A 3D SEDIMENT TRANSPORT MODEL FOR COMBINED WAVE-CURRENT FLOWS

Transcription

1 A 3D SEDIMENT TRANSPORT MODEL FOR COMBINED WAVE-CURRENT FLOWS Peifeng Ma 1 an Ole Secher Masen Accurate preiction of current velocity an bottom shear stress, which both can be significantly influence by win waves, is essential for seiment transport preictions in the coastal environment. Consequently win-wave effects must be taken into account in a numerical seiment transport moel for application in coastal waters. In the present stuy, elements of a large-scale 3D numerical coastal circulation an seiment transport moel are evelope to preict net, i.e. the wave-perio-average, seiment transport rates. The seiment transport components consiere are (i) be-loa transport; (ii) mean suspene loa seiment transport within the wave bounary layer, which is obtaine from an analytical solution; an (iii) suspene loa seiment transport above the wave bottom bounary layer, which is obtaine from a numerical moel. In all moel components win wave effects are accounte for through simple analytical moels. Thus, the roughness prescribe for the hyroynamic part of the numerical coastal circulation moel is the apparent roughness, i.e. the roughness experience by a slowly varying current in the presence of waves. Similarly, the reference concentration specifie for the seiment transport part of the numerical moel is obtaine from analytical solutions for suspene seiment concentrations within the combine wave-current bottom bounary layer. Stratification effects cause by suspene seiment are inclue in the large-scale numerical seiment transport moel. Results of iealize tests suggest that win wave effects can be pronounce, e.g. in some typical coastal scenarios seiment can only be mobilize when win waves are present an accounte for. It is also shown that stratification can significantly affect suspene seiment transport rates of fine seiments. Keywors: seiment transport; bottom bounary layer; wave-current interaction; seiment-inuce stratification 1. INTRODUCTION As illustrate in Figure 1, a typical coastal environment consists of win waves, slowly varying currents, such as tie- an win-riven flows, an seiment in the bottom an water column. Seiment can be move as be-loa transport on the seabe or stirre up into the water column by be shear stress an then transporte by a current as suspene loa transport. It is well-known that win waves can intensify the near bottom turbulence significantly ue to the limite thickness of the wave bounary layer. As a result, the bottom shear stress an turbulent mixing in the water column can be markely enhance by the presence of win waves resulting in more seiment being mobilize an move to upper layers of the water column where it is then mae available for transport by currents. Consequently, win wave effects must be inclue in numerical coastal circulation an seiment transport moels. Since coastal circulation moels cannot resolve the short time-scales governing the wave bottom bounary layer, the hyroynamic as well as seiment transport processes within the combine wave-current bottom bounary layer must be calculate separately to prouce physically realistic, slowly varying bounary conitions that are passe on to the numerical coastal circulation an seiment transport moel. WAVES CURRENTS SHEAR STRESS SUSPENSION Figure 1 Illustration of a typical coastal environment Many numerical seiment transport stuies have been conucte in the past few ecaes (e.g. Li an Amos, 001; Lesser et al., 004; Warner et al., 008). Win wave effects have been taken into 1 Singapore-MIT Alliance for Research an Technology, 1 CREATE way, #09-03 CREATE Tower, Singapore, Parsons Laboratory, Massachusetts Institute of Technology, Room 48-16C, 15 Vassar Street, Cambrige, MA, 0139, USA 1

2 COASTAL ENGINEERING 01 account in some moels, e.g. some wave-associate coefficients are introuce by Lesser et al. (004) to account for wave effects on seiment transport. Wave-current interaction moels (e.g. Masen, 1994) are introuce by Li an Amos (001) an Warner et al. (008) to preict combine wave-current shear stresses for seiment transport preictions. However, most numerical seiment transport moels treat the wave bounary layer as a portion of numerical omain, regarless of the much ifferent physics within the layer which usually cannot be resolve by coastal circulation moels. Density changes cause by suspene seiment have been inclue in some seiment transport moels (e.g. Lesser et al., 004; Warner et al., 008), but harly any information has been reporte on the importance of this seiment stratification effect. The objective of the present stuy is to evelop a three imensional numerical seiment transport moel for combine wave-current flows. A bottom bounary layer moel is introuce to account for win wave effects on current flow an seiment transport. The mean suspene seiment transport within the bounary layer is calculate separately an analytically. The seiment stratification effect is stuie. The paper is organize as follows. A moel escription, incluing flow moel, bottom bounary layer moel an seiment transport moel, is presente in Section. In Section 3, numerical experiments are conucte to test the moel. Finally conclusions are provie in Section 4.. MODEL DESCRIPTION In coastal waters, seiment is usually stirre up by the be shear stress an then transporte by a current. Be shear stress etermines the amount of seiment being mobilize an flow velocity etermines how far the seiment can be transporte. In the present moel, a bottom bounary layer moel is introuce to preict be shear stress in combine wave-current flows, incluing the total an seiment transport shear stresses. A numerical coastal circulation moel is use to compute ey viscosity an velocity profiles for the slowly varying currents above the bottom bounary layer, for which wave effects are incorporate through the specification of an enhance apparent bottom roughness, i.e. the roughness experience by a current in the presence of win waves. The be-loa transport rate an the reference concentration for suspene seiment concentration are compute from the so-calle skin friction or seiment transport shear stress. The mean suspene loa transport within the bottom bounary layer is solve analytically. The suspene loa seiment transport rate above the bottom bounary layer is compute by a numerical seiment transport moel that solves the unsteay, i.e. slowly varying, avection-iffusion equation to obtain the suspene seiment concentration corresponing to a specifie reference concentration preicte by the analytical bottom bounary layer moel that inclues the effects of win waves. The present moel is formulate in a terrain following coorinate system (x, y, σ) with x an y inicating the two irections in the horizontal plane an σ representing the scale vertical irection. The velocities in the three irections are enote by (u, v, w). The still water epth is h an surface elevation is enote by η. The water ensity an molecular viscosity are taken to be ρ = 1,05kg/m 3 an ν = 1.3x10-6 m /s, respectively. To be realistic, we assume the presence of ranom win waves which have a root mean square (rms) wave height H r an a representative wave perio T r or angular frequency ω r = π /T r an assume escribe by linear wave theory. Non-cohesive seiment with meian iameter an ensity ρ s =,650kg/m 3 is consiere in the present stuy..1 Hyroynamic Moel The Princeton Ocean Moel (POM), Blumberg an Mellor (1987), is use to preict 3D flow velocities (u, v, w) in the x, y an σ irections, respectively. POM is a primitive equation ocean moel base on hyrostatic an Bousinesq assumptions, in which the turbulence closure scheme of Mellor an Yamaa (198), hereafter referre to as MY, is incorporate to estimate ey viscosity K M an ey iffusivity for heat, K H. In the present stuy, we assume the suspene seiment iffuses in the same way as the heat an therefore take the seiment ey iffusivity K S = K H. As pointe out by Warner et al. (005), the MY scheme preicts a substantially smaller ey viscosity than analytical solutions an other turbulence closure schemes. This uner-preiction may have significant influence on suspene seiment concentrations causing it to ecay too rapily with istance from the be. To avoi the unerpreiction, the wall proximity function with open channel correction propose by Blumberg et al. (199) is use to replace the original wall function in the MY scheme. Since POM cannot resolve the small scale wave motion, only net or wave-perio average quantities are compute in the present stuy, e.g. the net be-loa transport rate an mean concentrations.

3 COASTAL ENGINEERING Bottom Bounary Layer Moel A wave-current interaction moel (Grant an Masen, 1979; Masen, 1994; Humbyr an Masen, 010) is incorporate into POM to account for win wave effects. The moel is base on a bilinear time-invariant ey viscosity moel that is proportional to the combine maximum wavecurrent shear velocity u *m within an to the enhance current shear velocity u *c above the bottom bounary layer, i.e. K M κ u* mz, z δ ( z) = κu* c z, z δ The moel is able to preict the physical, i.e. movable, bottom roughness k n, the apparent bottom roughness, i.e. the enhance bottom roughness experience by a current in the presence of waves, k na, the wave bounary layer thickness δ, the total current an wave shear velocities, u* c an u * wm, an the current an wave skin friction, or seiment transport, shear velocities u* cs an u * wms. The information require to implement the wave-current interaction moel inclue: water epth h; rms wave height H r an perio T r or angular frequency ω r ; current bottom shear velocity u *c or a reference current velocity u r at z = z r ; an seiment iameter. The solution proceure for the wavecurrent interaction can be ivie into three steps...1 Preiction of physical bottom roughness k n base on wave an seiment specification It is assume that win waves ominate flui-seiment interaction an therefore etermine the be conition, i.e. flat without or with seiment motion or ripple, so that the physical roughness can be preicte solely from wave information. To o this, the maximum wave bottom orbital velocity U bm or the excursion amplitue A bm = U bm / ω r is compute from rms wave height, perio an water epth. Then the skin friction Shiels Parameter ψ is obtaine from [ s g ] f U [ s g ] ψ = τ / ρ( 1) = 0.5 / ( 1) () wm w bm where g = 9.81m /s is the acceleration ue to gravity an s = ρ / ρ =.59 is the relative ensity of seiment. The wave skin friction factor meian seiment iameter, i.e. s f w is compute base on a bottom roughness equal to the (1) f w exp 7.0X 8.8, X = A / < 10 = exp X, X = A / bm bm (3) There will be no seiment motion if the skin friction Shiels Parameter ψ given by () is smaller than the critical value, ψ cr, which can be estimate by the formula propose by Herrmann an Masen (007) where ( ) ( ) /3 3/4 ψ cr = 0.095S * exp S* / 0 S* = s 1 g / 4ν. The be will be flat with no transport if ψ < ψ cr, ripple if ψ cr ψ 0.35 an in sheet flow conition if ψ > The threshol Shiels Parameter for sheet flow conition is taken to be 0.35, rather than the usual value of 0.7, because the Shiels Parameter compute using the significant wave orbital velocity, ψ s = ψ, has been foun to escribe onset of sheet flow for ranom waves. The threshol Shiels Parameter for initiation of motion is still taken as ψ cr in orer to be consistent with the critical value use in the calculation of be-loa transport rate an reference concentration for seiment suspension in combine wave-current flows. The physical roughness for ifferent be conitions can therefore be compute from (4) ψ < ψ, cr (no transport) n = n ( bm n ) ψ cr ψ max(, An ), ψ > 0.35 (sheet flow) k k, A / k, 0.35(ripple be) (5)

4 4 COASTAL ENGINEERING 01 where A n is a function of ψ suggeste to be 15 ψ by Masen (00) base on very limite ata. For ripple be conitions, the physical roughness in (5) can be calculate iteratively through the empirical relationship for the energy issipation factor establishe by Humbyr an Masen (010) ( ) f = 0.14S exp 4.94ψ (6a) e 0.5 * This energy issipation factor is a function of bottom roughness {( π ) 10 ( ) } fe = fw cos / log Abm / kn (6b) where f w is compute from (3) with X = Abm / kn. Therefore, the energy issipation factor f e can be calculate by (6a) base on the skin friction Shiels Parameter obtaine from (). Having f e, the physical roughness can be compute iteratively from (6b)... Preiction of total shear velocities, bounary layer thickness an apparent roughness With the physical bottom roughness k n obtaine as outline in Section..1, an the current shear velocity u *c, a wave-current interaction analysis can be performe to yiel the maximum wave shear velocity, u *wm. The general proceure is ( ) 1/ Cµ = 1+ cos φ µ + µ with µ = u / u (7a) * c * wm Cµ exp 7.0X µ 8.8, X µ = Cµ Abm / kn < 10 f = (7b) Cµ exp 5.61X µ 7.30, X µ = Cµ Abm / kn 10 u* wm = f / U (7c) bm where φ = φc φw is the angle between currents an waves, with φc an φw enoting the current an wave irections with respect to the x axis. In this wave-current interaction proceure, the current shear stress or shear velocity u*c is kept unchange since it woul be slowly varying. But this oes not mean the current shear stress is fixe for the entire simulation, since it varies slowly with time as preicte by the numerical circulation moel an/or slowly changing win wave conitions. Hence, the bottom bounary layer moel an the flow moel are fully couple. Once the iteration proceure in (7a-c) converges, we can obtain u* c an u * wm an the combine maximum shear velocity u * m = Cµ u * wm, from which we can calculate the bottom bounary layer thickness where δ = κ ω (8a) PAµ u* m / r ( ) bm n Aµ = exp.96 Cµ A / k 1.45 (8b) 1 Cµ /( Cµ µ ) P = ( Cµ / µ ) 16.3 (8c) The current velocity within an above the bounary layer can now be compute from ( ) ( ) u u z k z uc ( z) = u / κ ln 30 z/ k, z δ * c / κ * m ln 30 / n, δ * c na (9) where the apparent roughness, k na, is obtaine by matching the velocity at z = δ k u ( ) * c / u δ k * m = 30δ 30 / (10) na n The apparent roughness is the roughness experience by the current in the presence of win waves, which is use to compute the quaratic bottom friction factor

5 COASTAL ENGINEERING 01 5 ( ) CD = κ / ln 30 z1 / k na (11) where z is the first interior velocity gri level above the bottom in POM simulations. A logarithmic 1 velocity istribution near the bottom is assume to obtain (11). Through this bottom friction factor, C D, the numerical circulation moel captures the enhancement of bottom resistance inuce by win waves. The apparent roughness is the only parameter passing from the bottom bounary layer moel to the flow moel, which in turn will provie an upate current shear stress to the bottom bounary layer moel. The current bottom shear stress τ bc in POM is compute from the friction factor an the horizontal velocity U 1 at the level z = z 1 τ = ρ C U (1) bc D 1..3 Preiction of seiment transport shear stress To compute seiment transport rates, we nee to evaluate the shear stresses or shear velocities responsible for moving an entraining seiment. For sheet flow conitions, the shear stresses obtaine as outline above are the seiment transport shear stresses. For ripple be conitions, the total shear stresses obtaine in the preceing section consist of skin friction an form rag, an of these only the skin friction shear stress is responsible for seiment motion. Thus, when the be is ripple, the skin friction shear stress nees to be separate from the total shear stress. To o this, another wave-current interaction analysis is performe base on a reference current velocity an a skin friction roughness. For ripple be conitions, a reasonable skin friction roughness shoul be relate to the flow intensity rather than a constant multiple of the grain size. In orer to make a smooth transition from ripple be to sheet flow conitions, we aopt the roughness for sheet flow conition in (5) to calculate skin friction bottom roughness, i.e. we take k = (13) ns max(,15 ψ ) Unfortunately, no experimental ata are available to support a specific choice of the reference level for the reference current velocity to be use in the computation of the seiment transport shear stresses for combine wave-current flows. Thus we choose, somewhat arbitrarily but physically an computationally pleasing, the upper bounary of the wave bottom bounary layer as the reference level at which the reference current velocity is specifie, i.e. we have from (9) ( ) u = u / κ ln 30 δ / k at z = δ (14) r * c na r Hence the seiment transport shear velocities u* cs an u * wms can be compute from the following iterative proceure ( ) 1/ Cµ = 1+ cos φ µ + µ with µ = u / u (15a) f s s s s * cs * wms s C exp 7.0X 8.8, X = C A / k < 10 = exp C X, X = C A / k µ s s s µ s bm ns µ s s s µ s bm ns (15b) u* wms = 0.5 fsu bm (15c) u = C u (15) u * ms µ s * wms ln( z / δ ) 1 1 κu ln( δ / z ) = u + + r s r s 0s * cs * ms ln( δ s / z0 s ) 4 u* ms ln( zr / δ s ) (15e) where z 0 s = kns / 30 an the thickness δs is compute in the same way as δ in (8a-c) but with the skin friction parameters kns, µ s, Cµ s an u* ms. After the iteration proceure in (15a-e) converges, we can obtain the seiment transport shear stresses for the current, τ cs = ρu, an for the waves, * cs τ wms = ρu. * wms The total seiment transport shear stress can then be written as

6 6 COASTAL ENGINEERING 01 τbs ( t) = τ bsx ( t), τbsy ( t) = τ wms cosωrt cosφw + τ cs cos φc, τ wms cosωrt sinφw + τ cs sinφc (16).3 Seiment Transport Moel Instantaneous seiment transport rates are average over a win wave perio to yiel mean or net seiment transport rates. Three categories of seiment transport are consiere: (i) Net be loa transport, q B, which takes place below the reference level for mean concentration specification for the suspene loa, z = 7; (ii) Mean suspene loa transport within the wave bounary layer, 7 < z < δ, q S1, which is obtaine from analytical solutions for current velocity an mean seiment concentration within the wave bounary layer, since POM oes not resolve wave bounary layer scales; an (iii) Mean suspene loa transport above the wave bounary layer, z > δ, q S, which is obtaine by solving an avection-iffusion equation for the mean seiment concentration an the current velocity using POM. The instantaneous be-loa mass transport rate is compute by the moel propose by Masen (1991). ( τ bs αβ τ cr, β ) 8ρs τ bs qb ( t) = Max{ τ bs ( t) τ cr, β,0} 1.5 ρ ( s 1) g tanϕ tan β τ where β is the bottom slope in the irection of the instantaneous seiment transport shear stress an consiere positive when sloping up in the shear stress irection. In large scale coastal simulations, the be slope is usually very small, i.e. tan β sin β β << 1. The critical shear stress with slope effect is given by τ = τ ( + β ϕ ) with τ = ψ [ ρ( 1) ] cr, β cr 1 tan / tan s cr cr s g an ψ cr compute by (4). ϕs an ϕ m are static an ynamic friction angles for seiment which have values of 30 o an 50 o, respectively. Thus, the parameter α ( ϕ β ) ( ϕ β ) β = tan + tan / tan + tan 0.7. m The net be-loa transport rate can be compute by averaging (17) over a wave perio as q q ( ) B 1 T = T 0 B t t s The instantaneous be-loa transport rate for a pure sinusoial wave conition is q ( t) τ cosω t τ cosω t, which obviously leas to a perio-average be-loa transport of B wms r wms r zero. Thus, win waves can only give rise to a non-zero net be-loa transport when they are acting together with a current an/or on a sloping bottom. For example, if we assume wave shear stress to be ominant an the be slope to be small, the net be-loa transport was shown by Masen (00) to be 9u u u tan β / tanϕ. approximately given by qb * wms ( * cs * wms m ) To preict suspene loa transport rate above the be-loa transport layer, i.e. z > 7, a reference concentration at z = 7, relate to the excess seiment transport shear stress, was introuce by Masen (00) m bs (17) (18) C ( ) = 0.00ρ 0 max ( τ ( ) / τ, β 1 ),0 R t sc bs t cr at z = 7 (19) where c 0 = 0.65 is the volume concentration of seiment in the be. Again, only the mean reference concentration is consiere here, which is 1 T CR = CR ( t) t (0) T 0 expresse by ( ) R s 0 wms cr It can be shown that, unlike the net be-loa transport, linear win waves alone can prouce a significant mean reference concentration, e.g. the mean concentration for pure wave conitions can be C = / πρ c τ / τ 1 which obviously oes not vanish. Therefore, waveinuce bottom shear stress can play a very important role in preicting suspene loa transport an

7 COASTAL ENGINEERING 01 7 shoul not be neglecte, especially for wave ominate conitions which are common in most coastal waters. Since the large scale numerical moel oes not resolve wave bottom bounary layer scales, the mean concentration in this layer is compute analytically base on an ey iffusivity K S which is assume ientical to the ey viscosity, K M, given by (1). To obtain the analytical concentration solution, we assume the time scale of vertical iffusion in the bounary layer is much smaller than the scale of the slow current flow changes so that equilibrium can be assume. This assumption shoul be reasonable, since the wave bounary layer is thin an the turbulent mixing is very strong. For equilibrium, the vertical iffusion an the seiment settling effects are balance to yiel an analytical concentration solution within the wave bounary layer wf / κu* ( ) m C( z) = C z / 7 (1) R where w f is the seiment fall velocity. It is calculate from the formula propose by Jiménez an Masen (003) ( 1 ) / ( / ) w = s g + S () f n * n where n / 0.9 S* n = n s 1 gn / 4ν. Therefore the net suspene loa transport rate in the wave bottom bounary layer is an ( ) δ q S1 = uccz (3) 7 where the current velocity vector u ( z) c is given by (9). This integration of (3) can be performe analytically (e.g. Masen, 00) or numerically by iscretizing the bounary layer into a large number of layers. In the present stuy, the latter metho is aopte. The hyroynamic an seiment transport processes above the bounary layer are far more complex than their counterparts within the bottom bounary layer an are solve numerically in the present stuy. The slowly varying current velocity is compute from the numerical circulation moel, POM, whereas the mass concentration C(z) is obtaine by numerically solving the avection-iffusion equation for suspene seiment, DC + ( DuC) + ( DvC) + ( w wf ) C = t x y σ KS C C C + ( AH D ) + ( AH D ) σ D σ x x y y where D = h + η is the total water epth with η the free surface elevation preicte by POM, A H is the horizontal ey iffusivity which is assume the same as the horizontal ey viscosity an preicte by POM, an the vertical iffusivity for seiment, K S, is assume ientical to the heat iffusivity preicte by MY turbulence closure scheme in POM. Bounary conitions nee to be specifie in orer to solve (4). At lan bounaries, a no flux conition is applie, i.e. C / n = 0 with n enoting the outwar normal irection at the bounary. At open bounaries, concentration is specifie for inflow conition, i.e. C = Cin if un < 0 with un the flow velocity in the outwar normal irection, n, an an avection conition is applie for outflow, i.e. C / t + u C / n = 0 if u > 0. At the free surface, zero net seiment flux is assure by setting n n K C / σ + Dw C = 0 at z = h +η, whereas a seiment concentration, the reference concentration, is S f specifie as C = Cb at bottom. To obtain the reference concentration, C b, we first compute the mean concentration at the outer ege of the wave bounary layer, i.e. at z = δ, from the analytical wave bounary layer solution, (1), wf / κu* ( δ ) (4) m C = C( z = δ ) = C /7 (5) a R Physically, the bottom level in POM simulations is efine at the outer ege of the wave bounary layer, i.e. z = δ. However, ue to the staggere gri system use in POM, the bottom level for

8 8 COASTAL ENGINEERING 01 concentration is not at z = δ but a half gri spacing above it, i.e. at z = zb = δ + zb / with zb the spacing for the interior gri cell immeiately above the bottom. Assuming equilibrium conitions to apply within the bottom half of the first gri, the concentration C b can be obtaine analytically base on the reference concentration C a at z = δ an the linear ey iffusivity above the bounary layer given by (1). In this manner, the reference concentration for the large scale numerical seiment transport moel clearly accounts for win wave effects an it becomes ( δ ) wf / κu* c b a 1 b / at b C = C + z z = z (6) The ey iffusivity preicte by POM approaches zero at the bottom level which is actually at the outer ege of the wave bounary layer where the ey iffusivity accoring to the wave bounary layer moel shoul be κu*cδ. This relatively small ey iffusivity may prouce a large ifference in seiment concentration, since suspene seiment concentration is very sensitive to the iffusivity near the bottom. For example, if we assume a typical conition with w = 1 cm / s, u* =.5 cm / s, 3 3 δ = 5cm, z = 10cm an C = 1 kg / m, we obtain C = 0.5 kg / m from (6), an the concentration b a at the upper bounary of the first gri cell, z = δ + zb is roughly 0.33kg/m 3. If irectly using the ey iffusivity preicte by POM, the concentration at z = δ + zb preicte by the present moel 3 can be estimate analytically to be 0.5 kg / m. This unerestimates the concentration by about 4%. To avoi this uner-preictions an be more physically consistent, we a the constant ey iffusivity, κu *c δ, to the POM preicte ey iffusivity KSPOM over the entire water epth K = K + κu δ (7) S SPOM * c This ajustment may lea to some minor error in ey iffusivity near the surface as it oes not approach zero. However, this error shoul have very little influence on the preicte seiment concentration as it is usually quite small near the free surface. Having the flow velocity an the concentration, the suspene loa transport rate above the wave bounary layer is compute by h+ η q S = ucz (8) δ Suspene seiment will change the water-seiment mixture ensity an therefore prouce stratification, which in turn will affect the preictions of flow velocity an seiment concentration. The ensity of the water-seiment mixture can be compute from ρmix = ρ + C ( z)(1 1/ s) (9) A stable stratification, which woul generally be the case for seiment suspene in the water column since the concentration an therefore the mixture ensity woul ecrease with istance above the bottom, suppresses the turbulence an therefore weakens the turbulent mixing. This woul result in an increase in velocity an a reuction in seiment concentration. In contrast, an unstable stratification intensifies the flow turbulence an therefore results in more seiment transport. Due to the seiment settling effect, positive stratification shoul be more common. Unstable stratification may, however, occur in rapily varying flows when the near-be concentration, which respons nearly instantaneously to the reference concentration, ecreases so rapily that high seiment concentrations in upper layers have insufficient time to settle out of suspension. 3. MODEL TESTS In this section, a few iealize tests are performe to examine the moel s ability to preict win wave effects an seiment transport in combine wave-current flows. The computational omain is an open channel of 100km length, 10km with an 10m epth. In the numerical simulations, uniform horizontal gri cells of x = y = km an 40 non-uniform sigma layers with finer spacing near the be are use. The external an internal time steps are 6sec an 10sec, respectively. In the tests, Coriolis force is neglecte. A steay open channel flow is consiere which is riven by a prescribe constant volume ischarge at both open bounaries, equivalent to applying a epth- b f c

9 average velocity U 0.5 / ( 1 η / h) COASTAL ENGINEERING 01 9 = + m/s at the bounaries. This leas to U = 0.5m/s at the center of the omain where the surface elevation η = 0 after a steay state is establishe. The flow is uniform in the cross-channel irection. A ranom win wave conition with a rms height of H r = 1m an a representative perio of T r = 8sec is specifie. The waves are assume to propagate in the same irection as the current flow, i.e. this is a co-irectional combine wave-current flow case. Uniform seiment with a meian grain size of = 0.1mm or 0.mm is assume. This correspons to a critical shear stress of τ cr = 0.19 N / m or 0.0 N / m or a critical shear velocity of u* cr = 1.36 cm / s or 1.40 cm / s as compute from (4) an a fall velocity of w = 0.7 cm / s or. cm / s from (). For the inflow bounary conition, the seiment concentration is assume zero. This creates a transition region near the upstream bounary. The length of this transition region can be estimate approximately from L = Uh / KS / 0.01 = 5km. Eviently, this is a very short istance compare to the omain length of 100km, most of which will therefore be unaffecte by transition effects. The simulations are run for 3 ays. To avoi simulation crash, the forcing is impose graually with a linear ramp up uring the first 1hours. The results show that steay state is establishe within a few hours after the ramp up perio. All the results shown in this Section are obtaine at the center of the omain where the surface elevation is close to zero at a time corresponing to the en of the simulation, t = 7 hours. For the 0.1mm seiment, three simulations are performe: (1) Pure current case; () Combine wave-current case without stratification; (3) Combine wave-current case with stratification. For the 0.mm seiment, only simulations () an (3) are performe. Some preicte parameters for simulations () an (3) are liste in Table 1, incluing physical bottom roughness, apparent roughness, bounary layer thickness, wave an current shear velocities an reference concentrations. Seiment transport rates are liste in Table for the ifferent simulations. 3.1 Effects of Win Waves The vertical profiles of ey viscosity an horizontal velocity from simulations (1) an () for 0.1mm seiment are shown in Figure. For the pure current simulation, the physical roughness is chosen to be the larger one of the seiment iameter an 3.3ν/u *c, i.e. the simulation starts with k n = by assuming a rough turbulent flow, then checks an switches to smooth turbulent flow with k n = 3.3ν/u *c if < 3.3ν/u *c. The flow moel preicts a bottom shear velocity of 1.5cm/s, inicating a smooth turbulent flow conition an therefore the roughness is k n = 3.3ν/u *c = 0.8mm. When win waves are present, a ripple be is obtaine, leaing to a much larger physical roughness of k n = 1.66cm an an apparent roughness of k na =11.1cm (non-stratifie case in Table 1) an a bottom shear velocity of.7cm/s. As a result, the maximum ey viscosity (seen in Figure a) almost oubles from 0.01m /s to 0.0m /s ue to the presence of win waves. The current velocity profiles in Figure b show that the presence of win waves leas to a lower current velocity near the be ue to the larger flow resistance. The velocity at the first interior gri point is ecrease from about 0.3m/s for the pure current conition to 0.15m/s in the presence of win waves. Table 1. Parameters preicte by the moel for ifferent seiment in stratifie an non-stratifie flows. Diameter 0.1mm 0.mm Stratifie No Yes No Yes Parameters k n (cm) k na (cm) δ (cm) u *c (cm/s) u *cs (cm/s) u *wm (cm/s) u *wms (cm/s) C R (kg/m 3 ) C a (kg/m 3 ) f

10 10 COASTAL ENGINEERING 01 z (m) (a) Ey t=7th hour No Wave H r = 1m z (m) (b) t=7th hour No wave H r = 1m K M (m /s) u (m/s) Figure. Preicte vertical profiles of (a) ey viscosity an (b) current velocity for pure current an combine wave-current cases at the center of omain. Seiment size is 0.1mm. Due to the flat be conition, the seiment transport shear velocity for the pure current conition is equal to the total value of 1.5cm/s, which is very close to the critical shear velocity of 1.36cm/s for the 0.1mm seiment. This means that there is essentially no seiment transport for the pure current conition. In the presence of win waves the seiment transport shear velocities are calculate to be 1.4 cm/s an.8cm/s for current an waves, respectively, an are base on a reference current velocity of u r = 11.1cm/s at z r = 1.8cm an a skin friction roughness of k ns = 0.45mm. Hence, the seiment transport shear stress in this co-irectional flow case is τ bs ( t) = ( 0.8cosωt , 0) N/m. This gives a maximum instantaneous value of 0.95 N/m, which is much larger than the critical shear stress of 0.19N/m. The rather ramatic effect of the presence of win waves is seen in the reference concentrations, 7.3kg/m 3 an.7kg/m 3 (Table 1), at the lower an upper bounaries of the wave bounary layer. As shown in Table, the preicte transport rates are 0.01kg/m/s, kg/m/s an 0.5kg/m/s for the net be-loa transport, the mean suspene loa transport within an above the bounary layer, respectively. It can be seen that suspene loa transport in the water column is ominant as it is about 30 times the combine be-an suspene loa transport rate within the wave bounary layer. This is mainly cause by the small fall velocity of the fine seiment an strong turbulent mixing enhance by win waves. Due to the thin bounary layer an fine seiment, the mean suspene loa transport rate within the bounary layer is much smaller than that in the overlying water column, but it is still comparable to the net be-loa transport rate. Table. Transport rates (kg/m/s) of net be-loa, suspene loa within an above the w.b.b.l. for two types of seiment. The total transport rate q T is qt = qb + qs1 + qs. q B q S1 q S q T Stratifie No Yes No Yes No Yes No Yes Diameter 0.1mm 1.0x10-7.3x x x x x x x mm 6.x x x10-3.9x x10-1.3x10 -.5x10 -.x10-3. Effects of Seiment-Inuce Self-stratification for 0.1mm Seiments Seiment stratification effects are accounte for in the present stuy by consiering the stratification effects associate with the varying ensity of the water-seiment mixture as expresse in (9). The routine available in POM to account for stratification ue to heat an/or salinity graient is use to account for seiment stratification effects. In general, the stratification effect is incorporate by relating a turbulent kinetic energy (TKE) prouction to the vertical ensity graient. This leas to an enhance turbulence intensity by a positive vertical ensity graient an vice versa. In this steay flow case, the concentration ecreases all the way from the be to surface, leaing to stably stratifie conitions which mean that the turbulence intensity will be reuce an therefore turbulent mixing will be weakene. This is confirme by the ecrease current shear velocity of.4cm/s preicte by POM an liste in Table 1 for 0.1mm seiment when stratification is accounte for compare to.7cm/s for

11 COASTAL ENGINEERING the non-stratifie case. As a result, the current velocity within the bounary layer (seen in Figure 3a) shows quite a significant reuction, e.g. the velocity at the outer ege of the bounary layer is ecrease from 11cm/s to 8cm/s. It shoul be recalle that stratification effects within the wave bounary layer are not accounte for, an this is the reason why the wave shear velocity, as seen in Table 1, harly changes (it goes from 5.3cm/s to 5.cm/s). The reuce current shear velocity, in aition to the reuce mixing associate with stable stratification, makes the ey iffusivity above the wave bounary layer ecrease substantially as shown in Figure 4a, reucing the maximum value from about 0.07m /s to 0.015m /s. z (m) (a) t=7th hour Non-stratifie Stratifie z (m) (b) t=7th hour Non-stratifie Stratifie u (m/s) Concentration (kg/m 3 ) Figure 3 Analytical solutions of vertical profiles of (a) current velocity an (b) seiment concentration within the wave bounary layer at the center of omain. Seiment size is 0.1mm. z (m) (a) Ey t=7th hour Non-stratifie Stratifie z (m) (b) t=7th hour Non-stratifie Stratifie K S (m /s) u (m/s) (c) t=7th hour Non-stratifie Stratifie z (m) Concentration (kg/m 3 ) Figure 4 Preicte vertical profiles of (a) ey viscosity; (b) Velocity; (c) Concentration above the wave bounary layer at the center of omain. Seiment size is 0.1mm.

12 1 COASTAL ENGINEERING 01 Due to the reuce current shear velocity, the velocity graient near the bottom ecreases, but this tenency is counteracte by the reuce mixing ue to stable stratification which woul ten to increase the velocity graient. The combine effect of these processes is a ecrease of velocity near the bottom an, since the epth-average velocity has to be maintaine at 0.5m/s, a slight increase in the upper layers (seen in Figure 4b). For example, calculations suggest that the velocity for stratifie case is reuce by 7% (from 14.8cm/s to 10.9cm/s) at bottom to about 1% at m above the be an the ifference is less than about 1.% in the rest portion of the water column. Since a reuction in shear velocity leas to a smaller reference concentration an stable stratification reuces upwar mixing of seiments into the water column, the two effects mentione above both result in a ecrease in suspene seiment concentration. One can therefore expect a significant reuction in the seiment concentration an transport rates ue to self-stratification effects. The reference concentrations in Table 1 show that these are reuce by about 7% from 7.3kg/m 3 to 6.75kg/m 3 at z = 7. This very minor reuction is, of course, associate with the virtually unchange wave shear velocities, which ominate in the creation of the mean reference concentration. Consequently, the concentration within the bounary layer oes not show significant reuction as seen in Figure 3b. In contrast, the concentration above the wave bounary layer (shown in Figure 4c) is significantly ecrease by the stratification effect, e.g. the concentration at 1m above the be is reuce by about 68% from 0.8kg/m 3 to 0.09kg/m 3, although the ifference between non-stratifie an stratifie cases is only 13% at bottom. The transport rates in Table reveal that the net be-loa an mean suspene loa transport rates within the bounary layer ecrease only by roughly 30%. The relatively small reuction of net be-loa transport rate is because it epens only on the seiment transport shear stress which, as seen from Table 1, oes not change appreciably. The reason for the slight reuction of mean suspene loa transport rate within the bounary layer reflects primarily the reuction in current velocity (shown in Figure 3a), whereas the concentration (Figure 3b), preicte without consiering stratification, harly changes. However, the suspene loa transport rate above the bounary layer is significantly reuce as a result of stratification. The stratifie value of 0.1kg/m/s is only 4% of the 0.5kg/m/s preicte when stratification effects are neglecte. This significant influence of stratification effects above the bounary layer is mainly cause by the reuction of turbulent mixing associate with the stable stratification an, to a lesser extent, by the reuce current shear stress. 3.3 Effects of Seiment Diameter In the preceing sections, we iscusse win wave an stratification effects base on numerical results for 0.1mm seiment. To investigate the influence of seiment size on seiment transport, we perform two simulations, one with an the other without stratification effects, for seiment of 0.mm iameter. The same flow an wave conitions as those for the 0.1mm seiment are use, an results are liste in Tables 1 an. As shown in Table 1, the physical an apparent roughness are about 1cm an 5cm. These are significantly larger than those for the 0.1mm seiment, inicating larger ripples an therefore smaller ratios between skin friction shear velocities an total shear velocities. This is confirme by the results in Table 1, e.g. the ratio of skin friction an total current shear stress for non-stratifie case is 0.6 for 0.mm seiment compare to 0.46 for 0.1mm seiment. The reference concentrations at z = 7 are quite similar for the 0.1mm an 0.mm seiments; however, ue to its much larger fall velocity, the mean concentration at the outer ege of the bottom bounary layer, 0.75kg/m 3 for 0.mm seiment, is only 8% of the value for 0.1mm seiment. Due to the near-ientical critical shear stresses an much ifferent fall velocities for the two seiments, the ifference of net be-loa transport rates between the two seiments is smaller than that of suspene loa transport rates. As shown in Table, the ifference of net be-loa transport rates for non-stratifie case is about 38%, whereas the ifference between suspene transport rates above the bounary layer is as large as 97%. As for the stratification effect, the results in Table show that the stratification only results in a 1% reuction of total seiment transport rate for 0.mm seiment, i.e. much less than the 75% reuction foun for the 0.1mm seiment. The primary reason for this is that the concentration above the bounary layer is significantly reuce for the 0.mm seiment ue to its much larger fall velocity,.cm/s vs. 0.7cm/s, an therefore generates less stratification effect.

13 COASTAL ENGINEERING CONCLUSIONS In this paper, a three imensional seiment transport moel is propose for the computation of seiment transport in combine wave-current flows in coastal waters. The moel calculates three contributions to the total seiment transport rate: (i) net be-loa transport in the be-loa layer z < 7; (ii) mean suspene loa transport within the bottom bounary layer 7 < z < δ ; an (iii) suspene loa transport above the wave bounary layer z > δ. Win wave effects on flow an seiment transport are accounte for through an analytical bottom bounary layer moel. For the hyroynamics, the presence of win waves enhances the flow resistance by increasing the roughness, the so-calle apparent roughness, experience by a current in the presence of waves. For the seiment transport processes, win waves increase the bottom shear stress an the near-be turbulence an therefore mobilize an suspen more seiment than woul be the case if the waves were absent. In aition, the wave enhance turbulent mixing in the water column above the wave bounary layer iffuse more seiment up into the water column. In the water column above the wave bounary layer hyroynamics an seiment transport processes are solve numerically using POM an stratification effects inuce by suspene seiment are taken into account. Iealize tests were performe to examine the moel s ability to preict seiment transport in combine wave-current flows. The test results reveal that win waves have a pronounce effect on hyroynamics an suspene seiment transport for typical conitions of coastal waters, e.g. seiment may only be mobilize an transporte when win waves are present. The seiment transport rate within the wave bounary layer, which has been neglecte in most previous stuies, is more important for coarser seiment, e.g. it is about 0% the suspene loa transport above the bounary layer for 0.mm seiment compare to 1% for 0.1mm seiment case in the present stuy s computational example. Therefore, win wave effects shoul never to be neglecte in seiment transport moeling in coastal waters. The numerical tests suggest that the self-stratification effect is more significant for fine seiment as it causes 75% rop in total transport rate for the 0.1mm seiment case, but becomes less significant for coarser seiment as the reuction of total transport rate is only 1% for 0.mm seiment. Essentially, the stratification effect can be very significant when concentration graients are large, which usually happens for relatively fine seiment an strong flow conitions. In such circumstances, our results show that the self-stratification effect shoul be accounte for when moeling seiment transport in coastal waters. It is note that the present moel tests were performe for a steay co-irectional wave-current flow. Further tests must be conucte to test the moel s capabilities in more realistic flows, e.g. in unsteay flows an in the cases where waves an current are in ifferent irections. For unsteay, slowly varying flows, the applicability of the concentration bottom bounary conition, which assumes equilibrium conitions, nees to be teste. To more accurately preict the seiment transport in combine wave-current flows, the wave-associate suspene loa transport within the wave bounary δ layer, q S1 = ucz, with u ( t) an C ( t) enoting the time-varying wave-associate velocity an 7 concentration within the wave bounary layer, shoul be inclue. Consiering the significant influence of self-stratification on suspene transport above the bounary layer, it woul be pruent to examine the effect of stratification within the wave bounary layer, or in the extrapolation of the mean concentration from the outer ege of the wave bounary layer to the numerical moel s gri point where a reference concentration is specifie. We are currently pursuing extensions of the seiment transport moel along these lines as well as examining its performance in more realistic scenarios. ACKNOWLEDGMENTS The research escribe in this paper was fune in whole or in part by the Singapore National Research Founation (NRF) through the Singapore-MIT Alliance for Research an Technology s (SMART) Center for Environmental Sensing an Moeling (CENSAM). REFERENCES Blumberg, A.F., B. Galperin, an D.J. O Connor Moeling vertical structure of open-channel flows, Journal of Hyraulic Engineering, 118(H8), Blumberg, A.F., an G.L. Mellor A escription of a three-imensional coastal ocean circulation moel, Three Dimensional Coastal Ocean Moels, N. Heaps, E., Coastal Estuarine Science, Vol. 4, Amer. Geophys. Union, 1 16.

14 14 COASTAL ENGINEERING 01 Grant, W.D., an O.S. Masen Combine wave an current interaction with a rough bottom, Journal of Geophysical Research, 84(C4), Herrmann, M.J., an O.S. Masen Effect of stratification ue to suspene san on velocity an concentration istribution in uniirectional flows, Journal of Geophysical Research, 11, C0006, oi: /006JC Humbyr, C.J., an O.S. Masen Preicting movable be roughness in coastal waters, Proceeings of the International Conference on Coastal Engineering. No. 3(010), Shanghai, China. Paper #: seiment.6. Retrievable from Jiménez, J.A. an O.S. Masen A simple formula to estimate settling velocity of natural seiment, Journal of waterway, port, coastal an ocean engineering, 19, No., Lesser, G.R., J.A. Roelvink, J.A.T.M. van Kester, an G.S. Stelling Development an valiation of a three-imensional morphological moel, Coastal Engineering, 51, Li, M.Z. an C.L. Amos SEDTRANS96: the upgrae an better calibrate seiment-transport moel for continental shelves, Computers & Geosciences, 7(6), Masen, O.S. (1991) Mechanics of cohesionless seiment transport in coastal waters. Proceeings of Coastal Seiments 91, ASCE, Seattle, USA. 1:15-7 Masen, O.S Spectral wave-current bottom bounary layer flows. Proceeings of 4 th International Conference on Coastal Engineering, ASCE, Masen, O.S. 00. Seiment Transport Outsie the Surf Zone. In: Walton, T. (eitor), Coastal Engineering Manual, Part III, Coastal Processes, Chapter III-6, Engineer Manual , U.S. Army Corps of Engineers, Washington, DC. Mellor, G.L., an T. Yamaa Development of a turbulence closure moel for geophysical flui problems, Review of Geophysics an Space Physics, 0, Warner, J.C., C.R. Sherwoo, H.G. Arango, an R.P. Signell Performance of four turbulence closure moels implemente using a generic length scale metho, Ocean Moelling, 8, Warner, J.C., C.R. Sherwoo, R.P. Signell, C.K. Harris, an H.G. Arango Development of a three-imensional, regional, couple wave, current an seiment-transport moel, Computers & Geosciences, 34,