European Journal of Mechanics B/Fluids

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1 Europen Journl of Mechnics B/Fluids 41 (2013) Contents lists ville t SciVerse ScienceDirect Europen Journl of Mechnics B/Fluids journl homepge: Evolution of deep-wter wves under wind forcing nd wve reking effects: Numericl simultions nd experimentl ssessment Zhigng Tin,, Wooyoung Choi,c, Division of Ocen Systems Engineering, Kore Advnced Institute of Science nd Technology (KAIST), Dejeon , Repulic of Kore Nvl Architecture Deprtment, Exmr Offshore Compny, Houston, TX 77042, USA c Deprtment of Mthemticl Sciences, Center for Applied Mthemtics nd Sttistics, New Jersey Institute of Technology, Newrk, NJ 07102, USA r t i c l e i n f o s t r c t Article history: Received 4 Jnury 2012 Received in revised form 11 Ferury 2013 Accepted 1 April 2013 Aville online 11 April 2013 Keywords: Wter wves Wve reking Wind forcing The evolution of two-dimensionl dispersive focusing wve groups in deep wter under wind forcing nd wve reking effects is investigted numericlly nd mesurements collected from wind wve experiments re used to evlute the numericl simultions. Wind forcing is modeled y introducing into the dynmic oundry condition surfce slope coherent pressure distriution, which is expressed through Miles sher instility theory nd Jeffreys sheltering model. To ctivte Jeffreys model in simulting wves evolving under wind forcing, n ir flow seprtion criterion depending on wind speed nd wve steepness is proposed. Direct comprisons of the mesurements nd the simultions re mde y including the wind-driven current in the simultions. To simulte reking wves, n eddy viscosity model is incorported into system of nonliner evolution equtions to dissipte wve energy nd to predict surfce elevtion fter reking. For wve groups under no wind ction, the eddy viscosity model simultes well the energy dissiption in reking wves nd predicts well the surfce elevtion fter reking. Under the weker wind forcing condition, fter considertion of the wind-driven current, the numericl model produces stisfying predictions. As the wind forcing ecomes stronger, the disprity etween the experiments nd the simultions ecomes more evident while the numericl results re still regrded s cceptle. The reltive importnces of the Miles nd the Jeffreys models for wves under wind forcing re discussed through dditionl numericl tests Elsevier Msson SAS. All rights reserved. 1. Introduction Accurte prediction of the evolution of nonliner wter wves is crucil for ships nd offshore structures operting in severe se sttes where extreme events such s frek wves could occur. As ocen wves re conventionlly ssumed to e rndom, their description is typiclly through quntifiction of sttisticlly relevnt properties, e.g. wve spectrum, nd the phse of individul wves is disregrded. However, the phseverging pproch cnnot descrie locl wve kinemtics nd my not provide enough informtion to meet engineering design requirements. Recently, much progress hs een mde towrds deterministic prediction of ocen wves using phse-resolving nonliner wve models (e.g. [1,2]). Deterministic prediction of ocen wves is very chllenging tsk s the wvefield evolution involves mny complicted physicl processes, such s wve wve, wind wve, nd wve current Corresponding uthor t: Division of Ocen Systems Engineering, Kore Advnced Institute of Science nd Technology (KAIST), Dejeon , Repulic of Kore. E-mil ddress: wychoi1111@gmil.com (W. Choi). interctions. Wind lows over the se surfce nd trnsfers momentum nd energy to surfce wves through ir se interction. The generted wves my grow under continuous wind forcing nd their energy redistriutes mong different components through nonliner wve wve interction. Wve reking lso occurs nd dissiptes wve energy, prt of which my contriute to the genertion of surfce current. Some of the physicl processes involved, e.g. wind forcing nd wve reking, re not well understood yet nd, hence, the development of proper models remins chllenging. For the prediction of nonliner wter wve evolution, pseudospectrl method using symptotic expnsion ws developed y West et l. [3]. This method when comined with fst Fourier trnsform [4] hs een shown to e n ccurte nd effective tool to simulte non-reking irregulr wves, e.g. in [5,6]. Recently, the pseudo-spectrl method hs een further developed nd pplied to simulte energy dissiption in reking wves [7,8] nd to predict wve evolution under wind ction [9,10]. Simultion of energy dissiption due to wve reking cn e chieved through n eddy viscosity model sed on oundry lyer pproximtion nd dimensionl nlysis [7]. This model is further developed so tht it cn e implemented utomticlly in /$ see front mtter 2013 Elsevier Msson SAS. All rights reserved.

2 12 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) simultion of reking wve groups [8]. In the refined model, three sets of correltions etween pre-reking locl wve prmeters nd post-reking time nd length scles re identified ccording to experimentl mesurements. These connections re used to predict post-reking scles nd to estimte the mgnitude of the eddy viscosity sed on the pre-reking prmeters when wve reking occurs, indicted y criticl surfce slope S c (=0.95 in their numericl simultions). The model ws demonstrted to simulte well the totl energy dissiption due to wve reking in comprison with lortory experiments; in ddition, the surfce elevtion fter wve reking is predicted well for dispersive focusing wve groups nd irregulr wves chrcterized initilly y the JONSWAP spectrum. The influence of wind on the evolution of nonliner wter wves cn e modeled y introducing n externl surfce pressure in the dynmic free surfce oundry condition. One model ssumes tht the wind induced pressure is in phse with the wve surfce slope nd depends on the friction velocity on the surfce due to the wind forcing [11]. This model origintes from Miles sher flow instility theory [12] for wve genertion nd ssumes no ir flow seprtion. With this simple model, Bnner nd Song [11] studied the wind forcing effect on the performnce of wve reking criterion sed on locl energy convergence rte. In ddition, Klmikov [2] conducted the deterministic prediction of the evolution of nonliner wter wves under wind forcing using spectrl method. Neither of the numericl studies provided experimentl ssessment of the performnce of this wind forcing model. An lterntive model for wind forcing is sed on Jeffreys sheltering hypothesis [13]. This model lso involves wve slope coherent pressure on the free surfce, ut is supposed to e pplicle only to wves over which ir flow seprtes [9,10]. Although it hs een pplied to study extreme wves under wind ction, the sheltering model is still detle nd hs to e tested crefully with experiments. Tououl et l. [9] re the first to use the sheltering model to study the genertion of frek wves from dispersive focusing wve groups under wind forcing. Due to the focus of their study, s well s the high initil steepness of their wve groups, reproducing their experiments numericlly ws not trivil nd only limited ssessment of performnce of the sheltering model ws performed. Their susequent study [10] investigted the wind forcing effect on extreme wves. The influence of wind on possily sustining extreme wves due to ir flow seprtion is discussed through experimentl oservtions nd numericl simultions with similr sheltering model. Yn nd M [14] conducted CFD simultions on the interction of frek wves nd wind nd reveled tht the sheltering hypothesis my not descrie the pressure over frek wves ccurtely. Even if the sheltering model is ccepted s physiclly dequte one, condition under which ir flow seprtion occurs hs to e known efore this simple model is dopted in numericl simultions. In fct, the determintion of criterion for ir flow seprtion over wter wves hs een n interesting suject due to its importnce in the wind wve interction process. Wu [15] proposed tht ir flow seprtion over wter wves with following wind occurs when the friction velocity is greter thn the wve phse velocity. Lter, Bnner nd Melville [16] rgued tht ir flow seprtion occurs only in the presence of reking wves. They mnged to oserve ir flow seprtion over stedy reking crest y mens of smoke visuliztion nd lso non-seprted flow over n unroken, stedy wve. However, Kwi nd Weissmn [17,18] suggested the possiility of ir flow seprtion over non-reking wind wves nd supported their rguments with flow visuliztion experiments. Recently, Khrif et l. [10] used device composed of hot nd cold wires to detect ir flow seprtion over frequency focusing wve groups nd criticl locl wve slope, ( ζ / x), close to 0.35 ws determined to indicte the onset of ir flow seprtion. Khrif et l. [10] further noted tht the ir flow seprtion ws ccompnied generlly y reking wves. In numericl studies, Tououl et l. [9] chose criticl slope ( ζ / x) = 0.5 to predict the onset of ir flow seprtion. No experimentl support is provided for this criterion. Khrif et l. [10] dopted criticl locl wve slope, ζ / x 0.35 to indicte ir flow seprtion onset for their numericl simultions. The discrepncy in the criticl slope for ir flow seprtion my rise from the different wind forcing conditions considered in these studies. Therefore, n improved criterion, possily depending on oth locl wve slope nd wind speed, for the ppliction of the sheltering model shll e proposed. Overll, comprehensive experimentl evlution of the ovementioned numericl models for the evolution of nonliner wter wves under wind forcing is sprse. In this study, s first step towrds developing more roust wind wve interction models, we conduct lortory experiments of nonliner wter wves under wind ction to evlute the pplicility of two existing wind forcing models, i.e. Miles nd Jeffreys models, nd to ssess their performnce. In ddition, to resolve the oserved inconsistency on the ir flow seprtion criteri dopted in previous numericl simultions, seprtion criterion considering oth wind speed nd wve steepness is proposed. The rest of the study is orgnized s follows. After this introduction, detiled description of the numericl models is presented. Section 3 provides experimentl set-up, wve group genertion, nd wind forcing conditions. Experimentl nd numericl results, s well s their comprison, re given in Section 4. The lst section summrizes min findings nd concludes this study. 2. Numericl models For the prediction of the evolution of wter wves, West et l. [3] developed nonliner wve model sed on symptotic expnsion, where wve dynmics is governed y the following system of nonliner evolution equtions for the surfce elevtion, ζ, nd the velocity potentil Φ, on the free surfce [3,19,20]: ζ N t = Q n [ζ, Φ] nd Φ = t n=1 N R n [ζ, Φ]. (1) Here Q n nd R n re two nonliner opertors tht cn e written explicitly through recursion formuls nd N is the order of nonlinerity t which the originl infinite series on the right-hnd sides re truncted. This model cn e solved numericlly using n efficient pseudo-spectrl method sed on Fst Fourier Trnsform nd is known to e ccurte nd effective to predict non-reking irregulr wve evolution. However, the simultion cnnot provide relile predictions when wve reking occurs. In the presence of wve reking, the wve-induced flow ner the wter surfce ecomes turulent nd multi-phsed, nd no nlyticl description of the flow is possile. Nevertheless, the wve reking effect hs to e considered for resonle prediction of the evolution of wter wves. In ddition, modeling the wind forcing effect is necessry for nonliner wves under wind ction which trnsfers energy to wves, induces wve growth, nd ffects significntly the wve dynmics. To model these effects, the nonliner evolution equtions given y (1) re modified to ζ N t = Q n [ζ, Φ] + D ζ [ζ, Φ] Φ = t n=1 n=1 nd N R n [ζ, Φ] + D Φ [ζ, Φ] + Π [ζ, Φ], n=1 (2)

3 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) where D ζ nd D Φ re energy dissiptive terms due to wve reking nd Π my represent pressure term induced y the ction of wind over steep or reking wves Wve reking model In our previous studies [7,8], we developed n eddy viscosity model to simulte energy dissiption in two-dimensionl unstedy plunging rekers using oundry lyer pproch nd dimensionl nlysis: 2 ζ 2 Φ D ζ [ζ, Φ] = 2ν eddy nd D Φ [ζ, Φ] = 2ν x 2 eddy x, (3) 2 where the eddy viscosity ν eddy depends on reking strength nd cn e estimted through time nd length scles ssocited with reking event: ν eddy = α H rl r T r. (4) Here, T r is defined s the time when the wve crest egins to fll to the time when the surfce disturnce front is no longer ovious; L r is the distnce from incipient reking to where the ovious surfce disturnce ends; H r refers to the flling crest height [21]; α is proportionl constnt nd α = 0.02, s determined in [7]. We stress tht our wve reking model uses constnt eddy viscosity over finite sptil rnge of L r during finite period of T r for given wve reking event. However, the eddy viscosity ν eddy is determined dynmiclly nd vries depending on the reking strength of specific reking events. Note tht n lterntive formultion for the estimtion of the eddy viscosity ws proposed y Drzen nd Melville [22], who conducted mesurements of turulent mixing introduced y unstedy reking wves. While their interest lies in the eddy viscosity of the turulence generted y wve reking event (timescle of tens of wve periods), this study focuses on modeling energy dissiption due to wve reking within one or two wve periods susequent to reking. Detils of their study cn e found in [22]. Notice tht their eddy viscosity term hs not een modeled for ny deterministic wve model for the prediction of post-reking wve evolution nd, therefore, direct comprison etween the two eddy viscosity models is not strightforwrd. To implement the eddy viscosity model in numericl simultions, Tin et l. [8] used criticl surfce slope S c = ( ζ / x) c = 0.95 to indicte wve reking so tht the eddy viscosity model is put into effect nd three sets of correltions etween pre-reking prmeters nd post-reking scles (i.e. T r, L r nd H r ) to estimte the mgnitude of the eddy viscosity, s shown elow: κ L r = 24.3S 1.5, (5) ω T r = 18.4S + 1.4, (6) κ H r = 0.87R 0.3. (7) Here, S is locl wve steepness; k is locl wvenumer sed on zero crossings nd ω is the corresponding ngulr frequency ccording to the liner dispersion reltion; R = L /L C is horizontl wve crest symmetry. Here L is the distnce etween the crest tip nd the zero-crossing point immeditely ehind it nd L C is the distnce etween two consecutive zerocrossing points djcent to the reking crest. These prmeters cn e determined with simulted surfce profile just prior to wve reking, indicted y the criticl surfce slope S c exceeding 0.95 in the simultions. Detils regrding the definitions nd the implementtion scheme re referred to the previous work [8] Wind forcing model A physics-sed model for wind forcing effect on steep wves in terms of locl wve chrcteristics remins to e developed. However, typicl pproch to simulte the wind forcing effect is to introduce n externl surfce pressure distriution in phse with the wve surfce slope, i.e. wve slope coherent pressure, in the dynmic free surfce oundry condition. In this cse, Π [ζ, Φ] = P wind /ρ wter, where P wind is the wve slope coherent pressure due to wind. One of the models for P wind is expressed with the following eqution [12,11,2]: 2 ζ P wind = βρ u x, (8) where β is coefficient tht remins to e determined nd u is the friction velocity on the wter surfce due to wind. Coefficient β my e evluted through theoreticl nlysis sed on Miles sher flow instility theory [12]. Alterntively the coefficient cn e determined through the oserved wve growth rte in experiments nd field mesurements; e.g. Bnner nd Song [11] estimted tht β is pproximtely A second form of the wve slope coherent pressure origintes from the so-clled Jeffreys sheltering hypothesis [13]. The sheltering hypothesis ws developed to study the wve genertion y wind. The theory ssumes ir flow seprtion over wves nd tht wve genertion is minly due to wve slope coherent pressure. Recently, the hypothesis ws pplied to model the strong wind forcing effect on steep wter wves due to ir flow seprtion [9,23,10]. In this model, the forcing term is written s P wind = sρ (U 0 c) 2 ζ x, (9) where the sheltering coefficient s = 0.5 is reported in [9,10], ρ is ir density, U 0 is the wind speed, nd c is the locl wve phse velocity, which is determined through locl wvenumer nd the liner dispersion reltion. The locl wvenumer is defined y consecutive zero-crossings of the surfce elevtion. While the two forcing terms given y (8) nd (9) re similr in the sense tht oth re proportionl to the wve slope, we should remrk tht Jeffreys model is pplied only when the condition for flow seprtion is met, s discussed in Section 4. We note tht the wind friction velocity mesured in wve tnk is pproximtely 5% of the wind speed, i.e. u /U (e.g. [24,29]. In the cse of slow wves propgting under fst wind, U 0 c, the rtio of P wind from the two theories my e estimted s (P wind ) Jeffreys /(P wind ) Miles 400s/β 6, where s = 0.5 nd β = 32.5 re used. However, one hs to consider the fct tht ir flow seprtion is highly unstedy; it my occur rpidly nd only persist for very short time [25,26]. Therefore, the evlution of the overll effects from ir flow seprtion on wve evolution is non-trivil [27]; in ddition, the reltive importnce of the two mechnisms should depend on specific wind forcing conditions. Tououl et l. [9] conducted nlysis of the chrcteristic time scles ssocited with wind forcing nd found tht the sheltering mechnism plys more importnt role in their study (wind speed vries from 4 to 8 m/s nd pek wve frequency is 1 Hz). 3. Experiments Experiments re performed t the Kore Advnced Institute of Science nd Technology (KAIST) in two-dimensionl wind wve tnk with glss wlls nd removle, trnsprent plstic ceiling pnels. The tnk is 15 m long, 1.5 m wide, nd hs wter depth s used of 0.54 m. A servo-controlled piston-type wvemker nd uxiliry electronics re locted t one end of the tnk nd used to generte wter wves. At the sme end, twin-fn lower is instlled on the top of the tnk to generte following wind. A mximum wind speed of 10 m/s cn e produced. The distnce etween the clm wter surfce nd the ceiling pnels (i.e. ir pssge gp) remins 0.45 m in the experiments. At the other end

4 14 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) Fig. 1. Illustrtion of the two-dimensionl wind wve tnk (not to scle) nd mesurement devices. Tle 1 Specified prmeters for the wve groups. DF indictes dispersive focusing wve groups. f c is the center wve frequency nd f /f c is the frequency ndwidth specified. f p is the spectrl pek frequency. ε = Nk n n is the initil wve steepness specified in wve genertion (N = 128). Under no wind ction, DF 1 is nonreking group nd DF 2 is reking one. Note tht DF 1 remins non-reking under ll wind forcing conditions. Wve group f c (Hz) f p (Hz) f /f c ε = Nk n n DF DF of the tnk, wve sorer mde of loose nets nd stinless steel grids reduces wve reflection. A movle crrige is instlled on the top of the tnk nd provides work pltform. Fig. 1 illustrtes sketch of the experimentl set-up Wve group genertion Dispersive focusing wve groups re generted in the experiments, where the surfce elevtion, ζ, is descried s ζ (x, t) = N n cos(k n x ω n t φ n ). (10) n=1 Here, n is the mplitude of the nth wve component; k n is the wvenumer; ω n = 2πf n is the ngulr frequency nd f rnges from 1.0 to 2.4 Hz (center frequency f c = 1.7 Hz nd frequency ndwidth f = 1.4 Hz); N = 128 is the totl numer of frequency components; φ n is the initil phse to e determined. In ddition, x is the horizontl distnce downstrem of the wvemker with x = 0 eing the men position of the wvemker; time t is reltive to the initil motion of the wvemker (i.e. t = 0). The liner dispersion reltion is used to relte ω n nd k n. Wve steepness, ε n = k n n, for ech of the components is the sme nd cn e djusted. The phse φ n is determined so tht the wve groups focus t specified time t nd loction x, i.e. cos(k n x ω n t φ n ) = 1. Therefore, φ n = k n x ω n t + 2πm, where m = 0, ±1, ±2,.... By sustituting φ n into (10) nd setting x = 0, the surfce elevtion t the wvemker cn e otined s ζ (0, t) = N n cos [ k n x ω n (t t )]. (11) n=1 To generte the input signl to the wvemker, trnsfer function etween the wvemker stroke nd the surfce elevtion is first determined in clirtion tests; the trnsfer function is then pplied to Eq. (11) to otin the input to the wvemker. Note tht, in the clirtion tests, mesurements of the surfce elevtion in the trnsverse direction cross the wve tnk showed good twodimensionlity of generted sinusoidl wves nd focusing wve groups. To void n rupt motion of the wvemker nd the development of noise in the tnk, such s the cross tnk wves nd reflected wves from the sorer, window function is pplied to the produced input signl. Key prmeters of these wve groups re listed in Tle 1. Fig. 2. A smple imge (resized) nd the detected ir wter interfce (solid line) for dispersive focusing group under wind condition U 0 = 5 m/s Surfce elevtion mesurement We ttempt to mesure the surfce elevtion with cpcitncetype wve proes. However, it is found tht the wve proe supporting rod is distured nd virting in the presence of wind in the tnk. Therefore, wve proe mesurements my e rendered inccurte y the disturnce. Alterntively, surfce elevtion is mesured through high-speed imging. In the experiments, 15 W DPSS lser is used s the light source for illumintion. A thin lser light sheet is generted through series of optics nd is directed downwrd into the wter, in which fluorescent dye (Rhodmine 6G) is dissolved to improve the illumintion. A high-speed imger (Phntom V9.1 with 12 GB internl memory) mounted outside of the tnk is used to cpture the surfce elevtion nd to fcilitte the oservtion of evolution of the wve groups. The imger, equipped with 50 mm focl length Nikon lens, is positioned in front of the tnk with its xis oriented slightly downwrds for etter view of the illuminted ir wter interfce. Imges re cptured t 200 frmes per second (fps) nd the exposure time is set to one millisecond. The size of the field of view depends on specific set-ups t different loctions long the tnk, ut the length is pproximtely 50 cm (1600 pixels) nd the width is djustle (pproximtely 25 cm nd 800 pixels). Using precise plnr trget with known spcing, the sptil resolution is determined nd the imge distortion is shown to e negligile. The devices nd the mesurement set-up re illustrted in Fig. 1. For the imging t wve sttion long the tnk, the recording durtion is long enough to llow the entire wve group to pss y the wve sttion. Ech of the snpshots is then processed with the Imge Processing Toolox in MATLAB nd the ir wter interfce is determined through edge detection functions sed on the Cnny method. Fig. 2 provides n exmple of snpshot nd the detected wter surfce. Finlly, the surfce elevtion s function of time cn e otined y pplying the sme processing procedure to the series of the snpshots. Note tht the strtegy used to extrct surfce elevtion from the imging is sufficiently ccurte for low nd moderte steep wves; however, for very steep wves nd/or in the presence of wve reking, spurious surfce cn e detected due to illumintion relted issues. Therefore, filter (i.e. smooth function with roust locl regression method in MATLAB) is pplied to the temporl surfce elevtion to eliminte these ovious spurious surfce elevtions. Fig. 3 shows the temporl surfce elevtion of the two wve groups mesured t 2.84 m downstrem of the wvemker Wind forcing conditions In the experiments, the twin-fn lower is used to generte following wind. The wind entry is locted t 1.49 m downstrem of the wvemker, i.e. x = 1.49 m (herefter, x f is defined s the distnce reltive to the wind entry nd x f = x 1.49 m). In ech test, efore the opertion of the wvemker nd ny dt collection, the lower is run for t lest 10 min so tht sttisticl equilirium stte of the wind-generted wves cn e chieved in the tnk. We considered three wind forcing conditions, i.e. men free strem wind speed U 0 = 1.4, 3.2, nd 5.0 m/s. The men wind

5 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) Fig. 3. Surfce elevtions mesured t 2.84 m downstrem of the wvemker. No wind forcing is pplied. For clrity, n increment of 5 cm is pplied in the ordinte to seprte mesurements of the two wve groups. speed is mesured with portle nemometer (Flowtch, JDC Instruments) t the centerline (trnsverse direction of the wve tnk) 24 cm ove the clm wter level t fetch of 4.87 m. Notice the wind speed is mesured t verticl loction where the wind profile ecomes uniform in the verticl direction. Therefore, its distnce from the clm wter level is not so crucil s long s it is fr wy from the edge of turulent oundry lyer nd the vrition of the men wind speed is reltively little. The verging period for the mesurement is 30 s nd multiple redings re otined nd verged. Fig. 4 presents the men free strem wind speed s function of fetch (i.e. distnce downstrem of the wind entry) nd the trnsverse men wind velocity profile cross the tnk. Clerly, for given wind condition, the men wind velocity decreses significntly from the wind entry to out x f = 2.5 m, fter which it remins pproximtely constnt. In the trnsverse direction, the men wind velocity remins pproximtely constnt in the middle section (y ±40 cm) of the tnk. Verticl men wind velocities s function of distnce ove the men wter surfce re provided in Fig. 5. As shown, the ir flow is not fully developed t the shorter fetch (x f = 1.87 m) nd the verticl men velocity profile is significntly different from tht mesured t the longer fetches. We exmined the velocity profiles close to the men wter surfce (z 15 cm) t the longer fetches (x f = 4.87 m nd 7.87 m) nd found tht they follow well the so- Fig. 5. Verticl men wind velocity profiles s function of distnce ove the men wter surfce (i.e. z = 0). The results re mesured in the middle of the tnk in the trnsverse direction t three different fetches, i.e. x f = 1.87 m (circles), 4.87 m (sterisks), 7.87 m (squres). clled logrithmic lw: u(z) = 1 z u κ ln, z 0 where the von Krmn constnt κ is given y κ = 0.41 nd z 0 is the roughness length which cn e estimted through the mesured verticl velocity profile. The mesured profiles re then used to determine the friction velocities, s provided in Tle 2. The estimtions re comprle with those in [26,28] despite the difference of the experimentl fcilities nd the wind genertion technique. In ddition, the rtio of the friction velocity to the men speed, u /U t the fr downstrem loction (x f = 7.87 m), where the ir flow nd wind generted wves re fully developed, is consistent with typicl mesurements found in wve tnk (e.g. [24,29]). These wind forcing conditions re pplied to the focusing wve groups nd oservtions on the evlution of the groups re mde. The experimentl results will e used to evlute the performnce of existing numericl models of wind forcing nd wve reking effects. Fig. 4. () Men free strem wind velocity s function of fetch nd () trnsverse men wind velocity profile cross the tnk. Dt provided in () re mesured t 24 cm ove the men wter surfce t the center of the tnk in the trnsverse direction; () re mesured t the sme verticl distnce t x f = 4.87 m. y indictes the distnce from the middle of the tnk in the trnsverse direction.

6 16 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) Tle 2 Estimted friction velocity (u ) sed on wind profile mesurements (Fig. 4) nd u(z) the logrithmic lw, u = 1 κ ln z. Here, the von Krmn constnt z 0 κ = Note tht only mesurements close to the men wter surfce (z 15 cm) re used in the estimtion. U 0 (m/s) u (m/s) x f = 4.87 m x f = 7.87 m Numericl simultions nd results 4.1. Simultion set-up The wind forcing nd wve reking models, i.e. Eqs. (3), (8) nd (9), re incorported into the pseudo-spectrl model given y Eq. (2) to simulte the evolution of two-dimensionl wter wves. In the code, the right-hnd sides of Eq. (2) were truncted to the fifth order nd the nonliner evolution equtions of the system were solved numericlly with pseudo-spectrl method sed on the fst Fourier trnsform (FFT) nd fourth-order Runge Kutt method to integrte in time. A rief description of the numericl simultion set-up nd the genertion of initil conditions re provided here while the detils re referred to [8]. The simultions re conducted in numericl wve tnk 50 m long with the domin from 20 to 35 m corresponding to the physicl wve tnk. The numericl domin is discretized with 2 12 points nd time step of 0.01 s is used in the simultions. The simultion period, T, is s, which is of sufficient durtion for the wve groups to completely pss the lst wve sttion. Using liner wve theory, initil conditions, i.e. sptil vrition of the surfce profiles nd velocity potentils (from 0 to 20 m in the numericl domin) t the men wter level, re generted with surfce elevtion mesurements t the first wve sttion (see Fig. 3). As discussed in [6], this initiliztion scheme sed on liner theory introduces smll errors to the velocity potentil of n initil wvefield tht is reltively liner. In ddition, five point moving verge is pplied to the mesured surfce elevtion nd only the first 256 Fourier modes (up to 6.25 Hz) re used in the initil condition genertion. To mtch the surfce elevtion mesured t the first wve proe, the first-order model without viscous effects is solved over the sptil domin up to the loction of the first proe. In the reminder of the numericl tnk, the fifthorder model with the wve reking nd wind forcing models is solved. In ddition, n equivlent kinemtic viscosity, ν eqv = m 2 /s is pplied to the free surfce oundry conditions to ccount for the free surfce dmping nd the frictionl loss due to tnk side wlls nd ottom. The equivlent viscosity is determined such tht the totl energy predicted in the simultions mtches the mesurements for the non-reking wve group. Dommermuth [30] showed tht his high-order spectrl wve model initilized with liner theory could excite spurious high frequency wves nd nonliner terms need to e djusted to grow grdully over finite time period. Comined with our domin decomposition technique to solve liner nd nonliner models in their respective domin, our initiliztion scheme sed on liner theory gurntees numericl solutions for the surfce elevtion t the first wve proe loction to mtch exctly with the corresponding lortory mesurements. This cnnot e chieved with the djustment technique suggested y Dommermuth [30]. Although our initiliztion scheme needs to e improved, we remrk tht our numericl solutions show excellent greement with lortory mesurements in the sence of wind nd wve reking, s cn e seen lter, nd, therefore, the simple initiliztion scheme sed on liner theory is dopted for this study. To simulte reking wves under no wind ction, the criticl surfce slope S c = 0.95 is used to indicte wve reking nd the mgnitude of the eddy viscosity is estimted with Eq. (4), in which the post-reking scles re predicted with Eqs. (5) (7). The pre-reking wve prmeters in these equtions re determined with simulted surfce profile when the locl surfce slope just exceeds the criticl one. In the presence of the wind forcing, wves my rek t reduced steepness [31] nd the post-reking time nd length scles my demonstrte different chrcteristics from those under no wind ction; however, the sme reking model is dopted to simulte reking wves in wind forcing conditions in this study. As for the wind forcing, the surfce pressure distriution is pplied from the position of the wind entry (x = 1.49 m in the physicl tnk, corresponding to m in the numericl tnk) to the end of the tnk. Miles model, i.e. Eq. (8) with β = 32.5, is used in the sence of ir flow seprtion nd the sheltering model, i.e. Eq. (9) with s = 0.5, is pplied loclly if ir flow seprtion is predicted sed on the flow seprtion criterion discussed in the following section Air flow seprtion over steep wter wves Previous ttempts t the determintion of criterion for ir flow seprtion typiclly focused on the locl wve geometry only (e.g. wve steepness nd surfce slope); however, one my rgue tht plusile seprtion criterion my depend on oth wve steepness nd wind forcing condition, s oth ply importnt roles in the wind wve interction. In recent field mesurements, Doneln et l. [32] oserved full ir flow seprtion over nonreking wves in shllow lke nd they rgued tht the ir flow seprtion my depend on the force lnce over the wve crests. The lnce is s follows: the verticl grdient of pressure, proportionl to (U c)(k) 2, shll mtch the centripetl ccelertion, proportionl to ( 2 ζ / x 2 ), required to keep the stremlines in contct with the wve surfce. Here, U is the wind speed mesured t one hlf wvelength ove the wve crest nd 2 ζ / x 2 is pproximted with wve steepness, k, sed on deep wter Stokes expnsion. The study considered explicitly, for the first time, the wind speed effect on the ir flow seprtion over wter wve crests. More recently, Tin et l. [28] conducted flow visuliztion experiments nd oserved the ir flow structure over mechniclly generted wves. They identified conditions, i.e. proper comintions of locl wve steepness nd wind speed, under which ir flow seprtion is likely to occur. The lortory experimentl results re consistent with the field mesurements of Doneln et l. [32]. The dt in [32,10,28] re reprocessed nd plotted in Fig. 6. With the results, n ir flow seprtion criterion is proposed nd it is indicted with the lue solid line in the figure. The eqution for the criterion is given s follows (U 0 c) 59.3S for 0 < S < (12) c The sheltering model is dopted loclly wherever the ove criterion is stisfied. Note tht S is the wve steepness ccording to the experimentl dt, ut it is replced with the locl surfce slope in the numericl simultions in this study Wind-driven current The presence of wind forcing introduces thin surfce drift lyer, which my hve importnt effects on the evolution of the wve groups [31,33]. This lyer is of high vorticity nd the velocity profile depends strongly on depth [33]; however, for simplicity, the lyer my e modeled s uniform surfce current (e.g. [10]), which hs mgnitude of typiclly few percent of the wind

7 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) ζ (t) (cm) t (sec) ζ (t) (cm) 10 5 Fig. 6. Air flow seprtion criterion ccording to lortory experiments nd field mesurements. The sciss is wve steepness nd the ordinte is non-dimensionl reltive speed. U 0 is the free strem wind speed nd c indictes phse speed. Pluses: non-seprtion conditions; sterisks: conditions under which seprtion my or my not occur due to uncertinty in the flow visuliztion; circles: seprtion conditions; solid squres: seprtion conditions ccording to Doneln et l. [32]; dotted line: seprtion criterion used in [10]. The solid line indictes liner lestsqures fit of the Doneln dt nd Khrif dt (lower limit). The est fit is used the ir flow seprtion criterion in this study. speed. For exmple, the mesurement y Peirson nd Bnner [34] showed tht the men surfce drift velocity is in the rnge of 1% 2%, depending on the mesurement loctions (i.e. fetch, wve trough nd wve crest). However, uniform surfce current speed, U cur, is commonly ssumed in numericl simultions. Khrif et l. [10] used uniform surfce current of 2% of the wind speed (i.e. γ = U cur /U 0 = 2%) in their study of wind influence on extreme wve events. Yn nd M [27] conducted systemticlly numericl tests regrding proper wind-driven current. They found tht, depending on specific wind speed, γ = % produces stisfctory results nd γ = 0.5% in generl provides cceptle results for the predicted mximum surfce elevtion s function of spce for ll cses considered. In our numericl simultions, we lso include uniform surfce current to model the wind effects on the evolution of the wve groups. As discussed lter, wind-driven current of speed γ 1% provides resonle numericl results in comprison with our experimentl mesurements. Detils cn e found in the following sections Evolution of the wve groups under no wind ction Fig. 7() nd () provide comprisons of the predicted nd mesured surfce elevtions for the non-reking nd reking wve groups, respectively, in the sence of wind forcing. For the non-reking wve group, the predicted surfce elevtions mtch well the mesurement. For the reking group, the prediction lso grees well with the experimentl results, including the comprison downstrem of the wve reking region, which is locted etween the second nd the third wve sttions. Since the surfce elevtion is predicted well with the wve reking model, the sptil vrition of the totl energy is expected to e well predicted too. Fig. 7(c) shows the long time integrtion of the surfce elevtion squred, ζ 2 (t) which is proportionl to the totl energy pssing wve sttion ccording to liner wve theory. As expected, the totl energy s function of spce for oth non-reking nd reking wve groups re predicted well in the simultions, though the energy dissiption due to wve reking simulted with the eddy viscosity model ppers slightly smller c <ζ 2 (t)> (cm 2 ) t (sec) x (m) Fig. 7. Comprison of experimentl nd numericl results without wind forcing. () Surfce elevtion for non-reking wve group DF 1; dshed lines: experiment; solid lines: simultion; mesurement loctions downstrem of the wvemker re indicted in the figure, e.g m. () Sme s () ut for reking wve group DF 2; note tht wve reking occurs etween the second nd third wve sttions. (c) ζ 2 (t) is the long time integrtion of surfce elevtion squred; crosses: experiment; solid lines: simultion. thn the mesurement. Overll, the improved eddy viscosity model predicts well oth energy dissiption in reking events nd the surfce elevtion downstrem of reking for the wve groups. We now exmine the wve spectrum of the dispersive focusing wve groups. Similrly, good greement of the wve frequency spectrum etween the experimentl nd numericl results is expected. As shown in Fig. 8, the predicted mplitude spectr t different sttions long the tnk mtch well the mesurements. In the evolution process, noticele chnges in wve spectrum for oth the non-reking nd reking wve groups re oserved, which indictes tht the dispersive focusing is fr from liner superposition nd it undergoes strong nonliner process, consistent with the findings in [35]. In the previous study [35], compred to the experimentl results, the mgnitudes of the lower frequency wve components re found to e overestimted while the higher ones re underestimted in their numericl predictions. Tin et l. [35] ttriuted the discrepncy minly to reltively lrge effective kinemtic viscosity introduced to ccount for the viscous relted dissiption minly due to friction on the wve tnk (0.7 m wide) side wlls nd ottom. In this study, the wve tnk is much wider (1.5 m) nd smller effective kinemtic viscosity is pplied to the free surfce to ccount for the viscous relted dissiption. The good greement shown in Fig. 8 confirmed tht the discrepncy in the wve spectrum prediction in [35] is minly due to the lrge effective kinemtic viscosity used in their numericl simultions.

8 18 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) Fig. 9. Surfce elevtion of the wve groups under wind condition U 0 = 1.4 m/s: () the non-reking group DF 1 nd () the reking group DF 2. Dshed lines: experiment; solid lines: numericl results with γ = U c /U 0 = 0.9%. The friction velocity used in the simultion is u = 4.84 cm/s (men of the mesurements t two fetches in Tle 2). Fig. 8. Comprison of the mplitude spectrum for the groups under no wind forcing: () wve group DF 1 nd () wve group DF 2. Crosses re from mesurement nd solid lines re computed with simultion results. The mplitude spectrum is defined s A(f ) = 1 t+t ζ (t)e 2πift dt. T is chosen long enough tht T t the entire wve group is included within the durtion. For clrity, n increment of 0.25 cm is pplied to the ordinte to seprte the mplitude spectrum t different wve sttions Evolution of the wve groups under wind forcing Fig. 9 provides the experimentl nd numericl results for the wve groups under wind forcing condition U 0 = 1.4 m/s. In the simultions, uniform surfce current hs to e included to predict more ccurtely the evolution of the wve groups. We used three current speeds, i.e. γ = U cur /U 0 = 0.6%, 0.9% nd 1.2%, nd found tht they produce numericl predictions close to ech other in terms of oth mgnitude nd phse. As shown in Fig. 9, despite some locl disprities, the numericl models simulte well the evolution of the non-reking nd reking wve groups under this weker wind forcing condition (only the numericl results with γ = 0.9% shown). Note tht n ctive reking crest just occurs t the third sttion. Comprisons of the numericl nd experimentl results for the wind condition U 0 = 3.2 m/s re provided in Fig. 10. As shown, locl disprity in the comprison ecomes more noticele. Although not shown in the figure, we note tht numericl predictions with γ = 1.2% pper to provide etter mplitude estimtion while simultions with γ = 0.6% produce etter phse greement, compred with the numericl results with current speed γ = 0.9% which gree resonly well with the mesurements in terms of oth phse nd mplitude. We lso note tht wind-generted wves cn e oserved fr downstrem in the experiments; however, we include no surfce tension in the free surfce oundry conditions nd these short wind wves re not predicted well in the simultions. Fig. 10. Sme s Fig. 9 ut for wve groups under wind condition U 0 = 3.2 m/s. The friction velocity used in this simultion is u = 12.7 cm/s. As the wind speed further increses to U 0 = 5.0 m/s, locl disgreement etween the simultion nd the mesurement ecomes more evident, s shown in Fig. 11. It ppers tht the simulted surfce elevtion with γ = 0.9% demonstrtes cceptle mplitude estimtion lthough there is smll phse shift compred to the mesurement. For the reking group, n ovious disprity in the third wve sttion is oserved nd this disprity corresponds to n ctive reking pssing y the sttion. Note tht n ctive plunging reker lso presents t the fourth wve sttion. We lso exmined the mesured nd predicted mximum surfce elevtions s function of spce nd the results re shown in Fig. 12. The numericl prediction mtches well the mesurement for the non-reking wve group under no wind

9 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) Fig. 11. Sme s Fig. 9 ut for wve groups under wind condition U 0 = 5.0 m/s. The friction velocity used in this simultion is u = 22.2 cm/s. c Fig. 13. Long time integrtion of surfce elevtion squred, ζ 2 (t), s function of distnce. () U 0 = 1.4 m/s, () U 0 = 3.2 m/s nd (c) U 0 = 5.0 m/s. Crosses: experiment; solid lines: simultions with current speed γ = 0.9%. Fig. 12. Comprison of the mesured (symols) nd the predicted (lines) mximum surfce elevtions for the non-reking () nd the reking () wve groups. Crosses: experiment; solid lines: simultions with current speed γ = 0.9%. For clrity, n offset of 5 cm is pplied to the ordinte to seprte the wind forcing cse from the no wind ction cse. ction. The remining comprisons re not so good, ut still cceptle. Another ovious oservtion is tht the wind forcing hs delyed the wve focusing/reking process nd pushes the focusing/reking point further downstrem. Similr oservtions were reported in previous studies, e.g. [10]. They lso showed tht frek wves my e sustined longer y wind forcing (minly ir flow seprtion in the leewrd of the extreme crests) nd n symmetric ehvior of wve mplifiction in the focusing nd defocusing processes. The results shown in Fig. 12 do not confirm their oservtions, possily due to differences in wve group steepness, wind forcing condition, nd forcing durtion. We mke further comprisons y exmining the sptil vrition of the totl wve energy, which is proportionl to ζ 2 (t), nd the results re shown in Fig. 13. In generl, the totl energy for oth the non-reking nd reking wve groups propgting under wind forcing is well predicted with the numericl models. The mesured totl energy t the fourth wve sttion for the strongest wind forcing cse is higher thn the numericl prediction, minly due to the wind-generted wves in the experiments. Moreover, Fig. 14 presents comprison of the mesured nd the predicted wve mplitude spectr for U 0 = 3.2 m/s. In generl, the predictions t different loctions long the tnk gree well with the mesurement, despite some discrepncy t the higher frequency wve components. Overll, the disprity etween the mesured nd the predicted surfce elevtions ecomes more evident s the wind speed increses. However, when the wind-driven current is included in the simultion, the performnce of the numericl models is resonle for the weker wind forcing conditions. It my lso e regrded s cceptle for the strongest wind forcing condition, considering the simplicity of the wve reking nd the wind forcing models nd the simplifiction mde in the simultions, e.g. initil condition genertion using only liner wve theory nd the ssumptions of constnt wind speed, wind friction velocity, nd wind-driven current speed in given wind condition. For more ccurte wve predictions under strong wind forcing conditions, these ssumptions should e re-evluted to develop more relistic wind forcing model.

10 20 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) Fig. 15. Wve mplitude mplifiction fctors under two wind conditions. Dotted lines: no wind forcing; lue solid lines: Jeffreys sheltering model; green solid lines: Miles model; dshed lines: the comined model used in this study. Fig. 14. Sme s Fig. 8 ut for comprison of the mplitude spectr for the groups under wind U 0 = 3.2 m/s: () wve group DF 1 nd () wve group DF 2. Crosses re from mesurement nd solid lines re computed with simultion results (γ = 0.9%) Discussions It is instructive to understnd the roles of the two wind forcing models (i.e. Miles nd Jeffreys ) on the evolution of focusing wve groups. For low steepness wves nd wek wind forcing condition, ir flow seprtion my not occur; therefore, Miles model my e model relevnt to descrie wind forcing effect. As the wve steepness nd wind speed increse, ir flow seprtion my occur more frequently. However, the seprtion is highly unstedy process nd my persist for only short period [25,26] nd, therefore, the overll effect due to flow seprtion on wve evolution my e less significnt thn expected. For exmple, Yn nd M [27] found tht the overll effect of ir flow seprtion on the formtion of frek wves my e neglected. However, Khrif et l. [10] rgued tht frek wves due to dispersive focusing my e sustined longer cused y ir flow seprtion on the leeside of the extreme wve crests. Note tht ir flow my e fully seprted over wter wves nd persist for long durtion under extreme wind conditions [32], ut such conditions re eyond the scope of this study. Here, we conduct dditionl numericl tests to investigte the effects of the two wind forcing models on the evolution of focusing wve groups under moderte wind forcing conditions. The numericl set-up is lmost the sme s tht in Section 4.1 except for the following fcts. First, the fifth-order model is solved throughout the numericl domin insted of dividing the domin into one liner nd one nonliner (fifth order) region. Second, the kinemtic viscosity of wter t room temperture (ν = 10 6 m 2 /s) insted of the equivlent viscosity (ν eqv = m 2 /s) is used to minimize the non-reking dissiption effect. Third, the wind forcing is pplied throughout the numericl domin to chieve reltively long wind forcing period; the friction velocity is ssumed to e 5% of the wind speed nd the wind-driven current speed is tken s γ = 0.9%. The initil condition is generted from the nonreking wve group. Fig. 15 shows the mplitude mplifiction fctor, A mx (x n )/A ref, s function of spce for the wve group under four numericl wind forcing conditions: no wind, Miles model, Jeffreys sheltering model, nd comined Miles nd Jeffreys model. Here, A mx (x n ) is the mximum surfce elevtion predicted t loction x n in the numericl wve tnk nd A ref is reference mplitude, which is tken s the verge mximum surfce elevtion oserved etween x n = 22 m 22.5 m. As shown, the wind delys the wve focusing nd pushes it further downstrem. The mximum mplifiction fctor chieved under wind ction is greter thn tht under no wind forcing. In ddition, Miles mechnism produces greter increment in the mplifiction fctor thn the Jeffreys sheltering model for oth wind forcing conditions (U 0 = 5 m/s nd 7 m/s). Fig. 16 presents the totl potentil energy growth s function of time in the numericl wve tnk. Under no wind ction, the energy decreses slightly due to viscosity. In the wind condition of U 0 = 5 m/s, Miles model produces significnt energy trnsfer from wind to the wve group; on the other hnd, ir flow seprtion is predicted, ut the overll contriution due to the sheltering mechnism to wve energy growth is miniml. The comined model, therefore, provides results close to tht of Miles model. When the wind speed increses to 7 m/s, similr oservtions for Miles nd Jeffreys models re mde. However, when the two models re comined, the overll energy trnsferred to the wve group is much greter thn the simple summtion of those due to the two seprte mechnisms. In the focusing stge, Miles model is responsile for the wve energy growth; in the vicinity of the focusing point, the wves ecome strongly nonliner nd, under strong wind forcing, ir flow seprtion my occur continuously. The sheltering mechnism then contriutes gretly to the wve energy growth. According to this numericl investigtion, one my rgue tht Miles model my e considered proper for wves of moderte

11 Z. Tin, W. Choi / Europen Journl of Mechnics B/Fluids 41 (2013) Fig. 16. Totl potentil energy s function of time in the numericl wve tnk. Blue solid lines: no wind ction (for reference); red dsh dotted lines: Jeffreys model only; green solid lines: Miles model only; pink dshed lines: the comined model. steepness under wek to moderte wind forcing; however, for high steep wves under strong wind forcing, oth Miles nd Jeffreys mechnisms my hve to e considered. Note tht Peirson nd Grci [36] showed tht the growth rte prmeter, β, decreses systemticlly with increse wve steepness. In this study, constnt β is used, which my ffect the reltive importnce of the two wind forcing models. This should e n issue to e considered in future studies. The sclility nd pplicility of the numericl models, s well s the proposed ir flow seprtion criterion, to longer wves nd roder wve spectr (ocenic wves) should e exmined crefully. However, detiled studies on the prolem re non-trivil due to the limittion of the experimentl fcilities nd the complicted physicl processes involved. Collortive work involving rigorous theoreticl nlysis, high performnce numericl computtions, nd high qulity field mesurements re necessry to improve our understnding of the prolem. 5. Conclusions The evolution of two-dimensionl dispersive focusing wve groups in deep wter under wind forcing nd wve reking effects is investigted numericlly using wve prediction model sed on pseudo-spectrl method. In ddition, two-dimensionl wind wve experiments re conducted nd surfce elevtions t different wve sttions long the tnk re mesured with highspeed imging. Detiled mesurements of the wind conditions re lso performed. These mesurements re used to evlute the performnce of the numericl models. It is found tht the numericl model produces cceptle predictions for the evolution of the wve groups under the reking nd the wind forcing conditions considered in this study. In the numericl simultions, to model reking wves, n eddy viscosity model is incorported into system of nonliner evolution equtions for the surfce elevtion nd the free surfce velocity potentil to simulte energy dissiption due to wve reking nd predict surfce elevtions fter reking. Wind forcing is modeled y introducing surfce slope coherent pressure distriution in the dynmic free surfce oundry condition. The pressure term is expressed through Miles sher flow instility theory nd Jeffreys sheltering theory. To pply the two mechnisms to wves under wind forcing, i.e. comined wind forcing model, n ir flow seprtion criterion depending on wind speed nd wve steepness is proposed sed on lortory experiments nd field oservtions. In ddition, it is found tht wind-driven current hs to e considered in the simultion of wve evolution under wind forcing. For the wves nd wind forcing considered in this study, the mgnitude of wind-driven current eing 0.9% of the wind speed produces resonle predictions. Direct comprisons of the experimentl mesurements nd the numericl simultions re mde. For wve groups under no wind ction, the eddy viscosity model simultes well the energy dissipted in reking wves nd predicts well the surfce elevtion fter reking. The predicted wve spectr efore nd fter reking lso gree well with the mesurements. Under the weker wind forcing condition, the comined wind forcing model, fter considertion of the wind-driven current, produces stisfying prediction. As the wind forcing ecomes stronger, the disprity etween the experiments nd the simultions ecomes more evident, ut the numericl results re still regrded s cceptle, considering the use of the reltively simple wve reking nd wind forcing models. The reltive importnce of Miles nd the Jeffreys models re discussed through dditionl numericl investigtions. It is shown tht Miles model my e considered pproprite for wves of moderte steepness under wek to moderte wind forcing; however, for high steep wves under strong wind forcing, oth Miles nd Jeffreys mechnisms my hve to e considered. The wind forcing nd wve reking models considered in this study still need to e improved considerly for ocenic pplictions since these models hve een developed for longcrested wves y ssuming irflows re lminr or, t lest, neglecting turulent fluctutions in wind. Therefore, we stress tht cre must e tken when our findings re pplied to rel ocen conditions s the sptil nd temporl scles of turulence in the ocen re so different from those in lortory, in ddition to three-dimensionlity of rel ocen wves. For etter description of wve growth nd reking of ocen wves in wind, we should develop wve model with prmeteriztion for reking of short-crested wves long with turulent closure model vlid for wide rnge of sptil nd temporl scles. Although fr from complete, the present study is expected to help in development of such model. Acknowledgments The uthors grtefully cknowledge the support from the Kore Science nd Engineering Foundtion through the WCU progrm (Grnt No. R ). References [1] G. Wu, Direct simultion nd deterministic prediction of lrge-scle nonliner ocen wve-field, in: Ph.D. Disserttion, Msschusetts Institute of Technology, [2] A. Klmikov, Modeling wind forcing in phse resolving simultion of nonliner wind wves, in: Ph.D. Disserttion, Msschusetts Institute of Technology, [3] B.J. West, K.A. Brueckner, R.S. Jnd, D.M. Milder, R.L. Milton, A new numericl method for surfce hydrodynmics, J. Geophys. Res. 92 (1987) [4] W. Btemn, C. Swn, P. 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