VISCOUS FLOW FIELDS INDUCED BY A BREAKING SOLITARY WAVE OVER A SHELF

Journal of Marne Scence and echnolog, Vol. 3, No. 6, pp. 855-863 (05) 855 DOI: 0.69/JMS-05-060- VISCOUS FLOW FIELDS INDUCED BY A BREAKING SOLIARY WAVE OVER A SHELF Chng-Jer Huang,, Yen-sen Ln, and Chun-Yuan Ln Ke words: RANS equatons, partcle level set metd, soltar wave, splash-up, energ balance. ABSRAC hs work nvestgates the vscous flow felds nduced b a soltar wave passng over a shelf or a step. he proposed numercal model solves the unstead two-dmensonal Renolds Averaged Naver-Stokes (RANS) equatons and the turbulence equatons. he fnte-analtcal scheme s used to dscretze the dfferental equatons nvolved n the RANS model. he partcle level set metd s adopted to capture the complex free surface evoluton. Accurac of the proposed model n smulatng breakng soltar wave on a shelf s verfed b comparng numercal wave profles from the ncdent stage to the begnnng of et fall wth the expermental data. Followng verfcaton of the accurac of the proposed numercal model, the surface evoluton, knematc propertes and energ balance nvolved n a breakng soltar wave on the shelf are elucdated n detals. Numercal results ndcate that durng the overturnng of the soltar wave, maxmum veloct of the flud partcles occurs after the frst splash-up and before the second reattachment. I. INRODUCION Paper submtted /0/4; revsed 0/8/5; accepted 06/0/5. Autr for correspondence: Chng-Jer Huang (e-mal: chuang@mal.ncku.edu.tw). Department of Hdraulc and Ocean Engneerng, Natonal Cheng Kung Unverst, anan, awan, R.O.C. Coastal Ocean Montorng Center, Natonal Cheng Kung Unverst, anan, awan, R.O.C. Wave breakng s one of the most commonl observed features of water waves n the coastal zones. When waves break, the momentum of waves s transformed nto the ocean surface laer. Wave breakng thus plas a sgnfcant role n the dsspaton of wave energ. Gven the hgh complext of the phenomena assocated wth wave breakng, earler research focused manl on the evoluton of a breakng soltar wave on a contnental shelf, whch s represented b a vertcal step. Gorng (978) studed the reflecton and transmsson of a soltar wave passng over a shelf and the results were confrmed b expermental data. Losada et al. (989) measured the evoluton of a soltar wave at a step and classfed the evolutons nto four modes concerned manl wth the dstorton, fsson and breakng of the wave. Yasuda et al. (997) nvestgated the knematc propertes of overturnng soltar waves on a step b usng the potental flow model. Surface profles and veloct felds of the flow from the ntal state to the state when the ets that are eected from ther crests plunge nto the front faces were examned. Experments were also performed to verf the accurac of the numercal results for the temporal water surface elevaton before the breakng pont and the spatal water surface profles around the eected et. Despte ts contrbutons, ther stud dd not address the surface evoluton of the breakng soltar wave after the formaton of the eected et, such as re-attachment, splash-up and ar entranment. B solvng the Renolds averaged Naver-Stokes (RANS) equatons, Lu and Cheng (00) studed the evoluton of a soltar wave over a shelf. Both nonbreakng and breakng soltar waves were examned. he breakng waves were smulated b couplng the RANS equatons wth k turbulence equatons. Accordng to ther numercal results, the fsson processes for generatng the second and thrd soltons are qute dfferent for nonbreakng and breakng soltar waves. he aforementoned research focuses manl on the surface evoluton of the breakng soltar waves. However, the knematc behavor and energ balance of the flows assocated wth breakng waves are also crucal for elucdatng the mechansm of the breakng waves. Man expermental studes have explored the knematc behavor assocated wth breakng waves. B usng the Partcle Image Velocmetr technque, Chang and Lu (998) measured the flud partcle veloctes n the overturnng et of a breakng wave. Accordng to ther results, the maxmum flud partcle veloct at the tp of the overturnng et reached.68 tmes of the phase veloct calculated from the lnear wave theor. he complcated free surfaces nvolved n a breakng wave have been smulated usng numercal approaches such as the VOF metd and the smoothed partcle hdrodnamcs (SPH) metd. Recentl, a level set metd (LSM) s developed to capture the nterface between two fluds. he level set metd provdes an effectve means of computng the nterface separaton and combnaton, such as the moton of ar bubbles n water or fallng water drops n ar. However, numercal df-

856 Journal of Marne Scence and echnolog, Vol. 3, No. 6 (05) fuson ma occur as tme proceeds, subsequentl affectng the correct capturng of the nterface. Numerous studes have developed a more accurate and effcent soluton algorthm, referred to as the partcle level set metd, to capture the nterface accuratel, subsequentl mprovng the conservaton of mass n the flow doman (Enrght et al., 00). Wang et al. (009) developed a two phase flow model to smulate spllng breakng waves, n whch the level set metd was mplemented for retrevng the ar-water nterface. Accordng to ther results, surface elevaton, locaton of the breakng pont and undertow profles can be captured. Lubn et al. (0) smulated two-dmensonal breakng waves over a slopng beach b solvng the Naver-Stokes equatons, n ar and water, coupled wth the large edd smulaton (LES). her numercal results were compared wth the expermental observatons. hs work develops a numercal model to examne the surface evoluton, knematc propertes, and energ balance nvolved n a breakng soltar wave over a shelf. he numercal model solves the unstead, two-dmensonal Renolds Averaged Naver-Stokes (RANS) equatons and the turbulence equatons. he nterface between the ar and water phases was captured usng the partcle level set metd. II. GOVERNING EQUAIONS hs work develops a numercal model to stud the surface evoluton and knematc behavor nvolved n a breakng soltar wave on a contnental shelf. he contnental shelf s represented b a step wth a vertcal face nstalled n the computatonal doman. For an ncompressble, vscous flud, the contnut equaton n the Cartesan coordnate sstem s wrtten n tensor form as U x O and the unstead Renolds Averaged Naver-Stokes (RANS) equatons are U ( ) U P U uu U t x x x x x () () n whch the edd vscost s determned as k C (4) where C s an emprcal constant; k s the turbulent knetc energ; s the dsspaton rate of turbulent knetc energ; and s Kronecker s delta. In order to take the wall dampng effect nto account, k models for low Renolds number flows are adopted (Patel et al., 985), whch nvolve emprcal constants and addtonal terms expressed as follows: k k k U t x x x x U uu k U t x x x where U C f uu C f E k x k U E ( ) Launder and Spaldng (974) recommended the followng emprcal constants for a full turbulent flow,.e. C = 9; C =.44; C =.9; k = and =.3. Launder and Sharma (974) proposed the followng terms n the k model for low Renolds number flows n boundar laers to modf the general turbulent transport equaton: (5) (6) (7) k C f (8) where the dampng functon f depends on the turbulence Renolds number R accordng to where U denotes tme-averaged mean veloct of the flud, for two-dmensonal flows ranges from to ; x s the coordnates; t s tme; s denst; P s hdrodnamc pressure, whch equals the reducton of the hdrostatc pressure from the total pressure, and uu are the Renolds stress tensor. In the k model of turbulent flud flows, each Renolds stress s related to the correspondng mean rates of stran b an sotropc edd vscost as follows: U U uu k x x 3 (3) where and f 3.4 exp[ ] R /50 R k / (9) (0) f f R, 0.3exp ()

C.-J. Huang et al.: Vscous Flow Felds Induced b a Breakng Soltar Wave over a Shelf 857 III. LEVEL SE MEHOD he level set metd s a numercal scheme developed to treat the evoluton of nterfaces and shapes. One advantage of the level set metd s that one can perform numercal computatons nvolvng curves and surfaces usng an Euleran approach (wth a fxed Cartesan grd). In two dmensons, the level set metd represents a close curve Γ n the plane as the zero level set of a two-dmensonal auxlar functon,, Γ ( x, ) ( x, ) 0 () and then manpulates Γ mplctl through the functon. hs functon s called a level set functon. he sgned dstance functon s generall csen as the level set functon. In ths work, the nterface, Γ, between ar and water s the zero level set of a smoothed dstance functon ( x, t, ), n whch < 0 denotes the ar regon and > 0 refers to the water regon. Functon s defned as the sgned normal dstance from the nterface, Γ, and satsfes. Durng tme evoluton, can be vewed as a propert convected wth the flow feld. Hence, U t 0 x (3) However, numercal dffuson ma arse after a fnte amount of computatonal tme,.e., the level set ma become rregular and s no longer a dstance functon. hus, the level set functon must be re-ntalzed at each tme step to ensure that the level set functon mantans a smooth dstance functon. hs can be acheved b teratng the followng partal dfferental equaton to reach a stead state, and then replacng ( x, t, ) wth d( x,, τ ), d S( ) d (4) where s an artfcal tme and S( ) s a smoothed sgned functon expressed as S( ) (5) In the numercal computaton, the thckness of nterface, Δ, s csen at least three grd cells n the drecton normal to the nterface, Γ. hus, the level set functon remans a dstance functon wth convergng to a unt wtut changng ts zero level set. In numercal mplementaton, wever, conservaton of mass ma be volated durng the re-dstancng procedure. Enrght et al. (00) developed the partcle level set metd to enhance the mass conservaton propertes of the conventonal level set metd and to reduce the numercal dffuson. he level set functon near a free surface s adusted b usng Lagrangan marker partcles. IV. BOUNDARY AND INIIAL CONDIIONS AND NUMERICAL MEHOD Solvng the RANS and turbulent transport equatons requres approprate boundar condtons at all boundares of the soluton doman, as well as the ntal condtons at t 0 for the entre doman. he ntal condtons of veloctes, hdrodnamc pressure, and surface dsplacement are set to zero at t 0. he knematc condton requres that flud partcles move wth the free surface. hs concept can be descrbed n terms of the advecton of the level set functon, as descrbed earler n Eq. (3). he dnamc condtons along the nterface, Γ, are as follows: n O (6a) P P atm (6b) where denotes the mean veloct of the flud (U or V), the turbulent knetc energ (k), or the dsspaton rate of turbulent knetc energ (), and n s the drecton normal to the nterface, Γ, n /. Eq. (6a) can be satsfed b solvng the followng partal dfferental equaton, untl the stead state s acheved (Peng et al., 999), S( )( ) O (7) where S( ) s a smoothed sgned functon, as expressed n Eq. (5); s a fcttous tme; and the operator represents ( / x, / ). Furthermore, the computatonal doman s extended wth tme, such that the wave does not reach the downstream boundar of the computatonal doman. In the proposed numercal model, the governng equatons were dscretzed b means of a fnte-analtcal scheme. he coupled veloct and pressure felds were calculated usng the SIMPLER algorthm. he evoluton of level set metd was solved usng the fourth-order VD Runge-Kutta metd and ffth-order WENO scheme. Further detals on the generaton of ncdent soltar wave n a numercal wave flume and the assocated numercal schemes can be found n Huang and Dong (00) and Dong and Huang (004). V. VERIFICAION o confrm the accurac of the ncdent soltar wave and the assocated veloct feld of the flow generated n the computatonal doman, Fg. compares the numercal soltar

858 Journal of Marne Scence and echnolog, Vol. 3, No. 6 (05) H.5 0.5-0.5 H /h o = 0.5 - -0.75-0.5-0.5 0 0.5 0.5 0.75 t/(l eff /c) Fg.. Comparson of numercal soltar wave profle wth that gven from Boussnesq s theor; ( ) Numercal results, ( ) Analtcal results usng Boussnesq s theor. Boundar H = 0.34 m Boundar h o = 0.3 m R= 0.63 m Boundar P P3 P4 0.55 m 0.505 m Water Contnental Shelf Fg. 3. Schematc dagram of the expermental setup for measurng the evoluton of soltar wave over a shelf (Yasuda et al., 997 ). Boundar H *.5 0.5-0.5 8 6 4 - -0.75-0.5-0.5 0 0.5 0.5 0.75 t/(l eff /c) 0-30 0 30-30 0 30-30 0 30-30 0 30-30 0 30-30 0 30-30 0 30 U (cm/sec) Fg.. Horzontal veloct profles wthn the boundar laer nduced b a soltar wave at dfferent phases; (smbols) results from the proposed numercal model; (---) analtcal solutons obtaned from Huang and Dong (00). wave profle wth an ncdent wave heght of H /h o = 0.5 wth the theoretcal wave profle obtaned from Boussnesq s theor, H h K x ct (8) sec ( ) 3 where K 3 H /4 and c denotes the phase speed of the wave and equals to g( H h o ). Addtonall, h o s the stll water depth, and H and H denote the ncdent and local wave heght, respectvel. In Fg., the tme s normalzed b L eff /c, where L eff s the effectve wavelength of a soltar wave (Dean and Dalrmple, 995). Fg. compares the numercal and theoretcal rzontal veloct profles near the bottom boundar laer nduced b the soltar wave swn n Fg. at dfferent phases. he theoretcal rzontal veloct profles have been provded b Huang and Dong (00). he numercal grds used n the computatonal doman are x = 0. and = 5 except for near the wall regon, where 0 and 0 grds are unforml dstrbuted wthn 0 * < 5 and 5 * 0, respectvel, where * s defned as, 0.5 Kc /. Notabl, accordng to Fgs. and, the numercal wave profle and veloct feld of the flow generated n the computatonal doman are accurate. 0.6 0.5 0.4 0.3 0. 0. 0.5.5.0.5 t (sec) Fg. 4. Comparson between the numercal results and expermental data of the wave profle on a contnental shelf; smbols (,, ): wave profles recorded b the wave gauges P to P4, lnes ( ; - - -; ): numercal results; Δx = Δ = 06. o demonstrate the accurac of proposed numercal model n smulatng breakng soltar waves on a shelf, the numercal results of the wave profle, from the ncdent stage to the begnnng of wave overturnng, are compared wth the expermental data. Fg. 3 schematcall depcts the expermental setup of Yasuda et al. (997). he stll water depth, h o, s set to 0.3 m; the heght of the shelf s 0.63 m; and the ncdent soltar wave heght, H, s 0.34 m. Four wave gauges (P to P4) are dstrbuted near the leadng edge of the shelf to record the temporal water surface elevaton. o reduce the computatonal tme, the numercal wave flume s set up n a fnte doman of 0.5 m long and 0.6 m hgh usng 3500 00 unform computaton cells. he upstream boundar condton s appled to generate the desred ncdent soltar wave concdent wth that recorded at staton P n the experments,.e. wth a wave heght of 0.4 h o. he shelf s nstalled at 8.5 m (about 4 tmes that of the effectve wave length) awa from the wave paddle, and the wave probes are arranged at the same relatve locatons to the shelf as n Yasuda s experments. Fg. 4 compares the numercal and expermental water surface elevatons at wave gauges P to P4. he tme axs (t') n Fg. 4 s smpl csen to reflect the tme lag when the wave crest reaches varous wave gauges. he numercal results are computed wth x = = 06. he wave gauge P4 s placed n the vcnt of the breakng pont (B.P.). he lnes n Fg. 4 represent the numercal results and the smbols denote the expermental data. hs comparson reveals that evoluton of

C.-J. Huang et al.: Vscous Flow Felds Induced b a Breakng Soltar Wave over a Shelf 859.5.4.3.. 34.5 35.0 x/h o 35.5 36.0 (a).5.4.3.. 34.5 35.0 x/h o 35.5 36.0 (b).5.4.3.. 34.5 35.0 x/h o 35.5 36.0 (c) Fg. 5. Comparsons of the computed water surface profles wth the expermental records at the breakng pont (left curve) and at the et fall ntaton (rght curve) for varous grd cell szes; (a) Δx = Δ = 0, (b) Δx = Δ = 06, (c) Δx = Δ = 043; sold lne: numercal results, smbols: expermental data. the wave profles from the ntal ncdent stage to the begnnng of overturnng s properl smulated usng ths model. hs comparson ndcates also that before wave overturns, the grd cell sze x = = 06 s suffcentl fne to provde an accurate resoluton of the wave profles. Fg. 5 further compares the wave profles after wave breaks for varous grd cell szes. Wth x = = 0 n Fg. 5(a), the grd cell szes decrease to x = = 06 n Fg. 5(b), and to x = = 043 n Fg. 5(c). he tme step t vares wth the grd cell szes to make the Courant number, defned as max(ut/x, Vt/), less than one. B usng a hgh-speed vdeo camera, Yasuda et al. (997) obtaned the spatal wave profle around the eected et. wo wave profles were provded wth the earler one (.e. the left curve) beng that at the breakng pont, whle the latter one (.e. the rght curve) was at the begnnng of the et fall. However, Yasuda et al. (997) dd not provde the tme duraton between these two profles. he numercal wave profle at the breakng pont swn n Fg. 5 s dentcal to the expermental one, as verfed n Fg. 4. Moreover, the latter one s obtaned b smpl cosng the one closest to the expermental data at an tme. Fg. 5 ndcates that wth a fner grd cell sze, the overturnng wave t =.0 s t =.95 s t =.89 s t =.84 s t =.73 s t =.6 s t =.4 s t =.8 s t = 0.70 s 0 7 8 9 03456789034 x/h o Fg. 6. Evoluton of a breakng soltar wave on a shelf at dfferent tmes, where h o = 0.3 m, H /h o = 0.44, R/h o = 48. profle can be properl smulated b the proposed numercal model. Addtonall, the grd cell szes n Fg. 5(b) are fne enough to capture the expermental wave profles. Hence, n the latter computaton, the grd cell szes are set to x = = 06. VI. SURFACE EVOLUION AND KINEMAIC PROPERIES OF FLOWS Fg. 6 presents the evoluton of the breakng soltar wave on the contnental shelf from the ntal ncdent wave at t = 0.70 s, to t =.0 s, when the second splash-up occurs. he ncdent wave condtons and geometr of the shelf n Fg. 6 are the same as tse presented n Fg. 3. Fg. 6 ndcates that when the soltar wave propagates over the shelf, the leadng part of the wave rses due to the salng effect. For shallow water waves, the phase speed of the wave ncreases wth the water depth. Hence, the bulged porton of the wave propagates at a faster speed than the front part of the wave. hs propagaton causes the wave to steepen towards the front, at t =.73 s, eventuall topplng over at t =.84 s. hs topplng effect gves rse to the tpcal pcture of a plungng breaker. Owng to gravt, a seres of splash-up occurs subsequentl. Fg. 6 clearl reveals the reattachment and splash-up process of the overturnng waves at t =.89 s,.95 s, and.0 s. Accordng to Fg. 6, the proposed numercal model can elucdate complex phenomena nvolved n the wave breakng, such as the overturnng of wave, reattachments, and splash-ups.

Journal of Marne Scence and echnolog, Vol. 3, No. 6 (05 ) 860.6.4. 8.5 9 9.5 x/ (a) 0 0.5.6.4 0.93 g. 9 9.5 0 x/ (b) 0.5.6.4.3 g. 0 0.5 x/ (c).5.6.4..5 x/ (d).5.6.4..56 g.5 x/ (e) moves onsre graduall faster, and eventuall exceeds the speed of waveform, resultng n the curlng of the crest and the eventual breakng of waves. When the wave overturns and reattaches the free surface at t =.84 s (Fg. 7(c)), gravt seems to accelerate the eected et; n addton, the maxmum flow veloct ncreases to.30 Co at the leadng edge of the et. he eected water et then bumps aganst the undsturbed water surface, causng a water splash nto the ar, as swn n Fg. 7(d). Meanwhle, a vod forms as the et bumps nto the water. he splash-up seems to receve energ from the man flow, explanng wh the maxmum veloct occurs at the regon near the reattachment pont wth a hgh speed of.80 Co, whch s ver close to the value of.68 observed b Chang and Lu (997). Fgs. 7(e) and 7(f) sw the successve recurrence of the reattachment and splash-up. he flud wth the maxmum veloct of.8 Co n Fg. 7(d) decreases to.56 Co n Fg. 7(e) and to.53 Co n Fg. 7(f). VII. ENERGY BALANCE IN A BREAKING SOLIARY WAVE ON A SHELF.8 g 0.5 Fgs. 7(a) to (f) sw the contour maps of veloct felds nduced b the breakng soltar wave on a shelf at dfferent tmes to examne the knematc propertes of the overturnng waves. In Fg. 7 the abscssas are not fxed, but are csen to focus on the regon near the front of the waves. Notabl, Fg. 7 reveals that as the front of the wave evolved nto a vertcal shape at t =.73 s, Fg. 7(b), the maxmum flow veloct occurs at the top of the front wth a speed of 0.93 Co (Co = g ). From t =.73 s to.84 s, water at the wave crest.5 he last secton descrbed the evoluton of breakng soltar wave and ts knematc propertes on a shelf. he phscal phenomena nvolved n the breakng waves (e.g., reattachments, splash-ups, and ar entranment) cause energ dsspaton, whch s an mportant effect of the wave breakng and warrants further stud. otal energ of the water waves ( E total ) can be dvded nto the potental energ ( E pot ) and the knetc energ ( E kn ). he wave-nduced potental energ can be determned as follows. 3 E pot x H ( ) g d dx x 0 g d dx (9).6 where x and denote the nterval of ntegraton n the x and axes, respectvel, and represents the stll water depth. he knetc energ s.4.53 g..5.5 x/ (f) 3 3.5 Fg. 7. Contour maps of veloctes nduced b a breakng soltar wave on a shelf at dfferent tmes, n whch t = (a).6 sec, (b).73 sec (c).84 sec, (d).89 sec, (e).95 sec, and (f).0 sec. E kn x H ( ) (U V ) d dx (0) where U and V refer to the rzontal and vertcal tmeaveraged mean veloct components, respectvel, and H( ) denotes the smoothed Heavsde functon and s defned as

C.-J. Huang et al.: Vscous Flow Felds Induced b a Breakng Soltar Wave over a Shelf 86 E pot E E kn E E total E 0.6 0.4 0. 0.6..4.6.8.0. t (sec) Fg. 8. me evolutons of potental energ (), knetc energ ( ), and total energ () wthn the wle computatonal doman as a soltar wave propagates over a shelf under the same condtons as tse n Fg. 3. de pot dt de kn dt de total dt 5 5.0-5.0-5 0.6..4.6.8.0. t (sec) Fg. 9. emporal varaton of tme rate of change of potental energ (), knetc energ ( ), and total energ () as a soltar wave propagates over a shelf. he unt n the vertcal axs s Joule/(s m). he vertcal dashed lnes ndcate some of the tmes of wave profles swn n Fg. 6,.e. t = 0.70 sec,.8 sec,.6 sec,.84 sec,.89 sec, and.95 sec. O f Δ H( ) sn f Δ Δ Δ Δ f Δ () Secton III defnes the level set functon and the thckness. he reflecton coeffcent (K R ) and transmsson coeffcent (K ) are defned as offsre K E / E () R onsre K E / E (3) offsre where E denotes the ncdent wave energ, and E and onsre E are the energ of the wave propagatng n the offsre and onsre drectons, respectvel. he coeffcent of the energ dsspaton, K D, s then determned as follows. K K K (4) D R Fg. 8 presents the tme evoluton of the potental energ, knetc energ and total energ wthn the wle computatonal doman as a soltar wave propagates over a shelf under the same condtons as tse n Fg. 6. Notabl, Fg. 8 reveals that at the ntal state, the knetc energ of the wave s slghtl larger than the potental energ. However, as the wave propagates onto the shelf, the potental energ of the wave ncreases graduall and reaches the maxmum value at t =.4 s, as denoted b the frst vertcal dashed lne. hereafter, the potental energ decreases contnuousl. he decrease n the potental energ results n an ncrease n the knetc energ. he total energ decreases graduall due to the energ dsspaton caused b the nteracton of the wave and the shelf. he second vertcal dashed lne n Fg. 8 denotes the tme when the wave begns to break at t =.73 s. After the wave breaks, the potental energ decreases contnuousl, whle the knetc energ ncreases contnuousl due to the flow motons nduced b reattachments, splash-ups, and vod entranments. However, at the latter stage, e.g., t >.0 s, when two reattachments and two splash-ups have occurred, all three energes decrease contnuousl over tme. As s wdel recognzed, the wave breakng s accompaned b a sudden loss of energ, altugh no expermental data have demonstrated ths assumpton et. In Fg. 6, the wave breakng procedure begns when t >.73 s. Accordng to Fg. 8, no sudden loss of wave energ s assocated wth the wave breakng and, n most of the procedure of wave breakng, the knetc energ of the flud keeps ncreasng, whle the potental energ keeps decreasng. o further examne the energ varaton durng the wave breakng, Fg. 9 dsplas the temporal varaton of the tme rate of change of the potental energ, knetc energ, and total energ as a soltar wave propagates over a shelf. he unt n the vertcal axes of Fg. 9 s Joule /( s m). he vertcal dashed lnes n Fg. 9 ndcate some of the tmes of the wave profles swn n Fg. 6,.e., t = 0.70 s,.8 s,.6 s,.84 s,.89 s, and.95 s. Notabl, before the wave propagates onto the shelf (t < 0.9 s), the tme rate of change of the three energes remans unchanged wth values ver close to zero. he frst maxmum value of the tme rate of change of the potental energ appears at t =.8 s, correspondng to when the leadng part of the wave rses due to the salng effect. he frst maxmum value of the tme rate of change of the knetc energ appears at t =.6 s. Notabl, after the wave overturns and reattaches the free surface at t =.84 s, altugh Fg. 8 reveals no abrupt varatons n the tme evolutons of the potental, knetc, and total energ, Fg. 9 ndcates that the tme rate of change n the potental energ, knetc energ and total energ sgnfcantl var at some partcular nstance. For nstance, after the second reattachment, whch occurs at t =.95 s, de kn /dt ncreases abruptl and de pot /dt decreases rapdl. Immedatel after the second splash-up, t =.0 s, both de kn /dt and de total /dt declne abruptl, whle de pot /dt ncreases rapdl. Notabl,

86 Journal of Marne Scence and echnolog, Vol. 3, No. 6 (05) E total E E offsre E E onsre E 0.6 0.4 0. 0.6..4.6.8.0. t (sec) Fg. 0. me evoluton of reflected energ (E offsre /E, ), transmtted energ (E onsre /E, ), and total energ (E total /E, ) as a soltar wave propagates over a shelf. the maxmum negatve value of de total /dt appears at t =.03 s wth a value of 9.03 Joule /( s m), after the second splashup has occurred. As the soltar wave propagates over the shelf, total energ of the ncdent wave s dvded nto the transmtted energ and the reflected energ. o clarf the energ loss durng wave breakng, total energ s dvded nto two parts: the energ of the reflected wave (E offsre ) contaned n the regon from the wave paddle to the leadng edge of the shelf (x 7.7 h o ), and the energ of the transmtted wave (E onsre ) above the shelf from the leadng edge to the downstream of the wave tank (x 4.0 h o ). Fg. 0 presents the tme evoluton of the reflected energ (E offsre /E ) and the transmtted energ (E onsre /E ). Intall, 00% of the total energ was evaluated n front of the shelf. As the wave propagates over the shelf, approxmatel 4.8% of the ncdent wave energ remans n front of the shelf at t =. s. hus, accordng to Eq. (), the reflecton coeffcent K R s 0.48. Smlarl, the transmtted coeffcent K s 0.75. Upon completon of the computaton, around 87.3% of the ncdent wave energ remans n the computatonal doman. he energ dsspaton nvolved n the wle process s then.7%. Fg. 8 reveals that before wave breakng (t <.73 s), the dsspated energ s about 5%; whle at the end of computaton (t =. s), the totall dsspated energ ncreases to.7%. hese values ndcate that man part of energ dsspaton occurs n the srt perod after wave breaks. VIII. CONCLUSION hs work develops a numercal model to solve the unstead two-dmensonal Renolds Averaged Naver-Stokes (RANS) equatons and the k turbulence equatons for smulatng the evoluton of breakng soltar waves above a shelf, or a step. he partcle level set metd s adopted to capture the evolvng free surface, begnnng from the steepenng of the wave profle to the wave breakng and the successve reattachments and splash-ups. Based on the numercal results, we conclude the followng.. Numercal results ndcate that the developed numercal model can reveal the complex phenomena nvolved n a breakng soltar wave over a shelf, such as the overturnng of wave, reattachments of the eected et, and splash-ups.. he numercal results of wave profles near the breakng pont and at the stage wth an eected et have been swn to be dentcal to the expermental ones. 3. In the breakng soltar wave, the maxmum local flud veloct appears n the perod between the frst splash-up and the second re-attachment. he contour maps of flow veloctes near the breaker front ndcate that the maxmum local flud veloct s.8 gh o. 4. After the wave breaks, the potental energ frst decreases contnuousl and the knetc energ ncreases contnuousl; whle at the latter stage, the potental energ seems to approach a constant value, but the knetc energ decreases contnuousl n the same manner as that of the total energ. 5. Numercal results ndcate that n the plungng breakng wave, both the reattachment and the splash-up are normall accompaned b an abrupt change n the tme rate of change of knetc energ (de kn /dt) and the tme rate of change of potental energ (de pot /dt). ACKNOWLEDGMENS he autrs would lke to thank the Natonal Scence Councl, awan, for fnancall supportng ths research under Contract No. NSC98--E-006-5 -MY3 REFERENCES Chang, K. A. and P. L. F. Lu (998). 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