wave overtopping and induced currents at emergent low crested structures

www.elsevier.com/locate/coastaleng

Coastal Engineering 52

Wave overtopping and induced currents at emergent low

crested structures

Ivan Caceres a,*, Agustın Sanchez-Arcilla a, Barbara Zanuttigh b,

Alberto Lamberti b, Leopoldo Franco c

a Universitat Politecnica de Catalunya, Jordi Girona 1-3, 08304 Barcelona, Spainb University of Bologna, Italyc University of Roma 3, Italy

Available online 14 October 2005

Abstract

This paper deals with wave overtopping associated to low emergent detached breakwaters. It starts presenting the various

mechanisms which define the functional behaviour of detached breakwaters which are frequently overtopped. The emphasis is

on the role played by overtopping and how this contributes to the wave pumping in the area around the structure. The paper then

reviews the available formulations for predicting overtopping, considering the sensitivity of various equations to climatic

drivers and their suitability for low crested coastal structures. The paper then analyzes nearshore circulation models and how

overtopping discharges can be included. This is illustrated with a sample Q-3D model by analyzing the differences obtained

with/without overtopping. The emphasis is on the implications for the functional design of the structure and its morpho-

hydrodynamic impact. The main conclusion is the sensitivity of the functional design of such structures to the underlying

hydrodynamics processes and the importance to explicitly include overtopping for numerical simulations of the morpho-

hydrodynamic behaviour of low crested breakwaters.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Overtopping; Emergent; Low crested; Structure; Nearshore circulation; Numerical modelling

1. Introduction

Low crested structures (LCSs) offer a number of

potential advantages such as reduced visual and aes-

thetic impact and enhanced biological diversity. How-

ever, the functional design of such structures is

0378-3839/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.coastaleng.2005.09.004

* Corresponding author.

E-mail address: [email protected] (I. Caceres).

relatively difficult since they act as a partial filter for

incoming waves and a partial barrier for the associated

circulation field and resulting sediment fluxes. This

complexity is also illustrated by the high number of

variables (up to 14) participating in the functional

design of LCSs (Pilarczyk, 2003). Some of the inter-

acting forcing terms can be seen schematically in Fig. 1.

The partial filter activity of LCSs for incoming

waves depends on mean water level conditions, inci-

(2005) 931–947

Fig. 1. Schematization of wave-driven circulation fluxes for an alongshore uniform beach with an emerged LCS.

I. Caceres et al. / Coastal Engineering 52 (2005) 931–947932

dent wave parameters and structural geometry. The

shoaling/breaking process in front of a structure which

is frequently overtopped, increases the pumping of

wave fluxes over the detached breakwater.

Resulting sediment fluxes and morpho-dynamic

evolution are, therefore, a function of the wave and

circulation fields associated to the structure. Any

attempt to model (even understand) the actual mor-

pho-hydrodynamic impact of a LCS requires the expli-

cit consideration of terms driving the mentioned

pumping effect. However, there are a number of ways

to introduce those enhanced mass fluxes in wave, cur-

rent and sediment transport models, and also there are a

number of ways to compute the overtopped volume.

In the case of a LCS with the crest well above MSL

(or above an elevation of Hs/2) the shoreward transfer

of water mass and of waves past the crest can be

handled by numerical models based on a N–S equa-

tion solver, such as COBRAS (Liu and Lin, 1997) or

by using empirical formulae as described in Section 3

and in Van der Meer et al. (2005—this issue). In this

case, the wave agitation due to transmission is weak

and its effects (through second-order radiation stres-

ses) are assumed to be negligible compared to currents

induced by overtopping discharge.

In the case of a fully submerged structure (crest

below �Hs/2) standard wave averaged models can be

used to model the currents over the structure. These

models can also in this case be based on the N–S

equations (Losada et al., 2005—this issue) or, in order

to reduce the computational efforts, can use a set of

wave averaged nonlinear shallow water equations (as

e.g. the LIMCIR code used in this paper; see Johnson

et al., 2005—this issue) or some analytical solutions

of the same equations as the approximated model by

Bellotti (2004).

An alternative modelling strategy, able in principle

to deal with the two cases can be that of using a wave

resolving depth-integrated model such as those by

Brocchini et al. (2002) or Hu et al. (2000). It is

however clear that wave-averaged depth-integrated

models currently offer a reasonable compromise

between accuracy and computational costs. However,

a single model able to simulate the case in which the

LCS crest is above MSL and the case of submerged

LCS would be an attractive alternative.

Objectives of the paper are:

– to represent differences related to varying overtop-

ping probability between the most studied case of

emerged structures (i.e. rare overtopping) and LCS

(i.e. regularly overtopped structures) and the sub-

sequent effects on the overtopping discharge;

– to critically review available formulae for overtop-

ping discharge prediction;

– to analyze overtopping effects on nearshore circu-

lation at emergent structures through numerical

simulations.

I. Caceres et al. / Coastal Engineering 52 (2005) 931–947 933

In order to accomplish these aims, the paper is

organised as follows.

Section 2 briefly describes the most significant

process parameters as wave run-up and structure sub-

mergence through the analysis of the experimental

data derived from 3D tests at Aalborg University

(Kramer et al., 2005—this issue).

Existing formulations for the evaluation of bgreenwaterQ overtopping discharge, together with their lim-

itations, are recalled in Section 3.

Section 4 describes the Q-3D numerical tool devel-

oped at UPC (Caceres, 2004) that is adopted for

examining fluxes circulation around the structures.

First, the way the tool deals with overtopping fluxes

in the mass and momentum equations is presented;

then the model is validated by comparing numerical

and experimental data (Kramer et al., 2005—this

issue).

In Section 5, the numerical model is applied to

evaluate the functionality of a real LCS, the Altafulla

breakwater, by comparing the induced hydrodynamics

with and without overtopping.

2. Overtopping process

Overtopping refers to the process of water pas-

sing over a certain line, for instance the structure

crest. If the structure is not continuously submerged,

the process can be naturally interpreted and made

up of several events strictly related to the passage

of wave crests and characterized by a certain mass

and momentum (or volume and velocity). Overtop-

ping consequences are related to specific statistics

of the processes more relevant for the considered

effect: for instance the extreme momentum of one

overtopping crest may be responsible for damaging

exposed elements, whereas the mean volume per

unit time (or mean overtopping discharge) is

responsible for flooding. Nevertheless, the last is

the standard parameter for characterizing overtop-

ping intensity, probably because it is the easiest to

measure.

The total elapsed time and the total volume of

water passing over the structure are simply the sum

of elapsed time and passed volume within each wave.

This trivial observation is the basis of the relation

among mean overtopping discharge q, overtopping

probability Povt, mean overtopping volume VovtP

and

mean period of well-formed waves Tm:

q ¼P

waves VolumesPwaves Periods

¼ NovtdVovtP

NwdTm¼ Povt

VovtP

Tmð1Þ

where the overtopping probability Povt is the ratio

between the number of overtopping waves Novt and

the number of waves Nw.

Since every wave whose run-up exceeds the struc-

ture crest (and only these waves) cause overtopping,

the overtopping probability at the offshore edge of the

crest is equal to the probability that a single wave run-

up is higher than the crest.

For regular waves Povt is 0 if RuVF or 1 if RuNF.

For irregular waves, Van der Meer and Stam (1992)

suggested for run-up at rubble mound structures a

Weibull probability distribution, with parameters k1and k2 related to incident waves and slope character-

istics, from which the overtopping probability can be

derived:

Prob RuVzð Þ ¼ 1� exp � z=k1ð Þk2� �

ZPovt ¼ exp � F=k1ð Þk2� �

ð2Þ

where k1=0.4Hsisom�0.2cotga�0.2, som is the mean

wave steepness and a the structure offshore slope;

k2=3.0nm�0.75 for plunging waves (nmb2.5) and k2

¼ 0:52dP�0:3nPmffiffiffiffiffiffiffiffifficota

pfor surging waves (nmN2.5),

nm is the Iribarren number based on mean wave

period and P is the notional structure permeability.

Later, Van der Meer and Janssen (1995) provided a

simpler expression for the run-up distribution at dikes:

the shape parameter is constant k2=2, i.e. the distri-

bution is Rayleighian, and the location parameter is

proportional to incident wave height. The Rayleigh

distribution fits in any case rather well the distribution

of run-up.

In Eq. (1), Povt is the factor controlling the order of

magnitude of mean overtopping discharge and the

expression of the main effect of relative crest eleva-

tion F/Hs, as it can easily vary by orders of magnitude

as a consequence of modest variations of structure

crest elevation; see for instance Steendam et al.

(2004), where approximately:

qffiffiffiffiffiffiffiffiffigH3

s

p i0:1d10�FHsð Þ2 ð3Þ


It is remarkable that this trivial consequence of the

quasi-Rayleigh distribution of run-up was not focused

earlier, and most formulae do actually fit the right-

hand side of Eq. (3) with an exp-linear rather than a

parabolic pattern. The exp-parabolic pattern represents

a very rapid decrease to zero of overtopping discharge

with increasing crest elevation, that might be inter-

preted as a threshold.

The relation among overtopping frequency, volume

statistics and mean discharge is investigated in the

following based on data obtained (for the wave-by-

wave analysis, see Lamberti et al., 2004) from experi-

ments performed in Aalborg wave basin on perpendi-

cular and oblique layouts (for details of the

hydrodynamic tests, see Kramer et al., 2005—this

issue). Overtopping is initially referred to the seaward

side of the crest.

Fig. 2 shows, for emerged structures, the empiri-

cal Povt with varying wave intensity, represented

by the significant wave run-up Rus. Rus was eva-

luated following model (2), by assuming its exceed-

ing probability equal to 0.14 and accounting for

obliquity effects through the run-up reduction factor

mb=cos(|b|�108), in which b is the mean obli-

quity (mean angle between crests and structure

alignment).

Povt (invariably equal to 1 for regular wave experi-

ments) for irregular waves increases with Rus follow-

Fig. 2. Observed overtopping probability Povt versus the

ing more or less a ds-shapedT curve; the pattern and

values are in good agreement with predictions based

on Eq. (2).

Overtopping volumes can be well approximated by

a Weibull distribution, as in Van der Meer and Janssen

(1995). The mean value of such distribution VovtP

shows a more or less parabolic relation with the run-

up Ruovt, that is the median potential run-up of

waves causing overtopping (Fig. 3). Ruovt is derived

from (2) as the value giving exceeding probability just

half the overtopping probability. As suggested by

Pilarczyk (2000), when overtopping is rare, VovtP

is

almost proportional to (Ruovt�F)2; this implies a

fixed shape of the overtopping crests. The scaled

volumes however increase significantly as soon as

Povt exceeds 0.4–0.5.

The location parameter k1 of the volume distribu-

tion (typical overtopping volume) can be obtained

from the mean value as k1 ¼ VovtP

=C 1þ k�12

��.

The shape parameter k2 of the distribution in-

creases with increasing Ruovt (Fig. 4). Its values

range from something below 1 up to 3; the lowest

values refer to the case of drareT overtopping and the

highest values to very frequent overtopping. If due

attention is paid to difference in overtopping fre-

quency and parameter uncertainties, the lowest figures

are not substantially different from the value 3/4

suggested by Van der Meer and Janssen (1995).

relative wave run-up Rus/F for emergent structures.

Fig. 3. Mean overtopping volumes VovtP

scaled with (Ruovt�F) versus overtopping probability Povt, emergent and zero-freeboard structures.


In conclusion, for really low crested structures that

are overtopped by most waves, not just by the highest

(PovtN0.4 or F bk1), the crest level is so low in the

approximate Rayleigh distribution of run-up that the

shape of all overtopping characteristics distributions

become more symmetric and less variable (the shape

parameter in a Weibull distribution becomes greater)

and the overtopping wave crests become longer. The

combined model of a Weibull (or Rayleigh) distribu-

tion of run-up, that provides the overtopping prob-

Fig. 4. Shape parameter k2 of the Weibull volume distribution versus ov

ability, and a shape model for overtopping crests, as

the one suggested by Pilarczyk (2000), represents

properly the process for high and low freeboards if

the shape parameters are assumed slightly variable

with relative freeboard or overtopping probability.

The extrapolation below the aforementioned limits

of formulae adapted to high structures has two con-

trasting effects: the intrinsic limitation Povtb1 is not

recognized and the factor is overestimated; the

increasing length of overtopping crests (or duration

ertopping probability Povt, emergent and zero-freeboard structures.


of overtopping events) is not recognized and the mean

volume is underestimated. The total result on the

discharge is weak and may be confused with formulae

uncertainty.

When the crest is wide a significant part of the

overtopping volume is lost by percolation into the

crest. Water builds up the mean pressure in the rubble

mound and flows part offshore and part inshore. It is

questionable if this part should be included in the

mean discharge overtopping the structure; in any

case it should be made clear, when dealing with

wave overtopping, which position at the crest is

used to define wave overtopping, the seaward side

or the inner side of the crest.

Data on overtopping discharge from these experi-

ments are however not so numerous as necessary to

support a new formula for wave overtopping. In the

next section, empirical formulae for overtopping dis-

charge based on wide data sets are presented and

analyzed.

3. Existing formulation for overtopping discharge

Most of the formulations presented in this sec-

tion apply to emerged harbour structures or dikes,

which are seldom overtopped (very low Povt), and

rely on the empirical fitting of experimental data.

These formulae, based on small-scale data and

Froude scaling, are subject to model, scale and

wind effects (De Rouck et al., 2001) and their

application field is limited by the range of tested

parameters and configurations.

In general, the average overtopping discharge per

unit width of structure, q, is expressed as a function of

the standard parameters: Hs, significant wave height;

T, a characteristic wave period; r, spreading of short-

crested waves; b, angle of wave attack; F, structure

crest freeboard; h, water depth in front of structure;

structure geometry.

The presented formulations have been chosen

among the great variety of existing expressions,

and have been implemented in the LIMCIR circula-

tion model to include the overtopping effect in the

simulations.

First, as a classic yet still used formulation, the

Owen (1980) expression is considered. This formula-

tion for seawalls takes into account the effects of

roughness and of the structure slope and crest shape

through A1 and B1 experimental coefficients:

q

gHsTm¼ A1exp � B1

cf

F

TmffiffiffiffiffiffiffiffigHs

p�

ð4Þ

where Tm is the mean wave period at the toe of the

structure and cf is a reduction factor for slope rough-

ness (all other variables as defined previously).

Assuming a weir-type formula, Hedges and Reis

(1998) re-analyzed the data by Owen (1980) with the

aim of improving the prediction for large freeboards

and freeboards close to zero, obtaining the following

expression:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig CHsð Þ3

q ¼ A2 1� F

CHs

� B2

ð5Þ

discharge is null for F NCHs. Coefficient C relates the

maximum run-up in the wave attack to the incident

significant wave height (CHs=Rumax) and A2 and B2

are related regression coefficients whose value can be

found in the authors’ paper. By taking into account the

structure roughness, as considered in the CIRIA/CUR

manual, and referring to the modal maximum run-up

over 100 waves, C can be evaluated by

C ¼ 1:52 1:35np� �

cf if npb2C ¼ 1:52 3� 0:15np

� �cf if npN2

ð6Þ

Van der Meer and Janssen (1995) provide different

formulae for the overtopping discharge due to plun-

ging and surging waves, Eq. (7.1) and Eq. (7.2),

respectively. The reported regression coefficients are

adopted by Van der Meer (2002), and are valid for

dikes. Overtopping discharge can be evaluated as the

minimum of the two expressions.

qffiffiffiffiffiffiffiffiffiffiffigH3

m0

p ¼ 0:067ffiffiffiffiffiffiffiffiffitana

p cbnopexp

� 5:2 H 4:7ð Þ

� F

Hm0nopcbcfcbcv

!ð7:1Þ

qffiffiffiffiffiffiffiffiffiffiffigH3

m0

p ¼ 0:2exp � 2:6 H 2:3ð Þ F

Hm0cf cb

!ð7:2Þ

where Hm0 is spectral significant wave height at dike

toe (4ffiffiffiffiffiffim0

p), where nop is the breaker index at the

dike toe based on the spectral wave period, tana is


the structure slope and c are run-up reduction fac-

tors: cb for berms, cf for slope roughness, cb for

the angle of wave attack, cv for a wave-wall. The cfactor values can be found in Van der Meer (2002);

for traditional rocky structures they may be consid-

ered as a first approximation equal to 1, except cfthat is 0.55 for a two stones armour layer. Double

figures in brackets denote, respectively, average and

bcautiousQ design values (overestimating overtopping

discharge).

In order to interpret properly the formula, the

reader should realize that for the evaluation of run-

up the same report suggests a similar procedure, i.e.

run-up shall be evaluated as the minimum of the two

expressions (8.1) and (8.2):

Ru;2%=Hm0 ¼ 1:65 H 1:75ð Þdnpd cbcf cb ð8:1Þ

Ru;2%=Hm0 ¼ 4:0 H 4:3ð Þ � 1:5 H 1:6ð Þ=ffiffiffiffiffinp

qh id cfcb

ð8:2Þ

In case of small freeboard, Kofoed and Burcharth

(2002) suggest a discharge reduction factor in

(7.2, average formulation) for FHsV0:75 and surging

waves.

These formulae have been developed from data

sets including different structural types covering the

cases of: impermeable smooth, rough straight and

bermed slopes.

It must be stressed that some authors consider in

their formulations the influence of the angle of wave

attack, the permeability, the crest width and other

reduction factors. Since the performance of overtop-

ping formulations for LCS is rather an open question

(most expressions have been derived for fully

emerged structures which are only exceptionally over-

topped), the influence of the above mentioned factors

is outside the scope of this paper and has not been

further considered. The factors here considered are the

more directly relevant for overtopping predictions and

can be summarised as follows.

! Incident significant wave height, Hs; an increase in

wave height produces an exponentially rising over-

topping.

! Crest level or freeboard, F; the effect of the free-

board is reciprocal to that of the significant wave

height: lower freeboards give an exponentially

increasing discharge; it is however noted that the

associated overtopping variations are of a lower

magnitude than for similar changes in Hs.

! Roughness of the coastal structure. The roughness

reduction factor cf can be considered as an empiri-

cal coefficient that describe the effect on overtop-

ping of sloping and berm-structure with different

materials/permeability. The end result is a reduc-

tion of the overtopping discharge for increasing

permeability/roughness.

! Structure slope. The empirical coefficients A and

B that will have different values for Owen (sub-

index 1) and Hedges and Reis (sub-index 2),

depend basically on the structure profile. There

exist different values for slope angles ranging

from 1:1 to 1:5.

! Wave obliquity angle b and directional spreading.

Several studies have been carried out to analyze

their effects (Owen, 1980; De Waal and van der

Meer, 1992; Franco and Franco, 1999; Van der

Meer, 2002; Schuttrumpf et al., 2003) and different

results have been found. Most studies show that

run-up and overtopping decrease with increasing

wave obliquity; this is evident for long-crested

waves and b N408.

Other effects such as crest walls or crest berms may

be considered for harbour structures (Bradbury and

Allsop, 1988; Owen and Steele, 1991) but are of little

relevance for LCS.

The presented overtopping formulations have been

derived from hydraulic tests. Their application field is

therefore limited by the range of parameters and con-

figuration. On-going research aims at improving the

accuracy or number of factors controlling the over-

topping discharge (De Rouck et al., 2002). Since most

equations apply to emerged harbour structures which

are seldom overtopped, the presented formulations

should be considered as an order of magnitude for

the discharge above the LCS. The actual use of these

formulations for coastal problems should be done with

some caution since when trying to apply the presented

equations to limited freeboards the predictions tend to

diverge.

The selected formulations are, thus, only a first

attempt to introduce overtopping in nearshore circula-

tion simulations.


4. Overtopping in nearshore circulation modelling

Overtopping has been seldom considered in near-

shore circulation modelling. One exception is Gunar-

atna et al. (1998) in which the Fredsoe and Deigaard

(1992) formulation was used within a 2DH circulation

code. The discharge associated to the roller of a

spilling breaker (enhanced in the wave propagation

direction) was evaluated by:

q ¼ l1H2s cosbTm

ð9Þ

where l1 is a non-dimensional calibration coefficient.

In this paper, overtopping has been explicitly intro-

duced into the mass and momentum conservation

equations. This is illustrated by the LIMCIR code, an

advanced Q-3D circulation model solving the depth-

and time-averaged continuity and momentum equa-

tions while recovering a depth-averaged undertow.

The resulting partial differential equations are solved

with a staggered grid and an Alternating Direction

Implicit method that allows, at the end of each itera-

tion, to obtain a centered scheme in space and time

(Sanchez-Arcilla and Lemos, 1990; Caceres, 2004).

The resulting set of equations can be written as:

Continuity equation:

BhgiBt4

þ B

Bxjhþ hgið ÞUj

� �þ B

Bxj

Mj

q

� þ

BQj

Bxj

¼ 0 8 x; y; t4ð Þ ð10Þ

Momentum equations:

BUi

Bt4þ Uj

BUi

Bxjþ g

BbgNBxi

¼ 1

hþ bgNð Þ Fi � Bi þ Ci þ1

qBRij

Bxj

� � �

� Uj þMj

q hþ bgNð Þ

� B

Bxj

Mi

q hþ bgNð Þ

� � �

� BUi

Bxj

Mj

q hþ bgNð Þ

� � �� Uj þ

Qj

hþ bgNð Þ

!"

� B

Bxj

Qi

hþ bgNð Þ

!#� BUi

Bxj

Qj

hþ bgNð Þ

!" #

8 x; y; t4ð Þ; i ¼ 1; 2 ð11Þ

where Fi represents the wave shear stress including

huiwi as driving mechanism for currents (uiw are the

horizontal—i component—and vertical components

of the wave orbital velocity), Ci represents the wind

stress at the free surface, Bi the bottom shear stress,

Rij the turbulent Reynolds stresses, Mi are the wave-

induced mass fluxes associated to the wave field

(with or without breaking depending on the position

of the grid point inside the domain) and Qi repre-

sents the overtopping volume flux per meter length

of structure.

The closure sub-models are based on state of the

art formulations. The bed shear stresses are obtained

according to Madsen (1994) while the roller model is

based on Dally and Brown (1995). The eddy viscosity

is evaluated based on Nielsen (1985) and Osiecki and

Dally (1996) while the wave-induced mass flux can

be obtained from De Vriend and Stive (1987) or

Fredsoe and Deigaard (1992).

The overtopping term can be obtained from any of

the above-presented formulations for sloping or ver-

tical structures. The overtopped volume is bexternallyQintroduced into the continuity and momentum equa-

tions at the lee side of the structure. In this way, the

overtopped volume is considered in a manner analo-

gous to the wave-induced mass fluxes in both con-

tinuity and momentum equations.

As a first approximation, the amount of overtopped

water has been introduced in the mass-momentum

equations behind the LCS as if bextractedQ from the

front face of the structure.

The LIMCIR circulation model allows using dif-

ferent overtopping formulations: Owen (1980), Eq.

(4); Hedges and Reis (1998), Eq. (5). These expres-

sions are mostly valid for impermeable structures

although Hedges and Reis, Eq. (5), represents the

effect of structure permeability through an unspecified

coefficient C.

5. Model validation against laboratory results.

Application to multiple structures

This section analyzes flow patterns around two

round-heads with a gap in between. The configura-

tion corresponds to Layout 1 tests done at Aalborg

University (see Kramer et al., 2005—this issue,

Fig. 7).


From the 44 performed tests, the 12 emerged ones

were selected. The sample test presented here (Test

15) reproduces a narrow crest structure with a free-

board of 0.03 m (0.6 m in nature).

The Owen (1980) formulation (4) has been

employed using the conventional parameter setting

for a 1:2 slope structure (A1=0.00939, B1=21.6),

while the Hedges and Reis (1998) expression (5)

has been used employing the parameter setting

(A2=0.0079, B2=4.55 and C evaluated following

Eq. (6)). The roughness correction factor has been

set equal to 0.6, consistently with the rock armour

layer of d50=4.5 cm over a core with d50=3.4 cm (see

e.g. the CIRIA/CUR manual).

The numerical modelling suite has been run at

prototype scale. The wave conditions at the sea-side

boundary have been derived from the observed time

series at wave gauges 3 to 7 (Kramer et al., 2005—

this issue), adjusted using the appropriate shoaling

coefficient to btransformQ all values for the depth at

the offshore boundary.

Fig. 5. Wave propagation pattern numerically simulated by the LIMWAVE

m and Tm=6.44 s for normal wave incidence.

The structure-induced reflections were, however,

not modelled by the wave code and, although present

in the observations (in e.g. the wave gauges 1, 2, 9, 10

and 11) were not considered in the simulated wave or

current fields.

The absorbing end-beach and lateral boundaries

have been simplified in the numerical domain by

impermeable boundaries. Moreover, the LCS are

represented as impermeable breakwaters, neglecting

the transmission through the structures observed in the

hydraulic tests.

These discrepancies are expected to produce a

limited effect on the computed velocity fields, which

should be mainly driven by the radiation stress gra-

dients and wave-induced mass fluxes inside the com-

putational domain.

The obtained wave results (Fig. 5) agree reason-

ably with the observed ones (Table 1). The main

discrepancies (wave gauges 16 and 20) are considered

to be due to boundary-induced effects, associated to

the presence of the absorbing beach.

code under test 15 conditions. Incident wave conditions are Hs=1.62

Table 1

Summary of measured and modelled results for Test 15

Wave gauges LIMWAVE (m) Measured (m)

3, 4, 5, 6, 7 1.62 1.62

12 1.15 1.22

16 1.03 1.3

18 0.82 0.83

20 0.35 0.61

Table 2

Comparison between measured and simulated velocities (point III of

Test 15)

Measured (m/s) LIMCIR (m/s) RMAE

No overtopping 0.516 0.31 0.3898

Owen 0.62 0.19

V.d.Meer–Janssen 0.52 0.0097

Hedges–Reis 0.37 0.2706

Different overtopping formulations have been considered.


The corresponding circulation fields, without and

with overtopping appear to be bphysicallyQ consistent.Focusing on the velocities at point III (point B, F and IV

are either too close to the structure or directly affected

by wave reflection effects), it is clear that overtopping

enhances the return flow through the gap (19% velocity

increase when employing the Hedges and Reis, 1998

formulation; 67% for Van der Meer and Janssen’s,

1995; 100% for Owen’s, 1980). Moreover, the current

along the lee-side of the LCS is seen to increase 10% to

15% (Fig. 6) when considering overtopping.

The calculated results (Table 2) are clearly depen-

dent on the formulation used to evaluate overtopping

(Table 3). The LIMCIR code provides the best results

Fig. 6. Nearshore circulation field considering overtopping terms. This picture corresponds to the difference between circulation fields with and

without overtopping terms (Test 15).

when using the Van der Meer and Janssen (1995)

formulation (7.1) and (7.2). The Hedges and Reis

(1998) formulation (5) provides a better fit for higher

wave height conditions, while the Owen (1980) for-

mulation (4) overpredicts the experimentally mea-

sured results under the evaluated test conditions.

The obtained relative mean absolute error

(RMAE) obtained with the Van der Meer and Janssen

(1995) expression (7.1) and (7.2) is below 0.1 for the

circulation field in general (Test 15), as shown in

Table 2. The resulting fit can be considered as

bexcellentQ using the Van Rijn et al. (2003) proposed

thresholds. Also the Owen (1980) and Hedges and

Table 3

Different overtopping discharges obtained employing test 15

conditions

Formulation Q (m3/m/s)

Owen 460

V.d.Meer–Janssen 315

Hedges–Reis 95

Table 4

Selected wave height conditions (Tm=4 s) for the Altafulla LCS

simulations

Hs (m) nop Wave breaking Hs/h

1 2.49 Surging 0.25

2 1.76 Plunging 0.5


Reis (1998) equations improve the predictability capa-

city of the numerical code employed when LCS tests

are performed.

6. Application to a single LCS configuration. The

Altafulla field case

6.1. The Altafulla LCS

The Altafulla beach, located 80 km south of Bar-

celona (Fig. 7) in the North Western Mediterranean

coast of Spain, has a length of 2 km and is limited by

two rocky outcrops. The averaged slope is 1.6% and

the average d50 is about 210 Am. Because of sedi-

mentary losses, two artificial nourishment operations

were carried out in 1991 (158.000 m3 of sand sup-

plied) and in 1994 (252.000 m3 of sand supplied).

After the first nourishment in 1991, the LCS shown in

Fig. 7 was built.

This field case has been analyzed under bidealizedQconditions by using a constant slope (1.7%) simplified

bathymetry. The structure has been settled at 200 m of

the shoreline and the depth at the toe of the structure is

close to 4 m. The wave conditions are summarized in

Table 4. All cases have been simulated for three

Fig. 7. The Altafulla low crested breakwater in 2002.

different freeboards, F, equal to 0.3, 0.5 and 1.0 m.

The obtained results with and without considering

overtopping and for the studied wave conditions are

summarized in Table 5. This combination of test cases

covers most of the physics associated to the hydro-

dynamics around a LCS.

Although some of the analyzed cases did not fulfill

the theoretical criteria required by the selected over-

topping formulations, they have been included in the

analysis so as to obtain a more comprehensive picture

of the hydrodynamics around a LCS with and without

overtopping.

6.2. Numerical results

The main factor driving the circulation around the

LCS, before considering overtopping, is the wave dif-

fraction pattern around the structure (Fig. 8). The two

eddies generated by the wave diffraction pattern are

evident in all simulations. Although the pressure gra-

dients associated to the varying wave set-up are con-

servative forces and cannot generate a vortical

circulation (Dingemans, 1997), the obtained vortices

display a bclosedQ pattern and do not, therefore, violateKelvin’s circulation theorem. The resulting gradients in

radiation stresses generate, close to the shore, water

fluxes that converge towards the center of the sheltered

Table 5

Cross-shore and longshore velocity peaks at a transect 10 m up the

structure tip and for the different tested conditions

Hs F (m) U V

1 m 0.3 14.59% 6.53%

0.5 8.65% 4.02%

1 10.27% 2.01%

2 m 0.3 27.49% 8.18%

0.5 19.5% 6.37%

1 11.55% 3.17%


area. This converging flux shows also an offshore

component towards the structure (i.e. flowing towards

the offshore) and a diverging flux that flows parallel to

the lee side of the LCS. The inclusion of overtopping

increases the water pilling at the lee-side of the struc-

ture which modifies the longshore and cross-shore

pressure gradients and therefore the corresponding

velocities. This results in an intensification of currents

outgoing from the sheltered area.

A comparison of the circulation pattern with and

without overtopping appears in Fig. 9. When over-

topping is included the nearshore circulation pattern is

modified in shape and intensity. Although the main

Fig. 8. Nearshore circulation for the Altafulla LCS without considering ov

m and Tm=4 s).

apparent changes occur in the near field of the LCS

the inclusion of overtopping terms alter the overall

circulation pattern particularly in the sheltered area.

The onshore current associated to overtopping is seen

to spread towards both sides of the LCS. The con-

sideration of overtopping also increases the shoreward

velocity in front and over the structure since now an

onshoreward flux is allowed.

These velocity changes can be clearly observed in

the two cross-shore transects shown in Figs. 10 and

11, 10 m away from the structure and at the middle

section of the LCS (respectively).

The net result of overtopping is, therefore, an

increase of outgoing fluxes from the sheltered area

at both sides of the LCS. Both incoming and outgoing

fluxes tend to grow with increasing overtopping

volumes. The introduction of overtopping is also

seen to slow down the undertow towards the LCS.

All these phenomena are in accordance with the extra

mass and momentum of water introduced because of

overtopping and illustrates the considerable influence

that these terms have on the circulation pattern around

a LCS.

ertopping, and induced by normal wave incidence conditions (Hs=2

Fig. 9. Effect of overtopping on nearshore circulation for the Altafulla LCS. The figure highlights the difference between circulation fields with

and without overtopping (Fig. 8).


Fig. 11 shows the cross-shore velocity variation in

a cross-shore transect at the middle section of the

LCS. The increase in overtopping does not signifi-

cantly displace the undertow peak, although its result-

ing magnitude is decreased (approximately 10% for

the 0.3 m freeboard test).

As a summary, Table 5 shows the resulting max-

imum hydrodynamic variation in cross-shore (U) and

longshore (V) velocities associated to overtopping

terms.

The presented percentages refer to velocity incre-

ments with respect to the bno overtoppingQ situation.The variation in cross-shore velocities goes up to 27%

for waves of 2 m and a LCS freeboard of 0.3 m. The

obtained results also show the effect of increasing the

discharge rate (by increasing Hs or decreasing F).

Longshore velocities increase continuously with an

increasing overtopping discharges. Cross-shore velo-

cities show a more berraticQ pattern for the Hs=1 m

wave conditions. This shows the various competing

mechanisms (Sanchez-Arcilla et al., submitted for

publication) controlling the circulation pattern around

the LCS. The displacement of peak velocities can be

better appreciated for cross-shore velocities which can

be interpreted in terms of the opposing momentum

fluxes for these velocity components.

7. Conclusions

Overtopping discharge is decomposed in the two

factors: overtopping frequency and mean overtopping

volumes. The first is controlling the order of magni-

tude of overtopping discharge, the second is increas-

ing with increasing submergence.

Overtopping frequency can be derived by run-up

statistics following Van der Meer and Stam (1992),

whereas relevant scatter was found in predicting over-

topping discharge by means of existing formulae for

emerged structures.

Overtopping volumes are Weibull-distributed (Van

der Meer and Janssen, 1995) with shape coefficient

Fig. 10. Longshore (left) and cross-shore (right) velocity transects 10 m up the structure tip, for different free-boards F and with/without

overtopping. Incident wave conditions are Hs=2 m and Tm=4 s and perpendicular incidence. Owen’s formulation has been employed for

overtopping.


depending on freeboard (or overtopping probability)

and ranging from 1 (rare overtopping) to 3 (very

frequent overtopping).

Fig. 11. Cross-shore velocity transect in the middle of the LCS

structure, for different freeboards F and with/without overtopping.

Incident wave conditions are Hs=2 m and Tm=4 s and perpen-

dicular incidence. Owen’s formulation has been employed for

overtopping.

For small run-up events, overtopping is weak and

most of the water volume overtopping the offshore

edge is lost for percolation in the rubble mound; with

increasing run-up, overtopping discharge increases

and percolated fraction decreases following a ds-shapedT curve.

The performed numerical simulations and corre-

sponding intercomparisons with laboratory results

and field observations, show the complexity of LCS

functional design. Because of this, although low

crested detached breakwaters are a very promising

solution concept in coastal engineering, there seems

to be a need for further research to assess the various

mechanisms controlling the full morpho-hydrody-

namic behavior of these structures. In particular,

overtopping induces a clear enhancement of the cir-

culation field around the LCS, modifying the con-

ventionally presented circulation pattern controlled

by wave diffraction. The overtopping flux plays

thus a non-negligible role in controlling the intensity

of longshore and cross-shore currents near a LCS.

The performed simulations should be, however,

considered with some caution since the overtopping

formulations have been derived in general for higher

crested structures which are only seldom over-

topped. The expressions selected for this paper are

Q* and


therefore only intended to provide a first assessment

of the role played by this mechanism in nearshore

hydrodynamics. In spite of these uncertainties, it is

apparent that the introduction of overtopping affects

the full computational domain, i.e. down to the

shoreline. This implies that overtopping effects will

have a non-negligible role on the overall morpho-

dynamic evolution both for single-structure and sin-

gle-gap configurations. The derivation of design

rules for both configurations will require a more

extensive validation of a numerical suite such as

the one presented in this paper which should be

run for a wide range of construction and climatic

parameters.

A combination of numerical and hydraulic results,

which explicitly consider overtopping fluxes, will

allow an improved functional design of LCS and a

more frequent and reliable use of this coastal engi-

neering structure.

Notation

A1, A2, B1 and B2 Empirical coefficients

Bi Bottom shear stress

C Ratio of maximum run-up to the significant

height of incident waves

Ci Wind stress at the free surface

d50 Median sediment diameter

F Crest freeboard of the structure above the still

water level

FI Wave shear stress including huiwi as drivingmechanism for currents (uiw are the horizon-

tal—i component—and vertical components

of the wave orbital velocity)

Hm0 Spectral significant wave height at dike toe

Hs Significant wave height at the toe of the

structure

h* Wave breaking parameter

h Water depth in front of the structure

l1 Non-dimensional calibration coefficient

Mi Wave-induced mass fluxes associated to the

wave field

m0 Zero moment of spectrum

Novt Number of overtopping waves

Nw Number of well-formed waves

Povt Overtopping probability

Qi Overtopping

R* The dimensionless discharge and struc-

tural freeboard

q Mean overtopping discharge per unit length

RI Turbulent Reynolds stresses

Rus Significant wave run-up

Ruovt Wave run-up corresponding to the overtop-

ping probability associated to half crest

freeboard

Ru,2% Run-up level exceeded by 2% of the incident

waves

Sop Deepwater wave steepness associated to the

peak period

Som Mean wave steepness

Tm Mean wave period at the toe of the structure

Tp Peak period

U Cross-shore current velocity

V Longshore current velocity

VovtP

Mean overtopping volume

np Iribarren or surf-similarity parameter calcu-

lated using the period of the peak spectral

density

nop Idem but considered at the dike toe

cb Reduction factor for the angle of wave attack

cv Reduction factor for a wave-wall

cb Reduction factor for berm width

cf Reduction factor for slope roughness

a Structure slope

b Angle of wave attack

Acknowledgements

The authors acknowledge the DELOS research

project (contract no. EVK3-CT-2000-00041). The

authors wish also to thank the research staff in charge

of the experimental data acquisition and also to Mr.

Gonzalez-Marco for his assistance.

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