The BCI criterion for the initiation of breaking process in Boussinesq-type equations wave models

The Breaking Celerity Index (BCI) is proposed as a new wave breaking criterion for Boussinesq-type equations wave propagation models (BTE).The BCI effectiveness in determining the breaking initiation location has been verified against data from different experimental investigations conducted with incident regular and irregular waves propagating along uniform slope [Utku, M. (1999). “The Relative Trough Froude Number. A New Criteria for Wave Breaking”. Ph.D. Dissertation, Dept. of Civil and Enviromental Engineering, Old Dominion University, Norfolk, VA; Gonsalves Veloso dos Reis, M.T.L. (1992). “Characteristics of waves in the surf zone”. MS Thesis, Department of Civil Engineering, University of Liverpool., Liverpool; Lara, J.L., Losada, I.J., and Liu, P.L.-F. (2006). “Breaking waves over a mild gravel slope: experimental and numerical analysis”. Journal of Geophysical Research, VOL 111, C11019] and barred beaches [Tomasicchio, G.R., and Sancho, F. (2002). “On wave induced undertow at a barred beach”. Proceedings of 28th International Conference on Coastal Engineering, ASCE, New York, 557–569]. The considered experiments were carried out in small-scale and large-scale facilities. In addition, one set of data has been obtained by the use of the COBRAS model based upon the Reynolds Averaged Navier Stokes (RANS) equations [Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., and Sakakiyama, T. (2000). “A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions”. Proceedings of Coastal Structures ‘99, Balkema, Rotterdam, 169–174; Losada, I.J., Lara, J.L., and Liu, P.L.-F. (2005). “Numerical simulation based on a RANS model of wave groups on an impermeable slope”. Proceedings of Fifth International Symposium WAVES 2005, Madrid].Numerical simulations have been performed with the 1D-FUNWAVE model [Kirby, J.T., Wei, G., Chen, Q., Kennedy, A.B., and Dalrymple, R.A. (1998). “FUNWAVE 1.0 Fully Nonlinear Boussinesq Wave Model Documentation and User's Manual”. Research Report No CACR-98-06, Center for Applied Coastal Research, University of Delaware, Newark]. With regard to the adopted experimental conditions, the breaking location has been calculated for different trigger mechanisms [Zelt, J.A. (1991). “The run-up of nonbreaking and breaking solitary waves”. Coastal Engineering, 15, 205–246; Kennedy, A.B., Chen, Q., Kirby, J.T., and Dalrymple, R.A. (2000). “Boussinesq modeling of wave transformation, breaking and run-up. I: 1D”. Journal of Waterway, Port, Coastal and Ocean Engineering, 126, 39–47; Utku, M., and Basco, D.R. (2002). “A new criteria for wave breaking based on the Relative Trough Froude Number”. Proceedings of 28th International Conference on Coastal Engineering, ASCE, New York, 258–268] including the proposed BCI.The calculations have shown that BCI gives a better agreement with the physical data with respect to the other trigger criteria, both for spilling and plunging breaking events, with a not negligible reduction of the calculation time.     

Field measurements of wave overtopping at the rubble mound breakwater of Rome–Ostia yacht harbour

This paper describes a new station for full-scale measurement of wave overtopping at the Rome yacht harbour rubble mound breakwater in Ostia (Italy) and the results of the successful first measurement campaign carried out during the winter season 2003–2004. The equipment and the research activities were supported by the EU project CLASH, focusing on scale effects for wave overtopping at coastal structures. The site is characterized by a very small tidal range, a long shallow foreshore and depth-limited breaking waves which interact with a shallow sloping porous rock structure. Overtopping water is collected by a steel tank installed on the crown slab behind the parapet wall. The measurement of water level variation inside the tank by means of two pressure transducers allows the calculation of individual overtopping volumes. Incident waves, sea levels and wind are also measured. During seven independent storms, more than 400 individual overtopping events were recorded and about 86 h of valid data are available. This extensive dataset is presented, discussed and then used for comparison with two commonly used overtopping prediction formulae based on small-scale model tests showing their tendency to underestimate the prototype results. A strong correlation between the hourly mean overtopping discharge and corresponding maximum volume is also presented. The paper generally confirms the validity of the approach used in Troch et al. (2004) [Troch, P., Geeraets, J., Van de Walle, B., De Rouck, J., Van Damme, L., Allsop, W., Franco, L., 2004. Full-scale wave overtopping measurements on the Zeebrugge rubble mound breakwater. Coastal Engineering 51, 609–628] for field measurement of wave overtopping.     

A probability distribution for depth-limited extreme wave heights in a sea state

A model for the depth-limited distribution of the highest wave in a sea state is presented. The distribution for the extreme wave height is based on a probability density function (pdf) for depth-limited wave height distribution for individual waves [Méndez, F.J., Losada, I.J., Medina, R. 2004. Transformation model of wave height distribution. Coastal Eng, Vol. 50, 97:115.] and considers the correlation between consecutive waves. The model is validated using field data showing a good representation of the extreme wave heights in the surf zone. Some important statistical wave heights are parameterized obtaining useful expressions that can be used in further calculations.     

RANS modelling applied to random wave interaction with submerged permeable structures

This paper is the second part of the work presented by Garcia et al. [Garcia, N., Lara, J.L., Losada, I.J., 2004. 2-D numerical analysis of near-field flow at low-crested breakwaters. Coastal Engineering 51 (10), 991–1020]. In the mentioned paper, flow conditions at low-crested rubble-mound breakwaters under regular wave attack were examined, using a combination of measured data of free surface, bottom pressure and fluid velocities from small-scale experiments and numerical results provided by a VOF-type model (COBRAS) based on the Reynolds-Averaged Navier–Stokes (RANS) equations. This paper demonstrates the capability of the COBRAS model to reproduce irregular wave interaction with submerged permeable breakwaters. Data provided by the numerical model are compared to experimental data of laboratory tests, and the main processes of wave–structure interaction are examined using both experimental and numerical results. The numerical model validation is carried out in two steps. First, the procedure of irregular wave generation is verified to work properly, comparing experimental and numerical data of different cases of irregular wave trains propagating over a flat bottom. Next, the validation of the numerical model for wave interaction with submerged rubble-mound breakwaters is performed through the simulation of small-scale laboratory tests on different incident wave spectra. Results show that the numerical model adequately reproduces the main aspects of the interaction of random waves with submerged porous breakwaters, especially the spectral energy decay at the structure and the spectrum broadening past the structure. The simulations give good results in terms of height envelopes, mean level, spectral shape, root-mean-square height for both free surface displacement and dynamic pressure inside the breakwater. Moreover, large-scale simulations have been conducted, on both regular and irregular incident wave conditions. The overall pattern of the wave interaction with a large-scale submerged breakwater is adequately reproduced by the numerical model. The processes of wave reflection, shoaling and breaking are correctly captured. The good results achieved at a near prototype scale are promising regarding the use of the numerical model for design purposes.     

Numerical analysis of wave overtopping of rubble mound breakwaters

The paper describes the results of a two-dimensional (2-D) numerical modelling investigation of the functionality of rubble mound breakwaters with special attention focused on wave overtopping processes. The model, COBRAS-UC, is a new version of the COBRAS (Cornell Breaking Waves and Structures) based on the Volume Averaged Reynolds Average Navier–Stokes (VARANS) equations and uses a Volume of Fluid Technique (VOF) method to capture the free surface. The nature of the model equations and solving technique provides a means to simulate wave reflection, run-up, wave breaking on the slope, transmission through rubble mounds, overtopping and agitation at the protected side due to the combined effect of wave transmission and overtopping. Also, two-dimensional experimental studies are carried out to investigate the performance of the model. The computations of the free surface and pressure time series and spectra under regular and irregular waves, are compared with the experimental data reaching a very good agreement. The model is also used to reproduce instantaneous and average wave overtopping discharge. Comparisons with existing semi-empirical formulae and experimental data show a very good performance. The present model is expected to become in the near future an excellent tool for practical applications.     

Wave loads on rubble mound breakwater crown walls

Crown walls are primarily built to reduce wave overtopping of mound breakwaters. Several methods have been proposed to calculate wave loads on the crown wall, e.g., Iribarren and Nogales [Iribarren, R., Nogales, C., 1964. Obras Marı́timas. Dossat (Ed.), Madrid, 376 pp.], Jensen [Jensen, O.J., 1984. A Monograph on Rubble Mound Breakwaters. Danish Hydraulic Institute] and Günbak and Gökce [Günbak, A.R., Gökce, T., 1984. Wave screen stability of rubble-mound breakwaters. International Symposium of Maritime Structures in the Mediterranean Sea. Athens, Greece, pp. 2.99–2.112]. In this paper, a new method based on those previous results, and on further experimental work, using monochromatic waves, is presented. The application of the new method requires waves breaking on the armour layer; i.e., only broken waves will reach the crown wall. The method is extended to irregular waves via the hypothesis of equivalence introduced by Saville [Saville, T., 1962. An approximation of the wave run-up frequency distribution. Proc. 8th International Conference on Coastal Engineering, Mexico City] and is applied to the crown walls of Gijón and Bilbao breakwaters in Spain. The comparison of the probability force distributions obtained by the present method to that measured by Burcharth et al. [Burcharth, H.F., Frigaard, P., Berenguer, J.M., Gonzalez, B., Uzcanga, J., Villanueva, J., 1995. Design of the Ciervana breakwater, Bilbao. In: T. Telford (Ed.), Proc. 4th Coastal Structures and Breakwaters, Chap. 3. Institution of Civil Engineers] and Jensen (1984) is relatively good.     

2-D numerical analysis of near-field flow at low-crested permeable breakwaters

This paper describes the capability of a numerical model named COrnell BReaking waves And Structures (COBRAS) [Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. Journal of Fluid Mechanics 359, 239–264; Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. Journal of Waterway, Port, Coastal and Ocean Engineering 125, 322–330, Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., Sakakiyama, T., 2000. A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions. Proc. Coastal Structures '99, 169–174] based on the Reynolds Averaged Navier–Stokes (RANS) equations to simulate the most relevant hydrodynamic near-field processes that take place in the interaction between waves and low-crested breakwaters. The model considers wave reflection, transmission, overtopping and breaking due to transient nonlinear waves including turbulence in the fluid domain and in the permeable regions for any kind of geometry and number of layers. Small-scale laboratory tests were conducted in order to validate the model, with different wave conditions and breakwater configurations. In the present study, regular waves of different heights and periods impinging on a wide-crested structure are considered. Three different water depths were tested in order to examine the influence of the structure freeboard. The experimental set-up includes a flow recirculation system aimed at preventing water piling-up at the lee of the breakwater due to overtopping. The applicability and validity of the model are examined by comparing the results of the numerical computations with experimental data. The model is proved to simulate with a high degree of agreement all the studied magnitudes, free surface displacement, pressure inside the porous structure and velocity field. The results obtained show that this model represents a substantial improvement in the numerical modelling of low-crested structures (LCS) since it includes many processes neglected previously by existing models. The information provided by the model can be useful to analyse structure functionality, structure stability, scour and many other hydrodynamic processes of interest.     

Wave height parameter for damage description of rubble-mound breakwaters

In this paper it will be shown that the wave height parameter H₅₀, defined as the average wave height of the 50 highest waves reaching a rubble-mound breakwater in its useful life, can describe the effect of the wave height on the history of the armor damage caused by the wave climate during the structure's usable life.Using Thompson and Shuttler (Thompson, D.M., Shuttler, R.M., 1975. Riprap design for wind wave attack: A laboratory study on random waves. HRS Wallingford, Report 61, UK) data it will be shown that H₅₀ is the wave parameter that best represents the damage evolution with the number of waves in a sea state. Using this H₅₀ parameter, formulae as van der Meer (van der Meer, J.W., 1988. Rock slopes and gravel beaches under wave attack. PhD Thesis. Technical University of Delft) and Losada and Giménez-Curto (Losada, M.A., Gimenez–Curto, L.A., 1979. The joint effect of the wave height and period on the stability of rubble mound breakwaters using Iribarren's number. Coastal Engineering, 3, 77–96) are transformed into sea-state damage evolution formulae. Using these H₅₀-transformed formulae for regular and irregular sea states it will be shown how damage predictions are independent of the sea state wave height distribution.To check the capability of these H₅₀-formulae to predict damage evolution of succession of sea states with different wave height distributions, some stability tests with regular and irregular waves have been carried out. After analysing the experimental results, it will be shown how H₅₀-formulae can predict the observed damage independently of the sea state wave height distribution or the succession of sea states.     

Random wave-induced scour at the trunk section of a breakwater

This paper provides a stochastic method by which the random wave-induced scour depth at the trunk section of vertical-wall and rubble-mound breakwaters can be derived. Here the formulas for regular wave-induced scour depth provided by Xie [Xie, S.L., 1981. Scouring patterns in front of vertical breakwaters and their influence on the stability of the foundations of the breakwaters. Report. Department of Civil Engineering, Delft University of Technology, Delft, The Netherlands, September, 61 pp.] for vertical-wall breakwater and Sumer and Fredsøe [Sumer, B.M., Fredsøe, J., 2000. Experimental study of 2D scour and its protection at a rubble-mound breakwater. Coast. Eng. 40, 59–87] for rubble-mound breakwater are used. These formulas are combined with describing the waves as a stationary Gaussian narrow-band random process to derive the random wave-induced scour depth. Comparisons are made between the present method and the Sumer and Fredsøe [Sumer, B.M., Fredsøe, J., 2000. Experimental study of 2D scour and its protection at a rubble-mound breakwater. Coast. Eng. 40, 59–87.] random wave scour data for rubble-mound breakwater, as well as the Hughes and Fowler [Hughes, S.A., Fowler, J.A., 1991. Wave-induced scour predictions at vertical walls. ASCE Proc. Conf. Coastal Sediments vol. 91, 1886–1899] random wave scour data and formula for vertical-wall breakwater. A tentative approach to random wave-induced scour at a vertical impermeable submerged breakwater is also suggested.     

Three-dimensional investigations of wave overtopping on rubble mound structures

To study the influence of wave obliquity and directional spreading on wave overtopping of rubble mound breakwaters a total of 736 three-dimensional model tests were carried out at Aalborg University. The results of these tests are presented and analysed in this paper yielding a new empirical reduction factor to describe the influence of wave obliquity and directional spreading on the average wave overtopping discharges. The study shows that perpendicularly incident, long-crested waves result in conservative values of the overtopping discharge for the tested cross-section.     

The new wave overtopping database for coastal structures

Significant effort has been made to generate a homogeneous database on wave overtopping consisting of more than 10,000 irregular wave overtopping tests from more than 160 independent projects or test series, each described by means of 31 parameters. Many coastal structures, including dikes, rubble mound breakwaters, berm breakwaters, caisson structures and combinations have been considered and have been schematised for inclusion in the database. All these overtopping tests are represented by over 300,000 numbers in the database.     

Comparison of new large and small scale overtopping tests for rubble mound breakwaters

Many breakwaters are, due to functional requirements, designed for small wave overtopping discharges. From the EC-research projects OPTICREST and CLASH it is known that overtopping discharges determined from conventional Froude scale models of rubble mound breakwaters are smaller than measured in corresponding prototypes. The present study examines this scale effect by comparing overtopping discharges in small scale and large scale tests. The length scale ratio between the models was 5.7.     

Examination of the influence of several factors on longshore current computation with random waves

Influence of various factors affecting the longshore currents induced by obliquely incident random waves is examined through numerical calculation. Seven numerical models for random wave breaking process are found to yield large differences in the wave heights in the surf zone and longshore current velocities. The turbulent eddy viscosity formulation by Larson and Kraus [Larson, M. and Kraus, N.C. (1991): Numerical model of longshore current for bar and trough beaches, J. Waterway, Port, Coastal, and Ocean Eng., ASCE, 117 (4), pp. 326-347.] functions almost equal to that by Battjes [Battjes, J.A. (1975): Modeling of turbulence in the surf zone, Proc. Symp. Modeling Techniques, pp. 1050–1061.], but the formulation by Longuet-Higgins [Longuet-Higgins, M.S. (1970): Longshore current generated by obliquely incident sea waves, 1 and 2, J. Geophys. Res., 75 (33), pp. 6779–6801.] produces excessive diffusion of longshore currents into the offshore zone. The generation and decay process of the surface roller is indispensable in the longshore current analysis. The random wave transformation model called PEGBIS (Parabolic Equation with Gradational Breaker Index for Spectral waves) by Goda [Goda, Y. (2004): A 2-D random wave transformation model with gradational breaker index, Coastal Engineering Journal, JSCE and World Scientific, 46 (1), pp. 1–38.] produced good agreement with several laboratory and field data of longshore currents.     

A numerical model for wave propagation in curvilinear coordinates

Using the perturbation method, a time dependent parabolic equation is developed based on the elliptic mild slope equation with dissipation term. With the time dependent parabolic equation employed as the governing equation, a numerical model for wave propagation including dissipation term in water of slowly varying topography is presented in curvilinear coordinates. In the model, the self-adaptive grid generation method is employed to generate a boundary-fitted and varying spacing mesh. The numerical tests show that the effects of dissipation term should be taken into account if the distance of wave propagation is large, and that the outgoing boundary conditions can be treated more effectively by introduction of the dissipation term into the numerical model. The numerical model is able to give good results of simulating wave propagation for waters of complicatedly boundaries and effectively predict physical processes of wave propagation. Moreover, the errors of the analytical solution deduced by Kirby et al. (1994) [Kirby, J.T., Dalrymple, R.A., Kabu, H., 1994. Parabolic approximation for water waves in conformal coordinate systems. Coastal Engineering 23, 185–213.] from the small-angle parabolic approximation of the mild-slope equation for the case of waves between diverging breakwaters in a polar coordinate system are corrected.     

Scour below marine pipelines in shoaling conditions for random waves

This paper provides an approach by which the scour depth below pipelines in shoaling conditions beneath non-breaking and breaking random waves can be derived. Here the scour depth formula in shoaling conditions for regular non-breaking and breaking waves with normal incidence to the pipeline presented by Cevik and Yüksel [Cevik, E. and Yüksel, Y., (1999). Scour under submarine pipelines in waves in shoaling conditions. ASCE J. Waterw., Port, Coast. Ocean Eng., 125 (1), 9–19.] combined with the wave height distribution including shoaling and breaking waves presented by Mendez et al. [Mendez, F.J., Losada, I.J. and Medina, R., (2004). Transformation model of wave height distribution on planar beaches. Coast. Eng. 50 (3), 97–115.] are used. Moreover, the approach is based on describing the wave motion as a stationary Gaussian narrow-band random process. An example of calculation is also presented.     

Wave run-up and overtopping on smooth and rough slopes of coastal structures

A series of hydraulic model tests has been carried out in a glass wave flume to investigate the influences of wave height, wave period, wave steepness, surf similarity parameter, roughness, layer thickness and porosity on wave run-up and overtopping of 1:2 sloped impermeable and permeable breakwaters fronted by a 1:10 gentle, smooth beach slope. The analysis of results involves the correlation between the overtopping energy transfer with the relative wall height and the relationship between wave run-up and overtopping rate. Further, measured wave run-up and overtopping rates are compared with the results given in the Shore Protection Manual (1984), Automated Coastal Engineering System (1992)and results of other investigators.     

Numerical Simulation of Random Wave Overtopping of Rubble Mound Breakwater with Armor Units

Based on the open source code OpenFOAM,a three-dimensional model is presented for simulation of the interaction between waves and rubble mound breakwater with armor units.The armor units with their real geometries are depicted through computational grids.The volume-averaged RANS equation and the seepage equation containing nonlinear term are used to describe the percolation in the core and underlayer of the breakwater.Grids independence analysis are carried out,the horizontal and vertical grid size are recommended to take as one-fifteenth of the mean nominal diameter D₅₀ of the armor units and one-fifteenth of the wave height respectively.Random wave overtopping of rubble mound breakwater with armor units is simulated through the proposed model.The results show good agreement between the simulated and measured overtopping discharge rates for different types of armor units.The developed numerical model can be used to evaluate the random wave overtopping in design of rubble mound breakwater with artificial armor blocs.     

On the role of shoreline boundary conditions in wave overtopping modelling with non-linear shallow water equations

The note extends and completes the analysis carried out by Briganti and Dodd [Briganti, R., Dodd, N., 2009. Shoreline motion in nonlinear shallow water coastal models. Coastal Eng. 56(5–6) (doi:101016/j.coastaleng.2008.10.008), 495–505.] on the performance of a state of the art Non-Linear Shallow Water Equations solver in common coastal engineering applications. The case of bore-generated overtopping of a truncated plane beach is considered and the performance of the model is assessed by comparing with the Peregrine and Williams [Peregrine, D., Williams, S.M., 2001. Swash overtopping a truncated beach. J. Fluid Mech. 440, 391–399.] analytical solution. In particular the influence of shoreline boundary conditions is investigated by considering the two best performing approaches discussed in Briganti and Dodd [Briganti, R., Dodd, N., 2009. Shoreline motion in nonlinear shallow water coastal models. Coastal Eng. 56(5–6) (doi:101016/j.coastaleng.2008.10.008), 495–505.]. Different distances of the edge of the beach from the bore collapse point are tested. For larger distances, the accuracy of the overtopping modelling decreases, as a consequence of the error in modelling the tip of the swash lens and, consequently, the run-up. A sensitivity analysis using the numerical resolution is carried out. This reveals that the approach in which cells shallower than a prescribed threshold are drained and wave propagation speeds for wet/dry Riemann problem are used at the interface between a wet and a dry cell (referred as Option 2ea in [Briganti, R., Dodd, N., 2009. Shoreline motion in nonlinear shallow water coastal models. Coastal Eng. 56(5–6) (doi:101016/j.coastaleng.2008.10.008), 495–505.]) performs consistently better than the other.     

The phase difference effects on 3-D structure of wave pressure acting on a composite breakwater

In designing the coastal structures, the accurate estimation of the wave forces on them is of great importance. In this paper, the influences of the phase difference on wave pressure acting on a composite breakwater installed in the three-dimensional (3-D) wave field are studied numerically. We extend the earlier model [Hur, D.S., Mizutani, N., 2003. Coastal Engineering 47, 329–345] to simulate 3-D wave fields by introducing 3-D Navier–Stokes solver with the Smagorinsky's sub-grid scale (SGS) model. For the validation of the model, the wave field around a 3-D asymmetrical structure installed on a submerged breakwater, in which the complex wave deformations generate, is simulated, and the numerical solutions are compared to the experimental data reported by Hur, Mizutani, Kim [2004. Coastal Engineering (51, 407–420)]. The model is then adopted to investigate 3-D characteristics of wave pressure and force on a caisson of composite breakwater, and the numerical solutions were discussed with respect to the phase difference between harbor and seaward sides induced by the transmitted wave through the rubble mound or the diffraction. The numerical results reveal that wave forces acting on the composite breakwater are significantly different at each cross-section under influence of wave diffraction that is important parameter on 3-D wave interaction with coastal structures.