Estimating Irregular Wave Runup on Smooth, Impermeable Slopes

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1 Estimating Irregular Wave Runup on Smoot, Impermeable Slopes by Steven A. Huges PURPOSE: Te Coastal and Hydraulics Engineering Tecnical Note (CHETN) described erein provides new formulas for improved estimation of irregular wave runup on smoot impermeable slopes Te runup guidance is based on te recently introduced wave momentum flux parameter described in CHETN III-67 (Huges 2003) Sample calculations illustrate application of te formulas BACKGROUND: Estimates of maximum wave runup on smoot, impermeable sloping structures are necessary to determine weter overtopping will occur for a specified wave condition and water level Design formulas were originally developed based on teory and small-scale laboratory experiments using regular waves As laboratories acquired te capability to generate more realistic irregular waves, improved wave runup formulas were proposed based on wave parameters representative of te irregular wave train However, unlike regular waves tat result in a single value of maximum wave runup, irregular waves produce a runup distribution Tus, it was necessary for te runup formulas to determine a representative parameter of te wave runup distribution Presently, te most common irregular wave runup parameter is R u2% Tis parameter is defined as te vertical distance between te still-water level (swl) and te elevation exceeded by 2 percent of te runup values in te distribution In oter words, for every 100 waves running up a slope, two waves would ave a runup elevation exceeding te level estimated as R u2%. Irregular wave runup design guidance for smoot, impermeable slopes given in EM , Coastal Engineering Manual (CEM) is based on irregular wave runup experiments conducted by Arens (1981) and by de Waal and van der Meer (1992) Figure 1 reproduces Figure VI-5-3 from CEM Part VI-5 based on Arens data, and te corresponding runup formulas are reproduced as Equation 1 as follows: R H u2% mo 1.6 ξop for ξ op < 2.5 = ξ op for 2.5 < ξop (1) wit ξ op = tanα H mo Lop (2)

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3 Figure 1. Irregular wave runup guidance in EM were R u2% = vertical runup distance exceeded by 2 percent of runups H mo = zerot-moment energy-based significant wave eigt ξ op = deepwater Iribarren number based on peak period T p L op = deepwater wavelengt [=(g/2π) T 2 p ] g = gravitational acceleration T p = wave period associated wit peak spectral frequency tan α = structure slope Te scatter depicted in Figure 1 is more evident wen Arens (1981) actual data are plotted versus deepwater Iribarren number as sown in Figure 2 Data corresponding to milder slopes are clustered reasonably well for values of Iribarren number below about 3.0 At iger values of ξ op (representing steeper slopes and/or longer waves) scatter increases significantly Arens, et al. (1993) discussed reasons for te scatter and proposed modified equations to reduce te scatter sown in Figure 2 for nonbreaking wave conditions Predictive capability of Equation 1 is sown in Figure 3 were estimates of nondimensional 2 percent runup are plotted versus Arens (1981) observations tat were used to develop te guidance Variation about te solid line of equivalence indicates some lack of predictive prowess, and tus te need for improved design formulas. 2

4 Figure 2. Arens (1981) original 2 percent runup data plotted versus ξ op RUNUP EQUATION DEVELOPMENT: Huges (in preparation) presented a new nondimensional parameter representing te maximum dept-integrated wave momentum flux tat occurs in progressive water waves Te parameter, referred to as te wave momentum flux parameter, was defined as were ρg M F 2 max M F = dept-integrated wave momentum flux ρ = fluid mass density g = gravitational acceleration = water dept Because (M F ) max as units of force per unit wave crest lengt, it was argued tat maximum deptintegrated wave momentum flux would provide a good caracterization of wave processes at coastal structures 3

5 Figure 3. Comparison of Arens' (1981) data to predictions using Equation 1 Huges (in preparation) establised an empirical equation for estimating te wave momentum flux parameter for finite amplitude, nonlinear waves based on a numerical solution tecnique (Fourier approximation) Te resulting, purely empirical equation, was given as ρg M F 2 max = A 0 gt 2 A 1 (3) were A 0 = H (4) A 1 = H (5) 4

6 and H and T are te regular wave eigt and period, respectively More information and a sample calculation are given in CHETN III-67 (Huges 2003) Figure 4 depicts a simplification of wave runup geometry on a smoot impermeable slope at te point of maximum wave runup. At te instant of maximum runup te fluid witin te atced area in Figure 4 as almost no motion. Huges (in preparation) made te simple pysical argument tat te weigt of te fluid contained in te atced wedge area ABC (W (ABC) ) is proportional to te maximum dept-integrated wave momentum flux of te wave just before it reaces te toe of te structure slope, i.e., K P (M F ) max = K M W (ABC) (6) were K M is an unknown constant of proportionality, and K P is a reduction factor to account for slope porosity (K P = 1 for impermeable slopes). Te weigt of water per unit widt contained in triangle ABC sown in Figure 4 is given by W ( ABC) 2 ρg R tan α = 1 2 tanα tanθ (7) Figure 4. Maximum wave runup on a smoot impermeable plane slope 5

7 were R = maximum vertical runup from swl α = structure slope angle θ = unknown angle between swl and runup water surface (wic is assumed to be a straigt line) Substituting Equation 7 into Equation 6, rearranging, and dividing bot sides by 2 yields a new runup equation based on te dimensionless wave momentum flux parameter, i.e., R 2K p tanα M F = 2 tanα ρg KM 1 tanθ (8) For convenience te max subscript as been dropped from te wave momentum flux parameter. In te preceding runup equation, relative runup (R/) is directly proportional to te square root of te wave momentum flux parameter. (Note tat representing te runup sea surface slope as a straigt line is an approximation and may only be fully appropriate for waves on milder slopes were wave breaking occurs.) It proved convenient to treat te oter term on te rigt-and side of Equation 8 simply as an unknown constant times an unknown function of structure slope, bot to be determined empirically using laboratory data, i.e., R C F ( ) M F ρg = α 2 (9) Te success in applying Equation 9 to regular wave and breaking solitary wave runup on smoot, impermeable slopes (Huges, in preparation) prompted application of tis general form of te runup equation to te irregular wave runup data of Arens (1981). IRREGULAR WAVE RUNUP PREDICTION: Applying Equation 9 to irregular wave runup requires tat regular wave eigt and period (H and T) used to estimate te wave momentum flux parameter using Equations 3, 4, and 5 be replaced wit representative irregular wave parameters (H mo and T p ). Tis substitution does not imply tat an equivalence exists between values of wave momentum flux parameter calculated for regular and irregular waves, it only provides a convenient standard for application wit irregular waves wen establising empirical relationsips. Also note tat wen estimating te wave momentum flux parameter for regular waves, CHETN-III-67 recommends estimating te steepness-limited wave eigt to assure te specified wave condition is pysically realizable. For irregular waves, guidance is less clear for steepness-limited values of H mo, so be cautious wit estimates for irregular wave parameters tat approac te steepness-limiting condition of regular waves. For dept-limited wave eigt a good rule-of-tumb is H mo

8 Arens (1981) measurements for R u2% were normalized by water dept and plotted versus te wave momentum flux parameter as sown in Figure 5. Te data exibited two distinct trends tat seemed to be delineated by a value of local spectral steepness corresponding to H mo /L p = regardless of structure slope over te range of tested slopes. Tis steepness value appears to correspond to transition of breaker type from nonbreaking/surging waves for H mo /L p < to collapsing/plunging waves wen H mo /L p > Pysically, te data indicate tat nonbreaking/ surging waves need more wave momentum flux tan collapsing/plunging waves to acieve te same 2 percent runup level on te same slope and water dept. In oter words, collapsing/plunging waves ave more forward trust up te slope tan nonbreaking/surging waves. At te mildest slope (cot α = 4.0) te division was almost indistinguisable, and tis implies tat most of te waves in te distribution were breaking on te milder slope. Figure 5. Wave momentum flux parameter applied to Arens' (1981) original 2 percent runup data Te irregular wave runup data were separated into two groups, and data from eac group was fitted to Equation 9 to determine te unknown coefficient C and unknown function of structure slope F(α). Te resulting empirical runup equations are given as follows: for nonbreaking/surging waves (H mo /L p < ): R u2% ( 1.3cot ) M F = e α for 1.0 cot α ρg (10) 7

9 for collapsing/plunging waves (H mo /L p > ): R u2% ( 0.47cot ) M F = e α + for 1.5 cot α ρg (11) Te empirical slope functions introduce a relatively minor correction indicating slope is not too influential for te steeper slopes in te range of 0.25 < tan α < 1.0. Note tat Equation 11 is limited to slopes milder tan 1:1.5 wereas Equation 10 can accommodate slopes as steep as 1:1. Data for slope cot α = 1.01 and H mo /L p > did not follow te trend found for te oter slopes, and tus, were excluded from te empirical formulation. One possible explanation is tat tese sorter waves on te steep slope produced a runup wedge tat was not well approximated by te straigt-line water surface ypotesized in Figure 4. Figure 6 compares predictions based on Equations 10 and 11 to Arens observed 2 percent runup values. Wit te exception of data for slope cot α = 1.01 and H mo /L p > (sown by te X- symbol), te prediction is reasonable and exibits less scatter tan seen for te CEM metod sown in Figure 3. Figure 6. Comparison of Arens' (1981) data to predictions using Equations 10 and 11 8

10 Example: Irregular Wave Runup on Smoot, Impermeable Slopes Find: Te vertical runup distance from te swl wic is exceeded by only 2 percent of te waves (i.e., R u2% ) for structure slopes of 1:2 and 1:4 (tan α = 0.5 and 0.25). Given: = 20 ft Water dept at te toe of te slope T p = 9.0 s Wave period associated wit te spectral peak H mo = 8 ft Zerot-moment significant wave eigt g = 32.2 ft/s 2 Gravitational acceleration tan α = 0.5, 0.25 Structure slope Calculate Wave Momentum Flux Parameter: First calculate values of relative wave eigt and relative dept as H 8ft 20ft = = 0.4 and = = ft gt (32.2ft/s )(9s) Next, find te values of te coefficient A 0 and A 1 from Equations 4 and 5, respectively, i.e., H A0 = = (0.4) = H A1 = = (0.4) = Finally, te nondimensional wave momentum flux parameter is calculated from Equation 3 as 1 M = A = (0.0077) = 0.35 ρg F gt max A Determine Wic Runup Formula to Use: Local significant wave steepness H mo /L p is te criterion used to select te appropriate runup formula. Te linear wave dispersion relationsip is used to determine te local wave lengt L p associated wit peak spectral wave period T p and water dept at te structure toe. Tere are numerous ways to arrive at te local wavelengt of L p = 217 ft, and tis gives a local wave steepness of H L mo p 8ft = = ft 9

11 Because H mo /L p > , use runup Equation 11 Calculate Runup for 1:2 Slope: First ceck tat te structure slope falls witin te range of applicability for Equation 11, i.e., cot α = 1 = tanα = 2.0 wic is witin te range of 1.5 α cot α 4.0. Te nondimensional relative 2 percent runup is found as R u2% ( 0.47cot α) M F = e 2 ρg [ 0.47(2.0) ] { e } ( ) = = 1.44 (12) and te dimensional 2 percent runup is R u2% = 1.44 = 1.44 (20 ft) = 28.8 ft For comparison, te present CEM metod given by Equation 1 estimates runup to be R u2% = 30.2 ft. Calculate Runup for 1:4 Slope: After cecking tat te 1:4 structure slope falls witin te range of applicability for Equation 11, te nondimensional relative 2 percent runup is found as R u2% ( 0.47cot α) M F = e 2 ρg 0.47( 4.0) { e } ( ) = = 1.19 (13) and te dimensional 2 percent runup is R u2% = 1.19 = 1.19 (20 ft) = 23.8 ft Te present CEM metod given by Equation 1 estimates runup to be R u2% = 23.0 ft. 10

12 SUMMARY: Tis CHETN as described new empirical formulas for estimating te vertical runup distance above te swl tat will be exceeded by only 2 percent of te irregular wave runups on smoot, impermeable slopes. Te formulas are based on te ypotesis tat te weigt of water above swl at maximum runup is proportional to te maximum dept-integrated wave momentum flux occurring in a wave just before it reaces te toe of te impermeable plane slope. Irregular wave runup data of Arens (1981) were plotted versus te nondimensional wave momentum flux parameter, and two distinct trends were recognized corresponding to te predominant breaker type. Tese data were used to establis empirical runup formulas aving reasonable predictive capability. Structure slope as a relatively minor influence on te 2 percent runup for te range of slopes covered by te guidance. An example calculation illustrates application of te runup equations. ADDITIONAL INFORMATION: Tis CHETN is a product of te Scour at Inlet Structures Work Unit of te Coastal Inlets Researc Program (CIRP) being conducted at te U.S. Army Engineer Researc and Development Center, Coastal and Hydraulics Laboratory. Questions about tis tecnical note can be addressed to Ms. Jackie Pettway, Jackie.S.Pettway@usace.army.mil). For information about te Coastal Inlets Researc Program (CIRP), please contact te CIRP Tecnical Leader, Dr. Julie Rosati at Julie.D.Rosati@usace.army.mil. Beneficial reviews were provided by Mr. Dennis Markle and Dr. Jeff Melby, Coastal and Hydraulics Laboratory; and Mr. Jon Arens, retired Coastal and Hydraulics Laboratory and Sea Grant. Special tanks to Mr. Jon Arens for providing is original irregular wave runup data. REFERENCES Arens, J. P. (1981). Irregular wave runup on smoot slopes, CETA No , U.S. Army Corps of Engineers, Coastal Engineering Researc Center, Fort Belvoir, VA. Arens, J. P., Seelig, W. N., Ward, D. L., and Allsop, W. (1993). Wave runup on and wave reflection from coastal structures, Proceedings of Ocean Wave Measurement and Analysis (Waves 93) Conference, American Society of Civil Engineers, de Waal, J. P., and van der Meer, J. W. (1992). Wave run-up and overtopping on coastal structures, Proceedings of te 23rd International Coastal Engineering Conference, American Society of Civil Engineers, 2, Headquarters, U.S. Army Corps of Engineers. (2003). Coastal Engineering Manual, EM , Wasington, DC. Huges, S. A. ( 2003). Wave momentum flux parameter for coastal structure design, ERDC/CHL CHETN-III-67, U.S. Army Engineer Researc and Development Center, Vicksburg, MS. Huges, S. A. (in preparation). Wave momentum flux parameter for coastal structure design, Coastal Engineering, Elsevier, Te Neterlands. NOTE: Te contents of tis tecnical note are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of te use of suc products. 11