﬁrmation in literature (Victor (2012)), it can be concluded that parameters Tm−1,0 or α do not need to be included in the nonbreaking overtopping formula for smooth mild dike slopes with 1.5 ≤ cot(α) ≤ 3. 4.2. Smooth dike slope with storm wall

1.0E-01

1.0E-05

Rc/Hm0 1.0E+00

q/(g ·Hm0³)1/2

74

EurOtop average trendline 5% lower boundary 5% upper boundary new data UGent CLASH limited 255 tests exponential (data of this paper) exponential (CLASH limited 255 tests)

y = 0.2e-2.28x R² = 0.9435 y = 0.2e-2.29x R² = 0.866 y = 0.2e-2.6x

Fig. 7. Comparison between new data and CLASH database on a smooth dike slope.

Wave overtopping can be reduced by placing a vertical storm wall (with height hwall) on the dike slope. In this way incoming waves are projected upwards. Note that the crest level of the dike slope (dashed line in Fig. 9) can be maintained by placing the storm wall seaward on the slope (Fig. 9 left), or can be increased by placing the storm wall on top of the original crest (Fig. 9 right). The developed reduction factors for wave overtopping are of course generally applicable. The only difference is the situation on the right in Fig. 9 has a larger freeboard (larger X-value on the log-linear graph) in comparison with the X-value of the original smooth dike slope. The situation on the left of Fig. 9 has the exact same X-value and allows for direct comparison. In our experiments, the situation on the left in Fig. 9 was built in the wave ﬂume, to be able to keep the overtopping collector at the same level.

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

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Fig. 9. Smooth dike slope with a vertical wall with the deﬁnition of the freeboard Rc.

Slope angle of the smooth dike slope Mean spectral wave period Dimensionless freeboard Wall height Freeboard (top of structure to SWL) Dimensionless wall height Spectral wave height Water depth at toe of the structure Wave breaker parameter

cot(α) Tm−1,0 Rc/Hm0 hwall Rc hwall/Rc Hm0 d ξm−1,0

2 and 3 1.10–3.00 s 0.60–2.60 2, 4, 5, 6 and 8 cm 0.05–0.29 m 0.08–1.60 0.07–0.18 m 0.36–0.57 m 2.20–4.80

0

0.5

1

1.5

1.00E+00

2

2.5

Rc/Hmo

1.00E-01

1.00E-02

1.00E-03

3

3.5

smooth dike slope + wall 2cm smooth dike slope + wall 4cm smooth dike slope + wall 5cm smooth dike slope + wall 6cm smooth dike slope + wall 8cm Vertical structure smooth dike slope low freeboards steep slope 1:1 y = 0.2*exp(-2.28*x)

1.00E-04

y = 0.2e-2.28x 1.00E-05

Fig. 10. Data set on a smooth dike slope with storm wall — measured values.

0

0.5

1.00E+00

1.00E-03

1.5

2

2.5

3

3.5

Rc/Hmo/γ v smooth dike slope + wall COEVELD smooth dike slope + wall EUROTOP Vertical structure y = 0.2*exp(-2.28*x)

1.00E-01

1.00E-02

1

q/(gHmo³)1/2

Table 2 Summary of the characteristics of the tests on a smooth dike slope with storm wall (scale model values).

research by Tuan, which is only based on small wall heights hwall/ Rc b 0.5. Since none of the above methods are completely satisfying, and don't always offer an approach which is possible to combine with the expression of Eq. (2), new reduction factors are proposed in this paper. Similar to what is done in Fig. 8 for smooth dike slopes, the inﬂuence of the slope angle and the wave period is studied in the data set of

q/(gHmo³)1/2

A total of 117 tests were performed for a range of storm conditions and different heights of the storm wall in order to investigate its reductive capacity. Table 2 provides a summary of the parameters of the test program on this geometry. In Fig. 10, the overtopping results are grouped by the height of the wall and plotted in different symbols and colors. The different wall heights (2 cm to 8 cm) are shown in the legend. The reference formula (7) is plotted in a full red line. A clear reduction in overtopping volume can be noticed since all the data are located below the reference line. The lines for a vertical wall and a smooth dike slope cot(α) = 1 are also added to Fig. 10 for reasons of comparison, even though both are not applicable for the Belgian Coast. It is clear that the storm wall on top of a sloping dike does not increase the virtual slope that much that it can be considered as a vertical structure, but it still is highly reductive compared to the reference line from this research. The purple data points in Fig. 10 (overtopping over a smooth dike slope with storm walls of 8 cm) comes close to the steep dike slope cot(α) = 1. However, an increased virtual slope 1 V:1 H is not steep enough for Rc/Hm0 b 1, and too steep for Rc/Hm0 N 1.5. Calculating overtopping over a smooth dike slope with a storm wall as if it was overtopping over an increased virtual steeper slope does not give the best results. Note that the crest freeboard was not increased during the new tests, only the wall height was increased by moving the wall forward on the slope (Fig. 9 left). This allows for direct comparison between tests, since the value on the X-axis does not change: a lower overtopping discharge is found for a higher wall. At ﬁrst, the data are corrected by reduction factors as found in literature (formula (4) by EurOtop, (5) by Tuan and (6) by Coeveld), to see if they move closer to the black reference line. The approach by EurOtop in green symbols in Fig. 11 doesn't work, since only one general value is used and Fig. 10 already shows that there is a difference between the different wall heights which is not present in the γv according to EurOtop. The approach by Coeveld in red symbols in Fig. 11 is also not a good approach, probably due to the fact that this approach might not be valid when only a storm wall is present. The approach by Tuan in Fig. 12 works satisfying, but is less accurate for the highest walls where the overtopping discharge is overpredicted. That was to be expected, since those were outside the range of parameters from the

1.00E-04

1.00E-05 Fig. 11. Smooth dike slope with storm wall corrected by formula (5) EurOtop (green) or formula (7) Coeveld (red).

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0

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Rc/Hm0/γ v

Rc/Hmo/γ v TUAN 1.0E-01

1.00E-02

1.00E-03

q/(gHmo³)1/2

1.0E-02

q/(g·Hm0³)1/2

1.00E-01

1.0E-03

1.0E-04

smooth dike slope + wall 2cm TUAN smooth dike slope + wall 4cm TUAN smooth dike slope + wall 5cm TUAN smooth dike slope + wall 6cm TUAN smooth dike slope + wall 8cm TUAN Vertical structure y = 0.2*exp(-2.28*x)

1.00E-04

1.00E-05

y = 0.2e-2.28x

1.0E-05

Smooth dike + W2: corrected values Smooth dike + W4: corrected values Smooth dike + W5: corrected values Smooth dike + W6: corrected values Smooth dike + W8: corrected values Reference situation

Fig. 14. Data set on a smooth dike slope with storm wall — corrected values.

Fig. 12. Smooth dike slope with storm wall corrected by formula (6) Tuan.

smooth dike slope with storm walls. Only the conclusions are given below: - Slope angle: the difference between slope cot(α) = 2 and cot(α) = 3 is again minor, but in contrast to the dike slope without storm wall, the steepest slope gives the largest overtopping discharge. The reason for this is that on a steeper slope, the vertical velocity component of the run-up is larger compared to a milder slope, which leads to slightly larger overtopping discharges. Nevertheless, data analysis shows that the difference is again negligible, and no component α will be included in the formula. - Wave period: no difference between small and large wave periods was distinguished this time. To take the reducing effect of the storm wall into account, formula (7) is adjusted to formula (8) by introducing an inﬂuence factor γv, which is independent of slope angle and wave period as stated above. γv is calculated for every single data point, by isolation from formula (8). This value means for every data point how much it would have to be shifted on the X-axis of Fig. 10 to be exactly on the reference line and thus give perfect prediction of the overtopping discharge by using formula (8). q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:28 c Hm0 γv g H3m0

ð8Þ

γv is plotted versus a dimensionless parameter to ﬁnd a best ﬁtting curve. The dimensionless parameter plotted on the horizontal axis in Fig. 13 is the height of the storm wall divided by the crest freeboard 1.2 y = e-0.56x R² = 0.46

1 0.8 γv

0.6 0.4 trend line new data

0.2 hwall/Rc

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Fig. 13. Calculated γv as a function of hwall/Rc (left) and comparison of measured and calculated reduction factor γv.

hwall/Rc. It can be seen that the decrease of γ-value (or increase of reduction) slows down towards higher dimensionless wall-heights, which is better expressed by an exponential relationship than by a linear one. For hwall/Rc N 1.24 the reduction coefﬁcient becomes constant. There is no extra reduction when the wall is larger than the freeboard (i.e. water level is above the base of the wall). The reduction factor γv is deﬁned in formula (9). h γv ¼ exp −0:56 wall Rc γv ¼ 0:5

for for

hwall b1:24 Rc

hwall ≥1:24: Rc

ð9Þ

The data of Fig. 10 are corrected by means of formulae (8) and (9) and plotted in Fig. 14. This leads to a much better prediction of wave overtopping over smooth dike slopes with a storm wall, compared to the presentation in Fig. 10. All corrected points are now close to the reference line, which demonstrates the efﬁciency of formula (8). The exponential coefﬁcient 2.28/γv is taken as a normally distributed stochast with mean value 2.82 and standard deviation 0.41. The relative standard deviation becomes 0.15 which has the same order of magnitude as the relative standard deviation on formula (2) as mentioned in EurOtop (2007): 0.35/2.6 = 0.13. 4.3. Smooth dike slope with storm wall and parapet Wave overtopping can be further reduced, without increasing the height of the wall, by adding a “nose” to the vertical wall. This is further referred to as parapet. Due to the presence of the parapet, waves are not only projected upward, but also back towards open sea. A sketch of the tested geometry as well as the deﬁnition of the used parameters in the formulae are given in Fig. 15. 175 tests have been carried out on a smooth dike slope with storm wall and parapet, divided in 2 phases. In a ﬁrst phase the inﬂuence of the geometrical parameters such as the height of the wall and the nose (hwall and hn) and the angle ε of the parapet were investigated in order to ﬁnd an optimal geometry. The ranges of the parameters of the 92 tests from phase 1 are given in Table 3. In Van Doorslaer and De Rouck (2010) the results of phase 1 are analyzed in full detail. The main conclusions are given here. The data were plotted for the tested angles ε, and the different height ratios λ. The reduction of wave overtopping was found to be strongly dependent on both parameters.

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

77

Fig. 15. Sketch of a smooth dike slope with storm wall and parapet and deﬁnition of the used parameters.

A reduction factor γε describes the inﬂuence of the angle ε (Fig. 16 top left). One symbol in this graph represents the average of all data grouped by their nose angle (e.g. the 2nd red cross from the left is the average of all data points with wall 8 cm and parapet ε = 30°, their average γε-value is 0.61). The larger the angle ε, the better the waves are projected towards the sea resulting in smaller overtopping discharges. But the reduction doesn't increase linearly. An optimal angle with respect to reduction of wave overtopping and not too high uplift forces (which is outside of the scope of this paper) was found for ε-values of 30° to 45°. Although the parameter ε is the dominant geometric variable, wave overtopping also decreases when the nose of the parapet hn is more prominent, and thus when λ increases. A small descending trend is noticeable in Fig. 16 top right. Data were plotted versus λ, and a reduction factor γλ was deducted. Best reduction was achieved for λ ≥ 0.3. To determine γε and γλ, the data were grouped according to ε-value and λ-value separately. However, a storm wall with parapet always has both ε and λ as parameters, which makes independent separation of the data impossible: γε is inﬂuenced by λ and vice versa. Consequently, the multiplication of γε and γλ leads to an overestimation of the reduction. A curve ﬁtting between the measured data and γε∙γλ was drawn (see Fig. 16 bottom left), and formulae (11) and (12) are the ﬁnal formulae to predict the extra reducing effect of adding a parapet to a storm wall. Remark that γpar only takes into account this extra reducing effect of the parapet, as shown in Fig. 16 bottom right. Consequently γpar always has to be combined with γv of the vertical wall!

q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:28 c γ H 3 m0 v γpar g Hm0

! ð10Þ

Table 3 Summary of the test program on a smooth dike slope with storm wall and parapet (phase 1, scale model values). Slope angle of the smooth dike slope Mean spectral wave period Dimensionless freeboard Wall height Freeboard (top of structure to SWL) Dimensionless wall height Spectral wave height Water depth at toe of the structure Wave breaker parameter Height of the nose Height ratio parapet (hn/hwall) Nose angle parapet (in degrees)

cot(α) Tm−1,0 Rc/Hm0 hwall Rc hwall/Rc Hm0 d ξm−1,0 hn λ ε

2 1.10–1.45 s 0.60–2.40 2, 5 and 8 cm 0.09–0.18 m 0.11–0.90 0.08–0.15 m 0.44 m–0.53 m 2.47–2.92 1, 2 and 3 cm 0.125–1 15°, 30°, 45° 60°

For hwall/Rc ≥ 0.25: γpar ¼ 1:80 γ ε γ λ with −4

2

γ ε ¼ 1:53 10 ε − 1:63 10 γ ε ¼ 0:56 γ λ ¼ 0:75 − 0:20 λ

−2

ε þ 1

15 ≤ ε ≤ 50 ε ≥ 50 0:125 ≤ λ ≤ 0:6

if if if

ð11Þ For hwall/Rc b 0.25: γpar ¼ 1:80 γ ε γλ −0:53 with γε ¼ 1 − 0:003 ε γλ ¼ 1 − 0:14 λ

if if

15 ≤ ε ≤ 60 0:1 b λ ≤ 1

ð12Þ

γλ in (11) is not equal to one when λ is zero, and γpar in (11) or (12) is also not one for γε or γλ equal to one. The formula (11) and (12) cannot be used outside the mentioned intervals. In case λ or ε is really small, the parapet is not much more beneﬁcial than a vertical wall. In that case, the formula (7) for a vertical wall is recommended for a conservative design approach. In the second phase of the research on parapets, the inﬂuence of the wave period and slope angle is investigated on two optimal parapet geometries, based on 83 tests. These optimal parapets haveε = 30° or ε = 45°, keeping λ constant at 0.375. This leads to γpar = 0.79 respectively γpar = 0.70 for hwall/Rc ≥ 0.25 according to formula (11). A summary of the test program from phase 2 is given in Table 4. Data of the 2nd phase on dike slope 1:2 are plotted in Fig. 17, together with the data of phase 1 (green triangles in Fig. 17, which are the same green triangles as in Fig. 16 bottom right). Some conclusions can be drawn from this graph: - Data of phase 2 are amongst the data of phase 1, which is expected since apart from a larger wave period the range of parameters in phase 2 is similar to phase 1. - All data of both phase 1 and 2 are clearly below the reference line. In some tests with small freeboards, even lower overtopping discharges then vertical structures with equal dimensionless freeboards are noticed. This indicates that a storm wall with parapet is a very good measure to reduce wave overtopping for non-breaking waves over smooth dike slopes. - When looking at the data of slope angle 1:2 and ε = 30° (blue and red data points), the blue squares with the largest wave period clearly show more wave overtopping than the red diamonds. For the data slope angle 1:2 and ε = 45° (orange and purple data points) the same observations are made: the orange circles have the largest

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Fig. 16. Explanation of the analysis of γpar for the example case hwall/Rc ≥ 0.25.

Table 4 Summary of the test program on a smooth dike slope with storm wall and parapet (phase2, scale model values). Slope angle of the smooth dike slope Mean spectral wave period Dimensionless freeboard Wall height Freeboard (top of structure to SWL) Dimensionless wall height Spectral wave height Water depth at toe of the structure Wave breaker parameter Height of the nose Height ratio parapet Nose angle parapet (in degrees)

cot(α) Tm−1,0 Rc/Hm0 hwall Rc hwall/Rc Hm0 d ξm−1,0 hn λ ε

2 and 3 1.50–2.30 s⁎ 1.25–2.26 8 cm 0.16–0.29 0.28–0.50 0.09–0.18 0.36–0.51 m 2.20–4.61 3 cm 0.375 30° and 45°

⁎ Notice that the wave period in phase 2 was chosen larger in order to investigate the inﬂuence of this parameter on the overtopping discharge over the optimal parapet geometry.

(cot(α) = 2, Tm−1,0 = 1.64 s) or blue squares versus orange circles (cot(α) = 2; Tm−1,0 = 2.36 s) or black cross versus pink plus (cot(α) = 3; Tm−1,0 = 2.36 s), it can be seen that ε = 45° reduces a little bit more than 30°. This conﬁrms what was found in phase 1 in Fig. 16 top left. - When comparing data sets with the same nose angle ε and the same wave period, for example black cross versus orange circle (ε = 45°, Tm−1,0 = 2.36 s), there is hardly any difference. The mildest slope is overtopped the least since the run-up on the mildest slope has more horizontal velocity and less vertical velocity to overcome the structure. The inﬂuence is nevertheless again too small and the number of different slopes in this data set too limited to deduct an inﬂuence factor for the slope.

0

0.5

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1.5

2

2.5

3

1.00E+00

R c /Hm0 1.00E-01

1.00E-02

1.00E-03

1.00E-04

1.00E-05

1.00E-06

q/(g.Hm0³)1/2

wave period and give more wave overtopping than the purple triangles. For slope angle 1:3 no such comparison could be made, because all tests with wave period Tm−1,0 = 1.36 s lead to breaking waves and are therefore excluded in the current analysis. Based on the data on slope angle 1:2, there is a clear inﬂuence of the wave period for the geometry with parapet. This conﬁrms what was found in Kortenhaus et al. (2001); Pearson et al. (2004). It is visually observed in Pearson et al. (2004); Van Doorslaer (2008) that long waves, who have a larger volume of water under the crest of a wave, ﬁrst “ﬁll” the space underneath the parapet's nose, after which it acts as a normal storm wall which is more easily overtopped than a parapet. - When comparing data sets with the same slope angle and the same wave period, such as red diamonds versus purple triangles

1/2 par 30° T = 2.36 1/2 par 30° T = 1.64 1/2 par 45° T = 2.36 1/2 par 45° T = 1.64 1/3 par 30° T = 2.36 1/3 par 45° T = 2.36 y = 0.2*exp(-2.28x*x) data phase 1 Vertical Structure

Fig. 17. Data set on a smooth dike slope with storm wall and parapet — measured values.

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88 0

0.5

1

1.00E+00

1.5

2

2.5

3

3.5

Rc/Hm0/γ v / γ par / γ s0,par

1.00E-02

1.00E-03

1.00E-04

q/(g.Hm0³)1/2

1.00E-01

Table 5 Summary of the characteristics of the tests on a smooth dike slope with promenade (scale model values). Slope angle of the smooth dike slope Mean spectral wave period Dimensionless freeboard Length promenade Slope promenade Freeboard (top of structure to SWL) Dimensionless promenade length Spectral wave height Water depth at toe of the structure Wave breaker parameter

1/2 par 30° T = 2.36 corrected 1/2 par 30° T = 1.64 corrected 1/2 par 45° T = 2.36 corrected

79

cot(α) Tm−1,0 Rc/Hm0 B – Rc B/Lm−1,0 Hm0 d ξm−1,0

2 and 3 1.1–2.22 s 0.85–2.68 33.3, 66.7 and 100 cm 1% and 2% 0.10–0.28 m 0.045–0.5 0.07–0.17 m 0.28–0.53 m 2.2–4.2

1/2 par 45° T = 1.64 corrected

1.00E-05

1/3 par 30° T = 2.36 corrected 1/3 par 45° T = 2.36 corrected

1.00E-06

y = 0.2*exp(-2.28x*x) data phase 1 corrected

Fig. 18. Data set on a smooth dike slope with storm wall and parapet — corrected values.

Summarizing for the storm wall with parapet, it can be concluded that together with γv to include for the effect of the height of the storm wall, and γpar to include the reducing effect of the nose angle ε and location of inclination λ, also a correction factor γs0,par has to be included to account for the larger wave overtopping for larger wave periods, by means of the dimensionless wave steepness sm−1,0 (formula (14)). Formula (10) is now adapted to formula (13) to calculate the reduction through a parapet. q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:28 c γ H γ 3 m0 v par γs0;par g Hm0 γs0;par ¼ 1:33−10 sm−1;0

! ð13Þ

ð14Þ

Fig. 18 shows that by introduction of the correction factors γv, γpar and γs0,par, the data can be well predicted by using formula (13). The exponential coefﬁcient 2.28/(γv · γpar · γs0,par) is taken as a normal distributed stochastic variable, with a mean value of 3.62 with standard deviation 0.65. This gives a relative standard deviation of 0.18. 4.4. Smooth dike slope with promenade The typical geometry of a sea dike along the Belgian Coast consists of a dike slope with a wide promenade at crest level (Fig. 19). The freeboard Rc is deﬁned as the difference in height between the highest point

of the construction and the still water level. The slope of the promenade is included in Rc. The slope of the promenade in the test set-up was 1% or 2% to stimulate drainage on the promenade of overtopped water and rainfall back towards the sea (wave ﬂume). A total of 62 tests were performed on this geometry. The range of the parameters of the test program is given in Table 5. As mentioned in Section 2, EurOtop (2007) proposes a γb reduction factor to account for a berm around still water level. Eventhough this is not the case here, the data on the smooth dike slope with promenade are corrected by the γb from EurOtop (2007) in Fig. 20. In blue the data as measured, where it can be seen that all of them are located below the reference line, showing the reductive capacity of a promenade. In green, the same data are corrected by means of γb according to the EurOtop, 2007 formula. γb from EurOtop is too low (reduction overpredicted), bringing the green data points too high above the reference line. This shows that a new reduction factor especially for smooth dike slopes with promenades at crest level should be deducted. The blue data are analyzed, with extra attention paid to the wave period and slope angle. Similar conclusions as for the reference case (smooth dike slope) are found: - Slope angle: a minor difference in overtopping, where the mildest slope is overtopped slightly more due to the thicker layer thickness of the incoming wave on milder slopes. - Wave period: on slope 1:2 there is no difference, and on slope 1:3 there is a minor difference with the most overtopping for large wave periods.

Overall, both inﬂuences are neglectable and therefore not further considered here.

Fig. 19. Smooth dike slope with promenade and used parameters.

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0.5

1

1.5

2

2.5

3

3.5

1.00E+00

0

0.5

1.00E+00

R c /H m0 / γ b EUROTOP

1

1.5

2

2.5

3

3.5

R c /Hm0 / γ prom

1.00E-01

1.00E-02

1.00E-04 y = 0.2*exp(-2.28*x)

1.00E-03

q/(g.H m0³)1/2

1.00E-03

q/(g.H m0³)1/2

1.00E-01 1.00E-02

data corrected by EurOtop2007 form 5.29 data original

1.00E-05

Fig. 20. Data on a smooth dike slope with promenade (blue) corrected by EurOtop (2007) reduction factor for a berm (green).

1.00E-04

y = 0.2*exp(-2.28*x) smooth dike slope + B0.33 smooth dike slope + B0.66

1.00E-05

The data in Fig. 21 show that the presence of a promenade has a reducing effect on wave overtopping, slightly increasing with the length B of the promenade, but not as strong as a storm wall. A reduction factor γprom is ﬁtted through the data with the dimensionless promenade length as a parameter. Formula (7) is then transformed to (15) in case of wave overtopping over dike slopes with a promenade clearly above SWL: q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:28 c γ H 3 m0 prom g Hm0 γprom ¼ 1−0:47

B Lm−1;0

! ð15Þ

:

ð16Þ

All results of a smooth dike slope with promenade, grouped by the length of the promenade, are plotted together with the reference formula in Fig. 21 (B0.33 means a promenade length of 33.3 cm etc.). In Fig. 22 these values are corrected by means of formulae (15) and (16). All points are close to the reference line with little scatter. This

0

0.5

1

1.5

2

2.5

3

1.00E+00

R c /H m0

1.00E-02

1.00E-03

1.00E-04

Fig. 22. Smooth dike slope with promenade — corrected values.

proves that Eq. (16) describes the overtopping reduction accurately. The exponential coefﬁcient 2.28/γprom is taken as a normally distributed stochastic variable and has a mean value of 2.55 with standard deviation 0.19. This gives a relative standard deviation of 0.07. 4.5. Smooth dike slope with promenade and storm walll The physical process of a wave hitting a wall is different when a promenade is present in between the top of the dike slope and the wall. For this reason, it is not possible to just multiply the inﬂuence factors γprom (formula (16)) and γv (formula (9)) to account for the combined effect of a promenade and a wall. Therefore 136 model tests were performed on geometries with both a promenade and a wall. A sketch is given in Fig. 23. Rc again includes the slope of the promenade and the height of the storm wall. The range of the parameters of the test program is listed in Table 6. Again no clear inﬂuence of the wave period or the slope angle is noticed in the analysis: - Slope angle: mildest slope 1:3 is slightly more overtopped compared to 1:2, due to the thicker water layer. The inﬂuence is however too weak to include in the reduction factors. - Wave period: for both slopes there was just a little more overtopping measured for the longest wave periods. This is due to the larger layer

q/(g.H m0³)1/2

1.00E-01

3.5

smooth dike slope + B1.00

y = 0.2*exp(-2.28*x) smooth dike slope + B0.33

1.00E-05

smooth dike slope + B0.66 smooth dike slope + B1.00

Fig. 21. Smooth dike slope with promenade — measured values.

Fig. 23. Sketch of a smooth dike slope with promenade and storm wall.

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

Slope angle of the smooth dike slope Mean spectral wave period Dimensionless freeboard Freeboard (top of structure to SWL) Spectral wave height Length promenade Slope promenade Dimensionless promenade length Wall height Dimensionless wall height Water depth at toe of the structure Wave breaker parameter

cot(α) Tm−1,0 Rc/Hm0 Rc Hm0 B – B/Lm−1,0 hwall hwall/Rc d ξm−1,0

2 and 3 1.10–2.25 s 0.80–2.50 0.10–0.28 m 0.075–0.17 m 33.3, 66.7 and 100 cm 1%–2% 0.05–0.41 2, 4, 6 and 8 cm 0.07–0.80 0.36–0.55 m 2.26–4.80

0

0.5

1

1.0E+00

1.5

2

2.5

3

3.5

4

Rc /H mo

1.0E-01

1.0E-02

q/(g ·H mo³)1/2

Table 6 Summary of the characteristics on a smooth dike slope with promenade and storm wall (scale model values).

81

1.0E-03

y = 0.2e-2.28x

1.0E-04

1.0E-05

Smooth dike slope: measured data 1.0E-06

thickness of the water on the promenade as a consequence of the larger volume of water under the crest of longer waves. Also here, the inﬂuence will not be included in the reduction factors since the difference in overtopping measured was too small. The reducing effect of the combination of a wall and a promenade is stronger than the multiplication of both inﬂuences separately. γprom_v is smaller than γprom × γv, leading to smaller wave overtopping discharges, which makes this geometry a very efﬁcient measure to reduce wave overtopping (Fig. 24). A new inﬂuence factor γprom_v is introduced, which can be calculated by formula (18), and has to be incorporated in formula (17).

smooth dike slope + promenade + wall

Fig. 25. Smooth dike slope with promenade and storm wall — measured values.

0

0.5

1

1.5

2

2.5

3

3.5

4

1.0E+00

Rc /Hmo /γ prom_v 1.0E-01

ð17Þ v

ð18Þ

1.0E-03

with γprom as deﬁned in formula (16) and γv in formula (9). Note that this formula is only usable when both a promenade and a storm wall are present. When one of both is missing, formula (17) makes no sense and (15) or (8) should be used. Because 10 different combinations of a promenade (33.3 cm, 66.6 cm and 1 m) and a vertical wall (2 cm, 4 cm, 5 cm, 6 cm and 8 cm) at the end of the promenade were tested in order to investigate the combined effect, all the results are plotted in the same color, together with the reference data in Fig. 25. In Fig. 26 the blue values are corrected by means of formula (17) and (18), which leads to a high correlation to the trend line meaning that a good prediction is obtained. The exponential coefﬁcient 2.28/γprom_v has a mean value of 3.42

1.0E-04

v

1.0E-05

1.0E-06

Smooth dike slope: measured data smooth dike slope + promenade + wall corrected

Fig. 26. Smooth dike slope with promenade and storm wall — corrected values.

with standard deviation 0.46. A relative standard deviation of 0.13 is found. 4.6. Smooth dike slope with promenade, storm wall and parapet

1 0.8

prom_v

γ prom

¼ 0:87 γ prom γv

1.0E-02

q/(g ·H mo³)1/2

q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:28 c γ H 3 m0 prom g Hm0

!

0.6

y = 0.87x

0.4 0.2 0 0

0.2

0.4

0.6

γ prom × γ v

0.8

Fig. 24. Deduction of γprom_v as a function of γprom and γv.

1

Even though the combination of a promenade and a vertical wall is already a very efﬁcient measure, wave overtopping can be further reduced without increasing the height of the wall by adding a parapet to the wall. This combined effect has been investigated by means of 100 tests (Table 7). A sketch of this geometry is given in Fig. 27. It is not possible to simply multiply γprom_v and γpar since that would overestimate the actual reduction in wave overtopping (the calculated reduction factor γprom_v × γpar is lower than the measured one). This doesn't mean that adding the parapet will not further reduce the overtopping discharges; it means that the new parameter γprom_v_par is not as effective as the multiplication of γprom_v and γpar. A parapet intrinsically functions best directly on a slope, since it takes beneﬁt of the upward motion of the water to reﬂect it

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1

Table 7 Summary of the characteristics of the tests performed on a smooth dike slope with promenade, storm wall and parapet (scale model values). cot(α) Tm−1,0 Rc/Hm0 Rc Hm0 B – B/Lm−1,0 hwall hwall/Rc ε λ d ξm−1,0

2 and 3 1.25–2.25 s 0.7–1.9 0.08–0.24 m 0.08–0.17 m 33.3, 66.7 and 100 cm 1%–2% 0.04–0.40 4, 6, 8 cm 0.17–0.80 30°, 45° 0.25–0.375 0.40–0.55 m 2.15–4.77

prom_v_par

Slope angle of the smooth dike slope Mean spectral wave period Dimensionless freeboard Freeboard (top of structure to SWL) Spectral wave height Length promenade Slope promenade Dimensionless promenade length Wall height Dimensionless wall height Height ratio parapet Nose angle parapet (in degrees) Water depth at toe of the structure Wave breaker parameter

y = 1.19x

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

γ prom_v × γ par

1

Fig. 28. Deduction of γprom_v_par as a function of γprom_v and γpar.

back towards the sea. Following reduction factor can be concluded from Fig. 28: γprom

v par

¼ 1:19 γprom

v

γpar :

ð19Þ

Combining Eq. (18) with Eq. (19), the reduction factor can also be calculated as γprom

v par

¼ 1:03 γprom γv γpar :

ð20Þ

The underestimation of γprom_v × γpar almost neutralizes the overestimation of γprom × γv, and the ﬁnal reduction factor γprom_v_par seems to be 3% less efﬁcient than the product of all individual measures. This shows that reduction factors can't be just multiplied with each other without detailed study, especially in situations like these where the physics change between a wave overtopping a structure, and an overtopping bore on the promenade overtopping a storm wall. When one or more of the above parts are missing in the geometry, like no parapet or no berm, formula (19) or (20) cannot be used! The user should then use the correct geometry as mentioned in earlier paragraphs in this paper. The inﬂuences of slope angle and wave period on the overtopping of this geometry have been studied. The same conclusions as for the promenade with storm wall are valid: no inﬂuence of slope angle or wave period is clearly noticeable on dike slope with promenade, storm wall and parapet, despite the parameter γs0,par which accounts for the inﬂuence of the wave period on the geometry dike with storm

wall and parapet. The hydraulics on the geometry dike slope with promenade, storm wall and parapet are different, and make the overtopping over this geometry not strongly dependent on the wave period. The storm wall with parapet at the end of the promenade reﬂects the incoming water layer equally for long as for short waves with only very little difference due to the larger layer thickness on the promenade of long waves overtopping the dike. This is similar to the geometry dike slope with promenade and storm wall. The formula for overtopping over a dike slope with promenade, storm wall and parapet becomes: q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:28 c γ H 3 m0 prom v g Hm0

! ð21Þ par

with γprom_v_par as deﬁned in formula (20). For this geometry, 10 different combinations of a promenade and a storm wall with parapet at the end of the promenade were tested to investigate the combined effect. Therefore, all results on the geometry smooth dike slope with promenade, storm wall and parapet are plotted in the same color (blue), together with the reference data (red) in Fig. 29. In Fig. 30 the blue values are corrected by means of formula (20)

0

0.5

1

1.5

2

2.5

3

3.5

1.00E+00

Rc /Hmo

1.00E-02

1.00E-03

q/(g·H m0³)1/2

1.00E-01

1.00E-04 smooth dike slope 1.00E-05 Fig. 27. Sketch of a smooth dike slope with promenade, storm wall and parapet.

smooth dike slope + promenade + wall + parapet

Fig. 29. Smooth dike slope with promenade, storm wall and parapet — measured values.

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

0

0.5

1

1.5

2

2.5

3

3.5

1.00E+00

Rc/Hmo/γ prom_v_par

1.00E-02

1.00E-03

q/ (g·H m0³)1/2

1.00E-01

1.00E-04 smooth dike slope 1.00E-05

smooth dike slope + promenade + wall + parapet: CORRECTED

Fig. 30. Smooth dike slope with promenade, storm wall and parapet — corrected values.

and (21). As can be seen in Fig. 30, a good prediction is obtained. The exponential coefﬁcient 2.28/γprom_v_par is taken as a normally distributed variable, and has a mean value of 4.13 with standard deviation 0.59. The relative standard deviation is 0.14.

83

4.7. Stilling wave basin (SWB) A last measure proposed in this paper to reduce wave overtopping by modifying the existing crest of dike slopes, is the so-called stilling wave basin (Beels, 2005; Geeraerts et al., 2006). The SWB is made up of a seaward wall, a basin and a landward wall (Fig. 31). The seaward wall is partially permeable to allow the evacuation of the water in the basin. It may consist of a double row of shifted walls (Fig. 32) or a single wall with some gaps. This innovative crest design is based on the principle of energy dissipation: the incoming wave hits the seaward wall and is projected upward, then drops in the spilling basin before hitting the landward wall. At that moment, most of its energy is already dissipated. Consequently, the landward wall is overtopped less in comparison with an unmodiﬁed crest, even though the crest height has not been increased. Many geometrical variations of the SWB have been tested, with over 300 tests with non-breaking wave conditions. The range of hydraulic parameters and geometric variations is listed in Table 8, and is illustrated in Fig. 32 and 33. The front wall of the SWB varied in height from 48 mm to 144 mm, in which the 48 mm above the SWB ﬂoor was kept constant over all variations tested. In Fig. 32 a total wall height of 96 mm is shown; it's the 48 mm below the SWB ﬂoor which has been varied between 0 mm and 96 mm. At a Froude scale of 25, the ﬁxed 48 mm upper part becomes 1.2 m which is a perfect height to lean on, like a railing.

Fig. 31. Simple smooth dike slope (left) compared to a dike slope with SWB built in the crest (right).

Fig. 32. Side view of stilling wave basin (dimensions in mm), Geeraerts et al. (2006).

Table 8 Summary of the characteristics of the tests performed on a smooth dike slope with SWB (scale model values). Slope angle of the smooth dike slope Mean spectral wave period Dimensionless freeboard Freeboard (top of structure to SWL) Spectral wave height Length basin Slope basin Wall height Distance between front walls Water depth at toe of the structure

cot(α) Tm−1,0 Rc/Hm0 Rc Hm0 Lbasin – hfront wall – d

2; 2.5; 3 1.16 s–2.33 s 0.56–2.7 0.10–0.27 m 0.08–0.18 m 48, 36, 24 and 12 cm 2% 48, 72, 96, 120 and 144 mm 4 cm 0.30–0.52 m

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The two separate overlapping walls may be replaced by one wall with small gaps just above the ﬂoor of the basin. This has also been constructed in the city of Ostend, where the engineered and architectural design go hand in hand (Fig. 34). - The height of the front wall and the length of the basin have been studied. While the effect of γprom is smaller than the effect of γv (see Section 4.4), a similar conclusion can be drawn for the SWB: the variation of height of the front wall is dominant, while the effect of the basin's length is present but less pronounced. - The slope angle and the wave period have a minor inﬂuence on the reduction in wave overtopping. As for the vertical wall (γv) and promenade above SWL (γb), both inﬂuences are not strong enough to be included in the formula of the reduction coefﬁcient.

Fig. 33. Plan view of stilling wave basin (dimensions in mm), Geeraerts et al. (2006).

- The blocking coefﬁcient, which is the ratio between the open and the closed part of each row of shifted walls, has an important inﬂuence on the wave overtopping over the landward wall. An optimum between inﬂow (as low as possible) and outﬂow (as high as possible) was a subject of the study. A blocking coefﬁcient of 50% for the most seaward wall, and 65% of the 2nd row wall has been found optimal. To avoid that the wave ﬂows directly into the basin, 20% of each wall part of the ﬁrst row overlaps with a wall part from the second row. To encourage the drainage back towards the sea, the basin has been given a 2% slope.

Since so many variations in the geometry of the SWB are possible, one uniform reduction formula has not been determined. The blocking coefﬁcient, the distance in between the double row walls, the slope near the landward wall, the length of the basin and the height of the front wall all have their inﬂuence on the reduction of wave overtopping. The basic geometry (Lbasin = 48 cm, hfront wall = 96 mm) with the optimal blocking coefﬁcient of 50% (1st row) and 65% (2nd row) has been tested in full detail. A reduction factor of 0.48 is found for this geometry, and can be used to quantify wave overtopping over a dike slope with SWB. In case a speciﬁc geometry is required, the authors suggest determining the reduction capacity by means of scale model tests. q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:28 c Hm0 γSWB g H3m0

ð22Þ

γSWB ¼ 0:48 ð for the selected geometryÞ

ð23Þ

Landward wall constructed as steps

Frontwall with gaps Fig. 34. Stilling wave basin as constructed in the city of Ostend. Picture was taken during low tide ©airmaniacs.be.

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

85

5. Application of formulae in a case study The reductive effect of each measure is dependent on the hydraulic boundary conditions and the geometry of the structure. For example a promenade above the SWL will reduce the small overtopping volumes (large Rc/Hm0) better than the large overtopping volumes (small Rc/Hm0). Large overtopping volumes are best reduced by a combination of promenade and storm wall with or without parapet, or by an SWB. The SWB combines the effect of a promenade and a storm wall, but is capable of reducing the incoming energy even further by means of the double row of front walls and a spilling basin. This reﬂects in the low reduction coefﬁcient γSWB = 0.48 for the presented geometry, without increasing the crest height, whereas the combination of promenade and storm wall the original crest level increases with the height of the storm wall. An example is worked out below to demonstrate the reductive capacity for all proposed measures under 3 different wave heights. The other parameters, such as Rc, Tm−1,0 and cot(α) remain the same throughout the whole example. Based on the deterministic design approach according to EurOtop (2007), the basic formula to calculate wave overtopping discharge over a smooth dike slope for non-breaking waves is q R qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:3 C Hm0 g H3m0

ð24Þ

where the average discharge from formula (2) is increased by about a standard deviation as recommended in EurOtop (2007). For all reduction measures further on in this case study, the standard deviations deducted in this paper are not included. It is up to the designer which level of safety they want to achieve. cot(α) SWL Crest level Rc Tm−1,0 Hm0 Rc/Hm0 q

mTAW mTAW m s m – l/m/s

2 7.00 9.00 2.00 8.2 3.00 0.67 702.47

2.00 1.00 177.64

1.00 2.00 6.30

Wave overtopping over a smooth dike with crest level at +9.00 mTAW and water level at +7.00 mTAW is calculated by means of formula (24). A mean overtopping discharge of 6.3, 177.6 resp. 702.5 l/m/s is found for a storm with wave height 1.00, 2.00 resp. 3.00 m at the sketched smooth dike slope. When a storm wall is added to the slope of the dike, without increasing the freeboard Rc, a reduction factor γv has to be included in the exponential part of formula (24). The formula now becomes q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:3 C : H m0 γv g H3m0

With γv calculated using formula (9). The mean overtopping discharges reduces to 0.92, 67.75 resp. 369.46 l/m/s which is 6.9, 2.6 or 1.9 times less than without the storm wall and the same crest freeboard. Smooth dike slope with storm wall hwall = 1.25 m; hwall/Rc = 0.625 γv q Ratio

– l/m/s –

0.703 369.46 1.90

0.703 67.75 2.62

0.703 0.92 6.87

Further, a parapet nose is added to the same storm wall with ε = 45° and λ = 1/3. Again, no change in crest freeboard Rc. The average discharge can be calculated by means of formula (26).

q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:3 c γ H γ γs0;par 3 m0 v par g Hm0

! ð26Þ

With γv calculated using formula (9), γpar using formula (11) and γs0,par using (14). A reduction ratio of 17.5, 5.7 and 4.1 is achieved in comparison to a smooth dike slope under the same hydraulic conditions. The effect of a parapet is most prominent for larger dimensionless freeboards, which was

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K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

also concluded by Kortenhaus et al. (2001) in the SPP-project. Reductions up to a factor 10 and higher are possible, just like the study by Kortenhaus et al. (2003) and Pearson et al. (2004). Smooth dike slope with storm wall and parapet hwall = 1.25 m; hwall/Rc = 0.625 ε = 45°, λ = 1/3 γv*γpar*γs0,par q ratio

– l/m/s –

0.521 171.55 4.09

0.569 31.07 5.72

0.617 0.36 17.47

When a promenade at crest level is taken into account, it is explained in Section 4.4 to include a reduction factor γprom in the exponential part of the formula, which now becomes ! q Rc 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:3 : Hm0 γprom g H3m0

γprom is calculated by using formula (16). The promenade in this exercise is 20 m wide and has a 2% slope, leading to an increased crest height +9.04 mTAW. To maintain the Rc constant at 2 m (for direct comparison), the water level in this (theoretical) exercise is also increased by 4 cm. The table below shows the reduced mean overtopping discharges. The effect of a promenade is much lower than the effect of other measures. Nevertheless, the overtopping is reduced by a factor of 1.73, 1.38 resp. 1.28. Smooth dike slope with promenade Promenade width = 20 m, slope 2% γprom q ratio

– l/m/s –

0.91 549.40 1.28

0.91 128.79 1.38

0.91 3.64 1.73

When a storm wall of 1.25 m high is present at the end of the promenade, the crest height of the structure is increased to +10.29 mTAW. To maintain the same crest freeboard Rc = 2 m for reasons of comparison, the water level is also increased up to +8.29 mTAW. The mean overtopping discharge is now calculated using the formula q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:3 c γ H 3 m0 prom g Hm0

! v

with γprom_v according to formula (18). Smooth dike slope with promenade and storm wall Promenade width = 20 m, 2% slope Storm wall 1.25 m high. Rc = 2 m, hwall/Rc = 0.625 γprom_v q Ratio

– l/m/s –

0.56 189.71 3.70

0.56 26.13 6.80

0.56 0.15 42.05

This geometry is capable of reducing the wave overtopping discharge to a maximal level so far, with reduction ratio of 42.1, 6.8 and 3.7 compared to the discharge over smooth dike slopes with the same freeboard Rc. Therefore, this geometry is applied a lot at the Belgian coastline to reduce wave overtopping. The storm wall can be constructed as a mobile wall which is only set up when there is a ﬂood risk. When the high tide and storm surge have passed, this mobile wall can be deconstructed and the promenade regains its original function as a touristic promenade, without disturbing the open view at the sea. The discharge can be reduced even further, by adding a parapet to the above structure, with ε = 45° and λ = 1/3. The formula becomes q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:3 c γ H 3 m0 prom v g Hm0

! par

K. Van Doorslaer et al. / Coastal Engineering 101 (2015) 69–88

87

with γprom_v_par according to formula (19). Smooth dike slope with promenade and storm wall Promenade width = 20 m, 2% slope Storm wall 1.25 m high. Rc = 2 m, hwall/Rc = 0.625 Parapet (ε = 45°, λ = 1/3) γprom_v_par q Ratio

– l/m/s –

0.47 113.96 6.16

0.47 12.17 14.60

0.47 0.03 193.98

To conclude, also an SWB with the standard geometry as presented in Fig. 32 is included in this (theoretical) comparison. The average overtopping discharge now has to be calculated by using q ﬃ c pﬃﬃﬃﬃﬃﬃﬃﬃﬃ γ1 : ¼ 0:2 exp −2:3 HRm0 3 gHm0

SWB

Stilling wave basin Lbasin = 12 m hfront wall = 2.4 m γSWB q Ratio

– l/m/s –

0.480 133.42 5.27

0.480 14.70 12.08

0.480 0.04 145.96

The boundary conditions such as wave conditions, space, crest height, etc. will have to decide what kind of crest modiﬁcation can be applied to reduce wave overtopping. To conclude this example, overtopping over a rubble mound breakwater and a vertical caisson breakwater is added to this comparison. Since the γ-factor for the roughness of the rubble mound comes close to the values of reduction factors for SWB or dike slope with promenade and wall with/without parapet, similar reduction in wave overtopping is achieved. The vertical wall on the other hand reduces more than a dike slope with wall or parapet, since the slope is much steeper, but doesn't reduce wave overtopping as affective as an SWB or a dike slope with promenade and storm wall does. Rubble mound breakwater: γf = 0.50 q l/m/s ratio –

151.60 4.63

17.81 9.97

0.06 99.48

Caisson breakwater (vertical wall) q l/m/s Ratio –

127.52 5.51

24.04 7.39

0.12 54.60

6. Conclusions In order to investigate the reduction in wave overtopping over smooth dike slopes, over 1000 tests with non-breaking waves have been carried out on 7 different types of geometry, with many variations in both geometrical and hydraulic parameters for each geometry. The analysis of this large data set shows that each proposed reduction measure can be included in the wave overtopping formula to predict wave overtopping. Several reduction factors have been proposed in this paper, each of them applicable for a speciﬁc geometry and within the ranges of parameters given in the tables. Each individual reduction factor should be included in the formula (31) to calculate the average overtopping discharge for non-breaking waves. According to EurOtop, 2007, the deterministic design approach with exponential coefﬁcient 2.3 is used: q R 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 0:2 exp −2:3 C : Hm0 γreduction g H3m0

ð31Þ

The different reduction factors are: - γv (formula (9)) deﬁnes the reduction due to a storm wall on the crest, related to the height of the storm wall. - γpar (Formula (11) and (12)) gives the extra reduction that can be obtained by adding a parapet nose to the storm wall on the

-

-

-

-

crest. This inﬂuence factor is a function of the parapet angle ε and height ratio λ. It should always be combined with γv. The parapet is the only geometry where a clear inﬂuence of the wave period on the overtopping discharges was noticed. Therefore, also γs0,par (formula (14)) should be combined with γpar and γv. γprom (Formula (16)) is the reduction factor due to a promenade at the crest level. γprom is related to the length of the promenade, which is expressed dimensionless by means of the wave length. γprom_v (Formula (18)) represents a storm wall at the end of a promenade. This is a very efﬁcient reductive measure, which reduces wave overtopping more than multiplying their separate inﬂuence factors γprom and γv. γprom_v_par (Formula (19)) is the reduction factor when a promenade with storm wall and parapet are present. A parapet is less efﬁcient compared to when it's positioned at the dike slope, but still reduces the average overtopping discharge more than a storm wall without parapet at the end of a promenade. A stilling wave basin is also an efﬁcient way to reduce wave overtopping without increasing the crest level. The combined effect of (a double row shifted) storm walls, a promenade and an energy dissipating basin leads to a low reduction coefﬁcient γSWB. Due to the multiple geometric variables, a generic reduction formula was not deducted, but increasing the height of the front wall and choosing an optimal blocking coefﬁcient turned out to be the most efﬁcient ways to optimize the reduction of

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wave overtopping over the landward wall, which is in fact the crest of the original dike slope. While analyzing the data, for each geometry it was studied whether the slope angle of the dike slope and/or the wave period have their inﬂuence on the measured wave overtopping. - In general there was a minor inﬂuence of the slope angle 1:2 versus 1:3: for a normal dike slope or a dike slope with a promenade, the mildest slope was overtopped slightly more than steeper ones (least reﬂection on mildest slope and the thickness of the runup/ overtopping water layer is a bit larger). For dike slopes with an SWB, a storm wall or parapet at the crest, the steepest slope was overtopped more (steeper slope have more upward velocity in the runup, which allows more overtopping). For dike slopes with a promenade in combination with a storm wall with or without parapet, the same inﬂuence as without storm wall was found: slope 1:3 is slightly more overtopped than slope 1:2. However, the inﬂuence of the slope angle was in any of the 7 geometries too small to include in the overtopping formulae. - In most of the geometries the largest overtopping discharges were measured for tests with the largest wave period. For all geometries, except for the one with the parapet directly placed at the crest of the dike slope, the inﬂuence was however too small to include in the overtopping formula. For the case dike slope with storm wall and parapet, an extra parameter γs0,par is included in the overtopping formula, since the effect of the wave period was clearly noticeable in the results: long waves have the tendency to ﬁll the space under the parapet's nose with water, after which the parapet acts as a normal storm wall and is more easily overtopped compared to short waves who encounter a parapet. When a promenade is present in front of the parapet, this effect disappears.

Acknowledgments The authors acknowledge the assistance of the technical staff of the Coastal Engineering Laboratory at Ghent University. A large data set with many geometric variables was only possible to obtain with their effort. Master thesis students Tobias Boderé, Gilles Vanhouwe (Boderé and Vanhouwe (2010)), Sarah Audenaert and Valerie Duquet (Audenaert and Duquet (2012)) performed a great effort in data analysis during and after completion of their thesis. At last, discussions with prof. A. Kortenhaus and Dr. Ir. D. Vanneste are also acknowledged. References Allsop, N.W.H., Besley, P., Madurini, L., 1995. Overtopping performance of vertical walls and composite breakwaters, seawalls and low reﬂection alternatives, Final report of Monolithic Coastal Structures (MCS) project. University of Hannover. Audenaert, S., Duquet, V., 2012. Golfovertopping over zeedijken - krachten op stormmuur (Wave Overtopping Over Sea Dikes; Forces On a Storm Wall). (Master thesis). Ghent University.

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Crest modifications to reduce wave overtopping of non-breaking waves over a smooth dike slope

Crest modifications to reduce wave overtopping of non-breaking waves over a smooth dike slope

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