IMPROVEMENTS IN DESCRIBING WAVE OVERTOPPING PROCESSES

Transcription

1 IMPROVEMENS IN DESCRIBING WAVE OVEROPPING PROCESSES Steven Hughes 1, Christopher hornton 2, Jentsje van der Meer 3, and Bryon Scholl 4 his paper presents a new epirical relation for the shape factor in the Weibull distribution that describes the distribution of overtopping wave volues. his iproveent increases the applicable range of the Weibull distribution fro very low average overtopping discharges to large discharges resulting fro cobined wave overtopping and steady surge overflow at negative freeboards. he effect this iproveent has on wave overtopping siulation is also discussed. Measured axiu flow thicknesses, velocities, and discharges fro experients of cobined wave and surge overtopping are exained to learn ore about the variability of these key paraeters as a function of individual overtopping wave volues. A key finding is that wave volues containing the 2%-exceedance value of axiu velocity typically have axiu flow thicknesses well below the 2%-exceedance level, and viceversa. Furtherore, the 2%-exceedance hydrodynaic paraeters do not occur in the 2%-exceedance wave volues. Finally, epirical relationships are developed for several paraeters that showed strong trends. Keywords: wave overtopping; overtopping siulators; overtopping wave distribution; overtopping paraeters INRODUCION he agnitude of wave overtopping is a critical design paraeter for deterining the crest elevation of dikes, levees, breakwaters, and other structures. Flooding potential is deterined fro the quantity of overtopping water and stor duration, and a ajor concern for engineers is daage to the crest and landward-side slopes of earthen flood protection structures by hydrodynaic forces associated with severe wave overtopping. Epirical equations describing wave overtopping processes in ters of incident wave conditions, structure geoetry, and crest freeboard have been developed based on sall- and large-scale physical odel tests of coon structure geoetries. When wave conditions and freeboard reain relatively constant, wave overtopping can be represented by the ean discharge, q, and the distribution of individual overtopping wave volues, P V. However, the hydrodynaic forces exerted on the structure crest and landward-side slope depend largely on the distributions of instantaneous flow thicknesses and flow velocities that occur over the duration of each overtopping wave. Controlled full-scale testing of dike and levee slope resiliency can be achieved using Wave Overtopping Siulators (Van der Meer, et al. 2006; Van der Meer, et al. 2008) provided the details of wave overtopping hydrodynaics are understood and reasonably approxiated. Wave overtopping siulators replicate wave overtopping by releasing individual wave volues in such a anner that the axiu flow thickness and axiu velocity near the leading edge (tongue) of the released wave volue confor reasonably well to our present understanding of what these two paraeters should be for a given overtopping wave volue. As additional data becoe available fro experients, it is iperative that engineers incorporate the new data into existing data sets in order to iprove and refine the epirical relationships needed to conduct reliable siulations of wave overtopping at full scale. his paper presents a new epirical relation for the shape factor in the Weibull distribution that deterines the extree tail of the wave overtopping volue distribution. his iproveent results in a wave overtopping distribution that is ore suitable for severe wave overtopping, but still accurately reproduces the distribution for less severe overtopping conditions. he effect this iproveent has on wave overtopping siulation is also discussed. Measured axiu flow thicknesses, velocities, and discharges fro experients of cobined wave and surge overtopping are exained to learn ore about the variability of these key paraeters as a function of individual overtopping wave volue. WAVE OVEROPPING VOLUME DISRIBUION Previous Developents Van der Meer and Janssen (1995) investigated wave-only overtopping of sloping-front structures, and they found that the distribution of individual overtopping wave volues was well represented by 1 Engineering Research Center, Colorado State University, 1320 Capus Delivery, Fort Collins, CO 80523, USA. 2 Engineering Research Center, Colorado State University, 1320 Capus Delivery, Fort Collins, CO 80523, USA. 3 Van der Meer Consulting, BV, P.O. Box 423, 8440 AK Heerenveen, he Netherlands. 4 Engineering Research Center, Colorado State University, 1320 Capus Delivery, Fort Collins, CO 80523, USA. 1

2 2 COASAL ENGINEERING 2012 the two-paraeter Weibull probability distribution give by either Eq. 1 (cuulative probability distribution) or Eq. 2 (percent exceedance distribution). b V P V ( Vi V ) = 1 exp (1) a % ( ) exp V b P V i V = a ( 100% ) V (2) where P V is the probability that an individual wave volue (V i ) will be less than a specified volue (V), and P V% is the percentage of wave volues that will exceed the specified volue (V). he two paraeters of the Weibull distribution are the non-diensional shape factor, b, that helps define the extree tail of the distribution and the diensional scale factor, a, that noralizes the distribution. Van der Meer and Janssen (1995) forulated the scale factor as a = Γ 1 q ( 1+ 1/ b) Pov where q is the average wave overtopping discharge, is the ean wave period, P ov is the probability of overtopping waves for a given condition (nuber of overtopping waves divided by total incident waves), and Γ is the atheatical gaa function. Van der Meer and Janssen fit the Weibull distribution to 14 sets of overtopping easureents acquired for ild seaward slopes (1:3 and 1:4) spanning a range of 0.99 < R c /H 0 < 3.16, where R c is structure freeboard and H 0 is the incident energy-based significant wave height. hese values of relative freeboard represent low to oderate wave overtopping conditions. hey found a constant shape factor of b = 0.75 that gave a corresponding scale factor of (3) q a = (4) Pov Van der Meer (2010) odified Eq. 4 to better represent situations with a low nuber of overtopping waves, i.e., a = 0.8 q [ ( ) ] N ow Pov 0 (5) where N ow is the nuber of overtopping waves during the event given by the equation 2 R c N ow = Nw Pov = Nw exp ln 0.02 (6) Ru 2% with R u2% representing the elevation exceeded by 2% of the wave run-ups on the seaward slope. More recently Victor (2012) and Victor, et al. (2010) exained 364 tests fro the FlowDike experients for steeper seaward slopes (1:0.36 to 1:2.75) spanning a range of relative freeboard 0.11 < R c /H 0 < hese conditions represented heavier overtopping on steep seaward slopes. hey fit the Weibull distribution to the upper 50% of the volues, and their analysis deterined that the Weibull shape factor varied as a function of relative freeboard and seaward slope as R = exp c b ( cotα ) (7) H 0 with the scale factor given by Eq. 3 using b fro Eq. 7. Victor (2012) gave an approxiation of the scale factor, i.e.,

3 COASAL ENGINEERING q a = 1.13 tanh b (8) ( ) Pov Iproveent of the Overtopping Wave Volue Distribution Hughes and Nadal (2009) conducted 27 sall-scale laboratory experients of cobined wave overtopping and surge overflow for cases with negative relative freeboards -0.1 < R c /H 0 < -2.0 with a seaward slope of 1:4.25. hey had fit a Weibull distribution to all the overtopping wave volues and proposed epirical relationships for a and b. he individual wave volue data were re-analyzed for this paper using various percentages of the upper values to deterine a better Weibull fit to the extree values. It was concluded that the best fit to the extree tail of the wave volue distribution was obtained using the upper 10% of the values. Figure 1 illustrates a best-fit to one of the experients using the upper 50% of the values (upper plot) and the upper 10% of the values (lower plot). Upper 50% of Volues Upper 10% of Volues Figure 1. Best-fit of Weibull distribution to cobined wave and surge overtopping data. Fitting the Weibull distribution to the upper 10% of individual wave volues gave better representation of the extree values while sacrificing accuracy in the lower volue range of the distribution. his was considered acceptable because the operational range of wave overtopping siulators is linked to the larger waves in the distribution. he 27 best-fit values of Weibull shape factor fro the Hughes and Nadal (2009) cobined wave and surge overtopping experients were added to the 14 wave-only overtopping shape factors of Van der Meer and Janssen (1995) and the 364 values of Victor, et al. (2012) and plotted versus relative freeboard. he data are shown in Fig. 2. he dashed line is Victor, et al. s (2012) equation given by Eq. 7 with a seaward slope of 1:4. Equation 7 is being applied well outside the range of steep slopes on which it is based, and it over-predicts the shape factor for all the data points. However, for steep slopes and low relative freeboard, the Victor, et al. equation is preferred. he solid line in Fig. 2 is an epirical fit to the shape factor data given by the expression

4 4 COASAL ENGINEERING 2012 R = exp c b (9) H Figure 2. New Weibull shape factor, b, spanning a large range of relative freeboard. here is no physical significance to the for of Eq. 9 other than the fact that it represents the entire range of relative freeboard reasonably well, including the original data of Van der Meer and Janssen (1995). Also, Eq. 9 does not depend on the seaward slope angle despite the range of slope angles represented in the data. IMPAC ON WAVE OVEROPPING SIMULAION Capability of wave overtopping siulators is liited by the axiu wave volue required for a given set of overtopping conditions. A tentative estiate of the axiu wave volue in the Weibull distribution was given by Van der Meer and Janssen (1995) as 1/ ( V ) a [ ln( )] b ax = (10) N ow where N ow is the nuber of overtopping waves estiated by Eq. 6. Victor, et al. (2012) noted that larger values of the Weibull shape factor, b, result in relatively saller axiu wave volues. herefore, overtopping conditions having R c /H 0 < 2.0 will have shape factors greater than b = 0.75; and the estiated axiu wave volue required for wave overtopping siulation will be less than required by the original wave volue distribution. For exaple, the solid line in Fig. 3 is the theoretical wave volue distribution required for an average overtopping discharge of q = 370 l/s per using the original forulation with a constant value of b = he axiu required wave volue is /, but the axiu capacity of the wave overtopping siulator at Colorado State University is /. his overtopping condition was approxiated by adding additional wave volues at the siulator axiu to copensate for the inability to generate the theoretical axiu volues. he coproise is shown by the hatched area in Fig. 3.

5 COASAL ENGINEERING Figure 3. Wave overtopping distribution for siulating q = 370 l/s per. he dashed line in Fig. 3 shows the new wave volue distribution with scale and shape factors given by Eq. 3 and 9, respectively. he theoretical axiu wave volue for the new distribution is /, so this particular overtopping condition can now be siulated with uch less coproise. he ratio of the theoretical axiu wave volue calculated using the original distribution (Van der Meer s scale factor fro Eq. 5 and b = 0.75) to the theoretical axiu wave volue using the new distribution (scale factor given by Eq. 3 and shape factor by Eq. 9) is expressed as ( Vax ) ( V ) ax Orig New 1 Γ 1 + b 0.8 (1/ / b) [ ( N ) ] [ ln( N )] = (11) ow Equation 11 is plotted in Fig. 4 for various nubers of overtopping waves and for a specific value of Iribarren nuber, ξ -1,0 = 1.5. he CSU siulation depicted in Fig. 3 is indicated by the circle at R c /H 0 = hus, the new epirical expression for shape factor (Eq. 9) eans that wave overtopping siulators can replicate without coproise larger overtopping conditions than previously because the constraint of the largest required wave volue has been reduced, particularly at lower values of relative freeboard. INDIVIDUAL WAVE VOLUME PARAMEERS In addition to reproducing the distribution of individual overtopping wave volues and the average overtopping discharge, another iportant goal of wave overtopping siulation is to approxiate, to the extent possible, the hydraulic flow paraeters associated with individual overtopping wave volues. Of particular iportance are the axiu flow thickness (h ax ), axiu velocity (U ax ), and axiu discharge (q ax = velocity x thickness) at the leading edge of the wave, and the duration of overtopping ( o ) associated with each volue. hese paraeters are defined in Fig. 5. Measured tie series of flow thickness and flow velocity on the crest fro the sall-scale experients of Hughes and Nadal (2009) were analyzed in an attept to better describe the individual wave volue paraeters. he analyses of the cobined wave and surge overtopping experients were restricted to the nine tests with the lowest surge overflow (prototype equivalent of R c = ). he cobined overtopping cases represent an extree edge of the wave overtopping regie, so it was hoped that liiting the analyses to the lowest negative freeboard case ight better correspond to siilar analyses of wave-only overtopping. During overtopping with low negative freeboard, the levee ow

6 6 COASAL ENGINEERING 2012 crest and landward-side slope usually go dry between waves. However, every wave does overtop the levee structure. CSU Siulation Figure 4. Ratio of axiu wave volues required for wave overtopping siulation. Figure 5. Individual wave volue paraeter definitions.

7 COASAL ENGINEERING Figure 6 presents scatter plots fro one experient containing 226 individual overtopping wave volues. In ost of the plots the paraeter of interest is plotted against the corresponding wave volue, and trends are evident for U ax vs. V, q ax vs. V, and o vs. V. Lesser trend is shown for h ax vs. V, and there is a distinct lack of trend for U ax vs. h ax (upper-right plot in Fig. 6). Figure 6. Individual wave volue paraeters fro an experient with 226 waves. Representing Paraeters with a Raleigh Distribution Van der Meer, et al. (2010) questioned whether the distribution of individual wave volue paraeters were actually Rayleigh distributed. o test the Rayleigh hypothesis, paraeters fro each experient were rank-ordered; and the Weibull distribution was fit to the upper 10% of the values to deterine the Weibull shape factor, b. Shape factors equal to 2 correspond to a Rayleigh distribution. Fig. 7 presents best-fit shape factors as a function of relative freeboard for each of the wave volue paraeters. he horizontal line indicates the Rayleigh distribution. Six of the shape factors found for the paraeter U ax had values near b = 2. he three tests with larger shape factors had the lowest wave heights, and it is speculated that the steady overflow was doinating the axiu velocities in the overtopping waves for these cases. Maxiu flow thickness and axiu discharge were clustered about b = 2, indicating the Rayleigh distribution is perhaps a reasonable assuption for these paraeters. However, shape factors for overtopping duration were uniforly greater than b = 2, which iplies that durations cannot be approxiated well by a Rayleigh distribution. Siilar Weibull fits using the entire rank-ordered values rather than the upper 10% showed ore scatter and less conforance to the Rayleigh distribution. Interest lies in the extree values of the wave volue paraeters, so it was concluded that using the upper 10% of the values was warranted.

8 8 COASAL ENGINEERING 2012 Figure 7. Weibull distribution shape factors for individual wave volue paraeters. Characterization of the 2% Exceedance Paraeters Earlier experients (Schüttrupf, et al., 2002; Schüttrupf and Oueraci, 2005; and Van Gent, 2002) yielded epirical expressions for the axiu individual wave flow thickness (h 2% ) and axiu velocity (U 2% ) exceeded by only 2% of the overtopping wave axiu values. hese 2%- exceedance paraeters were related to incident wave conditions and dike freeboard. Van der Meer, et al. (2010) cobined the earlier data with additional data fro the sall-scale Flowdike experients (Lorke, et al. 2010a, 2010b), and a re-analysis of the data provided revised equations for the 2%- exceedance axiu flow thickness and 2%-exceedance axiu velocity on the crest of a sooth dike, i.e., U ( ) h2% = R u 2% R c (12) [ g ( R )] 1/ 2 u R 2% 0.35 cot 2% c = α (13) where α is the seaward-side slope angle, and g is gravitational acceleration. If indeed h ax and U ax are Rayleigh distributed, then the entire distribution of these two paraeters can be estiated fro the 2% values. Cobining Eqs. 12 and 13 yields U 2%.97 cot 1/ 2 ( g ) = 0 α h (14) 2% suggesting that a relationship exists between h 2% and U 2%. Individual overtopping wave paraeters fro the nine cobined wave and surge overtopping experients were exained to explore the relationship between axiu flow thickness and axiu velocity. Fro the rank-ordered lists of each of the paraeters, the 2%-exceedance values were deterined, and the corresponding values of the other paraeters were extracted fro the sae

9 COASAL ENGINEERING wave volue. For exaple, for the wave volue containing h 2% for an experient, the corresponding values of U ax and q ax were extracted. Likewise, for the wave volue containing U 2% for an experient, the corresponding values of h ax and q ax were extracted. Also, the values of h ax and U ax were taken fro the 2%-exceedance wave volues. Scatter plots of extracted values showed little trend when plotted against wave volue. Perhaps the ost interesting aspect of the 2%-exceedance values cae fro plotting the h 2% value versus the associated U ax fro the sae wave volue and the U 2% value versus the associated h ax fro the sae wave volue. Fig. 8 is a scatter plot showing pairs of h ax and U ax values fro just under 2,100 individual waves fro all nine experients. Overlain on the scatter are values of U 2% versus associated h ax and values of h 2% versus associated U ax. Figure 8. Maxiu flow thickness versus axiu velocity for individual wave volues. he nine data points represented by the larger squares show that axiu velocities (U ax ) associated with h 2% are well below typical values of U 2%. Conversely, the data points represented by the large diaonds show that axiu flow thicknesses (h ax ) associated with U 2% are well below typical values of h 2%. In other words, there is no reason to expect that 2%-exceedance values of flow thickness and velocity will occur in the sae wave volue. his coplicates specifying appropriate axiu velocity and flow thickness for individual wave volues when siulating wave overtopping. Interestingly, when values of h 2% are plotted versus values of U 2% fro the sae experient (but fro different wave volues); a definite trend is evident as shown by the large dots in Fig. 9. A fit to the nine data pairs in Fig. 9 is given by the equation 2%. 53 1/ 2 ( g ) U = 1 h (15) 2% which has the sae for as the derived Eq. 14. For the 1:4.25 seaward-side slope used in the Hughes and Nadal experients, the coefficient in Eq. 14 is 4.12, or 2.7 ties greater than the coefficient in Eq. 15. Equation 15 was obtained fro easureents at the landward side of the levee, whereas the velocity represented in Eq. 14 was acquired at the seaward edge of the dike crest. here could be soe

10 10 COASAL ENGINEERING 2012 velocity reduction across the levee crest, but not nearly enough to ake up the difference in coefficients. he difference in coefficients ay be due to the influence of the steady surge overflow in the Hughes and Nadal experients, or the decreased iportance of seaward-side slope for negative freeboards. Additional data and analyses are needed to better understand the 2%-exceedance wave volue paraeters. Figure 9. Values of h 2% versus U 2% fro the sae experient, but different wave volues. Maxiu Instantaneous Discharge for Individual Overtopping Waves All of the scatter plots of individual wave axiu paraeters versus wave volue showed varying degrees of correlation, but the best correspondence was seen for the axiu discharge (q ax ) versus individual wave volue as seen in Fig. 10. A best-fit of a diensionally hoogeneous equation with one free paraeter to the alost 2,100 data points resulted in 3 / 4 q ax = g V (16) where V is the individual wave volue per unit length of levee (e.g., units of 3 /). Equation 16 is shown as the solid line on Fig. 10. One possible use for a q ax vs. V relationship such as Eq. 16 ight be to operate a wave overtopping siulator by assuring that each wave has the correct q ax near the leading edge. However, any cobinations of flow thickness and velocity can give the sae axiu discharge; and this would ean that the leading edge velocity would ost likely be incorrect. An incorrect leading edge velocity would give incorrect shear stresses acting on the levee surface while the flow is still accelerating down the landward-side slope. After terinal velocity is reached farther down the slope, there possibly would be lesser effect caused by an incorrect initial velocity. More work is needed to develop this concept.

11 COASAL ENGINEERING Figure 10. Values of axiu discharge (q ax ) versus individual wave volue. Shape of Individual Overtopping Waves Hughes (2011) suggested siplified analytical power-curve fors for the instantaneous flow thickness and velocity in individual overtopping waves given by the expressions and a t h( t) = hax 1 for 0 t 0 (17) o b t u( t) = U ax 1 for 0 t 0 (18) o When the exponents a and b are unity, the decreases in flow thickness and velocity are linear (which turns out to be a reasonable assuption). Cobining Eqs. 17 and 18 gives the individual wave voluetric discharge per unit levee length (assuing h ax and U ax occur at the sae location), i.e., a+ b t t = hax U ax 1 = qax 1 for 0 o o q( t) t where the product of the peak flow thickness and peak velocity has been replaced with the peak discharge, q ax, and the exponent, =a+b. he volue (per unit levee length) of the individual overtopping wave is siply the integration of the discharge hydrograph with respect to tie, or 0 (19)

12 12 V COASAL ENGINEERING t q q t dt = q ax ( ) ax 1 dt = [ o t] dt ( o ) (20) = 0 Carrying out the integration and evaluating at the liits yields V = q ax o ( ) + 1 ( o t) ( + 1)( 1) 0 o = + 1 qax ( o ) ( + 1)( ) o (21) or V qax o = ( + 1) which is a siple expression for individual wave volue as a function of peak discharge, overtopping duration, and the unknown factor,. he product of axiu discharge and overtopping duration is plotted against individual wave volue for all nine experients in Fig. 11. he best-fit to the data was a diensionally nonhoogeneous equation (coefficient has units!) given by V 1.16 ( q ) ax o (22) = 0.43 (23) where V has units of 3 /, q ax has units of 3 /s per, and o has units of seconds. Rearranging Eq. 23 into the sae for as Eq. 22 gives the expression qax o V = (24) V which iplies that the exponent in Eq. 19 is a function of overtopping wave volue, i.e., 0.16 = a + b = 2.33V 1 (25) Equation 25 suggests that the discharge shape becoes ore concave with increasing wave volue, i.e., the exponent increases. Physically, this ight be interpreted as follows. When saller volues overtop, ost of the forward wave oentu is exhausted on the seaward slope. However, when large volues overtop, ore of the forward oentu is carried over onto the levee crest. Because oentu is related to velocity, it is feasible that the tie-varying wave velocity (Eq. 18) is ore concave which eans the exponent b in Eq. 19 is larger for large overtopping wave volues. Individual Wave Overtopping Duration Based on the correlations shown in Figs. 10 and 11, an expression can be derived for the overtopping duration ( o ) of individual wave volues. Solving Eq. 23 for q ax and equating to Eq. 16 yield the following diensionally non-hoogenous equation 0.41 o = 4.0 V (26) where o has units of seconds, V has units of 3 /, and g = /s 2 in Eq. 16. Equation 26 is shown on Fig. 12 along with the scatter plot of nearly 2,100 data pairs extracted fro the easured data. he wide scatter indicates soe of the uncertainty in deterining actual overtopping duration fro easured data, but the derived equation gives a reasonable trend. A best-fit two-paraeter power curve to the data gave a siilar expression, o = 3.9 V 0.46.

13 COASAL ENGINEERING Figure 11. Values of (q ax o ) versus individual wave volue. Figure 12. Overtopping duration o versus individual wave volue.

14 14 COASAL ENGINEERING 2012 SUMMARY Iproved estiation of wave overtopping paraeters provides engineers with better tools for designing resilient levees and dikes, including the full-scale siulation of wave overtopping. he Weibull distribution has been shown to give accurate estiation of the distribution of wave overtopping volues. However, the shape paraeter of the Weibull distribution was deterined for relatively low values of average overtopping discharge. New sall-scale easureents by Victor, et al. (2012), and reanalysis of the sall-scale cobined wave and surge overtopping of Hughes and Nadal (2009), resulted in an iproved epirical forula for the Weibull distribution shape that spans the range fro low overtopping with positive freeboards to huge overtopping with negative freeboards. he axiu wave volue estiated fro the new distribution is saller than previously thought, and this allows fix-volue wave overtopping siulators to reproduce larger average discharges before reaching the wave siulator s upper liit. Hydrodynaic paraeters of individual overtopping wave volues were exained using data fro nine tests of cobined wave overtopping and steady surge overflow having negative freeboard of R c = 0.29 (prototype equivalent). Analyses of axiu instantaneous flow thickness, velocity, and discharge of individual wave volues suggested that these axiu paraeters follow the Rayleigh distribution to soe degree. An iportant finding was that the 2%-exceedance values for flow thickness and velocity do not occur in the sae wave volue, and they do not occur in the 2%- exceedance wave volue. his coplicates specifying axiu flow thickness and velocity as a function of wave volue. he cobined wave overtopping data were used to deterine epirical equations for axiu instantaneous discharge (q ax ) and overtopping duration ( o ) as a function of individual wave volue. However, caution is urged before utilizing these equations because there was significant scatter about the central trends, and the data used to establish the equations represent an extree case relative to wave-only overtopping. he shape of the overtopping wave volue, as represented by the instantaneous discharge, appears to becoe ore concave for larger overtopping volues. It was hypothesized that this change in the discharge shape is caused by a greater aount of forward wave oentu overtopping the levee crest. his study has provided soe iproveents to the description of the wave overtopping physical process, but it is still apparent that additional research is needed to characterize individual overtopping waves ore accurately. Acknowledgents he data of Hughes and Nadel were originally obtained in experients funded by the New Orleans District of the U.S. Ary Corps of Engineers. A special thanks to Lander Victor and Peter roch for providing analyzed Weibull shape factor values fro 364 FlowDike experients. REFERENCES Hughes, S. A Adaptation of the levee erosional equivalence ethod for the Hurricane Stor Daage Risk Reduction Syste (HSDRRS), ERDC/CHL R-11-3, U.S. Ary Engineer Research and Developent Center, Vicksburg, Mississippi, 143pp. Hughes, S. A., and Nadal, N. C Laboratory study of cobined wave overtopping and stor surge overflow of a levee, Coastal Engineering, Elsevier, Vol 56. No. 3, pp Lorke, S., Brüning, A., Bornschein, A., Gilli, S., Krüger, N., Schüttrupf, H., Pohl, R., Spano, M., and Werk, S. 2010a. Influence of wind and current on wave run-up and wave overtopping, Hydralab FlowDike, Report 2009, 100pp. Lorke, S., Brüning, A., Van der Meer, J. W., Schüttrupf, H., Bornschein, A., Gilli, S., Pohl, R., Spano, M., Riha, J., Werk, S., and Schlütter, F. 2010b. On the effect of current on wave run-up and wave overtopping. Proceedings of the 32st International Conference Coastal Engineering, Aerican Society of Civil Engineers. Available at:

15 COASAL ENGINEERING Schüttrupf, H., Möller, J., and Oueraci, H Overtopping flow paraeters on the inner slope of seadikes, Proceedings of the 28th International Coastal Engineering Conference, World Scientific, Vol 2, pp Schüttrupf, H., and Oueraci, H Layer thicknesses and velocities of wave overtopping flow at seadikes, Coastal Engineering, Elsevier, Vol 52, pp Van der Meer, J. W Operating syste of the US wave overtopping siulator, Manual, Van der Meer Consulting B.W., he Netherlands, 53pp. Van der Meer, J. W., Snijders, W., and Regeling, E he wave overtopping siulator, Proceedings of the 30th International Conference Coastal Engineering, Aerican Society of Civil Engineers, Vol 5, pp Van der Meer, J. W., Steenda, G. J., de Raat, G., and Bernardini, P Further developents on the wave overtopping siulator, Proceedings of the 31st International Conference Coastal Engineering, Aerican Society of Civil Engineers, Vol 4, pp Van der Meer, J. W., and Janssen, W Wave run-up and wave overtopping at dikes, In: Kabayashi and Deirbilek (Eds.), Wave Forces on Inclined and Vertical Wall Structures, Aerican Society of Civil Engineers, pp Van der Meer, J. W., Hardean, B., Steenda, G. J., Schüttrupf, H., and Verheij, H Flow depths and velocities at crest and inner slope of a dike, in theory and with the wave overtopping siulator, Proceedings of the 32st International Conference Coastal Engineering, Aerican Society of Civil Engineers. Available at: Van Gent, M. R Wave overtopping events at dikes, Proceedings of the 28th International Coastal Engineering Conference, World Scientific, Vol 2, pp Victor, L Optiization of the hydrodynaic perforance of overtopping wave energy converters: experiental study of optial geoetry and probability distribution of overtopping volues, PhD dissertation, Departent of Civil Engineering, Ghent University, Belgiu. Victor, L., Van der Meer, J. W., and roch P Probability distribution of individual wave overtopping volues for sooth ipereable steep slopes with low crest freeboard, Coastal Engineering, Elsevier, Vol 64, pp