5.3 Run-up on straight smooth slopes

5.3 Run-up on straight smooth slopes#

5.3.1 Theoretical considerations#

The basis of the run-up calculation is Hunt’s formula from 1959:

(27)#\[ \frac{R_{u}}{H}= \xi \hspace{1cm} \textrm{or} \hspace{1cm} R_{u}= \sqrt{HL_{0}} \tan{\alpha} \]

The general shape of the formula can be understood physically from the flow velocities in a breaking wave []. For non-breaking waves (larger values of \(\xi\)) this does not apply. In this regime the run-up \(R_u/H\) tends to become more or less constant, or at least a much weaker function of \(\xi\).

Hunt’s formula has been derived for regular waves. In an irregular wave fields all individual waves are different, and therefore also in run-up of every wave is different. Instead of single parameters, the wave height, wave length and run-up are now stochastic parameters.

Like we describe the irregular wave by a single descriptor (usually \(H_{s}\), or \(H_{m0}\)), there is also a descriptor for the run-up by irregular waves. Usually the \(R_{u2\%}\) is used, which is the run-up value exceeded by \(2\%\)1

\cite{BATTJES1974_PhD} has shown that the linear relationship between \(\frac{R_{u}}{H}\) and \(\xi\) remains valid in the case of irregular waves, provided of course that the parameters are replaced by their equivalents for irregular waves, i.e. \(R_{u}\) by \(R_{u2\%}\), \(H\) by \(H_{m0}\) (or \(H_{s}\)) and \(\xi\) by \(\xi_{m}\) (or \(\xi_{m-1,0}\)). Modern literature gives preference to the use of \(\xi_{m-1,0}\) and that version of the fomula is given in equation (28). Battjes derived on purely theoretical grounds that the coefficient of proportionality should have a value between \(1.41\) and \(1.77\) in that case2.

The distribution of the incoming wave heights is commonly taken as a Rayleigh distribution, and this has been confirmed by numerous field observation. [] showed that the resulting distribution of run-up levels will also be a Rayleigh distribution. This result is used in many advanced applications such as the analysis of the distribution of individual overtopping volumes and the stability of the inner slope.

Relative Run-up as a function of the Iribarren number (EUROTOP2007)
Figure 5.2: Relative Run-up as a function of the Iribarren number (EUROTOP2007).

5.3.2 Prediction of run-up levels#

TAW formula#

For use in practical levee design these theoretical expressions have been validated and calibrated against extensive datasets of model tests and observations. A commonly used formula to calculate the run-up is as follows (\cite{TRGG2007, EUROTOP2007}):

(28)#\[ \frac{R_{u2\%}}{H_{m0}}= A \xi_{m-1,0} = 1.65 \xi_{m-1,0} \]

With a maximum of:

(29)#\[ \frac{R_{u2\%}}{H_{m0}}=4.0 -\frac{1.5}{\sqrt{\xi_{m-1,0}}} \]

Clearly the empirical value \(A = 1.65\) in equation (28) is within the theoretical prediction range of []. The two branches of this formula describe the behaviour for breaking and nonbreaking waves, respectively. This formula is valid for smooth, straight slopes. In other cases such as rough, permeable or bermed slopes this value has to be reduced. This will be elaborated in 5.5.

These formulas are plotted against the underlying dataset in Figure 5.2. It is clear that there is some scatter. The values given above describe the mean trend through the data, so they do not include a safety coefficient. For semi-probabilistic calculations a characteristic value should be used. For full probabilistic calculations the coefficients can be modelled as normally distributed stochastic variables. [] gives the following values:

Table 5.1: Coefficient values in (28) and (29).

General Shape \(\frac{R_u}{H} = A\cdot{\xi}\) and \(\frac{R_u}{H} = B - \frac{C}{\sqrt{\xi}}\)

Equation (28)

Equation (29)

A

B

C

Mean Value

1.65

4.0

1.5

Standard Deviation

0.11

0.28

1.05

Characteristic Value[^3]

1.75

4.3

1.6

An updated version of the maximum run-up formula (29) has been included in the current edition of [], but have not been included in this document, as the calculated results are relatively small.

Delft formula#

The ‘Delft formula’ is given here because it provides an easy way to obtain a first estimate of the run-up levels and can usually be calculated by heart, but should not be used for design:

(30)#\[ R_{u2\%} = 8\cdot H_{s}\cdot \tan{\alpha} \]

The formula was developed in The Netherlands in the context of the Zuiderzee works and is only valid for ‘typically Dutch’ wave conditions. Comparison with equation (28) shows that it provides the same prediction only for the case of breaking waves and a wave steepness of \(s_{m-1,0} = \left(\frac{1.65}{8}\right)^2 \approx 0.04\).

1

The choice of value of \(2\%\) is rather arbitrary. It was selected by Delft Hydraulics around 1940, because it gave a good optimum between accuracy and test duration of the waves.

2

Battjes did not use the modern definition of \(\xi_{m-1,0}\). The values given in [] have been re-calculated for this purpose.

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