A Directional Spectral Model for Combined Wave Refraction and Diffraction

Transcription

1 The oureeh OMIAR Worshop o Oea Models A Dreoal peral Model for Combed Wave Refrao ad Dffrao Ch Wa LI ad P. Z. LIN Deparme of Cvl ad ruural Eeer, POLYU, o Ko Deparme of Cvl Eeer, NU, apore ABTRACT A dreoal speral model for ombed wave refrao ad dffrao s developed hs sud. Besdes wave refrao, whh has bee osdered mos of he es dreoal speral models e.., he hrd eerao oea wave model b WAMDI Group, 988, wave dffrao s also luded he urre model, whh s epressed erms of omple wave heh des sperum. The dffrao effe s represeed b addoal dffuso erms wh maar dffusves, he oral form of whh a be derved from he mld slope equao. The model has bee valdaed aas omprehesve ase sudes for boh reular waves ad rreular waves. Numeral resuls are ompared o avalable aalal soluos ad epermeal daa for problems of wave dffrao, wave refrao, ad ombed wave refrao ad dffrao. The overall performae of he model s sasfaor. The model s a useful ool o smulae he rase sae of oea waves he oasal reo where boh wave refrao ad dffrao are sfa.. INTRODUCTION The dreoal wave speral model ha was developed based o he radave-rasfer equao has bee appled suessfull o oea wave model for more ha hr ears. Based o he eer balae oep, a eeral raspor equao erms of dreoal wave eer des spera a be derved. B lud approprae soure erms, he model a be used o smulae wave eerao, propaao, ad dsspao oeas. ar from he so-alled frs eerao wave model e.., Perso e al., 966, he model has bee ouousl mproved for see beer represeaos of olear wave eer rasfer. Overom he umeral dffules he frs ad seod eerao wave models e.., asselma, 976, he hrd eerao wave model WAMDI Group, 988 has bee prove o be a useful ool for model lo-erm ad lare-sale wave lmaes loball. Reel, effors have bee made o eed applaos of he hrd eerao wave model o oasal reos where wave brea ad wave dffrao ma beome Pred o reled paper -

2 The oureeh OMIAR Worshop o Oea Models - Pred o reled paper sfa. The dffrao effe ha a be easl luded b ma phase-resolv wave models, however, s o drel dervable from he speral eer balae equao. o far, here s o sadard wa of orpora dffrao a dreoal speral model. Wh he use of eoal equao, Ebersole 985 ad Lo 99 proposed a wa of ra wave ra dreo modfed b wave dffrao. uh a mehod, however, suffers dfful whe here ess a surfae-per obje suh as a breawaer, o whh he ra ra fals. Moreover, her models ol wor for sead sae suao. Reel, Rvero e al. 997 proposed a wa of orpora he dffrao effe o a oveoal speral wave model. New epressos for wave roup velo ad aular eer rasfer rae were derved from he eoal equao ha aes o aou wave dffrao effe. O he oher had, Booj e al. 997 proposed aoher model, whh a arfal dffuso erm s added o orre he erm of aular eer rasfer rae so ha he dffrao effe ould be osdered era deree. I hs sud, a ssema dervao ad eplaao of wave dffrao wll be aemped based o he mld slope equao, whh a be derved from he poeal flow heor e.., Berhoff, 97. The wave dffrao proess wll he be orporaed o a dreoal speral model based o he dervao from he mld slope equao. As a resul, he model a be used o smulae he rase sae of wave propaao, refrao, ad dffrao he oasal reo.. MODEL DECRIPTION or a moohroma wave ra, he me-depede mld slope equao whou urre reads Demas, 997; pp. 5: 0 η ω η η η, where η s he free surfae dsplaeme, ω s he aular freque, ω πf, s he wave umber, s he wave eler, s he roup velo, f s he freque. Assum η f,,,e f,,,, where s he wave heh fuo slowl var me ad spae ad s he phase fuo represe he fas var wave moo, ad subsu o equao, we have: 0 ω. If we assume ha everwhere, he wave heh fuo ma have o be a

3 The oureeh OMIAR Worshop o Oea Models Pred o reled paper -3 omple fuo o mae he assumpo of η f,,,e f,,, uforml vald whe dffrao proess s sfa. The wave heh fuo he oas boh wave amplude ad phase formao. The phsal wave heh s defed as. Uder hs rumsae, he las erm of aels ou. ubsu defos of / os ad / s o ad emplo he equal of ω, we have: 0 s s os os ω. 3 I osa waer deph ad a sead sae, equao 3 s a pal frs order resul of parabol appromao for wave dffrao, f we furher assume ha wave propaaes he -dreo ad wave heh vares muh more rapdl he -dreo ha he -dreo. I hs sud, se we are eresed he rase wave model, he frs order me dervave he real par of 3 wll be reaed. owever, for he eas mplemeao of umeral model, we do assume ha he emporal varao of wave heh s muh slower ha he spaal varao erms of seod order dervaves he maar par of 3. As a resul, equao 3 a be furher redued o he follow form wh he use of defos os ad s :. Equao s a rase adveo ad dffuso equao wh he adveo represe wave propaao ad he dffuso wave dffrao. I s oed ha b assum, we have eleed he hae of wave dreo due o wave dffrao. Isead, he dffrao effe s represeed b he dffuso of wave eer aross ad alo he ma wave propaao dreo. The over equao for he dreoal wave eer des sperum Ef,,,,, where s he ale of he propaa dreo, follows from he eeral eer oservao equao WAMDI Group, 988: E Q E E E E 5 where QE s he soure erm ha represes he eer rowh, dsspao, ad ehae amo dffere freque due o olear wave erao. Aord o Dea ad Dalrmple 99; pp. 0, he omple wave heh des sperum

4 The oureeh OMIAR Worshop o Oea Models - Pred o reled paper f,,,, s relaed o he wave eer des sperum Ef,,,, b he epresso E, where s he ojuae of. ubsu he above epresso o 5, we he have: Q. 6 The above equao a be spl o a ssem of wo equaos as follows: Q 7a Q 7b e equao 7a ad 7b are muuall dervable,.e., ompu ad 7a wll resul 7b ad vse verse, solv a oe of wo equaos would be suffe. I hs sud, he follow dervao wll be based o 7a. Requr ha 0, equao 7a s reas o he form: Q. 8 Compar equao 8 wh equao ad osder a sle b of wave ra wh a sle freque 8, s read o fd ha he wave propaao adveo erms equao 8 s eal he same as hose equao, whh s derved from he mld slope equao. owever, equao oas addoal dffuso erms ha represe wave dffrao effes, whle equao 8 solves wave refrao epll b lud he erms of eer rasfer amo dffere dreoal modes. or a wave ra wh he slowl bed wave ra, s jusfed o assume ha wave dffrao a dreoal speral model ould be modeled a wa smlarl o ha equao. Wh he added dffuso erms, equao 8 s he modfed o be:, P. 9 I he above equao, he wave dffrao proess s hadled b speal soure

5 The oureeh OMIAR Worshop o Oea Models erms besdes he oveoal oe P, Q /. The assumpo behd hs reame s ha he wave refrao ad dffrao are depede proesses wh he refrao be deermed ol b he var boom oporaph ad dffrao b he var wave heh dsrbuo. I should be sressed aa ha suh a reame does o pred eer rasfer amo dffere bs of waves due o dffrao. Isead, ol he orre amou of spread rae of wave eer due o dffrao s o be modeled. 3. NUMERICAL ALGORITM I he prese model, he wave heh des spera are represeed b he dsree wave modes. Thus, he oal sfa wave heh a era loao ad me s,, s he erao of he dsree wave heh des spera wh respe o boh freque f ad dreo,.e.,,, s m j f,,,, j f j. 0 The model solves he over equao 9 b us he operaor-spl mehod e.. L ad Mao, 99, whh reas he wave propaao, wave dffrao, wave refrao, ad wave rowh ad dsspao soure erms separael. The wave propaao s represeed b he frs fve erms o he lef-had sde of 9. I order o aheve beer aura, he equao s rerouped o be he summao of he oservave form mus a soure erm: / 0. I he urre model, Equao s furher spl o - ad -dreos ad he solved b he oservave La-Wedroff mehod, whh s seod-order aurae. The dffrao sep s represeed b: / / /. Equao s solved b us he Cra-Nolso mehod ha s a mpl sheme ad uodoall sable. I order o mprove he effe of he soluo, he alerae dreo mpl ADI mehod s used, whh furher spls he soluo proedure o he - ad -dreos. The sheme s aa seod order aurae. The wave refrao s he solved. mlarl o he reame for he wave propaao, he over equao s frs modfed o be he summao of he Pred o reled paper -5

6 The oureeh OMIAR Worshop o Oea Models -6 Pred o reled paper oservave form mus a soure erm as: 0 / / 3 /. 3 The oservave upwd sheme s hose o dsreze 3 beause a effevel suppress umeral dsperso ha auses arfal wles. all, he soure erm s solved b he forward me sheme as follows: 3 / 3 /, P. The umeral sheme s sable as lo as he Coura umbers defed he propaao ad refrao proesses are less ha u: os s, s, os ma. 5. NUMERICAL IMULATION Dffrao behd a sem-fe breawaer for a udreoal wave ra The dffrao of a udreoal wave ra, whh propaaes he -dreo wh he al wave heh of 0, behd a sem-fe breawaer osa waer deph s a lassal problem for es a wave dffrao model. Cera dsae awa from he breawaer, he asmpo soluo of he dffrao oeffe D ha a be epressed as he resel fuo s a ood appromao o he rue soluo e.., Pe ad Pre, 95: / 0 / π σ σ s e C D, 6 where σ σ π σ π σ π σ 0 0 os s / du u C du u. I hs sud, a udreoal lear wave ra wh he wave perod of T0s s se from he lef boudar. The sem-fe breawaer aled wh he eave -as s deploed osa waer deph of h0m. The orrespod waveleh s L9.3m. The mesh ssem of wh he rd sze of 30m s used o dsreze he ompuaoal doma. The wall s assumed o be a ompleel absorb boudar o elmae he refleed waves. I ure, he ra-sale plo wh oour les of

7 The oureeh OMIAR Worshop o Oea Models dffrao oeffe D s show. I s oed ha he wave dffrao effe eeds o he upsream area. The reaso s ha he urre model reas he dffuso proess he wave propaao dreo -dreo hs ase ad hus allows he baward eer dffrao, whh s epeed o aree wh phsal suaos. I he shadow reo, loal hae wave dreo ess as he wave heh s omple ad he spaal dsrbuo of he wave heh s o uform here. The wave dreo a be deermed from he rade of he phase fuo whh a be derved from he omple wave heh. ure ves more dealed omparsos of alulaed wave heh wh he asmpo soluos from 6 a hree seos behd he breawaer. arl ood areemes are obaed a all seos wh he beer omparsos aheved a farher seo awa from he breawaer. e he model does o requre he mmum resoluo for a waveleh, a eve oarser rd ssem ma be emploed prple. owever, order o aurael pred he dffrao paer, a few rds eeds o be used o resolve he oslla leh sale show ure, whh rows he dowsream dreo. Aoher umeral es s also performed ha eles dffrao erms he ompuao. I s foud ha a almos perfe sep fuo of D s obaed behd he breawaer. Ths daes ha he umeral dffuso of he urre model, whh has seod-order aura for wave propaao ompuao, s elbl small. Combed wave dffrao ad refrao behd a breawaer ap fro of a slop beah Reel, Yu e al. 000 odued a seres of epermeal sudes of rreular wave dffrao ad refrao behd a breawaer ap. I hs sud, we shall use wo ases of hem o furher valdae our model for smula ombed wave dffrao ad refrao. The dealed formao of epermeal seup should be referred o he oral paper. Impora parameers are provded here ha wll be used he umeral smulao. A hree-dmesoal wave bas of m 7m wh he osa deep waer deph of h 0 0.0m was used sud. Two breawaers were deploed 7.00m awa from he wave-maer wh a ap of 7.85m he eer. Two meers from he breawaer owards he shorele, a plae beah wh osa slope of s/50 was bul see ure 3. The JONWAP freque sperum ad power ose spread fuo were used o eerae rreular waves. Boh of wo ases hs sud have he de ale of 0 5. Oe of hem has wde spread ale wh β6 ad he oher arrow spread ale wh β9. Boh ases have he same pea freque f p z, freque pea ehaeme faor γ, ad Phllps umber α The uform meshes he -dreo wh 0.5m are used. The o-uform meshes he -dreo wh he Pred o reled paper -7

8 The oureeh OMIAR Worshop o Oea Models averaed ~0.5m are used o produe he ea value of breawaer ap. The o-uform aular resoluo wh m 5 ad he oal umber of dreoal modes of 8 s emploed. A osa me sep of 0.s s seleed for sabl osderao. ure shows omparsos of dffrao oeffe dsrbuo bewee umeral resuls ad laboraor measuremes for boh wde ad arrow spread ases. Comparsos have bee made erms of oour les he ere doma. The overall areemes for boh ases are reasoabl ood. The spread of wave eer b dffrao ad bed of wave ras owards o he shorele b refrao has bee aurael preded. I boh ases, wave heh rh behd he breawaer v of he ap has bee uderesmaed aa b he umeral model. or wde spread de waves, he model preds lower wave heh he eral area, ompared o epermeal daa. Whle for arrow spread waves, overesmaos of wave heh b he umeral model he same area are observed. The dffere behavors of he model wo ases are lel aused b he emplome of a raher smple boom fro model, whh he boom fro oeffe s se o be a osa depede of wave heh. We have performed he umeral epermes b ele boom fro ad he model he sfal overesmaes wave heh he eral area for boh ases. Ths mples ha aurae alulao of boom fro s mpora hs ase. Ths ssue s relaed o he aurae represeao of eeral soure erms ad wll be furher vesaed he fuure wors. 5. CONCLUION I hs sud, a ew form of dreoal speral wave model s developed erms of omple wave heh des speral. The model has he apabl of smula he ombed wave dffrao ad refrao. Wave dffrao s reaed as he wave heh dffuso hs model. The dffuso erms a be derved from he mld slope equao ad have he maar dffusv ha s proporoal o waveleh ad roup velo. The model has bee valdaed b omprehesve ase sudes for boh reular waves ad rreular waves. The model s useful for pred wave odos a reoal oasal area where boh wave refrao ad dffrao are sfa. ACKNOWLEDGEMENT Ths wor was parl suppored b a ra from he o Ko Poleh Uvers Proje No. A-PD7. REERENCE Berhoff, J. C. W., 97. Compuao of ombed refrao dffrao. Pro. 3 h I. Coasal E. Cof., ACE, New Yor, pp Pred o reled paper

9 The oureeh OMIAR Worshop o Oea Models Booj, N., olhujse, L.., Door, N., ad Kefebur, A. T. M. M., 997. Dffrao a speral wave model. Pro. 3 rd I. mp. o Oea Wave Measureme ad Aalss, WAVE 97, ACE, pp Dea, R. G. ad Dalrmple, R. A., 99. Waer Wave Mehas for Eeers ad ess. Advaed eres o Oea Eeer,, World ef. Demas M. W., 997. Waer Wave Propaao over Ueve Booms. Advaed eres o Oea Eeer, 3, World ef. Ebersole, B. A., 985. Refrao-dffrao model for lear waer waves. J. Waerwa, Por, Coasal ad Oea E., 6, asselma, K., Ross, D. B., Muller, O., ad ell, W., 976. A paramer wave predo model. J. Phs. Oeaor., 6, L, C. W. ad Mao M., 99. peral model of phoo-eeraed waves shallow waers. J draul. Res., 30, 6-6. Lo, J.-M., 99. A umeral model for ombed refrao-dffrao of shor waves o a slad. Oea E., 8 5, 9-3. Perso, W. J., T, L. J., ad Bear, L., 966. Compuer-based proedure for prepar lobal wave foreass ad wd feld aalses apable of us wave daa obaed from a speeraf. Pro. 6 h mp. Naval drod., pp Pee, W. G. ad Pre, A. T., 95. The dffrao heor of sea waves ad he sheler afforded b breawaers. Phl. Tras. Ro. o. A, 88, Rvero,. J., Arlla, A.., ad Car, E., 997. A aalss of dffrao speral wave models. Pro. 3 rd I. mp. o Oea Wave Measureme ad Aalss, WAVE 97, ACE, pp WAMDI Group, 988. The WAM model A hrd eerao oea wave predo model. J. Phs. Oeaor., 8, Yu, Y.-X., Lu,.-X., L, Y.., ad Wa, O. W.., 000. Refrao ad dffrao of radom waves hrouh breawaer. Oea E., 7, Pred o reled paper -9

10 The oureeh OMIAR Worshop o Oea Models ure Caulaed dsrbuo of dffrao oeffe ra sales ad oour les aroud a ompleel absorb sem-fe breawaer. ure. Comparsos of alulaed dffrao oeffe rles wh asmpo soluo sold le a a /L.6, b /L6.50, ad /L Pred o reled paper

11 The oureeh OMIAR Worshop o Oea Models ure 3. Epermeal seup for rreular waves pass hrouh a breawaer ap fro of a slop beah; dashed oours represe waer deph meer. ure. Comparsos of dffrao oeffe bewee umeral resuls sold les ad epermeal daa dashed les for a wde ad b arrow spread de waves behd he breawaer ap. Pred o reled paper -