Modelling of waves and currents around submerged breakwaters

Coastal Engineering 52 (2005) 949 – 969 www.elsevier.com/locate/coastaleng

Modelling of waves and currents around submerged breakwaters Hakeem K. Johnson a,*, Theophanis V. Karambas b, Ioannis Avgeris b, Barbara Zanuttigh c, Daniel Gonzalez-Marco d, I. Caceres d a

DHI Water and Environment, Agern Alle 5, DK-2970 Hørsholm, Denmark b Aristotle University of Thessaloniki, Greece c University of Bologna, Italy d Universitat Polite`cnica de Catalunya, Spain Available online 27 October 2005

Abstract Recent experimental data collected during the DELOS project are used to validate two approaches for simulating waves and currents in the vicinity of submerged breakwaters. The first approach is a phase-averaged method in which a wave model is used to simulate wave transformation and calculate radiation stresses, while a flow model (2-dimensional depth averaged or quasi-3D) is used to calculate the resulting wave driven currents. The second approach is a phase resolving method in which a high order 2DH-Boussinesq-type model is used to calculate the waves and flow. The models predict wave heights that are comparable to measurements if the wave breaking sub-model is properly tuned for dissipation over the submerged breakwater. It is shown that the simulated flow pattern using both approaches is qualitatively similar to that observed in the experiments. Furthermore, the phase-resolving model shows good agreement between measured and simulated instantaneous surface elevations in wave flume tests. D 2005 Elsevier B.V. All rights reserved. Keywords: Submerged breakwaters; Waves; Currents; Numerical modelling

1. Introduction Submerged breakwaters are commonly used for coastal protection on many eroding coasts. A desirable feature of submerged breakwaters (and low crested structures, in general) is that they do not interrupt the clear view of the sea from the beach. * Corresponding author. E-mail address: [email protected] (H.K. Johnson). 0378-3839/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2005.09.011

This aesthetic feature is important for maintaining the tourist value of many beaches and it is usually one of the considerations in using such structures for shoreline protection. The basic idea in the use of submerged structures is to reduce the wave energy reaching the beach, by triggering wave energy dissipation over the structure, and thus reduce sediment transport and the potential for coastal erosion. A proper understanding of the effect of submerged breakwaters on nearshore waves

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and currents is necessary for the calculation of sediment transport and morphological evolution in the vicinity of such structures. This is important in order to achieve a good functional design of the submerged structure for coastal protection. Traditionally, the study of the behaviour of low crested structures in terms of transmission, reflection, and wave overtopping has been done in flume experiments in which only one horizontal dimension is taken into account (Drei and Lamberti, 1999; Yamashiro et al., 1999; Kriezi et al., 1999; Gironella and Sanchez-Arcilla, 1999). There are fewer investigations about full 3D experimental studies, in which more detailed information is obtained (Chapman et al., 1999; Ilic et al., 1999). In this paper, our focus is on the modelling of waves and currents around submerged breakwaters. These structures result primarily in wave energy dissipation through the physical mechanisms of wave breaking and friction. The energy dissipation results in gradients in wave radiation stresses, which drives the mean flow pattern and wave setup. Two approaches are investigated and compared with recent laboratory measurements of waves and currents around submerged breakwaters. The first approach is a phase-averaged method in which a wave model is used to model the wave transformation and calculate radiation stresses, while a 2-dimensional depth averaged flow model is used to calculate the resulting wave driven currents using the radiation stresses computed by the wave model. The second approach is a phase resolving method in which a 2DH-Boussinesq-type model is used to calculate the waves and currents. The phase-averaged method has been widely used in the literature for modelling waves and nearshore currents in the coastal zone. Examples are Noda (1974) and Madsen et al. (1987). However, applications over submerged breakwaters are quite limited. Recent examples are van der Biezen et al. (1998); Christensen et al. (2003) and Lesser et al. (2003). The use of the phase resolving method for wavedriven circulation is more recent, especially over porous structures. Cruz et al. (1997) derived a set of 2DBoussinesq equations over a porous bed of arbitrary thickness and tested their applicability for refraction, diffraction and reflection around a submerged porous breakwater with an opening. Recently, Hsiao et al. (2002) presented a fully non-linear 2DH-Boussinesq-

type model for waves propagating over a permeable bed and compared model results with experimental data for the case of regular waves passing over a porous submerged breakwater. In this study wave evolution over porous submerged breakwaters is investigated with the use of a 2DH-Boussinesq-type model, following a procedure similar to that of Cruz et al. (1997). The goal of this study is to investigate the validity of the two presented approaches for modelling waves and currents in the vicinity of submerged breakwaters by comparison with recent laboratory data. This study is part of a larger effort to improve coastal morphological modelling in the vicinity of submerged breakwaters. The plan of the remainder of this paper is as follows: a description of the numerical models is presented in Section 2, followed by a brief description of the laboratory experiments in Section 3. The numerical model results are presented and discussed in Section 4. Lastly, a summary of the work carried out and conclusions are presented in Section 5.

2. Brief description of numerical models 2.1. Phase-averaged method In this study, two phase-averaged models developed at DHI Water and Environment, Denmark and Universitat Politecnica de Catalunya, Spain are used. These models are briefly described in the sections below. While the general approach of the two models is similar, the components adopted for representing wave propagation, wave breaking, friction and so on are different, so that the models can be regarded as two completely different examples from the 2DH phase-averaged family. An overview of these models and the phase-resolving Boussinesq model is presented in Table 1. 2.1.1. MIKE 21 phase-averaged model This model consists of a parabolic wave model (MIKE 21 PMS) for calculating integral wave parameters and radiation stresses, while the flow computations are carried out with a depth averaged flow model (MIKE 21 HD). Both models are part of the MIKE 21 modelling system.

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Table 1 Numerical models used in the study Phase-averaged

Phase-resolving

MIKE 21 model

LIM model

Boussinesq model

Wave model

Parabolic refraction and diffraction wave model

2DH Boussinesq wave model

Flow model Wave breaking Bottom friction in wave model Wave–current interaction Driving force in flow model Bottom friction in flow model

2DH flow model Energy dissipation given by modified Battjes and Janssen (1978) Quadratic friction law No Gradients in radiation stresses Wave–current bed shear (Fredsøe, 1984)

Flow in porous medium Shoreline boundary condition

No Flooding and drying

Irrotational wave number, eikonal equation, conservation of wave action Q3D flow model Energy dissipation given by Dally et al. (1984) Quadratic friction law No Gradients in radiation stresses Wave–current bed shear (Madsen, 1994) No Flooding and drying

MIKE 21 PMS is a refraction/diffraction model based on the parabolic approximation to the mild slope equation. It includes the wide-angle parabolic approximation equations (Minimax approximations) of Kirby (1986). The model accounts for the influence of shoaling, refraction, diffraction, forward scattering, breaking, bottom friction, frequency and directional spreading. Bottom friction dissipation is described using the expression by Dingemanns (1983) for random waves, while dissipation due to wave breaking is described using the theory of Battjes and Janssen (1978), hereafter referred as BJ78. For modelling wave dissipation over submerged breakwaters, it was found necessary to modify the wave breaking theory in order to get model parameters that sensibly depend on known wave and breakwater properties. The approach used in the modified wave breaking theory is to split wave breaking into two parts, namely: (1) depth limited breaking described with BJ78 theory and (2) steepness limited breaking described with an integrated form of the whitecapping dissipation term commonly used in spectral wind–wave models. The modified wave breaking theory is presented in detail in Johnson (in press). MIKE 21 PMS takes into account the frequency and directional randomness by decomposing the user-specified frequency-directional spectrum into discrete components, propagation of the components and linear superposition at each grid point to get the total energy, and lastly application of

Same as above Eddy viscosity term Quadratic friction law Yes Not applicable Not applicable Yes Fixed

wave energy dissipation formulation on the total energy. MIKE 21 HD is a two-dimensional depth-averaged hydrodynamic model for simulating water levels and depth-integrated fluxes driven by wave breaking (radiation stresses), wind, atmospheric pressure conditions and tide. The main features of the model are described in Abbott et al. (1973). In the calculations, a MIKE 21 tool program is used to calculate the enhanced bed resistance in combined waves and current using the method of Fredsøe (1984). The calculated resistance is used in the flow model. 2.1.2. LIM Phase-averaged model This model consists of a wave model (LIMWAVE) and a Q3D nearshore circulation model (LIMCIR). These models have been specifically adapted for the study of hydrodynamics around low crested structures (LCS) and have been validated with observations (Sa´nchez-Arcilla et al., 2004). The LIMWAVE model is a phase-averaged model based on the conservation of the wave action equation (for the wave height), the eikonal equation (for the phase information) and the irrotationality of the wave number vector (for the wave angle) as described in Ebersole (1985) and Liu (1990). This model incorporates depth limited breaking dissipation according to Dally et al. (1984) while the bottom friction dissipation is calculated using the expression given by Tolman (1992). The LIMWAVE model takes into account

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the directional randomness via the decomposition, propagation and later linear superposition of a Mitsuyasu directional spectrum (Mitsuyasu et al., 1975). The Q3D circulation model (LIMCIR) is based on the depth and time averaged mass and momentum conservation equations. It includes the driving effect of waves via the corresponding radiation stresses (Rivero and Sanchez-Arcilla, 1995 and Alonso, 1999), roller effects (Dally and Brown, 1995) and wave induced mass fluxes (De Vriend and Stive, 1987). The turbulence closure is calculated using the Osiecki and Dally (1996) algebraic formulation. The rest of the closure terms include a wind friction factor and Madsen (1994) formulation to estimate bed shear stress considering the co-existence of waves and currents. 2.2. Phase-resolving method The phase-resolving method is based on a 2DHBoussinesq-type model for wave evolution over porous submerged breakwaters, developed at Aristotle University of Thessaloniki, Greece. The model development follows a procedure similar to that of Cruz et al. (1997). 2.2.1. Governing equations A higher-order Boussinesq-type model, with improved linear dispersion characteristics is used to describe wave motion in the regions upstream and downstream of the breakwater (Karambas and Koutitas, 2002). Above the breakwater the model incorporates two extra terms accounting for the interaction between the waves over the structure and the flow within the porous structure, one in the continuity equation and one in the momentum equation respectively, following the approach of Cruz et al. (1997).

Table 2 Parameters in the long wave equation for porous medium Parameter

Authors

1 þ c 1u 1 þ cm u ¼ u u c m = added mass coefficient, c = empirical coefficient that accounts for the added mass m Cf a1 ¼ ; a2 ¼ pffiffiffiffi K K m = kinematic viscosity (10 6 m2/s), C f = dimensionless parameter, K = intrinsic permeability

van Gent, 1995

cr ¼

K¼

2 d50 du3 2

að1  uÞ a = empirical coefficient, d 50 = mean size of the porous material pffiffiffiffi 1u K Cf ¼ b u d50 b = empirical coefficient

Ward, 1964; Sollitt and Cross, 1976; Losada et al., 1995; Cruz et al., 1997 van Gent (1995, 1994), Burcharth and Andersen (1995) van Gent (1995)

In two-dimensional depth-averaged form the governing equations (continuity and momentum equations) are: gt þ j½ðh þ gÞu þ ujðhs us Þ ¼ 0 ð1Þ
2  h þ 2hg ut þ uju þ gjg ¼ j2 ut þ jhj2 ut 3  h2  3 uj u juj2 u þ jhx ut þ 3   þ hjgjut þ hjhuj2 u þ Bh2 ½j2 ut þ gj3 g þ j2 ðujuÞ þ 2Bhjhðjut u þ gj2 gÞ þ hj2 ðhs ust Þ 2 ð2Þ where u is the depth-averaged, horizontal velocity vector, g is the surface elevation, h is the water

Fig. 1. Definition of variables in the phase-resolving model.

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depth, B is the dispersion coefficient, us is the depthaveraged, seepage (fluid) velocity vector inside the porous medium, h s is the porous medium thickness and u is the porosity (Fig. 1). As suggested by Madsen and Sørensen (1992), B is set equal to 1/15, which gives the closest match to linear theory dispersion relation for h / L 0 as large as 0.5 (where L 0 is the deep water wave length). Eqs. (1) and (2) are solved in the region of the breakwater in conjunction with a depth-averaged Darcy–Forchheimer (momentum) equation describing the flow inside the porous medium. Assuming that O[(h s / L)2] V 1 the two-dimensional, depth-averaged momentum equation written in terms of the fluid velocity us (uD = uus, uD is Darcy velocity) reduces to (Cruz et al., 1997):

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An eddy viscosity formulation is adopted in order to simulate wave breaking (Kennedy et al., 2000) by introducing an eddy viscosity term in the right-handside of the momentum equation (Eq. (2)). The governing equations are finite-differenced utilizing a high-order predictor–corrector scheme as described in Wei and Kirby (1995). Wave generation is implemented inside the computational domain using the source function method as described by Wei et al. (1999), and adapted in the present model to be consistent with the Karambas and Koutitas (2002) equations.

3. Laboratory experiments 3.1. Wave basin experiments

cr ust þ us jus ¼  gjg  ua1 us  u2 a2 us jus j: ð3Þ Eq. (3) is known as the non-linear long wave equation for porous medium. On the right hand side of Eq. (3), the second and the third term are the friction terms. The second term is the laminar friction term, while the third term is the turbulent friction term. In Eq. (3), c r is inertial coefficient, a 1 and a 2 are porous resistance coefficients and u is porosity, see Table 2. A number of studies (Madsen, 1974; Vidal et al., 1988; van Gent, 1995) propose values for the nondimensional coefficients a, b and c depending on the material type and the length scale of the solid particles. Table 3 contains values for the coefficients a and b obtained by van Gent (1995). In the model, a, b and c are chosen as 1000, 1.1 and 0.34, respectively, as recommended by van Gent (1995). The corresponding values of c r, K, C f, a 1 and a 2 are calculated from Table 2 assuming u = 0.5. Table 3 Material tested and corresponding coefficients (van Gent, 1995) Material

D 50 (m)

u

a

b

Irregular rock Semi round rock Very round rock Irregular rock Irregular rock Spheres

0.0610 0.0487 0.0488 0.0202 0.0310 0.0460

0.442 0.454 0.393 0.449 0.388 0.476

1791 0 1066 1662 1007 2070

0.55 0.88 0.29 1.07 0.63 0.69

A series of 3D experiments on low crested structures were carried out in the 9.7 m x12.5 m wave basin at Aalborg University, Denmark. The experiments were aimed at describing wave conditions, wave setup, overtopping discharge conditions as well as current flow fields in the vicinity of such structures. Detailed description of the experimental layouts and the characteristics of the tests are provided by Zanuttigh and Lamberti (in press) and Kramer et al. (2005—this issue). Thus, only a brief description is given here. One of the layouts tested is a symmetric arrangement of two detached breakwaters separated by a gap (Fig. 2). For details on structure geometry, see the companion paper by Kramer et al. (2005—this issue). Two breakwater types were tested, one with a narrow berm and the other with a wide berm. The breakwaters were built using an armor layer of rock on a core with slopes of 1 : 2. The structures were built over a horizontal bottom and were backed by an absorbing beach consisting of crushed stones placed on a 1 : 5 slope. The breakwaters were subjected to tests at zero freeboard, positive freeboard (emerged) and negative freeboard (submerged). The tests include regular unidirectional waves and random directional waves with a mean wave direction normal to the breakwaters. JONSWAP spectral shape and directional spreading of 22.78 were used for the random waves.

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breakwaters, while absorbing structures (i.e., caissons filled in by sea-stones) were placed shoreward of the breakwater. The water level is sampled at a frequency of 40 Hz at the wave gauges marked 1 to 21, from which the wave height and mean water levels are determined. The current speed components are measured at the locations marked F, B, III and IV. Measurement errors during irregular tests due both to instrument and sampling difficulties can be up to 5%. A summary of the tests used in this paper is shown in Table 4. 3.2. Wave flume experiments

Fig. 2. Bathymetry for Layout 1, wide berm showing the locations of wave gauges (filled circles, numbering 1 to 21). Top: bathymetry cross-section through breakwater. Mean water level at 0.43 m.

Along the lateral boundaries (the two sides perpendicular to wave generator), impermeable wave guides were placed extending from the wave generator to the

Experiments on low crested structures were carried out in the wave and current flume of the Coastal Laboratory of the University of Cantabria (UCA), Spain, as a part of the research carried out for the DELOS project. The experimental set-up is described in detail by Garcia et al. (2004) and Losada et al. (2005—this issue). The wave and current flume of the UCA Coastal Laboratory is 24 m long, 0.60 m wide and 0.80 m high. Wave propagation over a rubble mound breakwater on a sloping beach was tested. The breakwater had a trapezoidal shape, while its crest width ranged between 0.25 and 1.0 m. Crest elevation (0.25 m from the floor), front and back slope angles (1 : 2) and rubble characteristics were maintained constant. The model had a two-layer armour of selected gravel and a gravel core. During the experiments water depth at the paddle was either 0.30, 0.35, or 0.4 m resulting in a freeboard of 0.05, 0.00 and  0.05 m, respectively. Several data sets corresponding to different regular and irregular wave conditions were used for verification of the Boussinesq model (Avgeris et al., 2004).

Table 4 Summary of tests used in this paper Test

Berm

Target wave conditions

Average (gauges 3 to 7)

H m0 (m)

T p (s)

H m0 (m)

T p (s)

19 21 33 35 37

Narrow Narrow Wide Wide Wide

0.103 0.054 0.122 0.054 0.103

1.81 1.32 1.97 1.32 1.81

0.120 0.065 0.140 0.065 0.085

1.81 1.32 1.97 1.32 0.80

Wave type

Regular Directional Jonswap Directional Jonswap Directional Jonswap Regular

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Table 5 Setup of Numerical models used in the study Phase-averaged

Wave model Grid spacings Friction

Breaking Time step Flow model Grid spacings Time step Bed resistance Eddy viscosity

Phase-resolving

MIKE 21 model

LIM model

Boussinesq model

Dx = Dy = 0.025 m k N = 0.3 cm (concrete floor) k N = 5 cm (breakwater) k N = 2 cm (gravel beach) Modified Battjes and Janssen Not applicable

Dx = Dy = 0.2 m k N = 0.3 cm (constant)

Dx = 0.1 m f = 0.003

Dally et al., 1984 Not applicable

Eddy viscosity (Kennedy et al., 2000) Dt = 0.0025 s

2DH flow model Dx = Dy = 0.05 m 0.056 s k N = 0.3/5/2 cm wave–current bed shear stress using Fredsøe (1984) Constant (0.006 m2/s)

Q3D flow model Dx = Dy = 0.2 m 0.134 s k N = 0.3 cm (constant) wave–current bed shear stress using Madsen (1994) Osiecki and Dally (1996)

Not applicable

For this paper, results for a test with irregular waves (incident H s = 0.07 m and T p = 2.4 s) are presented.

4. Numerical model results 4.1. Model setup 4.1.1. MIKE 21 phase averaged model The offshore boundary condition is calculated from the average of the measured wave heights at gauges 3 to 7 (Fig. 2), and adjusted using the calculated shoaling coefficient between the offshore boundary and the location of the gauges. The offshore boundary condition was calculated in this way in order to eliminate

differences between target wave conditions and the generated waves measured in front of the breakwater. A summary of the model set-up parameters for the three models is given in Table 5. In the flow model, the lateral boundaries are represented along their entire length as impermeable boundaries. This is correct from the wave generator to the breakwater; however, shoreward of the breakwater, this is not quite consistent with the experiments, where absorbing structures were present. The breakwaters are represented as impermeable structures. The offshore boundary (at the wave generator) is represented as an impermeable boundary (no-flow across the boundary). The shoreline boundary is represented as impermeable at dry points, where the location of

Fig. 3. Comparison of measured and simulated wave heights using standard Battjes and Janssen theory and the modified theory.

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the first line of dry points depends on wave set-up and the specified flooding depth. Due to wave set-up close to the shoreline, the first line of dry points will usually be shoreward of the original depth = 0 line. The measured significant wave heights (test 33) in the vicinity of the breakwater are compared with those obtained from numerical model simulations using the standard BJ78 theory (a = 0.83, c 1 = 1, c 2 = 0.8) and the modified theory (a = 0.83, c 2 = 1.4 for depth limited wave breaking) in Fig. 3. It is seen from Fig. 3 that the standard BJ78 theory gives lower wave heights (compared to measurements) on the breakwater crest and in the lee of the breakwater. On the other hand, the calculated wave heights with the modified theory are closer to the measurements. Unlike the case of sloping beach where BJ78 has been extensively used, here the waves are breaking on a very steep slope (1 : 2) on the breakwater, and it is also complicated by the submergence of the structure. Recent research shows that the wave transmission coefficients over submerged breakwaters can be adequately simulated using the modified Battjes and Janssen (1978) wave breaking model. The method is described briefly in Section 2.1.1. Investigations using several data-sets for submerged breakwater cases show that the ratio of limiting wave height to water depth H max / d (c 2) is related to the relative freeboard, F / H m0. The method is described in detail in Johnson (in press), where it is also shown that the calculated transmission coefficients using this approach are comparable with calculations with empirical wave transmission formulas. The MIKE 21 simulations are carried out using the modified breaking model and wave breaking parameters related to relative freeboard. 4.1.2. LIM phase averaged model Since the wave model (LIMWAVE) does not have a limitation in the number of nodes per wavelength ratio, the grid sizes were selected as Dx = Dy = 0.2 m. This minimizes the computational costs, while providing good agreement with the measurements. In this model the offshore and lateral boundary conditions were the same as those considered in Mike 21 PMS. The values of the dimensionless coefficients K D and C from Dally et al. (1984) for wave breaking criteria used in the simulations were K D = 0.1 and C = 0.4. These values which were specially tuned for these

simulations, differ slightly from the original ones proposed by the authors for irregular bathymetries (K D = 0.15 and C = 0.4). For bed friction (Tolman, 1992) the Nikuradse sand roughness K N, was set constant and equal to 0.3 cm throughout the domain. In the circulation model (LIMCIR) the offshore and lateral boundary conditions were the same as those used in Mike 21 HD. The shoreline boundary, which was also impermeable, was movable depending on the flooding capacity of wave set-up. In this model the eddy viscosity was not constant and it was computed according to Osiecki and Dally (1996). 4.1.3. Phase resolving model For the initial test with wave flume experiments, the time step used was D = 0.0025 s and the grid size Dx = 0.05 m. The porosity of the rubble mound was set equal to 0.5 while the characteristic diameter d 50 of the gravel was set equal to 2 cm. For the wave basin experiments, the numerical model is used to simulate the conditions at the laboratory scale. The time step used was Dt = 0.0025 s and the grid size Dx = 0.1 m. The porosity of the rubble mound was set equal to 0.5 while the characteristic diameter d 50 of the gravel was set equal to 3.5 cm. The specified target wave characteristics are used as surface elevation input. As in the phase-averaged models, the lateral boundaries are represented as impermeable boundaries along the entire length in the model. 4.2. Model results 4.2.1. Phase averaged models A typical example of the wave simulation results is shown in Fig. 4 for random waves (test 21). This shows a map of the significant wave heights in the basin (top panel) and the variation of the bathymetry and significant wave heights along a cross-section ( Y = 11 m). Fig. 4 illustrates the reduction of the wave height in the lee of the breakwater due to wave energy dissipation over the breakwater. In the areas behind the breakwater close to the gap, diffracted waves combine with the transmitted waves to give higher waves than in the areas far from the gap. Close to the shoreline, the waves shoal and break on the gravel beach. A typical plot of the simulated mean water level (contours) and flow pattern (vectors) is shown in Fig. 5.

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Fig. 4. Top panel: contours of significant wave heights (H m0) and wave direction vectors scaled with H m0 for test 21. Bottom panel: crosssection of bathymetry and significant wave height at alongshore distance, Y = 11 m. (Results from MIKE 21 simulation).

The main flow pattern is the onshore flow over the submerged breakwater and offshore flow (towards the wave generator) at the breakwater gap. This flow pattern is qualitatively similar to observations using dye

plumes during the laboratory experiment (Zanuttigh and Lamberti, in press). Near the shoreline, a secondary eddy can be seen in which water flows towards the lee of the breakwater. A similar flow pattern was observed by

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Fig. 5. Flow velocities (top panel) and contours of mean water level (bottom panel) for test 21 (MIKE 21 results). The still water level is 0.43 m above datum.

Haller et al. (2002) who carried out experiments on fixed barred beach with rip channels. In sensitivity tests with MIKE 21 model, it was found that the reproduction of the secondary eddy close to the shoreline is sensitive to the eddy viscosity coefficient used. The main flow pattern is much less sensitive, as this is always reproduced in the simulations.

Fig. 5 (bottom panel) shows a general set-down (reduction in mean water level) in front of the breakwater, and wave set-up leeward of the breakwater. The still water level is at 0.43 m above datum. A comparison of the measured and simulated wave and flow parameters is presented and discussed in Section 4.3.

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Fig. 6. Layout of the computational domain and location of wave gauges.

4.2.2. Phase resolving model The Boussinesq model results were first validated using data collected during the wave flume experiments. Evaluation of the numerical model was performed for the case of the submerged breakwater (freeboard =  5 cm) with 1 m crest width. Fig. 6 shows the layout of the computational domain and the locations of wave gauges used in the experiment. Figs. 7, 8 and 9 respectively show plots of the measured and simulated surface elevation time series over the frontal slope of the breakwater (gauge 5), breakwater crest (gauge 6) and over the shoreward slope of the structure (gauge 7).

Fig. 7. Computed and experimental free surface elevation (UCA flume experiment 275, case e, H s = 0.07 m, T p = 2.4 s, gauge 5).

Next the Boussinesq model is applied to the wave basin experiments. Fig. 10 shows the wave-induced current field, averaged over two wave periods for a regular wave test (test 19). The onshore flow over the submerged breakwater and the offshore flow at the gap can also be recognised. 4.3. Comparison of model results and measurements The numerical model results for four tests (tests 19, 21, 35 and 37; see Table 4) are compared with measurements. These tests include two cases from the narrow berm tests (19 and 21) and two cases from the wide berm tests (35 and 37). Furthermore, two of

Fig. 8. Computed and experimental free surface elevation (UCA flume experiment 275, case e, H s = 0.07 m, T p = 2.4 s, gauge 6).

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Fig. 9. Computed and experimental free surface elevation (UCA flume experiment 275, case e, H s = 0.07 m, T p = 2.4 s, gauge 7).

the cases are tests with regular unidirectional waves (19 and 37), while two are with irregular directional waves (21 and 35). The measured wave heights, wave set-up and current speed components in the vicinity of the breakwater are compared to numerical model results in Figs. 11 and 12. The overall agreement between measured and calculated values at all gauges is quantified by calculating the Wilmott index (Haas et al., 2003) and error function (Haller et al., 2002). The values of these parameters are tabulated in Table 6 for the three models. N X ðXck  Xmk Þ2 IW ¼ 1 

k¼1 N  X P P 2 jXck  X m j þ jXmk  X m j

denotes the average measured value. The summation is taken over all values of the same type (for example, wave heights), and N is the number of gauges. For perfect agreement, the Wilmott index (I w) is 1, while the error function (e) is 0. We consider the agreement between the model and measurement to be good if the Wilmott index is close to 1 and the error function is also close to 0. In general, the overall agreement for the wave heights is reasonably good for all the three models. However, the agreement with measurements is not quite good for wave setup and current speeds. For all the models, the error function (wave setup and current speeds) is typically close to 1 or greater. Possible reasons for the poor agreement with the measured wave setup and current speeds are discussed in the sections below. Typical variation of the wave heights and wave setup along a cross-shore transect ( Y = 11 m) is shown in Fig. 13 for the three models. A similar plot is shown in Fig. 14 for the longshore transect (X = 5.65 m). 4.3.1. Wave heights In general, the simulated wave heights with the modified BJ78 theory agrees fairly well with the measured data. In particular, the wave heights leeward of the structure (gauges 18 and 19–21) agrees quite well with the measurements. Thus, indicating that the amount of energy dissipated over the breakwater is reproduced quite well in the model. The main difference is at the breakwater crest (tests 19 and 21), where the model under-estimates the measured wave height.

ð4Þ

k¼1

2

N X

ðXck  Xmk Þ 6 6 k¼1 e¼6 6 X N 4 ðXmk Þ2

2

31=2 7 7 7 7 5

ð5Þ

k¼1

In Eqs. (4) and (5), X c denotes the computed values, X m denotes the measured values and Xm

Fig. 10. Wave-induced current field, averaged over two wave periods, of a regular wave test (test 19). dCoastlineT represents the shoreward end of the computational domain of the phase-resolving model.

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Fig. 11. Comparison of measured and simulated conditions for regular waves tests. Left panel: narrow berm case; right panel: wide berm.

The LIMWAVE results also agree fairly well with the measured wave heights for all tests. The main differences are found in tests 19 and 21 for positions located at breakwater crests. At these positions (gauges 13 and 14) the model under-estimates the observed wave heights. Additionally the LIMWAVE results behind the breakwaters fit very well with observations. The best agreement is found in the leeward side of the structure (gauges 18 and 19–21) for tests 37 and 21. The simulated wave heights with the Boussinesq model also agree fairly well with the measured data, except at the breakwater crest for tests 19 and 21. In

addition, the simulated wave heights are slightly lower (compared with the measurements) in the lee of the breakwater (gauges 18, 19–21). For all models, the main difference can be found at the breakwater crest (tests 19 and 21), where the simulated wave heights are generally lower than the measured wave height. The large measured wave height at this location was initially thought to be due to partial reflection from the breakwater. However, the Boussinesq model, which includes partial reflection, does not show the same tendency. One possibility is that the models do not reproduce the

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Fig. 12. Comparison of measured and simulated conditions for irregular waves tests. Left panel: narrow berm case; right panel: wide berm.

physics of wave breaking for these cases. However, it is doubtful that this is the case, when one considers that test 21 and 35 are identical, except for differences in the crest width of the submerged breakwaters (test 21 is for a narrow berm breakwater, while test 35 is for a wide berm breakwater). Since wave breaking depends on the seaward breakwater slope and the water depth over the breakwater crest, it is not expected that the measured wave heights for the two tests will be different at the wave gauges on the breakwater crest. What is expected is that the trans-

mitted wave height will be lower in test 35 because of the wider berm allowing more energy dissipation. The ratio of measured significant wave height to water depth (H / d) at the breakwater crest for test 21 is 1.41, while that for test 35 is 0.91. The reason for this difference is not clear, and may be due to measurement error. Furthermore, H / d value of 2.44 for test 19 appears quite high, and well above the usual range. Hence, it is concluded that the large wave heights at the breakwater crest for tests 19 and 21 may be due to measurement error.

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Table 6 Wilmott index and error function for the three models Wilmott index

Error function

MIKE 21 model

LIM model

Boussinesq model

MIKE 21 model

LIM model

Boussinesq model

Wave heights Test 19 Test 21 Test 35 Test 37

0.56 0.59 0.93 0.93

0.51 0.56 0.71 0.94

0.60 0.54 0.80 0.97

0.26 0.20 0.07 0.13

0.35 0.27 0.17 0.12

0.27 0.25 0.15 0.10

Wave setup Test 19 Test 21 Test 35 Test 37

0.57 0.85 0.77 0.64

0.62 0.18 0.21 0.31

0.54 0.31 0.31 0.53

1.70 0.82 1.25 1.48

0.68 4.72 4.08 3.53

1.44 3.24 4.52 2.01

Current speed Test 19 0.79 Test 21 0.85 Test 35 0.78 Test 37 0.84

0.59 0.79 0.89 0.93

0.81 0.80 0.57 0.91

1.06 0.95 1.05 0.86

0.83 0.59 0.44 0.37

0.76 1.14 1.96 0.62

4.3.2. Wave set-up and set-down The simulated wave set-down with the modified BJ78 theory agrees fairly well with the measurements in front of the breakwater and on the breakwater crest. However in the lee of the breakwater, the simulated set-up is large compared to the measurements. Similarly, the simulated wave set-down with the LIM model also agrees with the measurements in front of the breakwater. However while the LIM model gives a piling of water behind the breakwaters, the measurements do not provide such water accumulation. This is also reflected on the mean water level at the breakwater crest, which is also higher in LIM predictions than the measurements. The discrepancy between the simulated and measured wave set-up in the lee of the breakwater is related to the model representation of lateral boundaries (caisson absorbing structure) shoreward of the breakwater. In the experiments, the tendency for wave set-up is partly balanced by flow across the absorbing structure, due to differences in water levels in the basin and behind the caisson. It was observed in the experiments that a significant amount of water buildup occurred in the area behind the caissons. We note that in other laboratory experiments where the lateral boundaries are wall boundaries, a clear wave setup is observed leeward of the submerged structures (see for example, Haller et al., 2002).

In the numerical models, the absorbing structure in the experiment is modelled as a wall boundary. In order to investigate the influence of the lateral boundary condition, several sensitivity tests were carried out. In one of the tests, the LIM model was used to carry out a simulation in which the bed friction and eddy viscosity is artificially increased at a few (four) grid-points close to caisson structure (absorbing section of the lateral boundaries). However, this simulation (and similar model setups using MIKE 21) did not result in reduced wave set-up. Using MIKE 21, it was found that if sinks (extracting water from the model) are placed along the caisson structure, the calculated wave setup in the lee of the breakwater is reduced. This is illustrated in Fig. 15. For this calculation, a constant water level is maintained at the wave generator instead of an impermeable boundary. The sinks are used to allow water to escape through the absorbing section. In the laboratory, the pressure gradient (due to water level differences across the caisson is responsible for letting water through. The introduction of sinks modifies the general flow pattern and the intensity of the rip currents through the gaps. For the case of no sinks, the flow over the submerged breakwater is diverted towards the breakwater gap as rip currents. This gives the maximum rip current intensity. In the other extreme, if the sink strength is large enough, the flow

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Fig. 13. Typical variation of wave height (top panel) and wave setup along cross-shore transect Y = 11 m with the three models. Measured values indicated with filled circles.

over the breakwater is diverted towards the sinks, with little or no flow through the breakwater gap. Thus, the amount of flow leakage through the caisson structure significantly affects the measured flow velocities and wave setup. It should be noted that this type of flow leakage is only an issue in this laboratory setup, and is unlikely to arise in the field. Other differences between the numerical model and the experiments are the representation of the breakwaters and the gravel beach as impermeable in the numerical model. This means that the effect of

porosity in the breakwaters and the gravel beach are not included in the phase-averaged model simulations. The main effect of the submerged breakwater is the wave dissipation over it, which is dominated by wave breaking. This creates an imbalance in normal radiation stresses, which supports the water flow over the structure, tending to pile water on the leeward side. The effect of porosity may lead to flow retardation over the structure, but not significantly affecting the tendency for build-up on the lee side. Using the phase resolving model, it is found that the change in water

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Fig. 14. Typical variation of wave height (top panel) and wave setup (bottom panel) along longshore transect X = 5.65 m with the three models. Measured values indicated with filled circles.

level leeward of the structure is only slightly sensitive to the representation of the structures as porous or impermeable. In the case of the gravel beach, it is expected that if this is permeable enough, it may affect the ability of the gravel beach to support an increase in mean water level. However, this effect is expected to be relatively small, since the gravel beach is less permeable than the caisson-filled stones at the lateral boundaries. The simulated wave set-down with the phase resolving model is large compared with the measurements.

No wave set-up is predicted with this model in the lee of the breakwater. A test simulation was carried out assuming the structures are impermeable. In that test, wave set-up is not predicted in the lee of the breakwater, although the set-down was smaller. Another test simulation was carried out without a sponge layer, but with an eddy viscosity in a smaller region adjacent to the coastline (Fig. 16). This also shows a reduction in the wave set-down in the lee of the breakwater, but no wave setup. It is concluded that the phase-resolving model used here results in a rela-

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Fig. 15. Illustration of wave setup sensitivity to application of sinks along the absorbing lateral boundaries. Top panel: Wave setup. Bottom panel: Flow speeds. Calculations using MIKE 21 model.

tively large wave set-down. This may be related to the sponge layer boundary condition which is used to absorb the waves near the shoreline (Fig. 16). The sponge layer boundary condition makes it impossible for water to pile up in the lee of the breakwater. See also Belloti and Brocchini (2005—this issue) for a discussion on the effect of boundary conditions in long wave models. 4.3.3. Current speeds As shown earlier, the simulated flow pattern is in qualitative agreement with measurements as observed with dye plumes. In general, there is an onshore flow over the breakwater and a large offshore flow (rip currents) through the breakwater gaps. The simulated current speed components are large (by factor of

about 2) in the phase-averaged models (MIKE 21 simulation with modified BJ78 theory and LIMCIR). The higher current speeds in the models are consistent with the use of a wall boundary instead of the actual absorbing lateral boundary shoreward of the breakwater. By specifying sinks along the absorbing section of the lateral boundaries (caisson structure), the simulated current speeds are reduced significantly at the gap as discussed in Section 4.3.2. A comparison between the simulated results with and without sinks is shown in Fig. 13 (bottom panel) for test 21. For the phase resolving model, the main pattern of flow over the breakwater and rip current through the gap is reproduced. Furthermore, a secondary eddy close to the coastline (with flow of water into

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Fig. 16. Schematic illustration of wave absorption region (near the coastline) in the phase resolving model.

the lee of the breakwater near the coastline) is also calculated. This has a typically larger size than that calculated with the phase averaged models. In the phase-resolving model, the calculated current speeds are low (by factor of about 2) compared to measurements and the calculated wave set-down is generally large.

5. Summary and conclusions 1. Recent experimental data collected during the DELOS project have been used to test two approaches (phase-averaged and phase-resolving) for simulating waves, changes in the mean water level (set-up and set-down) and currents in the vicinity of submerged breakwaters. In the phase-averaged approach, a wave model is used to simulate wave transformation and calculate radiation stresses, while a flow model (2-dimensional depth averaged or quasi-3D) is used to calculate the resulting mean wave driven currents (wave-averaged). In the phase resolving approach used here, a 2DH-Boussinesq-type model combined with a depth-averaged Darcy–Forchheimer equation is applied to calculate the waves and intra-wave flow. 2. The phase-averaged models predict wave heights that are in reasonably good agreement with measured wave heights in the vicinity of the breakwater, provided the wave breaking sub-model is appropriately tuned for wave dissipation over the breakwater. The phase-resolving model also predicts wave heights that are in reasonably good agreement with the measured wave heights. Furthermore, the model shows reasonably good agreement between the instantaneous water levels and measurements in the flume experiments. 3. The phase-averaged models predict flow circulation pattern that are qualitatively in good agree-

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ment with the observed pattern in the basin experiments. The main flow pattern is the onshore flow over the submerged breakwater and offshore flow (towards the wave generator) through the breakwater gap. Near the shoreline, a secondary eddy can be seen in which water flows towards the lee of the breakwater. In sensitivity tests with MIKE 21 model, it was found that the reproduction of the secondary eddy close to the shoreline is sensitive to the eddy viscosity coefficient used, while the main flow pattern is less sensitive, as this is always reproduced in the simulations. 4. The calculated current speeds and wave set-up leeward of the breakwaters are higher in the phaseaveraged models than measured. It is concluded that the deviation from the measurements is due to differences in the lateral boundaries in the experiments and the numerical models. In the laboratory, an absorbing structure is placed along the lateral boundaries (leeward of the structures), whereas the lateral boundaries are treated as wall boundaries in the phase-averaged models. The absorbing structure leads to leakage of water from the basin, and hence a reduction of the wave setup and rip current through the breakwater gap in the laboratory. This is demonstrated by introducing sinks along the absorbing lateral boundaries to allow for flow leakage across the absorbing boundaries. 5. The phase-resolving model also predict flow circulation pattern that are qualitatively in good agreement with observed pattern in the experiments. The main pattern of flow over the breakwater returning in a rip current through the gap is reproduced. Furthermore, a secondary eddy close to the coastline is also calculated but with a typically larger size than that calculated with the phase averaged models. In the phase-resolving model, the calculated current speeds are low compared to measurements and the calculated wave set-down is generally large. 6. In this study, three models are used to calculate waves and currents in the vicinity of submerged rubble mound breakwaters in controlled laboratory experiments. The investigations show that the wave heights can be fairly reasonably predicted by all the models. However, the agreement between model calculations of wave setup and current speeds is not that good. It is shown that the absorbing lateral boundaries in the lee of the breakwater are mainly

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responsible for the deviation between measurements and calculations in the phase-averaged models. This gives some confidence in the ability of the models to reproduce waves and currents in the vicinity of submerged breakwaters. This is a necessary step for improving our understanding of coastal area morphological evolution around low crested coastal defence structures. Acknowledgements This study was supported financially by the Commission of the European Communities, Directorate General for Science, Research and Development under contract no. EVK3-CT-2001-00041 (DELOS). Hakeem Johnson is also co-sponsored by the Danish Technical Research Council (STVF) under the project bCoasts and Tidal InletsQ which is part of the framework programme bUtilisation and Protection of the coastal zoneQ. The authors are also indebted to their colleagues in this project for their contributions and discussions. References Abbott, M.B., Damsgaard, A., Rodenhuis, G.S., 1973. System 21, Jupiter. A design system for two-dimensional nearly-horizontal flows. Journal of Hydraulic Research 1. Alonso, D., 1999. Modelado 2DH/3D de la circulacio´n en puertos y estuarios. Internal ReportUniversitat Polite´cnica de Catalunya, Barcelona (In Spanish). Avgeris, I., Karambas, Th., Prinos, P., Koutitas, Ch., Belloti, G., Briganti, R., Brocchini, M., 2004. DELOS Internal Report, dFlow Description (WP 2.1), Final Phase-Resolving Boussinesq-Type Models (D42)T (August 2004). Battjes, J.A., Janssen, J.P.F.M., 1978. Energy loss and set-up due to breaking of random waves. Proceedings of the 16th Int. Conf. Coast. Eng.. Hamburg, Germany, pp. 569 – 587. Belloti, G., Brocchini, M., 2005—this issue. Swash zone boundary conditions for long wave models. Coastal Engineering. doi:10.1016/j.coastaleng.2005.09.003. Burcharth, H.F., Andersen, O.H., 1995. On the one-dimensional steady and unsteady porous flow equations. Coastal Engineering 24, 233 – 257. Chapman, B., Ilic, S., Simmonds, D., Chadwick, A., 1999. Physical model evaluation of the hydrodynamics produced around a permeable breakwater scheme. Proceedings of the International Conference on Coastal Structures, pp. 803 – 812. Christensen, E.D., Zanuttigh, B., Zyserman, J.A., 2003. Validation of numerical models against laboratory measurements of waves and currents around low crested structures. Proc. Coastal Structures 2003, Portland, Oregon, USA.

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