NEW PRACTICAL MODEL FOR SAND TRANSPORT INDUCED BY NON-BREAKING WAVES AND CURRENTS

Transcription

1 NEW PRACICAL MODEL FOR SAND RANSPOR INDUCED BY NON-BREAKING WAVES AND CURRENS Domini A. van der A 1, Jan S. Ribberink, Jebbe J. van der Werf 3 and om O Donoghue 1 Many existing pratial sand transport formulae for the oastal marine environment are restrited to limited ranges of hydrodynami and sediment onditions. his paper presents a new pratial formula for net sand transport indued by non-breaking waves and urrents, and urrents alone. he formula is based on the semi-unsteady, half wave-yle onept, with bed shear stress as the main foring parameter. Unsteady phase-lag effets between veloities and onentrations are aounted for, whih are espeially important for rippled bed and fine sand sheet-flow onditions. Reently reognized effets on the net transport related to flow aeleration skewness and progressive surfae waves are also inluded. he formula is alibrated against a large dataset of net transport rate measurements from osillatory flow tunnels and a large wave flume overing a wide range of flow and sand onditions. Good agreement is obtained between observations and preditions, and its validity is shown for bedload dominated steady flow onditions. Keywords: sediment transport formula; sheet flow, ripples; osillatory flow; non-breaking waves INRODUCION Sand transport models for the oastal marine environment are generally empirial formulas based on experiments with speifi ranges of hydrodynami and sediment onditions (e.g. Bailard 1981; Dibajnia and Watanabe 199; Ribberink 1998; Nielsen 6; Van Rijn 7). For example, many models have foused on veloity-skewed ( nd order Stokes) waves and do not take into aount the transport effets resulting from aeleration skewness (sawtooth shaped waves). Other models whih are speifially developed for high energy flat bed sheet-flow onditions are not able to predit transport rates under more moderate onditions when the bed is generally overed with ripples. Furthermore, models that are based on a quasi-steady assumption between the transport and veloity or bed shear stress are not able to predit transport rates that are affeted by phase differenes between the veloity and onentration fields. Under ertain onditions, these so-alled phase lag effets an dominate the transport, both in the rippled bed and sheet flow regimes. Moreover, nearly all the onventional wave-indued sand transport models an essentially only be applied to horizontally uniform osillatory flows; they do not aount for effets related to progressive surfae waves that further influene the net transport (Ribberink et al. ). he limited apabilities of many existing models are, to some extent, the result of the limited range of onditions ontained in the dataset that have been used for the development of these models. For this reason Van der Werf et al. (9) brought together existing transport rate measurements from a number of failities overing a wide range of wave and wave-urrent onditions. he database has reently been signifiantly extended with new transport data for aeleration-skewed osillatory flows (Van der A et al. 1; Silva et al. submitted) and data for progressive surfae waves obtained in a large wave flume (Shretlen et al. 9). his database is used for the development and alibration of the present pratial sand transport model. his work has been part of the SANOSS ollaborative researh projet. Initial stages of the development of the present transport model were reported by Van der Werf et al. (7). his paper presents the final model developed during the SANOSS projet. Model development has speifially foused on inluding the effets of aeleration skewness and progressive surfae waves. he paper is organized as follows. In the next setion the net transport formula is presented, with subsetions dediated to the aeleration skewness and progressive surfae wave effets. he following setion illustrates the performane of the new formula, inluding its apabilities to predit transport rates for steady flow onditions. he onlusions are presented in the final setion. 1 Shool of Engineering, University of Aberdeen, AB4 3UE, Aberdeen, UK Water Engineering and Management, University of wente, PO Box 143, Enshede, he Netherlands 3 Marine and Coastal Systems, Deltares (formerly Delft Hydraulis), P.O. Box 177, 6 MH, Delft, he Netherlands 1

2 COASAL ENGINEERING 1 MODEL Model onept he net transport model is based on a modified version of the half-yle model onept initially proposed by Dibajnia and Watanabe (199). In this onept the wave-averaged transport rate is essentially the differene between the amount of sand transported during the positive rest half yle and the amount of sand transported during the negative trough half yle. Unsteady phase lag effets are taken into aount by aounting for two ontributions to the amount of sand transport during eah half yle: sand entrained and transported during the present half yle and sand entrained during the previous half yle whih is transported during the present half yle, the latter being the phase lag ontribution. Eah half yle, the magnitude of the phase lag ontribution is governed by a phase lag parameter whih is the ratio of the fall time of the sand partiles and the period of the orresponding half yle, and thus represents the tendeny for a partile to remain in suspension. he main driving mehanism in Dibajnia and Watanabe s (199) model is the near bed (free-stream) veloity. he new model proposed in the remainder of this paper inludes the following main modifiations to Dibajnia and Watanabe s (199) original formulation: bed shear stress is the main driving parameter; the phase lag effets related to flow unsteadiness are modeled in a different way; the effets of the wave shape (veloity- and aeleration skewness) are inorporated; effets speifially related to progressive surfae waves are aounted for; the model is apable of dealing with waves and urrents under an angle. hese new modifiations will be presented in the three setions. First a desription of the basi model will be given, apable of dealing with veloity-skewed flows and urrents, followed by a setion on the additional modifiations regarding aeleration skewness. he last setion presents the modifiations related to progressive surfae waves. Although the omplete model is able to deal with waves and urrents under arbitrary angle, for brevity reasons we will fous in the present paper only on waves and ollinear urrents. For the omplete desription of the new formula in vetor notation, see Ribberink et al. (1). Model Formulation for Veloity-Skewed Flow plus Current Fig. 1 illustrates a near-bed veloity time-series at the edge of the wave boundary layer due to ombined wave-urrent motion: ut () = u + ut () (1) δ with u δ the net urrent at z = δ alulated from the measured urrent veloity by assuming a logarithmi veloity profile; in the present model we have taken δ =. m. Furthermore it is assumed that the orbital veloity time-series, ut (), and therefore the veloity harateristis ( u, u t, u rms, t, u, tu ), are known a priori. A harateristi orbital veloity amplitude û and harateristi orbital exursion amplitude a for the omplete wave yle are alulated from: uˆ = u () rms u ˆ aˆ = (3) π For the purpose of alulating the shear stress eah half yle (see next setion) we use a representative orbital veloity for the wave rest and wave trough, defined as the root-mean-square veloity assuming a sinusoidal flow: 1 u r = u (4) 1 u tr = u t (5) he harateristi half-yle veloities in the ase of an osillatory flow with urrent now beome:

3 COASAL ENGINEERING 1 3 u(t) u u δ u t t u tu t Figure 1. Example veloity time-series used as input to the formula. he subsripts and t represent the rest and trough diretion respetively. (6) ur = ur + u δ utr = utr + u δ (7) he degree of veloity skewness is expressed through the parameter R: R u = u + u t Similarly, the degree of aeleration skewness is expressed through the parameter β for whih: (8) u β = u + u t (9) with u and u t the amplitudes of the horizontal flow aeleration in rest and trough diretion respetively. For the onditions presented in the present paper R and β were known from measurements, and therefore the periods, t, u, tu ould be determined from standard shapes for veloity skewness ( nd order Stokes) and aeleration skewness (sawtooth time-series, see Ribberink et al. 1). However, for wider appliation the hydrodynami input parameters need to be derived from loal wave parameters and bathymetry. It is suggested to use the set of empirial formulae proposed by Elfrink et al. (6), whih enables the alulation of the near bed veloity time-series, inluding veloity and aeleration skewness, based on the wave height and period, water depth, and loal bed slope. he dimensionless net transport rate is given by: q Φ= s = 3 ( s 1) gd5 θ θ θ ( Ω +Ω ) + θ ( Ω +Ω ) t t t tt t θt θ t (1) where q s is the volumetri net transport rate per unit width, s = ρ s /ρ where ρ s and ρ are the densities of sand and water respetively, g is aeleration due to gravity, d 5 is the sand median diameter and θ the Shields parameter. here are four ontributions to the net sand transport: Ω represents the sand load that is entrained during the wave rest period and transported during the rest period, Ω t represents the sand load that is entrained during the wave rest period and transported during the trough period, Ω tt represents the sand load that is entrained during the wave trough period and transported during the trough period,

4 4 COASAL ENGINEERING 1 Ω t represents the sand load that is entrained during the wave trough period and transported during the rest period hese half-yle sand loads are multiplied by θ, representing a (non-dimensional) near-bed (frition) veloity, to obtain a non-dimensional half-yle transport rate. Both half-yle transport rates are weighted with their duration relative to the wave period ( / and t /, respetively). he load ontributions are alulated in the following manner: Ω Ω if P 1 = 1 Ω if P > 1 P (11) if P 1 Ω t = ( P 1) Ω if P > 1 P (1) Ωt if Pt 1 Ω tt = 1 Ω t if Pt > 1 Pt (13) if Pt 1 Ω t = ( Pt 1) Ω t if Pt > 1 Pt hus, in the equations above when the phase lag parameter P exeeds the ritial value of 1, there is an exhange of sand from one half yle to the next. he sand load entrained in the flow during eah half yle depends on the Shields parameter as follows (with subsript i either for rest or t for trough): (14) if θi θr Ω i = n m( θi θr ) if θi > θr (15) where the ritial Shields number, θ r, is alulated following Soulsby (1997) and the oeffiients m and n are alibration oeffiients. he phase lag parameter P i represents the ratio of a representative sediment stirring height and the sediment settling distane: if r w i s i = δsi αs if η = w i s P α η η > (ripple regime) (sheet flow regime) where η is the ripple height and δ si the sheet flow layer thikness. he sediment settling veloity, w s, is alulated aording to Soulsby s (1997) method assuming a grain size of suspended sand d s =.8d 5. he oeffiients α s and α r are used for model alibration, as desribed hereafter. (16) he Shields parameter for osillatory flow plus urrent is alulated following: θ = i 1 f u u wδ ir ir ( s 1) gd 5 (17) Following Madsen and Grant (1976) the frition fator f wδi is alulated as a linear ombination of the wave frition f wi fator and the urrent frition fator f δ :

5 COASAL ENGINEERING 1 5 ( 1 α) f = α f + f (18) wδ δ w with σ uδ α = σ u + uˆ δ (19) where σ = 3 is a alibration fator. he wave frition fator is defined following Swart (1974): f f w w.19 aˆ aˆ =.51exp 5.1 for > ksw k sw aˆ =.3 for k he wave roughness height k sw inludes the mobile bed roughness for sheet flow onditions and an additional form roughness in the presene of ripples: sw 5 5 sw () k = max{ d, d [ μ + 6( θ 1)]} + p. η / λ (1) with λ is the ripple length and p a alibration parameter p =.4. In the ase of sheet flow onditions the bed roughness is solved iteratively, sine the mean absolute Shields parameter θ depends on the bed roughness k sw (Ribberink, 1998). he parameter μ is introdued to reate an inreased roughness and sand load for fine sand onditions (d 5 <.15mm) and results from alibration tests. For fine sands μ = 6 whih linearly redues to μ = 1 for medium and oarser sands (d 5 >.mm). Using μ = 6 the predited roughness heights for fine sands still fall within the range of measured roughness heights by Wilson (1989) whih formed the basis of Ribberink s (1998) formulation (for more details see Ribberink et al., 1). he urrent-related frition fator is alulated assuming a logarithmi veloity profile: f δ.4 = ln ( 3 δ ksδ ) in whih the urrent-related roughness k sδ is alulated in a similar way as in Eq. 1: sδ 9 5 () k = max{3 d, d [ μ + 6( θ 1)]} + p. η / λ (3) Finally, the sheet flow layer thikness δ si in Eq. 16 is alulated following Dohmen-Janssen (1999): δ d si 5 5 θwi if d5.15 mm 3 ( 13 5) = 5 ( 1 d ) θ (..15) wi if.15 mm < d5 <. mm 13 θwi if d5. mm his equation differs slightly from Dohmen-Janssen s (1999) original formulation sine the multiplier for fine sands is redued from 35 in the original formulation to 5 in Eq. 4 in order to ompensate for the inreased mobile bed roughness for fine sands in the sheet-flow regime. In the development and alibration of the model using laboratory data, the measured ripple dimensions were used as input to avoid any unertainty related to applying a ripple preditor. In all other appliations of the model the ripple dimensions are predited using the ripple preditor of O Donoghue et al. (6), in whih the ripple height and ripple length are a funtion of the grain size d 5 and maximum mobility number Ψ max (= u /( s 1) gd 5 ). In the present model the ripple preditor is adjusted to allow for a smooth transition between the roughness in the ripple regime (Ψ max <19) and the sheet flow regime (Ψ max 4). (4)

6 6 COASAL ENGINEERING 1 1 fine sand medium sand 1 fine sand medium sand Φ (-) Φ (-) -1-1 (a) u rms (m/s) (b) u rms (m/s) Figure : Non-dimensional transport rate as funtion of u rms for a veloity-skewed osillatory flow with = 6.5s, R =.6 and β =.5 and fine sand with d 5 =.13mm and medium sand with d 5 =.5mm: (a) flat bed onditions; (b) with ripple preditor swithed on. Fig. illustrates the behaviour of the predited (non-dimensional) transport rates with u rms for a typial fine and medium sand. o highlight the effet of ripples on the transport rates, Fig. a shows the preditions by assuming a flat bed over the entire u rms range (ripple preditor swithed off), while in Fig. b the ripple regime is inluded in the transport rate preditions (ripple preditor swithed on). Fig. a shows that for medium sand the transport shows a quasi-steady behaviour, i.e. inreasing transport with inreasing veloity. For fine sand on the other hand, phase-lags effets in the sheet-flow layer dominate the transport rate, whih hanges diretion and beomes inreasingly larger in the negative diretion with inreasing veloity. Introduing ripples hanges the transport rate diretion in the lower u rms range for medium and fine sands, sine for both sands ripple-related phase-lag effets dominate the net transport. he behaviour illustrated in Fig. b is in agreement with observations from laboratory experiments as may be inferred from Van der Werf et al. (9). Aeleration skewness effets For osillatory flows with aeleration skewness the following additional transport mehanisms are inluded in the model: 1. Inreased bed shear stress asymmetry with inreasing aeleration skewness.. Inrease/derease of sediment settling time during the rest/trough period with inreasing aeleration skewness. 3. Inrease/derease of the effetive travel distane of the suspended load during rest/trough period. Inreasing aeleration skewness leads to larger peak bed shear stress under the strongly aelerating (rest) half-yle, beause the boundary layer has less time to develop before maximum veloity is reahed. Aordingly, under the weakly aelerating (trough) half yle the peak bed shear stress is redued beause the boundary layer has muh more time to grow until maximum veloity is reahed (Nielsen 199; Van der A et al. 8). his effet is inluded in the present model by defining separate frition fators for the wave rest and trough following the approah by Silva et al. (6): f wi f wi.19 iu aˆ i aˆ =.51exp 5.1 for k > sw ksw aˆ =.3 for k sw (5) For example, for a forward-leaning rest half-yle, ( u / ) < 1, whih leads to a larger frition fator and hene bed shear stress ompared to the equivalent symmetri (sinusoidal) half yle for

7 COASAL ENGINEERING 1 7 whih ( u / ) = 1 (thus Eq. 5 redues to Eq. ). he hoie of the power is fairly arbitrary, however Van der A (1) has shown that it yields very good agreement with the measurements of bed shear stress for aeleration-skewed flow. Inorporating this enhaned bed shear stress for aeleration-skewed flow affets two primary transport proesses in the model. Firstly, even though the magnitude of the free-stream veloity is equal in both diretions, an asymmetry is reated in the amount of sand that is entrained eah half wave yle (Eq. 15). Seondly, beause within the boundary layer the maximum veloity is larger under the rest half yle ompared to the trough half yle (Van der A et al., 8), the same amount of sand will travel further under the rest half yle ompared to the trough half yle. his effet is implied in the transport equation (Eq. 7), sine in the new model the travel distane is assoiated with the frition veloity (~ θ ) rather than the free-stream veloity. o aount for the inreased settling time during the rest half yle and dereased settling time during the trough half yle aused by aeleration skewness, the phase lag parameter given by Eq. 16 is replaed by: r ( i iu ) ws i = δsi αs if η = ( i iu ) ws P α η if η > (ripple regime) (sheet flow regime) thus reognising that with inreasing (forward leaning) aeleration skewness the settling time during the rest half yle inreases, leading to smaller P and the settling time during the trough half yle dereases, leading to larger P t. In ase of no aeleration skewness ( u ) = and ( t tu ) = t and Eq. 6 redues to Eq.16. he amounts of sand that are arried over to the next half yle, Ω t and Ω t, are further affeted by aeleration skewness, sine the sand remaining in suspension after half a wave yle is transported further when followed by a steep front half-wave yle ompared to a gradual front half-wave yle, sine for a given (short) duration after flow reversal, Δt, the travel distane uδt is larger. his effet is aptured by inreasing/dereasing the travel distane of the phase-lag loads Ω t respetively Ω t with inreasing aeleration skewness. he transport equation is therefore extended as follows: (6) q Φ= s = 3 ( s 1) gd5 θ t θt θ Ω + Ω t + θt t Ω tt + Ωt u θ tu θt (7) Again, when there is no aeleration skewness the effet disappears and Eq. 7 reverts bak to Eq. 1. he transport behaviour for a typial purely aeleration-skewed flow (i.e. sawtooth shaped) is shown in Fig. 3. For flat bed onditions (Fig. 3a) both the fine and medium sand transport rates inrease with inreasing u rms, albeit that the fine sand transport rates are muh larger ompared to the medium sand transport rates. In part this is due to the enhaned roughness for fine sands, but the main mehanism responsible are the phase lag effets related to aeleration skewness, whih now augment the positive net transport. his is in strong ontrast to the veloity-skewed flows (Fig. ), where the positive fine sand transport rates diminish and even beome largely negative. Similarly, introduing ripples (Fig. 3b) inreases the positive transport rates for both sands, again, in part due to the enhaned roughness but also due to ripple-related phase lag effets now augmenting the positive transport rates. Although no observations exist of transport rates by aeleration skewed flow in the ripple regime, the behaviour predited by model and shown in Fig. 3 for both sand sizes in the sheet flow regime agrees well with the transport behaviour for aeleration skewed flows measured by Van der A et al. (1).

8 8 COASAL ENGINEERING 1 1 fine sand medium sand 15 1 fine sand medium sand 5 Φ (-) Φ (-) (a) u rms (m/s) -15 (b) u rms (m/s) Figure 3: Non-dimensional transport rate as funtion of u rms for a aeleration-skewed osillatory flow with = 6.5s, β =.75, R =.5: (a) flat bed onditions; (b) with ripple preditor swithed on. Surfae waves For progressive surfae waves the influene of three additional flow mehanisms is brought into the transport model: 1. he wave-reynolds stress,. he Lagrangian grain motion, whih differs during the rest and trough period. 3. he vertial orbital veloity affeting the grain settling veloity near the bed. Following the approah as suggested by Nielsen (6), the wave-reynolds stress is added as additional (onshore) omponent to the Shields parameter for the wave rest and wave trough (subsript sw denotes surfae waves): θ = θ + θ (8) i,sw i wre he wave Reynolds Shields parameter, θ wre as follows: θ wre τ wre = ρ( s 1) gd with the wave Reynolds stress following Nielsen (6): 5 wδ 3 τ ˆ wre ρ αwu, aounts for the streaming related bed shear stress (9) f = (3) in whih α w = 4/(3π) =.44 and is the wave speed. his additional stress enhanes the rest shear stress but redues the trough shear stress and thus has a similar influene as an inrease in wave asymmetry (veloity skewness). For surfae waves sediment grains move with the wave during the wave rest and against the wave during the wave trough (Lagrangian motion). In this way they experiene a longer rest period,sw (= + Δ ) and therefore a longer onshore travel distane, but also a shorter trough period t,,sw (= t Δ t ) and thus a shorter offshore travel distane. As a result this will lead to an inrease of the net sand transport in the diretion of wave propagation (generally onshore). Δ depends on the ratio of the wave speed and the horizontal grain displaement during the half wave-yle (orbital diameter) d g : d g Δ = (31) Assuming a sinusoidal wave shape for the half-yle horizontal grain motion, the rest-period extension an be written as follows (for the derivation see Ribberink et al., 1):

9 COASAL ENGINEERING dg Δ = = ζ uˆ π During the wave trough, the period and orbital diameter are redued and the following expression follows: 1 dg t Δ = = ζ uˆ π + herein ζ is the ratio of the horizontal grain-veloity amplitude and the free-stream veloity amplitude. he magnitude of this ratio is estimated from detailed veloity measurements in the sheet flow layer arried out under full-sale surfae waves in a large wave flume (GWK, Hannover). It is shown for a range of onditions that the ratio is more or less onstant and an be roughly estimated as ζ =.55 (see Ribberink et al. 1). his value is adopted in the present model. he (Lagrangian) half-yle duration in the ase of surfae waves are now alulated using: (3) (33),sw = +Δ (34) t,sw = t Δ t (35) For surfae waves a vertial orbital veloity is present whih may affet the settling of grains. Although this veloity is small near the bed, it an still be of the order of magnitude of the settling veloity, espeially for fine sand and high waves. In those onditions the vertial orbital motion therefore diretly influenes the phase lag proess (see also Kranenburg et al. 1). In the present transport model we inlude this effet by enhaning the settling veloity of the rest sediment load and reduing it for the trough sediment load. In the deeleration phase of the rest flow the settling proess dominates and is magnified by the presene of a downward vertial orbital flow. Similarly the settling of the trough load is redued due to the presene of an upward vertial orbital flow. We use non-linear ( nd -order Stokes) wave theory to determine the vertial orbital veloity amplitude at elevation z near the bed wz ˆ ( ). he sand settling veloities at rest and trough are now orreted with the vertial veloity at level r i as follows: w = w + wˆ ( r ) (36) s s with w = max( w wˆ ( r),) (37) st s t r i εη = εδsi if η > (ripple regime) if η = (sheet flow regime) Note that the trough settling veloity, w st, is not allowed to beome negative, its minimum is therefore limited to zero. he fator ε is a alibration fator; in the alibrated model ε = 3. Inorporating Eqs in the phase lag parameter results in: η αr if η > (ripple regime) ( i,sw iu,sw ) wsi Pi = (39) δsi αs if η = (sheet flow regime) ( i,sw iu,sw ) w si he rest/trough period orretion as well as the settling veloity orretion have a similar influene, i.e. a redution of P and an inrease of P t, both leading to a (further) shift of the mean sand transport in onshore diretion. (38)

10 1 COASAL ENGINEERING 1 he net transport rate now beomes: q Φ= s = 3 ( s 1) gd5 θ Ω + Ω θ + θ Ω + Ω θ,sw,sw t,sw t,sw,sw,sw t t,sw t,sw tt t u,sw θ,sw tu,sw θt,sw (4) Note that when the flow is horizontally uniform osillatory flow, as in a flow tunnel, the flow is essentially a wave with infinite wave length and wave speed and Eq. 4 reverts to Eq. 7 sine and wz ˆ ( ) =. 15 fine sand medium sand 15 fine sand medium sand 1 1 Φ (-) 5 Φ (-) 5-5 (a) u rms (m/s) -5 (b) u rms (m/s) Figure 4: Non-dimensional transport rate as funtion of u rms for a progressive surfae wave with = 6.5s, R =.6, β =.5, h = 3.5m: (a) flat bed onditions; (b) with ripple preditor swithed on. Fig. 4 shows the transport rate behaviour with u rms for a typial progressive surfae wave. Note that any degree of aeleration skewness is omitted in this example, in order to enable omparison with the equivalent veloity-skewed osillatory flow in Fig.. Comparison of Fig. 4a with Fig. a shows that the real wave effets lead to signifiantly larger transport rates for the medium sand, approximately by a fator 1.5, while for the fine sands the effets lead to a hange of diretion of the net transport rate, in agreement, at least qualitatively, with the transport rates observed in the large wave flume experiments of Ribberink et al. () and Shretlen et al. (9). Roughly between.45 < u rms <.65m/s it seems that the fine sand transport rate is onstant, whih appears to be the net effet of a derease in transport due to offshore-direted phase-lag effets being in balane with the inrease in transport related to the onshore-direted surfae wave effets. Phase-lag effets related to ripples still lead to small negative transport rates for low orbital veloities in Fig. 4b, although experimental evidene for this behaviour still has to be obtained. It may be inferred from these results together with the behaviour shown in Fig. 3b that the transport rates beome positive throughout the entire u rms range when a ertain degree of aeleration skewness is present in the wave orbital veloity. Current Alone When no waves are present and a steady flow is the only driving mehanism for sand transport (α = 1), the Shields parameter beomes the Shields parameter for urrent alone: 1 fu δ δ θ δ = ( s 1) gd 5 (41) By applying Eq. 41 to the load formula (Eq. 15) the net transport rate in the diretion of the urrent beomes: q s Φ= = θδωδ 3 ( s 1) gd5 (4)

11 COASAL ENGINEERING 1 11 COMPARISON WIH MEASURED NE RANSPOR RAES able 1 presents an overview of the range of onditions overed by the database used for alibration of the new model. he database ontains measured net transport rates for a wide range of full-sale onditions in both the rippled bed and sheet flow regime, inluding osillatory flows with veloity skewness or aeleration skewness (or a ombination of both), osillatory flows with superimposed urrents and non-breaking (shoaling) surfae waves. he alibration proedure involved the following steps. First, the oeffiients in the phase lag parameter α s and α r were tuned to obtain the highest orrelation between measured and predited transport rates for the sheet flow regime and rippled bed regime respetively. he proportionality onstant m was found from least-square fitting a straight line with zero interept through all the measured and omputed transport rates. his proedure was repeated for different values of the proportionality onstant n. Next, the following oeffiients were alibrated to obtain the best agreement between preditions and measurements for seleted subdatasets: 1) the relative weight of the urrent frition fator and wave frition fator through oeffiient σ, ) the form roughness oeffiient for rippled bed onditions p, 3) the oeffiient for inreased mobile bed roughness for fine sands μ, and 4) for real wave onditions the level at whih the vertial orbital is alulated through oeffiient ε. Following the alibration of these data subsets the oeffiients m and n were realibrated using the entire database, resulting in m = 9.48 and n = 1. and phase-lag parameter oeffiients α r = 9.3 and α s = 8. able 1. Overview of range of flow onditions ontained in the SANOSS net transport rate database. Maximum veloity u max Wave period Median Grain size d 5 Degree of veloity skewness R Degree of aeleration skewness β Net urrent veloity u δ Number of veloity skewed flows Number of aeleration skewed flows Number of os. flows with urrent Surfae wave onditions otal number of onditions m/s s mm m/s Fig. 5 shows the omparison between all 6 measured transport rates in the database and the preditions by the alibrated model. It is shown that for nearly all the onditions the orret transport diretion is predited, the exeptions being a few onditions near the origin for whih the measured transport rates are very low (and the measurement errors are the largest). he majority of the onditions are predited within a fator of the measurements. he model performane for eah of the subgroups is summarized in able, whih lists the number of onditions ontained in eah data subset, N, the orrelation oeffiient, r, and the perentages of the preditions falling within a fator and fator 5 of the measurements. Overall very good agreement is obtained with 77% of the preditions falling within a fator of the measurements. It was mentioned before that during the development and alibration of the model, we have onstantly used the measured ripple dimensions as input to the model. able therefore also shows the performane of the model when the ripple dimensions are predited using O Donoghue et al. s (6) method. Obviously, inorporating a ripple preditor reates an extra soure of unertainty in the formula whih is refleted in some redution of the performane. able. Quantitative performane measures for the data subsets. Data (sub)set N r % fa % fa 5 Veloity skewed os flows Aeleration-skewed os.flows Os. flows + urrent Progressive surfae waves All All with ripple preditor

12 1 COASAL ENGINEERING 1 Calulated transport rate ( 1 6 m /s) veloity skewed os. flow aeleration skewed os. flow osillatory flow + urrent surfae waves Measured transport rate ( 1 6 m /s) Figure 5: Model performane for all 6 onditions ontained in the database. he solid line indiates perfet agreement, the dashed lines a fator differene. he model uses measured ripple dimensions. Finally, to test the performane of the alibrated model for steady flow transport rates, we have ompared the model preditions with an independent dataset of bedload net transport rates for steady flows (Guy et al., 1966; Van den Berg, 1986; Nnadi and Wilson, 199). he same datasets have been used by Ribberink (1998) for model development but were not used in the development or alibration of the present formula. he ombined dataset omprises 137 onditions with urrent veloities ranging between.3-.3 m/s and median grain sizes between mm. he results and quantitative performane measures in Fig. 6 illustrate that the new formula also is quite apable of prediting the net transport rates in bedload dominated onditions for steady flows. In fat, these results are not entirely surprising sine, for steady flows, the present formula is very similar to Ribberink s (1998) formula, whih gave very good agreement for alibration oeffiients n = 1.17 and m = 1.4. CONCLUSIONS A new pratial sand transport model is developed for non-breaking waves, non-breaking waves with urrents and urrents alone. he model is developed and alibrated on the basis of a large dataset of measured net transport rates for a wide range of hydrodynami and sand onditions typial for oastal areas. he formula is semi-unsteady based on the half-wave yle onept and aounts for the bedload and suspended load within the wave boundary layer in the rippled bed regime and sheet flow regime; for steady flow the model aounts for bedload only. Data and new insights from flow tunnel and large wave flume experiments onduted within the SANOSS projet have been used to inlude the effets of aeleration skewness and the effets related to progressive surfae waves. When the experimentally observed ripple dimensions are used, the new formula is able to predit 77% of the transport rates within a fator of the measured transport rates, and 93% within a fator 5

13 COASAL ENGINEERING Guy et al. (1966) Van den Berg (1986) Nnadi & Wilson (199) Calulated transport rate (m /s) r =.87 % fa = 87 % fa5 = Measured transport rate (m /s) Figure 6: Model performane for steady flow onditions. of the measurements. When applied in ombination with a ripple preditor these figures redue to 65% and 85% respetively. A omparison with an independent dataset of 137 bedload transport rates for a wide range of steady flow onditions resulted in 87% within a fator and 99% within a fator 5 of the measurements. he model is urrently being implemented in a ross-shore morphologial model to further test its validity and ompare its performane with existing transport formulae. Future work is aimed at i) further improvement of modeling surfae wave effets, and ii) extending the model to irregular wave and breaking wave onditions. ACKNOWLEDGMENS his work is part of the SANOSS projet ( SANd ransport in OSillatory flows in the Sheetflow regime ) funded by the UK s EPSRC (GR/889/1) and SW in he Netherlands (CB.6586). he first author would also like to thank the Royal Aademy of Engineering in the UK for the award of an International ravel grant to assist with ICCE 1 attendane. REFERENCES Bailard, J.A., An energetis total load sediment transport model for a plane sloping beah, Journal of Geophysial Researh, 86(C11), 1,938-1,954. Dibajnia, M. and A. Watanabe Sheet flow under nonlinear waves and urrents, Proeedings of 3 rd International Conferene on Coastal Engineering, ASCE, Dohmen-Janssen, C.M Grain size influene on sediment transport in osillatory sheet flow, phase lags and mobile-bed effets. Ph.D. thesis, Delft University of ehnology, he Netherlands. Elfrink, B., D.M. Hanes, and B.G. Ruessink. 6. Parameterization and simulation of near bed orbital veloities under irregular waves in shallow water. Coastal Engineering, 53, Guy, H.P., D.B. Simons, and E.V. Rihardson Summary of alluvial hannel data from flume experiments , US Geologial Survey, Prof. Paper 46-I, Washington, DC. Kranenburg, W.M., J.S. Ribberink, and R.E. Uittenbogaard. 1. Sand transport by surfae waves: an streaming explain the onshore transport? Proeedings of 3 nd International Conferene on Coastal Engineering, Shanghai, China.

14 14 COASAL ENGINEERING 1 Madsen, O.S., and W.D. Grant Sediment transport in the oastal environment. MI Ralph M. Parsons Lab., Rep. 9, Cambridge, USA. O'Donoghue,., J.S. Douette, J.J. van der Werf, and J.S. Ribberink. 6. he dimensions of sand ripples in full-sale osillatory flows. Coastal Engineering, 53, Nielsen, P Coastal bottom boundary layers and sediment transport. Advaned Series on Oean Engineering, vol. 4. World Sientifi. Nielsen, P. 6. Sheet flow sediment transport under waves with aeleration skewness and boundary layer streaming, Coastal Engineering, 53, Nnadi, F.N., and K.C. Wilson Motion of ontat-load partiles at high shear stress, Journal of Hydrauli Engineering. ASCE 118 (1). Ribberink, J.S Bed-load transport for steady flows and unsteady osillatory flows. Coastal Engineering, 34, Ribberink, J.S., C.M. Dohmen-Janssen, D.M. Hanes, S.R. MLean and C. Vinent,. Near-bed sand transport mehanisms under waves. Proeedings 7 th International Conferene on Coastal Engineering. ASCE, pp Ribberink, J.S., D.A. van der A, R.H. Buijsrogge. 1. SANOSS transport model A new formula for sand transport under waves and urrents. Report SANOSS_U_IR3, University of wente, he Netherlands. Shretlen, J.J.L.M., J.S. Ribberink, and. O Donoghue. 9. Sand transport under full-sale surfae waves. Proeedings of Coastal Dynamis 9, World Sientifi, paper 13. Silva, P.S.,. Abreu, D.A. van der A, F. Sanho, B.G. Ruessink, J.J. van der Werf, and J.S. Ribberink. submitted. Sediment transport in non-linear skewed osillatory flows: the ranskew experiments. Journal of Hydrauli Researh. Soulsby, R.L Dynamis of marine sands, homas elford Publiations, London, pp. 7. Swart, D.H Offshore sediment transport and equilibrium beah profiles. Delft Hydraulis Laboratory Publiation 131, Delft Hydraulis, he Netherlands. Van den Berg, J.H Aspets of sediment and morphodyanmis of subtidal deposits of the Oostersehlde (the Netherlands), Rijkswaterstaat, Communiations no. 43. Van der A, D.A. 1. Effets of aeleration skewness on osillatory boundary layers and sheet flow sand transport. PhD hesis, University of Aberdeen. Van der A, D.A.,. O Donoghue, A.G. Davies and J.S. Ribberink. 8. Effets of aeleration skewness on rough bed osillatory boundary layer flow. Proeedings of 31 st International Conferene on Coastal Engineering, World Sientifi, Van der A, D.A.,. O Donoghue, and J.S. Ribberink. 9. Sheet flow sand transport proesses in osillatory flows with aeleration skewness. Proeedings of Coastal Dynamis 9, World Sientifi, paper no 133. Van der A, D.A.,. O Donoghue and J.S. Ribberink. 1. Measurements of sheet flow transport in aeleration-skewed osillatory flow and omparison with pratial formulations, Coastal Engineering, 57, Van der Werf, J.J., J.S. Ribberink and. O Donoghue. 7. Development of a new pratial model for sand transport indued by non-breaking waves and urrents. Coastal Sediments 7, ASCE, Van der Werf, J.J., J.J.L.M. Shretlen, J.S. Ribberink, and. O Donoghue. 9. Database of fullsale laboratory experiments on wave-driven sand transport proesses, Coastal Engineering, 56, Van Rijn, L.C. 7. Unified view of sediment transport by urrents and waves. I: Initiation of motion, bed roughness, and bed load transport. Journal of Hydrauli Engineering, 33(6), Wilson, K.C Mobile-bed frition at high shear stress. Journal of Hydrauli Engineering, 115(6),