PREDICTIONS O CIRCULATING CURRENT IELD AROUND A SUBMERGED BREAKWATER INDUCED BY BREAKING WAVES AND SURACE ROLLERS Yoshmtsu Tama 1 Ths paper develops a quas-three-dmensonal nearshore current model whch accounts for excess shoreward volume fluxes due to waves and surface rollers. The model splts the water column at the wave trough level and solves two dfferent layers of DH momentum equatons. The frst layer covers above the trough level and determnes the shear stress at the trough level whle the the second layer covers over the entre depth and determnes the volume flux due to mean current components. Lnearly approxmated vertcal profle of the shear stress and correspondng turbulent eddy vscosty model enables the present model to yeld analytcal explct expresson of the horzontal velocty profle whch enhance computatonal effcency of the model. The extended model was appled to predctons of longshore current velocty on the long straght beach and crculaton current patterns behnd the detached submerged breakwater on the plane beach. Excellent agreement between model predctons and measurements supports the valdty of the model and mportance of quas-three-dmensonal features around the submerged break water. Keywords: submerged breakwater, Q3D nearshore current model, surface roller, wave breakng INTRODUCTION After the revson of the coastal law n 1999 n Japan, a submerged breakwater became one of preferred optons as coastal protecton measures. A submerged breakwater has advantages n that t has less mpact on the coastal landscapes and t also enhances water exchanges to keep the water qualty behnd the structure. Submerged-type breakwater however tends to cause more complcated crculaton current patterns and, as a result, these structure often faled to realze the expected coastal protecton functons. or example, breakng and broken waves on the submerged structure should transport excess amount of water toward the shore over the structure. The excess amount of shoreward water ncreases mean water level behnd the structure and may alter the forcng balances of the crculatng current feld behnd the breakwater. Excess amount of shoreward volume flux may also enhance the returnng undertow velocty near the water bottom. Wdely used two-dmensonalhorzontal (DH) nearshore current model cannot properly express such phenomena. In addton to such three-dmensonal phenomena, submerged-type breakwater may cause complex wave feld due to breakng, reflecton and dffractons of waves. Tme-dependent non-lnear dspersve wave model such as Boussnesq-type model may be one of preferred models to drectly express such complex wave-current co-exstng feld. Tama et al (007), for nstance, appled Boussnesq-type model for predctons of the crculaton current around the submerged breakwater and ponted out that the predctve sklls of the crculaton current are strongly dependent on the breakng wave models and most models fal to represent the observed crculaton current patterns. Besdes breakng model, depth-ntegrated two-dmensonal-horzontal (DH) tmedependent model also has drawbacks n that the model accounts only for the wave-assocated vertcal profles of the horzontal current velocty and does not compute the vertcal profles of the mean shear current velocty. urthermore, the model does not account for the shoreward volume flux due to broken waves and surface roller whch should have sgnfcant nfluence on the return flow velocty. Especally around the submerged breakwater, broken waves appear to transport extensve amount of water toward the shore near the water surface and the model may underestmate the return flow velocty under the wave trough level, whch compensates for the shoreward volume flux. As a model that keeps computatonal effcency and capablty of accountng vertcal profles of horzontal shear current velocty, quas-three-dmensonal (Q3D) models have been developed by varous researchers. Some of these Q3D model drectly computes horzontal velocty at arbtral elevaton under the approxmatng assumpton of hydrostatc pressure (e.g., Kurowa et al., 1997). Most of other Q3D models determne approxmate vertcal profle of the horzontal velocty, substtute t nto the horzontal momentum equatons and then ntegrate the obtaned momentum equatons n 1 Department of Cvl Engneerng, The Unversty of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, 113-8656, Japan 1
COASTAL ENGINEERING 01 the vertcal drecton to yeld the depth-ntegrated DH momentum equatons (e.g., Svendsen and Putrevu, 1994, Sanchez et al., 199). Smlar to the well-known DH nearshore current model, most of these Q3D models apply the phase-averaged momentum equatons wth wave radaton stress and dsperson terms wth emprcally determned dsperson coeffcents. Svendsen and Putrevu (1994) on the other hand ponted out n ther model, SHORECIRC that extra advecton terms whch are yelded by accountng for vertcal varaton of the horzontal current velocty, have smlar effect of horzontal dsperson terms. Tama and Madsen s (005) undertow model can be appled for estmatons of vertcal profles of both undertow and longshore current velocty when perodc or random waves were oblquely ncdent on a long straght beach. Besdes advecton effects smlar to SHORECIRC, the model accounts for the nfluence of surface rollers both n momentum and mass conservaton equatons. Influence of excess amount of volume flux due to broken waves and surface rollers may be one of crucal factors for better predctons of the mean shear current feld behnd the submerged breakwater. Ths study thus ams to extend Tama and Madsen s (005) model to the arbtrary Q3D problems and eventually apply t to the predctons of current feld around the submerged breakwater. MODEL DESCRIPTIONS The entre model conssts of three sub-models for predctons of: () wave shoalng and breakng, () surface roller evoluton and () Q3D nearshore current feld. Ths study apples Tama and Madsen s (006) broken wave and surface roller models. The Q3D nearshore current model nputs phase-averaged forces and volume fluxes assocated wth waves and surface rollers. gure 1 llustrates the basc concept of the present model. As shown n the gure, the model splts the water column nto two parts at the wave trough level and determne the DH phase-averaged momentum equatons vertcally ntegrated over the two dfferent layers: one above the wave trough level; and the other over the entre depth. More detals of the model are dscussed n the followng sectons. Governng Equatons Under the wave trough level, vertcal gradent of the horzontal velocty s determned by followng momentum equaton. gure 1. Defntons of the varables used n the model.
COASTAL ENGINEERING 01 3 T U z cz, cb, c h tr cb, z (1) where U s horzontal velocty components n the x drecton (=1,), s water densty, T s turbulent eddy vscosty dscussed later, cz, s phase averaged horzontal shear stress actng n the x drecton, cb, and c are horzontal shear stress respectvely at bottom and trough level and h tr s the water depth under the trough level. Vertcal profle of the shear stress dstrbutons s approxmately assumed to be lnear. Ths assumpton s vald f the wave-nduced forces ncludng surface rollers are domnant under the shallow water approxmatons. Lnear approxmaton moreover enables us to determne the shear stress at arbtrary elevaton only from the shear stresses at bottom and at wave trough level. The present turbulent eddy vscosty model follows the one proposed by Tama and Madsen (006), n whch, besdes general law-of-the-wall-type lnear eddy vscosty model, nfluence of excess turbulence caused by broken waves and by shear stress actng at the wave trough level s accounted for. Combned wth determned turbulent eddy vscosty model, equaton (1) enables us to determne the vertcal profle of the horzontal current velocty n an analytcal expresson, whch sgnfcantly mproves the computatonal effcency of the model. nally, as seen n gure 1, mean current velocty above the trough level was smply assumed to be constant wth ther quanttes represented by those determned at the wave trough level. The shear stress at the trough level, c s determned by ntegratng horzontal momentum equaton from the surface to the wave trough level: τ c shp, sw, sr, sc, swc, src, sv, () where the seven terms n the rght hand sde of equaton () are respectvely: () hydrostatc pressure force; () momentum force due to wave moton; () momentum force due to surface roller; (v) momentum force fucurrent-current nteracton; (v) momentum force due to wave-current nteractons; (v) momentum force due to current-surface roller nteractons; and (v) horzontal shear stress due to turbulence. Detaled expressons of each term are as follows. shp, ga (a) x E auˆ uˆ wˆ a x s sw s wˆ,, s x x x (b) R sr, (c) x au U sc, (d) x swc, a uˆ U x U uˆ x (e) A U sr Cn, Cn U s (f) CT x x src, U a Ts (g) x x sv,
4 COASTAL ENGINEERING 01 where s mean water level, E s wave energy, u ˆ s, and ŵ s are ampltude of the wave-assocated oscllatng current velocty at the surface respectvely n the x and z drecton R s radaton stress of the surface roller, U s the mean current velocty components n the x drecton, A sr s the crosssectonal area of the surface roller per each wave length, T s wave perod, C s wave phase velocty, n s a x -component of the unt vector of the wave propagatng drecton, Ts s a turbulent eddy vscosty near the surface and =1, follows the summaton conventon. Svendsen and Putrevu s (1994) model does not account for the thrd, ffth, and sxth terms n equaton (),.e., the momentum forces due to surface roller, wave-current and roller-current nteractons. Surface roller radaton stress tensor, R, and the cross-sectonal area of the roller, A sr, are respectvely determned by AsrC nn R Esr nn (3) CT A sr TEsr (4) C Smlar to the shear stress at the trough level, the shear stress at the bottom can also be determned by ntegratng the horzontal momentum equaton over the entre water depth,.e., q (5) t c, τ cb, bhp, bw, sr, bc, bwc, bsrc, bv, Respectve contrbutng factors of the forcng terms on the rght hand sde of equaton (5) are the same as the ones n equaton () whle ther expressons are slghtly dfferent snce the ntegraton range of the momentum equaton s dfferent. bhp, gh (5a) x S bw, (5b) x bc, au U qb, U0, x (5d) (5e) bwc, auˆ U au uˆ x bsrc, Asr n U nu s T x, (5f) U h T (5g) x x bv, where S s well-known wave radaton tensor q b s mean volume flux component under the trough level n the x drecton, U 0 s averaged mean current velocty under the trough level and T0 s depthaveraged turbulent eddy vscosty. Introducton of these two depth-averaged values assumes that fluctuatons of mean current velocty n the vertcal drecton are small relatve to ts depth-averaged velocty. As seen n equaton (5), the model keeps a tme-dervatve term of q c,, depth-ntegrated volume flux due to mean current components. Note that phase-averaged velocty components of waves and surface rollers should be constant n tme and thus the tme-dervatve term was only left for the
COASTAL ENGINEERING 01 5 depth-ntegrated mean current components. Whle the present model ams to determne the steady state condton the model keeps tme-dervatve term n equaton (5) and, as dscussed n the next secton, the model uses ths tme-dervatve term to teratvely estmate the modfcaton of the volume flux, q c,, untl t reaches to the steady state condtons. Tama and Madsen (006) ponted out that all of these phase-averaged terms related to wave and surface roller motons are determned as functons of phase-averaged energy of the waves and surface rollers and thus the wave model can be a smple and computatonally effcent phase-averaged-type model such as energy balance equatons. In ths study, however, we apply tme-dependent lnear mld slope equatons for evaluatons of the wave feld snce the partal reflected waves around the submerged break water may play sgnfcant role to determne the phase-averaged forcng feld. As dscussed n the prevous paragraph, depth-ntegrated DH momentum equaton (5) s used to determne the volume flux of the mean current component, q c, and the model stll needs to determne the bottom shear stress and mean water level. To estmate the bottom shear stress under wave current coexstng feld, the present model followed Tama and Madsen s (006) model. Tama and Madsen s (006) model, based on Madsen s (1994) wave-current bottom boundary layer model, determnes the phase averaged bottom shear stress as a functon of: () ampltude of near-bottom wave orbtal velocty; () wave perod; () angle between wave and mean shear stress; (v) equvalent bottom roughness; (v) depth-ntegrated volume flux, q c,. nally, mean water level s determned by applcaton of the depth-ntegrated mass conservaton equaton,.e., t x q c, qw q sr, (6) Computaton Procedures The present model dffers from the Tama and Madsen s (006) undertow model n that the model determnes the mean water level from the tme-dependent mass conservaton equaton. Numercal computaton procedures of the present Q3D nearshore current model are summarzed as follows: 1. Set mean water level,, mean current velocty component U s, and U 0 to be zero as ntal condtons.. Compute q c of the newer tme step from the dscretzed equaton of (5). Gradually ncrease waveassocated and surface roller-assocated terms from zero to the actual values as the numercal teratons proceed. 3. Estmate mean water level from (6) wth newly estmated q c,. 4. Apply Tama and Madsen s (006) bottom boundary layer model and estmate bottom shear stress as functons of newly determned q c. 5. Replace former varables by newly estmated ones and return back to the number,, and terate the same procedures untl estmated q c and cb, reach to the equlbrum state. MODEL APPLICATIONS The present extended Q3D model was appled to two dfferent cases: the frst case s when perodc or random waves were oblquely ncdent on the long straght beach; and the second case s when perodc waves were normally ncdent on the plane beach wth submerged detached breakwater nstalled. Longshore Currents and Undertow Velocty Profles on the Long Straght Beach The present model was frst appled to the expermental case presented by Hamlton and Ebersole (001), who utlzed ther Large-Scale Sedment Transport aclty (LST) to generate the longshore current when perodc or random waves were oblquely ncdent on the long straght beach. gure and gure 3 compare predcted and measured wave heght wave setup, vertcal profles of crossshore and longshore mean current velocty, and cross-shore dstrbutons of the depth-averaged longshore current velocty for the expermental cases of Test 6A-N and 8A-E, respectvely. Test 6A-N appled regular wave whle Test 8A-E appled random wave. In the fgure, wave heghts were
6 COASTAL ENGINEERING 01 computed by energy balance equaton wth breakng and broken dsspaton model presented by Tama and Madsen (006). In gure 3, dashed lne shows the computed results when the regular wave condtons were assumed nstead of random waves. Excellent agreements of these comparsons surely valdate the predctve sklls of the present model not only for wave propertes but also for cross-shore and longshore current veloctes both under regular and random wave cases. Phase-Averaged Current eld around the Detached Submerged Breakwater The present model was fnally appled to the case when regular waves were normally ncdent on the plane beach wth submerged breakwater. gure 4 shows overvew of the expermental condtons performed by Tama et al. (007). As seen n the fgure, ths experment ntroduces vertcal walls on both sdes of the plane beach and represents the long straght beach condtons wth the mrror-mages reflected at the walls. In addton, steeply slopng 1:1 edge of the submerged break water should cause partal reflectons. Under such condton reflected wave components should have sgnfcant nfluence on the nearshore current feld and thus phase-averaged energy balance equaton, whch cannot account for wave phase nformaton, may not be the optmum wave model to be appled n ths specfc case. Ths study therefore appled lnear mld slope equatons (Oonaka and Watanabe, 1987) to determne the wave and surface roller feld. ollowng Tama and Sato (010), Tama and Madsen s (006) broken wave model was ntroduced to the mld slope equatons so that wave attenuaton obtaned n ths model should be consstent wth that of the energy balance equaton model. As dscussed n the prevous secton, the present Q3D model requres nputs of phase-averaged wave propertes such as wave radaton stress and volume flux due to wave motons. Snce the tme-dependent mld slope equaton yelds nstantaneous surface water level, and horzontal velocte u, and volume flux, P, ths study followed Tama and Sato (010) and numercally computed phase-averaged radaton stress tensor, S, and wave-nduced volume flux q w, by gure. Comparsons of computed wave heght setup, vertcal profles of cross-shore and longshore current velocte cross-shore dstrbutons of depth-averaged longshore current veloctes wth measured data presented by Hamlton and Ebersole (001) (Test 6A-N, regular wave case).
slope1:0 COASTAL ENGINEERING 01 7 gure 3. Comparsons of computed wave heght setup, vertcal profles of cross-shore and longshore current velocte cross-shore dstrbutons of depth-averaged longshore current veloctes wth measured data presented by Hamlton and Ebersole (001) (Test 8A-E, random wave case). waves unform depth 3 1:1 Sub.B.W. 14 4 Sde Wall 84 30 1 4 Stll Shorelne unt : cm gure 4. Overvew of the expermental setup. S q 1 0 uu w dz (7) h w k u dz P (8) 0 tanh kh where k s wave number determned from wave perod and local water depth. ollowng Tama and Madsen(006), evolutons and dsspatons of the surface roller s determned through the energy balance equaton of the surface roller. Recallng the conceptual assumpton that a part of broken wave dsspaton energy should be provded to the growth of the surface roller the present model determned balance equaton of the surface roller energy, E sr, by
8 COASTAL ENGINEERING 01 gure 5 Cross-shore dstrbutons of computed wave heghts and shoreward volume flux due to wave and surface roller (SR) compared wth measured data (crcles) presented by Cox and Kobayash (1996) E t sr EsrCn x K h sr EsrC x P B P x (9) where K sr s coeffcent of the energy dsspaton of the surface roller and the second term n the rght hand sde of (9) s the energy producton term, whch corresponds to the nstantaneous energy dsspaton rate of the broken waves. Dsperson coeffcent of broken wave, B, s determned so that the phase averaged broken wave energy dsspaton rate corresponds to the that of Tama and Madsen s (006) broken wave model based on the energy balance equatons. In the numercal model, nstantaneous surface roller energy computed from (9) was stored and the present Q3D model appled the phase-averaged value of these obtaned nstantaneous E sr. gure 5 compares predcted and measured wave heghts and shoreward volume flux due to waves and rollers. Measured data were obtaned by Cox and Kobayash (1996) n ther one-dmensonal flume experment. In the fgure, observed total volume flux was nversely estmated by vertcally ntegratng measured undertow velocty under the wave trough level. As seen n the fgure, the present model well explans the wellknown feature that wave-assocated volume flux tends to underestmate the volume flux nsde the surf zone and the gap between measured and predcted volume flux nsde the surf zone s reasonably compensated by the volume flux due to surface roller. nally, the entre model was appled to Tama et al. s (007) expermental case wth ncdent wave condtons of H=3cm and T=0.8s. Ths s the case that Boussnesq model could not suffcently acheve the excellent predctve sklls of crculaton current patterns behnd the detached submerged breakwater (Tama and Madsen, 007). gure 6 compares predcted and measured wave heghts and mean current velocty felds. In the fgure, three dfferent models were appled for ther comparsons: () Boussnesq equatons wth ansotropc dsperson-type broken wave model (Tama et al., 007); () the present Q3D model; and () conventonal DH neashore current model. or comparson both Q3D and DH models nput the same radaton stress obtaned from the computatons based on the mld slope equatons. As seen n the comparsons of the wave heght lnear-based mld slope equaton tends to underestmate the wave heghts around the breakng pont. It should also be ponted out that the present mld slope equaton model also underestmates the concentraton of the wave heghts along the left vertcal wall where s located at the gap of submerged breakwaters n the mrror mages and thus strong return flow should be domnant. Ths study dd not account for wave-current nteracton n the wave model. Except these two feature the present model showed overall good predctve sklls of the broken wave heght especally behnd the breakwater. As ponted out by Tama et al. (007), dsperson-type broken wave model tends to over-smooth mean current components and thus the model tends to underestmate the velocty of horzontal crculatng current. Moreover, Boussnesq model does not express excess amount of shoreward
COASTAL ENGINEERING 01 9 volume flux due to surface rollers and thus the model requres larger shoreward mean current velocty to keep the same amount of shoreward volume flux as the ones of the Q3D model. Ths may be another reason why, compared to the measured data, the locaton of the crculaton was shfted toward the shore n the case of Boussnesq model. Smlar to the Boussnesq model, DH model has stronger shoreward current velocty on the submerged break water and fals to represent the crculatng current pattern. In contrast to these model Q3D model tends to have smaller shoreward current velocty above the submerged breakwater and successfully represent the ant-clockwse crculatng current pattern. It should also be ponted out that measured ant-clockwse crculatng current tends to have larger offshore-ward current velocty rather than shoreward current velocty. Ths asymmetrc feature s also observed n the predcted current patterns of the present Q3D model. Ths should be because the model requres larger seaward return flow velocty to compensate the volume flux due to the surface roller. CONCLUSIOS Tama and Madsen s (006) undertow model was extended for predctons of mean shear current feld on the arbtrary bathymetry. The present model was appled to expermental cases and showed excellent predctve sklls of mean current components. One of the most mportant dfference between gure 6 Comparsons of predcted and measured wave heghts and mean current behnd the submerged detached breakwater. Predctons are: Boussnesq model (BSQ); the present model (Q3D); and conventonal DH model wthout surface roller (DH).
10 COASTAL ENGINEERING 01 Q3D and DH models s n that Q3D s able to account for excess amount of shoreward volume flux due to waves and surface rollers whle DH s not. Influence of ths dfference was clearly seen n the predctons of crculatng current patterns around the detached submerged breakwater. urther nvestgaton should be performed to examne the more general valdty and lmtatons of the present model. ACKNOWLEDGMENTS Ths study was gratefully supported by Grant-n-Ad for Young Scentsts (A), No. 3686070, from Japan Socety for the Promoton of Scence (JSPS). REERENCES Cox, D.T. and N. Kobayash. 1996. Undertow profles n the bottom boundary layer under breakng wave Proc. 5th Int. Conf. on Coast. Eng., ASCE, 3194-306. Hamlton, D. G. and Ebersole, B. A. 001. Establshng unform longshore currents n a large-scale laboratory faclty, Coastal Eng., 4, 199-18. Kurowa, M., H. Noda and Y. Houch. 1997. A quas-three dmensonal model of wave-nduced current n the surf zone, Proc. of Coast. Eng., JSCE, vol.44, 151-155 (n Japanese). Madsen, O. S. 1994. Spectral wave-current bottom boundary layer flow Proc. 4th Int. Conf. on Coast. Eng., ASCE, 384-398. Ohnaka, S., A. Watanabe and M. Isobe. 1989. Numercal computaton of wave feld by the unsteady mld slope equaton extended to wave-current coexstng feld, Proc. of Coast. Eng., JSCE, 91-95 (n Japanese). Putrevu, U., and Svendsen, I. A. 199. A mxng mechansm n the nearshore regon, Proc. 3rd Int. Conf. on Coast. Eng., ASCE, 758-771 Sanchez-Arclla, A., Collado,., and A. Rodrguez. 199. Vertcally varyng velocty feld n Q-3D nearshore crculaton, Proc. 3rd Int. Conf. on Coastal Eng. 811-84. Svendsen, I.A. and U. Putrevu. 1994. Nearshore mxng and dsperson, Proc. Math. and Phys. Scence vol.445, 561-576. Tama, Y., and O. S. Madsen. 006. Modelng near-shore wave surface roller and undertow velocty profle J. of Waterway, Port, Coastal and Ocean Eng., Vol. 13, No. 6, 49-438. Tama, Y. and S. Sato. 010. Local concentraton of slowly varyng wave and current felds around the abruptly changng bottom slopes along the shore, Proc. 3nd Int. Conf. Coast. Eng., Tama, Y., S. Sato, T. Shmozono and M. Isobe. 007. Modelng of wave-nduced current around submerged detached breakwater Proc. of Int. Conf. on Coast. Str. 007, CD-ROM publshed by World Scentfc Publshng Corp. and East Meets West Productons.