wave overtopping of póvoa de varzim breakwater: physical and numerical simulations

Wave Overtopping of Póvoa de Varzim Breakwater:Physical and Numerical Simulations

Maria Graça Neves1; Maria Teresa Reis2; Inigo J. Losada, M.ASCE3; and Keming Hu4

Abstract: This paper compares the output from four methods used to predict the mean overtopping discharge at the root of the SouthBreakwater of Póvoa de Varzim Harbour �Portugal�: two numerical models AMAZON, based on solving the nonlinear shallow-waterequations, and COBRAS-UC, based on the Reynolds averaged Navier–Stokes equations, Pedersen’s empirical formula, and the neuralnetwork tool NN_Overtopping. Output from the methods is considered alongside data from two-dimensional �2D� physical model tests.The results show that rather accurate predictions can be obtained with COBRAS-UC. Agreement between the data and the AMAZONoutput depends principally on the impact of porosity on overtopping: agreement is very good when the impact is limited �for highovertopping discharges� and it worsens as porosity plays a more important role �for small discharges�. NN_Overtopping estimates themean discharges with a degree of accuracy similar to AMAZON. Pedersen’s results are those which deviate more from the data,substantially overpredicting the mean discharges for the conditions analyzed.

DOI: 10.1061/�ASCE�0733-950X�2008�134:4�226�

CE Database subject headings: Numerical models; Model tests; Empirical equations; Neural networks; Breakwaters; Portugal;Coastal structures; Coastal engineering.

Introduction

The safe use of coastal regions depends to a large extent on theperformance of coastal structures with regard to wave overtop-ping. Overtopping may cause fatalities and considerable eco-nomic damage: it may represent a hazard to people and vehicleson or close behind the overtopped structure, and to property, in-frastructure, and environment protected by sea defenses. Overtop-ping into harbors and marinas may affect the structural integrityof the sea defense, damage boats, and cause disruption to normalactivities.

At present, the most widely used tools for predicting overtop-ping of coastal structures are wholly empirical or semiempiricalformulas. The wholly empirical models �such as the models ofOwen 1980; Aminti and Franco 1989; Bradbury and Allsop 1988;Pedersen and Burcharth 1993; de Waal and van der Meer 1993;Franco et al. 1995; Allsop et al. 1995; Pedersen 1996; van der

1Research Officer, National Civil Engineering Laboratory, DHA/NPE,Av. do Brasil, 101, 1700-066 Lisbon, Portugal �corresponding author�.E-mail: [email protected]

2Research Officer, National Civil Engineering Laboratory, DHA/NPE,Av. do Brasil, 101, 1700-066 Lisbon, Portugal. E-mail: [email protected]

3Professor, Ocean & Coastal Research Group, Instituto de HidráulicaAmbiental, “IH Cantabria,” Univ. de Cantabria, Avda. de los Castros s/n,Santander 39005, Spain. E-mail: [email protected]

4Manager of Numerical Modelling, Royal Haskoning, Burns House,Harlands Ro., Haywards Heath RH16 1PG, U.K. E-mail: [email protected]

Note. Discussion open until December 1, 2008. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on February 5, 2007; approved on August 6, 2007. Thispaper is part of the Journal of Waterway, Port, Coastal, and OceanEngineering, Vol. 134, No. 4, July 1, 2008. ©ASCE, ISSN 0733-950X/
2008/4-226–236/$25.00.
226 / JOURNAL OF WATERWAY, PORT, COASTAL, AND OCEAN ENGINE

Meer et al. 1998; and Besley 1999� were developed by fittingdimensionless groups to data derived from physical model tests.The Hedges and Reis semiempirical model �Hedges and Reis1998, 2004; Bay et al. 2005; Reis et al. 2008� was derived fromconsideration of the unsteady flow of water over a weir and it wascalibrated using the results of physical model tests. Direct appli-cation of these models is limited to particular structural configu-rations and wave conditions, and even for these cases, there aregaps in data, particularly for the low overtopping discharges forwhich structures are usually designed.

A more accurate method for determining overtopping is basedon physical model tests. Despite the fact that they are moreexpensive and more time consuming than the use of formulas,they are used often, not only in the development of formulas butalso for prototype studies and for calibration/validation of nu-merical models of overtopping and training/validation of neuralnetworks. In physical model tests, an understanding of model andscale effects is crucial for the correct representation of the phe-nomenon. Results from field measurements or from large-scalelaboratory tests are rather rare and several studies have been per-formed only recently under the CLASH project to fill this gap andallow investigation of both model and scale effects �Kortenhauset al. 2005; de Rouck et al. 2005�.

Numerical models of overtopping may be more flexible thenempirical/semiempirical and physical models since, once cali-brated and validated, they can be applied reliably to a large rangeof alternative geometries and wave conditions. In recent years,due to the continuous increase in computer power, numericalmodels have been developed further and their use is becomingincreasingly attractive. Impressive advances have been reportedin the volume of fluid �VOF� method �Hsu et al. 2002;Kortenhaus et al. 2005; Lara et al. 2006b�, surface capturing �SC�approach �Ingram et al. 2003, 2004� and smoothed particle hydro-dynamics �SPH� models �Shao et al. 2006�.

Although the latest numerical models are comprehensive, their

ERING © ASCE / JULY/AUGUST 2008

use in practical engineering applications still has limitations.Some difficulties are related to their considerable demand oncomputer power and their need for calibration; others are relatedto the representation of certain structural characteristics, such asporosity. As a consequence, these models are often applied withregular waves only instead of random waves and, to the writers’knowledge, there is no report on the application of a SPH modelto a porous structure. However, simpler numerical models basedon the nonlinear shallow-water equations, in spite of their limita-tions principally relating to the shallow water assumptions, arealready being used for the purposes of design and flood forecast-ing, since wave trains of several thousand random waves aresimulated rapidly �Hu 2000; Hu et al. 2000; Hu and Meyer 2005�.

The use of artificial neural networks is also proving to be apromising way forward �Medina et al. 2003; Shareef 2004;Wedge et al. 2005�. Recently, NN_Overtopping, a prediction toolbased on neural network �NN� modeling, was developed as partof the CLASH European project �Coeveld et al. 2005; van Gentet al. 2005� to predict Froude-scaled mean wave overtoppingdischarges and the associated confidence intervals for a widerange of coastal structure types �such as dikes, rubble moundbreakwaters, and caisson structures�. In addition, prototype meanovertopping estimations, allowing for scale and model effects, areprovided. This tool is based on a database of about 8,400 testconditions, which originate from small-scale tests at many dif-ferent laboratories �Steendam et al. 2005�. Nevertheless, Coeveldet al. �2005� suggest that the reliability of the predictions shouldbe verified using dedicated physical model tests for the particularwave conditions and structure geometry under consideration.Thus, this method may still have limited application, despite itssimplicity and speed of application.

Despite the variety of existing tools for the determination ofovertopping of coastal structures, designers are still faced with thequestion of what tool�s� should be applied in specific projects. Forengineering case studies, often only mean values of the dischargeare required. Even so, to obtain a reliable estimate of this dis-charge, the method�s� used to compute it should have been fullyvalidated.

This paper compares results from different approaches usedto compute the mean wave overtopping discharge for a realstructure: the root of the South Breakwater of Póvoa de VarzimHarbour, Portugal. The methods comprise the use of two numeri-cal models; one empirical formula and a tool based on artificialneural network modeling. Thus, they cover a wide range of theavailable options. Output from the tools is compared with datafrom four tests carried out on a two-dimensional �2D� physicalmodel at the National Civil Engineering Laboratory, Portugal�Reis et al. 2006�.

The numerical models used are AMAZON �Hu et al. 2000�,based on solving the nonlinear shallow-water �NLSW� equations,and COBRAS-UC �Losada et al. 2008�, based on the Reynolds-averaged Navier–Stokes �RANS� equations. AMAZON has beenvalidated and extensively used to study the overtopping of dikes.One of the prime virtues of this kind of numerical model lies in itsspeed of calculation. A disadvantage is that it does not explicitlyaccount for porous flow �although the friction coefficient may beused to reduce the incident wave energy requiring model calibra-tion� and the wave conditions have to be input at a distance fromthe structure toe of approximately one wavelength �Hu and Meyer2005� or satisfy the shallow water restriction. COBRAS-UC hasbeen validated for random waves acting on submerged, permeablestructures at both model and prototype scales �Garcia et al. 2004;
Lara et al. 2006a� and recently for wave overtopping on rubble
JOURNAL OF WATERWAY, PORT, COASTA

mound breakwaters �Losada et al. 2008�. It can simulate all ofthe important hydrodynamic processes involved in wave overtop-ping of permeable complex structures. However, it is very timeconsuming and must be calibrated for porous flow.

The semiempirical formula chosen is the Pedersen expression�Pedersen 1996�, as it provides the best results for the structure/wave conditions considered �Brito 2006�, when compared withresults from other formulas presented in Besley �1999� and in theCoastal Engineering Manual �U.S. Army Corps of Engineers2003�.

The tool based on artificial neural network modeling applied inthis study is that developed as part of the CLASH project, i.e., theNN_Overtopping prediction tool mentioned above.

Following this introduction, the next section gives details ofthe case study. After that, a description is made of the results ofthe physical model tests, of the output from AMAZON andCOBRAS-UC, of the mean overtopping discharges calculatedusing Pedersen’s formula, and of the predictions from NN_Over-topping. Finally, a discussion of the results is presented and thepaper ends with the main conclusions which arise from the study.

Case Study: South Breakwater of Póvoa de VarzimHarbour

The Póvoa de Varzim Harbour is located on the west coast ofPortugal �Fig. 1�. It is sheltered by two breakwaters, the Northand the South Breakwaters. At present, preliminary studies arebeing undertaken on the rehabilitation of the South Breakwater.The rehabilitation works differ along its length, depending on thecurrent condition and on the functional requirements of eachstretch. The root of the breakwater protects the local NauticalClub building and, therefore, it is the stretch for which overtop-ping must be minimal. Any overtopping that occurs on this stretchmay cause unacceptable damage to the building and surroundingarea and disruption to local activities.

The current cross section of the root of the South Breakwateris basically a combination of a concrete vertical wall with a rockslope in front of it �Fig. 2�. The weight of the rock ranges from 10to 50 kN. The Nautical Club building is located immediately inthe lee of the breakwater.

The preliminary studies on the rehabilitation of the SouthBreakwater include two proposals for the cross section of theroot. The new cross sections were designed to reduce overtoppingto acceptable levels, even for extreme wave and water level con-ditions. In this study, only one of these two proposed solutionswas used �Fig. 2�. It has been obtained by adding a prism of75–100 kN rocks to the current cross section.

Results

Physical Model Tests

To verify the efficiency of the proposed solutions for the crosssection of the root of the South Breakwater, two-dimensionalphysical model tests of wave overtopping were performed at theNational Civil Engineering Laboratory �Lemos et al. 2006�. Inaddition to these tests, an extra 138 model tests were performedwith different equipment to further study the process of overtop-ping �Reis et al. 2006�. These 138 tests corresponded to four main
wave conditions repeated for several different test durations and
L, AND OCEAN ENGINEERING © ASCE / JULY/AUGUST 2008 / 227

wave trains. This paper concentrates on the results of four tests,covering the four main wave conditions, with duration, D, of300 s.

The tests were performed in a wave flume, which is approxi-mately 50 m long and has an operating width and an operatingwater depth of 0.80 m. The flume is equipped with a randomabsorbing piston-type wave maker.

The model was built and operated according to Froude’s simi-larity law: the geometrical scale was 1:50. The extra tests were allperformed for a still-water level of +4.5 m chart datum �CD�,which corresponded to the high-water level with setup. The seabed in front of the model breakwater was represented by differentslopes down to a level of −15.0 m �CD�, giving an operatingwater depth, h, in front of the wavemaker of 0.39 m. The samestructure configuration was used in all tests, in which the free-board, Rc, was 0.06 m and the depth, ds, at the toe of the structurewas 0.09 m �Fig. 3�.

Fig. 1. Case study locatio

Fig. 2. Current cross section of root of South Breakwater and propolevels relative to datum �CD�


The flume was equipped with six resistive-type wave gauges�Fig. 4�. The first array of three wave gauges was located in frontof the wavemaker and the second array in front of the structure.Within each array, the wave gauges were positioned so that inci-dent wave characteristics could be determined according to Man-sard and Funke �1980�.

The equipment used to collect the overtopping water was atank, located at the back of the structure, and the water was di-rected to the tank by means of a chute, 0.30 m wide �Fig. 4�. Agauge was used in the overtopping tank to measure the increase ofwater level within a test run, allowing determination of the meanovertopping volume. A computer collected and stored the data indigital format at a frequency of 40 Hz.

Random waves, conforming to the mean JONSWAP spectralform �with a peak enhancement factor of 3.3�, were employed inthe study, with model mean peak periods, Top, in front of thewavemaker ranging from about 1.7 to 2.3 s and with significant

Póvoa de Varzim Harbour

oss section used in this paper; values shown are for prototype, with

n and

sed cr


wave heights, Hos, ranging from approximately 0.08 to 0.14 m.First, each test condition was repeated for 12 different test dura-tions ranging from 300 to 3,600 s, in steps of 300 s. Second, forsome durations, each test was repeated several times, with differ-ent wave trains all conforming to the same JONSWAP spectrum,except that the JONSWAP seed was varied for each �Reis et al.2006�. Finally, in order to calibrate COBRAS-UC, four tests wereundertaken with regular waves with the wave height, H, and waveperiod, T, corresponding to the significant wave height, Hos, andpeak period, Top, of the irregular tests.

In Table 1, the mean overtopping discharges per meter lengthof structure, q, are presented for four tests with durations of300 s, together with the variability of q obtained for each ofthe four sets of tests. In this table, Hs and Tp are, respectively,the significant wave height and the peak period in front of thestructure.

The tests are numbered in order of increasing discharge. Thedifferences in overtopping are related to the different characteris-tics of the waves that approach the breakwater and their differentways of breaking. As expected, the variation about the mean wasgreater for small magnitudes of overtopping discharges.

AMAZON

The wave overtopping model AMAZON was run for the four testconditions in Table 1.

AMAZON uses a “nonreflective wave inlet boundary condi-tion” which is able to remove more than 98% of the energy of any

Fig. 3. Physical model of altern

Fig. 4. Sketch of wave flume and location o


waves reflected from the modeled structures at the seawardboundary. As a consequence, the seaward boundary can be setclose to the structure to avoid deep water conditions, whereAMAZON has limitations, since it is based on the NLSW equa-tions. According to Hu and Meyer �2005�, AMAZON producesgood results when its seaward boundary is located at a distancefrom the structure toe of approximately one wavelength, Ls,where Ls is the shallow water wavelength in depth ds at the struc-ture toe �Fig. 3� and it is calculated using the peak period of theincident waves, Tp �Ls=Tp

�gds, g is the acceleration due to grav-ity�. The location of the seaward boundary suggested by Hu andMeyer �2005� was adopted in this study and Gauge 4 �Fig. 4� wasat this position. Thus, the input to AMAZON was the incidentwave series obtained using the Mansard and Funke �1980�method applied to the data measured by Gauges 4, 5, and 6�Fig. 4�. It should be pointed out that this method has consider-able error when breaking occurs. However, in the four tests con-sidered here, only the largest waves have break during the testsbetween Gauges 4 and 6. Consequently, the input incident waveseries is likely to be somewhat different from the series in thephysical model.

The values of the relative water depth, d /Lop, at AMAZON’sseaward boundary ranged from 0.019 to 0.035, in whichLop�deep water wavelength corresponding to the peak of the in-cident wave spectrum and calculated, according to linear wavetheory, as Lop=gTop

2 /2�. Researchers have reported differentmaximum permissible values of d /Lop providing good results

cross section used in this paper

surement equipment used during extra tests

ative

f mea


when used with the NLSW equations, varying from approxi-mately 0.016 to 0.3 �Pullen and Allsop 2003; Hu and Meyer2005�.

AMAZON’s landward boundary was a full absorption bound-ary set 0.16 m behind the crest of the wall. A nonuniform com-putation grid was used: 0.01 m for the foreshore at the deeper partof the computational domain and for the area behind the verticalwall; 4 mm for the foreshore at the toe of the structure; and 2 mmfor the breakwater. The minimum water depth at each cell was setto 2�10−7 m. Any cell with a water depth below the minimumvalue was treated as dry and was removed from the computation.The total computational domain was 2.4 m long and the totalnumber of cells was 555.

The physical model geometrical characteristics of the fore-shore and of the breakwater’s envelope were reproduced withinAMAZON. However, since AMAZON does not explicitly ac-count for porous flow, the two different prisms of rock used in thebreakwater were not distinguished. The foreshore and the verticalwall were represented as impermeable and frictionless. The per-meability of the prisms of rock was accounted for through appli-cation of the bottom friction coefficient, f , only. Table 2 andFig. 5 show the values of f used and the variability which theyproduced in the mean overtopping discharge �shown in modeldimensions�. In this figure, qPM represents the mean overtoppingrate obtained from the physical model.

The numerical tests were run on a 2.2 GHz AMD Athlon 64with 1 GB of RAM. AMAZON took about 1 h to complete eachrun, which represented a 300 s physical model test.

COBRAS-UC

By taking the volume-average of RANS equations, Liu et al.�1999� presented a two-dimensional numerical model, nicknamedCOBRAS, to describe the flow inside and outside coastal struc-tures including permeable layers. Hsu et al. �2002� extended thepreliminary model by including a set of volume-averaged k-�turbulence balance equations. The movement of free surface istracked by the VOF method. In the VARANS equations, the in-terfacial forces between the fluids and solids have been modeled

Table 1. Mean Overtopping Discharges per Meter Length of Structure, q,and Range of q Obtained in 138 Tests

Test

Hos Top Hs

�m� �s� �m�

1 0.09 1.69 0.07

2 0.08 2.24 0.07

3 0.11 2.28 0.08

4 0.14 2.21 0.09

Table 2. Mean Overtopping Discharges per Meter Length of Structure,

TestPhysicalmodel No friction f =0.001 m f =0.

1 1.66 10−5 6.76 10−5 6.54 10−5 6.10

2 2.65 10−5 9.21 10−5 8.87 10−5 8.20

3 1.65 10−4 2.76 10−4 2.70 10−4 2.57

4 4.44 10−4 4.61 10−4 4.53 10−4 4.37


by the extended Forchheimer relationship, in which both linearand nonlinear drag forces are included. COBRAS-UC �Losadaet al. 2008� is a new version of the model developed at the Uni-versity of Cantabria in order to overcome some of the initiallimitations and especially to convert it into a tool for practicalapplication. Most of these modifications in the new versionCOBRAS-UC, have been founded on the extensive validationwork carried out for low-crested structures �Garcia et al. 2004;Losada et al. 2005; and Lara et al. 2006a� and wave breaking onpermeable slopes �Lara et al. 2006b� carried out with the model.The improvements cover the wave generation process; code up-dating and refactoring; optimization and improvement of the mainsubroutines; improvement of input and output data definition; andthe development of a graphical user interface and output dataprocessing programs. In this study, COBRAS-UC was used tocalculate the mean wave overtopping discharge for an emergedbreakwater, which is basically a combination of an impermeableconcrete vertical wall with a two-layer permeable rock slope infront of it.

The computational domain was 41 m long and 0.7 m high andit reproduced the full dimensions of the experimental flume plus a4 m sponge on the seaward side of the structure, to dissipateseaward-going waves �Fig. 6�. The source region was located atthe same distance from the structure as the paddle in the flume. Itwas designed based on the Lara et al. �2006a� rules.

The computational mesh was divided into three regions ofdifferent resolutions, corresponding to the wave generation zone,the breakwater vicinity, and the absorbing zone. The grid wasnonuniform in the x direction, with a minimum cell width, dx, of0.02 m in the structure zone �Fig. 6� and a cell width of 0.04 m inthe generation zone. In the y direction, the grid was uniform withdy=0.01 m in the whole domain. The total number of cells was1,421�71.

Eight sections were considered in the numerical flume: sixwere located at the same positions as the wave gauges in thephysical model, one was in the source region in order to controlthe wave generation, and another was on top of the structure tocompute the overtopping discharge.

ed for Each One of Four Physical Model Tests Considered in This Study

Tp q Range of q

�s� �m3 /s /m� �m3 /s /m�

1.75 1.66 10−5 0.90 10−5–2.83 10−5

2.24 2.65 10−5 2.65 10−5–8.44 10−5

2.33 1.65 10−4 1.15 10−4–2.02 10−4

2.93 4.44 10−4 3.43 10−4–4.69 10−4

ined with AMAZON as Function of Bottom Friction Coefficient, f

q3 /s /m�

AMAZON

f =0.01 m f =0.02 m f =0.03 m f =0.04 m

5.87 10−5 5.45 10−5 5.13 10−5 Error

7.88 10−5 7.25 10−5 6.75 10−5 Error

2.51 10−4 2.39 10−4 2.29 10−4 Error

4.29 10−4 4.14 10−4 4.01 10−4 Error

Obtain

q, Obta

�m

006 m

10−5

10−5

10−4

10−4


The location, dimensions, and characteristics of the experi-mental structure were reproduced in the computational domain.As mentioned previously, these included two porous layers. Tosimulate the flow into the porous medium, some parametersrequired calibration: the linear, �L, and nonlinear, �, friction co-efficients related to the linear and nonlinear drag forces, respec-tively. Following the procedures presented in Lara et al. �2006a�,the calibration of the porous flow parameters was done usingregular waves. The porosity, p, and the diameter of the rocks, D50

were the same as in the physical model �see Table 3�. The testssimulated 150 s in the physical model, and the values used in thecalibration runs were: �L=200 and � between 0.7 and 0.9 for thecore layer and between 0.8 and 1.1 for the armor layer. Thesevalues were within the parameter ranges tested in previous simu-lations reported in the literature.

The free surface computed by COBRAS-UC was compared tothat measured in the tests at the six gauges, in order to establishthe optimum values of the calibration parameters. Fig. 7 showsthe time series of free surface displacement from the model simu-lations and from the laboratory measurements at Gauges 4–6, i.e.,close to the structure, for a test with H=0.09 m and T=1.69 s.The chosen values are presented in Table 3. Although the pro-cesses of wave generation used in the laboratory and in the modelwere different, the agreement in the free surface for regular wavesis reasonable.

After calibration for porous flow, the values of the parameterswere kept constant for the four irregular wave tests, and were notused as tuning parameters.

Ideally, the irregular time series considered in numerical simu-lations should be identical to those generated in the laboratory.Unfortunately, no information was available on the free surfacevariations at the paddle. The alternative was to input the incidentwave conditions obtained from the Mansard and Funke methodapplied to the data measured at the three gauges close to the

Fig. 5. Variation of mean overtopping discharge per meter length ofstructure, q, obtained with AMAZON as function of bottom frictioncoefficient, f

Fig. 6. Sketch of computational grid used


paddle �Gauges 1–3, Fig. 4�. This method is not exact; conse-quently, a difference was expected in the surface time–series mea-sured in the laboratory and computed in the model. Furthermore,there were different procedures of wave generation in the labora-tory and in the model, as noted earlier. Consequently, due to therandom nature of the irregular waves simulated, the surface dis-placement could not be compared directly to the measured values.Instead, the mean overtopping discharges were compared. Table 4shows the values obtained using COBRAS-UC.

The numerical tests were run on a 2.2 GHz AMD Athlon 64with 1 GB of RAM and the average execution time was about22 h for 300 s simulations. It has to be pointed out that this com-putational time corresponds to a simulation of the completeflume.

Pedersen’s Formula

Since, at present, empirical formulas remain the most widelyused tools for predicting the overtopping of coastal structures, thePedersen �1996� formula was also applied in this study. This wasthe chosen expression as it provides better results for thestructure/wave conditions considered �Brito 2006� than other for-mulas presented in Besley �1999� and in the Coastal EngineeringManual �U.S. Army Corps of Engineers 2003�. However, thisexpression was developed for rock armored permeable slopeswith a berm in front of a crown wall �U.S. Army Corps of Engi-neers 2003�, whereas a vertical wall is used instead of a crownwall at the root of Póvoa de Varzim South Breakwater.

Pedersen’s formula may be written as follows:

q = 3.2 · 10−5Lom2

Tom�Hs

Rc�3 Hs

2

AcB cot ��1�

where Tom�deep water spectral mean period; Lom�deep waterwavelength corresponding to wave period Tom �Lom=gTom

2 /2��;Hs�incident significant wave height �defined as Hs=4�mo, inwhich m0�zeroth moment of the incident wave spectrum�;Rc�vertical distance from SWL to top of the vertical wall;Ac�armor crest freeboard defined as the vertical distance fromSWL to the armor crest; B�width of the armor crest berm, and��angle of the breakwater front slope measured from the hori-zontal �see Fig. 3�. Since the Hs value at the toe of the structurewas not measured during the tests, the value used here is the

Table 3. Values of Porous Flow Parameters Used in Model

Armorlayer p

D50

�m� �L �

Top 0.54 0.0332 200 1.1

Bottom 0.52 0.0219 200 0.8

BRAS-UC simulations �axes not scaled�
in CO

incident wave height obtained using the Mansard and Funke�1980� method applied to the data measured by Gauges 4–6 �de-scribed before for the AMAZON model�.

Strictly speaking, empirical formulas can only be applied toconditions for which they were developed. Table 5 shows theranges of applicability of Pedersen’s formula and the correspond-ing ranges for the four physical model tests considered in thisstudy. The table shows that, for the four tests analyzed, some ofthe values of the parameters used in the formula are outside theirvalid ranges. Consequently, the values of q calculated usingPedersen’s expression and shown in Table 6 should only be con-sidered as indicative values.

It is important to note that the reliable use of empirical formu-las, despite their simplicity and speed of application, is often lim-ited in practical cases because the values of some parameters inthe formulas fall outside their ranges of applicability: the complexgeometry of real structures is rarely accounted for in the devel-opment of formulas.

Table 4. Mean Overtopping Discharges per Meter Length of Structure, q,Obtained with COBRAS-UC

Test

q�m3 /s /m�

Physical model COBRAS-UC

1 1.66 10−5 2.89 10−5

2 2.65 10−5 8.70 10−5

3 1.65 10−4 1.69 10−4

4 4.44 10−4 4.06 10−4

Fig. 7. Time series of free surface displacement from laboratory andH=0.09 m and T=1.69 s


NN_Overtopping

Within the framework of the CLASH project, a tool based onartificial neural network modeling was developed: the NN_Over-topping prediction tool �van der Meer et al. 2005; van Gent et al.2005�. The values of the mean overtopping discharge, q,computed by this method are built on about 8,400 input-outputcombinations which originate from measurements from small-scale physical model tests for different structure geometries andwave conditions.

Table 5. Ranges of Applicability of Pedersen’s Formula andCorresponding Ranges for Four Physical Model Tests Considered inThis Study

Parameters inPedersen’s formula

Ranges of applicabilityof Pedersen’s formula

Ranges for the fourphysical model tests

Hs �m� 0.10–0.18 0.07–0.09

Tom �s� 1.07–1.94 1.36–1.64

Top �s� 1.20–2.20 1.69–2.28

som=Hos /Lom 0.02–0.06 0.023–0.034

�om=tan � / �som�0.5 1.1–5.1 2.72–3.29

Rc /Hs 0.7–3.6 0.7–0.9

Hs /Ac 0.5–1.7 4.6–5.8

Ac /B 0.3–1.1 0.05

cot � 1.5–3.5 2

h �m� 0.51–0.59 0.15

Rc �m� 0.11–0.37 m 0.06

Ac �m� 0.11–0.19 0.015

B �m� 0.18–0.36 0.3

AS-UC at Gauges 4–6 for test with regular waves characterized by
COBR

The method uses 15 input parameters, which include informa-tion about the geometry of the structure and the wave conditions.Coeveld et al. �2005� suggest that the reliability of the predictionsshould be verified using dedicated physical model tests for theparticular wave conditions and structure geometry under consid-eration. Thus, this method may have limited use, despite its sim-plicity and speed of application.

Table 7 shows the values of q calculated by the NN_Overtop-ping tool for the four tests considered in this study and the 95%confidence interval given by the quantiles q2.5% and q97.5%. Thesevalues were obtained with the incident wave conditions used pre-viously in the AMAZON model and in the Pedersen formula, asinput to the tool �see Table 1�, since, as referred before, the waveconditions at the toe of the structure were not measured during thetests.

Discussion

The mean overtopping discharge, q, at the root of the SouthBreakwater of Póvoa de Varzim Harbour was estimated for speci-fied wave conditions and water level. Different methods wereused: the empirical formula of Pedersen, physical modeling�qPM�, numerical modeling �AMAZON and COBRAS-UC�, andan artificial neural network tool �NN_Overtopping�. Fig. 8 pre-sents a synthesis of the results for each of the four tests analyzed.The four results for AMAZON correspond to a value of the bot-tom friction coefficient f =0.03 m, which provided the best over-all agreement with the physical model values. The ranges for thephysical model results are shown in Table 1 �minimum and maxi-mum values� and q /qPM represents the ratio obtained for each ofthe four specific test conditions.

Pedersen’s formula performs worst, especially for small valuesof q. Its results are always outside the range of q obtained in thephysical model, overestimating qPM by between two and 11 times.This was not unexpected: the values of some parameters for thefour physical model tests were outside the range of input param-

Table 6. Mean Overtopping Discharges per Meter Length of Structure, q,Obtained with Formula of Pedersen �1996�

Test

q�m3 /s /m�

Physical model Pedersen’s formula

1 1.66 10−5 1.84 10−4

2 2.65 10−5 2.08 10−4

3 1.65 10−4 4.05 10−4

4 4.44 10−4 9.02 10−4

Table 7. Mean Overtopping Discharges per Meter Length of Structure, q,and Confidence Intervals Obtained with NN_Overtopping Tool

Test

Physical model NN_Overtopping

q q q2.5% q97.5%

�m3 /s /m� �m3 /s /m� �m3 /s /m� �m3 /s /m�

1 1.66 10−5 4.75 10−5 8.91 10−6 2.29 10−4

2 2.65 10−5 8.15 10−5 8.87 10−6 6.4710−4

3 1.65 10−4 1.48 10−4 1.70 10−5 1.05 10−3

4 4.44 10−4 3.60 10−4 3.80 10−5 3.52 10−3


eters for which the formula was developed and the value of thewave height used in the formulas was not measured at the toe ofthe structure, as it should be.

Agreement between the physical model data and theAMAZON output is inconsistent for the four tests analyzed, withAMAZON overestimating the value of q for the smaller dis-charges �Tests 1 and 2� and predicting rather accurately for thehigher discharges �Tests 3 and 4�. The differences are mainlyrelated to the way the model characterizes the porous medium,i.e., simply by using the bottom friction coefficient; but the dif-ferences are also due to the fact that the wave series input to thenumerical model was obtained using the Mansard and Funkemethod and, consequently, it is likely to be somewhat differentfrom the incident wave series in the physical model.

The influence of the bottom friction coefficient on the resultsof Tests 3 and 4 is small �see Fig. 5� as, most of the time,the porous medium is submerged by substantial depths of waterwhen overtopping occurs, leading to a better performance ofAMAZON. In Test 2, the influence of the porous medium isstronger, since it is submerged only during a small number ofovertopping events. In Test 1, the submergence is even less fre-quent and the water depths over the structure even smaller. ForTests 1–3, not even a value of the friction coefficient as high as0.03 m is enough to simulate accurately the influence of the po-rous medium. The highest value of f considered, 0.04 m, led tonumerical instabilities in all tests. Nevertheless, despite its limi-tations, AMAZON’s results are generally within the range ofmean discharges obtained in the physical model, except forTest 1. Furthermore, it should be pointed out that AMAZON isvery computationally efficient.

COBRAS-UC generally provides the best performance. Asnoted earlier, the differences between the COBRAS-UC resultsand the measured discharges are mainly related to the differencesbetween the incident wave series for the numerical and physicalmodels. Nevertheless, the performance is very good: except forTest 2, the values of q /qPM are close to unity and, even for Test 2,the values of q are approximately within the range obtained in thephysical model test. Note that Test 2 has the widest range ofvalues of qPM, which suggests a particular sensitivity of q to theincident wave series for this test condition. Note that small over-topping discharges are generally very sensitive to the characteris-tics of the irregular wave field close to the structure.

The performance of COBRAS-UC encourages its use in realcase studies of overtopping. An advantage of the model is that,once correctly calibrated, the wave field in the whole computa-tional domain is accurately quantified, allowing its use as a nu-merical flume, complementing the physical model tests, andproviding an easy method for investigating the influence on theresults of geometrical modifications to the structure. The maindisadvantage is related to the computational effort even though ithas to be pointed out that the complete flume has been simulated.

The NN_Overtopping prediction tool overestimates the resultsfor the smaller values of q �Tests 1 and 2� and underestimates forthe higher values �Tests 3 and 4�. The results obtained for the fourtests considered provide the same level of agreement with thedata as AMAZON and a much better agreement than Pedersen’sformula. In fact, for Tests 2 and 3, the results are within the rangeof q measured in the laboratory, and the differences for Tests 1and 4 are small, despite the fact that the value of the wave heightused in the tool was not measured at the toe of the structure, as itshould be. Furthermore, NN_Overtopping is simple and can berapidly applied. However, the width of the confidence intervals
supports the contention that the results from the NN_Overtopping

tool must be used carefully and only for the conceptual design ofcoastal structures.

Conclusions

This paper provides estimates of the mean overtopping dischargefor specified wave conditions and water level at the root of theSouth Breakwater of Póvoa de Varzim Harbour, Portugal, usingdifferent available techniques: a physical model �Reis et al. 2006�,the two numerical models AMAZON �Hu et al. 2000�, andCOBRAS-UC �Losada et al. 2008�, the empirical formula ofPedersen �1996�, and the neural network tool NN_Overtopping�van Gent et al. 2005�. The cross section at the root of the SouthBreakwater is a combination of a concrete vertical wall with adouble-layer rock slope in front of it.

The results show that rather accurate predictions were obtainedwith the COBRAS-UC numerical model: once correctly cali-brated, COBRAS-UC quantified the wave field in the whole com-putational domain, including the porous flow and the nonlinearphenomena that occur in wave breaking. This allows the model to

Fig. 8. Values of q and q /q

be used a s a numerical flume, complementing the physical model


tests and providing an easy method for investigating the influenceon the results of geometrical modifications to the structure.

Agreement between the data and the AMAZON output de-pended on the importance of porosity in influencing overtopping:agreement was very good when the impact was small; it worsenedas porosity played a more important role. Considering the com-putational efficiency of the model, it would be worthwhile tointroduce some improvements to the code in order to account forporosity �Roig and King 1991�.

Pedersen’s results are those which deviated most from thephysical model results, with the formula substantially overpre-dicting the measured mean overtopping discharges for all the con-ditions analyzed.

The NN_Overtopping prediction tool showed good agreementwith the laboratory tests, with a degree of accuracy similar toAMAZON. It performed well despite its simplicity and speed ofapplication. However, Coeveld et al. �2005� note that the reliabil-ity of the predictions should always be verified using dedicatedphysical model tests for the particular wave conditions and struc-ture geometry under consideration.

For engineering case studies of overtopping of coastal struc-

puted by different methods
PM com
tures, methods should correctly represent the physical phenomena


involved in overtopping, including porous flow and the nonlinearphenomena associated with wave breaking. However, the morecomplete a model is, the more time it takes to run. In order tochoose the most appropriate method to be used in any particularcase study and in each phase of a project, a compromise willusually have to be made between accuracy and computationaleffort.

Acknowledgments

The writers wish to thank the Instituto Portuário e dos TransportesMarítimos �IPTM�, Portugal, for the authorization granted to pub-lish some of the results of the overtopping two-dimensional scalemodel tests undertaken for the South Breakwater root of Póvoa deVarzim Harbour. The writers also gratefully acknowledge the fi-nancial sponsorship of Dr. Reis’s postdoctoral studies byFundação para a Ciência e a Tecnologia, Portugal. Thanks also goto Terry Hedges, University of Liverpool, U.K., for his help.I. J. L. is indebted to Puertos del Estado for the funding providedfor the development of COBRAS-UC.

Notation

The following symbols are used in this paper:Ac � armor crest freeboard defined as vertical distance

from SWL to armor crest;B � width of armor crest berm;D � test duration;

D50 � nominal diameter of rocks;ds � water depth at toe of structure;dx � cell width in x direction;dy � cell width in y direction;

f � bottom friction coefficient;H � wave height;

Hos � deep water significant wave height;Hs � significant wave height in front of structure;

Lom � deep water wavelength based on spectral meanperiod;

Lop � deep water wavelength based on spectral peakperiod;

Ls � shallow water wavelength at toe of the structure;m0 � zeroth moment of incident spectrum;

p � porosity;q � mean overtopping discharge per meter length of

structure;qn% � quantile n% for mean overtopping discharge per

meter length of structure;qPM � mean overtopping discharge obtained from physical

model;Rc � freeboard;T � wave period;

Tom � deep water spectral mean period;Top � deep water peak period;Tp � peak period in front of structure;� � angle of breakwater front slope measured from

horizontal;�L � linear friction coefficient; and
� � nonlinear friction coefficient.

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wave overtopping of póvoa de varzim breakwater: physical and numerical simulations

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