2DV RANS-VOF MODELLING OF DEPTHS AND VELOCITIES OF OVERTOPPING WAVES AT OVERWASHED DIKES

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1 2DV RANS-VOF MODELLING OF DEPTHS AND VELOCITIES OF OVERTOPPING WAVES AT OVERWASHED DIKES Andrea N. Raosa, Barbara Zanuttigh 2, Javier L. Lara 3 and Steve Hughes 4 The purpose of this ontribution is the representation of real wave overtopping over sea dikes with the Rans-Vof ode (IH-2VOF) developed by the University of Cantabria. More speifi objetives are: to identify the real apaity of the IH-2VOF model in the predition of overflow and to determinate the auray of these preditions in order to provide designers with a generally appliable methodology to use the ode. The model is validated against experimental tests onduted by Hughes at the U.S. Army Engineer Researh and Development Center (ERDC). This analysis shows that the model tends to overestimate wave refletion and better results are obtained by introduing a porous layer around the struture as artifiial way to redue refleted wave energy. Keywords: overwashed dikes; real wave overtopping; numerial simulations; experiments; RANS-VOF model INTRODUCTION Vulnerability and resiliene of dikes and sea banks play a key role in the safety of oastal areas. Ideally these defenses should be designed with a rest height and width to prevent flooding for any limate senarios but the unertainty in limate onditions, worsened by limate hange, and the high osts require that a ertain overtopping level has to be aepted. Dike overtopping produes high-speed urrents and turbulene (Shuttrumpf and Oumerai, 25) that may indue damage of the protetion layers (for instane grass over, Van der Meer et al., 2) and expose at erosion the under layers in ase of soil dikes. If the overtopping persists for suffiient time, rest lowering and struture breahing may our (Hughes and Nadal, 29). Catastrophi onsequenes of this proess happened during the hurriane Katrina in New Orleans (ASCE Hurriane Katrina External Review Panel, 27). Aurate estimates of the statistis of overtopping waves in terms of flow depths, duration and espeially veloities for a set of limate onditions are needed and have to be ombined with onsolidated riteria for identifying tolerable overtopping threshold. The evaluation of wave-struture interation has been mainly investigated by means of physial model testing in laboratories (a.o. Shuttrumpf and Oumerai, 25; Van Gent, 22) and at prototype sale, ahieving in depth knowledge by means of the Wave Overtopping simulator, Van der Meer et al. (26). However the theories adopted so far for prediting overtopping depths and veloities along the dike still require analysis and verifiation (Van der Meer et al., 2). Aim of this ontribution is to verify the possibility to aurately predit wave overtopping and related fluxes over sea dikes with the RansVof ode (IH-2VOF) developed by the University of Cantabria (Lara et al., 28). The possibility to use a numerial model with high resolution for suh purposes would allow obtaining many results in a ontrolled environment regarding depths and veloities on the dikes, i.e. relevant parameters for a more resilient design. The paper struture is as follows. First the experimental tests used for omparison with the numerial model are presented. Then the numerial model and its set-up are desribed, inluding the validation proess to hek the model auray. The model preditive apaity is then examined by means of omparison with experimental results, suh as statisti distribution of wave overtopping depths and veloities (h2%, T2%, u2%) over and landward the dike. A sensitivity analysis is arried out to set the best modeling onfiguration. Numerial results for the best modeling parameters are finally ompared with the existing theoretial approah developed by Shuttumpf et al. (25). Dipartimento di Ingegneria Civile, Ambientale e dei Materiali DICAM, University of Bologna, Viale del Risorgimento 2, Bologna, 436, Italy 2 Dipartimento di Ingegneria Civile, Ambientale e dei Materiali DICAM, University of Bologna, Viale del Risorgimento 2, Bologna, 436, Italy 3 Environmental Hydraulis Institute IH Cantabria, University of Cantabria, C/ Isabel Torres nº 5 Parque Científio y Tenológio de Cantabria 39, Santander, Spain 4 U.S. Army Engineer Researh and Development Center (ERDC), Coastal and Hydraulis Laboratory (CHL), Viksburg, MS

2 2 COASTAL ENGINEERING 22 HUGHES TESTS Desription of the tests The overtopping disharge resulting from ombined storm surge overflow and wave overtopping of a levee with a trapezoidal ross setion was studied in :25 sale in the.9-m-wide wave flume at the U.S. Army Engineer Researh and Development Center (ERDC), Coastal and Hydraulis Laboratory (CHL) in Viksburg, MS. The tested levee ross setion repliated in the physial model is shown in Fig. in model-sale units. Fig. 2 shows the relative loation of the levee model in the wave flume. The levee rest was approximately 32 m from the wave board (left end of Fig. 2) with the rest elevation.6 m above the wave flume bottom. Seaward of the levee model setion there was a long : approah slope and a shorter :2 slope transition to the bottom of the flume. Surge and waves that overtopped the levee flowed into a reservoir (right end of Fig. 2), and a pump reirulated the water to the seaward end of the flume. Four apaitane-type wave gauges were plaed at the loations shown in Fig. 2. Gauge was loated over the horizontal bottom of the wave flume losest to the wave board. Gauges 2 4 were plaed as a three-gauge array near the toe of the levee model berm. Two of the pressure ells were loated on the levee rest, and the remaining five instruments were evenly spaed down the protetedside slope. Fig. 3 shows the loations of the pressure ells with spaing given in model-sale units. The purpose of the pressure ells was to measure flow thikness variations over the levee as a funtion of time. A fiber-opti laser Doppler veloimeter (LDV) was used to measure the horizontal omponent of flow veloity diretly above pressure ell P2 (see Fig. 3) near the rearward edge of the levee rest. This ross-flume loation was diretly above pressure ell P2. For eah experiment the vertial position of the LDV beam rossing was adjusted to an elevation approximately half of the water depth of the steady overflow. This vertial position was thought to provide a reasonable value for depth-averaged horizontal veloity, but the drawbak to this vertial loation was the loss of veloity signal during wave troughs when the water level fell below the elevation of the laser beams. Figure. Tested levee ross setion (model-sale units). Figure 2. Profile view of wave flume (model-sale units). The wave-marker is to the left side of the figure. Figure 3. Loations of flush-mounted pressure ell on levee (model-sale units).

3 COASTAL ENGINEERING 22 3 Simulated tests The numerial analysis is arried out for four different tests that are haraterized by different wave height, peak periods and struture submergene (see Table ). Table. Signifiant wave height (Hs), peak period (Tp), water depth (h) and submergene (R) for the four different tests simulated numerially. Hs (m) Tp (s) h (m) R (m) R R R R The work is organized into two different phases: one to hek the preision of the model and the other to verify the model preditive apaity. DESCRIPTION OF THE NUMERICAL MODEL Numerial models of fluid/wave-struture interations are inreasingly beoming a viable tool in furthering our understanding of the ompliated phenomena that govern the hydrauli response of breakwaters, inluding effets of permeability (Losada, 23). These models are based on different approahes, for instane: Lagrangian models with partile-based approahes suh as the Moving Partile Semi-Impliit method (Koshizuka et al., 25) and Smooth Partile Hydrodynamis (SPH) (Dalrymple et al., 29), Reynold Averaged Navier Stokes-Volume Of Fluid (RANS-VOF) models suh as those developed by Lara et al. (28) and Shi et al. (24). The IH-2VOF model (Lara et al. 28) has been developed by implementing various extensions to the RIPPLE model (Kothe et al, 99; originally designed to provide a solution of two-dimensional versions of the Navier-Stokes equations in a vertial plane with a free surfae), making it speifially appliable to the study of wave interation with oastal strutures. The model solves the 2DV Reynolds averaged Navier Stokes (RANS) equations based on the deomposition of the instantaneous veloity and pressure fields into mean and turbulent omponents, and the k equations for the turbulent kineti energy k and its dissipation rate. The influene of turbulene flutuations on the mean flow field is represented by the Reynolds stresses. A nonlinear algebrai Reynolds stress model is used to relate the Reynolds stress tensor and the strain rate of mean flow. The free surfae movement is traked by the volume of fluid (VOF) method, for only one phase, water and void. In order to repliate solid bodies immersed in the mesh, instead of treating them as sawtooth-shape, the model uses a utting ell method first presented by Clarke et al This tehnique uses an orthogonal strutured mesh in the simulations to save omputational ost. This approah defines the openness funtion to define the fration of volume of free spae in the ell. Aording to this definition is a solid ell (entirely oupied by the solid), is a fluid ell and is a partial ell. Variables in the ell or on the ell faes are redefined by the produt of the openness oeffiients times the original variables. To numerially implement this, partial ell oeffiients need to be defined at both the ell enters and boundaries:,,,, whih orrespond to the openness at the left, right, top and bottom, respetively. As a result, the governing RANS equations are redefined as follows: ui x i l r t b () and u u p ij t i u j x j i gi x x i j f b (2) The last term in equation (2) is a virtual fore. It appears beause the IH-2VOF model onsiders two different numerial tehniques to simulate moving bodies within the omputational

4 4 COASTAL ENGINEERING 22 domain. The first one is alled virtual fore method (see Mittal and Iaarino, 25) and the other one is the diret foring method (Mohd-Yusof, 997). VALIDATION OF THE NUMERICAL MODEL Model set-up In these preliminary tests the IH-2VOF model is used foring as offshore boundary ondition the same water levels measured by the gauge in front of the struture (at around 3 m from the wavemaker). Seaward boundary ondition measured water level at the laboratory gauge P2 Landward boundary ondition absorption The length of the numerial hannel is therefore shorter than the one in the laboratory sine the numerial simulation starts from Wave Gauge (WG) 2 on (plaed at a distane of m from the WM). Flume dimension 5.7 m x.8 m Mesh resolution. m x.5 m Results With this type of approah the representation of water levels along the hannel is very good. In Fig. 4 three graphs are reported: the first represents the water level at the WG in front of the struture (WG2), the seond at the WG sets on the rests of the struture at the offshore edge (P) and the last at the WG sets at inshore edge (P2). At the WG 2 the numerial water level (red line) reprodues almost perfetly the laboratory one (blue line) beause it is preisely the boundary ondition imposed. Figure 4. Trend of three water levels measured respetively at WG 2 (plaed in front of the struture), at the pressure gauge P (plaed on the rest of the struture) and at the pressure gauge P6 (plaed on the land side of the struture). Fig. 5 shows a omparison among measured and simulated overtopping volumes at gauge P (gauge at the rest offshore edge). It an be observed that the representation is overall optimal and both experimental and numerial data are very well approximated by a Weibull statisti distribution.

5 Non exeedane probability COASTAL ENGINEERING 22 5 In Table 5 (ase A ) the experimental and numerial signifiant results are ompared in terms of the wave height and peak period at the gauges WG and WG2 (i.e. the gauges between the wave maker and the struture). The more signifiant quantitative results (h2%, T2%, u2%) are reported in Table 6 (ase A ). The satter of the numerial results from the experimental data is very modest, suggesting that the RansVof model ensures a good preision in the representation of the overtopping proess. Overtopping volume at P Experimental Experimental weibull fitting Numerial Numerial weibull fitting Sorted overtopping volume per unit length, m*m Figure 5. Test R9 (ase A ): omparison of omputed and measured overtopping volumes at gauge P. USE OF THE MODEL FOR DESIGN PURPOSION Model predition apaity In order to hek the preditive apaity of the model, the numerial hannel is idential to the experimental one and a Jonswap spetrum haraterized by the same signifiant wave height, peak period and spetrum parameters as in the laboratory is imposed as offshore boundary ondition. Therefore the following model settings are adopted: seaward boundary ondition Hs and Tp (Jonswap spetrum) landward boundary ondition absorption k turbulene model flume dimension 32 m x.8 m mesh resolution ell width:..4 m ell height:. m Fig. 6 shows a omparison among the overtopped volumes measured in the laboratory (blue olor) and the numerial ones (red olor) for the test R4 at the gauge P. It is possible observe learly that the model underestimates the overtopped volumes espeially in orrespondene of the bigger volumes. As before, in Table 5 (ase B ) it is possible to see the omparison between the experimental and numerial signifiant wave height and peak period at the gauges WG and WG2. The results of u2%, h2% and T2% for the tests R4, R8 and R2 (haraterized by wave height, peak period, water deep and submergene shown in Table ) are reported in Table 6 (ase "B"): all the parameter values derived from the experiments are signifiantly higher than the numerial values. It is possible to onlude that in these tests the numerial approah annot apture the higher overtopping volumes. This poor approximation may be essentially justified by the lower inident wave height simulated in front of the struture (see Table 5, ase B ).

6 Non exeedane probability 6 COASTAL ENGINEERING 22 Overtopping volume at P Experimental Experimental weibull fitting Numerial Numerial weibull fitting Sorted overtopping volume per unit length, m*m Figure 6. Test R4 (ase B ): omparison of omputed and measured overtopping volumes at gauge P. Several simulations are then onduted by hanging the disretization of the domain. The best solution seems to be ahieved when the resolution of the numerial mesh equals the % of Hs in the horizontal diretion and the.5% of Hs in the vertial diretion. Even though this expedient, the variation of the results is modest. To improve the quality of the simulations, three different hypotheses an be made:. too high dissipation indued by the seleted turbulene model; 2. possible wave deay along the numerial flume; 3. too high refletion indued by the dike-obstale. The results of a sensitivity analysis to hek these three hypotheses and define a method to improve the performane of the numerial model are reported in the next setions. The simulations are performed first without inluding the turbulene model, then in a shorter flume (that is around 3 wave lenghts long in front of the struture); finally the struture is modifyied by inluding a omposite porous medium. SENSITIVITY ANALYSIS Simulations without turbulene To verify the dissipation indued by the turbulene model used in the RansVof ode ( k turbulene model), some simulations are repeated without turbulene. The same numerial flume desribed for ase B is here used again (idential to the experimental one). Also boundary onditions and mesh resolution are kept unhanged. Without turbulene (see Table 6, ase C ) the underestimation of the overtopped volumes and onsequent overestimation of the harateristi parameters h2%, T2%% and u2% are redued but not in a satisfatory manner. In fat the inident wave height simulated in front of the struture still remains lower that the experimental one (see Table 5, ase C ). Simulations with a shorter hannel If the problem is that a wave deay ours along the hannel, a way to redue this deay ould be the use of a numerial hannel shorter than the experimental one. Some simulations are repeated in a hannel whose length from the wave-maker to the struture approximately equals three times the maximum wave length. The following settings are therefore adopted in this ase: seaward boundary ondition Hs and Tp (Jonswap spetrum) landward boundary ondition absorption flume dimension 4 m x.8 m mesh resolution ell width:.. m ell height:.5 m

7 Non exeedane probability COASTAL ENGINEERING 22 7 The results of tests R4, R8 and R2 (haraterized by wave height, peak period, water depth and struture submergene in Table ) are shown in Table 6 (ase "D"). The representation of overtopped volumes does not improve signifiantly, the overestimation of the inident wave height being still high (Table 5, ase D ). Therefore this model setting does not allow to reah a final onlusion regarding the improvement of the model preditive apaity. However it an be onluded that shortening the hannel up to 3 times the peak wave length does not indue any relevant hange of the results, suggesting that it would be possible to redue in this way the omputational effort without loosing auray. Simulations without the struture To hek if the disrepany with experiments is due to wave refletion or to an internal numerial dissipation, a simulation without the struture is arried out. The simulation is therefore performed in the same onditions as in ase B (i.e. R4 with a numerial hannel perfetly idential to the experimental one and without turbulene): Mesh resolution ell width:..2 m ell height:. m The results obtained at the gauges set between the numerial wave maker and the struture are reported in Table 2: Table 2. Comparison between Hs and Tp obtained experimentally and numerially for the simulation onduted without the struture Hs (m) Tp (s) EXP NUM EXP NUM WG WG Overtopping volume at P Experimental Experimental weibull fitting Numerial Numerial weibull fitting Sorted overtopping volume per unit length, m*m Figure 7. Test R4 (test without struture): omparison of omputed and measured overtopping volumes at gauge P. As it an be seen in Fig. 7, the omputed and measured wave statistis at P perfetly math. This means that the problems with mismathing and dissipation should be related to wave refletion from the dike seaward slope or eventually to the way the run-up proess is reprodued. Simulations with a part of porous medium In order to redue wave refletion, the representation of the struture is hanged from perfetly impermeable to partially permeable. Different onfigurations are tested (see Fig. 8):. the seaward slope is replaed by a homogeneous porous medium (ase E ) 2. the seaward slope is replaed by a omposite porous medium (ase F ) 3. a thin porous layer is wrapped around the struture (ase G ) As a first attempt (ase E ) a single porous layer at the seaward-side is inserted. Then, as a seond attempt (ase F ), the seaward slope is replaed by three layers eah haraterized by different value of porosity and nominal diameter. This approah allows to reate a gradient of porosity: the

8 8 COASTAL ENGINEERING 22 lowest layer presents a greater porosity than the upper layers, in order to redue both wave refletion in the lower part and wave perolation in the upper part (i.e. to maintain wave run-up). Finally (ase G ) to avoid the disontinuity that alters the flow, the best solution seems to inlude a thin and homogenous layer of porous medium around the entire struture (seaward side, rest and landward side). The harateristis of eah porous layers for all the ases are shown in Table 3. The omputed and measured wave statistis at P are reported in Fig. 9 and Fig. for the ases E and F respetively. In Table 4 it is reported a omparison between u2% obtained at the gauge P2 with the onfiguration E and F (Test R4). The use of the porous medium as seaward slope of the struture redues the wave refletion but aelerates the flow on the rest due to the disontinuity between the permeable and impermeable parts of the struture. CASE E CASE F CASE G Figure 8. Geometrial sheme for the test with a single porous layer. Table 3. Charateristi of porous medium Case Thikness Porosity Nominal diameter E F G.5.. Table 4. Comparison of u2% at P2 between the ases E and F. EXP NUM Case E NUM Case F The use of a permeable seaward slope generates a disontinuity between the porous seaward side and the impermeable rest that alters the water flow. So in ase G the porous medium is set around the entire struture. This thin layer of porous medium (about / Hs), is haraterized (see Table 3) by a low porosity in order to limit the water perolation into the dike. Hene to avoid struture disontinuity that alters the flow, the best solution seems to inlude a thin and homogenous layer of porous medium around the entire struture (seaward side, rest and landward side).

9 Non exeedane probability Non exeedane probability Non exeedane probability COASTAL ENGINEERING 22 9 Overtopping volume at P Experimental Experimental weibull fitting Numerial Numerial weibull fitting Sorted overtopping volume per unit length, m*m Figure 9. Test 4 (ase E ): omparison of omputed and measured overtopping volumes at gauge P. Overtopping volume at P Experimental Experimental weibull fitting Numerial Numerial weibull fitting Sorted overtopping volume per unit length, m*m Figure. Test 4 (ase F ): omparison of omputed and measured overtopping volumes at gauge P. Overtopping volume at P Experimental Experimental weibull fitting Numerial Numerial weibull fitting Sorted overtopping volume per unit length, m*m Figure. Test 4 (ase G ): of omputed and measured overtopping volumes at gauge P. Keeping onstant all the other parameters, the results show a signifiant improvement in the performane of the model in reproduing the statistial distribution of the overflowing volumes in orrespondene of the gauge P at the offshore edge. Verifiation of the refletion oeffiient The refletion oeffiient obtained by the simulations is ompared in Fig. 2 with the data for smooth straight slopes that are inluded in the refletion database by Zanuttigh and Van der Meer (26). It an be observed that the value (yan filled-in irle) obtained by foring the model exatly with the measured levels in front of the struture (ase A ) falls perfetly in the range of the

10 y, m COASTAL ENGINEERING 22 experimental values as well as the values (red filled-in irles) derived with the inlusion of the porous layer (ase G ) and the use of the target wave spetrum as boundary onditions. The values obtained by the model in preditive set-up when the struture is fully impermeable (ase B ) are well above the experimental data, being around two times the measured values for the same Iribarren-Battjes parameter. Therefore the overestimation of wave refletion with the numerial ode derived from the previous analysis is onfirmed. Relevane of profile versus loal measurement Fig.3 reports, in the same graph, the omputed and the measured veloities for the ase G. The figure shows the veloity profile obtained with the introdution of the porous medium. The red, green and blue olours represent respetively the maximum, the minimum and the average veloity. The veloity values (maximum, minimum and averaged) at the measurement point are well approximated. The model allows to get the veloity profiles over the flow depth in time - usually not measured in the lab or at prototype sale. Peak veloities along the profile differ in some ases of a fator 2 from the u2% at the measurement point, highlighting the relevane of this modelling result for design purposes. Figure 2. Comparison between the refletion oeffiient obtained from test A (yan filled in irle), test B (red empty in irles), test G (red filled in irles) and Zanuttigh and Van der Meer s database (26). Veloity profile at P mean (numerial) max (numerial) min (numerial) mean (experimental) max (experimental) min (experimental) Veloity, m/s Figure 3. Test R4 (ase G ): omparison of omputed and measured horizontal veloity at gauge P2 test with porous medium.

11 COASTAL ENGINEERING 22 Table 5. Signifiant wave height (Hs) and peak period (Tp) at the gauges set in front of the struture. Case Hs Tp A R9 WG EXP NUM WG2 EXP NUM B R4 WG EXP NUM WG2 EXP NUM C R4 WG EXP NUM WG2 EXP NUM D R4 WG EXP NUM WG2 EXP NUM G R4 WG EXP NUM WG2 EXP NUM Table 6. Numerial and experimental u2%, T% and h2%. Case Test Comparison at P2 u2% T2% h2% A R9 EXP NUM B R4 EXP NUM R8 EXP NUM R2 EXP NUM R9 EXP NUM C R4 EXP NUM R8 EXP NUM R2 EXP NUM D R4 EXP NUM R8 EXP NUM R2 EXP NUM E R4 EXP NUM F R4 EXP NUM R8 EXP NUM R2 EXP NUM G R4 EXP NUM R8 EXP NUM R2 EXP NUM R9 EXP NUM

12 2 COASTAL ENGINEERING 22 COMPARISON BETWEEN NUMERICAL AND THEORY (SCHUTTRUMPF) Layer thikness on the rest of the dike On the dike rest, the layer thikness oordinate on the dike rest x. The x-oordinate on the dike rest B are ombined to a dimensionless parameter on the dike rest an be desribed by the following equation: h ( x ) h ( x ) i h depends on the width of the dike rest B and the x- exp x and the width of the dike rest h x x / B ( x / B ). The layer thikness ) x B (3) ( with:. 75. Fig. 4 shows the good agreement among the points extrated from the numerial simulations and the theoretial urve. Table 7 reports the values of rmse and the ratio between the standard deviation relative to the numerial urve and the standard deviation relative to the theoretial urve (see line evolution of layer thikness along the rest ). Overtopping veloities on the dike rest The variation of the overtopping veloities on the dike rest is derived from the Navier-Stokes equation and reads v v x exp 2 f h (4) with: v = overtopping flow veloity on the dike rest, x = oordinate on the dike rest, h = layer thikness, f = bottom frition oeffiient, v v x = overtopping veloity at the beginning of the dike rest. The frition oeffiient must be determined empirially, here the value used in Fig. 5 is.2. Table 7 shows the values of rmse and the ratio between the standard deviation relative to the numerial urve and the standard deviation relative to the theoretial urve (see line evolution of veloity along the rest ). Flow field on the landward slope of the dike Shuttrumpf developed an approah to desribe overtopping veloities on the landward slope and, as for the evolution of the veloity and the layer thikness along the rest, verified this theoretial approah by the small sale model test results. v B 2hB g sin f (5) where: h B = layer thikness at the beginning of the landward slope; v B = overtopping veloity at the beginning of the landward slope; f = bottom frition oeffiient; = landward slope angle. Fig. 6 shows the omputed and theoretial evolution of the veloities on the landward slope. Table 7 shows the values of rmse and the ratio between the standard deviation relative to the numerial urve and the standard deviation relative to the theoretial urve (see line evolution of veloity on the landward side ).

13 COASTAL ENGINEERING 22 3 Figure 4. Test R4 (ase G ): evolution of layer thikness on the dike rest. Figure 5. Evolution of the overtopping veloities on the dike rest. Figure 6. Evolution of the overtopping veloities on the landward side.

14 4 COASTAL ENGINEERING 22 Table 7. Values of rmse and ration between the standard deviation relative to the numerial urve and the standard deviation relative to the theoretial urve for the ase G. Case rmse σ N/ σ T Layers thikness over the rest Eq. (3) Veloities over the rest Eq. (4) Veloities on the landward side Eq. (5) R R R R R R R R R R R R CONCLUSIONS Numerial simulations with the RansVof model were performed and ompared with the tests provided by Hughes on impermeable submerged or zero freeboard dike. The model reprodues with high auray the physial proess of wave overtopping when it is fored with the measured levels obtained in the hannel. When used for prediting wave overtopping based on a Jonswap target wave spetrum, the model does not fully represent the overtopping volume distribution and tends to underestimate higher volumes, espeially for low waves. This an be only partially asribed to the mesh resolution and turbulent dissipation. Speifially best results are obtained with mesh resolution: % Hs horizontal and.5% Hs in vertial without turbulene model. Atually the disrepany among numerial and experimental distribution of overtopping volumes and overtopping statisti parameters (h2%, u2%, T2%) an be fully explained by the model overestimation of wave refletion. The inlusion of a porous layer around the struture redues wave refletion and therefore signifiantly improves - keeping onstant the other parameters - the performane of the model in reproduing wave overtopping statistis together with flow harateristis at the dike landward slope. To avoid disontinuity in water fluxes and ontemporarily redue wave refletion from the dike, the best solution seems to inlude a homogenous thin porous layer around the dike haraterised by low thikness (around / Hs) and low porosity (n=. to avoid perolation). The model represents very well u2% at the measurement point over the rest and at the same time allows to get the veloity profiles over the flow depth in time -usually not measured in the lab or at prototype sale. Peak veloities along the profile differ in some ases of a fator 2 from the u2% at the measurement point, highlighting the relevane of this modelling result for design purposes. The numerial results also fit very well the theoretial approah developed by Shuttrumpf and Oumerai (25) to represent flow depths and veloities over and landward the dike. REFERENCES ASCE Hurriane Katrina External Review Panel. 27. The New Orleans Hurriane Protetion System: What went wrong and why. Amerian Soiety of Civil Engineers, Reston, Virginia. 92. Clarke D.K., Hassan H.A., and M.D. Salas Euler alulations for multielement airfoils using artesian grids. AIAA Journal, 24, Dalrymple, R.A., Gómez-Gesteira M., Rogers B.D., Panizzo A., Zou S., Crespo A.J.C., Cuomo G., M. Narayanaswamy. 29. Smoothed Partile Hydrodynamis for Water Waves, in Advanes in Numerial Simulation of Nonlinear Waves Shi F., Q.Ma, ed., World Sientifi Press, ISBN Hughes, S.A., N.C. Nadal. 29. Laboratory study of ombined wave overtopping and storm surge overflow of a levee, Coastal Engineering, 56, Kothe D.B., Mjolsness R. C., M. D.J. Torrey. 99. RIPPLE: A omputer program for inompressible flows with free surfaes, Report LA-27, Los Alamos National Laboratory. Lara J.L., Losada I.J., R. Guanhe. 28. Wave interation with low-mound breakwater using a rans model. Oean Engineering, 56,

15 COASTAL ENGINEERING 22 5 Losada I.J. 23. Advanes in modeling the effets of permeable and refletive strutures on waves and nearshore flows. In: Chris Lakhan, V. (Ed.), Advanes in Coastal Modeling. Elsevier Oeanography Series, vol. 67. Mittal R., G. Iaarino. 25. Immersed boundary methods. Annual Review. Fluid Mehanis. Mohd-Yusof J Combined immersed-boundary/b-spline methods for simulations of flow in omplex geometries. CTR annual researh briefs, , Center for Turbulene Researh, NASA Ames/Stanford University. Shüttrumpf H., H. Oumerai. 25. Layer thiknesses and veloities of wave overtopping flow at sea dikes, Coastal Engineering, 52 (6), Shi F., Zhao Q., Kirby J.T., Lee D.S., S.N. Seo. 24. Modeling wave interations with omplex oastal strutures using an enhaned VOF model. In Pro. 29 th Int. Conf. Coastal Engineering, Lisbon, Van der Meer J. W., Hardeman B., Steendam S.J., Shüttrumpf H., H. Verheij. 2. Flow depth and veloities at rest and landward slope of a dike, in theory and with the wave overtopping simulator. ASCE, Pro. ICCE 2, Shanghai. Van der Meer, J.W., Bernardini, P., Snijders,W., E. Regeling. 26. The wave overtopping simulator. Proeedings of the 3th International Conferene on Coastal Engineering, vol. 5. World Sientifi, Van Gent M.R. 22. Wave overtopping events at dikes. Proeedings of the 28th International Coastal Engineering Conferene, vol. 2. World Sientifi,