A NUMERICAL INVESTIGATION OF SHEET FLOW UNDER NON-BREAKING AND BREAKING WAVES. Abstract

Transcription

1 Coatal Dynamc 2017 A NUMERICAL INVESTIGATION OF SHEET FLOW UNDER NON-BREAKING AND BREAKING WAVES Yeulwoo Km 1, Zhen Cheng 2, Tan-Jan Hu 1, Ryan S. Mera 1 and Jack A. Puleo Abtract An Euleran two-phae flow model for edment tranport, SedFoam (Cheng et al., 2017), fully coupled wth the urface wave model, wave2foam (Jacoben et al., 2011). The reultng olver, SedWaveFoam, can mulate edment tranport under urface wave propagaton wthout artfcal matchng between the urface wave feld and bottom boundary layer edment tranport. SedWaveFoam valdated wth meaured heet flow data drven by non-breakng monochromatc wave (Dohmen-Janen and Hane, 2002). Onhore tranport rate obtaned mlar to the meaured data. The onhore tranport rate about two tme larger than that obtaned n a hypothetcal ocllatng water tunnel. Ongong work ue the coupled model to mulate heet flow meaured under kewed/aymmetrc wave over a urf zone andbar (BARSED, Mera et al., 2017). Key word: edment tranport, heet flow, wave-drven edment tranport, numercal modelng, OpenFOAM. 1. Introducton Predcton of nearhore morphologcal evoluton of major mportance to enure the utanablty of coatal communte. However, predctng nearhore morphodynamc, partcularly durng major torm, ha been a challengng topc due to dffculte n acqurng detaled flow charactertc and edment dynamc n the thn bottom boundary layer. A comprehenve undertandng of the complex mechanm aocated wth wave-drven edment tranport reman ncomplete, partcularly regardng the relatve mportance among many nterconnected procee, uch a wave kewne, wave aymmetry and boundary layer treamng (e.g. Henderon et al., 2004; Ruenk et al., 2009; Yu et al., 2010). Conventonally, edment tranport rate drven by wave wa etmated baed on the concept of bed hear tre (Rbbernk, 1998). A number of tude have revealed that that under trong bed hear tre, mot of the tranport occur wthn the concentrated regon n heet flow (e.g. Horkawa et al., 1982; McLean et al., 2001; Rbbernk and Al-Salem, 1994). In addton, horzontal preure gradent recognzed a an mportant factor trggerng momentary bed falure under teep wave (Maden 1974; Sleath, 1999). Hence, parameterzaton of wave-nduced heet flow may requre the combned effect from the bed hear tre, horzontal preure gradent and boundary layer treamng (Anderon et al., n revew; Cheng et al., 2017; Foter et al., 2006; van der A et al., 2013). Many laboratory experment (e.g. Dohmen-Janen et al., 2001; O Donoghue and Wrght, 2004; Rbbernk, 1998; Rbbernk and Al-Salem, 1994) have been conducted to tudy heet flow n large ocllatng water tunnel (OWT, alo called U-tube). Ocllatory flow of the order of u = O(1) m/ and T = O(10) can be generated n uch OWT, and hence a dynamc calng effect can be mnmzed. Rbbernk and Al-Salem (1994) howed that the moble bed and edment affect turbulence ntenty n the ocllatory boundary layer. In addton, bed-load baed tranport formula obtaned from teady flow requre modfcaton for ocllatory flow due to change n effectve roughne and turbulence modulaton (Dohmen-Janen et al., 2001; Rbbernk, 1998). On the other hand, ntergranular nteracton ncreae flow retance n the heet flow layer, dependng on the gran ze (Dohmen-Janen et al., 2001). A a reult, typcal fxed-bed ngle-phae boundary layer model may not be approprate for 1 Coatal Engneerng Laboratory, Department of Cvl and Envronmental Engneerng, Unverty of Delaware, Newark, DE, USA. ykm@udel.edu 2 Appled Ocean Phyc and Engneerng, Wood Hole Oceanographc Inttuton (WHOI), Wood Hole, MA, USA. 1779

2 Coatal Dynamc 2017 modelng heet flow. The man lmtaton n OWT experment the lack of realtc urface wave effect. Dohmen-Janen and Hane (2002) reported detaled meaurement of heet flow under non-breakng urface wave. They found a gnfcant dfference n net edment tranport rate and heet layer thckne between OWT and wave flume experment under the ame maxmum non-dmenonal bed hear tre. A modfed Meyer- Peter and Müller type (Meyer-Peter and Müller, 1948) tranport model wa appled (Nelen, 2006; Nelen and Callaghan, 2003) to account for the ncreaed net tranport rate under the real wave. The reult ndcated that wave-nduced boundary layer treamng an mportant factor caung the ncreaed net edment tranport rate. An enhanced heet flow thckne wa alo oberved under trongly kewedaymmetrc wave on a andbar cret (Mera et al., n revew). Therefore, a numercal model whch can reolve the evoluton of free urface and the reultng wave feld, wave-nduced boundary layer, and wavedrven heet flow tranport needed for predctng nearhore edment tranport. Motvated by th need, edment tranport under perodc wave for flat bottom and barred beach condton are nvetgated by couplng an Euleran two-phae flow model, SedFoam (Cheng et al., 2017), wth a volume-of-flud (VOF) olver, InterFoam (Berberovc et al., 2009) facltated by a comprehenve wave generaton/aborpton toolbox, wave2foam (Jacoben et al., 2011). SedFoam reolve the full vertcal profle of edment tranport ung the Reynold-averaged Euleran two-phae flow equaton wth cloure of nter-granular tree and a k-ε turbulence model. The wave generaton toolbox, wave2foam, degned to work wth InterFoam, can generate/aborb varou type of urface wave. The fully coupled model, named SedWaveFoam, can be ued to mulate the effect of wave-nduced bottom hear tre, preure gradent, boundary layer treamng, and wave-breakng turbulence on the full vertcal profle of edment tranport wthout conventonal bedload/upended load aumpton. A a frt tep, the model valdated for heet flow under non-breakng monochromatc wave (Dohmen-Janen and Hane, 2002). The coupled model then appled to the andbar SEDment tranport experment (BARSED; Anderon et al., n revew; Mera et al., n revew) data to tudy the effect of teep wave on heet flow procee. 2. Numercal Model The numercal model baed on couplng/mergng everal extng olver n the open-ource CFD toolbox, OpenFOAM. For the ar and water phae, the flow olved by InterFoam (Berberovc et al., 2009), whch track the nterface of two ncompreble flud. Partcularly, the water-ar nterface are computed by trackng the volumetrc concentraton of the water phae, mlar to the well-known volumeof-flud method (Hrt and Nchol, 1981). Jacoben et al. (2011) enhanced th olver wth a comprehenve urface wave generaton/aborpton cheme and the reultng olver, called wave2foam, ha been appled to tudy urf zone procee (e.g. Jacoben et al., 2014). In the pat decade, the Euleran two-phae flow formulaton have been wdely ued to model edment tranport n heet flow condton (e.g. Dong and Zhang, 2002; Hu et al. 2004). Recently, A mult-dmenonal two-phae flow model for edment tranport, called SedFoam (Cheng et al., 2017), wa developed ung OpenFOAM. The SedFoam model wa valdated wth meaured data for everal ocllatory heet flow data et reported by O Donoghue and Wrght (2004) (ee alo Secton 3). Combnng thee two olver to a new olver for wave-nduced edment tranport applcaton become le dffcult under the framework of OpenFOAM. The governng equaton of th new two-phae flow olver for wave-nduced edment tranport are brefly preented. The ma conervaton equaton for the mxture of carrer flow (water-ar) phae and edment phae are derved by aumng no ma tranfer between the phae: u x t m m t u x m 0 0 (1), (2) 1780

3 Coatal Dynamc 2017 m n whch and are the volumetrc concentraton of the edment and water-ar mxture phae, m water ar m m repectvely, wth and 1. The varable u the edment velocty and u the carrer flud phae velocty whch can be wrtten a u u u u water water ar ar ar r m water u water ar m, (3) r r water ar where u repreent the relatve velocty between ar and water, defned a u u u. The varable r u obtaned by teraton ung the nterface compreon method (Klotermann et al., 2012) to mnmze the dffuon at the free urface whle retanng computatonal effcency. Equaton (2) can be rewrtten a The momentum equaton are wrtten a water ar r water water water u water ar u 0. (4) t x x uu m m m m m m m m m u j p m m j m g 3 M t x j x x j uu m water water ar ar where u j p p j m g 3 M t x j x x x j (5), (6) m, the edment denty, p the flud preure, and g the gravtatonal acceleraton. The flud tre nclude the um of vcou and Reynold tree, calculated wth a two-equaton k- turbulence model derved for two-phae flow, n whch edment can attenuate flow turbulence va a range of mechanm dependng on the partcle nerta (Cheng et al., 2017; Yu et al., 2010). The nter-phae momentum tranfer between the carrer flow and edment phae follow Newton 3 rd m m law, M M. Here, drag and buoyancy force are condered. The partcle normal tre, p, and hear tre, j, are modeled wth knetc theory of granular flow for moderate concentraton. For hgh edment concentraton tre aocated wth endurng contact and frctonal effect are ncorporated. More detal on model formulaton and numercal mplementaton are gven elewhere (Cheng et al., 2017; Jacoben et al., 2011; Klotermann et al., 2012). 3. Model Reult 3.1. Capablty of ndvdual model SedFoam wa valdated (Fgure 1) wth the meaured concentraton profle of medum and (d = 0.28 mm) obtaned under nuodal wave moton n an OWT wth wave perod T = 7.5 and maxmum velocty U max = 1.26 m/ (cae 7515 n O Donoghue and Wrght, 2004). The concentraton profle at two dfferent ntant (dentfed by ymbol n Fgure 1a) were elected for comparon. Supended edment concentraton notably dfferent between the wave cret (Fgure 1b) and flow reveral (Fgure 1c). Generally, good agreement wth meaured concentraton and flow velocty (not hown) were obtaned. More comprehenve model valdaton for other cae reported by O Donoghue and Wrght (2004), ncludng velocty profle and eroon depth are found n Cheng et al. (2017). A comprehenve undertandng of edment tranport n nearhore envronment cannot be acheved due to the lack of realtc nfluence from the free urface, although SedFoam capable of predctng edment tranport 1781

4 Coatal Dynamc 2017 under ocllatory flow n OWT. A a reult, the capablty of mulatng the evoluton of free urface and the reultng wave feld were ncluded n the model. Fgure 1. Model-data comparon of (a) tme ere of free-tream velocty at the wave cret and flow reveral: (b) t = 1.90 ; (c) t = The correpondng comparon of edment concentraton profle between the model reult (old lne) and meaured data (ymbol) at thee two ntant are hown n panel (b) and (c). The BARSED experment wa carred out n the large-cale wave flume at O. H. Hndale Wave Reearch Laboratory, Oregon State Unverty, USA. Wave2Foam wa teted wth the wave condton of Tral 14 (S1T7H60) from the BARSED experment (Fgure 2). Th tral had a wave perod of 7.0 and wave heght of 0.66 m at the toe of the profle, and wa choen nce t had the larget heet layer thckne (Mera et al., n revew). The model doman wa compred of 1321, 43, 73 grd pont n the treamwe (x), panwe (y), and vertcal (z) drecton, repectvely. The grd ze n the treamwe and vertcal drecton wa non-unform, wth a mnmum ze of 3 cm (n x) and 1 cm (n z). On the other hand, the grd ze n the panwe drecton wa unformly 4 cm. Standard Smagornky cloure wa ued for ubgrd vcoty. A total of 4.1 mllon grd pont were ued for th prelmnary mulaton. Twn-wre capactance wave gauge were deployed at 18 cro-hore locaton n BARSED (Fgure 2a). The wave breakng occurred rght on the andbar cret n the model, content wth the obervaton (Mera et al., 2017). The temporal evoluton of free-urface at the eaward-edge of the andbar alo hown n Fgure 2b. After about fve wave perod, the free urface elevaton reache qua-tatonary tate. Overall, the model able to predct the free-urface elevaton reaonably well. Baed on the wave hape, a teep wave velocty expected to drve bottom edment tranport although the breakng-wave-nduced turbulence doe not approach the bar cret for th partcular tral SedWaveFoam reult The heet flow experment under monochromatc nonbreakng wave reported by Dohmen-Janen and Hane (2002) wa mulated to tet the newly developed olver SedWaveFoam. The detaled phycal model etup and reult were explaned n Dohmen-Janen and Hane (2002), thu only a bref overvew of the etup dcued here. The heet flow experment wa carred out n the large wave flume (GWK), n Hannover, Germany. The water depth at the meaurement ecton (flat and bed) wa 3.5 m, elevated by 0.75 m relatve to the bottom of the wave paddle (h = 4.25 m). The and bed wa contructed of well-orted quartz wth a medan gran dameter of 0.24 mm. A multple tranducer array (MTA) wa deployed to meaure the bed level change. A conductvty concentraton meter (CCM) ytem wa ntalled under the and bed to obtan edment concentraton n the concentrated regon. Two et of acoutc backcatter enor (ABS) were ued to meaure the upended edment concentraton. The velocty of the carrer flud wa meaured ung an acoutc doppler velocmeter (ADV) m above the and bed. 1782

5 Coatal Dynamc 2017 Fgure 2. Meaured (red ymbol) and modeled (blue old lne) (a) profle of free-urface elevaton at t = 86.5 and (b) tme ere of free-urface elevaton at the eaward-edge of the andbar cret (x = 43.2 m). Fgure 3. The numercal flume wth (a) meh (down-ampled) and (b) whte urface (ar phae), blue urface (water phae), and red urface at x = 0 m (edment phae) baed on the experment etup (Dohmen-Janen and Hane, 2002). For vblty, vertcal cale tretched three tme Model etup The numercal model doman a 2D flume that 176 m long (~5 wave length, 5L) and 6.5 m deep wth a water depth of 3.5 m (Fgure 3). The edment pt of 2 m long and 0.1 m deep wa located 2L away from the nlet, conted of the and (d = 0.24 mm) wth the maxmum volumetrc concentraton of φ max = 0.61 n the edment bed. It hould be noted that the model doman wa mplfed for computatonal effcency 1783

6 Coatal Dynamc 2017 wth only the flat porton of the flume n the phycal experment mulated. The model doman wa compred of 8,994 and 1,116 ~ 1,216 grd pont n the treamwe (x) and vertcal (z) drecton, repectvely. The grd ze wa non-unform, wth a mnmum wdth of 1 cm and heght of 1 mm near the pt whle the grd ze 2 cm n wdth and 1 cm n heght away from the pt. Hence, a total number of 10.1 mllon computatonal grd pont wa ued for the mulaton (Fgure 3a). Except for the top atmopherc boundary (open boundary condton) and de wall (empty condton n OpenFOAM), all other boundare (bottom and end) were pecfed a wall wth no-flux boundare for calar quantte and velocty component normal to the wall. The wall-parallel velocty at the boundare wa et to 0 (no-lp). At each end of the wave flume, a relaxaton zone of 1L wa adopted for wave generaton/aborpton and to uppre reflected wave. In the preent etup, the wave heght of reflected wave wa le than 10 % of ncdent wave heght. Non-breakng monochromatc wave (H = 1.6 m, T = 6.5 ) were ent nto the doman ung 10 th order tream functon (Fgure 3b). Fgure 4. (a) A naphot at t = 45 for edment concentraton over the entre and pt. (b) tme ere of volumetrc concentraton of edment phae at the center (x = 0 m) of the and pt Model-data comparon The model can produce the expected temporal and patal evoluton of edment phae under the urface wave (Fgure 4). For example, Fgure 4a how the x-z plane naphot of the volumetrc concentraton of edment at the elected ntant (t = 45 ). Notceable cour oberved at the left edge of the pt (Fgure 4a), caued by the lack of edment flux uptream from the pt. Smlarly, notable accumulaton of edment oberved at the downtream edge of the pt. At the center of the pt (x = 0 m), a regon of flatbed occur. The tme ere of edment phae at the center of the pt preented n Fgure 4b. After the paage of a couple of wave, the edment concentraton evoluton ha approxmately reached a qua-teady tate, allowng the 5 th wave (t = 37.7 ~ 44.2 ) to be further analyzed for the model-data comparon. The temporal evoluton of free tream velocty for the 5 th wave at z = m above the bed n the center of the pt agree reaonably well wth meaured data (Fgure 5). Some dcrepance were found near the wave trough due to lmted nformaton of ncdent wave. The wave condton wa only gven at the wave paddle, hence everal dfferent type of ncdent wave (e.g. cnodal wave) have been teted to better match the 1784

7 Coatal Dynamc 2017 forcng condton. Fgure 5. Tme ere of free tream velocty (meaured data, ymbol; model reult, lne) at z = m n the center of the and pt (x = 0 m). The model-data comparon of wave-averaged (for the 5 th wave) edment concentraton profle preented n Fgure 6. It hould be reterated that the maxmum concentraton reported n the phycal experment 0.67 (Dohmen-Janen and Hane, 2002), whch wa lghtly larger than the typcal packng lmt for unform, phercal partcle probably owng to mxed gran ze n the edment. The preent model cannot account for the effect of mxed gran, and hence the modeled concentraton are mply multpled by a factor of 0.67/0.61 for the remander of model valdaton. The modeled edment concentraton agree well wth the meaured data (meaured by CCM, ; Fgure 6) n the moderate to hghly concentrated regon (φ > 10 2 ). However, the predcted concentraton n the dlute regon le atfactory (thoe meaured by ABS, +; Fgure 6). A demontrated later, the model reult ugget that the edment flux wthn the bottom 10 mm (φ > 10 3 ) account for 99.8% of the total tranport rate. Fgure 6. Wave-averaged edment concentraton profle between the meaured data (CCM, ; ABS, +) and model reult (old lne). Fgure 7 how the ntantaneou edment concentraton profle n the heet flow layer n the 5 th wave cycle. Under the wave cret (Fgure 7b) and trough (Fgure 7d), the ncreae of heet layer thckne apparent compared to thoe durng the flow reveral (Fgure 7a and 7c). The model reult ndcate that the drvng force of heet flow cloely related to near-bed velocty (ee Fgure 5), content wth the obervaton reported by Dohmen-Janen and Hane (2002). The agreement between the model reult and meaured data are good. Clearly, hgher vertcal numercal reoluton needed n the future run. 1785

8 Coatal Dynamc 2017 Fgure 7. Intantaneou edment concentraton profle of meaured data (ymbol) and model reult (lne) at (a) t = 37.7 (flow reveral), (b) t = 39 (cret), (c) t = 40.3 (flow reveral), and (d) t = 42.1 (trough). The aymmetry n the heet layer thckne more clearly oberved n the tme ere of edment concentraton at the dfferent vertcal elevaton (Fgure 8). The numercal reoluton n the vertcal drecton 1 mm, hence the modeled reult were lnearly nterpolated for comparon wth meaured data. Neglgble eroon wa oberved at z = 4 mm n both meaured data and model reult, ndcatng no and tranport below that level (tll bed level). In the pck-up layer (from z = 4 to 0 mm), the model reult under-predct the edment eroon at z = 1.4 mm. Overall, the numercal model (Fgure 8b) able to produce meaured temporal evoluton of edment concentraton n the heet layer reaonably well. Fgure 8. Comparon between (a) meaured and (b) modeled edment concentraton tme ere at dfferent elevaton n the heet flow layer. 1786

9 Coatal Dynamc 2017 Tme ere of net tranport rate (q = u φ dz) calculated by ntegratng the edment fluxe over the water column at a gven ntant (Fgure 9). The onhore-drected edment tranport evdent under the wave cret, whle offhore tranport wth maller magntude occur durng the wave trough. The reult are content wth the more gnfcant edment upenon under the wave cret (Fgure 8). Notably reduced onhore tranport durng the wave cret paage wa oberved for the 1DV mulaton ung SedFoam (Fgure 5; mlar to OWT etup). Th fndng ugget that the urface wave effect, probably due to boundary layer treamng, play an mportant role n drvng onhore edment tranport. The waveaveraged net tranport rate, Q = 1 T u φ dzdt, over the 5 th wave cycle obtaned from SedWaveFoam 54 mm 2, whch mlar to meaured data of 42.9 mm 2 ung bed level change (MTA) and 70.7 mm 2 ung edment flux (CCM and ABS). But thee value are gnfcantly larger than the 1DV SedFoam reult of 35 mm 2. Fgure 9. Tme ere of net tranport rate n 5 th wave cycle obtaned from SedWaveFoam. 4. Concluon and Future Work A mult-phae Euleran model for edment tranport under urface wave, SedWaveFoam, developed by mergng SedFoam wth InterFoam and wave2foam. Each model capable n reolvng heet flow edment tranport n OWT (O Donoghue and Wrght, 2004) and wave propagaton over a barred beach (Mera et al., n revew). The coupled model SedWaveFoam follow the mathematcal formulaton for the water-edment nteracton (Cheng et al., 2017) and for the nterface trackng of two mpreble flud (Jacobon et al., 2011; Klotermann et al., 2012). Newly developed SedWaveFoam frt valdated wth laboratory data of Dohmen-Janen and Hane (2002) for and tranport n heet flow condton under monochromatc non-breakng wave. The concurrence between modeled and meaured data how the potental of the model to nvetgate a wde range of edment tranport procee under realtc urface wave. The numercal model predct the heet flow charactertc n the concentrated regon reaonably well. However, the agreement n the dlute regon le atfactory. Through numercal experment, t found that urface wave effect drve more onhore tranport compared to that obtaned n the OWT. Hgher reoluton run and more detaled analy warranted to quantfy the mechanm reponble for the enhanced onhore tranport under urface wave. SedWaveFoam ha been ued to mulate the cae reported n the BARSED experment (Anderon et al., n revew; Mera et al., n revew). The wave teepne durng BARSED wa larger than thoe reported n Dohmen-Janen and Hane (2002) and mulaton reult can be further ued to quantfy boundary layer treamng and large horzontal preure gradent aocated wth the teep wave. Acknowledgement 1787

10 Coatal Dynamc 2017 Th tudy upported by NSF (OCE ; OCE ) of USA. Numercal mulaton preented n th tudy were carred out ung the Mll cluter at Unverty of Delaware, and the SuperMc cluter at Louana State Unverty va XSEDE (TG-OCE100015). Reference Anderon, D., Cox, D.T., Mera, R.S., Puleo, J.A., and Hu, T.-J,, n revew. Obervaton of wave-nduced pore preure gradent and bed level repone on a urf zone andbar. J. Geophy. Re. Ocean. Berberovc, E., van Hnberg, N. P., Jakrlc, S., Roman, I. V., and Tropea, C., Drop mpact onto a lqud layer of fnte thckne: Dynamc of the cavty evoluton, Phy. Rev., E79, Cheng, Z., Hu, T.-J. and Calanton, J., SedFoam: A mult-dmenonal Euleran two-phae model for edment tranport and t applcaton to momentary bed falure, Coatal Eng., 119, Dohmen-Janen, and Hane, D. M., Sheet flow dynamc under monochromatc nonbreakng wave, J. Geophy. Re., 107(C10), 3,149. Dohmen-Janen, C. M., Haan, W. N., and Rbbernk, J. S., Moble effect n ocllatory heet flow, J. Geophy. Re., 106(C11), 27,103-27,115. Dong, P., and Zhang, K., Intene near-bed edment moton n wave and current, Coat. Eng., 45(2), Foter, D. L., Bowen, A. J., Holman, R. A., and Natoo, P., Feld evdence of preure gradent nduced ncpent moton, J. Geophy. Re., 111, C Henderon, S. M., Allen, J. S., and Newberger, P. A., Nearhore andbar mgraton predcted by an eddydffuve boundary layer model, J. Geophy. Re., 109, C Hrt, C. W., and Nchol B. D., Volume of flud (VOF) method for the dynamc of free boundare, J. Comput. Phy., 39, Horkawa, K., Watanabe, A., and Kator, S., A laboratory tudy on upended edment due to wave acton. Proc. 18th Int. Conf. Coat. Eng., ASCE, Cape Town. Hu, T.-J., Jenkn, J. T., Lu, P. L.-F., On two-phae edment tranport: heet flow of mave partcle, Proc. R. Soc. London Ser. A Math. Phy. Eng. Sc,.460(2048), Jacoben, N. G., Fuhrman D. R., and Fredøe J., A wave generaton toolbox for the open-ource CFD lbrary: OpenFoamR, Int. J. Numer. Method Flud, 70(9), Jacoben, N. G., Fredøe, J., and Jenen, J. H., Formaton and development of a breaker bar under regular wave. Part 1: Model decrpton and hydrodynamc, Coatal Eng., 88, Klotermann, J., Schaake, K., and Schwarze, R., Numercal mulaton of a ngle rng bubble by VOF wth urface compreon, Int. J. Numer. Method Flud, 71(8), Maden, O. S., Stablty of a and bed under breakng wave, Proc. 14th Int. Conf. Coat. Eng., ASCE, Reton, Va. McLean, S. R., Rbbernk, J. S., Dohmen-Janen, C. M., and Haan, W. N., Sand tranport n ocllatory heet flow wth mean current, J. Waterw. Port Coat. Ocean Eng., ASCE, 127(3), Meyer-Peter, E., and Müller, R., Formula for bed-load tranport, Proc. 2nd Congre of the Int. A. Hydraulc Structure Re., Stockholm Mera, R., Puleo, J. A., Anderon, D., Cox, D. T., and Hu, T. -J., Large-cale expermental obervaton of wave-nduced edment tranport over a urf zone andbar, Proc. 18th Int. Conf. Coat. Dyn., Helngør, Denmark. Mera, R., Puleo, J. A., Anderon, D., Cox, D. T., and Hu, T. -J., n revew. Large-cale expermental obervaton of heet flow on a andbar under kewed-aymmetrc wave, J. Geophy. Re: Ocean. Nelen, P., Sheet flow edment tranport under wave wth acceleraton kewne and boundary layer treamng, Coatal Eng., 53, Nelen, P., and Callaghan, D. P., Shear tre and edment tranport calculaton for heet flow under wave, Coatal Eng., 47, O Donoghue, T., Wrght, S., Concentraton n ocllatory heet flow for well orted and graded and, Coat. Eng., 50(3), Rbbernk, J. S., Bed-load tranport for teady flow and unteady ocllatory flow, Coatal Eng., 34, Rbbernk, J. S., and Al-Salem, A. A., Sedment tranport n ocllatory boundary layer n cae of rppled bed and heet-flow, J. Geophy. Re., 99, 12,707-12,727. Ruenk, B. G., van den Berg, T. J. J., and van Rjn, L. C., Modelng edment tranport beneath kewed aymmetrc wave above a plane bed, J. Geophy. Re., 114, C Sleath, J. F. A., Condton for plug formaton n ocllatory flow, Cont. Shelf Re., 19(13), 1,643-1,664. Van der A., D. A., Rbbernk, J. S., van der Werf, J. J., O Donoghue, T., Bujrogge, R. H., and Kranenburg, W. M., Practcal and tranport formula for non-breakng wave and current, Coatal Eng., 76, Yu, X., Hu, T.-J., and Hane, D. M., Sedment tranport under wave group: Relatve mportance between nonlnear wavehape and nonlnear boundary layer treamng, J. Geophy. Re., 115, C