EMPIRICAL, SEMI-EMPIRICAL & NUMERICAL OVERTOPPING MODELS: A COMPARISON

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1 EMPIRICAL, SEMI-EMPIRICAL & NUMERICAL OVERTOPPING MODELS: A COMPARISON Maria Terea Rei 1, Keming Hu 2 and Terry S. Hedge 3 Abtrat: The aer omare the outut from four method ued to etimate the overtoing rate at eawall ubjet to wave ation: two wholly emirial model, a emi-emirial model and a numerial model. Data from hyial model tet are alo inluded. The wholly emirial model of Owen (1980) and of van der Meer and Janen (1995) were develoed olely through a roe of fitting dimenionle grou to data derived from hyial model tet. The Hedge and Rei (1998, 2004) emi-emirial model wa derived from onideration of the unteady flow of water over a weir (Kikkawa et al. 1968). Like the wholly emirial model, it wa alibrated uing the reult of hyial model tet. The numerial model AMAZON (Hu 2000) i a high-reolution two-dimenional finite volume model and it i baed on olving the non-linear hallow-water equation. The tudy overed a range of eawall front loe from 1:1 to 1:20, inident wave teenee from 0.01 to The reult how general agreement between the outut from the numerial and emi-emirial model and the data. Agreement with the wholly emirial model deend rinially on the value of the urf imilarity arameter, with thee model ubtantially overrediting diharge ome ondition. INTRODUCTION At reent, the mot-widely ued tool rediting wave overtoing of eawall are emirial (or emi-emirial) mulae baed on hydrauli model tet (eley 1999). However, diret aliation of thee mulae i limited to imle trutural onfiguration. Even imle truture, there are ga in data, artiularly at low overtoing diharge, the level whih many eawall are deigned. Numerial model of wave 1 Dr, National Civil Engineering Laboratory, Harbour and Maritime Struture Diviion, Av. do rail 101, Libon, Portugal, trei@lne.t 2 Dr, lak & Veath Euroe, Grovenor Houe, 69 London Road, Redhill, Surrey RH1 1LQ, United Kingdom, huk@bv.om 3 Eur Ing, Univerity of Liverool, Deartment of Civil Engineering, rownlow Street, Liverool L69 3GQ, United Kingdom, e22@liverool.a.uk 1 Rei, Hu and Hedge

2 overtoing may be le retrited, ine a validated model an be onfigured any truture within it overall range of aliability. Although imreive advane in the volume of fluid (VOF) and urfae aturing numerial heme have been reorted, ue of uh model in ratial engineering aliation i till very limited, due to their oniderable demand on omuter ower (Hu and Meyer 2005). Numerial model baed on the non-linear hallow-water equation, deite their limitation, have beome inreaingly attrative heme deign and flood eating, ine wave train of 1000 random wave (or more) are imulated raidly. The aer deribe four method ued to etimate the mean wave overtoing diharge at eawall: the emirial model of Owen (1980) and of van der Meer and Janen (1995), the emi-emirial model of Hedge and Rei (1998, 2004) and the numerial model AMAZON (Hu 2000) baed on the non-linear hallow-water equation. One motivation the tudy i that the non-linear hallow-water equation are theoretially mot valid ondition in whih the water deth and wave height are mall omared with the wavelength. In ontrat, emirial (and emi-emirial) overtoing model have uually been alibrated uing laboratory data rereenting torm ondition in whih the inident wave are tee and are eified in either dee or intermediate water deth. Conequently, it i ueful to etablih whether the model rodue imilar reult at the boundary between their uoed region of aliability. Outut from the different method i onidered alongide four et of data from hyial model tet: Owen (1980) data, the Hawke et al. (1998) data, data from Kanai Univerity (Mae et al. 2003) and the SHADOW data (ay et al. 2004). EMPIRICAL MODELS Introdution Variou overtoing model exit rediting the mean diharge when random wave overto eawall. Here, the mulae of Owen (1980) and of van der Meer and Janen (1995) are onidered. Thee model are both of an exonential m: Q* = A EXP( R*) (1) Q* = dimenionle mean overtoing diharge; R* = dimenionle freeboard; and A and = emirial oeffiient (non-dimenional). Note the inability of Eq. (1) to redit a zero overtoing rate, regardle of the ize of the dimenionle freeboard. Note alo that Q* and R* may be written in variou m (Hedge and Rei 1998). Owen' Model Owen model overtoing of a unimly loing eawall by erendiular wave attak may be exreed a: Q T gh m R A EXP γ rtm gh = (2) where Q = mean overtoing diharge (m 3 //m); g = aeleration due to gravity (m/ 2 ); H = ignifiant wave height at the toe of a ehore loing unimly eaward at 1:20 to 2 Rei, Hu and Hedge

3 a deth unaffeted by wave-breaking (m); T m = mean zero-roing wave eriod (); R = eawall freeboard (m); and γ r = redution fator aounting loe roughne (γ r 1, non-dimenional). Overtoing diharge were evaluated run of 100 wave. Note that, fixed value of R and H, the above equation ugget that the overtoing diharge alway inreae with an inreae in T m. However, thi i not the ae and Owen relationhi tend to overetimate overtoing rate well ondition (ay et al. 2004). Owen ued model eawall with mooth, unim front loe of 1:1, 1:2 and 1:4 to evaluate oeffiient A and. Aording to Owen (1980), oeffiient A and are valid the following range of ondition: 10 6 Q < T gh m < < 10 H L om 2 ; 0.05 < γ T < 0.055; 1.5 < r m d H R gh < 5.5 < 0.30 (3) 2 in whih Lom = gtm / 2π (m); and d = water deth at the toe of the eawall (m). Owen alo uggeted that it i oible to ue hi equation to extraolate reult when the dimenionle freeboard wa uh that the dimenionle diharge fell below aed on further reearh, eley (1999) revied Owen original value of A and and, following omletion of the SHADOW tet (ay et al. 2004), Allo and Pullen (2003) rovided further guidane, extending aliation of the model to hallower loe (Table 1). They alo extended the ue of the model from an uer value of the dimenionle freeboard of 0.30 to Table 1. Coeffiient Ue in Owen' Model Seawall Sloe Emirial Coeffiient Owen (1980) eley (1999) Allo and Pullen (2003) 1:1 A * :2 A :3 A :4 A :5 A :6 A :8 A :10 A :15 A * The number in bold are the one adoted later in thi tudy. 3 Rei, Hu and Hedge

4 Van der Meer and Janen Model A method etimating torm wave run-u and overtoing on ea dike wa develoed the Netherland by van der Meer and Janen (1995). aed on extenive laboratory teting, it onit of two overtoing mulae: one breaking wave and the other non-breaking wave. Four redution fator inororated in the mulae aount the influene of a berm, of a hallow ehore (deth-limited wave), of loe roughne and of obliquely inoming wave (both hort-reted and long-reted). The model ha two ontraint: one related to the aetable range of aliation of the breaking wave relationhi (the dimenionle freeboard, inluding the four redution fator, hould lie between 0.3 and 2) and the other related to the rodut of the four redution fator (whih hould be limited to a minimum value of 0.5). For a unimly loing eawall ubjet to erendiular wave attak and negleting the influene of a hallow ehore, the average of all of van der Meer and Janen obervation may be exreed a: Q gh Q gh ξ = EXP 5.2 tanα γ r R = 0.2 EXP 2.6 γ r H R ξ H ξ < 2 ξ > 2 (4) where H = ignifiant wave height near the toe of the eawall front loe (m); ξ = tanα / H / L i the urf imilarity (or breaker) arameter (non-dimenional); L o = gt 2 o / 2π (m); T = wave eriod orreonding to the eak of the inident wave etrum (); and α = angle of the eawall front loe meaured from the horizontal. THE HEDGES AND REIS (H&R) SEMI-EMPIRICAL MODEL The Hedge and Rei (1998) emi-emirial model i baed on an overtoing theory regular wave develoed by Kikkawa et al. (1968). Thi theory aume that the eawall at a a weir whenever the inident water level exeed the eawall ret level, and that the intantaneou diharge may be deribed by the weir mula. The Hedge and Rei (H&R) model extend the onet to random wave and may be written a follow: Q = A Q = 0 gr 3 R 1 γ rr 0 R γ rr R γ rr < 1 1 (5) where R = imum run-u on a mooth loe indued by the random wave during a torm (m). The reie value of R during any artiular torm annot be known a riori, and an etimate of it value ha to be made. Unle R > R /γ r, there i no overtoing aart from wind-blown ray. 4 Rei, Hu and Hedge

5 Like the wholly emirial model, oeffiient A and have been alibrated uing the reult of hyial model tet. Originally, Hedge and Rei (1998) ued Owen' data that uroe, in whih overtoing had been evaluated run of 100 wave loe of 1:1, 1:2 and 1:4. In thi ae, the mot robable imum run-u during eah run, i.e. the value not exeeded in 37% of the ae auming a Rayleigh ditribution of run-u wa: ( R ) = 1.52R (6) 37%,100 where R = ignifiant wave run-u (m), whih wa evaluated uing the equation given in the CIRIA/CUR (1991) manual. More reently, Mae et al. (2003) modified and extended the Hedge and Rei (1998) model to aount Jaanee data on run-u and overtoing. The new data (referred to later a the Kanai data) overed front loe a hallow a 1:20. Intead of uing the exreion given in the CIRIA/CUR (1991) manual to etimate R, Mae et al. (2003) ued a modified verion of Hunt equation, inororating wave et-u: R / H = ξ 0 < ξ 2.2 = ξ 2.2 < ξ 9.0 (7) = < ξ Combining Eq. (6) and (7) then give: ( R ) 37%,100 / H ξ 0 < ξ ξ 2.2 < ξ 9.0 (8) < ξ Furthermore, if the Rayleigh ditribution alie to run-u, then: ( R ) = R (9) 37%,100 1% in whih R n% = value exeeded by n% of all the individual run-u (non-dimenional). A value uh a (R ) 37%, 100 imly rovide an etimate of the atual imum run-u during a torm (i.e. an etimate of the minimum freeboard needed zero overtoing). However, thi etimate hould enure, at the very leat, that any overtoing whih doe our remain negligible. In thi onnetion, it i worth noting that ea defene in ontinental Euroe are often lanned with a freeboard equal to R 2% under deign wave ondition. Re-analyi of Owen data and the Kanai data, together with the ue of Eq. (7) intead of the CIRIA/CUR (1991) exreion, ha rovided udated value of oeffiient A and. Thi new analyi ha alo rovided oeffiient the hallower loe overed by the Kanai data (loe a hallow a 1:20). Uing (R ) 37%, 100 rovided by Eq. (8), the value of A and are now deribed by the following 5 Rei, Hu and Hedge

6 exreion, whih uerede the reliminary value given by ay et al. (2004): A = otα = = otα = otα 1 otα < otα 20 1 otα 8 8 < otα 20 (10) Thee exreion have ubequently been validated uing the data of Hawke et al. (1998) unim eaward loe of 1:2 and 1:4 and the SHADOW data (ay et al. 2004) loe of 1:2, 1:10 and 1:15. Inut arameter value overed by all alibration and validation tet had: R d 0.22 < ξ < 8.25; 0.06 < < 7.65; 0.93 < < (11) H H However, not all individual loe were overed by thi wide range of ondition. Conequently, individual oeffiient A and in the H&R model are not neearily valid thi full range. AMAZON NUMERICAL MODEL AMAZON i a two-dimenional eond-order finite volume numerial model baed on the non-linear hallow-water (NLSW) equation (Hu 2000). It emloy a modern uwind heme of the Godunov-tye with an HLL (Harten, Lax and van Leer 1983) aroximate Riemann olver, whih i aable of aturing bore wave and of imulating uerritial flow. Seond-order auray ha been ahieved by mean of MUSCL reontrution in onjuntion with a Hanok two-tage heme the time integration. The total variation diminihing (TVD) roerty of the heme i enured by alying a re-roeing loe limiter, a art of the MUSCL reontrution roe by whih ell interfae data are reontruted from ell entre data. In addition, AMAZON ue a non-refletive wave inlet boundary ondition, whih i able to remove at the eaward boundary more than 98% of the energy of any wave refleted from the modeled eawall. Conequently, the eaward boundary an be et loe to the eawall to avoid dee water, where AMAZON ha limitation. In fat, the ditane between the model eaward boundary and the toe of the truture may influene the model reult (Pullen and Allo 2003, Hu and Meyer 2005). AMAZON ha three limitation in modeling random wave overtoing: i) it aume hallow water; ii) broken wave are aroximated by bore (it ignore the detailed truture of breaking wave); and iii) vertial aeleration are ignored. The hallow water aumtion and the breaking aroximation limit it ue to relatively long wave (i.e. to water deth muh maller than the wavelength) and to mall amlitude (i.e. to wave height muh maller than the water deth). 6 Rei, Hu and Hedge

7 CONDITIONS TESTED The work arried out in thi tudy overed wave overtoing at eawall with mooth (γ r = 1) front loe of 1:1, 1:2, 1:4, 1:7, 1:10, 1:15 and 1:20, all with a ehore loe of 1:50. A JONSWAP etrum (with a eak enhanement fator of 3.3) wa ued to deribe the inident wave, having teenee of H /L o = 0.01, 0.02 and 0.03, with H = 1m. The value of R /H ranged between 0.5 and 4. oth Owen model and the H&R model, the inut wave ondition hould be eified at the toe of a ehore loe loated at a deth unaffeted by wave-breaking, whilt van der Meer and Janen model, the inut wave ondition relate to the toe of the eawall itelf. In the reent tudy, the wave ondition were aumed to be the ame at the two loation, owing to AMAZON requiring the eaward boundary to be within about one wavelength of the eawall. Differene between the two value were exeted to be mall in the abene of wave-breaking over the ehore. The value of T m ue in Owen model wa etimated from the relationhi T m = T /1.15. Coeffiient A and adoted hi model have been hown in bold in Table 1, while oeffiient A and in van der Meer and Janen model are imliit in Eq. (4). Coeffiient A and in the H&R model have been derived from Eq. (10). In AMAZON, 3000 random wave were generated. The eaward boundary wa loated at a ditane from the eawall toe of aroximately L, the hallow water wavelength in deth d orreonding to the eak wave eriod ( L = T gd ). The value of the relative water deth, d/l o, at the eaward boundary ranged from to Reearher have reorted on different imum ermiible value of d/l o roviding good reult when ued with the NLSW equation, varying from to 0.3 aroximately (Pullen and Allo 2003, Hu and Meyer 2005). The landward boundary wa et 10m behind the ret of the eawall and a non-unim omutation grid wa ued: 0.5m over the ehore, 0.2m the deeer art of the eawall loe and 0.1m the uer art of the eawall. The minimum water deth wa et to 10-5 m (any ell with a water deth below the minimum value i treated a dry and i removed from the omutation) and bottom frition wa ignored. COMPARISON OF OVERTOPPING MODELS Only a amle of the reult i reented here: the reult eawall with a front loe of 1:4 (Figure 1). However, the omarion reented below relate to the full et of reult analyed. Note that, tritly eaking, diret omarion between van der Meer and Janen model and Owen data, Hawke data and the Kanai data hould not be made, ine thi model require a ignifiant wave height at the toe of the eawall, wherea in thee three data et, wave height were meaured at the toe of the ehore loe. Equally, a diret omarion between the four data et and reult from AMAZON i not tritly orret, ine AMAZON ha not been run exatly the ame ondition a thoe teted during the hyial modeling. However, diret omarion between the data and the H&R and Owen model i oible. 7 Rei, Hu and Hedge

8 Q/(gH 3 ) Seawall Sloe 1:4 AMAZON: H/Lo=0.01 AMAZON: H/Lo=0.02 AMAZON: H/Lo=0.03 H&R model: H/Lo=0.01 H&R model: H/Lo=0.02 H&R model: H/Lo=0.03 Owen' data: 0.025<H/Lo<0.035 Hawke' data: 0.005<H/Lo<0.015 Hawke' data: 0.015<H/Lo<0.025 Hawke' data: 0.025<H/Lo< R /H Q/(gH 3 ) Seawall Sloe 1:4 AMAZON: H/Lo=0.01 AMAZON: H/Lo=0.02 AMAZON: H/Lo=0.03 Owen' model: H/Lo=0.01 Owen' model: H/Lo=0.02 Owen' model: H/Lo=0.03 Owen' data: 0.025<H/Lo<0.035 Hawke' data: 0.005<H/Lo<0.015 Hawke' data: 0.015<H/Lo<0.025 Hawke' data: 0.025<H/Lo< R /H Q/(gH 3 ) Seawall Sloe 1:4 AMAZON: H/Lo=0.01 AMAZON: H/Lo=0.02 AMAZON: H/Lo=0.03 Van der Meer' model: H/Lo=0.01 Van der Meer' model: H/Lo=0.02 Van der Meer' model: H/Lo=0.03 Owen' data: 0.025<H/Lo<0.035 Hawke' data: 0.005<H/Lo<0.015 Hawke' data: 0.015<H/Lo<0.025 Hawke' data: 0.025<H/Lo< R /H 4 Fig. 1. The mean overtoing rate rovided by the emirial, emi-emirial and numerial model together with Owen and Hawke data Generally eaking, there i a good agreement between the reult from the H&R emi-emirial model and the available data, not only the data ued in it alibration (the Owen data and Kanai data) but alo the data ued later it validation (Hawke 8 Rei, Hu and Hedge

9 data and the SHADOW data). Of the four method reented rediting overtoing, the H&R model i the one whih ha the mot onitent behaviour throughout the range of eawall harateriti and wave ondition teted. The reult rovided by the AMAZON numerial model deend trongly on the loation of the eaward boundary, with the dimenionle diharge inreaing a the loation aroahe the toe of the eawall. When the eaward boundary wa loated at a ditane from the eawall toe of one hallow water wavelength, AMAZON overredited the dimenionle diharge in relation to the H&R redition ξ >3, aroximately. For maller value of ξ, AMAZON and the H&R reult are very imilar. Owen wholly emirial model ytematially overredited the dimenionle diharge ξ >2.2 (both inide and outide it trit range of aliability). For ξ <2.2, Owen model behaved imilarly to the H&R model, with a general tendeny to lightly overredit the dimenionle diharge. For ξ >2, the van der Meer and Janen model gave imilar reult to the H&R model in the range 0.75<R /H <3 but alway gave higher value outide thi range (both inide and outide it trit range of aliability). For ξ <2, van der Meer and Janen model ytematially gave higher dimenionle diharge. The H&R model how that, a H /L o hange, there are hange in ξ whih rodue orreonding hange in R /H (ee Eq. 8) and, thu, in the overtoing rate. Conequently, it i exeted that R and the overtoing rate will inreae a H /L o dereae rovided ξ remain below 2.2. On the other hand, ξ >2.2, R and the overtoing rate will dereae a H /L o dereae. Thi behaviour aear to be onfirmed by the hyial model data and the reult from AMAZON and the wholly emirial model ξ <2.2. For ξ >2.2, the exeted behaviour i not o obviou from the hyial model data. Indeed, the reult from AMAZON and Owen model both ontradit thi exeted behaviour, with van der Meer and Janen model uggeting that the overtoing rate do not hange with H /L o in thi range (ee Eq. 4). CONCLUSIONS Comarion have been made of the outut from four model ued to etimate the overtoing rate at eawall ubjet to wave ation: two wholly emirial model, a emi-emirial model and a numerial model. Data from hyial model tet have alo been rovided. There wa general agreement between the emi-emirial model and the data and between the emi-emirial and numerial model. Agreement with the wholly emirial model wa inonitent and deended rinially on the value of the urf imilarity arameter: thee model ubtantially overredited diharge ome ondition, both inide and outide their trit range of aliability. ACKNOWLEDGEMENTS The author are indebted to HR Wallingd their ermiion to ue the wave overtoing data olleted by Mihael Owen and Peter Hawke. They thank Aoiate Profeor Hajime Mae of the Diater Prevention Reearh Intitute, Kyoto Univerity, 9 Rei, Hu and Hedge

10 roviding the data olleted by Kanai Univerity. The SHADOW data tet undertaken at HR Wallingd were ommiioned by the UK Deartment Environment, Food and Rural Affair (Defra) under ontrat FD2410 and FD2412. The author alo gratefully aknowledge the finanial onorhi of Dr Rei' otdotoral tudie by Fundação ara a Ciênia e a Tenologia, Portugal. REFERENCES Allo, N.W.H. and Pullen, T Wave Overtoing of Simle Embankment: Imroved Method. <htt:// emirial_model_guidane.df>. ay, I., Pullen, T., Hedge, T. and Shareef, M Wave Overtoing of Shallow Sloing Seawall: Extenion and Refinement of Emirial Predition Method. Proeeding of 29th ICCE, World Sientifi, Singaore (to be ublihed). eley, P Overtoing of Seawall: Deign and Aement Manual. R&D Tehnial Reort W 178, Environment Ageny, ritol. CIRIA/CUR Manual on the Ue of Rok in Coatal and Shoreline Engineering. Seial ubliation 83, Contrution Indutry Reearh and Inmation Aoiation, London. Harten, A., Lax, P.D. and van Leer, On Utream Differening and Godunov Tye Sheme Hyerboli Conervation Law. SIAM Review, 25(1), Hawke, P.J., Coate, T.T. and Jone, R.J Imat of i-modal Sea on eahe and Control Struture. Reort SR 507, HR Wallingd, UK. Hedge, T.S. and Rei, M.T Random Wave Overtoing of Simle Seawall: A New Regreion Model. Water, Maritime and Energy Journal, Pro. ICE, 130(1), Hedge, T.S. and Rei, M.T Aounting Random Wave Run-U in Overtoing Predition. Maritime Engineering Journal, Pro. ICE, 157(MA3), Hu, K High-Reolution Finite Volume Method Hydrauli Flow Modelling. PhD Thei. Centre Mathematial Modelling and Flow Analyi, Manheter Metroolitan Univerity, UK. Hu, K. and Meyer, D The Validity of the Non-Linear Shallow Water Equation Modelling Wave Runu and Refletion. Proeeding of Coatline, Struture and reakwater 2005 (to be ublihed). Kikkawa, H., Shi-igai, H. and Kono, T Fundamental Study of Wave Over-Toing on Levee. Journal of Coatal Engineering in Jaan, 11, Mae, H., Hedge, T.S., Shareef, M. and Nagahahi, S Wave overtoing mula gentle loe inororating wave runu. Proeeding of Coatal Engineering, JSCE, 50, (in Jaanee). Owen, M.W Deign of Seawall Allowing Wave Overtoing. Reort EX 924, Hydrauli Reearh Station, Wallingd, UK. Pullen, T. and Allo, N.W.H Ue of Numerial Model of Wave Overtoing: A Summary of Current Undertanding. <htt:// rojet/overtoing/num_model_guidane.df>. van der Meer, J.W. and Janen, J.P.F.M Wave Run-U and Wave Overtoing at Dike. Wave Fore on Inlined and Vertial Wall Struture, N. Kobayahi and Z. Demirbilek (Ed.), ASCE, New York, Rei, Hu and Hedge