Faculty of Engineering Technology
The effects of transitions on wave
overtopping flow and dike cover
erosion for flood defence reliability
Literature report
Vera M. van Bergeijk, MSc
May 2018
CE&M research report 2018R-002/WEM-002 ISSN 1568-4652
THE EFFECTS OF TRANSITIONS ON WAVE
OVERTOPPING FLOW AND DIKE COVER EROSION
FOR FLOOD DEFENCE RELIABILITY
Vera M. van Bergeijk, MSc
May 2018
Supervisors:
prof. dr. S.J.M.H. Hulscher dr. J.J. Warmink
Marine and Fluvial Systems University of Twente
CE&M research report 2018R-002/WEM-002 ISSN 1568-4652
Acknowledgements
This work is part of the research programme All-Risk, Implementation of new risk standards in the Dutch flood protection program (P15-21), which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO). I thank Wim Kanning for assistance with the probability of failure exercise and Marcel van Gent for his useful feedback. I would also like to thank my supervisors, Jord Warmink and Suzanne Hulscher, for their feedback and guidance.
Contents
1 Introduction 1 1.1 Context . . . 1 1.2 Background . . . 1 1.3 Research questions . . . 2 2 Wave overtopping 3 2.1 Overtopping flows . . . 4 2.1.1 Wave run-up . . . 42.1.2 Average overtopping discharge . . . 5
2.1.3 Wave overtopping volume . . . 7
2.2 Overtopping parameters . . . 9
2.2.1 Flow velocity and layer thickness . . . 9
2.2.2 Overtopping time and instantaneous parameters . . . 12
2.3 Summary . . . 14
3 Erosion due to wave overtopping 15 3.1 Initiation of erosion . . . 16
3.2 Erosion depth . . . 18
3.2.1 Maximum erosion depth . . . 18
3.2.2 Depth-dependency . . . 19
3.3 Empirical relations . . . 20
3.3.1 Flow velocity . . . 20
3.3.2 Shear stress . . . 21
3.3.3 Work . . . 22
3.3.4 Excess wave volume . . . 23
3.4 Conclusions . . . 24
4 Models 25 4.1 Idealised wave model and erosion model . . . 25
4.1.1 Flow velocity and layer thickness on the dike crest . . . 25
4.1.2 Flow velocity and layer thickness on the landward slope . . . 26
4.1.3 Erosion model . . . 29
4.1.4 Required parameters . . . 29
4.2 Numerical hydrodynamic models . . . 30
4.2.1 DualSPHysics . . . 30
4.2.2 OpenFOAM . . . 30
4.2.3 Model domain . . . 31
4.3 Coupling to erosion model . . . 32
4.4 Summary . . . 34
5 Transitions 35 5.1 Classification of transitions . . . 35
5.2 Experimental results of the wave overtopping simulator . . . 37
5.2.1 Cumulative load model accounting for transitions . . . 39
5.2.2 Changes in inclination . . . 39
5.2.3 Changes in cover type . . . 40
5.2.4 Embedded objects . . . 41
5.3 Summary . . . 42
6 Probability of dike cover failure due to wave overtopping 43 6.1 Reliability methods . . . 43
6.2 Example: Failure probability for the experiment at Millingen a/d Rijn . . . 44
6.2.1 The limit state function . . . 44
6.2.2 Calculation of parameters . . . 45
vi
6.2.4 Results . . . 46
7 Conclusions 49
1
Introduction
1.1 Context
This literature review is part of the PhD research of the author, which focusses on the effects of transitions on wave overtopping flow and dike cover erosion. The PhD research is part of the All-Risk program, which investigates flood risk and the reduction of flood risk by innovative measures such as flood defences. The program was initiated after the adaptation of a new probabilistic risk approach for the management of the flood defences by the Dutch Flood Protection program (HWBP) in 2014. Within the All-Risk program the reliability of the flood defences following the new approach are determined, and the PhD research of the author investigates the reliability of the flood defences related to wave overtopping.
1.2 Background
Two-thirds of the Netherlands is endangered of flooding without dikes (PBL, 2009). These highly populated areas are protected by the Dutch flood defence system, where earthen dikes are one of the main flood defence structures. However, the sea level rises and peak river discharges increase due to climate change, calling for more powerful flood defences. During storms, extreme high water levels can occur when a high tidal elevation occurs in combination with a positive storm surge. The surge is caused by a combination of the barometric effect of the low pressure system and wind set-up. A combination of high water levels in combination with increased wave heights during storms causes waves to overtop the dikes. Wave overtopping is one of the main mechanisms causing dike breach (Sharp et al., 2013). Waves that overtop the dike crest, run down on the landward slope and cause erosion on landward side of the dike. Frequent wave overtopping results in erosion of the dike cover. Once the cover is eroded, the core material of the dike structure starts to erode resulting in weakening of the dike and, in the end, in a dike breach (Oumeraci et al., 2005).
Figure 1: A photo of wave overtopping of a grass-covered dike at Hartlepool, UK. Source: HR Wallingford
A photo of wave overtopping during a storm is shown in Figure 1. Wave overtopping is a complex process to describe, since it is very random in time, space and volume. Only a percentage of the waves will overtop the dike crest. The shape of the wave transforms as it runs up the waterside slope of the dike, flows over the crest and runs down again on the landward side of the dike. The wave shape transformation makes it hard to describe wave overtopping theoretical. A few attempts of a theoretical description were made by Van Gent (2002), Schüttrumpf and Oumeraci (2005) and
2 1.3 Research questions
Dean et al. (2010), but some of these formulas were based on empirical models and include a lot of calibration constants. For practical use, empirical models like the formulas of the European Overtopping manual (EurOtop manual, Van der Meer et al., 2016) are used. Most empirical models are based on average quantities, like the average discharge or average wave overtopping volume. These quantities fail to describe the instantaneous character of wave overtopping. The instantaneous character of wave overtopping is simulated by detailed hydrodynamic models that calculate the overtopping volume and flow velocities of every wave separately. This is computational demanding, but numerical models are a valuable tool to calculate the erosional effect of wave overtopping, since turbulence plays an important role in the erosion due to wave overtopping. The flow velocities vary highly in time and the erosion of the dike cover is mostly caused by the impact of the maximum flow velocity during a short time period within the overtopping duration (Van der Meer et al., 2010). For this reason it is important to take the time dependency of the velocity into account, which requires a detailed hydrodynamic model.
1.3 Research questions
This literature review gives an overview of the current knowledge about wave overtopping and the erosional effects. The five research question that will be answered in this literature review are:
1. What parameters are important to describe wave overtopping and what methods are used to determine the amount of wave overtopping?
2. How can the erosion due to wave overtopping be described and how is the erosion affected by the dike cover quality?
3. What type of numerical models are used to describe the wave overtopping flow in detail? 4. What transitions occur at a dike and how do they influence the flow and erosion?
5. How is the probability of dike cover failure due to wave overtopping determined?
Chapter 2 describes the important parameters and the different methods that are used to determine
the amount of wave overtopping. These are necessary to describe the erosional effects of wave
overtopping, which are discussed in Chapter 3. Chapter 4 will give an overview of the hydrodynamic models that could be used for simulation of overtopping flow and how these hydrodynamic models can be coupled to a dike cover erosion model. It is already known that most erosion due to wave overtopping occurs at transitions of the dike structure, for example a transition in cover type from grass to asphalt in case of a road (Steendam et al., 2014; Bomers et al., 2018). These transitions are classified and their effect on the overtopping flow and the dike cover erosion are discussed in Chapter 5. Chapter 6 shows an example of the probability of failure due to dike cover erosion. When the dike cover erodes, the dike weakens and a breach might happen. The probability of failure related to dike cover quantifies the probability that the dike breaches due to the erosion of dike cover. The conclusions of the literature review are found in Chapter 7.
2
Wave overtopping
A typical dike structure is show in Figure 2. The core material of Dutch dikes consists of sand and clay (Wondergem et al., 1997). The clay layer is covered by grass on the crest and landward slope. The slope on the water side is usually enforced with stones or asphalt to withstand the forces of waves and currents. The height difference between the still water level (SWL) and the crest is called
the crest freeboard Rc. A berm can be located at the landward side or the waterside of the dike.
The toes of the dike structure are the points where the slope meets the foreshore or the protected lowland.
Figure 2: A typical structure of a dike protecting the lowland from the water. The crest freeboard Rc is the height difference between the still water level (SWL) and the crest.
A wave field consists of waves with various heights and periods. The wave field can be characterized
by the significant wave height Hs, which is defined as the mean of the highest one-third of the waves.
The spectral wave height Hm0 is determined from the variance energy spectrum E(f ), which shows
the distribution of energy over the frequencies (Holthuijsen, 2007). In most cases, the significant wave height is approximately equal to the spectral wave height, but the two wave heights are distinguished in the formulas in this study due to difference in the method of calculation. The spectral wave height is defined as
Hm0= 4
√
m0 (1)
with m0 the zeroth-order moment. The moments of the variance density spectrum are defined as
mn =
Z ∞
0
fnE(f ) df for n = .., −2, −1, 0, 1, 2, .. (2) The moments of the variance density spectrum also hold information about the frequencies in a wave
record and can be used to determine the spectral wave period Tm−1,0,
Tm−1,0=
rm
−1
m0
. (3)
The mean wave period Tm, defined as the mean of the wave periods, and the peak period Tp, which is
the inverse of the frequency with the highest energy density, are also used in the formulas describing wave overtopping.
When waves approach the dike from the offshore zone (zone 1, Figure 3), the wave shape trans-forms. The wave transformation from the foreshore to the landward slope is shown in Figure 3. The waves approach the dike over the foreshore, where the waves shoal and break (zone 2). In zone 3, the waves run-up and run-down over the waterside slope. The overtopping water can no longer flow
back to the waterside at the start of the crest, at point xC = 0. In that case the water flows over
the crest (zone 4) and runs down on the landward slope (zone 5). In Figure 3, the wave run-up
height Ru,2% [m] is indicated, which is the wave run-up height that is exceeded by 2% of the incident
4 2.1 Overtopping flows
exceeding this level is related to the number of incoming waves and not to the number that runs up the slope (Van der Meer et al., 2016).
The effect of wave overtopping can be separated in the water load, defined as the amount of water that overtops the dike, and the erosional effect on the landward slope. The water load is described by the wave run-up, the average discharge q or the overtopping volume V . The wave run-up determines
the condition for wave overtopping. When the run-up height Ru,2% is larger than the free crest
height Rc, the waves overtop. The amount of wave overtopping can be quantified by the average
discharge q or the overtopping volume V and is used to determine the tolerable overtopping for design criteria. Both methods will be described in this chapter. The erosional effects of wave overtopping are described in Chapter 3 and the overtopping parameters used to determine the erosion are given in Section 2.2 of this chapter.
Figure 3: Wave transformation during wave overtopping from the offshore zone (1), where the waves start
(2) shoaling and breaking, (3) run-up and run-down, (4) flow over the crest and (5) run down on the land ward zone. The wave run-up height Ru,2% is indicated in zone 4. The variables x0, xA, xC and xB are defined as the horizontal distance from the start of the waterside slope, the still water level of the waterside slope, the crest and landward slope, respectively. Figure adapted from Schüttrumpf and Oumeraci (2005).
2.1 Overtopping flows
2.1.1 Wave run-up
The formula of the wave run-up height Ru,2% orginates from the study of Hunt (1959). The
dimen-sionless run-up height Ru,2%/Hs is a function of the Iribarren number ζ, which describes the type
of wave breaking. The Iribarren number ζp based on the deep-water wave length L0 and the peak
period Tp is given by ζp= tan α q Hs/L0,p with L0,p= g T2 p 2π (4)
where g is the gravitational acceleration (9.81 m/s2) and α is the bed slope angle, in this case the
angle of the waterside slope. The Iribarren number ζmbased on the mean energy wave period Tm−1,0
is also used in various studies, defined as
ζm= tan α q Hm0/L0,m with L0,m = g T2 m−1,0 2π (5)
Several experiments and overtopping test have been performed since Hunt (1959). These data is used by Van Gent (2002), Pullen et al. (2007) and Van Damme (2016) to develop a wave run-up formula. The empirical relations for the dimensionless run-up height of the various studies are shown in Table 1. As seen in the empirical relations, the run-up depends on the sea state and the dike structure. For example, the wave height distribution and wave energy spectra at the toe of the dike
Table 1: Empirical relations of the dimensionless wave run-up height Ru,2%/Hsas function of the Iribarren number ζ of various studies.
The dimensionless run-up height Constants
Hunt (1959) Ru,2% Hs = c1ζp = c1 tan α √ Hs/L0,p wave spectra: c1= 1.5 regular waves: c1= 1.0 Van Gent (2002) Ru,2% Hs = c0γfζp for ζp ≤ c1/2 c0 γf(c1− 0.025 c2 1 c0ζp ) for ζp > c1/2 c0 γf = 0.7, c0 = 1.95, c1 = 5.2 EurOtop Manual
Van der Meer et al. (2016)
Ru,2% Hm0 = min 1.65 γβγfγbζm, γfγβ 4.0 −√1.5 ζm Van Damme (2016) Ru,2% Hs = 0.93 (−ln(0.02)) 1 2 = min γβγfγbζm, γ1.65fγβ 4.0 − √1.5 ζm
is determined by the foreshore and the water depth at the toe of the dike (Van Gent, 2002). When the waves approach the dike under an angle, the wave run-up and wave overtopping is reduced. This
effect is incorporated in formulas using the reduction parameter γβ. The reduction factor γf accounts
for the reduction of wave run-up and overtopping by the roughness of the waterside slope compared with a smooth waterside slope. A berm on the waterside slope can also lead to a reduction of the
wave run-up and wave overtopping, which is incorporated in the formulas by the reduction factor γb.
The empirical relations for the dimensionless run-up height in Table 1 are plotted in Figure 4 under
the assumption that the significant wave height Hs is equal to the spectral wave height Hm0, which
holds in most cases. The up height according to Hunt (1959) is significantly higher than the run-up height determined by the other formulas. The run-run-up height of Van Gent (2002) is comparable to Van Damme (2016), contrary to the run-up height of the EurOtop manual 2007 (Pullen et al., 2007) which is smaller than the other studies.
The empirical relations for the wave run-up height are determined using the mean value approach. This means that the formulas as given can be used for prediction or comparison with data using the mean values of the stochastic parameters (Van der Meer et al., 2016). For the design or assessment of dikes, the calculation needs to be more conservative and a safety factor needs to be taken into account. This is called the design or assessment approach and the uncertainty of the prediction is included using the mean value plus the standard deviation of the stochastic parameters. For example, the constant 1.65 in the wave run-up formula of Van der Meer et al. (2016) increases to 1.75 in case of the design or assessment approach.
2.1.2 Average overtopping discharge
The average discharge q is given in l/s per meter width and is used in design criteria to determine the tolerable amount of wave overtopping. The tolerable overtopping discharge is defined as the discharge that does not lead to significantly damage to the dike crest or landward side slope and is determined from the mean discharge over a time period. In the Dutch guidelines, the tolerable discharge depends on the soil and cover type of the dike, as well as the wave height. The tolerable average overtopping for the design of a dike with a grass-covered crest and landward slope are shown in Table 2. However, wave overtopping experiments on real Dutch dikes showed that for an average discharge below 30 l/s per m the grass cover of the landward slope never fails (Van der Meer, 2008). This means that the design values are very conservative.
6 2.1 Overtopping flows 0.5 0.6 0.7 0.8 0.9 1 2 2.5 3 3.5 Hunt (1959) van Gent (2002) EurOtop (2007) van Damme (2016) 1.7 1.8 1.9 2 2.1 2.2 2.3 2 2.5 3 3.5
Figure 4: The empirical relations of the dimensionless wave run-up height Ru,2%/Hsshown in Table 1 are plotted as function of the wave height Hsand the Iribarren number ζ. The relations are plotted for regular waves and using γβ= γb= γf = 1, Tm= 5 s, α = tan(1/4) and γf = 0.7 (used in experiments of Van Gent (2002)) . It is assumed that Hs= Hm0.
The formula for the average discharge was originally proposed by Van der Meer and Janssen (1995)
and is a Weibull-shaped function with the dimensionless overtopping discharge q/qgH3
m0as function
of the relative crest freeboard Rc/Hm0.
q q gH3 m0 = a exp − b Rc Hm0 c for Rc ≥ 0 (6)
The constants a, b and c are empirically determined from experimental data using the mean value approach and stated in Table 3 for the study of Van der Meer and Janssen (1995) and the EurOtop formulas of the manual in 2007 and 2016. The formulas can also be applied to vertical structures and
the reduction factor γvaccounts for the effects of a vertical wall on the waterside slope. The formulas
of Van der Meer and Janssen (1995) include the reduction factor for shallow water γh. The average
discharge of the three studies is plotted in Figure 5, again under the assumption Hs = Hm0. The
formulas for the average discharge by the EurOtop manuals show similar behaviour. The average discharge formula of Van der Meer and Janssen (1995) coincides with the EurOtop manuals in case of nonbreaking waves (ζ > 0), but is significantly lower in case of breaking waves (ζ < 2).
Table 2: Limits for wave overtopping used for structural design of dike with a grass-covered crest and
grass-covered landward slope. Values are adapted from Van der Meer et al. (2016).
Average discharge q (l/s per m) Max volume Vmax (m3 per m)
Maintained and closed grass cover
Hm0= 1 − 3 m
5 2-3
No maintained grass cover (e.g. open spots, moss)
Hm0= 0.5 − 3 m
0.1 0.5
Hm0< 1 m 5-10 0.5
Table 3: The empirical relations for the dimensionless overtopping discharge q/pgH3
m0 as function of the relative crest freeboard Rc/Hm0for various studies.
The dimensionless average discharge
Van der Meer and Janssen (1995)
q √ gH3 s = √0.06 tan αζp· exp −5.2 Rc ζp·HS·γb·γf·γβ·γh for ζp < 2 0.2 exp−2.6 Rc HS·γb·γf·γβ·γh for ζp > 2 EurOtop 2007 Pullen et al. (2007) q √ gH3 m0 = √0.067 tan αγb· ζm· exp −4.75 Rc ζm·Hm0·γb·γf·γβ·γv
with a maximum of: √ q
gH3 m0 = 0.2 · exp−2.6 Rc Hm0·γb·γf·γβ·γv EurOtop 2016
Van der Meer et al. (2016)
q √ gH3 m0 = √0.023 tan αγb· ζm· exp −2.7 Rc ζm·Hm0·γb·γf·γβ·γv 1.3
with a maximum of: √ q
gH3 m0 = 0.09 · exp −1.5 Rc Hm0·γb·γf·γβ·γv 1.3 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 10 -3
Van der Meer (1995) EurOtop (2007) EurOtop (2016) 1.4 1.6 1.8 2 2.2 2.4 0 0.2 0.4 0.6 0.8 1 1.2 10 -3
Figure 5: The empirical relations of the dimensionless average discharge q/pgH3
s shown in Table 3 are plotted as function of the wave height Hsand the Iribarren number ζ. The relations are plotted for regular waves and using γf = γβ= γb = γv= γh= 1, Tm= 5 s, Rc= 2 m and α = tan(1/4).
2.1.3 Wave overtopping volume
The overtopping volume V is defined as the amount of water that comes over the crest of the dike in
m3 per wave per meter width (Van der Meer et al., 2016). A mean overtopping discharge gives no
indication of how many waves overtop and how much water is overtopped by individual waves. Less overtopping waves, but with large overtopping volumes, cause more erosion than many overtopping waves with small volumes (Van der Meer et al., 2010), although the average discharge is the same. These two cases can be differentiated when the overtopping volumes is used to describe the water load. The distribution of individual wave volumes can be described by a probability density function. Franco et al. (1995) and Van der Meer and Janssen (1995) proposed the Weibull distribution to describe the overtopping volumes of individual waves. The exceedance distribution describes the
8 2.1 Overtopping flows
percentage of wave volumes PV that will exceed a specified volume V
PV(Vi≥ V ) = exp " − V a b# · (100%). (7)
where the scale factor a normalises the distribution and the non-dimensional shape factor b defines the extreme tail of the distribution. There have been various studies to determine the factors a and b, which are summarised in Table 4. Van der Meer and Janssen (1995) started with a constant shape factor b. However, the data of various experiments with a large range of wave heights and
crest heights showed a dependency of the shape factor b on the dimensionless crest freeboard Rc/Hm0
and the angle of the waterside slope α (Victor, 2012). Hughes et al. (2012) adapted the function of the shape factor b to get rid of the slope dependency. However, the function of Victor (2012) is preferred in cases of steep slopes and low relative freeboard. In all studies, the scale factor a depends on the average discharge q. This makes it not possible to use the discharge and volume method separately to determine the tolerable amount of wave overtopping. Van Damme (2016) used a Rayleigh distribution to describe the overtopping volumes that does not depend on the discharge.
Following Van Damme (2016), the Rayleigh exceedance distribution function FV for overtopping
waves is given by FV(V ) = exp − fvsin2α cdcosα V + 2hc fvsin2α cdcosα V 1/2 2 σ2H2 s2 · (100%). (8)
where σ = 0.658, cd is a parameter that account for the energy loss of the overtopping wave, fv is a
factor that describes the shape of the wave front, the crest height hc and the parameter is used to
determine the run-up height, see Table 1.
The percent exceedance as a function of volume is shown in Figure 6 for the different studies in Table 4 and Van Damme (2016). The same parameter values are used as in Figure 5, except the
crest freeboard Rc was decreased to 1.5 m. The other parameter values used are a significant wave
height Hs = 1.0 m, run-up height Ru,2% = 2.25 m, average discharge q = 4 l/s per m, number of
waves Nw = 1000, energy loss parameter cd = 0.8 and wave shape factor fv = 3.0. The percent
ex-ceedance determined from Van der Meer and Janssen (1995), Van der Meer et al. (2010) and Hughes et al. (2012) are similar. The percent exceedance by Victor (2012) is developed for cases with a steep slope and low relative freeboard. The formulas are plotted outside this range, which can be an explanation for the difference between Victor (2012) and the other studies. Van Damme (2016)
Table 4: The values of the Weibull scale factors a and shape factor b of various studies to describe the
percent exceedance of wave volumes.
a b
Van der Meer and Janssen
1995 0.84 (q Tm/Pov) 0.75
Van der Meer et al. 2010 0.84 + 1.2 (NwPov)−0.8 (q Tm/Pov) Pov = exp[−( √ ln 0.02Rc/Ru2%)2] 0.75 Victor 2012 1.13 tanh(1.32 b)(q Tm/Pov) exp(−2.0 RC Hm0) + 0.56 + 0.15 cot α Hughes et al. 2012 -h exp(−0.6 Rc Hm0) i1.8 + 0.64
uses a Rayleigh distribution instead of a Weibull function resulting in a higher percent exceedance for larger wave volumes.
The water load for design criteria are expressed in a maximum overtopping volume Vmax. Using
the percent exceedance function, the chance of exceeding the maximum overtopping volume can be calculated for several wave conditions. The overtopping limits for design criteria are found in Table 2.
0
5
10
15
10
-110
010
110
2Van der Meer (1995) Van der Meer (2010) Victor (2012) Hughes (2012) Van Damme (2016)
Figure 6: The percent exceedance of overtopping volumes of various studies summarized in Table 4 and
Van Damme (2016). The parameter values used are γf = γβ = γb = γv = 1, Tm = 5 s, Hs = 1.0 m,
Ru,2% = 2.25 m, Rc= 1.5, α = tan(1/4), q = 4 l/s per m, Nw= 1000, cd = 0.8 and fv= 3.0.
2.2 Overtopping parameters
2.2.1 Flow velocity and layer thickness
Wave overtopping is described in more detail using expressions for the flow velocity u and layer thick-ness h of the overtopping tongue. The shape of the overtopping tongue changes significantly from the waterside slope (A), to the crest (C) and the landward slope (B) resulting in separate formulas
for the layer thickness (hA, hC, hB) and flow velocity (uA, uC, uB) at the three locations. Several
re-searchers have studied the flow velocity and layer thickness of the overtopping tongue. Their results
are summarized in Table 5. The variables xA, xC and xB are defined as the horizontal distance from
the start of the waterside slope, the crest and landward slope, respectively (see Figure 3).
The changes in the flow velocity and the layer thickness over the dike are illustrated in Figure 7
for a crest width Bc of 2.5 m and a symmetric dike with equal waterside and landward slope
α = ϕ = tan−1(1/4) and horizontal lengths of 5 m. The flow velocity and layer thickness at the waterside slope, crest and landward slope are plotted using the formulas of Van Gent (2002), Schüttrumpf and Oumeraci (2005) and the EurOtop manual (Van der Meer et al., 2016), because these are the only studies with a formula for the crest and the landward slope. Van Damme (2016) also has formulas for all three locations, but the formula for the crest and the landward slope consist of many unknown parameters. An attempt was done to plot the formulas of Van Damme (2016) but the parameters turned out to be very sensitive resulting in unphysical results for the chosen
param-10 2.2 Overtopping parameters T able 5: F orm ulas for the la y er-thic kne ss of the o v ertopping tongue h and the depth-a v eraged flo w v elo cit y of the o v ertopping tongue u ofv arious studies for the w atersi de slop e (h A , uA ), the crest (h C , uC ) and the landw ard slop e (h B , uB ). All studies use th e la y er thic kness h2% and flo w v elo c it y u2% exceeded b y 2% of the inciden t w a v es, ho w ev er th e studies of Sc hüttrumpf use th e exceedan ce of 50%, indicated b y hA, 5 and uA, 5 . The v ariables xA , xC and xB are defi ned as the horizon tal distance from the start of the w aterside slop e, the c rest and landw ard slop e, resp ectiv ely (see Figure 3). Study W aterside slop e (w a v e run-up) Crest Landw ard slop e Sc hüttrumpf 2001 uA, 5 (x A ) = 0 .75 q π g ( Ru, 2% − tan αx A ) 2 -V an Gen t 2002 -hC ∗ = h 0 Ru, 2% − Rc γf , uC ∗ = u 0 q g ∗ ( Ru, 2% − Rc ) γf sea w ard sid e: h 0= 0 .15 , u 0= 1 .3 land side: h 0= 0 .1 , u 0= 1 .7 γf ,c 1+0 .1 Bc /H s ) Dep endin g on xC hC (x C ) = hA exp (− 0 .4 xC /B c ) uC (x C ) = uA exp (− 0 .5 fc xC /h C ) hB (x B ) = hC uC α/β +( uC − α/β ) exp( − 3 αβ 2 cos ϕ xB ) uB (x B ) = α β + (u C − α )β exp( − 3 α β 2cos ϕ xB ) α = 3 √ g sin ϕ , β = 3 p fL / 2 hC uC Sc hüttrumpf 2005 hA, 5 (x A ) = 0 .028 Ru, 2% tan α − xA uA, 5 (x A ) = 0 .94 q g ( Ru, 2% − tan αx A ) Hs hC (x C )/h C (0) = exp (− 0 .75 xC /B c ) uC (x C )/u C (0) = e xp( − f xC 2 hC ), f = 0 .0058 uB (x B ) = uC + k2 tanh( k1 t/ 2) 1+ uC tanh( k1 t/ 2) /k 2 t( xB ) = − uC g sin ϕ + q u 2 C g 2 sin 2 ϕ + 2 xB g hB = uC hC uB , k1 = p 2 f g sin ϕ/h B , k2 = k1 hB /f Bosman 2007 -hC (x C )/h C (0) = 0 .81 exp (− 15 xC /γ c L0 ,m ) uC (x C )/u C (0) = e xp (− 0 .042 xC /γ c hC ) -V an der Meer 2010 -hC = 0 .13( Ru, 2% − Rc ) uC (x C ) = 0 .35 cot α exp(1 .4 xC L0 ,m ) p g (R u, 2% − Rc ) -Hughes 2012 -uC = 1 .53 √ g hC -EurOtop 2016 hA (x A ) = ch (R u, 2% − tan (α )x A ) uA (x A ) = cu p g (R u, 2% − tan (α )x A ) hC = 2h 3 A uC (x C )/u C (0) = e xp( − 1 .4 xC /L 0 ,m ) same as Sc hüttrump f (2005) V an Damme 2016 hA (x A ) = ch Ru, 2% − tan( α ) xA tan( α ) uA (x A ) = q 0 .9961 g Ru, 2% − tan( α ) xA Ru, 2% hC (x C ) = h∗ + (R u, 2% − h∗ ) exp 2 h∗ ( qt − S q ) xC g ( h 3 ∗− R 3 c) h 2 ∗ = − cf q 2 qt − S q , S = 1 ρ dρ dt , qt = ∂ q ∂ t . hB (x B ) = h∗ + d∗ exp (3 g sin ϕh ∗ 2 − qt h 2)∗ cos ϕ xB h 3 ∗− d 3 c d∗ = (h B (x B = 0) − h∗ ), dc = q 2/ cos ϕ
0
2
4
6
8
10
12
0
2
4
6
0
2
4
6
8
10
12
0
0.2
0.4
Van Gent (2002)
Schüttrumpf (2005)
EuroTop (2016)
Figure 7: The transformation of the flow velocity u and layer thickness h as function of the horizontal
distance x for a symmetric dike with a crest width Bc = 2.5 m, slope angles α = ϕ = tan−1(1/4) and slope widths of 5 m. The formulas of three studies (see Table 5) are plotted using mean wave period Tm= 5 s, wave height Hs= 1.0 m, crest height Rc = 1.5 m, run-up height Ru,2% = 2.25 m and roughness reduction factor γf = 1. The parameters used in the formula of Van Gent (2002) are set to fc =0.05 and fL= 0.005, and the EurOtop parameters (Van der Meer et al., 2016) are set to ch= 0.2 and cu=1.4.
eters. The formulas of Van Gent (2002) show a rapid decrease in flow velocity over the crest, which
is due to unknown value of fc that is set to 0.05 in the figure. The difference between the formulas
of Schüttrumpf and Oumeraci (2005) and Van der Meer et al. (2016) is caused by a difference in the empirically determined constants in the run-up formulas. Also, the run-up formulas of Van der Meer et al. (2016) describe the 2% exceedance flow velocities and layer thickness leading to higher flow velocities and higher layer thickness compared to the formulas of Schüttrumpf and Oumeraci (2005) that describe the 50% exceedance flow velocity and layer thickness.
The plotted formulas show a decrease of the layer thickness and flow velocity on the waterside slope. The overtopping wave is slowed down due to the roughness of the slope and gravity works in opposite direction of wave run-up. The wave volume is not conserved during run-up, since part of the wave does not reach the crest and runs down the waterside slope. The layer thickness and flow velocity decrease over the crest due to bottom friction. The average discharge is not conserved over the crest, because the 2% exceedance flow velocity and layer thickness do not occur at the same time and for the same waves. The layer thickness decreases on the landward slope, however the thickness becomes almost constant at the end of the landward slope. The flow velocity increases at the landward slope, because the force of gravity works in the same direction as the flow resulting in acceleration of the flow. The acceleration decreases over the distance because friction of the bed slows down the flow till a balance is reached between the momentum of the flow and the frictional force. This results in
12 2.2 Overtopping parameters
A formula of the terminal velocity is obtained in case of t → ∞. In this case, the flow velocity on the landward side described by Schüttrumpf and Oumeraci (2005) becomes
u∞ =
s
2 hBg sin ϕ
f (9)
with f = 0.0058 a friction factor that was empirically determined by Schüttrumpf and Oumeraci (2005). This formula of the terminal velocity is also used by Hughes and Nadal (2009) and the EurOtop Manual.
2.2.2 Overtopping time and instantaneous parameters
Flow velocity and layer thickness of individual overtopping waves vary over time. Expressions for
the time-varying flow velocity u(t) and layer thickness h(t) depend on the overtopping duration T0,
also named the overtopping time. Due to transformation of the overtopping tongue with distance,
the overtopping duration of an individual wave T0 is location dependent. Following Bosman et al.
(2008), the dimensionless 2% exceedance overtopping time at the start of the crest is given by
T0,2%(xC = 0) Tm−1,0 = 1.15 s Ru,2%− Rc 2g (10)
As shown in Figure 7, the flow velocity decreases on the crest leading to an increase in the overtopping time described by T0,2%(xC) T0,2%(xC= 0) = 1 + 2.8 xC L0,m ! (11) The overtopping time is used to find expressions for the time-dependent flow velocity u(t) and layer thickness h(t). Measurements of wave overtopping at dikes showed that u(t) and h(t) are represented by a saw-tooth like shape (Van der Meer et al., 2010; Hughes et al., 2012), see Figure 8 for an example of the flow velocity. The saw-tooth shape is simplified to an idealised shape, as shown in Figure 8. Hughes (2011) used a simple mathematical power-curve to describe the velocity u(t) and layer thickness h(t) for an idealised wave,
h(t) = hmax 1 − t T0 c for 0 ≤ t ≤ T0 (12) u(t) = Umax 1 − t T0 d for 0 ≤ t ≤ T0 (13)
with hmax and Umax the maximum layer thickness and the maximum flow velocity at the leading
edge of the wave, respectively. The exponent factor c and d need to be determined from experimental data. For the layer thickness h(t), the equation describes an exponential decay to the power c from
hmax at t = 0 s to zero at t = T0 s. The discharge q(t) is related to the flow velocity u(t) and layer
thickness h(t) as
q(t) = u(t) · h(t) = hmaxUmax
1 − t T0 c+d = qmax 1 − t T0 m for 0 ≤ t ≤ T0 (14)
This equation was fitted to experimental data of wave volume, maximum discharge qmax and
over-topping duration by Hughes et al. (2012) to find an expression for the exponent of the discharge m,
m = c + d = 2.33 V0.16− 1 (15)
Figure 8: The flow velocity u(t) as a function of the time t for a realistic time series, which is based on a
measurement series of Hughes (2011). The time-dependency of the velocity is simplified by Hughes (2011) to an exponential decay from Umax at t = 0 s to zero at t = T0 s, with T0 the overtopping duration.
The overtopping volume per unit width of an individual wave can be determined by integrating the discharge V = Z T0 0 q(t) dt = V = Z T0 0 qmax 1 − t T0 m dt = qmaxT0 m + 1 (16)
Using Equation 15 and Equation 16, the maximum discharge can be written as
qmax =
2.33 V1.16
T0
(17) The empirical fit of Hughes et al. (2012) for the maximum discharges and wave overtopping volumes of individual waves resulted in
qmax= 0.184
√
g V3/4 (18)
Combining both equations for the maximum discharge results in an expression for the overtopping time for an individual wave with overtopping volume V ,
T0= 4.0 V0.41 (19)
which is close to the empirical fit of from measurements of overtopping duration and volumes by Hughes et al. (2012),
T0 = 3.9 V0.46 (20)
The expressions of Hughes et al. (2012) deviate more from the empirical relation between overtopping time and volume found by Van der Meer et al. (2010),
T0= 4.4 V0.3. (21)
The difference between the two empirical relations might be caused by the measurement location. The empirical fits of Hughes et al. (2012) are based on measurements at the end of the crest, around 3/4 of
the crest width Bc, while the empirical fit of Van der Meer et al. (2010) was based on measurements
on different locations at the crest. Despite these differences, all three expressions show that the overtopping duration depends non-linearly on the volume.
14 2.3 Summary
2.3 Summary
Wave overtopping can be described by averaged quantities, which are useful to determine if wave overtopping will occur and to determine the water load related to overtopping. Significant wave
overtopping occurs when the wave run-up height Ru,2% exceeds the crest height Rc. The water load
is described by the average overtopping discharge or the maximum overtopping volume. The limits of the average overtopping discharge and maximum overtopping volume in the Netherlands are shown in Table 2, which are used for the design of dikes with a grass cover. In the next chapter, the erosional load of wave overtopping is described. The erosion due to wave overtopping is a complicated process which asks for a more detailed description of wave overtopping. The layer thickness and flow velocity as function of the distance were introduced as wave overtopping parameters. These parameters are not only location dependent, but also time dependent. The time-dependent saw-tooth like shape of the overtopping parameters over time for individual waves are important for determining the erosion, since erosion only occurs during the time period when a threshold is exceeded.
3
Erosion due to wave overtopping
Erosion of the dike cover occurs when the forces induced by the overtopping waves exceed the strength of the dike cover. The strength of the dike cover depends on the structure of the cover. Figure 9 shows a schematic view of the grass cover. The turf layer is characterised by a high root density and is elastic under moist conditions (Hoffmans, 2012). The roots prevent wash out of small clayey aggregates by connecting the aggregates. The root density decrease with depth, so the underlying clay layer is stiffer and less permeable (Hoffmans, 2012). The grass strength near the surface is determined by the root reinforcement, while in the deeper clay layer the strength is dominated by cohesion (Hoffmans, 2012). First, the initiation of erosion is described. Erosion starts when the loading forces exceed the dike cover strength, described by an erosion threshold. The amount of erosion can be described in an analytical way using the erosion depth, where the amount of erosion is a velocity in m/s resulting in a bed level update. Aguilar Lopez (2016) defined dike cover failure when the erosion depth exceeds the thickness of the grass cover, which was set to 10 cm. However, there are also various empirical methods which determine an erosional limit or a damage criterion. Failure is defined as exceedance of the erosional limit or the damage criteria, where the damage criteria differentiate between the erosion at one location, damage at various location and failure of the dike cover. The different methods to determine the amount of erosion are discussed in this chapter, as well as the definitions of failure.
Figure 9: (a) The structure of a grass cover. Source: Wondergem et al. (1997). (b) The root model of a
well-anchored root. The mean root tensile strength σrootis resolved into a component parallel (σroot,h) and perpendicular (σroot,v) to the shear zone using the angle of shear rotation θ. Adapted from Hoffmans (2012)
Table 6: Indicative values of the root area ratio Aroot(0)/A1, the critical mean grass normal stress at the
surface σgrass,c(0) and the critical depth-averaged flow velocity Uc computed using the equation for turf aggregates in Table 7 with r0=0.2 and Ψc= 0.03 (Hoffmans, 2012).
Grass quality Aroot(0)/A1 σgrass,c(0) (kN/m2)
Uc (m/s) with pw = −10 kN/m2 Uc (m/s) with pw= 0 kN/m2 Very poor < 2 × 10−4 < 3.0 < 3.0 < 6.2 Poor 2 × 10−4− 4 × 10−4 3.0 − 5.3 3.0 − 4.0 6.2 − 6.8 Average 4 × 10−4− 5 × 10−4 5.3 − 7.5 4.0 − 4.7 6.8 − 7.2 Good > 5 × 10−4 > 7.5 > 4.7 > 7.2
16 3.1 Initiation of erosion
3.1 Initiation of erosion
The initiation of erosion is described based on a turf element model following Hoffmans (2012). The strength of the grass surface is determined by the root reinforcement, which can be expressed in
terms of the mean root tensile strength σroot. The threshold for vertical motion is given by the mean
grass normal stress σgrass,
σgrass=
Aroot
A1
(σroot,v+ σroot,htan φ) = σroot
Aroot
A1
(cos θ + sin θ tan φ) (22)
where Aroot/A1 is the root area ratio (number of roots per m2), φ is the internal friction angle, θ is
the angle of shear rotation, the parallel component σroot,h and perpendicular component σroot,v of
the mean grass normal stress σgrass with respect to the the shear zone (see Figure 9). The threshold
of grass strength is given by the critical mean grass normal stress σgrass,c,
σgrass,c(z) = σgrass,c(0) ez/λref with σgrass,c(0) =
Aroot(0)
A1
σroot,c (23)
with λref a reference height that varies between 5 cm and 10 cm and Aroot(0)/A1 the root area ratio
near the surface (see Table 6 for typical values). The critical mean grass normal stress near the
surface σgrass,c(0) varies between 13.45 × 106 N/m2 and 85.10 × 106 N/m2 for different grass types
(Hoffmans, 2012).
Figure 10 shows the forces working on a turf element with length scales Lx, Ly and Lz in the x, y
and z direction respectively. Wave overtopping results in pressure fluctuations related to turbulent
motion. The relative turbulence intensity r0 and the depth-averaged flow velocity U0 are related as
ro= α0
u∗
U0
(24)
with the bed shear velocity u∗ and the coefficient α0 = 1.2. The maximum pressure fluctuation pm
[N/m2] is written as
pm= αττ0 = ατρ(u∗)2= ατ α−20 ρ (r0U0)2 (25)
with the density of water ρ, the coefficient ατ = 18 and the mean bed shear stress τ0. The pressure
fluctuations try to lift the element. The lift force Fp is written as
Fp= pmLxLy. (26)
Table 7: Formulas for the critical shear stress τc and the critical depth-averaged flow velocity Uc for clayey and turf aggregates from Hoffmans (2012), with the representative size of the aggregate near the bed dclay, the relative density ∆ = (ρs− ρ)/ρ, the characteristic size of detaching aggregates da = 0.004 m, the pore water pressure pw and the fatigue rupture strength of clay Cf, M = 53Cclay,c= 0.035 c with cohesion c.
The critical shear stress τc The critical depth-averaged flow velocity Uc
Clayey aggregates
Saturated clay τc= Ψc[(ρs− ρ)gdclay+ Cclay,c] Uc= 1.2 r
−1
0 [0.012 (∆gda+ 0.6Cf,M/ρ)]1/2
Clayey aggregates
Structured soil τc= ΨcCclay,c Uc= 1.2 r
−1
0 [ΨcCclay,c/ρ]1/2
Turf aggregates τc= Ψc[4τgrass,c+ σgrass,c(−λref)]
= 2.9 σgrass,c(0) Uc= 2.0 r
−1
Figure 10: Forces acting on a turf element, with the maximum lift force Fp, critical friction force Fcworking on the side walls, the critical mean tensile force Ft and the submerged weight of the soil Fw.
The turf element is unstable when the load is large than the strength, so
Fp≥ Fw+ 4Fc+ Ft (27)
where the strength is determined by the submerged weight of the soil Fw, the critical frictional force
Fc and the critical mean tensile force Ft.
Fw= (1 − p)(ρs− ρ)gLxLyLz
4Fc = 2(1 − p)(Cclay+ τgrass,c)(Lx+ Ly)Lz
Ft = (1 − p)[Cclay+ σgrass,c(z = −Lz)]LxLy
with the density of soil ρs, the porosity p ≈ 0.4, the critical rupture strength of clay Cclay and the
critical mean grass shear stress τgrass,c. Assuming Lx= Ly = Lz = −z this reduces to
pm≥ σsoil(z) = −(1 − p) [(ρs− ρ)gz − 4(Cclay+ τgrass,c) − (Cclay+ σgrass,c(z))] (28)
The critical condition for motion is usually expressed in terms of the bed shear stress. The turf
aggregates start to moved when the mean bed shear stress τ0 exceeds the critical mean bed shear
stress τc. Using z = −λref, the critical condition for motion is given by
τ0 ≥ τc = Ψc[(ρs− ρ)gλref + 4(Cclay+ τgrass,c) + (Cclay+ σgrass,c(−λref))] (29)
with the critical Shields parameter Ψc = α−1τ (1 − p) = 0.033.
Table 7 shows the formulas for the critical shear stress τc and the critical depth-averaged flow
ve-locity Uc for clayey aggregates and turf aggregates. Structured clay has cracks, because the clay
shrinks when it becomes dry. Cracking of the clay can damage the roots and the strength of the clay
aggregates decreases, resulting in a lower value for τc and Uc(Hoffmans, 2012). In case of turf
aggre-gates, the effect of the suction pressure in the roots is included in the formulas by including the pore
water pressure pw. The pore water pressure is negative and increases the strength in unsaturated
soils (Hoffmans, 2012). In case of a saturated grass cover, the pore water pressure is zero. Table
6 shows values for the critical mean grass normal stress at the surface σgrass,c(0) and the critical
depth-averaged flow velocity Uc of grass. Values of the critical mean rupture strength of clay Cclay
18 3.2 Erosion depth
Table 8: Indicative values of the critical mean rupture strength of clay Cclayand the critical depth-averaged flow velocity Uccomputed using the equation for clayey aggregates in structured soil in Table 7 with r0=0.2
and Ψc = 0.03. These values correspond to structured soil on river dikes and high turbulent conditions (Hoffmans, 2012).
Quality of clay Critical mean rupture strength Cclay (kN/m2) Critical flow velocity Uc (m/s)
Sandy - < 0.2 Poor < 0.15 0.2 − 0.4 Average 0.15 − 0.33 0.4 − 0.6 Good 0.33 − 0.75 0.6 − 0.9 Very good > 0.75 > 0.9 3.2 Erosion depth
The erosion depth z is determined from the mass balance,
dz(t) dt + 1 1 − p dS(T0) dx = 0 (30)
where the sediment transport S depends on the transport parameter T0 = µτ0/τc with µ the bed
form factor. The mass balance can also be written in terms of the deposition rate D and the pick-up rate E, also called the erosion rate.
dz(t) dt +
1
1 − p(E(T
0) − D(T0)) = 0 (31)
Hoffmans (2012) used the formula of Partheniades for the erosion rate of cohesive sediment
E = Mp τc
(τ0− τc) (32)
with Mp a sediment coefficient varying between 0.00001 − 0.0005 kg/(s m2). Verheij et al. (1995)
introduced the overall strength parameter CE (ms)−1 to represent the magnitude of erosion of clay
and grass
CE =
Mp
τc
(33)
Table 9 shows values of the strength parameter CE, the critical shear stress τc and the critical flow
velocity Uc for different types of soil from Verheij et al. (1995). Hoffmans (2012) expressed CE in
terms of the critical depth-averaged flow velocity based on dimensional considerations resulting in
CE =
αCg cv,0
(daUc)2
(34)
with the consolidation coefficient cv,0 which has a default value of 10−3 m2/s, the coefficient αc =
5 · 10−8 and the characteristic size of detaching aggregates da= 0.004 m.
3.2.1 Maximum erosion depth
The maximum erosion depth zm is based on the Breusers time-dependent scour equation (Breusers,
1966) zm= −λ t t1 γt (35)
with
t1 =
2(1 − p)2λ2γt
Sm(t1)
(36)
where λ is the characteristic length scale, γt is a coefficient, t1 is the characteristic time at which
zm = −λ and Sm(t) is the sediment transport at the location of maximum erosion. Hoffmans
(2012) wrote the sediment transport in terms of the depth-averaged flow velocity and the critical
flow velocity. The coefficient γt is 0.8 for 3D flow (Van der Meulen and Vinjé, 1975), however the
difference in erosion depth between γt = 0.8 and γt= 1.0 is small (Hoffmans, 2012). Assuming γt = 1
and highly turbulent flow conditions, the maximum erosion depth of clay of one overtopping wave is
zm= −
(ωU02− Uc2)T0
Esoil
(37)
with the turbulence coefficient ω, the overtopping duration T0 and the erosion parameter Esoil is
given by Esoil = αsoil U2 c √ ∆gda (38)
where αsoil is a coefficient.
3.2.2 Depth-dependency
Equation 37 only holds for erosion of the cover, so for depths smaller than 0.1 m. Valk (2009) adapted the formula for greater depths by adding a depth-dependency factor for the strength terms.
zm= −
(ωU2
0 − Uc(d)2)T0
Esoil(d)
(39)
and writing the erosion parameter Esoil in terms of the strength parameter CE
Esoil = αsoilCE−1(d) (40)
with d the depth of the cover and αsoil = 1. The turbulence coefficient ω is related to the relative
turbulence intensity r0 as
ω = 1.5 + 5 r0 (41)
Table 9: Values of the strength parameter CE, the critical shear stress τc and the critical flow velocity Uc for different types of soil (Verheij et al., 1995).
Soil type CE (m/s)−1 τc (N/m2) Uc(m/s) Grass Good Average Poor Very Poor 0.01 · 10−4 0.02 · 10−4 0.03 · 10−4 < 0.5 · 10−4 125 − 250 50 − 125 25 − 75 5 − 25 5.0 − 8.0 3.0 − 5.0 2.0 − 4.0 1.0 − 2.0 Clay Good Average/structured Poor 0.5 · 10−4− 1 · 10−4 1 · 10−4− 3 · 10−4 3 · 10−4− 5 · 10−4 1.5 − 5.0 0.5 − 1.5 0.3 − 0.5 0.7 − 1.0 0.5 − 0.7 0.3 − 0.5 Sand - > 10 · 10−4 0.1 − 0.2 0.15 − 0.3
20 3.3 Empirical relations
The value of the turbulence intensity r0 depends on the bed characteristics. Valk (2009) estimated
r0 = 0.15 for a horizontal bed (dike crest or berm) and r0 = 0.2 for the landward slope of a dike,
while Bomers (2015) used a value of 0.17 and 0.1 for the crest and the landward slope, respectively. The maximum erosion depth was written in terms of the shear stress by Bomers (2015), resulting in
zm= −
(ωτ0− τc(d))T0
Esoil(d)
. (42)
Valk (2009) also adapted the formula for the critical shear stress of Hoffmans (2012) to make it depth-depended, resulting in
τc(d) = α−1τ [(ρs− ρ)gda+ f τclay(d) + σroots(d)] (43)
with the pressure fluctuation coefficient ατ = 18, a factor for the clay cohesion f , the clay cohesion
τclay as a function of depth
τclay(d) = τclay,0(1 + αcsd) (44)
and the grass strength σroots as a function of depth
σroots(d) = σroots,0exp(−βd) (45)
where Valk (2009) assumed a linear increase of the clay cohesion with coefficient αcs and an
expo-nential decrease of the root density with coefficient β. The parameter values used by Valk (2009) are
da = 0.004 m, f = 0.21 and β = 22.32. Two values of αcs are reported αcs= 20 in case of large clay
cohesion (Valk, 2009) and αcs = 1.75 when the bonding forces of the clay aggregates are reduced
(Bomers, 2015).
3.3 Empirical relations
Various empirical erosion limits are based on observations during wave overtopping experiments. The different erosion indices and their erosional limits are discussed in this section. The erosion indices are divided in four bases: flow velocity, shear stress, work and excess wave volume.
3.3.1 Flow velocity
The first erosion indices are based on flow velocity. Dean et al. (2010) introduced the erosion index
Eu based on the excess velocity over the critical value Uc,u, which is given by
Eu= Ku j=8
X
j=1
(Uj− Uc,u)tj [m s−1] (46)
with Kuthe velocity erosional coefficient, and Uj and tj are combinations of the depth averaged flow
velocity and duration for acceptable erosion due to steady wave overtopping. Dean et al. (2010) used
the combinations of U and t reported by Hughes (2007), in which case Uj increases from 1.0 m/s to
4.5 m/s in steps of 0.5 m/s. These eight combinations of U and t are used to determine Uc,u and the
erosion limit Eu/Ku for three grass cover strength: good, average and poor. The results are shown
in Table 10. A dike cover should be able to withstand the erosion limit during a storm. According to Dean et al. (2010), the dike cover has failed when the erosional limit is exceeded. In case of good grass quality, the criterion for acceptable erosion would be
Nc X n=1 (Un− Uc,u)∆tn ≤ Eu Ku ≤ 22.10 × 103 m (47)
Table 10: The threshold velocities, erosional limits and velocity errors for the three bases for various grass
cover strengths with ατ = βW =12ρfF (Dean et al., 2010).
Grass cover
strength Basis
Threshold
depth-averaged velocity (m/s) Erosion limit
Standard error in velocities (m/s) Good Velocity Shear stress Work Uc,u = 1.93 Uc,τ = 1.88 Uc,W = 1.80 Eu/Ku = 22.10 × 103 m Eτ/Kτατ = 1.18 × 105 m2/s EW/KWβW = 4.92 × 105 m3/s2 1.70 0.76 0.38 Average Velocity Shear stress Work Uc,u = 1.43 Uc,τ = 1.39 Uc,W = 1.30 Eu/Ku = 15.55 × 103 m Eτ/Kτατ = 0.67 × 105 m2/s EW/KWβW = 2.29 × 105 m3/s2 0.98 0.38 0.12 Poor Velocity Shear stress Work Uc,u = 0.94 Uc,τ = 0.89 Uc,W = 0.76 Eu/Ku = 13.01 × 103 m Eτ/Kτατ = 0.408 × 105 m2/s EW/KWβW = 1.03 × 105 m3/s2 0.81 0.26 0.04 3.3.2 Shear stress
Dean et al. (2010) introduced an erosional index based on the shear stress, using the relation between the bed shear stress τ and flow velocity u
τ = 1
2ρfFu
2 (48)
with fF the bottom friction coefficient. The erosion due to shear stress Eτ is given by
Eτ = Kτ j=8 X j=1 (τj− τc)tj = 1 2ρfFKτ j=8 X j=1 (Uj2− U2 c,τ)tj [m2s−1] (49)
with Kτ the shear stress erosional coefficients and Uc,τ the threshold velocity associated with the
shear stress. The erosion limit Eτ/Kτατ for the three grass cover strength are shown in Table 10
with ατ = 12ρfF (Hughes, 2011). In case of a good grass cover quality, the criterion for acceptable
erosion would be Nc X n=1 (Un2− U 2 c,τ)∆tn≤ Eτ Kτατ ≤ 1.18 × 105 m2/s (50)
Van der Meer et al. (2010) performed overtopping experiments on real dikes and observed that erosion was caused by the impact of the maximum flow velocity during a short time period within the overtopping duration. For this reason, Van der Meer et al. (2010) omitted the overtopping duration and introduced an erosional index called the cumulative hydraulic load D given by
D = Nc X i=1 (Ui2− U 2 c) [m2s−2] (51)
which sums over the number of waves Nc for which holds Ui > Uc. From the experiments, Van
22 3.3 Empirical relations
Table 11: The empirically determined erosional limits for the cumulative hydraulic load index from Van
der Meer et al. (2010) and Steendam et al. (2012).
Van der Meer et al. (2010) Steendam et al. (2012)
Initial damage 500 m2/s2 500 m2/s2
Damage at various locations 1000 m2/s2 500 m2/s2 −1500 m2/s2
Failure Mole holes: 3500 m
2/s2
Normal slope: > 6000 m2/s2 > 3500 m
2/s2
categories: initial damage, damage at various locations and failure of the slope. The erosional limits using the cumulative hydraulic load index are shown in Table 11. This table also includes the em-pirically determined values from Steendam et al. (2012), which are slightly different from the values found by Van der Meer et al. (2010).
The definition of the cumulative load used by Van der Meer et al. (2010) and Steendam et al.
(2012) only includes the largest waves (Ui > Uc). Hoffmans (2012) introduced a cumulative load
model that takes all the overtopping waves Nov into account, using a reference depth λref = zm and
Equations 37 and 38, i=Nov X i=1 Ui2= CdamageUc2 [m2s−2] (52) with Cdamage = 2αsoilλref ω T0 √ ∆gda (53)
Assuming da = 0.004 m, T0=3.5 s, αsoil = 77 · 103„ ∆ = 1.65, ω = 10, Hoffmans (2012) defined the
three erosional limits based on the erosional depth indicated by λref:
• initial damage / slit erosion in case λref ≤ 0.05 m : Cdamage≤ 850
• damage at various locations in case 0.05 < λref ≤ 0.10 m: 850 < Cdamage ≤ 1700
• failure of dike slope λref > 0.10 m: Cdamage> 1700
3.3.3 Work
Dean et al. (2010) introduced an erosional index based on the work W . Hughes (2011) found that
the work index is actually the same as the stream power Ps, which could be expressed as
W = Ps= τ · U =
1
2ρfFU
3 (54)
According to Hughes (2011), stream power is the rate of doing work, so it is a measure of the available energy for the transport of sediment. The energy comes from the potential or position energy, which is converted to kinetic energy when the water flows (Knighton, 1999). The erosion due to excess
work EW is given by EW = KW j=8 X j=1 (Wj− Wc)tj = 1 2ρfFKW j=8 X j=1 (Uj3− Uc,W3 )tj [m3s−1] (55)
with KW a work erosional coefficient and Uc,W the threshold velocity associated with the work. The
erosion limit EW/KWβW for the three grass cover strength are shown in Table 10, with βW = 12ρfF
(Hughes, 2011). In case of plain grass-good cover the criterion for acceptable erosion would be
Nc X n=1 (Un3− Uc,W3 )∆tn≤ EW KWβW ≤ 4.92 × 105 m3/s2 (56)
3.3.4 Excess wave volume
The excess wave volume VE is the overtopping wave volume per unit width for which the threshold
is exceeded. This is determined from the integral over the discharge over a duration Tc when the
discharge exceeds the critical discharge qc (Hughes, 2011).
VE = Z Tc 0 q(t) − qcdt = Z Tc 0 qmax 1 − t T0 m dt − Z Tc 0 qcdt (57)
where Equation 14 is used for q(t). Using t = Tc and q(Tc) = qc, an expression for the duration Tc
is found, qc = qmax 1 − Tc T0 m → Tc = T0 " 1 − q c qmax 1/m# (58)
Integrating Equation 57 and using the expression of Tc results in
VE = T0 m + 1 qmax− qc− m qc " 1 − q c qmax 1/m#! (59)
Rearranging and using Equation 16 for the total wave volume VW gives
VE = VW " 1 − (m + 1) q c qmax + m q c qmax m+1m # (60)
Hughes (2011) adapted the erosional work index EW to a erosional index based on excess wave
volume VE. Using the terminal velocity (Equation 9) and q = u · h, the flow velocity Un is written
in terms of the discharge qn
qn=
fFUn3
2g sin ϕ (61)
The erosional limit of the excess wave volume index is given by
VET(t) = Nc X n=1 (qn− qc)∆tn ≤ EW KWβW f F 2g sin ϕ for qn> qc (62) where qc= fFUc,W3 2g sin ϕ (63)
Using Equation 60, the total excess wave volume can also be written as
VET(t) = Nc X n=1 VW n " 1 − q cTon VW n + 2 33/2 q cTon VW n 3/2# ≤ E W KWβW f F 2g sin ϕ (64)
under the assumptions m = 2 and Equation 16 is used to express qmaxin terms of VW nand Ton, which
24 3.4 Conclusions
3.4 Conclusions
The empirical erosion models use four bases: flow velocity, shear stress, work and excess wave vol-ume. Dean et al. (2010) determined the threshold velocities based on experiments and the errors for different quality of grass cover of the first three bases. Dean et al. (2010) found that the work index has the smallest error. For lower flow velocities, the three bases are equally good, but the error in the work index is significantly smaller for higher flow velocities. However, the work base as defined by Dean et al. (2010) is a physical quantity that cannot be measured. Hughes (2011) identified that the work base of Dean et al. (2010) actually is stream power per unit area. Hughes (2011) in-troduced the excess wave volume method as empirical model with a more intuitive variable than work. The empirical erosion models are independent of time and only take the large waves into account, except the model of Hoffmans (2012). In some cases, a more detailed and a time-dependent erosion model is preferred. The erosion model of Valk (2009) for the maximum erosion depth is the most suitable erosion model for predicting dike cover erosion. This model was also used by Bomers et al. (2018), and the use of shear stress as base makes it possible to compare the results with the empir-ical damage criteria of Dean et al. (2010), Van der Meer et al. (2010), Steendam et al. (2012) and Hoffmans (2012). However, the model of Valk (2009) is not validated and was part of a master thesis work, which is not peer-reviewed.
