Modeling Breaking Waves and Wave-induced Currens wih Fully Nonlinear Boussinesq Equaions K.Z. FANG,2*, Z.B. LIU,2 he Sae Key Laboraory of Coasal and Offshore Engineering Dalian Universiy of Technology Linggong Road 2, 624 Dalian 2 Key Laboraory of Waer-Sedimen Sciences and Waer Disaser prevenion of Hunan Province Changsha Universiy of Technology Chiling Road 45, 44 Changsha CHINA *kfang@dlu.edu.cn Absrac: A Boussinesq-ype wave model is developed o numerically invesigae he breaking waves and wave-induced currens. All he nonlinear erms are reained in he governing equaions o keep fully nonlineariy characerisics and i hence more suiable o describe breaking waves wih srong nonlineariy in he nearshore region. The Boussinesq equaions are firsly exended o incorporae wave breaking, moving shoreline and boom fricion, and hen solved numerically using finie difference mehod. Using well documened experimenal daa as a reference, numerical experimens are conduced o invesigae he effec of unable parameer values on he compued resuls. The developed model is used o simulae breaking waves and wave-induced currens over complex bahymeries and he numerical resuls are compared agains he measuremens. Key-Words: - Breaking waves, Wave-induced currens, Boussinesq-ype wave model, Numerical simulaion. Inroducion Originaing from he pioneering conribuions made by Boussinesq [], Koreweg and de Vries [2], and Peregrine [3], recen research effors have provided a solid heoreical background for a new generaion of he so-called Boussinesq-ype wave equaions. Wih he increase of he compuaion abiliy, numerical models based on his kind of equaions have proven o adequaely describe mos of he waer wave phenomena aking place in he nearshore zone, see recen reviews by Kirby [4]. The developmen of Boussinesq-ype equaions usually involves wo parameers, namely µ=(raio of ypical waer deph o wave lengh) and ε (raio of ypical wave ampliude o waer deph). µ denoes he dispersion and he limi µ= represens he nondispersive limi. ε characerizes he nonlineariy and he limi ε= represens he linear limi. These wo parameers are usually assumed small ( ) in deriving Boussinesq-ype equaions, referring o weakly dispersive and weakly nonlinear regimes. In he recen pas, grea sudies have been made o improve he linear properies of Boussinesq-ype equaions. Since he weak dispersion (µ ) is he mos criical limiaion for many applicaions, mos of work has focused on exending models applicabiliy range o deep waer. Abundan research resuls have been published and he applicabiliy range of he equaions has been grealy improved (e.g., Madsen e al. [5]; Madsen and Sørensen[6]; Nwogu[7]; Madsen and Schäffer [8]; Madsen e al. [9]; Gobbi and Kirby []; Zou and Fang []). The linear shoaling propery could no be ignored considering he applicable range of waer deph is enlarged for many new forms of Boussinesq-ype equaions. In many improved models (Madsen and Sørensen [6]; Zou [2]; Gobbi and Kirby []; Zou and Fang []), aenion has also been given o he shoaling propery of equaions. Despie heir improved linear properies, he exended Boussinesq equaions are sill resriced o siuaions wih weak nonlineariy (i.e., ε ). In many pracical cases, however, he effecs of nonlineariy are oo large o be reaed as a weak perurbaion o a primarily linear problem. For example, ε approaches for he wave moion near he breaking poin(due o he effec of shoaling) and can be assumed a small value anymore. And hus exensions are required in order o obain a compuaional ool which is locally valid in he E-ISSN: 2224-347X 3 Volume 9, 24
viciniy of a seep, almos breaking or breaking wave cres. Wei e al. [3] relaxed he limiaions of weak nonlineariy (ε ) and reained all he nonlinear erms in he equaions, hus creaing a fully nonlinear Boussinesq-ype model a he order of O(µ 2 ). Laely, Gobbi and Kirby [], Zou and Fang [] derived fourh-order Boussinesq-ype equaions wih fully nonlinear characerisics. For nonbreaking waves wih srong nonlineariy, hese sudies have shown ha fully nonlinear models could presen beer numerical resuls han weakly nonlinear models. This conclusion is also valid for a soliary wave shoaling up o he breaking poin [3], and for breaking waves [4]. Though Boussinesq-ype models have already been brough ino he family of operaional coasal wave predicion models, he applicaions of such models wih fully nonlineariy in surf zone is quie limied ye, excep FUNWAVE2D model [5]. This model, based on he fully nonlinear version of Boussinesq equaions presened by Wei e al. [3], has been exensively esed for heir applicabiliy in modeling nearshore waves and currens [3]-[9]. Wave breaking, moving shoreline and oher mechanisms are common physical phenomena in surf and swash zone bu approximaely reaed in Boussinesq-ype models, and hus many unable parameers are inroduced. Previous sudies also underline he scarciy of he deailed invesigaion of parameer values on he numerical resuls. Addiionally he applicabiliy of Boussinesq-ype equaions, formulaed in oher forms, in surf zone has received considerably less aenion ye. The requiremen for furher invesigaion are furher amplified for 2D breaking waves as hey always induce relaively long scale wave moions, such as mean currens, which have been long recognized o play a key role in coasal ecology and morphology. Usually phase-averaged ype models are adoped o simulae mean currens, he effec of shor waves is considered by compuing he radiaion sress for a separae run of a waveaveraged model, where he effec of nonlineariies such as he ineracion beween waves, waves and currens, wave asymmery and ec. can only be included in an approximae manner. While he Boussinesq-ype models have he capabiliies of modeling nonlinear shor wave moions and fully coupled wave-curren ineracion [6][7]. All hese advanages enable he Boussinesq approach, especially hose wih fully nonlineariy characerisics, a promising and an alernaive ool for he sudy of low-frequency moions, superior o he phase-averaged ype models. The presen paper develops a 2D wave breaking model, based on a se of fully nonlinear Boussinesqype equaions. In secion 2, he mahemaical model is described. In secion 3, he numerical experimens are conduced o invesigae he effec of parameers on he compued resuls. And he validaed model is used o simulae nonlinear evoluion of breaking waves and wave-induced currens over plane and barred beaches. The final conclusions are drawn in secion 4. 2 Mahemaical Model 2. Governing Equaions The second order Boussinesq equaions derived by Zou [2] are expressed in erms of free surface elevaion η and deph-averaged velociy u as η + ( du ) = () u + ( u ) u+ g η + G = h [ ( hu) ]/2 2 2 h ( u) / 6 + Bh [ ( u + g η)] (2) 2 2 + B2 [ ( h u + gh η)] + R 2 2 2 2 G = { h [( u) u u B3 ( u u) /]}/3 ηh u where he subscrip denoes he parial derivaive wih respec o ime, = (, ) is he wo x y dimensional gradien operaor, h is he sill waer deph and d=h+η is local waer deph, g he graviy acceleraion. B and B 2 are se o be 29/885, 2/59 respecively afer maching equaions dispersion o a Padé[2,2] approximaion of he exac linear dispersion and opimizing he shoaling propery in medium waer deph limi. While B 3 =5.3 is chosen o ge opimum nonlinear propery wihin he applicable range of he equaions. R=R b +R f +R s is he exended erm for wave breaking, boom fricion and subgrid mixing and will be deailed in he nex subsecion. However in he above equaions, nonlinear erms are only reained up o he order of O(εµ 2 ) under he assumpion of ε hus he equaions only possess weak nonlineariy. Afer relaxing his limiaions and reaining all he nonlinear erms, he nonlinear erm G has he form of 2 2 2 2 G = { d [( u) u u B3 ( u u) / ] }/ 3 2 2 + ηd[( u) u u] / 3 ηd u (4) η(2 h + η) ( u )/3 Hereafer,()-(3) are named weakly nonlinear model while, equaions ()-(2) wih G defined by (4) are named fully nonlinear model. 2.2 Exend Governing Equaions o Surf and Swash Zone E-ISSN: 2224-347X 32 Volume 9, 24
Energy dissipaion due o wave breaking is reaed by inroducing an eddy viscosiy erm(r b ) ino he momenum equaions, wih he viscosiy srongly localized on he fron face of he breaking waves (Kennedy e al. [4]; Chen e al. [6] ) T Rb = ( h + η) [ ν ( h + η)( u+ ( u ) )] (5) 2 where he eddy viscosiy is defined as ν=bcbr du, wih C br =.2 being he defaul breaking srengh coefficien. The parameer B conrols he occurrence of he wave dissipaion and is given by *, η 2η * * * B = η / η, η < η 2η *, η η (6) The breaking crieria changes in a linear rend once breaking evens occur F * * η, T η = I F I * (7) η + ( η η ), < T * T T* is he ransiion ime wih he defaul value 5(h/g) /2, is he ime where breaking occurs, and is he duraion of wave breaking. I F η = (.65.35) gh and η =.5 gh are he criical values for wave breaking iniiaion and cease. Though defaul values have been proposed (see [4][5]), he previous research also shows ha hey also vary wihin a large range depending on he specific forms of he governing equaions. In addiion o he energy dissipaion due o wave breaking, subgrid-scale urbulen processes associaed wih surf zone eddies may become an imporan facor influencing he flow paern of he wave-generaed curren field(chen e al., [6]; Nwogu [2]). The urbulence ha occurs in regions wih large gradiens in he horizonal velociies is herefore simulaed by he Smagorinsky ype subgrid model [6][7][2], inroduced in he momenum equaions, o accoun for he effec of he resulan eddy viscosiy on he underlying flow. The subgrid mixing erm(r s ) has he same form as Eq.(5), bu he eddy viscosiy due o he subgrid urbulence has he form of 2 2 2 /2 υ s = Cm x y[( Ux) + ( Vy) + ( Uy + Vx) /2] (8) in which U and V are he velociy componens of he ime-averaged underlying curren field(in he presen sudy hey are esimaed every wo wave period), x and y are he grid spacing in he x(cross shore) and y (longshore) direcions respecively, and C m is he mixing coefficien whose values have been assumed ranging beween. and 2.. The boom shear sress is given by a quadraic erm wrien in he form of: R f = Cf ( h+ η) uu (9) where C f is he fricion coefficien wih defaul value range.-.. I should be noed ha R f is inversely proporional o local waer deph. So his erm is imporan for wave ransformaion in shallow waer and hence conribues favorable o he nearshore circulaion paerns. 2.3 Numerical Scheme and Boundary Condiions The equaions are solved numerically by using a finie-difference mehod on recangular grid sysem. Higher order finie difference formula is used o approximae boh spaial and emporal derivaives in he equaions. While ime inegraion is made by using he 3rd order Adams-Bashforh predicor and a 4h order Adams-Moulon correcor. This numerical scheme is exensively used for solving Boussinesq models and he deails are referred o [5]. The heoreical analysis of he sabiliy condiions for his algorihm is provided in [22]. By applying he Von Neumann sabiliy analysis o he presen Boussinesq model, he similar expressions can be obained. For he predicor algorihm, i is sable when he Couran number is smaller han or equal o uniy, and for he correcor algorihm, he Couran number is smaller han or equal o.5. In he following numerical simulaions, he ime sep and space sep saisfy his sable condiion. A he offshore boundary, he relaxaion zone mehod for generaing non-reflecive waves (Bingham and Agnon [23]) is used. Our numerical experimens show ha his mehod is more effecive o generae high nonlinear and large period waves han he inernal wave generaion mehod embodied in FUNWAVE2D, which may be due o he fac ha he inernal source funcion is derived using linearized Boussinesq equaions. As slo mehod is used, he res hree boundaries are reaed as closed solid walls, and sponger layer is placed a he shoreward end of he domain o absorb any ougoing wave energy. However for longshore curren simulaions, periodic laeral cross-shore boundaries are imposed, following he suggesion from Chen e al. [6]and Chen and Svendsen [24]. The enire compuaional domain is reaed as an acive fluid domain by using permeable sea-bed echnique, where he beach is considered porous or conaining narrow slos, hus he moving shoreline is considered, he deails of his mehod is referred o [4] or [6]. E-ISSN: 2224-347X 33 Volume 9, 24
3 Numerical Resuls and Discussions Excep specially addressed elsewhere, a Caresian coordinae sysem is adoped wih origin locaed on he sill waer plane (SWL) wih x axis increases offshore ward, y axis direcs longshore direcion and z axis poining verically upwards. And all he imeaveraged quaniies are obained afer he seady sae of wave field are reached. 3. Normally Inciden Soliary Wave Shoaling Over Plane Beaches To illusrae he necessiy of incorporaing fully nonlinear erms in he Boussinesq equaions, he shoaling of soliary waves over plane beaches is simulaed. This example is a good benchmark es as he wave heigh o waer deph raio reached prior o breaking is exremely high and hus nonlineariy is srong. In addiion, soliary waves propagae as an isolaed pulse wihou exra background disurbances and hus allowing clearly examining he model performances. Two differen slopes of :35 and : are used in he compuaions and he numerical resuls from weakly nonlinear model and fully nonlinear model are ploed in Figure. Where he numerical soluion from FNPF model(fully Nonlinear Poenial Flow, Wei e al. [3]) is also presened as he analyical soluions o he problem. he fully nonlinear model presens much beer resuls. The discrepancy beween wo models is amplified when he beach slope becomes mild as he shoaling effec is srenghened hereby. 3.2 Normally Inciden Regular Waves Breaking Over Plane Beaches D periodic waves breaking over plane beaches will be considered. Firs he experimenal daa for breaking cnoidal waves from Ting and Kirby [25] is used o invesigae he role of each of he breaking parameers. To avoid oher effecs on he numerical resuls, boom fricion and moving shoreline boundary are no considered. The numerical experimens show ha he numerical resuls are no sensiive o η F and hus defaul value.5 (gh) /2 is used. The value of η I could be easily deermined by maching he breaking poin o he measuremen, which is found o be.8(gh) /2. Three values of breaking srengh C br =.6,.2,.8 are used for simulaion and he compued resuls are ploed in Figure 2(a). Where as expeced, small breaking srengh fails o effecively dissipae wave heigh afer wave breaking. While he values of.2 and.8 primarily show he same resuls, hence he defaul value.2 is used in his paper. Keep C br =.2, hree simulaions are done using T*=(2.5,5.,7.5) (h/g) /2 and he numerical resuls are given in Figure 2(b). Three values primarily presen he similar resuls excep sligh difference near he breaking poin. Hence we also use he recommended value T*=5.(h/g) /2 for D breaking waves. Fig. Wave profiles of shoaling soliary waves over plane beaches. In he figure he lengh/heigh, ime variables are scaled by h and (h /g) /2, respecively. While H and h are soliary wave heigh and waer deph over fla boom respecively. x = is defined a beach oe and x increases shoreward while = is defined a he ime when soliary wave cres reaches beach oe. I could be seen from he figure ha in he relaively deep waer, he weakly nonlinear and fully nonlinear models have almos he same performance as he nonlineariy is no so srong. However as he nonlineariy increases under he effec of shoaling, he differences beween wo models become apparen. The weakly nonlinear model fails o predic boh wave ampliude and wave phase, while Fig.2 The effec of breaking srengh coefficien C br (a) and breaking duraion parameer T* (b) on he numerical resuls. E-ISSN: 2224-347X 34 Volume 9, 24
Cases h (m) H(m) L(m) T(s) Breaking ype x(m) (s) A2.36.67.44. Spilling.2. 67.36.67 2.87.67 Spilling.2. 34.36.43 3.53 2. Plunging.2. 44.36.39 4.52 2.5 Spilling-plunging.2. 54.36.36 6.2 3.33 Spilling.2. Table : Wave condiions and simulaion parameers for D regular waves breaking over plane beaches. (h is he waer deph over fla boom, H, L and T are he inciden wave heigh, wave lengh and wave period, respecively.) Anoher five ess for regular waves breaking over plane beaches [26], covering a wide range of breaker ypes, are considered for validaion. Wave characerisics for all waves esed and he simulaion parameers are lised in Table. The compued resuls including wave heigh and mean waer level are presened in Figure 3. In he figure he increase of wave heigh during shoaling process and he iniiaion of wave breaking, he subsequen decrease of wave heigh are well reproduced and he seup rend are also prediced well. While for all he ess (including hose in Figure 2), he wave heighs in he inner surf zone end o be over-prediced by he model in he case of breaking poin are capured. This phenomenon are also found by oher researchers, [4][27]for example, when hey carried ou Boussinesq-ype simulaion. The discrepancy, in our opinion, is caused by he inrinsic limiaion of using he eddy viscosiy mechanism o approximaely mimic wave breaking in he model. Considering he empirical reamen of wave breaking, he agreemens beween he numerical resuls and measuremens are reasonably good. H/, MWL (cm) H/, MWL (cm) H/, MWL (cm).5 67.5..5.2.25.3.35.4 deph(m) A2.5.5.5..5.2.25.3.35.4 deph(m) 34.5..5.2.25.3.35.4 deph(m) H/, MWL (cm) H/, MWL (cm).5.5 44.5..5.2.25.3.35.4 deph(m) 54.5..5.2.25.3.35.4 deph(m) Fig.3 The comparison of wave heigh and mean waer level beween numerical resuls(solid line) and experimen daa(symbols) for five cases of regular waves breaking over plane beach. 3.3 Obliquely Inciden Waves Breaking on Plane Beaches Visser [28] has conduced experimens o invesigae he obliquely inciden waves breaking on plane beach, in he experimens free surface elevaion and wave-induced longshore currens are deailed colleced and hey will be used here for model validaion. A snapsho of he 3D compued free surface elevaion and phase-averaged curren field for case 4 (see Table 2) of Visser experimens [28] are presened in Figure 4(a) and Figure 4(b) respecively. I is seen from his figure ha he modelled wave cres becomes narrow and asymmeric while wave rough becomes fla during shoaling process. The increase and decrease of wave heigh before and afer breaking even are well reproduced. The longshore currens are quie seady and uniform alongshore, indicaing he proper and efficiency of he numerical implemenaion, especially he rea of periodic laeral cross-shore boundary condiion, relaxaion zone mehod for wave generaion, and his will definiely increase he confidence of obaining reliable resuls using he model. E-ISSN: 2224-347X 35 Volume 9, 24
Fig.4 The insananeous surface elevaion(a) and he mean curren filed(b) for case 4 of Visser s experimen. Also using case 4, numerical experimens are conduced o invesigae he parameer values effec on he compued resuls and he corresponding resuls are ploed in Figure 5. Firs here values of boom fricion coefficien C f =.3,.6 and. are used. Clearly, wave heigh and mean waer level are no very sensiive o his parameer while he ampliude of longshore currens is grealy conrolled by his parameer. The increase of C f resuls in apparenly he decrease of o correcly predic he locaion of he longshore currens cres. While he large value only provides marginal improvemens of wave heigh in surf zone and ends o cause longshore currens biased offshore. Then we keep C f =.6, C br =.2 and here values of subgrid mixing C m =.5,. and 2. are used for simulaion. As seen in he figure, large value resuls in oscillaions of surface elevaions while mean waer level and longshore currens are no very sensiive o his parameer. Large variaion of C m, he longshore curren ampliude. Numerical even over one order of magniude is observed o experimens also show ha neglecing boom fricion(c f =) causes compuaion blow up while he exremely large value of C f (=.) deerioraes he compued wave heah near he breaking poin. Then C f =.6 is kep consan and here values of breaking srengh C br =.6,.2 and.8 are used for simulaion and he compued resuls are also presened in Figure 5. Clearly, small value fails o resul in a few percen changes. This is in consisen wih he conclusions drawn by Mendonca e al. [29]. Chen e al. [6] also found he ampliude of dissipaion caused by subgrid urbulence is smaller compared agains hose by breaking, hough he former spreads seaward away he breaking poin while he laer is srongly localized in surf zone. In his paper, C m =. is used. effecively decrease he wave heigh in surf zone and Fig.5 The effec of boom fricion(lef), breaking srengh(middle) and subgrid mixing(righ) on he compued wave heigh(a), mean waer level(b) and mean curren profile(c). E-ISSN: 2224-347X 36 Volume 9, 24
Cases Beach slope H(cm) h (cm) T(s) α Breaker ype x (cm) y (cm) (s) 2. 9.5 39.9. 3.5 Plunging 5. 7.24.2 4.5 7.8 35..2 5.4 Plunging 5. 9.24.2 5.5 7. 34.8.85 5.4 Plunging 7.5 3.3.2 Table 2: Experimen cases of Visser [27] and simulaion parameers (where h is he waer deph over fla boom, H and T are he inciden wave heigh and wave period, respecively. α is he angle of incidence wave.) Via he above process, he unable parameers are deermined as C br =.2, C f =.6, C m =.. To es he applicable range of hese parameers, he res wo cases(case 2 and case 5) of Visser experimens (see Table 2) are simulaed and he numerical resuls are compared agains he experimenal daa in Figure 6. Generally he numerical resuls and experimenal resuls are in good agreemens. Wave heigh increases due o nonlinear shoaling and decrease afer breaking are well prediced. The variaion rend of he mean waer level is also capured by he model. The wave heigh however in he inner surf zone is also overesimaed, jus as found in secion 3.2 for D breaking waves. The disribuion of longshore currens profile, including he magniude and he locaion of he cres value are generally in good agreemens wih he experimenal daa. I is also noeworhy o menion ha hree cases considered here belong o plunging breaker while he eddy viscosiy mehod is originally designed for spilling breaker wih mild energy dissipaion. The compued mean curren field are shown in Figure 7. For wo cases considered, he compued resuls show high longshore uniformiy of longshore currens. Comparing he resuls for cases 2, 4 and 5, he effec of differen inciden waves on he final mean curren field also could be observed. Fig.6 The comparison of wave heigh(a), mean waer level(b) and longshore curren profile(c) beween compued resuls and experimenal daa for case 2(lef) and case 5(righ) of Visser s experimens. E-ISSN: 2224-347X 37 Volume 9, 24
Fig.7 The mean curren field for Case 2(lef) and Case 5(righ) of Visser s experimens. 3.4 Normal Inciden Waves Breaking on Barred Beaches The experimenal daa of breaking waves on a barred beach, colleced by Haller e al. [3], are widely used for model validaion[8][2]. In he experimens, a longshore uniform bar, incised by wo rip channels, is superposed on a plane beach o mimic he ypical feaure of a real barred beach. The spaial variaion of he wave-induced excess momenum flux will finally drive a ime-varying circulaion paern rip curren sysem. The experimen is believed o reproduce he main feaures of he bahymery-conrolled rip curren in real naure. Neglecing he small variaions from longshore uniformiy in he planar beach and considering sligh asymmery abou he cener axis of he physical wave basin, he idealized op half of he bahymery is used for simulaion, see Figure 8 for he deails. Numerical simulaions were carried ou for es B which corresponds o a normally inciden regular wave wih period, T=s and heigh, H=.48m on a consan waer deph.363m. The grid-space and ime-sep incremens are x=.5m, y=.m and =.s. Simulaion duraion is 3s and he compued daases from las 2s are used for saisical analysis. The compued wave heigh and mean waer level(mwl) along hree cross-shore ransecs are ploed in Figures 9 and, respecively. The experimenal daa and he numerical resuls from FUNWAVE2D are also presened. The agreemens beween he presen model and measuremens are good. Wave breaking poin, wave heigh decrease afer wave breaking and he corresponding variaion of MWL(sedown and seup) are well capured by he model. Especially he delayed wave breaking in rip channel due o relaively deep waer in he channel, wave heigh increase in he offshore ward of he rip channel due o he ineracion of propagaion waves and offshore direced curren are simulaed. The wave heigh is grealy overprediced by presen model and FUNWAVE2D in he offshore region of rip channel(y=4.6m), which is due o he delayed insabiliy moion of rip curren on an ideal bahymery, as also explained in Chen e al. [8]. Excep he sligh accurae resul of seup in inner surf zone, he presen model and FUNWAVE2D primarily have he same performance. Fig.8 The 3D(op) and conour(boom) map of barred beach wih rip channel (wih he origin defined a he lower lef corner). E-ISSN: 2224-347X 38 Volume 9, 24
Hrms (cm) Hrms (cm) Hrms (cm) 8 6 4 2 y=.m Barred beach.5.5 2 2.5 3 3.5 4 4.5 x (m) 8 6 4 2 y=4.6m Rip channel.5.5 2 2.5 3 3.5 4 4.5 x (m) 8 6 4 2 y=7.6m Barred beach.5.5 2 2.5 3 3.5 4 4.5 x (m) Fig.9 The comparisons of wave heigh beween numerical resuls from presen model(solid line), from FUNWAVE2D(sars) and measuremens(circles). Mwl(mm) Mwl(mm) Mwl(mm) 5 y=.m Barred beach -5.5.5 2 2.5 3 3.5 4 4.5 x(m) 5 y=4.6m Rip channel -5.5.5 2 2.5 3 3.5 4 4.5 x(m) 5 y=7.6m Barred beach -5.5.5 2 2.5 3 3.5 4 4.5 x(m) Fig. The comparisons of mean waer level beween numerical resuls from presen model(solid line), from FUNWAVE2D(sars) and measuremens(circles). U(m/s).2 x=m V(m/s).2 x=m U(m/s) U(m/s) U(m/s) -.2 2 3 4 5 6 7 8 9.2 x=.25m -.2 2 3 4 5 6 7 8 9.2 x=2.3m -.2 2 3 4 5 6 7 8 9.2 x=3m -.2 2 3 4 5 6 7 8 9 Fig. The comparisons of cross-shore mean curren (U) beween numerical resuls from presen model (solid line), from FUNWAVE2D(dash-doed line) and measuremens(circles). V(m/s) V(m/s) V(m/s) -.2 2 3 4 5 6 7 8 9.2 x=.25m -.2 2 3 4 5 6 7 8 9.2 x=2.3m -.2 2 3 4 5 6 7 8 9.2 x=3m -.2 2 3 4 5 6 7 8 9 Fig.2 The comparisons of longshore mean curren (V) beween numerical resuls from presen model (solid line), from FUNWAVE2D(dash-doed line) and measuremens(circles) E-ISSN: 2224-347X 39 Volume 9, 24
Fig.3 The comparisons of mean curren field from presen model(lef), from FUNWAVE2D(middle) and experimens(righ). bahymery (wih sric symmery) insead of The longshore mean curren(u) and he crossshore mean curren(v) are compared agains he experimenal bahymery (wih sligh asymmery), as he insabiliy moion of rip curren is delayed on experimenal daa and he numerical resuls from he ideal bahymery [8]. FUNWAVE2D in Figures and Figure 2, More insigh ino he generaion and moion of respecively. Generally he simulaed mean curren rip curren could be obained by examining he componens agree well wih he measuremens. ime-dependen variaion of compued quaniies, Firs, he variaion rend of he offshore direced such as free surface elevaion, mean curren filed mean curren (rip curren) is well capured. The and voriciy field (defined as V over-predicion of U nearshore shoreline (x=3m) x -U y and esimaed every wo wave period in simulaion). The and he under-esimaion of U in rip channel numerical resuls a four momens, i.e., =2s, 8s, (x=.25m and x=2.3m) are observed for he 2s and 8s, are shown in Figure 4. A he presen model and FUNWAVE2D, which may be iniial sage =2s, he compued wave cress are due o he use of ideal bahymery insead of primarily parallel o he shore wih he wave heigh experimenal bahymery, as also argued by Chen e in he rip channel is higher han ha on he bar al. [8]. Secondly, he variaion of longshore profiles, which is due o he fac ha he dephinduced breaking is delayed in rip channel. curren, which have he opposie sign and consis of rip feeder, are also prediced well by he model. Meanwhile he vorex rings are seen o be The compued mean curren field is shown in generaed on he seaward bar-channel inersecs. Figure 3 and compared agains he measuremens And he offshore direced mean curren also could and he numerical resul from FUNWAVE2D. The be observed beween clockwise and ani-clockwise flow field prediced by he numerical model is roaion vorex. Wih he subsequen waves similar o he measured velociy field. The main breaking on he bahymery (=8s, 2s, 8s), he componens of a rip curren sysem, namely, rip vorex rings are released ino deep waer and he curren, rip feeder and rip neck are reproduced by srengh of rip curren grows gradually. The process he model. Two numerical models also predic a of generaion and release of vorex rings is repeaed shoreward mean flow and secondary circulaion subsequenly. Addiionally a secondary circulaion close o he shoreline, which however is no paern close o he shoreline is also found o feed apparen in measuremens due o he sparse he rip curren. The increase of he wave heigh a measuremens. The measured rip curren shows he region where he rip curren and inciden waves bias oward he boom, while he numerical resuls coexis are apparen, denoing he applicabiliy of from he presen model and FUNWAVE2D show Boussinesq-ype model in describing he much symmery abou he cener axis and he ineracion beween waves and currens. compued rip curren penerae farher o deep waer. This is also aribued o he use of ideal E-ISSN: 2224-347X 4 Volume 9, 24
Fig.4 The insananeous free surface elevaion(lef) and vecor field, vorex(righ) a four momens. 4 Conclusions A numerical model, based on he fully nonlinear Boussinesq equaions, is developed o simulae he nonlinear wave breaking waves and wave-induced longshore currens wih he main conclusions are drawn below. () The numerical resuls of shoaling of soliary waves demonsrae he necessiy of incorporaing E-ISSN: 2224-347X 4 Volume 9, 24
fully nonlinear characerisics in he equaions. The model has shown iself o accuraely predic nonlinear wave ransformaion in he surf zone. Breaking phenomena and associaed mean currens are prediced boh qualiaively and quaniaively for a variey of D and 2D ess over complex bahymeries. (2) For waves breaking over plane beaches, he model is observed o over-predic wave heighs in he inner surf zone, which is in consisen wih oher researchers. I suggess ha proper mehod o deal wih wave breaking in a Boussinesq-ype model sill needs furher invesigaion. (3) Parameer analyses are conduced for waves breaking on plane beaches. For normally inciden waves, he numerical resuls are found less sensiive o breaking relaed parameers(excep for iniiaion parameer) and he variaion only resul in marginal difference near wave breaking poin and hus recommended defaul values are adoped. For oblique inciden waves, we found ha he mean waer level is no sensiive o he variaion of parameers, while he profiles of mean longshore curren grealy depend on he breaking srengh and boom fricion, while subgrid mixing has a negligible effec. (4) The main feaures of a rip curren sysem could be qualiaively and quaniaively reproduced by he model. The discrepancy beween he numerical model and measuremens are mainly aribued o he use of ideal bahymery insead of experimenal bahymery. As he firs sep of developing a fully nonlinear model and validaion aim, only ime-averaged quaniies are considered in he presen paper. Bearing in mind ha, however, Boussinesq-ype wave models belong o phase-resolving ype and hey describe inra-wave properies. Though no pursued in presen sudy, furher invesigaion on inra-wave properies, coupled wave-wave, wavecurren ineracion and also heir conribuion o he polluan/sedimen ranspor are sill in process. 5 Acknowledgmens The auhors graefully acknowledge he financial suppor from he Naional Naural Science Foundaion of China Gran no. (598), Key Laboraory of Waer-Sedimen Sciences and Waer Disaser prevenion of Hunan Province (Gran No. 23SS2 and 22SS2). References: []. Boussinesq J., Theorie des ondes e des remous qui se propagen le long d'un canal recangulaire horizonal, en communiquan au liquide conenu dans ce canal des viesses sensiblemen pareilles de la surface au fond, J. Mah. Pures Appl., Series.2, no.7, 872, pp.55-8. [2]. Koreweg D.J. and G. de Vries, On he change of form of long waves advancing in a recangular canal and on a new ype of long saionary wave, Phil. Mag., Series5, no.39, 895, pp.422-443. [3]. Peregrine D.H., Long waves on a beach, J. Fluid Mech., vol.27, no.4, 967, pp.83-827. [4]. Kirby J.T., Boussinesq models and applicaions o nearshore wave propagaion, surf zone processes and wave-induced currens, in Advances in Coasal Modeling, Oceanogr. Ser., V.C. Lakhan, Ed., Elsevier, New York, 23, pp.- 4. [5]. Madsen P.A., Murray R. and Sørensen O.R., A new form of Boussinesq equaions wih improved linear dispersion characerisics, Coasal Engineering, vol.5, no.4, 99, pp.37-388. [6]. Madsen P.A. and Sørensen O.R., A new form of he Boussinesq equaions wih improved linear dispersion characerisics, Par 2. A slowlyvarying bahymery, Coasal Engineering, vol.8, no.3-4, 992, pp.83-24. [7]. Nwogu O., An alernaive form of he Boussinesq equaions for nearshore wave Propagaion, Journal of Waerway, Por, Coasal, and Ocean Engineering, vol. 9, no.6, 993, pp. 68 638. [8]. Madsen P.A. and Schäffer H.A., Higher order Boussinesq-ype equaions for surface graviy waves derivaion and analysis, Phil. Trans. R Soc Lond A, vol.356, no.794, 998, pp.323 386. [9]. Madsen P.A., Bingham H.B. and Schäffer H.A., Boussinesq-ype formulaions for fully nonlinear and exremely dispersive waer waves: derivaion and analysis, Phil. Trans. R Soc Lond A, vol.459, no.233, 23, pp.75 4. []. Gobbi M.F., Kirby J.T. and Wei G., A fully nonlinear Boussinesq model for surface waves. Par 2. Exension o O(kh) 4, Journal of Fluid Mechanics, vol.45, no.4, 2, pp.8 2. []. Zou Z. and Fang K., Alernaive forms of he higher-order Boussinesq equaions: Derivaions and validaions, Coasal Engineering, vol.55, no.6, 28, pp.56-52. [2]. Zou Z., A new form of higher order Boussinesq equaions, Ocean engineering, vol.27, no.5, 2, pp.557-575. [3]. Wei G., Kirby J.T., Grilli S.T. and Subramanya R., A fully nonlinear Boussinesq model for surface waves: Par I. Highly nonlinear E-ISSN: 2224-347X 42 Volume 9, 24
unseady waves, Journal of Fluid Mechanics, vol.294, 995, pp.7 92. [4]. Kennedy A.B., Chen Q., Kirby J.T. and Dalrymple R.A., Boussinesq modeling of wave ransformaion, breaking, and runup. Par I: D, Journal of Waerway, Por, Coasal, and Ocean Engineering, vol.26, no., 2, pp.39 47. [5]. KIRBY J. T., WEI G., e al. FUNWAVE. Fully nonlinear Boussinesq wave model documenaion and user s manual, Research Repor No. CACR-98-6, Cener for applied Coasal Research, Universiy of Delaware, Newark, 998. [6]. Chen Q., Kirby J.T., DalrympleR.A., Kennedy A.B., Chawla and Arun, Boussinesq Modeling of Wave Transformaion, Breaking, and Runup. II: 2D, Journal of Waerway, Por, Coasal, and Ocean Engineering, vol.26, no., 2, pp.48-56. [7]. Chen Q., Kirby J. T., Dalrymple R. A., Shi F. and Thornon E. B., Boussinesq modeling of longshore currens, Journal of Geophysical Research, vol.8, no. c, 23, pp.-26. [8]. Chen Q., Boussinesq modeling of a rip curren sysem, Journal of Geophysical Research, vol.4, no.c9, 999, pp.267-2637. [9]. Mil-Homens J., Fores C. J.E.M. and Pires- Silva A. A., An evaluaion of wave propagaion simulaions over a barred beach wih a Boussinesqype model, Ocean Engineering, vol.37, no., 2, pp.236-25. [2]. Zou Z., Higher order Boussinesq equaions, Ocean Engineering, vol.26, no.8, 999, pp.767-792. [2]. Nwogu O.K., Numerical predicion of rip curren on barred beaches, Proceedings of he 4h Inernaional Symposium on Ocean Wave Measuremen and Analysis, San Francisco, USA, 2, pp.396-45. [22]. Lin, P. and C. Man. A saggered-grid numerical algorihm for he exended Boussinesq equaions. Applied Mahemaical Modelling, vol. 3, no.2, 27, pp. 349-368 [23]. Bingham H.B. and Agnon Y., A Fourier- Boussinesq mehod for nonlinear waer waves, Eur. J. Mech. B/Fluids, vol.24, no.2, 25, pp.255-274. [24]. Chen Q. and Svendsen I.A., Effecs of cross-shore boundary condiion errors in nearshore circulaion modeling, Coasal Engineering, vol.48, no.4, 23, pp.243-256. [25]. Ting F. C. K., and Kirby J. T., Dynamics of surf-zone urbulence in a spilling breaker, Coasal Eng., vol.27, no.3-4, 996, pp.3 6. [26]. Hansen J. B. and Svendsen I. A., Regular waves in shoaling waer: Experimenal daa, Tech. Rep. ISVA Ser., 2, Technical Univ. of Denmark, Denmark, 979. [27]. Zaleski A.R., A numerical model for breaking waves in he surf zone, Disseraion, Technical Universiy of Denmark, 27. [28]. Visser P. J., Laboraory measuremens of uniform longshore curren, Coasal Engineering, vol.5, no.5-6, 99,pp.563-593. [29]. Mendonca A., Brocchini M., Besio G.., Piaella A., Flow mixing soluions of a Boussinesq-ype model: owards a dispersive HLES, Proc. 7h Asian Compuaional Fluid Dynamics Conference, Bangalore, India. 27, pp. 89. [3]. Haller M.C., Dalrymple R.A. and Svendsen I.A. Experimens on Rip curren dynamics and nearhsore circulaion, Research Repor No. CACR- -4, Cener for applied Coasal Research, Universiy of Delaware, Newark, 2. E-ISSN: 2224-347X 43 Volume 9, 24