466 010,(4):466-477 DOI: 10.1016/S1001-6058(09)60078-0 NUMERICAL SIMULAION OF EXREME WAVE GENERAION USING VOF MEHOD * ZHAO X-zeng RIAM, Kyushu Unversty, Fukuoka, Jaan State Key Laboratory o Coastal and Oshore Engneerng, Dalan Unversty o echnology, Dalan 11604, Chna, E-mal: xzengzhao@gmal.com HU Chang-hong RIAM, Kyushu Unversty, Fukuoka, Jaan SUN Zhao-chen State Key Laboratory o Coastal and Oshore Engneerng, Dalan Unversty o echnology, Dalan 11604, Chna (Receved October 14, 009, Revsed Arl 5, 010) Abstract: Numercal smulatons o extreme wave generaton are carred out by usng the Volume O Flud (VOF) method. Extreme waves are generated based on wave ocusng n a -D numercal model. o valdate the caablty o the VOF-based model descrbed n ths artcle, the roagaton o regular waves s comuted and comared wth the theoretcal results. By adjustng the hases o wave comonents, extreme waves are ormed at gven tme and gven oston n the comutaton. he numercal results are comared wth theoretcal solutons and exermental data. It s concluded that the resent model based on the VOF technque can rovde accetably accurate numercal results to serve ractcal uroses. Key words: extreme wave, Volume O Flud (VOF), wave ocusng, wave tank, regular wave 1. Introducton he extreme wave (also called rogue wave, reak wave, huge wave) s an extremely large water wave n ocean and may cover all the sea area. Deste ts low robablty o occurrence, such a wave may lead to damage o shs and oshore latorms [1]. Many accdents were reorted that may be traced to extreme waves [], n whch shs were broken down and a great number o lves (about 55 durng 1969-1994) were lost. As the ocean ndustral exloraton moves towards deeer sea and harsher envronments, extreme waves would become a real threat and, thereore, an * Project suort by the Natonal Natural Scence Foundaton o Chna (Grant No. 50779004). Bograhy: ZHAO X-zeng(1979-), Male, Ph. D., Research Fellow ssue o ntensve studes [3-6]. Recent studes show that the occurrence o extreme waves may be related to wave energy ocusng ncludng a number o actors: wave-wave nteracton, wave-current nteracton, bathymetry, wnd eect, sel-ocusng nstabltes, and drectonal eects. hese related varous mechansms o ormaton o extreme waves were revewed n detals by Khar and Pelnovsky [7] and Dysthe et al. [8]. Due to the lmtaton n wave tank length, none o exstng hyscal lumes has ever been long enough to create a ull-scale extreme wave wth long-tme evolutons [9, 10]. In laboratory, a relatvely easy way to generate extreme waves s usng wave ocusng. he wave ocusng n model tests and smulatons s not a new concet and s aled n studes such as Re.[11,1], where attentons were ad on the characters o the extreme wave wthout consderng the structures. Along wth laboratory exerments, numercal
467 smulatons are useul tools or desgnng coastal structures as well as or understandng natural hydrodynamc rocesses n the eld o ocean engneerng. A number o numercal methods have been develoed over the ast ew decades [13-17] or the nterace modelng, among whch Volume O Flud (VOF) methods were extensvely emloyed to redct the wave moton, wave dstorton and the nteractons between the waves and the marne structures. Snce the VOF method made ts rst aearance and was mlemented wth the SOLA-VOF code, the VOF method has been mroved n several asects and the VOF-based model has been aled to solve varous roblems n the elds o castng, dynamcs o dros, thn lm, ol sllng, sray deoston, meltng rocess n metallurgcal vessels, sh hydrodynamcs, etc. For recent studes wth the VOF-based model to smulate extreme wave roblems, one may quote [18-0]. It should be onted out, as ar as roblems assocated wth extreme waves are concerned, the VOF-tye models are not yet wdely aled. hs work s motvated by the requrement o develong a mathematcal model to nvestgate the survvablty o oshore and ocean structures due to extreme wave macts. In ths artcle, a Numercal Wave ank (NW) s constructed or extreme wave generaton based on VOF method and the roosed model s valdated. In the numercal model, a two-equaton k turbulence model s ncororated nto the low solver. Extensve valdaton comutatons have been carred out ste by ste. Regular wave roagaton wth a range o wave length and wave steeness s rst checked by comarng to the theoretcal solutons. he extreme waves are generated at gven ostons and tmes by adjustng the comonent wave hases. Some o the numercal results are comared wth the exerments erormed n Dalan Unversty o echnology by one o the authors [1]. Fnally, some artcular eatures o the ocused extreme waves are also dscussed, to clary the mechancal background o the extreme wave wth resect to Benjamn-Fer Index (BFI) related wth nonlnear eects.. Mathematcal model.1 Governng equatons and boundary condtons We consder a -D turbulent low eld wth a ree surace. he numercal smulaton s carred out on a -D wave lume wth a lat bottom as shown n Fg.1. A ston-tye wavemaker located at x =0 s used to generate the ncdent waves. he water s assumed to be an ncomressble, vscous lud. he governng equatons are u x =0 (1) and the Reynolds tme-averaged Naver-Stokes equaton u u u 1 u t x y x xx + uj + v = + g + ν ρ j () where u, =1, are the velocty comonents along the coordnate axes x, t s tme, s the hydrodynamc ressure, ρ s the water densty, ν = μ/ ρ the knematc vscosty coecent o the water, g s gravty acceleratons. In what ollows, the coordnate axes and the velocty wll be wrtten nterchangeably as ( x, y ) or ( x 1, x ) and ( u, v ) or ( u 1, u ). For smulatons o hgh Reynolds number o lows, the Reynolds averaged orm o Eqs.(1)-() should be used. hen the molecular vscosty ν s relaced by the turbulent vscosty ν t n the N-S equaton. κ ν t = C μ (3) where C μ =0.09, κ s the turbulent knetc energy and s the dssaton rate o the turbulent knetc energy, whch can be solved by the two-equaton k turbulence models to account or the eect o turbulence as ollows. Fg.1 Schematc dagram o the numercal wave tank κ κ + uj = Dκ + P t x j κ + u j = D + C P C t x κ j ( ) 1 κ (4) (5)
468 where 1 u u j P = t + ν ν t κ, D = x j x κ x j σ D t = x j σ ν x j C =1.43 1, C =1.9, σ k =1.0, σ t =1.3, κ κ, x j In addton to the boundary condtons at the ree surace, the no-sl boundary condtons are satsed at the bottom o the lume. he downstream boundary condton, to be descrbed n the ollowng secton, s used to dam out the outgong waves. he ntal condtons o the veloctes, hydrodynamc ressure, and surace dslacements are set to zero at tme t =0.. Numercal mlementaton..1 Doman dvson he rogram comutes the ree surace, the velocty and ressures n three derent, couled domans: (1) wave generaton boundary, n whch a ston or a la tye wavemaker can be smulated, () man doman, or wave roagaton and wave-structure nteracton, and (3) damng doman, or dssatng wave energy and reducng wave relectons. he domans are brely dscussed as ollows. (1) Wavemaker boundary Accordng to the lnear wavemaker theory, the ncdent velocty U can be calculated by the ollowng equaton or a seced wave. ω U = η Τ ω ( ) (6) where η s the exected water wave elevaton and ω s the wave requency, ( ω ) s the transer uncton or a ston tye wave maker and can be calculated by the ollowng equaton or two-dmensonal waves. can be ound n Re.[,3]. () Man doman he doman s next to the wave-maker boundary. In ths doman, the desred waves are ormed, the actual wave data o nterest are comuted wth consderaton o the lud structure nteracton. (3) Damng doman In ths regon, the horzontal dmensons o the grd cells are relatvely large and are ncreased by means o grd stretchng. In the drecton o the wave roagaton the sze o the cells s ncreased by a actor β 1. β = 1 corresonds to the unorm grd. he damng doman, next to the man doman, s used to avod dsturbance o relectons o waves reachng the end o the tank. In the resent numercal model, a unorm vertcal grd sacng nterval along the lume s adoted wth a unorm horzontal grd sacng nterval n the man doman, whle wth varable horzontal grd sacng ntervals n the damng doman... Flow solver A ractonal ste scheme s used to solve the N-S equatons. he ntermedate velocty s comuted rst, then a ressure s obtaned by solvng the Posson equaton derved by enorcng the contnuty constrant, and the nal velocty s udated by smle algebrac oeratons. he ntermedate veloctes û n+1 are comuted by solvng the ollowng sem-mlct equatons: n+1 n n u u 1 u u = + + g Δt ρ x x n+1 n+1 j n+1 ν u j (8) n+1 he ressure s then comuted by solvng the Posson equaton as ollows: n+1 n+1 ρ u = + Δt x n (9) n+1 he velocty u are nally calculated by smle algebrac oeratons: ( ω) ( kh ) cosh 1 = kh +snh kh (7) u = u n+1 n+1 n+1 n Δt ρ x x (10) where h s the water deth to be smulated and k s the wavenumber corresondng to ω. A tme ram s suermosed on the wavemaker moton at the start o the smulaton. A more detaled dscusson on the tme ram o the wavemaker moton he Successve Over-Relaxaton (SOR) method s used to solve Eq.(10). Once the ressure s comuted rom the Posson equaton, the velocty eld can be udated by Eq.(10)...3 VOF method
469 he VOF method s used to descrbe the comlcated ree surace, wth a volume o lud uncton F to dene the water regon. he hyscal meanng o the F uncton s the ractonal volume o a cell occued by water, wth unty to reresent a cell ull o water and zero or a cell wth no water. Cells wth F values between zero and unty must then contan the ree surace. he tme deendence o F s governed by the ollowng equaton: ( uf) F + =0 t x (11) here are a number o algorthms such as the Donor-accetor, YOUNGS method, YOKOI method [17] etc. or the treatment o the advecton term n Eq.(11). In ths study, the soluton algorthm roosed n the orgnal VOF method s emloyed. 3. Comutatonal results and dscussons o demonstrate the erormance o the roosed numercal model, the numercal results o waves and extreme waves are comared wth theoretcal solutons and exerments. 3.1 Generaton o regular wave In the ollowng art, the regular waves s generated by usng three grds as shown n able 1, where Nx and Ny are the number o grds n x- and y-drectons, Dx and Dy are the mnmum grd sacng ntervals n x- and y-drectons, resectvely. hree wavelengths are chosen to test the numercal model or short waves and long waves. he comutatons are comared wth lnear wave solutons and the second order Stokes solutons [4]. he lnear analyss s suorted wthn the rame o the lnear wave-maker theory. he numercal results are comared wth the Stokes soluton n order to evaluate the eects due to the nteracton o wave comonents n the wave tran. Comutaton condtons are resented n able. Wave condtons smulated here wll be used as the man art o extreme wave comonents n the next art. Case 1 resonds to the eak requency wave and Cases 3 and 4 gve the range o the wave comonents n the ocused wave grou. surace elevaton or Case 1 at x/ h =5 between the smulatons and the theoretcal soluton, where the smulatons were carred out by usng three derent grds as shown n able 1. Fgure (a) shows the comarson between the smulatons and the lnear solutons, whle Fg. (b) shows the comarson between the smulatons and the second order Stokes solutons. here s almost no derence by usng three derent grds. However, or the comutaton usng Grds 3, the wave heght s a lttle underestmated. he smulatons agree well wth the second Stokes solutons, whle the wave roles can not be redcted by the lnear wave solutons. able Wave condtons h (m) H (m) (s) L (m) (ka) Case 1 Case Case 3 Case 4 0.4 0.4 0.4 0.4 0.096 0.03 0.03 0.064 1.0 1.0 0.769.63 1.94 1.94 0.9 5.00 0.156 0.05 0.11 0.04 able 1 Comutaton grds Dx Dy Nx Ny Grd 1 0.04 m 0.0 m 65 6 Grd 0.06 m 0.03 m 45 18 Grd 3 0.08 m 0.04 m 31 13 Fgure resents a comarson o the ree Fg. Wave elevaton or Case 1 Smulatons o regular waves wth low steeness
470 are llustrated n Fg.3 or Case usng Grds 1, and 3. he theoretcal solutons are also resented n Fg.3 or comarson. One can notce that the smulated results usng three derent grds agree well wth the theoretcal solutons wth a mnor derence between the lnear soluton and the second order Stokes soluton, where nonlnearty eects can be neglected. ree waves or relatvely long waves [4]. Madsen [4] ound that the moton o the wavemaker was rescrbed by Eq.(6), the generated wave would take ths orm () 1 ( ) () 3 a ( kx t) η = η + η + η = cos ω + ( ) ( ) a cos k x ωt + a sn k x ω t (1) where a ( +coshkh) ka coshkh = 4 snh kh 3 (13) a 1 a (tanh) kh 3 n1 = h 4snh kh tanh kh n, n 1 kh = 1+ snhkh (14) Fg.3 Wave elevaton or Case Fg.4 Comarson between smulaton and lnear soluton or Case 3 Fgures 4 and 5 show the smulaton results o short wave Case 3 and long wave Case 4 usng Grds 1, and 3, as comared wth the theoretcal solutons. In Fg.4, t can be notced that the numercal model under-redcts the wave heght or Grds and 3, due to the too coarse grds wth much numercal dssaton. For long waves, there are great dscreances between the smulated results and the theoretcal solutons. In Fg.5(a), the dscreancy between the lnear solutons and the comutatons s manly due to the wave-wave nonlnear nteractons. Whle n Fg.5(b), the dsagreement s manly due to the lnear wavemaker theory that generates undesred Fg.5 Smulated regular wave or Case 4 In Eqs.(13) and (14) k and k denote the wave number o the undamental wave and the ree wave, resectvely. he rst term on the rght hand
471 sde o Eq.(1) s the rst-order wave comonent. he second term reresents the Stokes second-order waves travelng at the same seed as the rst-order wave. he thrd term s the second-order ree wave that roagates accordng to the dserson relatonsh 4 = gk tanh ω k h (15) Hence, nothng s done, the generated waves are contamnated by an unaccetably large ree second harmonc that wll serously dsturb the wave roles and the wave energy. Addng a second-order wavemaker moton could be a useul technque to elmnate the second-order ree waves roduced by a snusodally movng () wavemaker χ. he moton o a ston-tye wave generator s exressed as results usng HOS method [1] and the redcted ree surace roles usng Eq.(1). It can be noted that the resent VOF numercal results are consstent wth the comutatons usng HOS method and those redcted by usng Eq.(1). hs ndcates that the snusodally movng wavemaker does generate second-order ree waves. By adjustng the moton o the wavemaker accordng to Eq.(16), the comarsons among the resent comutatons, the HOS numercal results, and redcted ree surace roles (Eq.(17)) are shown n Fg.7. It can be notced that the resent numercal results agree well wth HOS results and the theoretcal solutons. Moreover, the second-order ree waves were no longer there. () 1 ( ) a χ = χ + χ = χ0 snω t hn1 3 1 4snh kh n sn ω t (16) where χ 0 s the amltude o the wavemaker corresondng to the lnear wavemaker theory. he generated waves are second-order Stokes waves wth a ermanent orm and wll be exressed as Fg.6 Comarson among HOS results, resent VOF comutatons and redcted ree surace roles (Eq.(1)) generated by snusodally movng wavemaker ( ) () 1 ( ) ( ) η x, t = η + η = acos kx ω t + π a cosh kh 3 L snh kh ( t) ( cosh kh + ) cos kx ω (17) he above aroxmate wavemaker theory s lmted 3 to waves wth U 1 = al / h < 8 π /3 o very the accuracy o the resent numercal scheme, the numercal solutons or ths wave maker roblem are shown n the ollowng gures. he comutaton was conducted or the case o a wave channel wth a water deth h = 0.38 m and a erod =.75s. he moton o the wavemaker was taken to be snusodal wth an amltude χ 0 =0.061m. hese values corresond to an Ursell number Ur =7., whch s aroxmately the valdty lmt o the wavemaker theory. he wave seres recorded at x = 4.9 m are shown n Fg.6 and are comared wth the numercal Fg.7 Comarson among HOS numercal results, resent VOF comutatons and redcted ree surace roles (Eq.(17)) by wavemaker wth second-order moton o summarze, we can clam that the numercal model can well redct the generaton o short and long waves, and waves o small and large steeness. It ndcates that the numercal model can be aled to smulate the wave grou comosed o a range o wave comonents. In the next secton, the extreme wave wll be generated by usng the above model and numercal results are comared wth the theoretcal results and exermental data. 3. Generaton o extreme wave In the revous secton, we resent detals o the grd renement and the generaton o regular waves ncludng short and long waves. In ths art, the extreme wave s generated to very the ecency and accuracy o the resent model. he numercal smulatons reroduce the hyscal exerments
47 Fg.8 Comarsons between exermental data and comutatons or derent requency wdths descrbed by Zhao [1]. he exermental nvestgatons were carred out n a wave lume o 50 m long, 3.0 m wde and 1 m dee at the State Key Laboratory o Coastal and Oshore Engneerng, Dalan Unversty o echnology, Chna and the water deth n the exerment was set to 0.4 m. he waves were generated by a ston wavemaker and wave relectons were absorbed by a 4 m oam layer laced at the downstream end o the lume. Wave Gauges (WG) were used to measure the surace elevaton along the lume. In the study by Zhao [1], derent cases were nvestgated wth derent nut amltudes, derent eak requences and requency wdths. he ocusng area was set to be rom 13.8 m to 17.0 m downstream o the wavemaker, where 1 wave gauges were nstalled. In the numercal smulatons, smlar domans are adoted. o save comutatonal resources the doman s shortened to 5 m. he ocal doman s set to be rom 9 m to 10 m away rom the wavemaker. For the urose o urther study o the mact o extreme wave on structure, the nut ocal oston s adjusted to make wave comonents ocusng at x =9.5m, n whch the box s laced or urther study. able 3 Wave condtons or extreme waves A Case c ( mn, max ) d (m) 5 0.0 0.83 (0.68, 0.98) 0.3 6 0.06 0.83 7 0.0 0.83 8 0.06 0.83 9 0.0 0.83 10 0.06 0.83 (0.53, 1.13) 0.6 (0.38, 1.8) 0.9 Case studes are carred out n a two-dmensonal stuaton wth a constant water deth o h = 0.4 m.
473 Derent requency wdths d = 0.3, 0.6, 0.9 are consdered. he eak erod takes values rom Fg.9 Comarsons between exermental data and comutatons or derent eak erod 1.0 s to 1.6 s wth d = 0.6. Wave condtons n the numercal model are shown n able 3. Other comutaton condtons are as ollows: the number o wave comonents: N = 9, the water deth: h = 0.4 m, the ocusng oston and tme: x =9.5m and t = 0 s, the total tme o ntegraton t = 30 s. he ocusng ont s taken to be where the surace elevaton takes a maxmum value and the surace elevatons n the ollowng gures are shted n order to reach the ocusng ont at the same tme. Fgure 8 shows the ree surace elevatons at the ocal ont or three derent requency wdths d =0.3, d = 0.6 and d = 0.9, resectvely. Comutatons are carred out or two nut wave amltudes ( A = 0.0m, 0.06m), wth Grd 1 and Grd. he measured tme seres o the ree surace elevaton are also shown n Fg.8 or comarson. It s shown that the calculatons agree well wth the exermental data. Wth resect to the eect o the requency wdth, t s noted that the derence between the maxmum ocused wave heght and the sdeband-wave heght ncreases wth the ncrease o the requency wdth. hereore, n ractcal alcatons, the wave sectrum should be careully selected. able 4 Wave condtons or extreme waves Case A (m) ( mn, max ) 11 0.0 1 0.06 1.0 (0.70, 1.30) 13 0.04 14 0.08 1.4 (0.41, 1.01) 15 0.04 16 0.08 1.6 (0.33, 0.93)
474 o urther valdate the numercal model, the smulaton results o ocused wave grous wth derent eak erods are comared wth exermental results. he ree surace elevatons at the ocal ont or three derent eak erods are dslayed n Fg.9, n whch two nut ocusng amltudes are consdered usng grd wth a xed requency wdth d = 0.6. he numercal condtons are lsted n able 4. From Fg.9, one can see that wth derent eak erods the numercal results agree well wth exermental data, excet or some derence wth the trough, where the wave trough n the exerment s deeer than that o comutatons. (BFI) to analyze the numercal results. Frstly, the numercal smulatons are carred out wth ncreasng nut amltudes. Fgure 10 shows the ocused surace elevatons or three nut wave amltudes ( A = 0.0 m, 0.038 m, 0.055 m). he eak erod = 1. s and requency wdth d =0.9 and Grd are chosen. In each gure, the surace elevaton measured at the ocusng ont s comared wth the lnear soluton obtaned rom the lnear sum o the ndvdual wave comonents. Fg.10(a) sees a good agreement between the smulatng data and the lnear soluton. In ths case, the amltude o the ndvdual wave comonents s less than 1 mm (.e. a = 0.0 / 0.9 m ) and, consequently, the nonlnear wave-wave nteractons are almost non-exstent. Fgs.10(b) and 10(c) show cases o larger nut amltudes, where the ncreased wave amltude roduces an ever greater devaton rom the lnear soluton. he wave crest at the ocusng oston becomes hgher and narrower, whle the adjacent wave troughs become wder and less dee. he nonlnearty thereore creates a steeer wave enveloe. he maxmum crest elevaton s shown n Fg.11. he surace elevaton at the ocusng ont s comared between the smulaton and the lnear nut. he horzontal axs denes the lnear nut soluton A, and the vertcal axs descrbes the crest measured at the ocusng oston. It shows that the devaton between the smulatng data and the lnear nut soluton ncreases wth the nut amltude, whch conrms agan that the ocusng o wave comonents roduces a hghly nonlnear wave grou, n whch the nonlnearty ncreases wth the nut wave amltude. Fg.10 Surace role at the ocal ont 3.3 Eects o nonlnearty he revous sectons concern wth the generatons o regular and extreme waves. he grd renement tests are also dscussed. he numercal results are comared wth the theoretcal soluton and the exermental data wth good agreements. In ths secton the eect o wave nonlnearty on ocused waves are studed wth ncreasng nut amltudes and also wth ntroducton o the Benjamn-Fer Index Fg.11 Crest elevatons or derent amltudes he nonlnearty o the wave acket can be descrbed n terms o a global, sectrum-based wave steeness,, dened as 9 = Ak = k a (18) =1 where a s the amltude o the wave comonents.
475 he wavenumber related wth the eak requency s. Wth the eak requency k held constant, vares wth the nut ocusng amltude A. As shown n Fg.10 and Fg.11, k ncreases as the nut amltude ncreases wth the eak requency k ket constant and t can be understood easly that the dscreances between the smulated ocusng amltudes and the nut amltudes ncrease wth the ncreasng nut amltude. Wth resect to the eak requency, s not very useul. We ntroduce the BFI to analyze the numercal results. It s well known that a weakly nonlnear, unorm, dee water wave tran suers an nstablty known as Benjamn-Fer nstablty, whch aears n the orm o growng modulatons. he rato o steeness to bandwdth s known as the BFI dened by Janssen [5], and s exressed n terms o requency bandwdth as BFI = δω ω (19) where, δω and ω are, resectvely, the wave steeness, the bandwdth and the eak requency n the requency sectrum. BFI was suggested as an ndcator or the robablty o occurrence o extreme waves and lays a key role n the nhomogeneous theory o wave-wave nteractons A smlar arameter was ntroduced and dscussed by Onorato [6] n the context o extreme waves n random sea states. BFI s adoted here and the relaton between Asm / A and BFI s establshed to ndcate the eect o wave nonlnearty, where A sm s the smulated ocusng amltude. Calculatons show that the ocusng amltudes decrease wth decreasng BFI. Four wave cases are consdered wth ket constant s 0.6. able 5 lsts the numercal condtons. able 5 shows that the BFI decreases wth the ncrease o the eak erod. Fgure 1 shows the relaton between Asm / A and BFI. he arameter Asm / A ncreases wth the ncrease o BFI. It s concluded that the wave grous wth large BFI are more severely aected by the nonlnearty than those o small BFI, as s consstent wth what was ound by Janssen [5] n modulaton wave-trans. he relaton between Asm / A and eak erod s shown n Fg.13 and also another wave steeness (=0.086) s consdered wth the same other condtons as shown n able 5. he sold lne, corresondng to = 0.6, s hgher than that corresondng to = 0.086. hs s undoubtedly due to the derence o the wave steeness. For a constant steeness, Asm / A decreases wth the ncrease o eak erod due to the decreasng BFI dscussed n Fg.1. Fg.1 Relaton between Asm / A and BFI able 5 Wave condtons or extreme waves Case 17 18 19 0 A (m) n (s), max ) ( mn 6.0 7.9 9.8 11.6 1.0 1. 1.4 1.6 (0.70, 1.30) (0.53, 1.13) (0.41, 1.01) (0.33, 0.93) BFI 0.43 0.35 0.30 0.7 Fg.13 Relaton between Asm / A and eak erod o urther nvestgate the behavor o ocused waves, the ower sectra o the smulated wave elevatons are obtaned based on the Fourer transorm, as shown n Fg.14 or two eak erods = 1.0 s and = 1.6 s. Fgures 14(a), 14(b) show the sectral roertes o our wave cases corresondng to =0.086 and = 0.6, resectvely. For each
476 case, two normalzed ower sectra are shown: the rst s based on the nut sgnal set to the wavemaker, the second corresonds to the surace elevaton smulated at the ocusng oston. In Fg.14(a) ( =0.086), or the case = 1.0, the shae o the sectra kees almost unchanged; whle or the case =1.6, the smulaton shows a small energy transer to the low-requency and hgh-requency comonents at the ocusng locaton, but the transer o energy s not sgncant. In contrast, Fg.14(b) ( = 0.6 ) shows a sgncant transer o energy nto the lower and hgher harmoncs at the ocusng oston, esecally, or the case = 1.6. For the case =1.0, the transer o energy manly covers the hgher harmoncs, whle or =1.6, more energy s transerred nto the lower as well as the hgher harmoncs, whch can be exlaned by the act that Asm / A decreases wth the ncrease o eak erod as shown n Fg.13. In Fg.14(b), the sectra at the ocusng oston exhbt essental derences rom the ntal sectra close to the wavemaker. A comarson between Fgs.14(a) and 14(b) llustrates that steeer wavetrans ( = 0.6 ) have more energy o transer than those o = 0.086. VOF method s develoed. Secc care s ad to the grd dvson and the moton o the wavemaker. he model s valdated by solvng several -D roblems nvolvng the generatons o regular waves and the ocusng extreme waves. Comutatons are comared wth other numercal results, theoretcal solutons and exermental data and a good agreement s ound. Based on the numercal results, the ollowng conclusons can be drawn: (1) Short and long regular waves are generated. By usng a second order wavemaker theory, the second order ree waves are deleted and long waves are generated n ermanent orms. () Extreme waves are generated, and the comarson o the surace elevatons wth exermental data shows that the wavemaker generaton o extreme waves can be smulated by the numercal scheme wth satsactory accuracy. (3) BFI s ntroduced to analyze the numercal results. It s ound that BFI can be used to analyze the nonlnearty o wave ackets, where the steeness does not work. he results o ths study show that the model can handle the regular and extreme wave generatons. Further nvestgatons are necessary or the resent model to smulate the mact o extreme waves on structure. Reerences Fg.14 Sectral roertes o derent and 4. Conclusons In ths artcle, a numercal model based on the [1] SAI C. H., SU M. Y. and HUANG S. J. Observatons and condtons or occurrence o dangerous coastal waves[j]. Ocean Engneerng, 004, 31(5-6): 745-760. [] LAWON G. Monsters o the dee (the erect wave)[j]. New Scentst, 001, 170(97): 8-3. [3] SUN Z. C., ZHAO X. Z. and ZHANG Y. F. et al. Focusng models or generatng reak waves[c]. Proceedngs o the Eghth ISOPE Pacc/Asa Oshore Mechancs Symosum. Bangkok, haland, 008, 3-38. [4] BUNNIK., VELDMAN A. and WELLENS P. Predcton o extreme wave loads n ocused wave grous[c]. Proceedngs o the 18th (008) Internatonal Oshore and Polar Engneerng Conerence. Vancouver, BC, Canada, 008, 3-38. [5] SUN Y-yan, LIU Shu-xue and LI Jn-xuan. An mroved numercal method or calculatons n wave knematcs[j]. Journal o Hydrodynamcs, 009, 1(6): 86-834. [6] ZHAO X-zeng, SUN Zhao-chen and LIANG Shu-xu. Ecent ocusng models or generaton o reak waves[j]. Chna Ocean Engneerng, 009, 3(3): 49-441. [7] KHARIF C., PELINOVSKY E. Physcal mechansms o the rogue wave henomenon[j]. Euro. J. Mech. B/Fluds, 003, (6): 603-634. [8] DYSHE K., KROGSAD H. E. and MULLER P. Oceanc rogue waves[j]. Annual Revew o Flud
477 Mechancs, 008, 40(1): 87-310. [9] DUCROZE G., BONNEFOY F. 3-D HOS smulaton o extreme waves n oen seas[j]. Nat. Hazards Earth Syst. Sc., 007, 7(1): 109-1. [10] WU G. Y., LIU Y. M. Studyng rogue waves usng large-scale drect hase-resolved smulatons[c]. Proc. 14th, Aha Hulko a Wnter Worksho, Rogue Waves, Honolulu, Hawa, USA, 005, 101-107. [11] LIU S. X., LI J. X. and HONG K. Y. Exermental study on mult-drectonal wave ocusng and breakng n shallow water[c]. Proc. 15th Int. Oshore and Polar Eng. Con. (ISOPE). Seoul, Korea, 005, 486-49. [1] ZHAO X. Z., HU C. H. and SUN Z. C. Numercal smulaton o ocused wave generaton usng CIP method[c]. Proc. 0th Int. Oshore and Polar Eng. Con. (ISOPE). Bejng, Chna, 010, 596-603. [13] GONG Ka, LIU Hua and WANG Ben-long. Water entry o a wedge based on SPH model wth an mroved boundary treatment[j]. Journal o Hydrodynamcs, 009, 1(6): 750-757. [14] ZHENG Kun, SUN Zhao-chen and SUN Ja-wen et al. Numercal smulatons o water wave dynamcs based on SPH methods[j]. Journal o Hydrodynamcs, 009, 1(6): 843-850. [15] HU C. H., KASHIWAGI M. A CIP-based method or numercal smulatons o volent ree-surace lows[j]. J. Mar. Sc. echnol., 004, 9(4): 143-157. [16] ZHAO X-zeng, SUN Zhao-chen and LIANG Shu-xu. A numercal method or nonlnear water waves[j]. Journal o Hydrodynamcs, 009, 1(3): 401-407. [17] YOKOI K. Ecent mlementaton o HINC scheme: A smle and ractcal smoothed VOF algorthm[j]. Journal o Comutatonal Physcs, 007, 6(): 1985-00. [18] HUIJSMANS R., GROESEN E. Coulng reak wave events wth green water smulatons[c]. Proc. 14th Int. Oshore and Polar Engneerng Conerence ISOPE. 004, 1: 366-373. [19] CLAUSS G., SCK R. and SEMPINSKI F. et al. Comutatonal and exermental smulaton o nonbreakng and breakng waves or the nvestgaton o structural loads and motons[c]. 5th Internatonal Conerence on Oshore Mechancs and Arctc Engneerng. Hamburg, Germany, 006, 1-9. [0] MOCAR O. E., SCHELLIN. E. and JAHNKE. et al. Wave load and structural analyss or a jack-u latorm n reak waves[j]. Journal o Oshore Mechancs and Arctc Engneerng, 009, 131(): 0160. [1] ZHAO X X-zeng. Exermental and numercal study o reak waves[d]. Ph. D. hess, Dalan: Dalan Unversty o echnology, 008(n Chnese). [] ZHAO X. Z., SUN Z. C. and LIANG S. X. A numercal study o the transormaton o water waves generated n a wave lume[j]. Flud Dyn. Res., 009, 41(3): 035510. [3] ZHAO X. Z., HU C. H. and SUN Z. C. et al. Valdaton o the ntalzaton o a numercal wave lume usng a tme ram[j]. Flud Dyn. Res., 010, 4(4): 045504. [4] MADSEN O. S. On the generaton o long waves[j]. J. Geohys. Res., 1971, 76(36): 867-8683. [5] JANSSEN P. A. E. M. Nonlnear our-wave nteractons and reak waves[j]. J. Phys. Oceanogr., 003, 33(4): 863-899. [6] ONORAO M., OSBORNE A. R. and SERIO M. et al. Freak waves n random oceanc sea states[j]. Phys. Rev. Lett., 001, 86(5): 5831-5834.