2 Existing Formulae for Average Wave Overtopping of Sloping Structures ...3
2.1 Owen (1980) ...4
2.2 Correction factor of Besley (1999) to formula of Owen (1980) ...4
2.3 Van der Meer & Janssen (1994) ...5
2.4 Pedersen (1996) ...5
2.5 Hebsgaard et. al. (1998) ...6
3 Existing Knowledge on Influence of Wave Direction and Directional Spreading on Wave Overtopping ...6
4 Existing Knowledge on Horizontal Distribution of Wave Overtopping Behind the Breakwater Crest ...7 5 Test Setup...7 5.1 Cross Sections...8 5.2 Overtopping Tank ...9 5.3 Layout in Basin ...10 6 Test Series...13 7 Data Analysis ...15
8 Average Overtopping – Comparison of Measurements to Existing Formulae and Data...16
8.1 Owen (1980) ...16
8.2 Owen (1980) and Besley (1999) ...19
8.3 Van der Meer & Janssen (1994) – Non-Breaking Waves...21
8.4 Van der Meer & Janssen (1994) – Breaking Waves...24
8.5 Van der Meer & Janssen (1994) with correction of Besley (1999) ...27
8.6 Hebsgaard et. al. (1998) ...29
8.7 Comparison with Data of Pedersen (1996) ...32
9 New Formula for Direction Reduction Factor ...38
10 Spatial Distribution of Overtopping...41
11 Conclusions...42
1 Introduction
The international CLASH project of the European Union (Crest Level Assessment of coastal Structures by full scale monitoring, neural network prediction and Hazard analysis on permissible wave overtopping, www.clash-eu.org) under contract no. EVK3-CT-2001-00058 is focussing on wave overtopping for different structures in prototype and in laboratory (see De Rouck et al., 2002). The main scientific objectives of CLASH are (i) to solve the problem of possible scale effects for wave overtopping and (ii) to produce a generic prediction method for crest height design or overtopping assessment.
Based on a database of laboratory data on wave overtopping (≈10,000 tests) a neural network technique is used for producing a generic prediction method. In the first database (≈7,000 tests) the following white spots were detected where additional tests could improve the generic prediction method:
Influence of surface roughness/permeability
Effect of obliqueness, short-crested waves and directional spreading Influence of roughness around still water level
Low steepness (s0p < 0.01) Influence of Gc and Ac Angle of berm
Toe details
The first two were considered as the most important ones. Besides from those two, tests have been performed with low steepness and reshaping breakwaters at Ghent University and Aalborg University respectively. The reshaping breakwater is a type of breakwater where the information on overtopping is very limited. The additional tests are described in the following four reports:
D24 – Part A: Effect of obliqueness, short-crested waves and directional spreading D24 – Part B: Influence of surface roughness/permeability
D24 – Part C: Low steepness tests
D24 – Part D: Reshaping breakwater tests
This report deals with the laboratory tests performed to give additional information on obliqueness, short-crested waves and directional spreading performed at Aalborg University, 2004.
2 Existing Formulae for Average Wave Overtopping of
Sloping Structures
Wave overtopping is very unevenly distributed in time and space because the amount of water overtopping the structure varies considerably from wave to wave. In fact the local overtopping discharge from a single wave can be more than 100 times the mean overtopping discharge. Nevertheless, most information on overtopping is given as the time averaged overtopping discharge (Q) expressed in m3/s per meter structure length. Design of structures is therefore normally based on average overtopping discharge.
(
b R)
a Q= ⋅exp − ⋅ (1) b R a Q= ⋅ − (2)in which Q is a dimensionless average overtopping discharge and R is a dimensionless freeboard. The following five overtopping models are described in the following and used to check the new data:
Owen (1980)
Owen (1980) with correction of Besley (1999) Van der Meer & Janssen (1994)
Pedersen (1996)
Hebsgaard et. al. (1998)
2.1 Owen
(1980)
Many authors refer to the formula given by Owen (1980). ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ γ ⋅ ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ m m r C m m T g H R B A H g T Q 1 exp 0 0 (3) which could be rewritten as:
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ γ ⋅ π ⋅ ⋅ − ⋅ = π ⋅ ⋅ r m m C m m s H R B A s H g Q 1 2 exp 2 0 0 0 3 0 (4)
where γr is a surface roughness/permeability reduction factor which should be taken as 0.50-0.55 for simple slopes with two layers of rock as armour.
The value of the coefficients A and B is given in Table 1.
Slope A B 1:1 0.008 20 1:1.5 0.010 20 1:2 0.013 22 1:3 0.016 32 1:4 0.019 47
Table 1: Coefficients in formula of Owen (1980) for non-depth limited waves and straight slopes.
2.2
Correction factor of Besley (1999) to formula of Owen (1980)
Owen (1980) produced typical values of the roughness coefficient upon the relative run-up performance of alternative types of construction. These coefficients were originally derived for simple slopes but can conservatively also be applied to bermed slopes [Besley (1999)].
Armoured seawalls often include a crest berm that will dissipate significant wave energy and thus reduce overtopping. Owen (1980) does not take this into account in the equation and hence
overtopping discharges are over predicted. Besley (1999) performed additional tests to include the effect of a rock armoured crest berm in the formula of Owen (1980). Besley (1999) found that the permeable crest could be taken into account by multiplying the following reduction factor (Cr) to the overtopping discharge calculated by Eq. 3:
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ − ⋅ =min 3.06 exp 1.5 ;1 0 m C r H G C (5)
Which will say no reduction when Gc/Hm0 < 0.75.
2.3
Van der Meer & Janssen (1994)
Van der Meer & Janssen (1994) is a widely used model to calculate mean overtopping discharge. Two different cases are considered; the case when waves will break on the structure (ξ0p<2) and the case where they will not (ξ0p>2). The formulae are based on data from Owen and from Delft Hydraulics. The structure of the formula for breaking waves is very similar to that of Owen (1980). Non-breaking waves (ξ0p>2): ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ − ⋅ = ⋅ γ 1 6 . 2 exp 2 . 0 0 3 0 m C m H R H g Q (6) Breaking waves (ξ0p<2):
( )
( )
⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ ⋅ − ⋅ = ⋅ ⋅ α α γ 1 tan 2 . 5 exp 06 . 0 tan 0 0 0 3 0 p m C p m s H R s H g Q (7)The reduction factor (γ) is taken as a product of reduction factors: β γ γ γ γ γ = r ⋅ b ⋅ h⋅ (8)
Which takes into account the influence of surface roughness/permeability, berm, water depth and wave direction. The minimum value of any combination of the γ-factors is 0.5. In some later publications a minimum value of 0.4 is suggested due to limited information on all combinations. Distinction between breaking and non-breaking waves is made based on ξ0p. Doing this, there is a discontinuity between both formulae. This was overcome by the formulae presented by Van der Meer et. al. (1998) where the lowest value of Q calculated from Eq. 6 and 7 should be used.
The formulae does not include the effect of a permeable crest, thus the correction factor of Besley (1999) can improve the formulae as demonstrated later in this report.
2.4 Pedersen
(1996)
Pedersen (1996) made an extensive research on overtopping of crown wall rubble mound breakwaters. The total test number is 373 with overtopping events in 284 of the tests. All tests are with head on waves (flume tests). Pedersen (1996) gives the following formula to calculate the average overtopping discharge:
( )
( ) cot 10 2 . 3 3 5 5 2 0 armour f G A R H L T Q C C C s m m ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ − α (9)Where f(armour) = 1 for rock. The advantage of this formula compared to the formula of Van der Meer & Janssen (1994) is the inclusion of the two parameters Ac and Gc describing the crown geometries. The disadvantage of the formula is that it is very conservative for small overtopping discharges.
2.5
Hebsgaard et. al. (1998)
Hebsgaard et. al. (1998) presented on the basis of a model test study the following formula to calculate overtopping on rubble mound breakwaters:
(
)
( )
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ β ⋅ ⋅ + ⋅ ⋅ α ⋅ ⋅ = ⋅ ⋅ m r c C p m H G R k k s H g Q 1 cos 35 . 0 2 ) ( cot exp ) ln( 1 0 3 . 0 2 1 0 3 0 (10)The following values are recommended by Hebsgaard et. al. (1998) for k1 and k2:
Pure rubble mound structures: k1 = -0.3 and k2 = -1.6
Rubble mound structure with superstructure (both high and low): k1 = -0.01 and k2 = -1.0 These coefficients have been selected so the formula fits the data when applying a roughness factor (γr) of 0.55 for rock in two layers as suggested by Van der Meer & Janssen (1994) and others. The influence of the wave direction is based on results from 6 different angles of wave attacks (0,10,20,30,40 and 50 degrees) with long crested waves.
The formula includes the Gc parameter but the Ac parameter is not included.
3 Existing Knowledge on Influence of Wave Direction and
Directional Spreading on Wave Overtopping
The influence of angled wave attack on the overtopping discharge of sloping structures is studied by various authors (i.e., Herbert and Owen (1995), Pilarczyk and Zeidler (1996) and Herbert et al. (1994)). For long-crested waves, the mean overtopping discharge is greatest for angles of wave attack varying from 0° to 20°, whereas for short-crested waves, the mean overtopping discharge stays fairly constant in the range of angles between 0° and 30°, before falling to smaller values for larger angles.
Allsop (1994) found the effect of recurved walls smaller for oblique wave attack than for perpendicular wave attack.
Banyard and Herbert (1995) studied the performance of recurve walls under angled wave attack. The recurved crown walls exhibited similar behaviour in both short and long-crested seas. Under angled wave attack significant increases in overtopping discharges can occur. The largest measured increases in overtopping were over six times that predicted for normal wave attack. Banyard and
wall’s discharge factor, Df, rather than the angle of attack. discharge factor, Df, is the ratio of the discharge overtopping the recurve wall to the discharge which would have occurred if the recurve wall had been absent The greatest increases in overtopping occurred when the discharge factor was low. For Df > 0.31 however, discharge under angled wave attack decreased.
Van der Meer & Janssen (1994) and Van der Meer et al. (1998) give the following reduction factors for angled wave attack:
Short crested waves: γβ =1−0.0033⋅β (11)
Long crested waves:
(
)
⎪ ⎩ ⎪ ⎨ ⎧ ° > ° ≤ ≤ ° ° − ° ≤ ≤ ° = 50 6 . 0 50 10 10 cos 10 0 0 . 1 2 β β β β γβ for for for (12)
For long-crested waves, the reduction factor stays fairly constant in the range of angles between 0° and 30°, before falling to smaller values for larger angles. For short-crested waves the reduction factor diminishes slowly with increasing angle of wave attack.
Sakakiyama and Kajima (1998), carried out experiments on a seawall in a multidirectional wave basin. Multidirectional waves give significantly smaller overtopping rates than uni-directional waves for normal incident condition, while oblique waves give the smaller overtopping rates.
Hebsgaard et. al. (1998) tested 6 different angles of wave attacks (0,10,20,30,40 and 50 degrees) with long crested waves and found that the following expression for the reduction factor (γβ):
( )
βγβ =cos (13)
4 Existing Knowledge on Horizontal Distribution of Wave
Overtopping Behind the Breakwater Crest
The intensity of wave overtopping behind a breakwater decreases very quickly with the horizontal distance from the breakwater crest. Based on model tests and prototype measurements Juul Jensen and Sørensen (1979) suggested an exponential decay of the overtopping intensity:
β / 0 10 ) (x q x q = ⋅ − (14)
where q is the intensity at a distance x, and q0 is the intensity for x = 0. The parameter β is a constant and equal to the distance for which the overtopping intensity decreases by a factor of 10. By integrating Eq. 14 from zero to infinity the total amount of overtopping (Q) can be calculated if the constant β is known. Hence the following expression for q0 can be derived:
( )
β 10 ln 0 ⋅ =Q q (15)5 Test
Setup
5.1 Cross
Sections
Two typical cross sections were tested, one with rock and one with cubes. Further four crest freeboards (Rc) were tested by varying the water depth, thus no rebuilding needed when changing freeboard. The four freeboards (Rc) tested were 0.07m, 0.10m, 0.13m and 0.16m, and the corresponding water depths (h) were 0.545, 0.515, 0.485m and 0.455m. The materials used for the breakwater are listed in Table 2.
Armour Rock Armour Cubes Filter Core
Weight W50 [kg] 0.228 0.146 7.6·10-3 0.39·10-3
Density ρ [kg/m3] 3060 2280 2670 2710
Nominal diameter Dn,50 [m] 0.042 0.040 0.014 ≈ 0.005
fg = Dn,85/Dn,15 1.30 1.00 1.43 1.47
Length ratio, l/b 1.96 1.00 2.08
Table 2: Material properties.
Figure 1 shows grain curves of the used materials.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 10.0 20.0 30.0 40.0 50.0 60.0 Dn [mm] P( D <D n) Frontside Armour Filter Core
Figure 1: Grain curves for used materials.
The front slope was 1:2 and the rear slope was 1:1.5 which is very typical for both rock and cubes. The thickness of the armour layer was 0.09m corresponding to approximately 2×D50 for the rock case and 2.25×Dn for the cubes. The cubes were placed randomly with 1080 cubes per meter width, which corresponds to a porosity of 42%. The thickness of the filter layer was 0.030m. The crown width (Gc) was 0.13m corresponding to approximately 3 ×Dn,50.
130 1:2 165 49 5 61 5 130 1:1.5 h=0.545m h=0.485m h=0.515m h=0.455m DIFFERENT WATER LEVELS
Figure 2: Tested cross-section.
5.2 Overtopping
Tank
The overtopping tank was 0.38m wide and had five chambers so the distribution of overtopping water behind the breakwater could be measured. The distribution was measured at the top level of the super-structure as shown on Figure 3 and 5.
430
145 145 145 145 145
Overtopping tank (0.38m wide)
DIFFERENT WATER LEVELS
h=0.455m h=0.515m h=0.485m h=0.545m
Figure 3: Overtopping tank.
Each chamber collects water from an area with a length of 0.145m. The main part of the overtopping water was collected in the first chamber. To give reasonable accuracy on the other four chambers the lower part of the chamber was narrower than the upper part collecting the water as shown in Figure 3.
Permanent pumps were installed in chamber 1 and 2. The data acquisition software controlled these two pumps, so they started and stopped automatically and the state of the pumps was stored in the data file. To measure the amount of overtopping water a water surface elevation gauge was installed in each chamber. The overtopping tank and super-structure was anchored to the bottom as shown in Figure 4.
Figure 4: Construction phase.
Figure 5: Overtopping tank.
5.3
Layout in Basin
The tests were performed in the shallow water basin at Aalborg University, Department of Civil Engineering. This basin is 12 m long 17.8 m wide and 1.0 m deep. The paddle system is a snake-front piston type with a total of 25 actuators with a stroke length of 1.2 m, enabling generation of short-crested waves. The laboratory has designed the wave basin and wave generation software in house.
Generating short crested waves with an angle other than perpendicular to the wave maker will lead to an asymmetrical spreading function and/or asymmetrical amount of spurious waves. As a
Pumps
Water surface
elevation gauges
consequence the structure was built in the basin at three different angles in order to test a large angle interval.
5 degrees (testing 0 and 10 degrees). 25 degrees (testing 25 degrees).
52.5 degrees (testing 45 and 60 degrees).
No refraction and shoaling of waves on the foreshore took place as a flat bottom was used in all tests. The flat bottom limits testing to non-breaking waves on the foreshore. The test section is 5m wide due to the very oblique wave attack in layout 3 (see Figure 8).
From visual observations the overtopping looks very uniform in layout 1 and 2. In layout 3 it did not look as uniform as in layout 1 and 2, especially the first 2-3 meters from the end nearest the wave maker looked non-uniform. The overtopping tank was not located at the centre of the structure in layout 2 & 3 but shifted slightly towards the end (see Figure 7 and 8), where the overtopping looked more uniform.
Passive absorption was placed at the end and at both sides of the basin.
Wavepaddle, max. position
Wavepaddle, max. position
Figure 7: Layout in basin for testing 25°.
Wavepaddle, max. position
Figure 8: Layout in basin for testing 45° and 60°.
At the roundheads of the structure, some larger armour stones were used. Not enough cubes were available to cover the whole front of the structure. The cubes covered 3 meters, the remaining two meters (a band in each end) were covered by rock.
The waves were measured just in front of the structure, with an array of 7 wave gauges, placed as shown in Figure 6-8. The array was orientated so it in all tests had the same orientation to the mean direction of the generated waves.
Figure 9: Layout in basin for testing 0° and 10°.
6 Test
Series
The total number of tests performed is 736. The length of the test series is around 1500 waves. Both long crested and short crested waves were tested, which will give information on the influence of directional spreading on overtopping discharges and on distribution of overtopping behind the breakwater.
The waves were in all tests generated from the JONSWAP spectrum with a peak enhancement factor (γ) of 3.3 using a white noise filtering method.
In case of short crested waves the Longuet-Higgins cosines spreading function with the following definition is used:
( )
(
(
)
)
⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ + Γ + Γ ⋅ = − 2 ) ( cos 1 2 1 2 , 2 1 2 2 0 f s s f D s s θ θ π θ (16)Two values of the spreading parameter s is tested (5 and 10) corresponding to a standard deviation (σ) of 34.5° and 25.0°.
A fairly wide range of wave steepness was tested as shown in Figure 10. 3-6 different wave heights were tested on each cross section.
Figure 10: Overview of sea states.
Target overtopping discharges is 0.05 l/ms to 10 l/ms in prototype scale, which corresponds to 0.0002 l/ms to 0.05 l/ms in laboratory tests assuming a scale of 1:35.
The Reynolds number (
ν = ,50 0
Re Dn gHm ) is in the range 2.8·104 to 5.2·104. The armour stability index (Hs/∆Dn,50) was in the range 0.6 to 1.8 for rock and 0.9 to 3.2 for cubes.
An overview of the tests performed is given in Table 3.
Direction
[deg.] Spreading (s) Freeboard (R[m] c) Water depth (h) [m] No. of Tests H[m] m0 [s] Tp
0 None 0.07 0.545 22 0.053-0.108 0.91-1.90 0 None 0.10 0.515 20 0.069-0.129 0.98-1.90 0 None 0.13 0.485 21 0.072-0.138 0.98-1.97 0 None 0.16 0.455 15 0.085-0.140 1.14-1.90 0 5 0.07 0.545 22 0.051-0.105 0.98-1.83 0 5 0.10 0.515 15 0.064-0.123 1.00-1.65 0 5 0.13 0.485 20 0.078-0.125 1.11-1.97 0 5 0.16 0.455 14 0.086-0.134 1.11-1.83 0 10 0.07 0.545 0 - - 0 10 0.10 0.515 8 0.070-0.127 1.00-1.42 0 10 0.13 0.485 11 0.080-0.126 1.02-1.90 0 10 0.16 0.455 7 0.087-0.136 1.14-1.90 10 None 0.07 0.545 12 0.057-0.107 0.88-1.83 10 None 0.10 0.515 8 0.082-0.128 1.11-1.38 10 None 0.13 0.485 12 0.081-0.138 1.11-1.90 10 None 0.16 0.455 7 0.091-0.134 1.14-1.83 10 5 0.07 0.545 22 0.048-0.106 0.93-1.77
10 5 0.10 0.515 16 0.077-0.126 1.11-1.60 10 5 0.13 0.485 18 0.074-0.127 0.98-1.90 10 5 0.16 0.455 14 0.087-0.139 1.11-1.90 10 10 0.07 0.545 0 0.000-0.000 0.00-0.00 10 10 0.10 0.515 6 0.077-0.126 1.11-1.83 10 10 0.13 0.485 0 - - 10 10 0.16 0.455 0 - - 25 None 0.07 0.545 21 0.065-0.117 0.95-1.90 25 None 0.10 0.515 18 0.081-0.138 1.05-1.71 25 None 0.13 0.485 21 0.079-0.139 1.07-2.05 25 None 0.16 0.455 17 0.090-0.151 1.09-1.77 25 5 0.07 0.545 21 0.062-0.118 1.02-1.83 25 5 0.10 0.515 17 0.071-0.129 1.14-1.77 25 5 0.13 0.485 19 0.082-0.139 1.11-2.05 25 5 0.16 0.455 16 0.085-0.148 1.11-1.90 25 10 0.07 0.545 0 - - 25 10 0.10 0.515 0 - - 25 10 0.13 0.485 0 - - 25 10 0.16 0.455 0 - - 45 None 0.07 0.545 27 0.070-0.121 0.97-1.90 45 None 0.10 0.515 18 0.093-0.142 1.16-1.90 45 None 0.13 0.485 18 0.092-0.151 1.16-1.90 45 None 0.16 0.455 12 0.110-0.164 1.25-1.97 45 5 0.07 0.545 25 0.065-0.116 1.00-1.90 45 5 0.10 0.515 19 0.079-0.138 1.19-1.83 45 5 0.13 0.485 19 0.095-0.147 1.22-1.97 45 5 0.16 0.455 17 0.103-0.153 1.28-1.97 45 10 0.07 0.545 0 - - 45 10 0.10 0.515 0 - - 45 10 0.13 0.485 9 0.104-0.146 1.22-1.90 45 10 0.16 0.455 0 - - 60 None 0.07 0.545 26 0.070-0.130 0.91-1.83 60 None 0.10 0.515 15 0.100-0.153 1.22-1.71 60 None 0.13 0.485 11 0.124-0.156 1.22-1.97 60 None 0.16 0.455 2 0.148-0.152 1.38-1.90 60 5 0.07 0.545 26 0.064-0.131 1.00-1.90 60 5 0.10 0.515 18 0.092-0.148 1.22-1.71 60 5 0.13 0.485 15 0.104-0.143 1.19-1.90 60 5 0.16 0.455 12 0.113-0.149 1.35-1.77 60 10 0.07 0.545 0 - - 60 10 0.10 0.515 0 - - 60 10 0.13 0.485 6 0.117-0.144 1.35-1.90 60 10 0.16 0.455 1 0.137 1.90
Table 3: Overview of test series.
7 Data
Analysis
All signals are filtered using an analog lowpassfilter with a cut-off frequency of 8Hz. All signal analysis is done with the WaveLab software package [http://hydrosoft.civil.auc.dk/wavelab]. A digital filter with cut-off frequencies of 1/3·fp and 3·fp is applied to the wave signals. The BDM method was applied to calculate the directional spectrum and the incident frequency domain wave parameters.
Time domain analysis has been performed on total signals (incident + reflected waves) to check wave height distributions. This analysis showed that for the tests with small waves the wave height distribution more or less follows the Rayleigh distribution. For the higher waves the wave height distribution deviates from the Rayleigh distribution. This was also expected as the Rayleigh distribution only is valid for deep water waves. In the test series with the highest waves and the lowest water depth some wave breaking was observed for the highest waves in the test series.
8 Average Overtopping – Comparison of Measurements to
Existing Formulae and Data
In this chapter the measured mean overtopping discharge is compared to the following 5 formulae: Owen (1980)
Owen (1980) with correction factor of Besley (1999)
Van der Meer & Janssen (1994) - Non-breaking & breaking waves
Van der Meer & Janssen (1994) with correction factor of Besley (1999) (Non-breaking waves)
Hebsgaard et. al. (1998)
Furthermore the present data is compared to the data of Pedersen (1996) in different plots.
8.1 Owen
(1980)
Many authors refer to the measurements and formula of Owen (1980). The formula of Van der Meer and Janssen (1994) is also based on the measurements of Owen (1980). The formula of Owen (1980) should be valid for 0.05 < R* < 0.3.
In the following plots the data are compared to the formula of Owen (1980) for the five tested directions. In the plots the shape of the data point indicates if it cubes or rock and the color indicates the directional spreading. One can observe a lot of scatter in these plots but there seems not to be any difference between cubes and rock.
Figure 11: Comparison of present data and formula of Owen (1980) for β = 0 degrees.
Figure 13: Comparison of present data and formula of Owen (1980) for β = 25 degrees.
Figure 15: Comparison of present data and formula of Owen (1980) for β = 60 degrees.
8.2
Owen (1980) and Besley (1999)
In the following five figures the reduction factor suggested by Besley (1999) to take into account the effect of a crest berm is included in the formula of Owen (1980). The correction clearly is an improvement of the formula as less scatter is observed. However the data is still far from γ = 0.5 -0.55 as given by Owen (1980) for rock.
Figure 17: Comparison of present data and formula of Owen (1980) and Besley (1999) for β = 10 degrees.
Figure 19: Comparison of present data and formula of Owen (1980) and Besley (1999) for β = 45 degrees.
Figure 20: Comparison of present data and formula of Owen (1980) and Besley (1999) for β = 60 degrees.
8.3
Van der Meer & Janssen (1994) – Non-Breaking Waves
In the following five plots the present data are compared with the formula of Van der Meer & Janssen (1994) for non-breaking waves. Van der Meer & Janssen (1994) recommends using the formula for non-breaking waves when ξ0p>2. In the present case the irribarren number (ξ0p) lies in the range 2.03<ξ0p<4.27, which means non-breaking waves in all tests. For non-breaking waves
Van der Meer & Janssen (1994) found little influence of the wave period on the overtopping discharge, and this parameter is thus not included in the formula.
For β = 0° the data points the data points lie in the range γ =0.3 to γ =0.45. Van der Meer & Janssen (1994) recommend γr =0.55 with all other γ-values equal to 1 in the β = 0° case. This means the formula of Van der Meer & Janssen (1994) for non-breaking waves over predicts the overtopping discharge by a factor of 10-100 in the present case.
Van der Meer & Janssen (1994) state the minimum value of any combination of γ-values is 0.5. In the present case the γ-value is clearly lower for all five directions with a minimum around γ =0.2 for the most oblique wave attack and long crested waves.
Figure 21: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) for β = 0 degrees.
Figure 22: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) for β = 10 degrees.
Figure 23: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) for β = 25 degrees.
Figure 24: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) for β = 45 degrees.
Figure 25: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) for β = 60 degrees.
8.4
Van der Meer & Janssen (1994) – Breaking Waves
Even though it is the formula for non-breaking waves that should be used as demonstrated in the last section, the formula for breaking waves is also used for evaluation. Sometimes when it in principle should be non-breaking waves (ξ0p>2) much less scatter can be observed in the plot for breaking waves and visa versa. In the next five plots the present data are plotted against the
breaking waves model of Van der Meer and Janssen (1994). One can easily see that more scatter is on the data in these plots compared to the non-breaking waves plots.
The formula of Van der Meer & Janssen (1994) for breaking waves has a structure which is very similar to the one of Owen (1980). Therefore these plots look very similar and the same amount of scatter is observed for these two formulae.
Figure 26: Comparison of present data and breaking waves formula of Van der Meer & Janssen (1994) for β = 0 degrees.
Figure 27: Comparison of present data and breaking waves formula of Van der Meer & Janssen (1994) for β = 10 degrees.
Figure 28: Comparison of present data and breaking waves formula of Van der Meer & Janssen (1994) for β = 25 degrees.
Figure 29: Comparison of present data and breaking waves formula of Van der Meer & Janssen (1994) for β = 45 degrees.
Figure 30: Comparison of present data and breaking waves formula of Van der Meer & Janssen (1994) for β = 60 degrees.
8.5
Van der Meer & Janssen (1994) with correction of Besley (1999)
In this section the data are plotted agains the Van der Meer formula for non-breaking waves with the correction factor of Besley (1999) to include the effect of the permeable crest. The correction factor of Besley has showed to be an improvement for the Owen (1980) formula and the following five figures shows that this is also the case for the Van der Meer & Janssen (1994) formula.
Figure 32: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) with correction factor of Besley (1999) for β = 10 degrees.
Figure 33: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) with correction factor of Besley (1999) for β = 25 degrees.
Figure 34: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) with correction factor of Besley (1999) for β = 45 degrees.
Figure 35: Comparison of present data and non-breaking waves formula of Van der Meer & Janssen (1994) with correction factor of Besley (1999) for β = 60 degrees.
8.6
Hebsgaard et. al. (1998)
The following five figures show comparison of measured overtopping discharge and calculated by the formula of Hebsgaard et. al. (1998). The formula of Hebgaard et. al. (1998) is adjusted to be in
0.55 for 2 layer of rock. Hebsgaard et. al. (1998) give no value for pure cubes, but recommend γr = 0.65 for antifer cubes. In the graphs below γr = 0.55 is applied for both cubes and rock.
For the present data the model of Hebsgaard et. al (1998) is superior to the model of Owen (1980) and Van der Meer & Janssen (1994). Without modifying any constants in the model, it gives a very good fit to the present data. More or less all measured overtopping discharges lies within a factor of 10 of the estimated value, at least for no and small obliqueness.
For large obliqueness it could easily be seen that the formula is fitted to experiments with long-crested waves only, as the formula fits these data well but poorly fits the data with short-long-crested waves. As demonstrated in Chapter 9 this could easily be improved by using a different direction reduction factor.
Figure 37: Comparison of present data and formula of Hebsgaard et. al. (1998) for β = 10 degrees.
Figure 39: Comparison of present data and formula of Hebsgaard et. al. (1998) for β = 45 degrees.
Figure 40: Comparison of present data and formula of Hebsgaard et. al. (1998) for β = 60 degrees.
8.7
Comparison with Data of Pedersen (1996)
In this section the present data are compared to the data of Pedersen (1996). The comparison is done by plotting the present data & the data of Jan Pedersen against the following formulae:
Owen (1980)
Van der Meer & Janssen (1994) - Non-breaking & breaking waves
Van der Meer & Janssen (1994) with correction factor of Besley (1999) (Non-breaking waves)
Hebsgaard et. al. (1998) Jan Pedersen (1996)
In the Owen (1980) plot more scatter is observed on the present data compared to the data of Pedersen (1996). The reason for this is not completely clear. The data of Pedersen (1996) correspond to γ ≈ 0.35 wheras the present data correspond to γ ≈ 0.27.
Figure 41: Comparison of present data and data of Pedersen (1996) in the Owen (1980) plot (Only data with Ac = Rc and β = 0 degrees).
The correction of Besley (1999) to the formula of Owen (1980) is an improvement for both data sets. More scatter is observed on the present data in this plot too.
Figure 42: Comparison of present data and data of Pedersen (1996) in the Owen (1980) and Besley (1999) plot (Only data with Ac = Rc and β = 0 degrees).
All present data corresponds to non-breaking waves in the Van der Meer & Janssen (1994) formula. The data of Pedersen (1996) includes both breaking and non-breaking waves, however only data where Ac = Rc is included in the plot and these are all non-breaking waves.
In the non-breaking plot of Van der Meer & Janssen (1994) the data of Pedersen (1996) corresponds to a higher reduction factor (γ ≈ 0.45). But still far from the value of γ = 0.55 which is recommended by Van der Meer & Janssen (1994). This means the formula of Van der Meer & Janssen (1994) for non-breaking waves over predicts the overtopping discharge also for the data of Pedersen (1996). A little less scatter is observed for the data of Pedersen (1996) compared to the present data.
Figure 43: Comparison of present data and data of Pedersen (1996) in the
Van der Meer & Janssen (1994) plot for non-breaking waves (Only data with Ac = Rc and β = 0 degrees).
In the breaking waves plot of Van der Meer & Janssen (1994) less scatter is observed on the data of Pedersen (1996) compared to the present data. The data of Pedersen corresponds to a reduction factor γ of approximately 0.25.
Figure 44: Comparison of present data and data of Pedersen (1996) in the
Including the correction factor of Besley (1999) in the non-breaking waves formula of Van der Meer & Janssen improves the formula as less scatter is observed. The data of Pedersen (1996) corresponds to γ ≈ 0.5 and the present data to γ ≈ 0.42 with the correction factor.
Figure 45: Comparison of present data and data of Pedersen (1996) in the Van der Meer & Janssen (1994) and Besley (1999) plot for non-breaking waves (Only data with Ac = Rc and β = 0 degrees).
As mentioned in section 8.6 the present data fits very well with the formula of Hebsgaard et. al. (1998). Unfortunately the formula does not fit as well to the data of Pedersen, in most cases Pedersen (1996) measured more overtopping than calculated by the formula. However, the estimated values are within a factor of 10 from the measured in most cases. More scatter is observed on the data of Pedersen (1996) compared to the present data in this plot.
Figure 46: Comparison of present data and data of Pedersen (1996) in the Hebsgaard et. al. (1998) plot (Only data with Ac = Rc and β = 0 degrees).
The formula of Pedersen (1996) shows very good agreement between the present data and the data of Pedersen (1996) both in mean value and amount of scatter. This demonstrates that the formula of Pedersen (1996) has some good characteristics. However, there is one import bad characteristic of the formula of Pedersen (1996), which is the large over prediction of small overtopping discharges due to linear fitting instead of logarithmic.
Figure 47: Comparison of present data and data of Pedersen (1996) in the Pedersen (1996) plot (Only data with β = 0 degrees).
9 New Formula for Direction Reduction Factor
Results from the tests with long crested waves more or less agree with the formula of Hebsgaard et. al. (1998) for all directions. The formula for the reduction factor for wave direction in the formula of Hebsgaard et. al. (1998) is:
( )
βγβ =cos (17)
For short crested waves and very oblique wave attack the formula of Hebsgaard et. al. (1998) over predicts the reduction factor. This is due to the fact that the formula of Hebsgaard et. al. (1998) is calibrated with an obliquity up to 50 degrees and long-crested waves only.
By analyzing the data it was found that the following expressions for the reduction factor gives a much better fit to the data compared to Eq. 17:
⎪ ⎩ ⎪ ⎨ ⎧ = σ = β ⋅ − = σ = β ⋅ − = σ ∞ = β ⋅ − = γβ ) 35 ( 5 0059 . 0 1 ) 25 ( 10 0068 . 0 1 ) 0 ( 0077 . 0 1 o o o s for s for s for (18) where β is the mean wave direction in degrees. This could be generalised by the following
approximation (β and σ in degrees):
(
σ)
βγβ =1− 0.0077−4.6⋅10−5⋅ ⋅ (19)
In Figure 48 this expression is compared to present data and the expression of Hebsgaard et. al. (1998) and Van der Meer and Janssen (1994). It can be seen that the formula given in Eq. 19 is a much better fit to the present data compared to the other two formulae.
Figure 48: Comparison of wave direction reduction factor formulae and best fit to present data.
The formula of Hebsgaard et. al. (1998) is modified to include the direction factor given in given in Eq. 19. In the next five figures the modified formula is evaluated against the present data. It is
easily observed that inclusion of the new direction factor improves the formula especially for β=60°.
Figure 49: Comparison of present data and the modified Hebsgaard et. al. (1998) formula for β = 0 degrees.
Figure 50: Comparison of present data and the modified Hebsgaard et. al. (1998) formula for β = 10 degrees.
Figure 51: Comparison of present data and the modified Hebsgaard et. al. (1998) formula for β = 25 degrees.
Figure 52: Comparison of present data and the modified Hebsgaard et. al. (1998) formula for β = 45 degrees.
Figure 53: Comparison of present data and the modified Hebsgaard et. al. (1998) formula for β = 60 degrees.
The improvement can also be observed by comparing correlation coefficients as shown in Table 4. Mean direction
[deg.]
Hebsgaard et. al. (1998) direction factor Proposed direction factor 0 R = 0.9122 R = 0.9122 10 R = 0.9202 R = 0.9195 25 R = 0.8401 R = 0.8370 45 R = 0.8408 R = 0.8591 60 R = 0.6546 R = 0.8048
Table 4: Correlation between data and formula of Hebsgaard with original and modified direction factor.
10 Spatial Distribution of Overtopping
The distribution of overtopping water behind the breakwater was measured in 5 chambers as shown in Figure 3. The distribution was measured at the top level of the crown wall.
Chamber number 1 which is located directly behind the crest collects around 90% of the total amount of overtopping water, which is also shown in the Figure 54 below. The Figure shows the distribution from all tests, and large amount of scatter is observed on the percentage of water collected in chamber 3, 4 & 5, as very small volumes of water is collected in these chambers. Actually the performance of the overtopping tank could have been improved by making these two chambers narrower in the bottom part not only in the length but also in the width.
Figure 54: Distribution of overtopping behind crest (probability distribution from all tests).
The distribution between the five chambers is in average as follows: Chamber 1: 89% Chamber 2: 7.9% Chamber 3: 2.0% Chamber 4: 0.10% Chamber 5: 0.09%
11 Conclusions
736 tests have been performed to give more information on the influence of wave direction and directional spreading on wave overtopping. The conclusions can be summarized as follows:
− Two layers of rock and two layers of cubes result in more or less the same amount of overtopping, which means identical roughness/permeability factor should be applied for both armour types.
− The case with β = 60° shows a big difference on overtopping discharge between long and short crested waves, where short crested waves give the most overtopping. This was also visually observed from the tests with short-crested waves and β = 60°, as the waves attacking the breakwater more perpendicular lead to much more overtopping than those attacking with the mean direction. For the other cases (0° ≤ β ≤ 45°) the difference between long and short crested waves are much less pronounced.
− The formula of Owen (1980) and the breaking wave formula of Van der Meer & Janssen gives very similar results for the present data, but with some deviations for the data of Pedersen
(1996). The data correspond to a significantly lower roughness factor than suggested by Owen (1980), as the data corresponds to γr between 0.2 and 0.4.
− In relation to the formulae of Van der Meer & Janssen (1994), all tests performed correspond to non-breaking waves. The results from tests with head-on wave attack correspond to a roughness factor (γr) between 0.3 and 0.45 which is a significantly lower value than suggested by Van der Meer & Janssen (1994). Further a lot of scatter is on the data in the plot of Van der Meer & Janssen (1994).
− Both the Owen (1980) and the Van der Meer & Janssen (1994) formula does not take into account the effect of a permeable crest berm. Besley (1999) suggested a correction factor to take into account this effect. The correction factor is a clear improvement of the formula of Owen (1980) and of Van der Meer & Janssen (1994). However the data correspond in both cases still to a lower roughness factor than suggested by Owen (1980) & Van der Meer & Janssen (1994). Their roughness factors are based on run-up measurements, and are maybe not identical for overtopping.
− Very good agreement exists between the present data and the formula of Hebsgaard et. al. (1998). The formula of Hebsgaard et. al. (1998) is fitted to long-crested waves only. Therefore the formula fits badly to the measurements with short-crested waves and very oblique wave attack. A modified direction factor has been proposed to take in to account the directional spreading, which improved the formula a lot for very oblique wave attack.
− The present data has been compared to the data of Pedersen (1996) who made an extensive study on overtopping of crown wall structures. Very good agreement between the present data and the data of Pedersen (1996) was found in the plot of Pedersen (1996) and fair agreement in the plot of Owen (1980, Owen (1980) and Besley (1999), Van der Meer & Janssen (1994) and Hebsgaard et. al. (1998).
− The intensity of wave overtopping behind the breakwater decreases very rapidly with the horizontal distance from the breakwater crest.
12 References
ALLSOP, N. W. H. (1994): Overtopping of seawalls under random waves. MAST advanced course, Bologna
BANYARD, L and HERBERT, D M. (1995): The effect of wave angle on the overtopping of seawalls. HR Wallingford, Report SR396.
BESLEY, P. (1999): Wave Overtopping of Seawalls – Design and Assessment Manual. HR Wallingford, R&D Technical Report W178.
DE ROUCK, J., VAN DER MEER, J.W., ALLSOP, N.W.H., FRANCO, L. and VERHAEGHE, H. (2002): Wave overtopping at coastal structures: Development of a database towards up-graded prediction methods. Proc. ICCE 2002, ASCE, 2140-2152.
HEBSGAARD, M., SLOTH, P. and JUUL, J. (1998): Wave overtopping of rubble mound breakwaters. ICCE 1998 – Paper No. 325.
HERBERT, D. M., ALLSOP, N. W. H. and OWEN, M. W. (1994): Overtopping of seawalls under random waves. Proc. 24th ICCE, Kobe, p. 1130-1142.
HERBERT, D.M. and OWEN, M. W. (1995): Wave overtopping of sea walls - further research. Proc. of coastal structures and breakwaters, p. 81-92.
JUUL JENSEN, O. and SØRENSEN, T. (1979): Overspilling/overtopping of rubble-mound breakwaters. Results of studies, useful in design procedures. Coastal Engineering 3 (1979), p. 51-65.
OWEN, M.W. (1980): Design of sea walls allowing for wave overtopping, Rep. EX924, HR Wallingford.
PEDERSEN, J. (1996): Wave Forces and Overtopping on Crown Walls of Rubble Mound Breakwaters – An Experimental Study. Series Paper 12, Hydraulics & Coatal Engineering Laboratory, Department of Civil Engineering, Aalborg University.
PILARCZYK, K. and ZEIDLER, R. (1996): Offshore breakwaters and shore evolution control A. Balkema, Rotterdam, p. 96-117.
SAKAKIYAMA, T., KAJIMA, R. (1998). Scale effects on wave overtopping of seawall covered with armor units. Proc. ICCE 1998, Copenhagen.
VAN DER MEER, J. W. and JANSSEN, J. P. F. M. (1994): Wave run-up and wave overtopping at dikes. Wave forces on inclined and vertical wall structures, ASCE.
VAN DER MEER, J. W., TÖNJES, P. and DE WAAL, J. P. (1998). A code for dike height design and examination. Breakwaters Conference, London.
