Overtopping Distribution behind the Crest

During the experiments the average overtopping discharge was determined for 8 horizontal positions in total, 6 of which were positioned behind the crest. The spatial distribution of overtopping for each experiment conducted will be investigated. A comparison will be drawn with 3 existing prediction methods, being 1) Juul Jensen (1984), 2) Van Kester (2009), and 3) Lioutas et al. (2012).

In Figure 5.3-1 the measured spatial distribution of overtopping is depicted. The aim for each prediction method for 𝐶𝑟 is to adjust the horizontal position of a data point on a dimensionless horizontal axis such that clustering of the data is achieved.

FIGURE 5.3-1: SPATIAL DISTRIBUTION OF OVERTOPPING IN THE EXPERIMENTS

In Figure 5.3-1 the horizontal position is depicted in consistence with Lioutas et al. (2012), i.e. x = 0 at the landward end of the crest.

5.3.1 COMPARISON WITH JUUL JENSEN (1984)

Juul Jensen chose to describe the spatial decay of overtopping with an exponential function. The method that was recommended can be written as a reduction factor Cr, see

Equation 5.3-1.

𝐶𝑟 = 10−𝑥𝛽 Equation 5.3-1

In Figure 5.3-2 a comparison is drawn between the method of Juul Jensen and the measured values of Cr. The measured value of Cr was calculated as 𝑞(𝑥)_𝑞 𝑚𝑒𝑎𝑠

𝑏𝑐,𝑚𝑒𝑎𝑠. x is the horizontal distance from the end of the crest.

FIGURE 5.3-2: COMPARISON WITH JUUL JENSEN

The prediction method seems adequate to provide an upper limit. There’s a large amount of scatter however. It is quite clear that other parameters play a major role. Apparently not solely the horizontal distance from the end of the crest is important for the distribution of the overtopping discharge.

Note: For this prediction method no goodness-of-fit is provided since only a reduction factor is calculated and no overtopping discharge.

5.3.2 COMPARISON WITH VAN KESTER (2009)

Van Kester introduced a reduction factor for the total overtopping based on the wave energy flux. 𝐶𝑟 =𝑞(𝑥)_𝑞 𝑏𝑐 = exp �−1.64 ∙ 10 5_∙ 𝑥 𝐻𝑚0∙ 1 (𝐻∗_𝑇∗₎3� Equation 5.3-2

Here H*T* is supposed to be a dimensionless form of the wave energy flux. The

horizontal position from the end of the crest, x, is made dimensionless as shown in Equation 5.3-4. 𝐻∗_𝑇∗₌𝐻𝑚0 𝑅𝐶 ∙ 𝑇 ∙ � 𝑔 𝑅𝐶 Equation 5.3-3 𝑥∗₌ 𝑥 𝐻𝑚0∙ 1 (𝐻∗_𝑇∗₎𝑛 Equation 5.3-4

5.3OVERTOPPING DISTRIBUTION BEHIND THE CREST 93

The prediction method is compared with the measured data in Figure 5.3-3.

FIGURE 5.3-3: COMPARISON WITH VAN KESTER

The prediction method does not have a good fit with the measured data. This

corresponds with Van Kester’s own remark that irregular waves lead to considerably larger values of x*_{. Moreover the dimensionless parameter x}* _{does not lead to reduction}

of scatter in the data. A rough trend can be discerned; larger values of x* _{lead to a lower}

value of 𝑞(𝑥)_𝑞 𝑏𝑐.

Figure 5.3-4 presents a similar graph as was made for the prediction method of Juul Jensen. This graph shows that the prediction method of Van Kester underestimates the overtopping discharge for quite a few experiments. Furthermore the chosen method leads to very much scatter in the data.

FIGURE 5.3-4: COMPARISON WITH VAN KESTER

Note: For this prediction method no goodness-of-fit is provided since only a reduction factor is calculated and no overtopping discharge.

5.3.3 COMPARISON WITH LIOUTAS ET AL. (2012)

Here the found expression for total overtopping is taken as a starting point. The γc term

as introduced by Lioutas is applied to the proposed expression.

5.3OVERTOPPING DISTRIBUTION BEHIND THE CREST 95

FIGURE 5.3-5: COMPARISON WITH LIOUTAS

The chosen 𝛾𝑐 results in clear grouping of the experimental data based on location. Measurements that were taken at a location quite far from the crest are considerably underestimated by the prediction method.

5.3.4 OVERALL SPATIAL DISTRIBUTION OF OVERTOPPING

None of the above mentioned prediction methods adequately predicts the

measurements done during the experiments. Therefore a new expression was sought for the spatial distribution of overtopping, which uses the same approach as Lioutas et al. (2012) chose. A parameter γc is introduced to enlarge the dimensionless freeboard and

thus make the prediction for the dimensionless overtopping discharge lower. The following requirements were posed that the expression should meet:

1. For 𝑥 ≤ 0 no reduction should be taken into account. 2. lim𝑥→∞𝛾𝑐= 0.

3. A good fit with the data should be established, especially for the larger

overtopping discharges. The data cloud should not show any grouping based on horizontal position.

After data analysis an expression was found that makes use of a dimensionless number 𝑥/𝐻𝑚0, see Equation 5.3-6. To facilitate design, the reference point 𝑥 = 0 is now placed at the seaward edge of the crest. Theoretically there should be a difference in the spatial decay over the armour layer and over the backfill. However this difference could not be discerned from the data.

𝛾𝑐 = exp �−0.14 ∙_𝐻𝑥

This expression leads to some reduction of scatter in the data. No grouping based on distance from the crest could be discerned. Furthermore the expression results in either a good fit with the data, for large overtopping discharges, or overestimation of

overtopping, i.e. conservative design. The expression was found after data analysis and is entirely based on curve fitting; it does not have a theoretical basis.

FIGURE 5.3-6: COMPARISON WITH PREDICTION METHOD

The plotted prediction method is printed below. 𝑞(𝑥) �𝑔 ∙ 𝐻𝑚03 = 0.2 ∙ exp �−2.6 ∙_𝐻𝑚0𝑅𝐶 _{𝛾𝑏∙ 𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔}1 _{∙ 𝛾𝛽∙ 𝛾} 𝑐� Equation 5.3-7 𝛾𝑏 = ⎩ ⎨ ⎧1 − 0.4 ∙ 𝑟𝐵∙ (1 − 𝑟𝑑𝑏) for 𝑑𝐵 ≥ 0 exp �_𝐻−0.43 ∙ 𝐵 𝑚0∙ 𝜉𝑚−1,0� for 𝑑𝐵 𝐴𝑐 ≈ −0.5 Equation 5.3-8 𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔= ⎩ ⎨ ⎧ 𝛾𝑓 for 𝜉𝑚−1,0< 1.8 𝛾𝑓+ �𝜉𝑚−1,0− 1.8� ∙1 − 𝛾_4.2𝑓 for 1.8 < 𝜉𝑚−1,0< 6 1.0 for 𝜉𝑚−1,0> 6 Equation 5.3-9 𝛾𝑐 = exp �−0.14 ∙_𝐻𝑥 𝑚0� Equation 5.3-10

For deterministic design Equation 5.3-11 can be used. 𝑞(𝑥) �𝑔 ∙ 𝐻_𝑚03 = 0.2 ∙ exp �−2.25 ∙ 𝑅𝐶 𝐻𝑚0 1 𝛾𝑏∙ 𝛾𝑓,𝑠𝑢𝑟𝑔𝑖𝑛𝑔∙ 𝛾𝛽∙ 𝛾𝑐� Equation 5.3-11

5.3OVERTOPPING DISTRIBUTION BEHIND THE CREST 97

A large difference exists between the term found by Lioutas et al. (2012) and the term applicable to the current data set, see Figure 5.3-7. Rather than an improvement to Lioutas et al. (2012), the here presented expression serves as an illustration of the influence of the experiment setup. A particular peculiarity is that – in spite of an

increased drainage capacity of the backfill – the 𝛾𝑐 term of Lioutas et al. (2012) does not lead to conservative design. One would expect the overtopping discharge to decay more quickly with horizontal position.

FIGURE 5.3-7: COMPARISON CURRENT γc AND THE γc TERM PROPOSED BY LIOUTAS ET AL.

(2012)

Further research – using an alternative experiment setup – will be required to provide a more accurate picture of the spatial distribution of overtopping for irregular waves. The experiment setup is discussed extensively in Chapter 6.

5.3.5 CONCLUSIONS FOR 𝑞(𝑥)

Comparison of the experimental data of the current research and that of Lioutas (2010) learned that the spatial distribution of overtopping is presumably strongly influenced by the used experiment setup. This influence is discussed separately in Chapter 6.

For the conditions as simulated during the experiments, the following conclusions can be drawn.

1. The spatial distribution of overtopping is associated with quite a lot of seemingly random behaviour.

2. The method of Juul Jensen (1984) does not accurately predict the experimental data. However the method can be used as an upper limit, i.e. conservative design. 3. The method of Van Kester (2009) does not accurately predict the experimental

data. The chosen dimensionless parameter does not lead to reduction of scatter in the data.

4. The 𝛾𝑐 term of Lioutas et al. (2012) leads to considerable grouping in the data based on horizontal position. The method is not conservative for the

experimental data of the current research.

5. A different 𝛾𝑐 term is presented, see Equation 5.3-12. This expression does result in reduction of scatter in the data and is conservative for the current data set.

𝛾𝑐 = exp �−0.14 ∙_𝐻𝑥 𝑚0�

Equation 5.3-12

6. The difference between the 𝛾𝑐 term of Lioutas et al. (2012) and Equation 5.3-12 is thought to originate from the difference in experiment setup.

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