TOWARD IMPROVING PREDICTION OF SEDIMENT TRANSPORT OVER WAVE-INDUCED RIPPLES

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1 Ai R. and Tanaka H. (211) Tohoku Jounal of Natual Diate Science, Vol. 47, pp TOWARD IMPROVING PREDICTION OF SEDIMENT TRANSPORT OVER WAVE-INDUCED RIPPLES Rafik ABSI 1, Hitohi TANAKA 2 ABSTRACT Sediment tanpot ove wave-induced ipple i a vey complex phenomenon whee availale model fail to povide accuate pediction. Fo coatal engineeing application, the 1-DV advection-diffuion equation could e ued with an additional paamete α elated to the poce of votex hedding aove ipple (Ai, 21). The aim of thi tudy i to povide imple pactical analytical tool. An analytical eddy vicoity pofile wa validated y DNS data of tuulent channel flow (Ai et al., 211). In thi tudy, we will how that: (1) the peiod-aveaged eddy vicoity in ocillatoy ounday laye could e decied y thi imple analytical fomulation; (2) The hape of the vetical pofile i validated y peiod-aveaged eddy vicoity of aeline (BSL) k-ω model (Suntoyo and Tanaka, 29) fo inuoidal and aymmetic wave; (3) The vetical eddy vicoity pofile depend on the wave non-lineaity paamete and equie theefoe a pecific caliation. 1. INTRODUCTION Coatal one ae of high vulneaility to natual haad/diate. Wave, cuent and tide make coatal one aea changing due to eoion and depoition of ediment. Thu, an undetanding of ediment tanpot poce in coatal one i of cucial impotance fo accuate pediction of coat line evolution and ea-ed change. Howeve, the modeling of coatal ediment tanpot need a compomie etween two type of model: detailed mathematical model and engineeing appoache. Thi compomie i impoed y on the one hand the accuacy of pediction and on the othe hand the uaility in pactical application (Ai 211). In coatal engineeing pactical accuate engineeing model which take into account the moe impotant involved phyic, ae needed. In the engineeing appoach, the net (aveaged ove the wave peiod) total ediment tanpot i otained a the um of the net ed load and net upended load tanpot ate (Fedoe and Deigaad 1992). Fo upended load, the net and tanpot i defined a the um of the net cuent-elated and the net waveelated tanpot component. The wave-elated upended tanpot component equie computation of the time-aveaged upended ediment concentation (SSC) pofile and it integation in the vetical diection (van Rijn 27). Computation of SSC need the ediment diffuivity which i elated to the eddy vicoity y the paamete β (i.e., the invee of the tuulent Schmidt nume). t Fo modeate wave condition and/o deep wate, wave ipple can e fomed on the ea ottom. If the ipple ae elatively teep (η /λ.12, whee η i the ipple height and λ i the ipple wavelength), the mixing cloe to the ed i dominated y coheent, peiodic votex tuctue. Aove ippled ed, the mixing in the nea ed laye i dominated y the mechanim of votex hedding which entain ediment. The aim of ou tudy i to impove the pediction of SSC ove ipple y uing imple analytical tool which take into account the moe impotant involved phyic, fo pactical ue in coatal engineeing. 2. TIME-AVERAGED CONCENTRATIONS OVER WAVE-INDUCED RIPPLES Sediment diffuivity ε decie the dioganied diffuive poce. The poce of votex fomation and hedding at flow eveal aove ipple i a elatively coheent phenomenon. The aociated ective ediment entainment poce may alo e chaacteied a coheent, intead of a pue dioganied diffuive poce epeented in the claical gadient diffuion model (Thone et al. 22). Nielen (1992) indicated that oth ective and diffuive mechanim ae involved in the entainment pocee. In the comined ection-diffuion fomulation, the teady tate advection-diffuion equation i given y dc w c F (1) d 1 EBI, Int. Polytech. St-Loui, Cegy Univeity, 32 Bd du Pot, 9594 Cegy-Pontoie Fance.ai@ei-edu.com 2 Depatment Civil Eng., Tohoku Univeity, Aoa, Sendai , Japan tanaka@tunami2.civil.tohoku.ac.jp

2 The epective tem in (1) epeent downwad ettling, upwad diffuion (given y gadient diffuion F dc / d ) and upwad ection F. The upwad ection tem F wa given y Thone et diff al. (22) a F w c F ( ), whee F() i a function deciing the poaility of a paticle eaching height aove the ed (Nielen 1992). Thone et al. (29) wote F w c whee w w w and c ae w w peiodic component epectively of concentation and vetical velocity and the ovea denote time aveaging. It i poile to wite (1) in the fom of a diffuion equation. The time-aveaged (ove the wave peiod) advection-diffuion equation i given theefoe y (Ai 21) dc w c (2) d whee and i a paamete elated to ective ediment entainment poce aociated to the poce of votex hedding aove ipple 1/ 1 F w c F /. With the upwad ection w c F ( ), ecome equal to 1/ 1 c / c F ( ), while with F w c w w (Thone et al., 29) 1 / 1 w c w c. The condition of Sheng and Hay (1995) w c / w c. 2 / w w w w how theefoe that when the ective tanfe i vey mall (aove low teepne ipple), 1 and theefoe (Ai, 211). Fom equation (1) and (2), it i poile to wite and theefoe 1 F / F diff coheent votex hedding (elated to 1 F F diff => 1, while F F => 1 and theefoe diff. Thi equation how that depend on the elative impotance of F ) and andom tuulence (elated to F diff ). When F F diff. Ai (21) popoed the following equation d 2 ln c w d (4) 2 2 d d (3) Eq. (4) povide a link etween upwad concavity/exity of concentation pofile (in emi-log plot) and inceaing/deceaing of deceaing. Inceaing allow an upwad ex concentation pofile. allow upwad concave concentation pofile, while In ode to allow adequate pediction of upended ediment tanpot, it i impotant to undetand inteaction etween ediment paticle and tuulence of fluid flow. The tuulent diffuion of upended ediment i given y t (5) whee = invee of the tuulent Schmidt nume, decie the diffeence etween diffuivity of momentum (diffuion of a fluid paticle ) and diffuivity of ediment paticle. It hould depend on the paticle Stoke nume (Ai et al. 211). Howeve, fo implicity and in ode to allow analytical analyi, we uggeted a imple equation exp C / ; whee = the value of cloe to the ed and C

3 = coefficient (Ai 21). Thi (y) pofile inceae with fo C > and deceae fo C <. We ued an analytical eddy vicoity given y t C u e (6) whee u = the fiction velocity (m/), δ = the ounday laye thickne (m), κ = the Kaman contant (=.41) and C a paamete =1.12 (Hu and Jan 1998, Ai 2, Ai 21). Uing the -function and eddy vicoity (6), the ediment diffuivity i given theefoe y B A e (7) whee A u (m/) and B C C y 1 D exp / h / (m). Ai (21) uggeted an empiical function fo given ; whee D and h ae two paamete. Tet cae: Fine and coae ediment ove ippled ed in the ame flow (McFetidge and Nielen, 1985) Maximum value of the fee team velocity, wave peiod, mean depth of the flow, oital amplitude o nea-ed flow emi-excuion, mean ipple height, mean ipple wavelength, equivalent oughne k 25 /, fiction facto f.237 k / a. 52 (Souly, 1997), mean magnitude of the w m.5 fiction velocity in the wave cycle u.763 f / 2 U (Davie, 1986) and w u / ae given epectively in tale 1. Tale 1. Flow paamete U ( cm / ) T () h (m ) a m (cm ) (cm ) (cm ) (cm ) f u ( cm / ) (cm ) k w Figue 1: Time-aveaged concentation pofile ove wave-induced ipple. Symol: meauement (McFetidge and Nielen, 1985), ( ) fine; ( ) coae; Cuve: olution of Eq. (2) (Ai 21).

4 Figue 1 how time-aveaged concentation of fine ( ) and coae ( ) ediment upended y wave ove ipple. The peent method allow a good deciption of concentation pofile fo oth fine (dahed line) and coae and (olid line). The paamete wee choen to give a good fit. Fo an eddy vicoity given y:.258 exp( 1.12 /.15 ), the paamete ae fo fine ediment:.97 1, , 1. 1 t C and theefoe.25 exp( /.22 ) and fo coae ediment: C and theefoe.17 exp( /.75 ) with 1 43 exp /.2. The value C. 438 fo fine ediment could e elated to an inaccuate etimation of the ounday laye thickne. Fo coae ediment, the pofile of (Ai 21, olid line in figue 7) how the effect of paamete which indicate that votex hedding occu at <.15m. Howeve in ode to allow pactical ue fo pedictive pupoe, the method need caliation. Befoe caliating paamete of α and β, we need to ae and validate the eddy vicoity pofile given y Eq. (6). 3. ANALYTICAL EDDY VISCOSITY FORMULATION Eq. (6) wa ued a an empiical equation. Howeve in ode to allow moe geneal ue, we need a deepe theoetical analyi. Steady plan channel flow: analyi y DNS data In the equiliium egion +>5, the tuulent kinetic enegy (TKE) i given y k u exp( C / ) k (Neu and Nakagawa 1993). Since in the inne egion the teamwie velocity pofile i given y the loglaw, it i poile to wite a mixing length a l exp( C / ) and theefoe Eq. (6) fo eddy m vicoity. Figue 2 how TKE pofile given y two analytical olution (Neu and Nakagawa 1993, Ai 28) and eddy vicoity pofile (white dahed line) given y Eq. (6). In figue 2, vaiale with the upecipt of + ae thoe nondimenionalied y the fiction velocity and the kinematic vicoity a u / ; k k / u ; / t t. Compaion with DNS data (data of Iwamoto 22, Iwamoto et al. 22, Hoya and Jiméne 26) how that Eq. (6) povide accuate deciption of DNS in the equiliium egion (Ai et al. 211). k Figue 2: Tuulent kinetic enegy (left) and eddy vicoity (ight) pofile in plan channel flow fo diffeent fiction Reynold nume. Symol: DNS data; Line: analytical (Ai, 28; Ai et al., 211). Ocillatoy flow: analyi y a two-equation model Eq. (6) fo eddy vicoity wa validated fo the cae of teady plane channel flow. Howeve fo ue in wave ounday laye, we need to ae thi equation fo the cae of ocillatoy flow. Eq. (6) i theefoe analyed y the aeline (BSL) k-ω model. Thi model allow accuate pediction of velocity pofile in ocillatoy ounday laye (Suntoyo and Tanaka 29). Figue (3.a) peent tempoal and patial vaiation of dimenionle eddy vicoity fo a inuoidal wave. Figue (3.) how compaion etween peiod-aveaged eddy vicoity otained y BSL k-ω model

(ymol) and analytical pofile of Eq. (6) (dahed line). Even if the eddy vicoity i highly time-dependent (figue 3.a), the peiod-aveaged dimenionle eddy vicoity (Figue 3.

5 (ymol) and analytical pofile of Eq. (6) (dahed line). Even if the eddy vicoity i highly time-dependent (figue 3.a), the peiod-aveaged dimenionle eddy vicoity (Figue 3.) ha a hape which i well decied y the analytical pofile given y Eq. (6) fo / h <.6 (figue3.) whee h i the wate depth o the ditance fom the wall to the axi of ymmety o fee uface. Figue (3.c) peent tempoal and patial vaiation of dimenionle eddy vicoity fo aymmetic wave. Figue (3.d) how compaion etween peiod-aveaged eddy vicoity otained y BSL k-ω model (ymol) and analytical pofile of Eq. (6) (dahed line). Even fo the cae of aymmetic wave, the peiodaveaged dimenionle eddy vicoity ha a hape which i well decied y Eq. (6) fo / h <.5 (figue3.d). Figue (3.) and (3.d) how that the peiod-aveaged eddy vicoity pofile fo inuoidal wave i diffeent fom the pofile fo aymmetic wave. Thi indicate that the peiod-aveaged eddy vicoity pofile hould depend on the wave non-lineaity paamete given y N i =U c /û, whee U c i the velocity at wave cet and û i the total velocity amplitude. We need theefoe a pecific caliation fo paamete of Eq. (6) uing full-ange equation of fiction coefficient (Tanaka and Thu 1994) and wave ounday laye thickne (Sana and Tanaka 27). (a) () (c) (d) Figue 3: Dimenionle eddy vicoity; Left: Tempoal and Spatial Vaiation; Right: Peiod-aveaged dimenionle eddy vicoity; Top: inuoidal wave; Bottom: aymmetic wave. 3. CONCLUSIONS The main concluion of the peent tudy ae: - A modified advection-diffuion equation with an additional paamete α elated to the poce of votex hedding aove ipple allow a good deciption of upended ediment concentation pofile - Fo pactical application the peiod-aveaged eddy vicoity could e decied y a imple analytical fomulation

6 - The hape of the analytical peiod-aveaged eddy vicoity fomulation wa validated y BSL k-ω model fo inuoidal and aymmetic wave - Peiod-aveaged eddy vicoity pofile depend on the wave non-lineaity paamete and equie theefoe a pecific caliation. ACKNOWLEDGMENTS The fit autho i gateful fo the financial uppot povided y Japan Society fo the Pomotion of Science (JSPS), within the FY21 JSPS Invitation Fellowhip Pogam fo Reeach in Japan (No. S- 1168). REFERENCES Ai, R. 2, Dicuion of Caliation of Buinge-Aya type of eddy vicoity model paamete, J. Wateway Pot Coatal Ocean Eng., ASCE, Vol. 126, No. 2, pp Ai, R. 28, Analytical olution fo the modeled k equation, Jounal of Applied Mechanic, ASME, Vol. 75, No. 4, 4451, 4 p., doi:1.1115/ Ai, R. 21, Concentation pofile fo fine and coae ediment upended y wave ove ipple: An analytical tudy with the 1-DV gadient diffuion model, Advance in Wate Reouce, Elevie, Vol. 33, No. 4, pp Ai, R. 211, Engineeing modeling of wave-elated upended ediment tanpot ove ipple, Coatal Sediment 11, Miami, Floida, USA, May 2-6, edited y Julie D Roati, Ping Wang and Tiffany M Roet, Wold Scientific Pulihing, pp Ai, R., S. Machandon, and M. Lavade, 211, Tuulent diffuion of upended paticle: analyi of the tuulent Schmidt nume, Defect and Diffuion Foum, Tan Tech Pulication, Vol , pp Davie, A. G. 1986, A model of ocillatoy ough tuulent ounday laye flow, Etuaine Coatal Shelf Sci., Vol. 23, pp Fedoe, J. and R. Deigaad, 1992, Mechanic of coatal ediment tanpot, Wold Scientific, 369 p. Hu, T.W. and C.D. Jan, 1998, Caliation of Buinge-Aya type of eddy vicoity model paamete, J. Wateway Pot Coatal Ocean Eng., ASCE, Vol. 124, No. 5, pp Hoya, S., and J. Jiméne, 26, Scaling of velocity fluctuation in tuulent channel up to Re τ = 23, Phy. Fluid, Vol. 18, Iwamoto, K., 22, Dataae of fully developed channel flow, THTLAB Intenal Repot No. ILR-21, Dept. Mech. Eng., Univ. Tokyo. Iwamoto, K., Y. Suuki, and N. Kaagi, 22, Reynold nume effect on wall tuulence: towad effective feedack contol, Int. J. Heat Fluid Flow, Vol. 23, pp McFetidge, W. F. and P. Nielen, 1985, Sediment upenion y non-eaking wave ove ippled ed, Technical Repot No. UFL/COEL-85/5, Coat Ocean Eng Dept, Univeity of Floida. Neu, I., and H. Nakagawa, 1993, Tuulence in Open-Channel Flow, A. A. Balkema, ed. Nielen, P. 1992, Coatal ottom ounday laye and ediment tanpot, Wold Scientific, 324 p. Sana, A. and H. Tanaka, 27, Full-ange equation fo wave ounday laye thickne, Coatal Engineeing, Vol. 54, pp Sheng, J., and A.E. Hay, 1995, Sediment eddy diffuivitie in the neahoe one, fom multifequency acoutic ackcatte, Cont. Shelf Re., Vol. 15, No. 2-3, pp Souly, R. L. 1997, Dynamic of Maine Sand, 249 pp., Thoma Telfod Pul., London. Suntoyo, and H. Tanaka, 29, Effect of ed oughne on tuulent ounday laye and net ediment tanpot unde aymmetic wave, Coatal Engineeing, Vol. 56, No. 9, pp Tanaka, H. and A. Thu, 1994, Full-ange equation of fiction coefficient and phae diffeence in a wavecuent ounday laye, Coatal Engineeing, Vol. 22, pp Thone, P.D., J.J. William, and A.G. Davie, 22, Supended ediment unde wave meaued in a lagecale flume facility, Jounal of Geophyical Reeach, Vol. 17, No. C8, 3178, 1.129/21jc988. Thone, P. D., A.G. Davie, and P.S. Bell, 29, Oevation and analyi of ediment diffuivity pofile ove andy ippled ed unde wave, Jounal of Geophyical Reeach, Vol. 114, No. C223. van Rijn, L.C. 27, United view of ediment tanpot y cuent and wave II: Supended tanpot, Jounal of Hydaulic Engineeing, ASCE, Vol. 133, No. 6, p

one primary direction in which heat transfers (generally the smallest dimension) simple model good representation for solving engineering problems

one primary direction in which heat transfers (generally the smallest dimension) simple model good representation for solving engineering problems CHAPTER 3: One-Dimenional Steady-State Conduction one pimay diection in which heat tanfe (geneally the mallet dimenion) imple model good epeentation fo olving engineeing poblem 3. Plane Wall 3.. hot fluid