Proceedings of the 7 th International Conference on HydroScience and Engineering Philadelphia, USA September 10-13, 2006 (ICHE 2006) ISBN:

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1 Proceedings of the 7 th International Conference on HydroScience and Engineering Philadelphia, USA September 1-13, 6 (ICHE 6) ISBN: Drexel University College of Engineering Drexel E-Repository and Archive (idea) Drexel University Libraries The following item is made available as a courtesy to scholars by the author(s) and Drexel University Library and may contain materials and content, including computer code and tags, artwork, text, graphics, images, and illustrations (Material) which may be protected by copyright law. Unless otherwise noted, the Material is made available for non profit and educational purposes, such as research, teaching and private study. For these limited purposes, you may reproduce (print, download or make copies) the Material without prior permission. All copies must include any copyright notice originally included with the Material. You must seek permission from the authors or copyright owners for all uses that are not allowed by fair use and other provisions of the U.S. Copyright Law. The responsibility for making an independent legal assessment and securing any necessary permission rests with persons desiring to reproduce or use the Material. Please direct questions to archives@drexel.edu

2 BOUSSINESQ MODELING OF WAVE RUN-UP AND OVERTOPPING Ioannis Avgeris 1, Theophanis V. Karambas and Panayotis Prinos 3 ABSTRACT In this study, wave overtopping of coastal structures in the surf zone is investigated numerically. Simulations are performed with the use of a Boussinesq-type model. The model incorporates high-order equations with improved dispersion characteristics. These equations are capable of modeling dispersive wave propagation even in deep water conditions. Wave breaking and bottom friction are also included in the model while a linear extrapolation technique is used to describe wave run-up on steep slopes. Model results are evaluated using experimental measurements conducted in wave flumes. Tests involving wave run-up on a plane beach and wave overtopping of permeable and impermeable breakwaters are considered. The analysis demonstrates that the model results had good agreements with the experiments except for some deficiencies in cases of complex flow structures. 1. INTRODUCTION Wave overtopping has big impacts on the stability and functionality of coastal defense structures, which poses a considerable risk to the integrity of the protected infrastructure and to human life. The nature of wave overtopping is complex and involves phenomena such as wave breaking, reflection, and turbulence thus modeling of this process is a demanding task. At early stages, research efforts focused on physical modeling using experimental facilities, which resulted in a number of empirical formula that relate overtopping discharge with the layout of the structures and the incident wave characteristics. An extensive review of the available overtopping formula is given by Soliman (3). Furthermore, numerous researchers have developed numerical models to study wave overtopping, mainly based on the Non-Linear Shallow Water Equations (NSWE). A recent development is to solve these equations with finite-volume, shock-capturing schemes combined with approximate Riemann solvers (Dodd, 1998, Hu et al., ). Stansby (3) also provided numerical simulations of wave run-up and overtopping over a trapezoidal hump using a finite-volume NSWE model that incorporated Boussinesq terms. On the other hand, Reynolds Averaged Navier-Stokes equations (RANS) combined with the Volume of Fluid (VOF) method present an alternative approach with promising results. Liu et al. (1999) studied the overtopping of a caisson breakwater, protected by armour units using a RANS-VOF model 1 PhD Student, Hydraulics Laboratory, Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, 541 4, GREECE (iavgeris@civil.auth.gr) Associate Professor, Department of Marine Sciences, University of the Aegean, Mytilini, 811, GREECE (karambas@marine.aegean.gr) 3 Professor, Hydraulics Laboratory, Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, 541 4, GREECE (prinosp@civil.auth.gr)

3 coupled with a non-linear k-ε turbulence model. Soliman (3) applied the same model to simulate the irregular wave overtopping tests conducted by Van der Meer and Janssen (1995). Recently, Shao et al. (6) compared an incompressible Smooth Particle Hydrodynamics (SPH) model with the experimental data of Cox and Ortega () and the numerical results of Hu et al. () and Soliman (3). In this study, a high-order Boussinesq model is developed to simulate wave run-up and overtopping. A previous version of the present model has been tested successfully for analyzing wave interaction with low-crested breakwaters (Avgeris et al., 4). The Boussinesq model includes additional frequency dispersion terms thus it lacks the inherent deficiency of the NSWE equations in the seaward region. Near the shoreline the non-linear dispersive contributions become very small compared to the convective contributions. The governing equations are discretized using a high-order finite differences scheme. A moving boundary technique that utilizes linear extrapolation is used to track the shoreline. Wave breaking (eddy viscosity formulation) and bottom friction (quadratic law) are also included in the model. The numerical model is employed to simulate wave run-up on a plane beach and wave overtopping of permeable and impermeable breakwaters in comparison with experimental data available in the literature.. DESCRIPTION OF THE MODEL The high-order Boussinesq equations are coupled in the region of the structures with a Darcy-Forchheimer equation in order to describe wave interaction with the porous flow. The governing equations in one-dimensional form are written: ( d ζ) u ( d u ) ζ t x x ( d + d ζ) s s φ 3 = (1) u u ζ u d u d ζ u + u + g = + d + t x x 3 x t x x t x x t 3 d u u u ζ u d u + u d d u x x x x x t x x 3 3 u ζ ( u u ) + Βd + g + x 3 x t x x 3 d u ζ ds us 1 u s + Βd + g φd d + + s x x t x x x t x t where u = depth-averaged, horizontal velocity, ζ = surface elevation, g = acceleration of gravity, d = water depth, B = dispersion coefficient = 1/15, u s = depth-averaged, seepage (fluid) velocity inside the porous medium, d s = porous medium thickness and φ = porosity. Under the assumption that O [(d s /L) ] << 1 (L = wave length), the depth-averaged, Darcy- Forchheimer equation expressed in terms of the fluid velocity u s (u d = φu s = Darcy velocity) reads: () us us ζ cr + u s = g φα1us φαu s u t x x s (3)

4 which is refered as the non-linear long wave equation for porous medium. In case of impermeable structures, equation (3) is omitted from the model and the last terms of equations (1) and () become zero. In equation (3), c r = inertial coefficient, given by (van Gent, 1995) 1 φ c = 1+c = 1+γ r m φ (4) where c m = added mass coefficient, γ = empirical coefficient that accounts for the added mass and α 1, α = porous resistance coefficients which are estimated from the following relationships (Sollitt and Cross, 197) ν α 1 =, α = K C f K (5) where ν = kinematic viscosity (1 1-6 m /sec), C f = dimensionless parameter and K = intrinsic permeability (m ). The following empirical formula proposed by van Gent (1995) are used for the calculation of C f and K 5 3 d φ K = (6) α 1 ( φ) C f 1 φ K = β (7) φ d 5 where α, β = empirical coefficients and d 5 = the mean diameter of the porous material. In the simulations involving permeable structures, the values of 1, 1. and.34 are chosen for the empirical coefficients α, β and γ respectively. The governing equations are discretized using FDM and solved utilizing a high-order predictor-corrector scheme that employs a third-order explicit Adams-Bashforth predictor step and a fourth-order implicit Adams-Moulton corrector step (Wei and Kirby, 1995). The corrector step is iterated until the desirable convergence is achieved. Waves are generated inside the computational domain using the source function method (Wei et al., 1999). This method employs a mass source term in the continuity equation that acts on a limited source region and it is adapted to be consistent with the form of equations used in the present work (Memos et al., 5). Wave breaking is incorporated in the model by adopting the eddy viscosity formulation. The eddy viscosity coefficient, which is a function of both space and time, is calculated from the relationships given by Kennedy et al. (). A linear extrapolation technique proposed by Lynett et al. () is employed to model wave run-up. This technique tracks accurately the shoreline (i.e., the boundary between the dry and the wet region) inside the computational domain according to the comparison of the total water depth, h (h = d+ζ) at each grid point near the shore to an empirical parameter, δ. If h < δ, the physical variables at that point are extrapolated from the seaward neighboring wet points; otherwise model equations (1)-() are solved at the point. The value of δ should be small in order to avoid numerical instabilities. In the following simulations that involve breaking waves, δ is chosen equal to H/1, where H is the incident wave height, as recommended by Lynett et al. ().

5 Assuming that the shoreline at a certain time step is located somewhere between grid points n and n+1, the linear extrapolation relationships used to calculate the surface elevation ζ and the velocity u at the dry region are P n+1 = Pn Pn 1 (8) P n+ = 3Pn Pn 1 (9) where P represents both u and ζ, subscripts n+1 and n+ denote the two dry points landward of the shoreline and subscripts n and n-1 the two wet points seaward of the shoreline. It should be noted that the derivatives of u and ζ are not calculated at the dry points; however, the values of u and ζ at these points are used to determine derivatives at the neighboring wet points. Finally, bottom friction is described by the quadratic law τ b 1 = f wu u (1) d + ζ where f w = bottom friction coefficient, typically in the range of 1-3 and 1 -, depending on the Reynolds number and the bed material. 3. COMPARISON WITH EXPERIMENTS 3.1 Solitary Wave Run-Up Synolakis (1987) studied experimentally the run-up and run-down of breaking and non breaking solitary waves on a plane beach with 1:19.85 slope. The experiments were conducted in a wave flume with glass sidewalls and dimensions m x.61 m x.39 m. Still water depth in front of the sloping bed was. m. Figure 1 shows snapshots of the computed surface elevation in comparison with the experimental data for a breaking solitary wave with amplitude ratio H/d =.8. Model predictions are good both in the surf and swash zone simulating well run-up. The collapse of the breaking bore occurs between the non-dimensional times t' = and t' = 5 (t' = t(gd) 1/ ). The maximum run-up at time t' = 45 is also accurately predicted. Bottom friction coefficient is set equal to in the simulations presented. ζ /d t'= x/d ζ / d t'= x/d Figure 1 Solitary wave run-up on a 1:19.85 beach. Comparison of normalized surface elevation (H/d=.8, experiment, model ).

6 .4.4 ζ/d.. -. t'= x/d ζ / d.. -. t'= x/d Figure 1 (continued) Solitary wave run-up on a 1:19.85 beach. Comparison of normalized surface elevation (H/d=.8, experiment, model ). 3. Overtopping of a Porous Breakwater The experimental tests of Vidal et al. () are also used to test the current model. The tests were conducted in a 4 m long,.6 m wide and.8 m high flume and modeled wave propagation over a rubble mound breakwater. The trapezoidal breakwater which consisted of an armour layer of selected gravel and a gravel core was built over a horizontal bottom, at the top of a 1: slope. Crest elevation from the bottom (.5 m), front and back slope angles (1:) and rubble characteristics were maintained constant while its crest width ranged between.5 and 1. m Water depth at the paddle was either.3 m, or.35 m, or.4 m resulting in a freeboard of.5 m,. and.5 m respectively. To assess free surface evolution and run-up on the beach, 15 resistive wave gauges were installed along the flume. Validation of the numerical model is performed for the case of the zero-freeboard breakwater with.5 m crest width. In the simulations, the center of the source function coincides with the wave paddle and the porosity of the structure is assumed uniform (φ =.5, d 5 =.5 m). Figure shows the layout of the computational domain. The vertical lines in this figure indicate the location of wave gauges 1-8. Sponge Layer Wave Gauges Sponge Layer Source function center Figure Layout of the computational domain. Figure 3 presents comparatively computed and recorded free surface elevation at wave gauges 5-8 for a regular wave case with target wave characteristics H =.1 m and T = 1.6 sec. Figure 4 also shows a comparison of the computed and experimental spectra at gauge 8 for the same case. The overall agreement between model results and data is satisfactory considering the complicated mechanisms (wave breaking, wave-porous media interaction) involved in the test. It can be seen from Figure 4 that the total transmitted wave energy is predicted quite well. However, the decomposition of the leading wave into shorter waves behind the structure is not so accurately described, especially at gauge 7.

7 .6 Gauge 5.6 Gauge Gauge 7.6 Gauge Figure 3 Comparison of free surface elevation at gauges 5, 6, 7 and 8 (Η =.1 m, Τ = 1.6 sec, experiment, model ) S (m /Hz) f (Hz) Figure 4 Comparison of transmitted spectra (Η =.1 m, Τ = 1.6 sec, experiment, model ).

8 3.3 Overtopping of a Trapezoidal Object Finally, the performance of the model is validated against the experimental data of Stansby and Feng (4) (data available from the IAHR Wave Database: In these tests, regular waves overtopping an impermeable trapezoidal obstacle in a wave flume were considered. The flume was 13 m long,.3 m wide and.5 m high. The obstacle which had 1: slopes and a.1 m horizontal crest was placed at the top end of a 1: sloping beach. On the landward side, the bed was horizontal. Both the bed and the obstacle consisted of plastic material. Water depth at the paddle was either.36 m or.34 m while crest elevation was.378 m. Incident wave characteristics (H =.1 m, T =.39 sec) resulted in plunging breakers which became bores before overtopping. Data acquisition involved 13 wave gauges along the flume, 1 were located over the sloping bed and the rest over the slopes and the crest of the obstacle as shown in Figure 5. A digital PIV system was also employed to measure velocity vectors in the case of the higher water level. The mean depth was maintained constant with the use of a recirculation system that returned overtopping flow back into the flume, close to the paddle. Numerical simulations are performed for the case of.36 m depth. In the model, free surface elevation recorded at gauge 1 is used as input for calculating the source function while a sponge layer dissipates outgoing waves. Furthermore, the criterion used to track the shoreline is checked at the grid points around the crest of the obstacle to determine if the extrapolation relationships or the governing equations are applicable. The selected value of bottom friction coefficient (1 1-3 ) corresponds to smooth channel bed Figure 5 Experimental layout. Dimensions in m. In Figure 6, computed and experimental surface elevation are compared at gauges 5, 8, and At gauge 5, the maximum wave height was measured thus it coincides with the breaking point while gauges 1 and 11 are placed inside the collision zone where incident waves interact with the reflected waves. Reasonably good agreements are obtained with some differences in detail. Breaking wave height is slightly underpredicted at gauge 5. Moreover, the computed wave speed at this location is higher leading to a small phase shift between the model and the experiment. A phase shift appears also at gauge 8; however, in this case wave height prediction is more accurate. Differences are more distinct at gauges 1 and 11. At gauge 1, the secondary peak that appears in the experiment is also evident in the model but with a different frequency while the computed waveset up is considerably smaller. At gauge 11, although mean water level prediction is better, computed wave pattern deviates notably from the experiment. Stansy and Feng (4) pointed out that turbulent air entrainment and complex vortical structures were observed in this area resulting from the collision of shoreward propagating bores with bores reflected from the structure. PIV measurements also revealed significant vertical variation of velocity with high velocities in the upper water column and almost stagnant water below. Considering that the Boussinesq equations are

9 depth-averaged and that they are derived under the irrotational flow assumption, it is expected that the complicated nature of the flow in this area could not be captured in detail. Measurements of the water volume gathered behind the structure were not performed; however, overtopping rate can be estimated indirectly by the time variation of velocity and depth of water over the crest at its junction with the landward slope. At this location where the bed slope changes, the flow is critical with depth-averaged velocity equal to the phase speed c, c = (gd) 1/. According to Stansby and Feng (4), the velocity over the crest obtained directly from PIV measurements was close to the calculated phase speed with the relative error less than 5%. Thus measuring elevation (depth) at this point can be considered as indicative of overtopping flow rates and hence volumes. From the comparison at gauge 1, it is estimated that peak overtopping level is overpredicted approximately 5% percent. The wave pattern is described quite well at gauge 13, but the experiment lags the model. In Figure 7, model surface profiles at 4 successive phases in a wave period (t/t=.1, t/t=.16, t/t=. and t/t=.4) during overtopping are presented. A video file (overtopping.mpg) accompanying the present paper is also provided, showing wave evolution on the sloping bed close to the structure and the overtopping process. The duration of the simulation is 7 wave periods Gauge Gauge Gauge Gauge Figure 6 Comparison of free surface elevation at gauges 5, 8, 1, and 11 (Η =.1 m, Τ =.39 sec, experiment, model ).

10 .3.5 Gauge Gauge Figure 6 (continued) Comparison of free surface elevation at gauges 1 and 13 (Η =.1 m, Τ =.39 sec, experiment, model ) t/t x (m) Figure 7 Computed wave profile during overtopping at four phases in a wave period. 4. CONCLUSIONS A numerical model has been developed to simulate the wave run-up and wave overtopping over permeable and impermeable structures. The model is based on high-order Boussinesq equations which are solved in conjunction with a Darcy-Forchheimer porous flow equation. The processes of wave breaking, bottom friction and wave run-up are treated well in the model using established techniques. Numerical results are compared with experimental data for three cases involving a) solitary wave run-up on a plane beach, b) wave overtopping of a porous breakwater with zero-freeboard, and c) wave overtopping of an impermeable obstacle located at the end of a sloping bed. Comparisons

11 for the first case show that the model predicts accurately wave profile in the surf and swash zone as well as maximum wave run-up. Wave transformation and wave energy transmission behind the breakwater are also described efficiently in the second case. In the last case, numerical results show good agreement with the experimental data in terms of breaking wave height evolution over the sloping bed. However, computed surface profile in the area where incident and reflected waves interact is less accurate due to the complexity of the processes involved. The comparative analysis also demonstrates that the model reproduces quite effectively the process of overtopping although peak overtopping level is overpredicted. ACKNOWLEDGMENTS This work is part of the Research Project Pythagoras II Wave Run-Up and Overtopping of Coastal Structures funded by the Greek Ministry of National Education and Religious Affairs. REFERENCES Avgeris, I., Karambas, Th.V. and Prinos, P. (4). Boussinesq Modeling of Wave Interaction with Porous Submerged Breakwaters, In: Proceedings of 9 th International Conference on Coastal Engineering, ASCE, pp Cox, D.T. and Ortega, J.A. (). Laboratory Observations of Green Water Overtopping a Fixed Deck, Ocean Engineering, 9, pp Dodd, N. (1998). Numerical Model of Wave Run-Up, Overtopping, and Regeneration, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 14, No., pp Hu, K., Mingham, C.G. and Causon, D.M. (). Numerical Simulation of Wave Overtopping of Coastal Structures Using the Non-Linear Shallow Water Equations, Coastal Engineering, 41, pp Kennedy, A.B., Chen, Q., Kirby, J.T. and Dalrymple, R.A. (). Boussinesq Modeling of Wave Transformation, Breaking, and Runup. I: 1D., Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 16, No. 1, pp Liu, P.L.-F., Lin, P., Chang, K.-A. and Sakakiyama, T. (1999). Numerical Modeling of Wave Interaction with Porous Structures, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 15, No. 6, pp Lynett, P.J., Wu, T.-R. and Liu, P.L.-F. (). Modeling Wave Runup with Depth-Integrated Equations, Coastal Engineering, 46, pp Memos, C.D., Karambas, Th.V. and Avgeris, I. (5). Irregular Wave Transformation in the Nearshore Zone: Experimental Investigations and Comparison with a Higher Order Boussinesq Model, Ocean Engineering, 3, pp Shao, S.D., Ji, C., Graham, D.I., Reeve, D.E., James, P.W. and Chadwick, A.J. (6). Simulation of Wave Overtopping by an Incompressible SPH Model, Coastal Engineering, 53, pp Soliman, A. (3). Numerical Study of Irregular Wave Overtopping and Overflow, PhD Thesis, The University of Nottingham, United Kingdom. Sollitt, C.K. and Cross, R.H. (197). Wave Transmission through Permeable Breakwaters, In: Proceedings of 13 th International Conference on Coastal Engineering, ASCE, pp Stansby, P.K. (3). Solitary Wave Run Up and Overtopping by a Semi-Implicit Finite-Volume Shallow-Water Boussinesq Model, Journal of Hydraulic Research, Vol. 41, No. 6, pp

12 Stansby, P.K. and Feng, T. (4). Surf Zone Wave Overtopping a Trapezoidal Structure: 1-D Modelling and PIV Comparison, Coastal Engineering, 51, pp Synolakis, C. (1987). The Runup of Solitary Waves, Journal of Fluid Mechanics, 185, pp van der Meer, J.W. and Janssen, W. (1995). Wave Run-Up and Wave Overtopping at Dikes, In: Wave Forces on Inclined and Vertical Structures, Kobayashi & Demirbilek, eds., ASCE, pp van Gent, M.R.A. (1995). Wave Interaction with Permeable Coastal Structures, PhD Thesis, Delft University, Delft, The Netherlands. Vidal, C., Lomonaco, P., Migoya, L., Archetti, R., Turchetti, M., Sorci, M. and Sassi, G. (). Laboratory Experiments on Flow Around and Inside LCS Structures. Description of Tests and Data Base, DELOS EU Project, Internal Report. Wei, G. and Kirby, J.T. (1995). Time-Dependent Numerical Code for Extended Boussinesq Equations, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 11, No. 5, pp Wei, G., Kirby, J.T. and Sinha, A. (1999). Generation of Waves in Boussinesq Models Using a Source Function Method, Coastal Engineering, 36, pp