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Proc. Coastal Dynamics 2001
1
NUMERICAL MODEL SIMULATIONS OF WAVE PROPAGATION AND
WAVE RUN-UP ON DIKES WITH SHALLOW FORESHORES
Marcel R.A. van Gent1 and Neelke Doorn1
ABSTRACT: Shallow foreshores considerably affect wave propagation
and hence wave run-up on coastal structures. This concerns for instance
the evolution of wave height distributions and wave energy spectra
between deep water and the toe of coastal structures. Based on numerical
model investigations and physical model tests it was concluded that wave
run-up and wave overtopping can best be described using wave
conditions at the toe of the coastal structures, in particular using the wave
period Tm-1,0 (Van Gent, 1999, 2000). With this wave period the effects of
arbitrarily-shaped wave energy spectra on wave run-up and wave
overtopping can be taken into account. In this paper estimates of wave
conditions, including estimates of this wave period Tm-1,0, are obtained
using a spectral wave model and a time-domain Boussinesq-type wave
model. Especially the applied Boussinesq-type wave model appeared to
be appropriate for conditions with severe wave breaking on a shallow
foreshore where also a considerable amount of energy is transferred to
lower frequencies. Also wave interaction with the dike itself was
modelled numerically.
1. INTRODUCTION
This paper describes comparisons between test results from two-dimensional
physical model investigations (Van Gent, 1999, 2000) and numerical model
computations. Details of this study are reported in Van Gent and Doorn (2000). In the
model tests use was made of a schematisation of a foreshore with a dike on which
also prototype measurements were performed (Petten Sea-defence).
In Van Gent (1999, 2000) it was described that wave run-up and wave overtopping on
dikes, especially for dikes with shallow foreshores, can best be described using wave
conditions at the toe of the structures. The spectral wave period Tm-1,0, based on the
spectral moments m-1 and m0 (Tm-1,0 = m-1/m0, where mn=
!
0
∞f n S(f) df ), appeared to
1)WL | Delft Hydraulics, P.O. Box 177, 2600 MH Delft, The Netherlands.
e-mail: marcel.vangent@wldelft.nl and neelke.doorn@wldelft.nl
Proc. Coastal Dynamics 2001
2
Figure 1 Schematised foreshore and dike in model tests (scale 1:40).
Figure 2 Measured wave height evolution over the foreshore.
Figure 3 Measured wave height evolution over the foreshore.
FORESHORE - PETTEN
-2 0
-1 6
-1 2
-8
-4
0
4
8
12
16
20
-1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 0
F O R E S H O R E (m )
ELEVATION (m
)
MEASURED
SCH EM A TISED
1:30 1:25 1:20 1:100 1:25
1:4.5
1:20
1:3
MP3 BAR MP5 MP6
W AVE HEIGHTS O N FORESHORE
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
-1200 -1000 -800 -600 -400 -200 0
FORESHO RE (m )
WAVE HEIGHT (m)
-2 4
-2 0
-1 6
-1 2
-8
-4
0
4
8
12
16
20
24
28
32
36
40
FORESHORE ELEVATION (m)
H2%
Hs
1:30
1:25 1:20 1:100 1:25
MP3 MP5 MP6
DEEP BAR TO E
ENERG Y D EN S ITY SPECTRA (1.01)
0
5
10
15
20
0.0 0.1 0.2 0.3
FREQUENCY (Hz)
ENERGY DENSITY (m
2/Hz
)
DEEP
MP3
MP5
MP6
TOE
Proc. Coastal Dynamics 2001
3
be the most appropriate characteristic wave period. For predicting wave run-up and
wave overtopping it is therefore suitable to make use of numerical models which can,
besides the wave height, also predict this wave period at the toe of structures. Two
sophisticated wave models are used for this purpose, a spectral wave model and a
time-domain wave model. Also a time-domain model is used to simulate wave
interaction with the dike itself.
The shallow foreshore used in this study affects the waves considerably before they
reach the toe of the dike. The severe wave breaking and surfbeat phenomena make
that wave propagation over the shallow foreshore is complex, which is also the case
for the wave motion on the structure. Therefore, this situation is considered as
difficult to accurately model numerically. However, valuable predictions can already
be obtained from the present versions of the models.
Two numerical models have been applied to model the wave propagation over this
foreshore and one numerical model has been applied to model wave motion on the
structure. The models applied for wave propagation over the shallow foreshore are a
spectral wave model (SWAN; Ris, 1997 and Ris et al., 1998) and a time-domain
Boussinesq-type model (TRITON; Borsboom et al., 2000, 2001). The model applied
for modelling wave motion on the structure is a time-domain model based on the
non-linear shallow-water wave equations (ODIFLOCS; Van Gent, 1994, 1995).
2. PHYSICAL MODEL TESTS
Figure 1 shows the foreshore as modelled in the flume (scale 1:40) which is a
schematisation of the foreshore measured in prototype. This figure also shows at
which locations wave conditions were measured during the tests.
Conditions that occurred in 6 storm periods for which prototype measurements are
available were studied. Also conditions to study the influence of several parameters
such as wave height, wave steepness, spectral shape and water level were considered.
Figure 2 gives an example of the measured wave height evolution over the foreshore
showing clearly the reduction in wave height due to wave breaking at the bar and the
shallow part in front of the dike. Figure 3 shows an example of measured wave
energy spectra at several locations on the foreshore. From deep water towards the toe
of the structure the amount of energy in the short waves reduces while the amount of
wave energy in the long waves increases. Both phenomena are due to (severe) wave
breaking, dissipating short wave energy and transferring energy to lower frequencies.
For comparison with numerical model results (Sections 3 and 4), use was made of
tests without the structure in position such that the total surface elevations do not
contain large contributions of reflected waves.
3. SPECTRAL WAVE MODEL
The applied spectral model is a one-dimensional version of the model SWAN (Version
30.75; Delft University of Technology), which is described in Ris (1997) and Ris et
al. (1998). This model simulates propagation of short waves. It does not model
processes where bound low-frequency energy becomes free due to wave breaking,
Proc. Coastal Dynamics 2001
4
Figure 4 Comparison between measured and computed wave heights Hm0.
Figure 5 Comparison between measured and computed wave periods Tm-1,0.
Figure 6 Evolution of differences (averaged over all 20 tests) in wave
heights and wave periods over the foreshore.
Hm0
0
1
2
3
4
5
6
7
01234567
Hm0 COM PUTED
Hm0 MODEL-TESTS
MP3
BAR
MP5
MP6
TOE
Tm -1,0
0
3
6
9
12
15
0 3 6 9 12 15
Tm-1,0 COM PUTED
Tm-1,0 MODEL-TESTS
MP3
BAR
MP5
MP6
TOE
DIFFEREN C ES A T DIFFERENT LOCATIONS
-3 0
-2 5
-2 0
-1 5
-1 0
-5
0
5
10
15
20
25
30
35
40
-1200 -1000 -800 -600 -400 -200 0
FORESHORE (m )
AVERAGE ABSOLUTE ERROR (%)
-2 4
-2 0
-1 6
-1 2
-8
-4
0
4
8
12
16
20
24
28
32
36
40
FORESHORE ELEVATION (m
)
Differences H m 0
Differences Tm -1,0
Differences Tp
1:30
1:25 1:20 1:100 1:25
MP3 MP5 MP6
D EEP BAR TO E
Proc. Coastal Dynamics 2001
5
which is a relevant process for situations with shallow foreshores. For the modelling
of wave breaking (depth-induced wave breaking and whitecapping), wave set-up,
bottom friction and triad wave-wave interaction the default settings were used.
Quadruplet wave-wave interactions were not modelled. The constant grid spacing
was 10 m. The spectral resolution was 73 within the frequency range between 0.04
Hz and 0.35 Hz (
"
f /f =0.03) and 60 in the directional sector between -7.5 and +7.5
degrees. The settings were used without calibration to this specific application.
For each of the 20 tests the measured and computed wave energy spectra at five
positions on the foreshore were used for comparisons (Figure 1). The wave energy
spectra at the position DEEP are the measured wave energy spectra that were used as
incident wave energy spectra for the numerical model. The wave parameters are
based on energy between the frequencies 0.04 and 0.3 Hz. This means that for
instance energy in the low frequency-waves is filtered away.
Figure 4 shows the comparisons for the wave energy levels characterised by the wave
height Hm0. On the first part of the foreshore (MP3, BAR and MP5) the computed wave
heights are slightly too high, while at the end (MP6 and TOE) they are somewhat lower
than the measured values due to an overprediction of wave energy dissipation. At the
TOE the absolute differences are 13% on average. Analysis of the shapes of the wave
energy spectra indicated that the numerical model shows too much wave energy
transfer to the higher frequencies. This kind of inaccuracies directly affects the values
for Tm-1,0. As can be seen in Figure 5 the computed values for Tm-1,0 are systematically
too low (21% on average). Figure 6 shows the evolution of the average differences
between measured and computed wave parameters over the foreshore, for Hm0, Tm-1,0
and also for the peak wave period Tp.
The results yield for some tests a too high transfer of wave energy to higher
frequencies; for some tests this results in a rather large amount of wave energy in a
peak with a frequency twice the main peak. Although the energy in these peaks is
overpredicted, in locations further landward the dissipation of energy reduces the
amount of energy in these peaks again. Nevertheless, at the toe of the structure the
numerical model shows wave energy spectra where the wave energy is still
distributed in peaks at the original deep-water peak and a peak with a frequencies
twice this peak, while the measurements show more flat wave energy spectra.
Considering the complex foreshore with a large amount of wave energy dissipation
between deep water and the toe of the structure, the model appears to be able to
predict the average wave energy levels in the short waves rather accurately but
provides less accurate estimates of the wave periods at the toe of the structure.
4. TIME-DOMAIN BOUSSINESQ-TYPE WAVE MODEL
The applied Boussinesq-type model is the two-dimensional wave model for wave
propagation in coastal regions and harbours TRITON (WL | Delft Hydraulics), which is
described in Borsboom et al. (2000, 2001). This efficient model simulates wave
propagation and wave breaking in the time-domain which also allows for simulation
of processes where bound low-frequency energy becomes free due to wave breaking.
This is a relevant process for situations with shallow foreshores. Wave breaking is
implemented based on a new method where wave breaking is modelled as an eddy-
Proc. Coastal Dynamics 2001
6
Figure 7 Comparison between measured and computed wave heights Hm0.
Figure 8 Comparison between measured and computed wave periods Tm-1,0.
Figure 9 Evolution of differences (averaged over all 20 tests) in wave
heights and wave periods over the foreshore.
Hm0
0
1
2
3
4
5
6
7
01234567
Hm0 COM PUTED
Hm0 MODEL-TESTS
MP3
BAR
MP5
MP6
TOE
Tm -1,0
0
3
6
9
12
15
03691215
Tm-1,0 COM PUTED
Tm-1,0 MODEL-TESTS
MP3
BAR
MP5
MP6
TOE
DIFFEREN C ES A T D IFFEREN T LO C A TIONS
-3 0
-2 5
-2 0
-1 5
-1 0
-5
0
5
10
15
20
25
30
35
40
-1 2 0 0 -1 0 0 0 -8 0 0 -6 0 0 -4 0 0 -2 0 0 0
F O R E S H O R E (m )
AVERAGE ABSOLUTE ERROR (%)
-2 4
-2 0
-1 6
-1 2
-8
-4
0
4
8
12
16
20
24
28
32
36
40
FORESHORE ELEVATION (m
)
Differences Hm 0
Differences Tm -1 ,0
Differences Tp
1:30
1:25 1:20 1:100 1:25
MP3 MP5 MP6
DEEP BAR TO E
Proc. Coastal Dynamics 2001
7
viscosity model in combination with a surface roller, similar to the method applied by
Kennedy et al. (2000). For the determination of the eddy viscosity use is made of the
concept of surface rollers, as also applied by Schäffer et al. (1992). In contrast to
many other existing models for wave breaking, this breaker model in TRITON allows
for modelling of severe wave breaking as is the case in the present applications.
For each of the 20 tests the measured and computed wave energy spectra at five
positions on the foreshore were used for comparisons (Figure 1). The measured time
signals at the position DEEP are used as incident waves for the numerical model. At
this seaward boundary the surface elevations were prescribed while at the landward
boundary (toe of the structure) an open weakly reflecting boundary was used. At this
open boundary the waves can leave the computational domain, using the long-wave
assumption to assess the phase velocity of the outgoing waves. The space step and
time step were
"
x =1.0 m and
"
t=0.06 s. The wave parameters are again based on
energy between the frequencies 0.04 and 0.3 Hz, based on computations with 500
waves per wave condition. This means that, similar to the approach in Section 3,
energy in the low frequency-waves was filtered away such that the comparisons
focus on the short waves.
Figure 7 shows the comparisons for the wave height Hm0. On the first part of the
foreshore (MP3, BAR and MP5) the computed wave heights are slightly too low, while
at the end (TOE) they are somewhat higher than the measured values due to an
underprediction of wave energy dissipation. At the TOE the absolute differences are
9.6% on average. Figure 8 shows the comparison between measured and computed
wave periods Tm-1,0; the differences between the measured and the computed Tm-1,0 at
the TOE are relatively small (4% on average). Figure 9 shows the evolution of the
average differences between measured and computed wave parameters over the
foreshore, for Hm0, Tm-1,0 and also for the peak wave period Tp.
The general impression from examining the wave energy spectra is that the time-
domain model simulates both the spectral shapes and the energy levels rather
accurately for these conditions with significant energy dissipation due to severe wave
breaking. Also the energy shift to the lower frequencies is modelled surprisingly well.
The accuracy of the predictions for the wave periods is higher than obtained with the
spectral wave model, though at the cost of higher computational efforts. The model is
considered as suitable to provide estimates of the relevant parameters for wave run-
up and wave overtopping on dikes with shallow foreshores.
5. WAVE INTERACTION WITH DIKE
The model applied here is the time-domain model ODIFLOCS (Delft University of
Technology) which simulates wave motion on coastal structures (Van Gent, 1994,
1995). Perpendicular wave attack on structures with frictionless impermeable slopes
is simulated by solving the non-linear shallow-water wave equations. Steep wave
fronts are represented by bores. Use is made of an explicit dissipative finite-
difference scheme (Lax-Wendroff) (Hibberd and Peregrine, 1979). Similar models
have been shown to predict well wave reflection and wave run-up on impermeable
rough slopes (Kobayashi et al, 1987).
Proc. Coastal Dynamics 2001
8
In the computations the slope below the berm was extended to a depth equal to the
depth of the trough between the bar and the toe of the structure. This extension to
deeper water was used because the depth at the toe is so small that in some situations
at this position there is only a little amount of water present for a part of the wave
cycle; it would be hard or impossible to place the incident wave boundary at such a
position. The space step and time step were
"
x =0.8 m and
"
t=0.01 s. The slope was
modelled frictionless. The computed wave run-up levels exceeded by 2% of the
incident waves are those with waterlayers thicker than 0.1 m. This corresponds to the
required thickness in the model tests to obtain measured wave run-up levels.
Three sets of computations were performed with three different ways to obtain
incident waves in the computations (500 waves per wave condition): a) Measured
surface elevations of incident waves, b) Computed surface elevations derived from
the spectral foreshore model (Section 3) and c) Computed surface elevations from the
time-domain foreshore model (Section 4). In these three sets of computations only
the surface elevation of the incident waves is prescribed while the corresponding
velocity is computed based on the long-wave assumption. Time-series from the
results of the spectral wave model are generated by using the assumption of ‘random-
phases’ and assuming a deep-water (Rayleigh) wave height distribution (which leads
to an overprediction of the extreme wave heights (e.g., H2%).
Data set differences
∆
z2%
standard
deviation
a) Measured incident waves 9.7% 4.4%
b) Incident waves from spectral foreshore model 18.2% 6.4%
c) Incident waves from time-domain foreshore model 2.7% 7.4%
Table 1 Comparison between measured and computed wave run-up levels.
Table 1 shows the differences between measured and computed wave run-up levels
for each of the three data-sets. The first data-set with measured surface elevations as
incident waves leads to an underprediction of wave run-up levels of about 10%. The
underprediction of the computed wave heights by the spectral wave model used in
Section 3 (about 13%) leads to an total underprediction of 18% in wave run-up
levels. The overprediction of the computed wave heights by the time-domain wave
model used in Section 4 (about 10%) leads to an average difference of less than 3%
in wave run-up levels. This accurate prediction is the result of counteracting errors
because the foreshore-model shows too little dissipation while the structure-model
shows too much dissipation. Applying the two models together leads therefore to
relatively accurate predictions of the wave run-up levels for the present data-set.
To gain insight on the influence of low-frequency energy on wave run-up the
correlation is analysed between measured time-signals at the toe of the dike and the
computed time-signals of the wave run-up point. The measured time-signals are
filtered such that only the low-frequency waves remain (<0.04 Hz). For the six
conditions studied here, those which correspond to the tests where prototype storms
are reproduced, approximately 30% of the total wave energy is present in these low-
frequencies. Figure 10 shows a strong positive correlation (method by Bendat and
Piersol, 1986) between the two types of signals which means that high wave run-up
Proc. Coastal Dynamics 2001
9
levels coincide with high surface elevations of low frequency waves at the toe. This
shows that low-frequency energy is an important part of the wave energy which
determines wave run-up.
Figure 10 Correlation between low-frequency waves and wave run-up.
6. CONCLUSIONS
Based on the investigations described in this paper the following conclusions can be
drawn:
• The spectral wave model, applied for wave propagation of short waves over the
foreshore (SWAN), yields valuable insight in the evolution of wave energy spectra
over the foreshore. It also shows that the computed energy levels in the short
waves are rather accurately predicted, considering the rather extreme energy
dissipation in the tests. The wave parameters Hm0 and Tm-1,0 at the toe of the
structure are both underpredicted (13% and 21% respectively), using the default
settings of this numerical model. Modifications of the numerical model settings
for this kind of applications might improve the results. Further improvements of
this model could be dedicated to decrease wave energy transfer to higher
frequencies and to increase wave energy transfer to lower frequencies.
• The time-domain wave model applied for wave propagation over the foreshore
(TRITON) shows accurate results for the wave parameters Hm0 and Tm-1,0 at each
position. The deviations at the toe of the structure remain below 10% and 5%
respectively (based on the energy in the short waves). The evolution of the wave
energy spectra is rather accurately simulated despite the extreme energy
dissipation. Also the energy transfer to lower frequencies is clearly present. The
model to include wave breaking in this Boussinesq-type model appears to be
effective in reducing the wave energy without significant loss of accuracy in the
simulation of wave energy spectra. Further validations of this model include 2DH-
situations with angular wave attack, directional spreading and non-uniform depth-
contours.
• The time-domain wave model applied for the simulation of wave interaction with
the dike (ODIFLOCS) shows that accurate results on wave run-up levels can be
obtained if use is made of measured surface elevations of the incident waves (on
average 10% underpredictions of the wave run-up levels). The use of incident
waves based on numerical results from the spectral wave model (SWAN) doubles
the mean differences (18%) because both numerical models lead to too much
−20 0 +20
−1
0
1
correlation coeff.
time lag (s)
storms 1−6
Proc. Coastal Dynamics 2001
10
wave energy dissipation. The use of incident waves calculated by the time-domain
wave model (TRITON) reduces the differences significantly (on average less than
3%) because the numerical models lead to counteracting errors. Applying the two
models together led to relatively accurate predictions of the wave run-up levels for
the present data-set.
ACKNOWLEDGEMENTS
The research presented here is partly performed within the European research project
MAST-OPTICREST (contract MAS3-CT97-0116) and partly in co-operation with
Rijkswaterstaat-DWW. Prof. dr. J.A. Battjes of the Delft University of Technology is
gratefully acknowledged for his review of the report on which this paper is based.
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