LARGE-WAVE SIMULATION OF TURBULENT FLOW INDUCED BY WAVE PROPAGATION AND BREAKING OVER CONSTANT SLOPE BED

Transcription

1 LARGE-WAVE SIMULATION OF TURBULENT FLOW INDUCED BY WAVE PROPAGATION AND BREAKING OVER CONSTANT SLOPE BED Gerasmos A. Koloythas 1, Aggelos S. Dmaopoulos and Athanassos A. Dmas 1 In the present study, the three-dmensonal, ncompressble, turbulent, free-surface flow, developng by the propagaton of nonlnear breang waves over a rgd bed of constant slope, s numercally smulated. The man objectve s to nvestgate the process of spllng wave breang and the characterstcs of the developng undertow current employng the large-wave smulaton (LWS) method. Accordng to LWS methodology, large velocty and freesurface scales are fully resolved, and subgrd scales are treated by an eddy vscosty model, smlar to large-eddy smulaton (LES) methodology. The smulatons are based on the numercal soluton of the unsteady, threedmensonal, Naver-Stoes equatons subject to the fully-nonlnear free-surface boundary condtons and the approprate bottom, nflow and outflow boundary condtons. The case of ncomng second-order Stoes waves, normal to the shore, wth wavelength to nflow depth rato λ/d Ι 6.6, wave steepness H/λ 0.05, bed slope tanβ = 1/35 and Reynolds number (based on nflow water depth) Re d = 50,000 s nvestgated. The predctons of the LWS model for the ncpent wave breang parameters - breang depth and heght - are n very good agreement wth publshed expermental measurements. Profles of the tme-averaged horzontal velocty n the surf zone are also n good agreement wth the correspondng measured ones, verfyng the ablty of the model to capture adequately the undertow current. Keywords: numercal smulaton, Naver-Stoes equatons, turbulent flow, spllng breang, surf zone, undertow current INTRODUCTION Wave breang, one of the major near-shore processes, s responsble for the development of wavegenerated currents that result nto the ntaton and development of sedment transport. Therefore, the nvestgaton of wave breang, as a phenomenon nterconnected wth a seres of sgnfcant coastal processes, s of great nterest. Wave breang taes place when wave heght and steepness become very large as the water depth becomes shallower. For the case of spllng breang, a vortex structure, usually called "surface roller", s formed under the collapsng wavefront just after breang. As mentoned above, closely related to wave breang s the generaton of wave-nduced currents,.e., the longshore current, whch s developed only when the drecton of the breang wave s oblque to the shorelne, and the cross-shore current, whch s nown as the undertow current. Both of them are developng n the surf zone,.e., the coastal area where wave energy dsspaton occurs after breang. The undertow current owes ts exstence to the mean shear stress feld, developng n the surf zone, n order to balance the pressure gradent and the momentum fluxes due to the wave set-up and the wave heght dsspaton. Close to the bottom, the current s offshore drected, whle near the free surface s orented towards the shore, ensurng that the total cross-shore water flux s zero. The wave propagaton and breang n the coastal zone has been nvestgated by several researchers for many years, by use of numercal and physcal models. As for the numercal smulaton of spllng wave breang, there are two broad categores of models based on the treatment of the surface roller: those that ncorporate emprcal models, often called surface roller (SR) models, and those that smulate the surface roller as part of the turbulent flow, nduced by the wave propagaton, utlzng one of the turbulence modelng methods,.e., the Reynolds-Averaged Naver-Stoes (RANS) equatons and the large-eddy smulaton (LES) method. The SR model, n whch ncpent breang s defned by emprcal crtera, may be coupled wth Boussnesq (Brgant et al. 004; Madsen et al. 1997; Schäffer et al. 1993; Veeramony and Svedsen 000) or Euler (Dmas and Dmaopoulos 009) equatons gvng very good results for the normal to the shore wave breang, but t cannot be easly extended to the case of oblque to the shore wave breang. RANS models, where all turbulent scales are treated by a closure model, have been appled for the case of two-dmensonal turbulent flow durng spllng breang (Bradford 000; Ln and Lu 1998; Torres-Freyermuth et al. 007). LES models, where only the large scales of turbulence are resolved whle the small ones are modeled, have been used for two-dmensonal (Heu et al. 004; Zhao et al. 004) and three-dmensonal (Chrstensen and Degaard 001; Chrstensen 006) flows. LES method requres more computatonal resources than RANS, but generally acheves better results for the 1 Department of Cvl Engneerng, Unversty of Patras, Patras, 6 500, Greece HR Wallngford Lmted, Howbery Par, Wallngford, Oxfordshre, OX10 8BA, Unted Kngdom 1

2 COASTAL ENGINEERING 01 dynamcs of spllng breaers. Both methods requre the use of a scheme for capturng the free-surface elevaton, such as the Volume of Flud (VOF) method and the Marer and Cell (MAC) method. Recently, Dmaopoulos and Dmas (011), n order to nvestgate the turbulent flow nduced close to the free surface by the oblque wave breang over a constant slope bed, employed the Large-Wave Smulaton (LWS) method coupled wth ther nvscd flow solver (Dmas and Dmaopoulos 009). Ther model was valdated by comparson of ther numercal results to correspondng expermental measurements presented n Tng and Krby (1994; 1996), whch was also used for the calbraton of the LWS model. The novelty of the present study s the couplng of the LWS methodology (Dmas and Falows 000) wth a numercal solver of a three-dmensonal vscous flow, n order to calculate the wavenduced currents n the surf zone, the generaton of whch s substantally affected by the presence of the bed shear stress. Specfcally, numercal smulatons of the three-dmensonal, free-surface flow, nduced by the propagaton of nonlnear breang waves, normal to the shorelne, over a constant slope, rgd bed, are presented. The LWS methodology s based on the decomposton of the flow varable scales (velocty, pressure and free-surface elevaton), to resolved (large) and subgrd (small) scales. One of the man objectves of ths wor, s to overcome some of the lmtatons of the aforementoned studes, manly arsng by the use of nvscd (Euler) and depth-averaged (Boussnesq) models. For example, nvscd models are not able to capture correctly the profle of the undertow current, snce they gnore vscous effects close to the bed, whle Boussnesq models are ncapable of calculatng flow varable profles over the depth. In the followng sectons, the flow equatons, the man features of the LWS methodology, the numercal method, the smulaton results and the man conclusons, are presented. FLOW EQUATIONS The three-dmensonal, ncompressble free-surface flow, for a flud of constant vscosty, s governed by the contnuty u x 0 (1) and the Naver-Stoes equatons u u p 1 u t x x x x u j j Red j j () where, j = 1,,3, t s tme, x 1, x are the horzontal coordnates, x 3 s the vertcal coordnate, postve n the drecton opposte to gravty, u 1, u and u 3 are the correspondng velocty components, p s the dynamc pressure and Re d s the Reynolds number. Equatons (1) and () are expressed n dmensonless form wth respect to the nflow depth d I, the gravty acceleraton g and the water densty π, therefore Re d = (gd I ) 1/ d I /ν, where v s the nematc water vscosty. For vscous flow, the nematc and the normal stress dynamc boundary condtons at the free surface are, respectvely, d u u u 3 1 dt t x1 x (3) p u 1 u u 1 u u Fr Red x3 x1 x3 x1 x x3 x where η s the free-surface elevaton, Fr s the Froude number, whch under the present dmensonless formulaton s equal to one, whle n Eq. (4) the atmospherc pressure s consdered equal to zero. The shear stress dynamc boundary condton at the free surface, s expressed for each of the horzontal coordnates x 1 and x 0 (4)

3 COASTAL ENGINEERING 01 3 u1 u 3 u1 u x1 x1 x3 x x x1 u u 1 3 u u x1 x3 x1 x1 x x3 x (5) u u 3 u1 u x x x3 x1 x x1 u u 3 u u x x3 x x1 x x3 x1 respectvely. In addton, the no-slp and non-penetraton boundary condtons at the bottom are d d u u 0, u u x1 x (6) (7) d d u u u x1 x (8) respectvely, where d s the bottom depth measured from the stll free-surface level. Gven that the free surface s tme-dependent, the Cartesan coordnates are transformed, n order for the computatonal doman to become tme-ndependent, accordng to the expressons x3 d s1 x1, s x and s3 d (9) where -1 s 3 1. In the transformed doman, s 3 = 1 corresponds to the free surface and s 3 = -1 to the bottom. By applcaton of Eq. (9), the contnuty and Naver-Stoes equatons (1) and () are transformed, respectvely, u u u s d s s 3 r (10) u 1 s3 u u u u u u3 r u t d t s3 s d s3 s3 1 u 4 r 1 u R V Red ss d s3 where = 1,, hereafter, and p p p R r, R s d s d s u r r u V r r d ss3 s d s3 s3 (11) (1) (13) r 1s3 1s3 d s s (14)

4 4 COASTAL ENGINEERING 01 LWS methodology As aforementoned, the large-wave smulaton (LWS) method s based on the applcaton of a volume flter to the velocty components and pressure, as n LES, and an exclusve surface flterng operaton for the free-surface elevaton. Therefore, each flow varable, f, s decomposed nto resolved, f, (large) and subgrd, f ', (small) scales, n a manner llustrated n Fg. 1 for the decomposton of the free-surface elevaton, η. wave propagaton drecton Fgure 1. Decomposton of free-surface elevaton for the case of a spllng breaer. The flterng operaton s appled on Eqs. (10) and (11), resultng nto the contnuty and Naver- Stoes equatons for the resolved scales of the flow, whch, respectvely, are u u u s d s s 3 r (15) where u 1 s3 u u u u u u3 r u t d t s3 s d s3 s3 4 r 1 R V T 1 u u Red ss d s3 3 1 s3 3 1 s3 d T s d s d s d s s (16) (17) ncludes the subgrd scale (SGS) terms,.e., the eddy SGS stresses and the wave SGS stresses, whch, respectvely, are u u u u (18) j j j u u u u u u p p 3 s s t t s s The eddy SGS stresses appear both n the LES and the LWS method, whle the wave SGS stresses appear exclusvely n the LWS method. The transformaton, gven by Eq. (9), and the flterng procedure are also appled successvely to the boundary condtons (3) - (8) at the free surface (s 3 = 1) and the bottom (s 3 = -1), resultng nto the transformed boundary condtons for the resolved scales. It must be noted that the flterng procedure taes nto to account some smplfyng assumptons, whch are specfed analytcally n Dmaopoulos and Dmas (011). In addton to ths, the SGS terms that result from the flterng of the vscous terms of Eq. (11), are consdered to be neglgble compared to the rest of the terms of the same equaton. In the present study, the eddy and wave SGS stresses, appearng n Eq. (16), are computed by use of Smagornsy eddy-vscosty models (Rogallo and Mon 1984). Specfcally, the model for the eddy SGS stresses s (19)

5 COASTAL ENGINEERING 01 5 j Sj C S Sj (0) where C = 0.1 s the model parameter, set accordng to the usual practce n LES method, Δ = (Δ 1 Δ Δ 3 ) 1/3 s the smallest resolved scale based on the grd sze, S j s the stran-rate tensor of resolved scales and 1/ S j j S j 1 u u j sj s S S s ts magntude. The model for the wave SGS stresses, based on the one presented n Dmas and Falows (000), s gven as j Sj C S Sj (1) () where C η s the model parameter and S j η s a modfed stran-rate tensor of resolved scales S S j 3 j s (3) where δ j s the Kronecer delta. NUMERICAL METHOD The flow smulatons are based on the numercal soluton of the transformed Naver-Stoes equatons, whch s acheved by use of a fractonal tme-step scheme for the temporal dscretzaton, and a hybrd scheme for the spatal dscretzaton. The hybrd scheme ncludes central fnte dfferences, on a unform grd wth sze Δs 1, for the dscretzaton along the streamwse drecton s 1, a pseudo-spectral approxmaton method wth Fourer modes along the spanwse drecton s, and a pseudo-spectral approxmaton method wth Chebyshev polynomals along the vertcal drecton s 3. The transformed Naver-Stoes equatons (16) can be wrtten n the form v t T 1 T A j j v V T Re d (4) where v u 3 r u s the transformed velocty feld, A ncludes all nonlnear terms, transformed gradent operator, p 0.5v j v j s the transformed pressure head and T j s the T j s the transformed Laplacan operator. The tme-splttng scheme for the temporal dscretzaton, n whch each tme-step conssts of three stages, acheves the calculaton of the velocty feld at the next tme step successvely, by addng the correspondng correctons of each of the three stages to the feld of the prevous tme-step. Moreover, the dynamc pressure feld s obtaned n the second stage of each tme step, whle the free-surface elevaton s calculated from the nematc boundary condton at the end of each tme step. At the frst stage of each tme-step, the nonlnear term, A, the SGS term, T, and the vscous term, V, of the transformed equatons of moton (4) are treated explctly by an Euler scheme. At the second stage, an mplct Euler scheme s used for the treatment of the pressure head term, T j, of Eq.(4), whch results nto a generalzed Posson s equaton for by satsfyng the transformed contnuty equaton as well. The transformed dynamc (normal stress) free-surface condton and nonpenetraton bottom condton are mposed at ths stage. At the thrd stage, the remanng vscous terms, T v, of Eq. 4, are treated by an Euler mplct scheme satsfyng the transformed dynamc (tangental j stress) free-surface and bottom condtons. Accordng to the hybrd scheme for the spatal dscretzaton, each flow varable f (veloctes and pressure) s approxmated by the followng expresson

6 6 COASTAL ENGINEERING 01 M/1 N ms f s1, s, s3, t f mn s1, texp Tn s3 mm / n0 L (5) where f mn s the Chebyshev-Fourer transformaton of f, M s the number of Fourer modes, L = M Δ, s the length of the computatonal doman n s, and T n s the Chebyshev polynomal of order n, and N s the hghest order of the Chebyshev polynomas. The transformatons between physcal and spectral space are performed by a Fast Fourer Transform algorthm (Press et al., 199). Applcaton of Eq. (5) for the dscretzaton of Eq. (4), leads to the formaton of a system of algebrac equatons, wth the general form A f b, for each of the transformed flow varables. The aforementoned system may be dvded nto M ndependent subsystems ( Am f m b m, one for each Fourer mode), whch can be solved n parallel, due to the decouplng of the Fourer modes. Each subsystem s solved at each tme-step, usng an teratve generalzed Gauss-Sedel method. The matrx of coeffcents Am s band dagonal, and s decomposed once at the begnnng of the computaton by usng the LU-decomposton method. In the present wor, the propagaton, transformaton and spllng breang of ncomng secondorder Stoes waves over a constant slope bed s smulated. As shown n the setch of the computatonal doman (Fg. ), a flat bed regon of length L I and constant depth d I, whch ensures the development of the ncomng waves, s followed by the nclned regon of the bed. Then, a flat regon of length L E and constant depth d E << d I (the formulaton allows the outflow depth d E to be small but nonzero) s consdered n order to smulate the swash zone of a coastal area, where the waves are completely damped. For ths reason, two overlappng zones are placed n the outflow regon: a wave absorpton zone of length L A L E, whch ensures that waves are not reflected by the outflow boundary (Dmas and Dmaopoulos, 009), and a velocty attenuaton (slowdown) zone of length L D,. For the numercal soluton of Eq. (4), a reduced value of Re d s used wthn the slowdown zone, whch corresponds to an ncreased value of the nematc vscosty. x 3 x x 1 L E d E d I L I Absorpton slowdone zones Inflow regon Inclned bed Fgure. Setch of the computatonal flow doman. RESULTS The valdaton of the nvscd verson of LWS methodology, coupled wth the Euler equatons, was performed by Dmaopoulos and Dmas (011), and one of the man results of ther wor, was the calbraton of the parameter C η, used n the model for the wave SGS stresses. The chosen value, C η = 0.4, resulted from the comparson of ther numercal results to correspondng expermental measurements, presented n Tng and Krby (1994; 1996), for the case of spllng wave breaers, propagatng normal to the shorelne over a beach of constant slope tanβ = 1/35. In the present study, the accuracy and effcency of the vscous verson of LWS methodology, coupled wth the Naver-Stoes equatons, s nvestgated, performng numercal smulaton for the normal to the shorelne propagaton, transformaton and spllng breang of ncomng second-order

7 COASTAL ENGINEERING 01 7 Stoes waves over a bed of constant slope tanβ = 1/35. Our numercal results are, also, compared to the expermental measurements conducted by Tng and Krby (1994), whle the suggested value of C η = 0.4 (Dmaopoulos and Dmas 011) s adopted. It must be noted that all of the numercal results presented n ths study are spanwse averaged. The expermental flow parameters (Tng and Krby 1994) for the case of spllng breang are summarzed n the followng: wave nflow depth, d I = 0.4 m, wave heght and perod, H I = 0.15 m and Τ = s, respectvely, whch correspond to wave heght and wavelength at deep water, H ο = 0.17 m and λ ο = 6.45 m, respectvely. In our case, the wave parameters at deep water are dentcal to those of Tng and Krby (1994), but a larger nflow depth d I = 0.7 m s consdered, snce the Stoes wave theory s utlzed for the ncomng waves, whch are of heght H I = m. The parameters of the ncomng waves are rendered dmensonless by d I, and g, and the resultng values are H I = 0.168, Τ = and λ = The Irrbaren number s ξ ο = tanβ(λ ο /Η ο ) 1/ = 0., whch corresponds to a spllng breaer of medum strength, whle a value of Re d = 50,000, s consdered. In the slowdown zone, a value of Re d dvded by 100, s utlzed. The total length of the computatonal doman s L = 60, the flat nflow regon has length L I = 15, whle the swash zone s of length, L E = 11.05, and depth, d E = The wave absorpton zone and the slowdown zone have lengths L Α = 11 and L D =, respectvely. The numercal parameters are: Δ 1 = 0.04, N = 18, M = 3, Δ = 0.0 and Δt = In Fg. 3, snapshots of the resolved free-surface elevaton,, at several tme nstants after 0 wave perods, are presented and compared to the expermental measurements of maxmum (wave crest) and mnmum (wave trough) values of the free surface elevaton (Tng and Krby 1994). The numercal model predcts accurately the breang depth d b = 0.8, whch corresponds to the poston x 1 = 40., but underestmates the breang heght, as ndcated by the devaton of about 9%, of the breang freesurface elevaton, 0.176, calculated by the LWS model, from the correspondng expermental one, b b Ths predcton s stll better than the one n Ln and Lu (1998), Bradford (000) and Chrstensen (006). At the outer surf zone (40 < x 1 < 44), the numercal model underestmates the wave heght dsspaton, as opposed to the nner surf zone (x 1 > 44), where the predcton of the model for the heght dsspaton s very good. Generally, the numercal results for the mnmum free-surface elevaton, are n a very good agreement wth the trough envelope of the expermental data. In the outer coastal zone, the numercal results for wave shoalng agree adequately wth the correspondng expermental data, however, LWS predcts a monotonc wave heght ncrease durng shoalng. In the surf zone, the numercal results predct very well the wave setup. Fgure 3. Snapshots of the resolved free-surface elevaton durng shoalng and n the surf zone, over bed of constant slope (tanβ = 1/35). Symbols correspond to the expermental free-surface envelope presented n Tng and Krby (1994). In Fg. 4, three typcal snapshots of the spanwse vortcty dstrbuton, ω, n the surf zone, are presented at the tme that ncpent wave breang occurs, and at tme nstants t = 0.5T and 0.8Τ after

8 8 COASTAL ENGINEERING 01 ncpent breang. Negatve vortcty s generated at the breang wavefront durng ncpent wave breang (wave crest at x 1 = 40.), whch corresponds to clocwse recrculaton of the flud. The strength of the surface roller ncreases, as the spllng breaer propagates towards the nner surf zone, untl t reaches ts full development (Fg. 4b), and subsequently, as ndcated n Fg. 4c, vortcty s advected and dffused n the roller wae. The vortcty dstrbuton close το the slopng bed, due to the bed shear stress, s also shown n Fg. 4, wth ts ampltude beng consderably larger (up to 10 tmes) than the one developng n the surface roller. Fgure 4. Vortcty feld n the surf zone, at three tme nstants durng wave perod, T. Snapshot (a) corresponds to ncpent breang, (b) at tme nstant t = 0.5T and (c) at t = 0.8T after ncpent breang. Dashed contour lnes correspond to negatve vortcty, whle sold lnes to postve one. Accordng to the LWS methodology, wave breang and dsspaton n the surf zone are coupled wth the generaton and combned acton of the eddy and wave SGS stresses. The dstrbuton of the wave stress 13, whch s the most sgnfcant SGS stress n terms of ts magntude, n the surf zone at two tme nstants (at the tme of ncpent breang and 0.5T later), s presented n Fg. 5. It s ndcated that the development of the surface roller (see Fg. 4) s connected to the contnuous ncrease of the magntude of wave stress 13 at the breang wavefront, whch taes place durng a tme nterval Δt =

9 COASTAL ENGINEERING Τ after the ncpent breang, correspondng to the regon 1 d/d b For tme greater than 0.5T after breang, ts strength attenuates gradually before t vanshes n the nner surf zone (x 1 > 45). Smlar behavor s also exhbted by the other SGS stresses on the x 1 -x 3 plane, whle stresses n the x drecton are an order of magntude smaller. As mentoned n the ntroducton, the development of the undertow current, resultng from wave breang, s constraned by the fact that the perod-averaged cross-shore water flux s zero. Fg. 6 presents a typcal perod-averaged velocty feld n the surf zone where t s ndcated that the numercal model s able to capture the occurrence of the undertow current. The perod-averaged velocty dstrbuton ndcates the presence of an onshore drected current, whch s due to the mechansms of the Euleran drft and the surface roller, n the upper layer of the water depth, and an offshore drected current,.e., the undertow current, near the bed, whch balances the onshore flux. Very close to the bed, a steady current of wea strength, the so-called wave boundary layer streamng, exsts offshore to the breang regon and part of the outer surf zone, and s drected towards the shorelne. Fgure 5. Snapshots of the wave SGS stress, 13, that correspond to (a) and (b) of Fg. 4. Note that the frst one ncludes two wave crests (at x 1 40 and 44.5), whle contour lnes are plotted at equal ntervals from 0 to wth a spacng of Fgure 6. Perod-averaged and spanwse-averaged velocty dstrbuton n the surf zone. Breang occurs at x In Fg. 7, LWS-predcted profles of the undertow current at four postons n the surf zone are compared to correspondng expermental measurements presented n fgure 5 of Tng and Krby (1994) for the case of spllng breaer. The perod-averaged horzontal velocty, U 1, s normalzed wth respect

10 10 COASTAL ENGINEERING 01 to the breang depth, d b. Overall, the LWS predcton s deemed adequate, snce the order of magntude as well as the gradent of the numercal profles, agree well wth the expermental ones. A sgnfcant devaton related to the depth of the mnmum of the velocty s observed between numercal and measured profles, as s ndcated n Fgs. (c) and (d), whch may be attrbuted to the large dfference between our Re d value and the one of the experments, whch s about 7 tmes larger. Fnally, n Fg. 8, the evoluton of the resolved free-surface elevaton and the bed shear stress, τ b, dstrbuton at several tme nstants durng a wave perod, are presented. The ampltude of the bed stress varaton s found to be substantally ncreased over the slopng bed, especally n the surf zone, becomng up to sx tmes larger (at the regon around x 1 = 4.5) than the correspondng ampltude n the flat regon (x 1 < 15). The magntude of τ b decreases n the nner surf zone, followng the wave heght attenuaton, whle the decrease of the wavelength wth decreasng water depth s ndcated by the boldlned snapshot. The poston of maxmum bed stress does not concde wth the breang poston, where the maxmum free surface elevaton s located, presentng a phase dfference of about 0.5T. Fgure 7. Normalzed perod-averaged horzontal velocty profles at four postons n the surf zone, compared to the correspondng expermental data (symbols) presented n fgure 5 (c)-(e) of Tng and Krby (1994).

11 COASTAL ENGINEERING Fgure 8. Snapshots of the resolved free-surface elevaton and the bed shear stress dstrbuton. The bold lnes correspond to the same tme nstant, whle bed slope starts at x 1 = 15. CONCLUSIONS A numercal model for the smulaton of wave propagaton and spllng breang over a constant slope bed s presented. The model s formed by the couplng of the large-wave smulaton (LWS) method wth a numercal solver of the three-dmensonal Naver-Stoes equatons. Accordng to the LWS methodology, the wave and eddy subgrd (SGS) stresses are modeled, by use of eddy-vscosty models, and then appled to the vscous flow solver, n order to capture wave breang and wave energy dsspaton n the surf zone. In general, the LWS model predctons related to the characterstcs of ncpent breang (breang depth - heght) are very well. The valdaton of our results s based on the comparson wth correspondng expermental measurements conducted by Tng and Krby (1994). The development of the surface roller n the breang wavefront s connected to the ncrease of the strength of the SGS stresses n the outer surf zone and ther successve decrease at shallower depths close to the shore. The perod-averaged velocty feld n the surf zone predcts very well the qualtatve characterstcs of the undertow current generated along the cross-shore drecton by the wave breang, whle the quanttatve comparson to the correspondng expermental data (Tng and Krby 1994) s good. Fnally, t s found that the magntude of the bed shear stress ncreases substantally n the surf zone becomng up to sx tmes larger than the correspondng one n the nflow flat regon. ACKNOWLEDGMENTS We than the Research Commttee of the Unversty of Patras, and partcularly the "K. Karatheodor Program" for fundng ths wor under contract D.168. REFERENCES Bradford, S.F Numercal smulaton of surf zone dynamcs, Journal of Waterway, Port, Coastal Ocean Engneerng, 16(1), Brgant, R., Musumec, R.E., Bellott, G., Brocchn, M., and E. Fot Boussnesq modelng of breang waves: descrpton of turbulence. Journal of Geophyscal Research, 109, C Chrstensen, E.D Large eddy smulaton of spllng and plungng breaers, Coastal Engneerng, 53(5 6), Chrstensen, E.D., and R. Degaard Large eddy smulaton of breang waves, Coastal Engneerng, 4(1), Dmaopoulos, A.S., and A.A. Dmas Large-wave smulaton of three-dmensonal, cross-shore and oblque, spllng breang on constant slope beach, Coastal Engneerng, 58(8),

12 1 COASTAL ENGINEERING 01 Dmas, A.A., and A.S. Dmaopoulos A surface-roller model for the numercal smulaton of spllng wave breang over constant slope beach, Journal of Waterway, Port, Coastal, and Ocean Engneerng, 135(5), Dmas, A.A., and L.T. Falows Large-wave smulaton (LWS) of free-surface flows developng wea spllng breang waves, Journal of Computatonal Physcs, 159(), Heu, P.D., Tanmoto, K., and V.T. Ca Numercal smulaton of breang waves usng a twophase flow model, Appled Mathematcs Modellng, 8(11), Ln, P., and P.L.-F. Lu A numercal study of breang waves n the surf zone, Journal of Flud Mechancs, 359, Madsen, P.A., Sørensen, O.R., and H.A. Schäffer Surf zone dynamcs smulated by a Boussnesq type model. Part I. Model descrpton and cross-shore moton of regular waves, Coastal Engneerng, 3(4), Press, W.H., Teuolsy, S.A., Vetterlng, W.T., and B.P. Flannery Numercal Recpes n Fortran 77, Cambrdge Unversty Press, 963 pp.. Rogallo, R.S., and P. Mon Numercal smulaton of turbulent flows, Annual Revew of Flud Mechancs, 16, Schäffer, H.A., Madsen, P.A., and R. Degaard A Boussnesq model for waves breang n shallow water, Coastal Engneerng, 0(3 4), Tng, F.C.K., and J.T. Krby Observaton of undertow and turbulence n a laboratory surf zone, Coastal Engneerng, 4(1-), Tng, F.C.K., and J.T. Krby Dynamcs of surf-zone turbulence n a spllng breaer, Coastal Engneerng, 7(3-4), Torres-Freyermuth, A., Losada, I.J., and J.L. Lara Modelng of surf zone processes on a natural beach usng Reynolds-Averaged Naver Stoes equatons, Journal of Geophyscal Research, 11, C Veeramony, J., and I.A. Svendsen The flow n surf-zone waves, Coastal Engneerng, 39(1 ), Zhao, Q., Armfeld, S., and K. Tanmoto Numercal smulaton of breang waves by a multscale turbulence model, Coastal Engneerng, 51(1),