COUPLING VOF/PLIC AND EMBEDDING METHOD FOR SIMULATING WAVE BREAKING ON A SLOPING BEACH

Transcription

1 498 Journal of Marine Science and Technology, Vol. 3, No. 4, pp (15) DOI: 1119/JMST44- COUPLING VOF/PLIC AND EMBEDDING METHOD FOR SIMULATING WAVE BREAKING ON A SLOPING BEACH Tai-Wen Hsu 1, 3, Chih-Min Hsieh, Chin-Yen Tsai 3, and Shan-Hwei Ou 3 Key words: spilling breaker, VOF/PLIC, embedding mehod, wave breaking. ABSTRACT A -D numerical model was developed o simulae wave breaking on a sloping beach. The model solves he Reynolds Averaged Navier-Sokes (RANS) equaions coupled wih he k-ε urbulence closure model. To rack free surface configuraions, he volume of fluid wih piecewise linear inerface calculaion (VOF/PLIC) is employed. An embedding mehod (EB) is also used o rea comple boom opography. Coupling hese wo mehods for simulaing wave breaking on a sloping beach, good agreemen beween numerical resuls and laboraory observaions is found for spilling breaker. Wave breaking characerisics is in erms of free surface profiles, mean velociies and wave heighs based on he numerical resuls. Turbulence ranspors under wave breaking were also invesigaed. I. INTRODUCTION When waves raveling over a sloping beach and ener waer ha is approimaely as deep as he wave heigh, hey become unsable and hen break. Much of he energy is dissipaed afer wave breaking in he nearshore ha provides naural forces o generae nearshore currens. These currens are of grea imporance in ha hey combine wih waves o ranspor beach sedimen and are a significan facor in conrolling he morphology of he beach. In addiion, urbulence induced by breaking waves could cause wide spreading of he diffusion Paper submied 5/18/14; revised 1/7/14; acceped 1/4/14. Auhor for correspondence: Tai-Wen Hsu ( whsu@mail.ncku.edu.w). 1 Deparmen of Habor and River Engineering, Naional Taiwan Ocean Universiy; Research Cener for Ocean Energy and Sraegies, Keelung, Taiwan, R.O.C. Deparmen of Mariime Informaion and Technology, Naional Kaohsiung Marine Universiy, Kaohsiung, Taiwan, R.O.C. 3 Deparmen of Hydraulic and Ocean Engineering, Naional Chen Kung Universiy, Tainan, Taiwan, R.O.C. processes for suspended paricles. Therefore, i is desirable o sudy he processes of wave energy dissipaion in he surf one which include wave heigh decay, he movemen of bores, wave-induced seup and sedown and longshore curren as well as rip curren. In he pas few years, due o he advance of compuing echnology, he numerical wave ank (NWT) has been widely applied in simulaing wave breaking. Currenly, here are four basic ypes of numerical models have been used o simulae wave breaking of waer waves: (1) non-linear shallow waer wave equaion, () Boussinesq ype models, (3) Laplace equaion, and (4) Navier-Sokes equaions. The firs hree approaches are based on poenial flow, while he las approach is caegoried ino real flow. For eample, Zel (1991) developed he Boussinesq-ype equaions in Lagrangian form o simulae wave run-up of nonbreaking and breaking soliary waves. A numerical model was developed by Kobayashi e al. (1987) on he basis of he finie ampliude shallow-waer equaions o predic wave reflecion and runup on rough slopes. However, he inviscid wave model may fail o capure imporan phenomena regarding he ineracion of viscous fluids wih solid srucures. So far, only a few numerical models have been proposed for simulaing breaking waves ravelling over a slopping beach. To sudy he process of viscous flow ineracing wih submerged breakwaers, he Navier-Sokes equaions wih fully nonlinear free surface boundary condiions are considered in his sudy. The presen sage of using Navier-Sokes equaions o simulae breaking waves can be classified ino hree caegories (Zhao e al., 4): (1) direc numerical simulaion (DNS) of Navier-Sokes equaions conaining urbulen flow (Pei e al., 1994); () numerical soluion of he Reynolds-Averaged Navier-Sokes (RANS) equaions (Lemos, 199; Lin and Liu. 1998a, 1998b; Hsiao and Lin, 1; Xie, 13; Xie, 14); and (3) numerical soluion of he space-filered Navier-Sokes equaions (Zhao and Tanimoo, 1998; Chrisensen and Deigaard, 1; Zhao e al., 4; Lubin e al., 6; Lakehal and Liovic, 11; Lubin e al., 11). Among hem, DNS demanding a sufficienly refined grid sie o accoun for all he lengh-scales of he urbulen flow is oo compuaionally epensive o apply in a large coasal waers. The space-

2 T.-W. Hsu e al.: Coupling VOF/PLIC and Embedding Mehod for Simulaing Wave Breaking on a Sloping Beach 499 filered mehod uses a cell volume filer o describe he subgrid urbulence in which he Smagorinsky subgrid scale (SGS) Reynolds sress and large eddy scale (LES) are implemened in he model. Alhough SGS model shows remarkable resuls of breaking waves, however, he compuaions are very laborious and ime consuming. Insead, he RANS model coupled wih urbulence closure models are commonly used o simulae he evoluion of breaking waves of kinemaics and urbulence in he surf one. I has he advanage of compuaional efficiency and reliabiliy o simulae wave breaking processes. Based on he review given above, progress in developing a numerical model for wave breaking processes on a sloping beach has been seadily performed. Gueyffier e al. (1999) developed volume of fluid (VOF) mehod o a second-order accuracy. Tsai e al. (8) applied his echnique o simulae spilling breaker ravelling over a sloping beach. In his paper he VOF mehod using piecewise linear inerface calculaion (PLIC), designaed as VOF/PLIC, has been employed o rack he free surface. In he model, a Lagragian advecion is used o adven he inerface segmens and calculae he corresponding volume flu in he cell. The wall boundary condiion is described by he embedding (EB) mehod developed by Ravou e al. (3), in which he solid boundary is accouned for by adding a force field o he flow phase in he compued cells ha are fully or parially occupied by he solid phase. Therefore, here is no need o impose boundary condiions on he body surface since he velociy componens are made vanish wihin he bodies as a par of he soluion. A D numerical model was developed o predic flow characerisics on a sloping beach for spilling breaker. The boundary condiions were improved based on VOF/PLIC and EB. The model is eensively eamined wih he eperimenal daa for a rain of waves breaking over sloping beaches. Flow characerisics of wave breaking propagaion over sloping profiles is presened and discussed. Numerical resuls are in erms of he free surface profiles, he mean velociies, he urbulen kineic energies and he eddy viscosiies. II. MODEL DESCRIPTION 1. Governing Equaions Waves propagaing over a sloping beach is considered in his invesigaion. The Caresian coordinae sysem is uilied as shown in Fig. 1, where h is he quiescen waer deph in fron of a sloping beach, H is he inciden wave heigh. The RANS model developed by Lin and Liu (1998a; 1998b) is widely used for solving ocean problems. This model solves D RANS equaions for he mean flow field combined wih he k- urbulence closure. The governing equaions which describe he mean quaniies of he flow field for unseady incompressible urbulen flows are he Reynolds-averaged equaions consising of he coninuiy equaion for incompressible flow and momenum equaions wrien as follows, Coninuiy equaion: Momenum equaion: U W U U U P U W ( ) U W W W P ( ), (1) ( U ) ( W ) k, 3 () U W W U W k ( ) ( ) g, 3 (3) where and are he horional and verical coordinaes in a fied Caresian sysem, is he ime, U and W are he mean velociy componens in he - and -direcions, P is he pressure, g is he graviaional acceleraion, is he eddy viscosiy, is he molecular viscosiy, ( /, / y ) is he gradien operaor, and k is he urbulen kineic energy. The Reynolds sress closure model is he k- model for resolving he eddy viscosiy in he presen sudy. The eddy viscosiy is represened by he urbulen kineic energy k and energy dissipaion rae ε using he Boussinesq assumpion, ha is Ck, (4) The ranspor equaions of he urbulen kineic equaion (TKE) and energy dissipaion equaion (EDE) are used as he closure equaions: k k k U W kprod, (5) k U W Cl Prod C, (6) k k where Prod erm represens he producion of urbulen kineic energy epressed as U W U W Prod, (7)

3 5 Journal of Marine Science and Technology, Vol. 3, No. 4 (15) where k,, C, C 1 and C are empirical coefficiens and are aken o be k = 1., = 1.3, C u =.9, C 1 = 1.44 and C = 1.9 as suggesed by Rodi (198) and Hsu e al. (4).. Iniial and Boundary Condiions The iniial condiion and boundary condiions are specified o solve he boundary value problems. A he beginning of he flow simulaion he velociy componens U and W are boh se o ero hroughou he whole flow field. The hydrosaic pressure is uilied for he firs sage of he pressure field. For he urbulence quaniies field, he logarihmic disribuion of mean angenial velociy in he urbulen boundary layer is applied, where he values of k and are assumed as funcions of disance from he boundary and he mean velociy ouside he viscous layer. On he free surface, he ero-gradien boundary condiions are imposed for boh k and ε, i.e. k/n = and /n = (Launder, 1989). The specificaion of inflow boundary condiions require more careful reamens o avoid singular erms eising in he EDE when k =. Following Lin and Liu (1998a), i is necessary o seed a small amoun of k as he iniial disurbance. The iniial value is imposed by seing k U p /, where U p = A 1 C p, U p and C p are he horional mean velociy and wave celeriy on he upsream boundary. The value of A 1 is somewha arbirary and is aken o be as suggesed by Lin and Liu (1998a; 1998b). Four boundary condiions are considered in his sudy, including he upsream, downsream, free surface and solid boundaries. In he numerical compuaions he free-surface displacemens and he velociy componens of wo kinds of periodic waves are given as he inflow condiions a he upsream end. The firs kind of periodic wave is seleced as he co-sinusoidal wave based on linear wave heory. The oher inciden wave is he Cnoidal (Cn) wave which is a nonlinear wave profile. The wave profile reads (Isobe e al., 1987) hs h h A 3 n ncn n L T, (8) where = /L is he wavenumber and L is he wavelengh. The horional and verical velociy componens are given by U gh B 3 m h n nm cn n m h L T, (9) W gh sn dn L T L T 3 m1 h n h n Bnm cn n1 m1 1 4 L m h L T, (1) where A n and B nm in Eqs. (8) o (1) are coefficiens deermined based on he heory of Isobe e al. (1987); cn, sn, and dn are Jacobian ellipic funcions, respecively. The non-slip boundary condiions is applied on he solid boundary for he mean flow field. On he free surface boundary, one kineic and wo dynamic boundary condiions are used which are wrien in he form, h s hs U W, U U n S, n a h s (, ), (11) a h s (, ), (1) U n P Sn, a h s (, ), (13) n where he subscrips n and denoe he ouward normal and angenial direcions, respecively. 3. Inerface Calculaion Like oher sudies of free surface flows, we also encouner difficulies in reaing he free surface boundary condiions of wave breaking on a sloping beach. A suiable way of finding he free surface is of grea imporance in he numerical calculaion. The piecewise linear inerface calculaion (PLIC) mehod presened by Gueyffier e al. (1999) which was developed on he basis of VOF is adoped in his sudy o rack he rapidly varying waer surfaces during wave breaking. The VOF/PLIC is epanded o he second-order in order o rea he inerface wih Lagrangian advecion of he inerface more precisely. This mehod was firs implemened in Tsai e al. (8) compuaion of wave breaking on a sloping beach. Based on Hir and Nichols (1981), we define a VOF funcion, F(,, ), which represens he fracional volume of fluid occupied on every cell, which is described by he following equaion F U F W F, (14) As menioned above he algorihm of VOF/PLIC by Gueyffier e al. (1999) is used for describing waer surface configuraion wih higher-order accuracy. The procedure for his algorihm is divided ino wo seps: a reconsrucion sep and a propagaion sep. The firs sep is he deerminaion of he orienaion of he segmen and he area occupied by waer in a surface cell wih he known volume fracion F. For he second sep, a Lagragian advecion echnique is uilied o advec he inerface segmens and evaluae he corresponding volume flues in he fluid cell. For he solid boundary condiion, a fleible immersed boundary mehod was proposed by Peskin and Prin (1993) o enforce no-flow boundary condiions on he boundary using discree dela funcions on boundaries. However, i has been

4 T.-W. Hsu e al.: Coupling VOF/PLIC and Embedding Mehod for Simulaing Wave Breaking on a Sloping Beach 51 shown o lose accuracy in rigid boundaries (Lai and Peskin, ). The oher alernaive is he embedding (EB) mehod developed by Ravou e al. (3). The model represens he solid body by adding a force field o he fluid momenum equaions in he compuaional cells ha are fully or parial occupied by he solid phase. Deails of he EB mehod are referred o Ravou e al. (3). An advanage of he EB mehod is ha he compuaions are performed on a Caresian grid wihou he need o fi he comple boundaries. For his reason, EB is firs applied in he presen invesigaion o simulae wave breaking and energy decay on a sloping beach. We rewrie he momenum equaions by adding he virual force erm as follows. U Rhs f, (15) W Rhs f, (16) where f and f are he horional and verical direcion virual force/body force erms. Rhs and Rhs are he horional and verical direcion convecive and diffusive erms of he momenum equaions, respecively. The hree-sep ime-spli scheme is used o advance he flow field. Firs he velociy is sepped from he nh ime level o he firs inermediae level by solving he adveciondiffusion equaions wihou he pressure and virual force for he momenum equaion. Subsequenly, his sep can be wrien in he forms U U n W W n Rhs n Rhs n, (17), (18) A his sage, one reas he sysem as a binary fluid. One phase is simply he ordinary fluid ouside he rigid body while he oher is he srucure phase iself, wihin he velociy is epeced o vanish. To eacly idenify he cells assigned o each phase, a volume fracion field is defined as he fracion of he area of each cell occupied by he srucure. By imposing he body force erm ( f, f ) in hose cells ha are parially or fully occupied by he srucure, we modify he velociy field o make i vanish in he srucure. Namely, he presence of he body force in he -and -momenum equaion implies he updae equaion, respecively, U U W W f, (19) f, () where U and U are inermediae value of he velociy before and afer his fracion sep. The embedding mehod deermines he force f o make he updae velociy U vanish wihin he srucure. To make he velociy vanish inside he srucure bu remain unchanged in he fluid, he case of he velociy componen (U, W), We se ( U ) U U f U (1 ) U, (1) ( W ) W W f W (1 ) W, () where and are he fracional volume of srucure, i.e. 1 inside he srucure (, ) ~ 1, on he srucure surface, (3) in he flow domain Thus, in cell (i,k) inside he srucure, he velociy along he -direcion is canceled ou since (i,k) = 1. In he boundary cells, he updae velociy is parially modified since (i,k) is beween ero and one. Ouside he srucure (i,k) = and U (i,k) is idenical o U (i,k) so ha he flow remains unchanged. By insering Eq. (1) ino Eq. (19), he value of he body force in he horional direcion is deduced: (, ik) U (, ik) f (, ik), (4) Subsiuing Eq. () ino Eq. (), he value of he body force in he verical direcion is deduced: (, ik) W (, ik) f (, ik), (5) Following Hsu e al. (4), he RANS equaions were solved numerically by a finie volume mehod wih a saggered sysem. Velociy componens are defined a he midpoins of a cell face o ensure mass conservaion, he pressure, he urbulen kineic energy, he dissipaion rae of energy and he wave profile are defined a he cener of he cells. For a deailed descripion of he numerical procedure, he readers may refer o Hsu e al. (4). In order o achieve he numerical sabiliy and accuracy, appropriae mesh incremens, and mus be seleced in he sandard finie difference approimaion of VOF mehod. For accuracy, he grid sies should be chosen small enough o resolve he emporal and spaial variaions of all flow characerisics. However, he VOF/PLIC combined wih EB mehod provide fleibly larger grid sies o raise numerical efficiency.

5 5 Journal of Marine Science and Technology, Vol. 3, No. 4 (15) H h SWL wave guages (a) (b) (c) (d) (m) Fig. 1. Definiion diagram of periodic wave rain over a sloping bed. (a), (b), (c) and (d) denoe four locaions for measuring emporal variaions of waer surface elevaions for waves ravelling over a sloping beach. III. VERIFICATIONS Numerical simulaions of breaking waves on a sloping beach are conduced o evaluae he performance of he presen model. For he comparison wih he oher models, eperimenal daa ses were used o compare wih numerical resuls. Fig. 1 shows he schemaic skech of waves propagaing over a sloping beach, where he beginning of he sloping bed is locaed a = m. Ting and Kirby (1994) performed eperimens o invesigae he evoluion of a spilling breaker of Cn wave over a sloping bed. The inciden wave heigh and period are H = 5 m and wave period T =. s wih a consan waer deph of m for waves propagaing a sloping beach. Wave breaking occurs a breaking poin = 6.4 m wih a spilling breaker on a 1/35 slope. The emporal variaion of waer surface elevaions were measured a 4 various locaions (a)~(d) (see Fig. 1) over he bed. In he numerical simulaion, he compuaional domain is 8 m long and m high wih he grid sies =. m and =.75 m in VOF mehod, and minimum grid sies =. m, =.1 m in VOF/ PLIC ogeher wih EB mehod. The simulaions were conduced for 6 s of waves. Fig. shows comparison of he simulaed and measured insananeous waer surface elevaions from he shoaling region o he bore region. I is noed ha he model resuls agree very well wih he eperimenal measuremens. Noe ha he VOF/PLIC echnique used in he presen model predics a beer resul han he VOF mehod a he bore region. In Fig., we also noice ha he wave profiles presens shorer and higher wave cress and longer and flaer wave roughs afer wave breaking. In such condiion, much wave energy has been released a he breaking poin where wave heigh suddenly decreases. Fig. 3 shows comparison of he compued and measured wave cres elevaions and rough depressions wih Ting and Kirby s daa, where in numerical resuls given by Bradford s () RNG model and Zhao e al. s (4) SGS model are also ploed for comparison. I is seen ha he predicaed wave cres profile by he presen model agrees well wih η (m) η (m) η (m) η (m) (a) =.5 m - (b) = 4.5 m - (c) = 6.5 m - (d) = 9.5 m /T Fig.. Comparisons of simulaed (solid line) and measured (open circles, Ting and Kirby (1994) waer surface elevaions a he shoaling regions (a)-(c) and he bore region (d). surface elevaion (m) Eperimenal daa (Ting and Kirby, 1994) RNG model (Bradford, ) SGS model (Zhao e al., 4) Presen model 4 Fig. 3. Compued and measured spaial disribuion of wave cres elevaions and wave rough depressions of he spilling breaker. eperimens ecep for he values in he region = 5.5~6.4 m. In his case, he wave breaking poin occurs a = 6.4 m and he bore region happens beween = 7.5 m and = 9.11 m. The compued breaking poin by he presen model is locaed a b = 6.1 m, which is very close o he measured breaking poin. Noably, he RNG model predics he wave breaking far early han ha observed in he eperimen, and also underesimaes he wave cress in he surf one. The compued breaking poin by SGS model is a = 6.16 m, which is also very close o he observed breaking poin. However, he model also overesimaes surface elevaions near he breaking poin as well as in he bore region. This resul indicaes ha he VOF/PLIC could improve numerical schemes for describing he waer surface profile wih higher-order accuracy. The presen model accuraely predics he wave heigh a he inner surf one in ha he surface profile presens a more peaked cres and flaer rough. The phase averaged horional and verical velociies compared wih laboraory measuremens are shown in Fig. 4. A he region of he wave breaking and he surf one comparisons are made a wo locaions of = 7.75 m and 9.11 m a differen verical locaions = - m, -.6 m and -. m. The general agreemen beween he simulaed resuls and measured daa is good for SGS and he presen model. However, i seems ha he accuracy of he presen model is 6 8 1

6 T.-W. Hsu e al.: Coupling VOF/PLIC and Embedding Mehod for Simulaing Wave Breaking on a Sloping Beach 53 (a) (b) (c) (d) u (m/s) w (m/s) u (m/s) w (m/s) - 1 Z = - m (s) Z = -.6 m Z = -. m (s).8-1 (s) Fig. 4. Comparisons of phase averaged horional and verical velociies a iniial breaking = 7.75 m (a), (b) and bore region = 9.11 m (c), (d). The verical locaions are, from lef o righ, = -; -.6; -. m. Lines: he presen model; circles: eperimenal daa from Ting and Kirby (1994); solid lines: presen model; dash lines: SGS model of Zhao e al. (4). beer han SGS owards he free surface. This is mainly due o he fac ha numerical resoluion of VOF/PLIC on he surface is higher. Zhao e al. (4) showed ha he wall boundary condiion needs o be implemened o ge accurae underow profiles. Here he EB mehod described by Ravou e al. (3) is used. Fig. 5 depics he compued and measured underow profiles a si differen measured locaions. Noably he presen model reproduces he eperimenal daa favorable in which he magniude and profile shape are well represened. The RNG model in general seems o underesimae he underow and fails o predic he underow profile. The SGS model shows general agreemen beween numerical and eperimenal resuls. However, we noe ha he SGS model overesimaes he underow profile a four locaions shown in Figs. 5(b), (c), (d) and (f). This comparison indicaes ha he accuracy of he RANS model seems o be significanly improved for he predicion of underow using EB mehod o he spilling breaker case. IV. RESULTS AND DISCUSSION In his secion, we presen numerical resuls and make some discussion on mean flow field and urbulen ranspor mechanism under wave breaking on a sloping boom using he presen model. Ineresing issues are physical propery induced (-ζ )/h (-ζ )/h (a) = m (b) = 7.75 m U/(gh) 1/ U/(gh) 1/ U/(gh) 1/ (c) = m (d) (e) (f) = m = 9.11 m = 9.75 m Fig. 5. Comparisons of simulaed and measured underows. Solid line: presen model; dashed lines: SGS model (Zhao e al., 4); solid cross line (Bradford, ); open circles: eperimenal daa from Ting and Kirby (1994). From (a) o (f) a locaions = 6.665; 7.75; 7.885; 8.495; 9.11; 9.75 m. by a spilling breaker, including he flow field, normalie vorices, urbulen kineic inensiy, urbulen dissipaion and eddy viscosiy. Four snapshos a differen wave phases, i.e. /T =, 5,.5 and.75, are demonsraed in he spaial domain o give a precise descripion of some ineresing phenomena in he breaking processes. Fig. 6 displays he spaial variaions of he mean flow field. The mean velociy is normalied by he wave celeriy c = gh. A /T =, a wave cres jus passes he breaking locaion ( = 6.4 m). I is shown ha he roller acs as he recirculaing flow in he fron of he urbulen bore. A je is ejeced from he cres of he breaking wave and impacs in he very upper par of he face of a wave. In he roller region, he horional fluid paricle velociy is srong and roughly equal o he local phase velociy. Fig. 7 shows he normalied mean voriciy field gh under he spilling breaker, where = (U/ W/) is he voriciy srengh. I is noiced ha a vore sars o form a he oe of he wavefron righ before wave breaking. This vore is furher srenghened and conveced o almos he whole cres region. The urbulen inensiy is defined as I ( uu ww ) k (6) where u and w are he velociy flucuaions in he and direcion, respecively. Following Lin and Liu (1998a, 1998b), he urbulence inensiy is normalied by wave celeriy c =

7 54 Journal of Marine Science and Technology, Vol. 3, No. 4 (15).7 /T = = 6.4 m /T = /T = /T = Fig. 6. Simulaed velociy fields for spilling wave breaker over a sloping beach a differen wave phases. (a) /T = ; (b) /T = 5; (c) /T =.5; (d) /T =.75. The velociy is normalied by gh..7 /T = = 6.4 m /T = /T =.5 /T = Fig. 7. Simulaed normalied voriciy for spilling wave breaker over a sloping beach a differen wave phases. (a) /T = ; (b) /T = 5; (c) /T =.5; (d) /T =.75. The normalied voriciyis ( U/ W/ )/ gh gh. The snapshos of urbulen inensiy are presened in Fig. 8. Fig. 8 shows ha he urbulen inensiy is generaed jus before he breaking poin and is higher in he roller region and forward o he face of he wave. Under he spilling breaking wave, he urbulen kineic energy coninues o dissipae in he bore region. This is due o he high shear raes a he wave fron, which generaes significan levels of k a he lower fron face of he wave. As he wave propagaes forward, he urbulence kineic energy gradually decreases bu wih very similar paerns of urbulence disribuion. Noe ha he urbulen inensiy is hen conveced and diffused o he forward face of he wave, remaining in considerable amoun as he wave moves ino he surf one. The urbulen dissipaion is presened in Fig. 9. I is seen ha he urbulen dissipaion akes place above he wave rough and a he wavefron, wih no urbulen dissipaion under he wave cres. The urbulen dissipaion displays a long ail a he back face of he wave when he wave approaches he shore. The eddy viscosiy is an imporan parameer measuring he miing rae of momenum. Fig. 1 demonsraes he spaial disribuion of he eddy viscosiy. The eddy viscosiy has been normalied by he molecular viscosiy. The eddy viscosiy decreases wih he shoaling of waer deph afer wave breaking. Since he eddy viscosiy is proporional o he lengh scale and urbulen inensiy. When compared wih he urbulen inensiy, he eddy viscosiy decreases faser in he surf one. I is noed ha mean voriciy, urbulen inensiy, urbulen dissipaion, and eddy viscosiy are mainly concenraed in he region very close o breaking wave frons. In oher regions, hese four quaniies are raher small, which suggess ha he mean flow is almos a poenial flow wih lile influence from he breaking processes.

8 T.-W. Hsu e al.: Coupling VOF/PLIC and Embedding Mehod for Simulaing Wave Breaking on a Sloping Beach 55.7 /T = = 6.4 m /T = /T = /T = Fig. 8. Simulaed normalied urbulen inensiy (k) 1/ /(gh) 1/ for spilling wave breaker over a sloping beach a differen wave phases. (a) /T = ; (b) /T = 5; (c) /T =.5; (d) /T = /T = = 6.4 m /T = /T = /T = Fig. 9. Simulaed normalied urbulen dissipaion for spilling wave breaker over a sloping beach a differen wave phases. (a) /T = ; (b) /T = 5; (c) /T =.5; (d) /T =.75. The normalied urbulen dissipaion is ε/g(gh) 1/ V. CONCLUDING REMARKS A wo-dimensional urbulen model was developed o simulae spilling wave breaker in he surf one by direcly solving Reynolds Averaged Navier-Sokes (RANS) equaions and he coninuiy equaion. The VOF wih piecewise linear inerface calculaion (VOF/PLIC) which was developed o a second order accuracy is adoped o rack he free surface configuraions on a Caresian grid. On he oher hand, he presen model is improved on he solid boundary using embedding (EB) mehod of Ravou e al. (3). The comple boom opography is accouned for using EB in which he solid boundary is represened by adding a force o he flow phase in he compued cells ha are fully or parially occupied by he solid phase. The advanage is ha i is unnecessary o impose boundary condiions on he body surface. By applying he developed model o he problems of spilling wave breaking on a sloping boom, we found ha he model resuls compare very well wih eperimenal measuremens as well as oher RANS model resuls in which he mesh incremen are chosen sufficienly small. Numerical resuls are in erms of waer surface elevaions, mean paricle velociies, wave heigh disribuions and underow profiles. In general, he presen model showed favorable agreemens o he eperimens and SGS model resuls by Zhao e al. (4). The improvemen is especially significan on he free surface and underow profile near he boom under he spilling breaker. As noed in secion 3, VOF is a sep funcion ha is sill needed o choose small enough o resolve he emporal and spaial variaions of flow characerisics for wave breaking. However, he presen model could provide a fleible course grid sysem o lower compuaional epense for he breaking wave problem. Deailed analysis of numerical resuls also showed ha he urbulen inensiy and voriciy are primarily locaed above he wave rough. The urbulen inensiy is conveced and diffused o he forward face of he wave, and coninues o

9 56 Journal of Marine Science and Technology, Vol. 3, No. 4 (15).7 /T = = 6.4 m /T = /T = /T = Fig. 1. Simulaed normalied eddy viscosiy for spilling wave breaker over a sloping beach a differen wave phases. (a) /T = ; (b) /T = 5; (c) /T =.5; (d) /T =.75. The eddy viscosiy is normalied by molecular viscosiy ν /ν. dissipae while he breaking wave moves owards he shore. Afer he recovery of wave energy, he second wave breaking akes place and he voriciy in he bore region is larger han ha in oher regions. The eddy viscosiy shows he same paern of he urbulen inensiy beween he breaking and role region. The numerical model accuraely reproduces he wave breaking, he recovery and he complicaed dynamics generaed under spilling breaking waves. A deailed descripion is provided in erms of free surface, velociy field, and urbulen energy ranspor. Fuure sudy on numerical simulaions of he plunging breaker is underaken and will be repored. ACKNOWLEDGMENTS This research was suppored by he Naional Science Council and he Landmark Projec of Naional Cheng-Kung Universiy, Taiwan, under he Grans No. NSC-96-E MY3 and A16. REFERENCES Bradford, S. F. (). Numerical simulaion of surf one dynamics. Journal of Waerway, Por, Coasal and Ocean Engineering, ASCE 16, 13. Chrisensen, E. D. and R. Deigaard (1). Large eddy simulaion of breaking waves. Coasal Engineering 4, Gueyffier, D., J. Li, A. Nadim, R. Scardovelli and S. Zaleski (1999). Volume-of-fluid inerface racking wih smoohed surface sress mehods for hree-dimensional flows. Journal of Compuaional Physics 15, Hir, C. W. and B. D. Nichols (1981). Volume of fluid (VOF) mehod for dynamics of free boundaries. Journal of Compuaional Physics 39, 1-5. Hsiao, S. C. and T. C. Lin (1). Tsunami-like soliary waves impinging and overopping an impermeable seawall: eperimen and RANS modeling. Coasal Engineering 57(1), 18. Hsu, T. W., C. M. Hsieh and R. R. Hwang (4). Using RANS o simulae vore generaion and dissipaion around impermeable submerged double breakwaers. Coasal Engineering 51, Isobe, M., H. Nishimura and K. Horikawa (1987). Epressions of perurbaion soluions for conservaive waves by using wave heigh. Proceeding of he 33rd Annual Conference, JSCE, Japan, Kobayashi, N., A. K. Oa and I. Roy (1987). Wave reflecion and run-up on rough slopes. Journal of Waerway, Por, Coasal and Ocean Engineering, ASCE 113(3), Lai, M. C. and C. S. Peskin (). An immersed boundary mehod wih formal second-order accuracy and reduced numerical viscosiy. Journal of Compuaional Physics 16, Lakehal, D. and P. Liovic (11). Turbulence srucure and ineracion wih seep breaking waves. Journal of Fluid Mechanics 674, Launder, B. E. (1989). Second-momen closure and is use in modeling urbulen indusrial flows. Inernaional Journal for Numerical Mehods in Fluids 9, Lemos, C. M. (199). Wave Breaking, Springer. Lin, P. and P. L. F. Liu (1998a). A numerical sudy of breaking waves in he surf one. Journal of Fluid Mechanics 359, Lin, P. and P. L. F. Liu (1998b). Turbulence ranspor, voriciy dynamics, and solue miing under plunging breaking waves in surf one. Journal of Geophysical Research 13, Lubin, P., S. Vincen, S. Abadie and J. P. Calagirone (6). Three-dimensional large eddy simulaion of air enrainmen under plunging breaking waves. Coasal Engineering 53, Lubin, P., S. Glockner, O. Kimmoun and H. Branger (11). Numerical sudy of he hydrodynamics of regular waves breaking over a sloping beach. European Journal of Mechanics-B/Fluids 3(6), Peskin, C. S. and B. F. Prin (1993). Improved volume conservaion in he compuaion of flows wih immersed elasic boundaries. Journal of Compuaional Physics 15, Pei, H. A. H., P. Tonjes, M. R. A. van Gen and P. ven den Bosch (1994). Numerical simulaion and validaion of plunging breakers using a D Navier-Sokes model. Proceeding of 4h Inernaional Conference on Coasal Engineering, Kobe, ASCE, Ravou, J. F., A. Nadim and H. Haj-Hariri (3). An embedding mehod for bluff body flows: ineracions of wo side-by-side cylinder wakes. Theoreical and Compuaional Fluid Dynamics 16, Rodi, W. (198). Turbulen models and heir applicaions in hydraulics - a sae of he ar review. Inernaional Associaion for Hydraulic Research, Delf, 14. Ting, F. C. K. and J. T. Kirby (1994). Observaion of underow and urbulence in a laboraory surf one. Coasal Engineering 4, Tsai, C. Y., T. W. Hsu, S. H. Ou and Y. J. Jhu (8). A D numerical simulaion by VOF/PLIC mehod on wave breaking over a sloping boom. Proceeding of 31h Inernaional Conference Coasal Engineering, Hamburg,

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