10. Overtopping

One of the failure mechanisms of a breakwater is overtopping. Overtopping is defined as the amount of water passing over the crest of the structure per unit of time, in practice the discharge is often expressed per meter width of the breakwater as q [\(l/s/m\)].

Warning

The specific discharge per meter width, q, must be given in \(l/s/m\), this is automatically converted to \(m^3/s/m\)

The EurOtop manual is the result of extensive research and experimental studies on overtopping. All formulas in this module can be found in the second edition of this manual. The following structures are defined: coastal dikes, rubble mound breakwaters, vertical walls and vertical composite walls

10.1. Rubble mound

Rubble mound breakwaters are characterized by a core with some porosity or permeability, covered by a sloping porous armour layer consisting of large rock or concrete armour units (for example Xbloc). However, the formula for coastal and river dikes can be used for a wider range of slopes, and therefore allows for more flexible input parameters. Since the formula for a rubble mound breakwater is a simplified case of the coastal or river dike, the formula for the dike is implemented. The definitions of the variables are presented in Figure 10.1.

Figure 10.1: schematisation of a rubble mound breakwater with definitions of variables

10.1.1. Formula

The general formula for the average overtopping discharge on a slope (dike, levee, embankment) implemented is the mean value approach (EurOtop, 2018). EurOtop (2018) advises against the use of a mean value approach for design or assessment purposes. Instead, EurOtop (2018) advises to increase the average discharge by one standard deviation for a design or assessment. Therefore, 1 is the default setting for the safety parameter.

breakwater.core.overtopping.rubble_mound(Hm0, q, xi_m_min_1, alpha, beta, gamma_b, gamma_v, gam_star, armour_layer, layers=1, permeability='permeable', safety=1, Gc=None, Dn50=None, limit=True)

Compute the crest freeboard of a rubble mound breakwater

Computes the crest freeboard of a rubble mound breakwater using equation 5.10 and 5.12 from EurOtop (2018).

\[\frac{q}{\sqrt{g \cdot H_{m 0}^{3}}}=\frac{0.023}{\sqrt{\tan \alpha}}\gamma_{b} \cdot \xi_{m-1,0} \cdot \exp \left[- \left(2.7 \frac{R_{c}}{\xi_{m-1,0} \cdot H_{m 0} \cdot \gamma_{b} \cdot \gamma_{f} \cdot \gamma_{\beta} \cdot \gamma_{v}}\right)^{1.3}\right]\]

with a maximum of

\[\frac{q}{\sqrt{g \cdot H_{m 0}^{3}}}=0.09 \cdot \exp \left [-\left(1.5 \frac{R_{c}}{H_{m 0} \cdot \gamma_{f} \cdot \gamma_{\beta} \cdot \gamma^{*}}\right)^{1.3}\right]\]

The reliability of the first equation is given by \(\sigma\) (0.023) = 0.003 and \(\sigma\) (2.7) = 0.20, and for the second equation by \(\sigma\) (0.09) = 0.0135 and \(\sigma\) (1.5) = 0.15.

Parameters:

• Hm0 (float) – the spectral wave height [m]
• q (float) – mean overtopping discharge per meter structure width [l/s per m]
• xi_m_min_1 (float) – \(\xi_{m-1.0}\), the surf-similarity parameter computed with the energy wave period \(T_{m-1.0}\) [-]
• alpha (float) – Angle of the front slope [rad]
• beta (float) – the angle of wave attack [rad]
• gamma_b (float) – influence factor for a berm [-]
• gamma_v (float) – influence factor for a vertical wall [-]
• gamma_star (float) – overall influence factor for a storm wall on slope or promenade [-]
• armour_layer (str) – name of the material of the armour layer, supported materials: Smooth, Rock, Cubes, Antifers, HARO, Tetrapods, Dolose, Accropode I, Xbloc, CoreLoc, Accropode II, Cubipods
• layers ({1, 2}, default: 1) – number of layers in the armour layer, required if the armour layer is made out of: Rock, Cubes or Cubipods
• permeability ({'permeable', 'impermeable'}, default: 'permeable') – permeability of the armour layer, required if the armour layer is made out of Rock
• safety (float, optional, default: 1) – safety factor of the design, positive values increase the safety of the design by increasing the mean value of the model constants with the number of standard deviations specified. In accordance with the recommendation from EurOtop (2018), the default value is set to 1 standard deviation.
• Gc (float, optional, default: None) – Effect of armoured crest width [m]
• Dn50 (float, optional, default: None) – nominal diameter of the armour units on the crest [m]
• limit (bool, optional, default: True) – If True, the discharge will be set to the limit for zero discharge in case the given discharge is below this limit. If False, the discharge will not be changed.

Returns:

Rc (float) – the crest freeboard of the structure [m]

10.1.2. Influence factors

The influence of roughness elements, oblique wave attack, berms, etc. are taken into account by introducing influence factors

breakwater.core.overtopping.gamma_f(armour_layer, xi_m_min_1, layers=None, permeability=None, placement=None)

Influence factor for the permeability and roughness of the slope

Computes the influence factor on roughness with table 6.2 from EurOtop (2018). These values are derived for 2.8 \(\leq \xi_{m-1.0} \leq\) 4.5, in case of larger surf-similarity parameters the influence factor for roughness is increased using equation 6.7 from EurOtop (2018)

Type of armour layer	\(\gamma_f\)
Smooth impermeable surface	1.00
Rock (1 layer, impermeable core)	0.60
Rock (1 layer, permeable core)	0.45
Rock (2 layers, impermeable core)	0.55
Rock (2 layers, permeable core)	0.40
Cubes (1 layer, flat positioning)	0.49
Cubes (2 layers, random positioning)	0.47
Antifers	0.50
HARO	0.47
Tetrapods	0.38
Dolos	0.43
Accropode I	0.46
Xbloc, CoreLoc, Accropode II	0.44
XblocPlus	0.45
Cubipods (1 layer)	0.49
Cubipods (2 layers)	0.47

Parameters:

• armour_layer (str) – name of the material of the armour layer, supported materials: Smooth, Rock, Cubes, Antifers, HARO, Tetrapods, Dolos, Accropode I, Xbloc, XblocPlus, CoreLoc, Accropode II, Cubipods
• xi_m_min_1 (float) – the surf-similarity parameter [-]
• layers ({1, 2}, optional, default: None) – number of layers in the armour layer, required if the armour layer is made out of: Rock, Cubes or Cubipods
• permeability ({'permeable', 'impermeable'}, optional, default: None) – permeability of the armour layer, required if the armour layer is made out of Rock
• placement ({'flat', 'random'}, optional, default: None) – placement of the armour layer, required if the armour layer is made out of Cubes

Returns:

gamma_f (float) – the influence factor for roughness

Raises:

Keyerror – If the armour layer is not in table 6.2 from EurOtop (2018)

breakwater.core.overtopping.gamma_beta(beta)

Influence factor for oblique wave attack

Computes the influence factor for oblique wave attack with equation 5.29 from EurOtop (2018).

\[\gamma_{\beta} = 1 - 0.0033 \mid \beta \mid\]

with a maximum of \(\gamma_{\beta} = 0.736\) for \(\mid \beta \mid > 80\)

Parameters:beta (float) – the angle of wave attack [rad] Returns:gamma_beta (float) – the influence factor for oblique wave attack

Note

The formulas for \(\gamma_b, \gamma_v, \gamma_*\) have not yet been implemented. These must thus be computed by hand.

10.2. Vertical and Composite Vertical

The vertical and composite vertical walls are comparable structure, the difference between is the depth in front of the vertical wall. A composite vertical wall is fronted by a berm or toe mound below the water level, whereas a vertical wall is not fronted by such a berm, in that case d = h. In figure 10.2 a definition sketch of both structures is presented.

Figure 10.2: schematisation of a vertical or composite vertical wall with definitions of variables

More than one equation have been derived to compute the overtopping discharge, or crest freeboard. Therefore, a general formula is implemented which automatically classifies the structure in order so that the correct formula is used.

10.2.1. General Formula

The general formula is an implementation of the decision chart from EurOtop (2018). In figure 10.3 the implemented decision chart is depicted with the references to the individual formulas.

Figure 10.3: decision chart for prediction of overtopping discharge for a vertical or composite vertical wall

breakwater.core.overtopping.vertical(Hm0, q, h, d, L_m_min_1, s_m_min_1, safety=1, logger=None, limit=True)

Compute crest freeboard for vertical and composite vertical walls

Compute the crest freeboard, Rc, of a vertical or composite vertical wall for a given mean overtop discharge. The function is an implementation of the decision chart, figure 7.2, from EurOtop (2018). The function determines if the input classifies as a vertical or composite vertical wall, if breaking is possible and if the structure has a low freeboard. Based on the classification the function calls the corresponding function and computes the freeboard.

Parameters:

• Hm0 (float) – the spectral wave height [m]
• q (float) – mean overtopping discharge per meter structure width [l/s per m]
• h (float) – water depth in front of the toe of the structure [m]
• d (float) – water depth above the toe of the structure [m]
• L_m_min_1 (float) – \(L_{m-1.0}\), spectral wave length in deep water [m]
• s_m_min_1 (float) – \(s_{m-1.0}\), wave steepness with the spectral wave length (\(L_{m-1.0}\)) [-]
• safety (float, optional, default: 1) – safety factor of the design, positive values increase the safety of the design by increasing the mean value of the model constants with the number of standard deviations specified. In accordance with the recommendation from EurOtop (2018), the default value is set to 1 standard deviation.
• logger (dict, optional, default: None) – dict to log messages, must have keys ‘INFO’ and ‘WARNINGS’
• limit (bool, optional, default: True) – If True, the discharge will be set to the limit for zero discharge in case the given discharge is below this limit. If False, the discharge will not be changed.

Returns:

Rc (float) – the crest freeboard of the structure [m]

10.2.2. Formulas

breakwater.core.overtopping.vertical_deep(Hm0, q, safety=1, limit=True)

Rc if the foreshore does not have an influence

Compute the crest freeboard for a vertical or composite vertical wall if the foreshore does not have an influence. Implementation of equation 7.1 from EurOtop (2018).

\[\frac{q}{\sqrt{g \cdot H_{m 0}^{3}}}=0.047 \cdot \exp \left[ -\left(2.35 \frac{R_{c}}{H_{m 0}}\right)^{1.3}\right]\]

The reliability of the equation is given by \(\sigma\) (0.047) = 0.007 and \(\sigma\) (2.35) = 0.23.

Parameters:

• Hm0 (float) – the spectral wave height [m]
• q (float) – mean overtopping discharge per meter structure width [l/s per m]
• safety (float, optional, default: 1) – safety factor of the design, positive values increase the safety of the design by increasing the mean value of the model constants with the number of standard deviations specified. In accordance with the recommendation from EurOtop (2018), the default value is set to 1 standard deviation.
• limit (bool, optional, default: True) – If True, the discharge will be set to the limit for zero discharge in case the given discharge is below this limit. If False, the discharge will not be changed.

Returns:

Rc (float) – the crest freeboard of the structure [m]

breakwater.core.overtopping.vertical_no_breaking(Hm0, q, safety=1, limit=True)

Rc if no possibility for breaking waves

Compute the crest freeboard for a vertical or composite vertical wall if there are no breaking waves. Implementation of equation 7.5 from EurOtop (2018)

\[\frac{q}{\sqrt{g H_{m 0}^{3}}}=0.05 \exp \left(-2.78 \frac{R_{c}}{H_{m 0}}\right)\]

The reliability of the equation is given by \(\sigma\) (0.05) = 0.012 and \(\sigma\) (2.78) = 0.17.

Parameters:

• Hm0 (float) – the spectral wave height [m]
• q (float) – mean overtopping discharge per meter structure width [l/s per m]
• safety (float, optional, default: 1) – safety factor of the design, positive values increase the safety of the design by increasing the mean value of the model constants with the number of standard deviations specified. In accordance with the recommendation from EurOtop (2018), the default value is set to 1 standard deviation.
• limit (bool, optional, default: True) – If True, the discharge will be set to the limit for zero discharge in case the given discharge is below this limit. If False, the discharge will not be changed.

Returns:

Rc (float) – the crest freeboard of the structure [m]

breakwater.core.overtopping.vertical_normal(Hm0, q, h, s_m_min_1, safety=1, limit=True)

Rc for a vertical wall if normal freeboard and breaking waves

Compute the crest freeboard for a vertical wall if there is a possibility for breaking waves, but the freeboard is not low. Implementation of equation 7.8 from EurOtop (2018)

\[\frac{q}{\sqrt{g H_{m 0}^{3}}}=0.0014\left(\frac{H_{m 0}} {h s_{m-1,0}}\right)^{0.5}\left(\frac{R_{c}}{H_{m 0}} \right)^{-3}\]

The reliability of the equation is given by \(\sigma\) (0.0014) = 0.0006

Parameters:

• Hm0 (float) – the spectral wave height [m]
• q (float) – mean overtopping discharge per meter structure width [l/s per m]
• h (float) – water depth in front of the toe of the structure [m]
• s_m_min_1 (float) – \(s_{m-1.0}\), wave steepness with the spectral wave length (\(L_{m-1.0}\)) [-]
• safety (float, optional, default: 1) – safety factor of the design, positive values increase the safety of the design by increasing the mean value of the model constants with the number of standard deviations specified. In accordance with the recommendation from EurOtop (2018), the default value is set to 1 standard deviation.
• limit (bool, optional, default: True) – If True, the discharge will be set to the limit for zero discharge in case the given discharge is below this limit. If False, the discharge will not be changed.

Returns:

Rc (float) – the crest freeboard of the structure [m]

breakwater.core.overtopping.vertical_low(Hm0, q, h, s_m_min_1, safety=1, limit=True)

Rc for a vertical wall if low freeboard and breaking waves

Compute the crest freeboard for a vertical wall if there is a possibility for breaking waves, and the freeboard is low. Implementation of equation 7.7 from EurOtop (2018)

\[\frac{q}{\sqrt{g H_{m 0}^{3}}}=0.011\left(\frac{H_{m 0}}{h s_{m-1,0}}\right)^{0.5} \exp \left(-2.2 \frac{R_{c}}{H_{m 0}} \right)\]

The reliability of the equation is given by \(\sigma\) (0.011) = 0.0045

Parameters:

• Hm0 (float) – the spectral wave height [m]
• q (float) – mean overtopping discharge per meter structure width [l/s per m]
• h (float) – water depth in front of the toe of the structure [m]
• s_m_min_1 (float) – \(s_{m-1.0}\), wave steepness with the spectral wave length (\(L_{m-1.0}\)) [-]
• safety (float, optional, default: 1) – safety factor of the design, positive values increase the safety of the design by increasing the mean value of the model constants with the number of standard deviations specified. In accordance with the recommendation from EurOtop (2018), the default value is set to 1 standard deviation.
• limit (bool, optional, default: True) – If True, the discharge will be set to the limit for zero discharge in case the given discharge is below this limit. If False, the discharge will not be changed.

Returns:

Rc (float) – the crest freeboard of the structure [m]

breakwater.core.overtopping.composite_normal(Hm0, q, h, d, s_m_min_1, safety=1, limit=True)

Rc for a composite wall if normal freeboard and breaking waves

Compute the crest freeboard for a composite vertical wall if there is a possibility for breaking waves, but the freeboard is not low. Implementation of equation 7.14 from EurOtop (2018)

\[\frac{q}{\sqrt{g H_{m 0}^{3}}}=1.3\left(\frac{d}{h}\right)^{0.5} 0.0014\left(\frac{H_{m 0}}{h s_{m-1,0}}\right)^{0.5}\left( \frac{R_{c}}{H_{m 0}}\right)^{-3}\]

The reliability of the equation is given by \(\sigma\) (0.0014) = 0.0006.

Parameters:

• Hm0 (float) – the spectral wave height [m]
• q (float) – mean overtopping discharge per meter structure width [l/s per m]
• h (float) – water depth in front of the toe of the structure [m]
• d (float) – water depth above the toe of the structure [m]
• s_m_min_1 (float) – \(s_{m-1.0}\), wave steepness with the spectral wave length (\(L_{m-1.0}\)) [-]
• safety (float, optional, default: 1) – safety factor of the design, positive values increase the safety of the design by increasing the mean value of the model constants with the number of standard deviations specified. In accordance with the recommendation from EurOtop (2018), the default value is set to 1 standard deviation.
• limit (bool, optional, default: True) – If True, the discharge will be set to the limit for zero discharge in case the given discharge is below this limit. If False, the discharge will not be changed.

Returns:

Rc (float) – the crest freeboard of the structure [m]

breakwater.core.overtopping.composite_low(Hm0, q, h, d, s_m_min_1, safety=1, limit=True)

Rc for a composite wall if low freeboard and breaking waves

Compute the crest freeboard for a composite vertical wall if there is a possibility for breaking waves, and the freeboard is low. Implementation of equation 7.15 from EurOtop (2018)

\[\frac{q}{\sqrt{g H_{m 0}^{3}}}=1.3\left(\frac{d}{h} \right)^{0.5} 0.011\left(\frac{H_{m 0}}{h s_{m-1,0}} \right)^{0.5} \exp \left(-2.2 \frac{R_{c}}{H_{m 0}}\right)\]

The reliability of the equation is given by \(\sigma\) (0.011) = 0.0045.

Parameters:

• Hm0 (float) – the spectral wave height [m]
• q (float) – mean overtopping discharge per meter structure width [l/s per m]
• h (float) – water depth in front of the toe of the structure [m]
• d (float) – water depth above the toe of the structure [m]
• s_m_min_1 (float) – \(s_{m-1.0}\), wave steepness with the spectral wave length (\(L_{m-1.0}\)) [-]
• safety (float, optional, default: 1) – safety factor of the design, positive values increase the safety of the design by increasing the mean value of the model constants with the number of standard deviations specified. In accordance with the recommendation from EurOtop (2018), the default value is set to 1 standard deviation.
• limit (bool, optional, default: True) – If True, the discharge will be set to the limit for zero discharge in case the given discharge is below this limit. If False, the discharge will not be changed.

Returns:

Rc (float) – the crest freeboard of the structure [m]