Estimation of Combined Wave and Storm Surge Overtopping at Earthen Levees

Transcription

1 May 008 Estiation of Cobined Wave and Stor Surge Overtopping at Earthen Levees by Steven A. Hughes PURPOSE: This Coastal and Hydraulics Engineering Technical Note (CHETN) provides epirical equations for estiating several paraeters of unsteady flow resulting fro the cobination of steady stor surge overflow and overtopping of irregular waves at a trapezoidal-shaped earthen levee. Equations are given for the average overtopping discharge and the cuulative probability distribution of instantaneous overtopping discharge. On the landward-side slope, epirical equations can be used to estiate the ean flow depth, ean flow velocity, root-ean-square wave height, and velocity associated with the overtopping wave front. Worked exaples illustrate application of the epirical equations. INTRODUCTION AND BACKGROUND: Earthen levees are used extensively in the United States to protect populations and infrastructure fro periodic floods and high water due to stor surges. Overtopping of levees and dikes produces fast-flowing, turbulent water velocities on the landward-side slope that can daage the protective grass covering and expose the underlying soil to erosion. If overtopping continues long enough, the erosion ay eventually result in loss of levee crest elevation and perhaps breaching of the protective structure. Econoics often dictate levee designs with crown elevations having a risk that soe wave/surge overtopping will occur during extree events. In addition, crown elevations for older levee systes ay have been established without coplete inforation about possible water levels that ight occur during extree events. Even levees that presently have sufficient freeboard to withstand all but the ost extree stors ay becoe vulnerable to wave overtopping and stor surge overflow in the future if sea level continues to rise at projected rates. The cost and engineering challenges of raising all levees to elevations where overtopping will be within tolerable liits ay be insurountable. Levees that cannot be raised reain vulnerable, and the landward-side levee slopes ust be protected with soe type of strengthening alternative such as turf reinforceent, soil strengthening, or hard aroring. Assessent of levee reliability and design of slope protection alternatives requires estiates of wave overtopping and stor surge overflow that will occur for a specified set of stor paraeters. This CHETN provides overtopping estiation equations for the case of steady overflow due to a still water level above the levee crown elevation cobined with irregular wave overtopping. The overtopping flow is unsteady in tie and spatially nonunifor, as illustrated in Figure 1. As each overtopping wave passes over the levee crown, it has a soewhat triangular-shaped discharge distribution with a axiu discharge at the leading edge that is several ties greater than the tieaveraged discharge. If the stor surge elevation above the levee crown is sall and/or the waves are large, the levee crown and landward-side slope will go dry during wave troughs. This fluctuating discharge ay contribute to accelerated erosion of the landward-side levee slope.

2 Report Docuentation Page For Approved OMB No Public reporting burden for the collection of inforation is estiated to average 1 hour per response, including the tie for reviewing instructions, searching existing data sources, gathering and aintaining the data needed, and copleting and reviewing the collection of inforation. Send coents regarding this burden estiate or any other aspect of this collection of inforation, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Inforation Operations and Reports, 115 Jefferson Davis Highway, Suite 104, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to coply with a collection of inforation if it does not display a currently valid OMB control nuber. 1. REPORT DATE MAY 008. REPORT TYPE 3. DATES COVERED to TITLE AND SUBTITLE Estiation of Cobined Wave and Stor Surge Overtopping at Earthen Levees 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) U.S. Ary Engineer Research and Developent Center,Coastal and Hydraulics Laboratory,3909 Halls Ferry Road,Vicksburg,MS, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR S ACRONYM(S) 1. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unliited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 11. SPONSOR/MONITOR S REPORT NUMBER(S) 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Sae as Report (SAR) 18. NUMBER OF PAGES 14 19a. NAME OF RESPONSIBLE PERSON Standard For 98 (Rev. 8-98) Prescribed by ANSI Std Z39-18

3 May 008 Figure 1. Sequence of wave overtopping on a scale-odel levee. LABORATORY EXPERIMENTS: Laboratory experients of wave overtopping cobined with steady surge overflow were conducted at a noinal prototype-to-odel length scale of 5-to-1 in a 3-ft-wide wave flue at the U.S. Ary Engineer Research and Developent Center s Coastal and Hydraulics Laboratory. The tested levee cross section replicated in the physical odel is shown in Figure. Seaward is on the left side of the figure, and the diensions correspond to full-scale units. This cross section was typical of the Mississippi River Gulf Outlet that experienced severe overtopping during Hurricane Katrina. Figure. Levee cross section tested in physical odel (full-scale diensions). Prototype-scale target wave and surge paraeters for testing were three surge elevations (S = +1.0, +3.0, and +5.0 ft above levee crest), three significant wave heights (H 0 = 3.0, 6.0, and 9.0 ft), and three peak wave periods (T p = 6, 10, and 14 sec). This gave a total of 7 unique conditions for cobined wave and surge overtopping. Instantaneous discharge over the levee crest was estiated at a location near the landward-side edge of the levee crest. Coincident values of water depth (easured with a pressure gauge) and horizontal velocity (easured with a laser Doppler velocieter) were ultiplied at each instant in tie to deterine the tie series of instantaneous discharge. This estiate assued that velocity was horizontal and constant throughout the water colun at that location. Incident wave characteristics were easured at a three-gauge array located seaward of the levee. For overtopping, the freeboard is defined as the difference between the levee crest elevation, h c, and the still-water elevation, h S, i.e., R c = (h c - h S ). When the water elevation is higher than the levee crest, R c < 0, which is referred to as negative freeboard. AVERAGE DISCHARGE FOR COMBINED WAVE AND SURGE OVERTOPPING: For each of the 7 experients, an average was calculated over the portion of the instantaneous discharge tie series that included both wave overtopping and steady overflow. This average represents the average overtopping discharge for that particular cobination of negative freeboard

4 May 008 and incident irregular waves. It is custoary in overtopping research to express the average wave overtopping discharge in ters of a diensionless paraeter defined as where q Q = (1) g 3 H 0 Q = diensionless average discharge due to cobined waves and surge q = average discharge due to cobined waves and surge per unit crest length g = acceleration of gravity H 0 = energy-based significant wave height Figure 3 plots the diensionless cobined wave/surge average overtopping discharge versus the relative (negative) freeboard for all 7 experients. The indicated surge levels in the plot legend are the average of the negative freeboards deterined for all nine experients at each noinal surge level. Figure 3. Diensionless cobined wave/surge average discharge versus relative freeboard. As shown in Figure 3, the easureents gave a nice trend with increasing relative freeboard, and the solid line is a best-fit epirical equation given by the forula 3

5 May 008 q R Q = = + () g H c H 0 This best-fit equation had a correlation coefficient of and a root-ean-square (RMS) percent error of 0.1. (Note that R c ust be specified as a negative nuber so the ratio in brackets will be positive.) Peak spectral wave period had negligible influence in the deterination of Q for the range of periods tested in the odel. Like any epirical equation, application of Equation should be liited to the range of tested paraeters. In particular, seaward (flood-side) levee slopes different fro 1:4.5 could influence the wave overtopping, but seaward slope effects should decrease as surge level increases. For zero freeboard (R c = 0), Equation yields a constant diensionless average wave overtopping rate that falls near the iddle of the range of experient results given by Schüttrupf et al. (001). This is encouraging; however, estiates for zero freeboard should be ade using Schüttrupf s equations as presented in the Overtopping Manual (Pullen et al. 007) rather than Equation. PROBABILITY DISTRIBUTION OF INSTANTANEOUS DISCHARGE: During cobined wave and surge overtopping, each wave will have instantaneous peak discharge that can be several ties the average overtopping rate, q. These peak discharges, while short in duration, ay be responsible for uch of the erosion of levee soil or instability/failure of aroring alternatives. Thus, prediction of the instantaneous overtopping distribution is needed for ore effective design of levee slope protection alternatives. The tie series easureents of overtopping flow discharge acquired near the landward-side edge of the levee crest were analyzed to deterine the cuulative distribution of instantaneous discharge over the levee crest. The analysis revealed that still-water elevation above the levee crest (i.e., the steady overflow discharge) is the ost iportant hydrodynaic paraeter, and it ost closely controls the scale of the cuulative probability distribution. Wave height and period appear to influence the shape of the distribution extree tail. The easured cuulative discharge distributions were fit to the Weibull cuulative probability distribution given by the expression b q ( q q = *) 1 exp (3) c P * where c is the scale factor and b is the shape factor of the distribution. The scale factor, c, has units of discharge per unit length, whereas the shape factor, b, is diensionless. Equation 3 gives the probability that a value of discharge will be below the specified discharge, q *. The corresponding distribution of percent exceedance is given by q b % ( q > q* ) = 100 exp (4) c P *

6 May 008 The best-fit procedure produced values of the distribution paraeters, b and c. Figure 4 sho an exaple of a good fit (upper plot) and a ediocre fit (lower plot) of the Weibull cuulative distribution to the easured distributions (given as percent exceedance distributions). The curves are plotted with a sei-log ordinate to better present the extree end of the exceedance probability. Figure 4. Exaple best fits of Weibull cuulative probability distribution to instantaneous discharge. The curves in Figure 4 give the percent occurrence that the overtopping discharge will be above a given value. For exaple, in the upper plot of Figure 4, for 10 percent of the tie the discharge will be above q 3 ft 3 /sec per foot, and the discharge will exceed q 36 ft 3 /sec per foot for 1 percent of the tie. The average wave/surge overtopping for this experient was q = 13.1 ft 3 /sec per foot. 5

7 May 008 The diensionless shape factor, b, exerts control over the extree tail of the distribution, and the tail is sensitive to sall differences in the shape factor. Through inspection, and after several attepts using diensionless cobinations of the wave height, wave period, and steady surge discharge, the best epirical result for the shape factor was the relationship where q s is estiated fro the broad-crest weir forulation as 8.10 qs b = (5) g H 0 Tp 3 / 3 / 3 / q s = g Rc = g Rc (6) 3 Figure 5 sho the best-fit curve for the shape factor given by Equation 5 along with the plotted values of b obtained fro the easured data. The correlation coefficient for the best fit was 0.9. Figure 5. Best-fit equation for Weibull paraeter b. The three data points in Figure 5 with values of b greater than 3 deviated significantly fro the data trend. These points cae fro experients with the surge level at approxiately +4.3 ft and noinal wave heights around 3 ft (scaled fro odel to prototype). A possible explanation for the observed deviation ight be the effect of wave deforation by the strong steady overflow as waves reach the levee. The nonlinear interaction between the waves and the coincident current at the levee 6

8 May 008 crest is ore pronounced at the higher surge level, and saller waves would be affected ore than larger waves. The resulting saller wave heights would reduce the extree values of overtopping discharge, giving higher values for the shape factor, b. The ean of the Weibull cuulative distribution is equal to the average overtopping discharge, q, and it can also be expressed in ters of the b and c paraeters as (e.g., Goda 000) 1 q = c Γ 1 + (7) b where Γ is the atheatical gaa function. Equation 7 was used to estiate q using the best-fit values of the Weibull b and c paraeters for all experients, and coparison with values of q deterined directly fro the discharge tie series showed excellent agreeent. Therefore, Equation 7 can be rearranged to give an expression for the scale factor, c, i.e., q c = 1 Γ 1 + b (8) The gaa function can be accurately calculated over the range of shape factor, b, found for these experients (0.5 < b < 4) by the best fit of a cubic equation given by 3 Γ ( x) = x 0.07 x x (9) which is valid for 1.5 < x < 3. This approxiation can be used for calculations when the gaa function is not available. In suary, the cuulative probability distribution of instantaneous discharge over the levee crest due to cobined wave and surge overtopping can be estiated using Equation 3 or 4 with the Weibull shape factor, b, deterined fro Equation 5, and the Weibull scale factor, c, estiated using Equation 8 as a function of q (fro Equation ). For an application in which calculation of the gaa function is not available, use Equation 9 within the stated range of applicability. FLOW PARAMETERS ON THE LANDWARD-SIDE LEVEE SLOPE: Steady overflow on the steep landward-side slope of a levee is supercritical with slope-parallel velocities increasing down the slope until a balance is reached between the oentu of the flow and the frictional resistance force of the slope surface. Once terinal velocity is reached (ean velocity is constant), the steady flow can be analyzed using the Chezy or Manning equation. Flow down the landward-side slope caused by cobined waves and surge overtopping is unsteady and ore difficult to analyze thoroughly. In this section epirical equations are given to characterize several representative paraeters of wave-related unsteady flow on the landward-side levee slope. Average Flow Depth on Landward-Side Slope. The average flow depth, η, perpendicular to the slope was calculated at four pressure gauge locations that were equally spaced down the landward-side slope. Coparison aong the four gauges indicated the values of η were nearly constant for each experient, so an average was taken to represent the ean flow thickness on the 7

9 May 008 landward-side slope. A correlation was sought between η and the hydrodynaic forcing, and a reasonable result is shown in Figure 6. Values have been scaled to prototype diensions using a length scale ratio of 5-to-1. The solid line is a linear best-fit equation given by the siple epirical expression q =.6 3 gη (10) This equation had a correlation coefficient of 0.988, and an RMS percent error of Figure 6. Average overtopping discharge versus ean flow thickness on landward-side slope (prototype scale). If the ean velocity on the landward-side slope is defined as v = q /η, then v =.6 gη (11) which bears reseblance to the Chezy equation. The Chezy equation for wide channels (hydraulic radius is approxiately equal to flow depth, d ) and steady flow (friction slope is equal to landward-side levee slope, θ ) is given as 8

10 May 008 v = sinθ f F g d (1) for steep slopes. In Equation 1 the Chezy coefficient was replaced with a function of the Fanning friction factor given by the expression C Z g = (13) f F Under the assuption the Chezy equation is an appropriate odel to approxiate the average of this rapidly varying unsteady flow situation, it is hypothesized that the constant in Equations 10 and 11 is a function of both levee slope and a representative friction factor, as given by the first radical ter in Equation 1. The Chezy coefficient C Z varies with flow thickness on the slope; therefore, the friction factor, f F, also varies continuously during the unsteady overtopping flow. A representative friction factor for these experients was found by assuing that the constant in Equation 11 is equal to the first radical ter in Equation 1. This yielded a value of f F = However, at this point there is no ethod for estiating siilar representative friction factors for other slope roughness, so little benefit is gained by including friction factor at this tie. It is assued, however, that the representation of structure slope in Equation 1 is reasonable. For the 1-on-3 slope in these experients, the constant in Equations 10 and 11 was deterined to be.6 = 3.96 sin θ. Substituting for the constant in Equation 10 and rearranging gives a tentative equation for the ean flow depth, i.e., 9 1/ 3 and the ean flow velocity equation becoes the following 1 η 0.4 ( ) / 3 = q g sin (14) θ ( q sin ) 1/ 3 v =.5 g θ (15) The constants in Equations 14 and 15 are related to the slope roughness, and the equations are applicable only for landward-side slopes having roughness siilar to that of the laboratory experients. Friction factors for grass levee slopes should not be too uch higher than those of the experients, but slopes arored with riprap or siilar aterial will have significantly higher representative friction factors. More work is needed to deterine appropriate representative friction factors for a range of slope roughness. Estiation of H rs on the Landward-Side Slope. Values of root-ean-square wave height, H rs, perpendicular to the slope at the four pressure gauge locations on the landward-side slope were averaged based on the observation that little variation was shown over this down-slope distance. A relationship was sought that expressed H rs in ters of other paraeters that could be estiated. Figure 7 sho the best correlation of those attepted. Wave period was found to have only

11 May 008 arginal influence, and soe of the observed scatter is possibly explained by wave period variation. The solid curve represents the best fit of a one-paraeter exponential function given as H η rs R = 3.43 exp H c 0 (16) This best-fit equation had a correlation coefficient of and an RMS percent error of When applying Equation 16, keep in ind that R c ust be entered as a negative nuber, and the equation should not be applied for cases in which R c 0 (surge level is at or below the crest elevation). Figure 7. Estiation of H rs on the landward-side slope as a function of η. The effect of levee landward-side slope is included in Equation 16 through the estiation of the ean flow depth, η, using Equation 14. However, this is still tentative and ay not be correct, so caution is urged until additional verification of Equation 16 is forthcoing. Also, note that H rs /η is less than unity when R c /H 0 is less than -1.. In these cases, the landward-side slope does not go dry during passage of the wave trough, and the ean flow depth is greater than the crest-to-trough length represented by H rs. 10

12 May 008 Once H rs is estiated, the Rayleigh distribution can be used to obtain reasonable estiates for H 1/3 and H 1/10 according to the equations H 1 / 3 = H rs and H1/ 10 = H rs (17) Coparisons between easured characteristic wave heights and those predicted by the Rayleigh distribution using easured values of H rs were perfored for all laboratory data. An excellent prediction was found for H 1/3, and the prediction for H 1/10 was quite good with a little ore scatter and a tendency to overpredict slightly. However, the Rayleigh distribution overpredicted the easured values of H 1/100 by as uch as 5 to 40 percent, indicating that the highest waves do not confor well to the Rayleigh distribution. Nevertheless, the reasonable fit of the Rayleigh distribution is a useful finding because it allo uch of the wave height distribution on the landward-side slope to be characterized in ters of the RMS wave height, H rs. Estiation of Wave Front Velocity on the Landward-Side Slope. Flow velocity was not easured directly on the landward-side slope during the laboratory experients. However, it was possible to ake a rough estiate of the speed of the wave front by dividing the distance between the first and last pressure gauges by the tie it took the wave front to ove over that distance. This approxiate wave front velocity represents an average over the distance between the four pressure gauges. In fact, the flow ight have been accelerating over this reach. The accuracy of wave front velocity estiates using this technique is liited by the data sapling rate of 50 Hz used in the experients. One data point shift either way is a relatively large change in estiated velocity. For this reason, the wave front velocity analysis was restricted to the first and last gauges to axiize the distance between gauges, and thus, to reduce the potential for error. Figure 8 plots the wave front velocity versus the velocity of a bore having a frontal depth equal to H rs. Values of H rs were the averages taken over the four pressure gauges. The discrete jups in velocity on the plot ordinate represent tie shifts of four, five, six, and seven tie steps (highest to lowest velocity). The actual wave front velocities of soe points are probably soewhere between the levels, and this could reduce the scatter soewhat. The straight line shown in Figure 8 is the best fit given by the equation v = g (18) w H rs where v w is the wave front velocity. This best fit had a correlation coefficient of and an RMS percent error of The coefficient in Equation 18 is constant for this particular data set; however, it ay be a function of slope angle and surface roughness even though Equation 16 for estiating H rs explicitly includes slope and iplicitly includes friction effects for sooth slopes. 11

13 May 008 Figure 8. Estiated wave front velocity deterined fro the first and last pressures gauges. Exaple: Cobined Wave/Surge Overtopping Flow Paraeters Find: The flow paraeters associated with wave overtopping cobined with overflow by a surge elevation that is 1.3 ft above the levee crest elevation. The levee surface is considered sooth with roughness coparable to grass or turf reinforceent ats. Given: H 0 = 6 ft = zeroth-oent significant wave height T p = 8 sec = wave period associated with the spectral peak h c = 17.0 ft = levee crest elevation h S = 18.3 ft = stor surge elevation R c = -1.3 ft = freeboard [= h c - h S ] tan α = 1/4 = seaward-side levee slope tan θ = 1/3 = landward-side levee slope [θ = 18.4 deg] g = 3. ft/sec = acceleration due to gravity 1

14 May 008 The seaward-side levee slope is not used in these calculations, but it is wise to ake sure the slope is not too different fro the 1-on-4.5 seaward-side slope used in the physical odel study on which these equations are based. Calculate Average Overtopping Discharge. The diensionless average overtopping discharge is found using Equation. Q Q R c = H ( 1.3ft) = ft = And, the diensional overtopping is calculated by rearranging Equation 1 to yield 1.58 q = g H 3 0 Q = 3 (3.ft/sec )(6ft ) q = 6.75ft /sec (Note the discharge units square feet per sec are equivalent to cubic feet per sec per foot, or cfs/ft.) Calculate Instantaneous Overtopping Discharge Cuulative Distribution. The Weibull percent exceedance cuulative probability distribution is given by Equation 4. The first step in deterining the distribution paraeters is estiating the steady overflow discharge (Equation 6) due to the surge elevation exceeding the crest elevation. q s = g R c 3 / = ft/sec 1.3ft 3 / q s = 4.58ft /sec (Note that q /q s = 1.47.) Now, calculate the Weibull distribution shape factor, b, using Equation q b = g H s 0 T p ft /sec = 8.10 (3.ft/sec )(6ft)(8sec) = 1.1 The Weibull scale factor, c, is given by Equation 8. For convenience, evaluate the gaa function in Equation 8 using Equation 9 with x = 1 + 1/b = 1.91, i.e., Γ( x) = x 0.07 x Γ(1.91) = (1.91) 0.07 (1.91) Γ(1.91) = x (1.91) 3 13

15 May 008 Substituting for the gaa function in Equation 8 yields q 6.75ft /sec 6.75ft /sec = = = = 7.0ft Γ 1 + Γ 1 + b 1.1 c The percent exceedance cuulative probability distribution for this exaple becoes P q* q > q* ) = 100 exp 7.0ft % ( 1.1 /sec As an exaple, the percentage of instantaneous overtopping discharge exceeding q * = 15 cfc/ft is given by P ft /sec 15ft /sec) 100 exp q > = 7.0 ft /sec % ( = /sec 10.0% In other words, 10 percent of the instantaneous discharge values are greater than. ties the average overtopping discharge. Calculate Flow Paraeters on the Landward-Side Levee Slope. Under the assuption that the levee surface roughness is sall (siilar to the sooth slope used in the laboratory tests), the ean flow thickness perpendicular to the landward-side slope is estiated fro Equation 14 as 1/ 3 1 η = 0.4 sin g θ q η = 0.4 ( ) / 3 1 = 0.4 ( 6.75ft /sec) /3 1/3 4/3 /3 ( 0.46s /ft )( 3.57ft /s ) = 0.66ft 1/ 3 o (3.ft/sec )sin (18.4 ) The ean flow velocity on the landward-side slope can be estiated using Equation 15, but an easier ethod is using the definition / 3 v q = η 6.75ft /sec = = 10.ft/sec 0.66ft The RMS wave height (perpendicular to the levee slope) is estiated fro Equation 16 as H H rs rs R = η 3.43 exp H = 1.8ft c 0 1.3ft = (0.66ft) 3.43 exp 6ft 14

16 May 008 Because the wave heights on the landward-side slope appear to follow the Rayleigh distribution except at the extree tail, other representative wave heights can be estiated fro Equation 17, i.e., H 1 / 3 = H rs = (1.8ft) =.6ft H 1 /10 = 1.80 H rs = 1.80 (1.8ft) = 3.ft Finally, a rough estiate of the velocity of the overtopping wave front on the landward-side slope is given by Equation 18 as v v w w = 3.85 g H = 9.3ft/sec rs = 3.85 (3.ft/sec )(1.8ft) The velocity of the wave front is alost 3 ties greater than the ean velocity on the landward-side slope. Most iportantly, the wave front velocity estiate should be considered preliinary, and the argin for error is significant due to the ethod by which the equation was developed. Flow Paraeter Variation for Sae Average Overtopping Discharge. Different cobinations of incident wave height and freeboard can produce the sae value of average overtopping discharge, q. Table 1 sho overtopping flow paraeters calculated for three additional cases that produced the sae average discharge (q = 6.75 ft /sec) deterined above. Table 1 Additional Paraeter Estiates for the Sae Value of q H 0 (ft) R c (ft) q (ft /sec) q s (ft /sec) b c (ft /sec) η (ft) v (ft/sec) H rs (ft) v w (ft/sec) REMARKS: All four of the cases given in the table above have different cobinations of incident significant wave height and levee freeboard, although the average overtopping discharge, q, was the sae. The ean flow paraeters on the landward-side slope (η and v ) were also identical. However, the paraeters of the Weibull distribution of instantaneous discharge (b and c), the RMS wave height (H rs ), and the velocity of the wave front (v w ) were quite different aong the cases. The difference of wave height and wave front velocity on the landward-side slope suggests that characterizing levee safety in ters of an upper liit for average overtopping discharge ay not be prudent unless it can be shown that levee erosion and aror stability is ore a function of the ean flow characteristics than a function of the variations in the unsteady overtopping flow. SUMMARY: This CHETN suarized new equations for average overtopping discharge and distribution of instantaneous discharge associated with cobined wave and stor surge overtopping of levees. Equations are also given for ean flow thickness, RMS wave height, ean velocity, and 15

17 May 008 velocity of the wave front down the landward-side slope. An exaple proble illustrates application of the new equations. These equations are intended for use in preliinary design. The epirical equations are based on sall-scale laboratory experients featuring a levee with a seaward-side slope of 1:4.5, and the equations ay not be applicable for levees having different seaward-side slopes. It is hypothesized that seaward-side slope ay not be as iportant for cobined wave and surge overtopping as it is for wave-only overtopping. Additionally, several assuptions were invoked during equation developent. First, it was assued that unsteady overtopping flow velocities over the levee crest are horizontal and unifor throughout the depth at each instant in tie. Second, the equations for the landward-side slope include slope angle and a constant that is likely a function of soe representative friction factor characterizing slope roughness. Applicability of these equations for landward-side slopes different fro 1-on-3 is uncertain, and the equations will give incorrect estiates where slope surface roughness is not relatively sooth. Finally, the flo down the landward-side slope in the sall-scale experients had little if any air entrainent. This is certainly not the case for siilar flo at full scale. How this aeration scale effect alters the paraeters estiated by the epirical equations is not yet known. ADDITIONAL INFORMATION: This CHETN is a product of the Affordable Levee Strengthening and New Design Work Unit being conducted at the U.S. Ary Engineer Research and Developent Center, Coastal and Hydraulics Laboratory, and sponsored by the Departent of Hoeland Security Levee Strengthening and Daage Mitigation Progra. Questions about this technical note can be addressed to Dr. Steven A. Hughes (Voice: , Fax: , eail: Steven.A.Hughes@usace.ary.il). Beneficial revie were provided by Norberto C. Nadal and Willia C. Seabergh. This docuent should be cited as: Hughes, S. A Estiation of cobined wave and stor surge overtopping at earthen levees. Coastal and Hydraulics Engineering Technical Note ERDC/CHL CHETN-III-78. Vicksburg, MS: U.S. Ary Engineer Research and Developent Center. REFERENCES Goda, Y Rando seas and design of aritie structures. d ed. Singapore: World Scientific Publishing. Pullen, T., N.W.H. Allsop, T. Bruce, A. Kortenhaus, H. Schüttrupf, and J. W. van der Meer EurOtop wave overtopping of sea defences and related structures: Assessent anual. Schüttrupf, H., J. Möller, H. Oueraci, J. Grüne, and R. Weissann Effects of natural sea states on wave overtopping of seadikes. Proceedings of the 4th International Syposiu Waves 001, Ocean Wave Measureent and Analysis. Aerican Society of Civil Engineers, : NOTE: The contents of this technical note are not to be used for advertising, publication, or prootional purposes. Citation of trade naes does not constitute an official endorseent or approval of the use of such products. 16