Soft shore protection methods: The use of advanced numerical modelsin the evaluation of beach nourishment

Theofanis V. Karambas a, Achilleas G. Samaras b,n

a Department of Civil Engineering, Aristotle University of Thessaloniki, University Campus, Thessaloniki 54124, Greeceb CIRI—EC, Fluid Dynamics Research Unit, University of Bologna, Via del Lazzaretto 15/5, Bologna 40131, Italy

a r t i c l e i n f o

Article history:Received 16 May 2013Accepted 27 September 2014Available online 22 October 2014

Keywords:Beach nourishmentNumerical modellingBoussinesq modelSediment transport

a b s t r a c t

Beach nourishment is one of the worldwide most common soft shore protection methods. However, thedesign of these projects is usually based on empirical equations and rules, leaving large margins of errorregarding their expected efficiency. In the present work, an advanced wave and sediment transportnumerical model is developed and tested in the evaluation of beach nourishment. Non-linear wavetransformation in the surf and swash zone is computed by a non-linear breaking wave model based onthe higher order Boussinesq equations, for breaking and non-breaking waves. The new Camenen andLarson (2007), transport rate formula for non-cohesive sediments (involving unsteady aspects of thesand transport phenomenon) is adopted for estimating the sheet flow sediment transport rates, as wellas the bed load and suspended load over ripples. Suspended sediment transport rate is incorporated bysolving the 2DH depth-integrated transport equation. Model results are compared with experimentaldata of both profile (cross-shore) and planform morphology evolution; the agreement between the twois considered to be quite satisfactory.

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1. Introduction

Beach nourishment comprises the placement of good qualitysand on a beach in order to extend the width of a specific coastalstretch or mitigate erosive phenomena; the nourishment locationin the cross-shore direction may vary in different beaches from thefirst dune row to the surf zone depending on the scope of theproject (i.e. long- or short-term design), the morphological char-acteristics and the wave/wind regime of every beach.

The first nourishment project in the US was that of ConeyIsland, NY, in 1923 (Dornhelm, 1995). The first recorded attemptin Europe was that of Estoril, Portugal, in 1950; a nourishmentproject in Norderney island, Germany, followed closely after that,with the practice expanding eventually to the rest of Europe untilthe early ’80s (Hamm et al., 2002). On the other hand, it was notuntil the early ’90 s that beach nourishment was introduced as apractice in China (Cai et al., 2011).

Nowadays, periodic nourishment is considered worldwide tobe an effective soft-engineering method for both protection andrestoration of beaches facing erosive phenomena. According torecent estimates (Hamm et al., 2002) the total annual rate ofnourishment in Europe adds up to 28 Mm3, roughly equal to the

respective value for all the Federal projects in the US (i.e. projectswith the involvement of the US Army Corps of Engineers).Although similar estimates are not available for China, Cai et al.(2011) attempted a comparison proposing the “volume per unitlength” index. The specific index for China (based on data for themain nourishment projects in the country between 1990 and2009) was found to be less than half of the respective value forEuropean countries, with the projects being smaller in length andtotal amount of material used, as well.

For a long time the implementation of beach nourishmentwas essentially based on engineering experience and empirical rules;however, a series of manuals are currently available to supportdecision-making, design and maintenance in nourishment pro-jects (see Hamm et al., 2002 for a detailed list). Furthermore, andparticularly for the design of the beach fill and the evaluation of itsperformance, the use of numerical models has gradually replacedempirical equations and guidelines. Capobianco et al. (2002)present a detailed description of the model classes, the modellingsituations and the elements that are the main sources of uncer-tainty in nourishment design. The range of the approaches in usevaries from shoreline evolution models, to profile evolutionmodels, multi-layer models, and 2D/3D models; exemplary refer-ence can be made to the recent work of van Duin et al. (2004),Larroudé (2008), Shibutani et al. (2009), Kuroiwa et al. (2010), Pan(2011), Van Rijn et al. (2011), Grunnet et al. (2012) and Zhang et al.(2012).

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Ocean Engineering

http://dx.doi.org/10.1016/j.oceaneng.2014.09.0430029-8018/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ39 51 2090551; fax: þ39 51 2090550.E-mail addresses: karambas@civil.auth.gr (T.V. Karambas),

achilleas.samaras@unibo.it (A.G. Samaras).

Ocean Engineering 92 (2014) 129–136

Models based on the Boussinesq equations have come a longway since the pioneering work of Peregrine in the ’60s(Peregrine, 1967), gradually incorporating breaking effects andcoupling with sediment transport modules. Zelt (1991) andKarambas and Koutitas (1992) were among the first to incorpo-rate breaking effects using an eddy viscosity analogy, withKennedy et al. (2000) further improving the representation ofwave height decay and setup in the swash. Schäffer et al. (1993)used a different approach based on the Svendsen (1984)roller concept, further improved by Madsen et al. (1997); thespecific approach set the basis for the later development ofBoussinesq models including vorticity effects as well (Briganti etal., 2004; Musumeci et al., 2005; Veeramony and Svendsen,2000). Recently, Antuono et al. (2009) proposed an innovativeapproach on the representation of nonlinear dispersive waterwaves, and Antuono and Brocchini (2013) a novel approach forthe description of wave propagation and flow circulation in thenearshore.

Given that the design of modern nourishment projects shouldbe based on a reliable physical representation of both the hydro-and morpho-dynamic processes in the nearshore and swash, thecoupling of Boussinesq models with sediment transport modulesemerged as a need in the respective field. The work of Rakha et al.(1997), Karambas and Koutitas (2002), Karambas (2002, 2003,2012) and Karambas and Karathanassi (2004) confirmed thesuitability of the specific approach. The coupled models incorpo-rate nonlinear breaking and non-breaking irregular wave propa-gation (deep to shallow water and swash), and comprise theprediction of quasi-3D currents and long waves, using the respec-tive information (i.e. breaking wave induced turbulence, near bedvelocity asymmetry/acceleration, swash zone dynamics, etc.) tofeed complex sediment transport formulae (Wenneker et al.,2011).

In the present work, an advanced phase-resolving nonlinearwave, sediment transport and bed morphology evolution 2DHmodel (developed by the authors) is tested against profile andplanform evolution experimental data for beach nourishmentevaluation. Regarding the coastal hydrodynamics, the nonlinearwave transformation in the surf and swash zone is computed by anonlinear breaking wave model based on the higher orderBoussinesq equations for breaking and non-breaking waves.Regarding the bed and suspended sediment load, the transportrate formula proposed by Camenen and Larson (2005, 2007,2008) is adopted, which comprises unsteady aspects of the sandtransport phenomenon (see also the recent formula evaluation byPostacchini et al., 2012). The innovation of this work is thevalidation of a new Boussinesq-type morphology model, understeep slope conditions and with the presence of a submergedbreakwater. The new model is an improved version of the modelof Karambas and Koutitas (2002), with the incorporation of a new,well-validated in literature, transport rate formula (after Camenenand Larson, 2007) and a new swash zone transport simulation(after Karambas, 2006). The Camenen and Larson (2007) formulaincorporated shear stress directly into bed load transport,as opposed to the Dibajnia and Watanabe (1992) formula usedin the original model version of Karambas and Koutitas (2002).Moreover, the suspended load in the formula is related to bothwave breaking dissipation and bottom shear stress, as opposed tothe model of Karambas and Koutitas (2002), where suspendedload was only related to wave breaking dissipation. Applicationsin the cross-shore direction reproduce experiments presented byDette et al. (2002) and Di Risio et al. (2010), the later referring toboth unprotected and protected nourished profiles by a sub-merged breakwater. Applications in the planform reproduceexperiments presented by Karasu et al. (2008) and Chonwattanaet al. (2005).

2. The wave and morphology evolution model

2.1. The phase-resolving wave model based on higher-orderBoussinesq-type equations

Models based on the Boussinesq equations have been provencapable of reproducing successfully the phenomena that affectwave-induced morphology of coastal areas. The classical Boussi-nesq equations have been extended so as to be able to includehigher order nonlinear terms, which can describe better thepropagation of highly nonlinear waves in the shoaling zone.The linear dispersion characteristics of the equations have beenimproved as well, in order to describe nonlinear wave propagationfrom deeper waters (Zou, 1999).

Based on the aforementioned velocity profile, the followinghigher order Boussinesq-type equations for breaking and non-breaking waves can be derived (Zou, 1999; Karambas and Koutitas,2002; Karambas and Karathanassi, 2004):

ζtþ∇ hUð Þ ¼ 0 ð1Þ

Utþ1h∇Mu �1

hU∇ðUhÞþg∇ζþG¼ 1

2h∇ ∇� dUtð Þ� �

�16h2∇ ∇� Ut½ �þ 1

30d2∇ ∇� Utþg∇ζ

� �� �

þ 130

∇ ∇� d2Utþgd2∇ζ� �h i

�d∇ðδ∇� UÞt�τb

hþE ð2Þ

where Mu is defined as:

Mu ¼ dþζ� �

u2oþδ c2�u2

o

� � ð3Þ

and G as:

G¼ 13∇ d2 ∇� Uð Þ2�U� ∇2U� 1

10∇2 U� Uð Þ

� ��12ζ∇ ∇� dUtð Þ� �

ð4Þ

In Eqs. (1–4) the subscript t denotes differentiation withrespect to time, d is the still water depth, U is the horizontalvelocity vector U¼(U,V) with U and V being the depth-averagedhorizontal velocities along the x- and y-directions, respectively,ζ is the surface elevation, h is the total depth (h¼dþζ), g is thegravitational acceleration, τb¼(τbx, τby) is the bottom friction term(shear stress components approximated by the use of the quad-ratic law according to Ribberink (1998)), δ is the roller thickness(determined geometrically according to Schäffer et al. (1993)), E isthe eddy viscosity term (according to Chen et al., 1999), and uo isthe bottom velocity vector uo¼(uo, vo) with uo and vo being theinstantaneous bottom velocities along the x- and y-directions,respectively.

The Boussinesq-type equations with the improved nonlinearityand linear dispersion characteristics in deeper water, are accurateto the third order O(ε2,εσ2,σ4) (Zou, 1999); the nonlinearityand dispersion parameters are defined as ε¼Α/d and σ¼d/L0,respectively, where A¼characteristic wave amplitude andL0¼characteristic wave length.

2.2. The sediment transport module

Sediment transport is usually divided into bed load, suspendedload and sheet flow. The modeling concepts presently used for theprediction of each one vary, from empirical transport formulas tomore sophisticated bottom boundary layer models. In the presentwork bed load transport (namely qsb) is estimated with a quasi-steady semi-empirical formulation developed by Camenen andLarson (2005, 2007, 2008), for an oscillatory flow combined with a

T.V. Karambas, A.G. Samaras / Ocean Engineering 92 (2014) 129–136130

superimposed current under an arbitrary angle; it is defined as:

Φb ¼

qsb;wffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis�1ð Þgd350

p ¼ anffiffiffiffiffiffiffiffiffiffiffiffiffiffiθcw;net

pθcw;mexp �b

θcr

θcw

�

qsb;nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis�1ð Þgd350

p ¼ anffiffiffiffiffiffiffiθcn

pθcw;mexp �b

θcr

θcw

�

8>>>><>>>>:

ð5Þ

where the subscripts w and n denote the wave direction andthe direction normal to the wave direction, respectively, s¼ρs/ρ isthe relative density between sediment (ρs) and water (ρ), g is thegravitational acceleration, d50¼median grain size, aw, an, b is theempirical coefficients (Camenen and Larson, 2007), θcw,m, θcw arethe mean and maximum Shields parameters due to wave-currentinteraction, θcn is the current-related Shields parameter in thedirection normal to the wave direction, and θcr is the criticalShields parameter for the inception of transport. The net Shieldsparameter θcw,net in Eq. (5) is given by:

θcw;net ¼ 1�apl;b� �

θcw;on� 1þapl;b� �

θcw;of f ð6Þwhere θcw,on and θcw,off are the mean values of the instantaneousShields parameter over the two half periods Twc and Twt

(Tw¼TwcþTwt, where Tw is the wave period and the indices cand t refer to crest and trough, respectively), and αpl,b is acoefficient for the phase-lag effects (Camenen and Larson, 2007).The Shields parameter in Eq. (6) is also defined as in Camenen andLarson (2007).

The suspended sediment load qss for the wave direction andthe direction normal to the wave directions is obtained fromEqs. (7) and (8), respectively (Camenen and Larson, 2007):

qss;w ¼Ucw;netcRεws

1�exp �wshε

�� ð7Þ

qss;n ¼ Uc sin φcRεws

1�exp �wshε

�� ð8Þ

where cR is the reference concentration at the bottom, ε is thesediment diffusivity, and Ucw,net is the net mean current.

The bed reference concentration cR is written as follows basedon the analysis of a large data set of sediment concentrationprofiles (Camenen and Larson, 2007):

cR ¼ 3:51�3exp �0:3dnð Þθcw;mexp �4:5θcr

θcw

�ð9Þ

where dn is the dimensionless grain size, defined by:

dn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis�1ð Þg=v23

qd50 ð10Þ

Sediment diffusivity is taken equal to the eddy viscositycoefficient, which is related to the energy dissipation due to wavebreaking according to Karambas and Koutitas (2002). Phase-lageffects in the suspended concentration due to ripples, are alsoincorporated according to Camenen and Larson (2007). The swashzone transport rate is estimated according to Karambas (2006).

The nearshore morphological changes are calculated by solvingthe conservation of sediment transport equation (Leont’yev(1996)):

∂zb∂t

¼ � ∂∂x

qx�ε qx�� ��∂zb

∂x

�� ∂∂y

qy�ε qy�� ��∂zb

∂y

�ð11Þ

where zb is the local bottom elevation, qx(¼qs,xþqb,x), qy(¼qs,yþqb,y) are the volumetric sediment transport rates along the x- andy-horizontal directions, respectively, and ε is the coefficient setequal to 2.0. The second term in each of the parentheses in theright part of Eq. (11) represents the diffusion term, an additionalgravitational term which reflects the effect of local bottom slopeon sediment transport (see also Karambas and Koutitas, 2002). Itshould be noted that for simulations in the presence of

breakwaters (as in the application of Section 3.2), Eq. (11) is notsolved in the breakwater region, while at the landward side of thestructure a “wall” boundary condition (qn¼0, where n is thedirection normal to the boundary) is introduced.

3. Methodology and applications

3.1. Methodology

In order to evaluate the model and to study the effect of beachnourishment on the morphology of coastal areas, a series of applica-tions are conducted, comparing model results with laboratorymeasurements of profile and planform morphology evolution. Themethodology adopted for the series of model applications, can beencoded into the steps described in the following. First, the initialbathymetry is inserted into the wave model (see Section 2.1) in orderto estimate the wave and current fields. These fields are afterwardsused by the sediment transport module (see Section 2.2) to calculatethe sediment transport rates. Finally, bathymetry is updated by thesediment transport module solving the equation of the conservationof sediment transport (Leont’yev, 1996) for the previously calculatedtransport rates. The procedure is repeated for a user-specified timeperiod or until a state of morphologic equilibrium is reached. For thethree-dimensional applications following the work of Chonwattanaet al. (2005) that was based on the use of representative waves (seeSections 3.3 and 4.2 in the following), the aforementioned procedurewas repeated successively for each of these representative wavesbased on their frequency of occurrence.

3.2. Two-dimensional applications

In the context of the MAST III—SAFE Project a series of experi-ments were carried out in the Grosser Wellenkanal in Hannover(Dette et al., 1998a, 1998b), the largest wave flume in the world witha length of 330 m, a depth of 7 m and awidth of 5 m. The objective ofthe project was to study the development of the underwater profileand sand losses from the beach in order to contribute to thedevelopment of recommendations for coastal protection and beachnourishment design. The beach consisted of well-sorted sand with ad50¼0.3 mm median diameter. The underwater profile from 4mabove the flume floor was shaped to the Bruun equilibrium profilefollowing d¼0.12x2/3 (see Bruun, 1954; Dean, 1977), where d¼waterdepth and x¼distance from shore. The beach slope above still waterlevel varied from 1:20 to 1:5. The Test Series reproduced in thepresent paper are B2 (1:10 dry beach slope), C2 (1:5 dry beach slope)and F1 (1:3 dry beach slope), under storm surge – erosive –

conditions (TMA spectrum with zero-moment wave heightHmo¼1.20 m and period Tm¼5.5 s) with and a 1 m increase in waterlevel (Dette et al., 2002).

Di Risio et al. (2010) performed a series of experiments in theframework of the Si.Co.R.A. Project (Regione Abruzzo, 2006), inorder to evaluate the design of prospective beach nourishmentprojects along the coasts of the Abruzzo Province in Italy. In a 45 mlong and 1.5 m wide wave flume Di Risio et al. (2010) tested theperformance of both unprotected and protected (by a submergedbreakwater) nourished beaches, representing prototype conditionsat a length scale n¼12; the offshore depth was set to 0.70 m.The beach consisted of well-sorted fine sand with a d50¼0.12 mmmedian diameter. The slope of the surf zone was set equal to1:110; the beach-face slope was set equal to 1:7. Along the beachprofile, water depth varied from 0.36 m at the deepest point to0.20 m at the toe of the steep nourished profile; the emergedbeach was set to a level equal to þ0.12 m from SWL and the profilewas connected to the offshore water depth by a steel ramp. For theprotected beach tests, the rubble-mound submerged breakwater

T.V. Karambas, A.G. Samaras / Ocean Engineering 92 (2014) 129–136 131

freeboard was set to �0.12 m from SWL, the landward slope toebeing at �0.31 m; the crest berm width was 0.80 m, the seawardslope 1:3 and the landward slope 1:2. The Test Series repro-duced in the present paper are AC-W1 (offshore wave heightHoff¼0.222 m, significant wave height HS¼0.210, peak periodTp¼2.28; accretive conditions) and ER-W3 (offshore wave heightHoff¼0.283 m, significant wave height HS¼0.260, peak periodTp¼1.5; erosive conditions) for both the protected and unprotectedprofiles.

3.3. Three-dimensional applications

Karasu et al. (2008) performed a series of three-dimensionalexperiments in order to investigate beach nourishment perfor-mance. The experiments were carried out at the 30 m long, 12 mwide and 1.2 m deep basin of the Hydraulics Laboratory, CivilEngineering Department, Karadeniz Technical University, Trabzon,Turkey. The still water depth was fixed to 0.75 m; the beachconsisted of well-sorted sand with a d50¼0.18 mm median dia-meter and the profile slope was set to 1:15. The trapezoidalnourishment fill length was 2.15 m (including the tapered endsextending 0.22 m on each side); the crest width was 0.30 m andthe base width 0.40 m. The fill back was set 0.20 m behind theshoreline with the berm height varying among the experiments.The Experiment reproduced in the present paper is No. 9 (incidentwave height H¼4 m, wave period T¼1 sec, berm heightB¼0.08 cm, median grain size of the fill d50¼0.18 mm).

Chonwattana et al. (2005) studied the morphological evolutionin a groin compartment due to wave-induced sediment transportfor Sangjun beach in Thailand, applying a 2D hydrodynamicmodel and a 3D morphological model on the basis of represen-tative waves. Model results were compared with high-qualityfield measurements of bed morphology before (December 2000;see Fig. 8a) and after (June 2001; see Fig. 8b) the construction ofthe fishtail groins in April 2001. It is noted that the fishtail groinsare depicted in Fig. 8a in order to facilitate visual comparisonwith Fig. 8b, even though they were not constructed at the time.In the present work, bed morphology evolution was simulatedusing the set of three representative waves (i.e. three equivalentwaves on an annual basis), calculated by Chonwattana et al.(2005) and presented in Table 1. A simple refraction method wasapplied to calculate the wave characteristics at the offshoreboundary of the computational domain. Three different wave-induced current fields were generated resulting to three differentpatterns of sediment transport. First, for representative wave no.1,the initial bathymetry was inserted into the Boussinesq model.The bathymetry was updated according to the sediment move-ment, taking into account the annual frequency corresponding tothe representative wave no.1. Then, the new bathymetry wasinserted to the Boussinesq model, using representative wave no.2characteristics as input. Finally, representative wave no.3 inputconditions were applied, resulting to a new morphology evolu-tion. This procedure was repeated until the final time wasreached.

4. Results and discussion

4.1. Two-dimensional applications

Fig. 1a shows two snapshots (Tp/2 time frame) of the computedwave propagation for the Test Series B2 of Dette et al. (2002);Fig. 1b and c shows, also for B2, the comparison between thecomputed and measured transport rates and profiles, respectively.Figs. 2 and 3 show the comparison between the computed and mea-

Table 1Characteristics of the representative waves used (adopted by Chonwattana et al.,2005).

Nos. Height Hο

(m)Period T(s)

Incident angle θο(deg)

Annual frequency f(%)

1 0.69 3.93 �45.68 19.672 0.56 3.92 �1.59 68.033 0.55 3.82 44.90 12.30

Fig. 1. (a) Computed wave propagation and comparison between the computedand measured (b) transport rates and (c) profiles for the Test Series B2 of Detteet al. (2002).

Fig. 2. Comparison between the computed and measure profiles for the Test SeriesC2 of Dette et al. (2002).

T.V. Karambas, A.G. Samaras / Ocean Engineering 92 (2014) 129–136132

sured profiles for the Test Series C2 and F1 of Dette et al. (2002),respectively. The agreement between model results and measure-ments is overall quite satisfactory. The small shift in transport rates inthe swash zone �x¼230–250 m for B2 (see Fig. 1b) explains therespective shift in profile evolution there (see Fig. 1c). How-ever, swash zone erosion is well predicted for all B2 (see Fig. 1b),C2 (see Fig. 2) and F1 (see Fig. 3) Test Series. Moreover, the barformation is also predicted well, with small divergences betweenthe computed and measured profiles at �x¼210–225 m for B2 (seeFig. 1b), �x¼205–225 m for C2 (see Fig. 2) and �x¼215–225 m forF1 (see Fig. 3).

Fig. 4a and b shows the comparison between the computed andmeasured profiles for the AC-W1 Test Series of Di Risio et al.(2010), for the unprotected and protected nourished beaches,

respectively. Profile evolution is well predicted for both theprotected and unprotected cases, with the divergence betweenmodel results and measurements being small. It should be notedthat for the protected case, the model run – and consequentlythe profiles were compared – only for the part inshore of thesubmerged breakwater. The sole exception to the overall goodagreement for both cases is the top of the beach berm, wheremeasurements indicate the formation of a relatively steep accre-tive profile, while model results indicate a mild shape that is moresimilar to the initial profile. This is attributed to the fact that themodel does not predict the onshore directed sediment transport atthe top of the berm.

Fig. 5a shows two snapshots (Tp/2 time frame) of the computedwave propagation for the unprotected case of Test Series ER-W3 ofDi Risio et al. (2010); Fig. 5b and c shows, also for the ER-W3unprotected case, the comparison between the computed andmeasured transport rates and profiles, respectively. It should benoted that transport rates are expressed in the dimensionless form

Fig. 3. Comparison between the computed and measure profiles for the Test SeriesF1 of Dette et al. (2002).

Fig. 4. Comparison between the computed and measured profiles for the TestSeries AC-W1 of Di Risio et al. (2010) for (a) the unprotected and (b) the protectednourished beaches.

Fig. 5. (a) Computed wave propagation and comparison between the computedand measured (b) transport rates and (c) profiles for the Test Series ER-W3 of DiRisio et al. (2010) for the unprotected nourished beach.

T.V. Karambas, A.G. Samaras / Ocean Engineering 92 (2014) 129–136 133

of Φ, as defined by:

Φ¼ Q=ð1�pÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis�1ð Þgd350

q ð12Þ

where Q is the total sediment transport rate (Q¼qssþqsb) andp¼beach porosity. The agreement between model results andmeasurements is overall quite satisfactory for this Test Series aswell, as profiles practically overlap along most of the nourishedbeach. The discrepancy in the dimensionless transport ratebetween data and model results at �x/Lmax¼0.25 (i.e. near thecrest of Φ), results in a respective discrepancy in bed morphologyevolution. However, the observed discrepancies in bed morphol-ogy evolution are smaller than those of the respective transportrates in general (see also Karambas and Koutitas, 2002); this canbe justified considering that profile evolution is affected by thespatial derivative of Φ, as well as by the diffusion term in theequation of the conservation of sediment transport (Eq. (9)).

Fig. 6a shows two snapshots (Tp/2 time frame) of the computedwave propagation for the protected case of Test Series ER-W3 ofDi Risio et al. (2010); Fig. 6b and c shows, also for the ER-W3protected case, the comparison between the computed andmeasured transport rates (expressed in the dimensionless formof Φ; see Eq. (11)) and profiles, respectively. There appears to be adiscrepancy between Φ values inshore and near the breakwater(�x/Lmax¼0.5–0.7) reaching �ΔΦ¼0.06 at maximum, in contrastto what happens at the beach-face where the divergence isinsignificant. The aforementioned discrepancy results in discre-pancies in bed morphology evolution at the toe of the structure.However, it is only reasonable that a small transport rate (and arespectively small spatial derivative) will lead to small morpholo-gical changes, as the model predicts in accordance with bedevolution measurements. Overall, profile evolution is well pre-dicted with minimal discrepancies along the entire profile, the soleexceptions being the one mentioned above and the beach-face,where experiments indicate a profile slightly advanced than theone calculated by the model.

4.2. Three-dimensional applications

Fig. 7a shows the comparison between the initial and finalmeasured bathymetries for Experiment no. 9 of Karasu et al.(2008); Fig. 7b shows the comparison between the computedand measured final bathymetry for the same Experiment. All

Fig. 6. (a) Computed wave propagation and comparison between the computedand measured (b) transport rates and (c) profiles for the Test Series ER-W3 ofDi Risio et al. (2010) for the protected nourished beach.

Fig. 7. Comparison between (a) the initial and final measured bathymetries forExperiment No. 9 of Karasu et al. (2008) and (b) the computed and measured finalbathymetry for the same Experiment.

T.V. Karambas, A.G. Samaras / Ocean Engineering 92 (2014) 129–136134

contours in Fig. 7a and b represent bed elevations expressed inmeters. It should be noted that – different from two-dimensionalapplications – in Fig. 7 x-axis denotes the direction alongshore andy-axis the direction cross-shore. The agreement of the results isoverall satisfactory; the divergences in bathymetric contours aremainly located closer to the right field boundary and are moresignificant for the 0.00 m, 0.06 m and 0.08 m contours. Modelresults, in general, indicate a milder evolution than the onemeasured; this is attributed to the model inability to reproducewell longshore ondulations.

Fig. 8c shows the breaking wave- induced current field forrepresentative wave no. 1 (see Table 1); Fig. 9 shows a snapshot ofthe computed free surface elevation for the same representativewave, depicting the refraction, diffraction, partial reflection, break-ing and dissipation after breaking of the obliquely incident waves.Fig. 8d shows the comparison between the measured bathymetryafter the groin construction (also shown in Fig. 8b) and thecomputed bathymetry by the model for the set of three represen-tative waves adopted by Chonwattana et al., 2005; see also Section3.3 and Table 1). Field measurements (Fig. 8a and b) are evident ofthe rapid morphology evolution that followed the construction(April 2001), as within only two months (June 2001) the initiallystraight shoreline between the groins transformed into a bay-shaped beach. Model results (Fig. 8d) are in quite close agreementwith field measurements, indicating a good representation of thecurrent-induced sediment movement towards the groins by themodel (see also Fig. 8c). As also noted for the other three-

dimensional application, model results indicate a milder evolutionthan the one measured.

5. Conclusions

In the present paper an advanced phase-resolving nonlinearwave, sediment transport and bed morphology evolution 2DHmodel (developed by the authors) is described and tested againstprofile and planform morphology evolution measurements in both

Fig. 8. Measured bed elevation (in m from MSL) of Sangjun beach (a) before (December 2000) and (b) after the construction (June 2001) of the fishtail groins; (c) breakingwave- induced current field for representative wave no.1 (He¼0.69 m, T¼3.93 s, θo¼�45.68 deg); (d) comparison between the measured bathymetry after the groinconstruction and the computed bathymetry by the model for the set of three representative waves adopted by Chonwattana et al. (2005).

Fig. 9. Snapshot of the computed free surface elevation for the representative waveno.1 (He¼0.69 m, T¼3.93 s, θo¼�45.68 deg).

T.V. Karambas, A.G. Samaras / Ocean Engineering 92 (2014) 129–136 135

the laboratory and the field. The objective of the specific work wasto test the model’s performance in the evaluation of beachnourishment; the laboratory tests used were selected accordinglyfrom the previous research attempts of Dette et al. (2002) and DiRisio et al. (2010) for the two-dimensional applications, and ofKarasu et al. (2008) and Chonwattana et al. (2005) for the three-dimensional applications, all for non-cohesive sediments. Thecomparison with the Di Risio et al. (2010) measurements alsocomprises experiments for nourished profiles protected by sub-merged breakwaters.

Comparative analysis of the results confirms the model’scapability to calculate both cross-shore (1DV) and planform(2DH) morphology evolution, as the agreement between modelresults and measurements is overall satisfactory and the diver-gences observed small and in limited parts of the studied profiles/planforms. The above support the use of advanced numericalmodels in the evaluation of beach nourishment, as the latter hasbecome a modern engineering solution for eroding beachesaround the world. Moreover, the present work is one of the fewusing wave models based on higher-order Boussinesq equations thatincorporate detailed descriptions of nonlinear waves and currents,suitable for feeding complex sediment transport formulae.

Acknowledgments

The authors would like to thank the anonymous reviewers fortheir constructive comments and suggestions.

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