Coastal Engineering 55 (2008) 1102–1112

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Coastal Engineering

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Characteristics of turbulent boundary layers over a rough bed under saw-tooth wavesand its application to sediment transport

Suntoyo a,b,⁎, Hitoshi Tanaka b, Ahmad Sana c

a Department of Ocean Engineering, Faculty of Marine Technology, Institut Teknologi Sepuluh Nopember (ITS), Surabaya 60111, Indonesiab Department of Civil Engineering, Tohoku University, 6-6-06 Aoba, Sendai 980-8579, Japanc Department of Civil and Architectural Engineering, Sultan Qaboos University, P.O. Box 33, AL-KHOD 123, Oman

⁎ Corresponding author. Department of Civil EngineerAoba, Sendai 980-8579, Japan.

E-mail addresses: suntoyo@oe.its.ac.id, suntoyo@kasetanaka@tsunami2.civil.tohoku.ac.jp (H. Tanaka), sana@sq

0378-3839/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.coastaleng.2008.04.007

A B S T R A C T

A R T I C L E I N F O

Article history:

A large number of studies Received 14 August 2007Received in revised form 30 March 2008Accepted 4 April 2008Available online 21 May 2008

Keywords:Turbulent boundary layersSheet flowSediment transportSkew wavesSaw-tooth waves

have been done dealing with sinusoidal wave boundary layers in the past.However, ocean waves often have a strong asymmetric shape especially in shallow water, and net ofsediment movement occurs. It is envisaged that bottom shear stress and sediment transport behaviorsinfluenced by the effect of asymmetry are different from those in sinusoidal waves. Characteristics of theturbulent boundary layer under breaking waves (saw-tooth) are investigated and described through bothlaboratory and numerical experiments. A new calculation method for bottom shear stress based on velocityand acceleration terms, theoretical phase difference, φ and the acceleration coefficient, ac expressing thewave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determinedempirically from both experimental and baseline k–ω model results. The new calculation has shown betteragreement with the experimental data along a wave cycle for all saw-tooth wave cases compared by otherexisting methods. It was further applied into sediment transport rate calculation induced by skew waves.Sediment transport rate was formulated by using the existing sheet flow sediment transport rate data underskew waves by Watanabe and Sato [Watanabe, A. and Sato, S., 2004. A sheet-flow transport rate formula forasymmetric, forward-leaning waves and currents. Proc. of 29th ICCE, ASCE, pp. 1703–1714.]. Moreover, thecharacteristics of the net sediment transport were also examined and a good agreement between theproposed method and experimental data has been found.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Many researchers have studied turbulent boundary layers andbottom friction through laboratory experiments and numericalmodels. The experimental studies have contributed significantlytowards understanding of turbulent behavior of sinusoidal oscillatoryboundary layers over smooth and rough bed (e.g., Jonsson and Carlsen,1976; Tanaka et al., 1983; Sleath, 1987, Jensen et al., 1989). Thesestudies explained how the turbulence is generated in the near-bedregion either through the shear layer instability or turbulence burstingphenomenon. Such studies included measurement of the velocityprofiles, bottom shear stress and some included turbulence intensity.An extensive series of measurements and analysis for the smooth bedboundary layer under sinusoidal waves has been presented by Hinoet al. (1983). Jensen et al. (1989) carried out a detailed experimentalstudy on turbulent oscillatory boundary layers over smooth as well asrough bed under sinusoidal waves. Moreover, Sana and Tanaka (2000)

ing, Tohoku University, 6-6-06

n1.civil.tohoku.ac.jp (Suntoyo),u.edu.om (A. Sana).

l rights reserved.

and Sana and Shuy (2002) have compared the direct numericalsimulation (DNS) data for sinusoidal oscillatory boundary layer onsmooth bed with various two-equation turbulence models and, aquantitative comparison has been made to choose the best model forspecific purpose. However, these models were not applied to predictthe turbulent properties for asymmetric waves over rough beds.

Many studies on wave boundary layer and bottom friction asso-ciated with sediment movement induced by sinusoidal wave motionhave been done (e.g., Fredsøe and Deigaard, 1992). These studies haveshown that the net sediment transport over a complete wave cycle iszero. In reality, however oceanwaves often have a strongly non-linearshape with respect to horizontal axes. Therefore it is envisaged thatturbulent structure, bottom shear stress and sediment transport be-haviors are different from those in sinusoidal waves due to the effectof acceleration caused by the skew-ness of the wave.

Tanaka (1988) estimated the bottom shear stress under non-linearwave by modified stream function theory and proposed formula topredict bed load transport except near the surf zone in which theacceleration effect plays an important role. Schäffer and Svendsen(1986) presented the saw-tooth wave as a wave profile expressingwave-breaking situation. Moreover, Nielsen (1992) proposed a bottomshear stress formula incorporating both velocity and acceleration

Fig. 1. Definition sketch for saw-tooth wave.

1103Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

terms for calculating sediment transport rate based on the King's(1991) saw-tooth wave experiments with the phase difference of 45°.Recently, Nielsen (2002), Nielsen and Callaghan (2003) and Nielsen(2006) applied a modified version of the formula proposed by Nielsen(1992) and applied it to predict sediment transport rate with variousexperimental data. They have shown that the phase differencebetween free stream velocity and bottom shear stress used to evaluatethe sediment transport is from 40° up to 51°. Whereas, manyresearchers e.g. Fredsøe and Deigaard (1992), Jonsson and Carlsen(1976), Tanaka and Thu (1994) have shown that the phase differencefor laminar flow is 45° and drops from 45° to about 10° in theturbulent flow condition. However, Sleath (1987) and Dick and Sleath(1991) observed that the phase difference and shear stress weredepended on the cross-stream distance from the bed, z for the mobileroughness bed. It is envisaged that the phase difference calculated atbase of sheet flow layer may be very close to 90°, while the phasedifference just above undisturbed level may only 10–20° and thephase difference about 51° as the best fit value obtained by Nielsen(2006) may be occurred at some depth below the undisturbed level.

More recently, Gonzalez-Rodriguez and Madsen (2007) presenteda simple conceptual model to compute bottom shear stress underasymmetric and skewed waves. The model used a time-varyingfriction factor and a time-varying phase difference assumed to be thelinear interpolation in time between the values calculated at the crestand trough. However, this model does not parameterize the fluidacceleration effect or the horizontal pressure gradients acting on thesediment particle. Moreover, this model under predicted most ofWatanabe and Sato's (2004) experimental data induced by skewwaves or acceleration-asymmetric waves.

Hsu and Hanes (2004) examined in detail the effects of waveprofile on sediment transport using a two-phase model. They haveshown that the sheet flow response to flow forcing typical ofasymmetric and skewed waves indicates a net sediment transport inthe direction of wave propagation. However, for a predictive near-shore morphological model, a more efficient approach to calculate thebottom shear stress is needed for practical applications. Moreover,investigation of a more reliable calculation method to estimate thetime-variation of bottom shear stress and that of turbulent boundarylayer under saw-toothwave over rough bed have not been done as yet.Bottom shear stress estimation is the most important step, which isrequired as an input to the practical sediment transport models.Therefore, the estimation of bottom shear stress from a sinusoidalwave is of limited value in connection with the sediment transportestimation unless the acceleration effect is incorporated therein.

In the present study, the characteristics of turbulent boundary layersunder saw-tooth waves are investigated experimentally and numeri-cally. Laboratory experiments were conducted in an oscillating tunnelover rough bed with air as the working fluid and smoke particles astracers. The velocity distributions were measured by means of LaserDoppler Velocimeter (LDV). The baseline (BSL) k–ω model proposed byMenter (1994) was also employed to and the experimental data wasused for model verification. Moreover, a quantitative comparisonbetween turbulence model and experimental data was made. A newcalculation method for bottom shear stress is proposed incorporatingboth velocity and acceleration terms. In this method a new accelerationcoefficient, ac and a phase difference empirical formula were proposedto express theeffect ofwave skew-nesson thebottomshear stressundersaw-toothwaves. The proposed ac constantwas determined empiricallyfrom both experimental and the BSL k–ω model results. The newcalculation method of bottom shear stress under saw-tooth wave wasfurther applied to calculate sediment transport rate induced by skew orsaw-tooth waves. Sediment transport rate was formulated by using theexisting sheet flow sediment transport rate data under skew waves byWatanabe andSato (2004).Moreover, the acceleration effect on both thebottom shear stress and sediment transport under skew waves wereexamined.

2. Experimental study

2.1. Turbulent boundary layer experiments

Turbulent boundary layer flow experiments under saw-toothwaves were carried out in an oscillating tunnel using air as theworking fluid. The experimental system consists of the oscillatoryflow generation unit and a flow-measuring unit. The saw-tooth waveprofile used is as presented by Schäffer and Svendsen (1986) bysmoothing the sharp crest and trough parts. The definition sketch forsaw-tooth wave after smoothing is shown in Fig. 1. Here, Umax is thevelocity at wave crest, T is wave period, tp is time interval measuredfrom the zero-up cross point to wave crest in the time variation of freestream velocity, t is time and α is the wave skew-ness parameter. Thesmaller α indicate more wave skew-ness, while the sinusoidal wave(without skew-ness) would have α=0.50.

The oscillatory flow generation unit comprises of signal controland processing components and piston mechanism. The pistondisplacement signal is fed into the instrument through a PC. Inputdigital signal is then converted to corresponding analog data througha digital–analog (DA) converter. A servomotor, connected through aservomotor driver, is driven by the analog signal. The piston mecha-nism has been mounted on a screw bar, which is connected to theservomotor. The feed-back on piston displacement, from one instantto the next, has been obtained through a potentiometer that com-pared the position of the piston at every instant to the input signal,and subsequently adjusted the servomotor driver for position at thenext instant. The measured flow velocity record was collected bymeans of an A/D converter at 10 millisecond intervals, and the meanvelocity profile variation was obtained by averaging over 50 wavecycles. According to Sleath (1987) at least 50wave cycles are needed tosuccessfully compute statistical quantities for turbulent condition. Aschematic diagram of the experimental set-up is shown in Fig. 2.

The flow-measuring unit comprises of a wind tunnel and onecomponent Laser Doppler Velocimeter (LDV) for flow measurement.Velocity measurements were carried out at 20 points in the verticaldirection at the central part of the wind tunnel. The wind tunnel has alength of 5 m and the height and width of the cross-section are 20 cmand 10 cm, respectively (Fig. 2). These dimensions of the cross-sectionof wind tunnel were selected in order to minimize the effect ofsidewalls on flow velocity. The triangular roughness having a height of5 mm (a roughness height, Hr=5 mm) and 10 mm width was pastedover the bottom surface of the wind tunnel at a spacing of 12 mmalong the wind tunnel, as shown in Fig. 3. Moreover, it was confirmedthat the velocity measurement at the center of the roughness and atthe flaking off region around the roughness has shown a similar flowdistribution as shown in Jonsson and Carlsen (1976).

These roughness elements protrude out of the viscous sub-layer athigh Reynolds numbers. This causes a wake behind each roughnesselement, and the shear stress is transmitted to the bottom by thepressure drag on the roughness elements. Viscosity becomes irrelevant

Table 1Experimental conditions for saw-tooth waves

Case T (s) Umax (cm/s) v (cm2/s) α am/ks Re S ks/zh

SK1 4.0 398 0.145 0.314 168.9 6.96×105 25.3 0.15SK2 4.0 399 0.147 0.363 169.3 6.89×105 25.4 0.15SK3 4.0 400 0.147 0.406 169.8 6.93×105 25.5 0.15SK4 4.0 400 0.151 0.500 169.8 6.75×105 25.5 0.15

Fig. 2. Schematic diagram of experimental set-up.

1104 Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

for determining either the velocity distribution or the overall drag onthe surface. And the velocity distribution near a rough bed for steadyflow is logarithmic. Therefore the usual log-lawcan be used to estimatethe time variation of bottom shear stress το(t) over rough bed as shownby previous studies e.g., Jonsson and Carlsen (1976), Hino et al. (1983),Jensen et al. (1989), Fredsøe and Deigaard (1992) and Fredsøe et al.(1999). Moreover, some previous studies (e.g., Jonsson and Carlsen,1976; Hino et al., 1983; Sana et al., 2006) also have shown that thevalues of bottom shear stress computed from the usual log-lawand themomentum integral methods gave a quite similar, especially by virtueof the phase difference in crest and trough values of the shear stress.Nevertheless, this usual log-lawmaybe under estimated by asmuch as20% up to 60% in accelerating flow and overestimated by as much as20% up to 80% in decelerating flow, respectively, for unsteady flow asshown by Soulsby and Dyer (1981). The usual log-law should bemodified by incorporating velocity and acceleration terms to estimatethe bed shear stress for unsteady flow, as given by Soulsby and Dyer(1981).

Experiments have been carried out for four cases under saw-toothwaves. The experimental conditions of present study are given inTable 1. The maximum velocity was kept almost 400 cm/s for all thecases. The Reynolds number magnitude defined for each case hassufficed to locate these cases in the rough turbulent regime. Here, v isthe kinematics viscosity, am/ks is the roughness parameter, ks,Nikuradse's equivalent roughness defined as ks=30zo in which zo isthe roughness height, am=Umax/σ, the orbital amplitude of fluid justabove the boundary layer, where, Umax, the velocity at wave crest, σ,the angular frequency, T, wave period, S (=Uo/(σzh)), the reciprocal ofthe Strouhal number, zh, the distance from the wall to the axis ofsymmetry of the measurement section.

Fig. 3. Definition sketch for roughness.

2.2. Sediment transport experiment

The experimental data from Watanabe and Sato (2004) foroscillatory sheet flow sediment transport under skew waves motionwere used in the present study. The flow velocity wave profile was theacceleration asymmetric or skew wave profile obtained from the timevariations of acceleration of first-order cnoidal wave theory byintegration with respect to time. These experiments consist of 33cases. Three values of the wave skew-ness (α) were used; 0.453, 0.400and 0.320. Moreover, the maximum flow velocity at free stream, Umax

ranges from 0.72 to 1.45 m/s. The sediment median diameters ared50=0.20 mm and d50=0.74 mm and the wave periods are T=3.0 s andT=5.0 s.

3. Turbulence model

For the 1-D incompressible unsteady flow, the equation of motionwithin the boundary layer can be expressed as

AuAt

¼ �1qApAx

þ 1qAsAz

ð1Þ

At the axis of symmetry or outside boundary layer u=U, therefore

AuAt

¼ AUAt

þ 1qAsAz

ð2Þ

For turbulent flow,

sq¼ v

AuAz

�PuVvV ð3Þ

The Reynolds stress �qPuVvV may be expressed as �q

PuVvV¼

qvt Au=Azð Þ, where νt is the eddy viscosity.And Eq. (3) became,

sq

vþ vtð ÞAuAz

ð4Þ

For practical computations, turbulent flows are commonly computedby the Navier–Stokes equation in averaged form. However, theaveraging process gives rise to the new unknown term representingthe transport of mean momentum and heat flux by fluctuatingquantities. In order to determine these quantities, turbulence modelsare required. Two-equation turbulence models are complete turbu-lence models that fall in the class of eddy viscosity models (modelswhich are based on a turbulent eddy viscosity are called as eddyviscosity models). Two transport equations are derived describingtransport of two scalars, for example the turbulent kinetic energy kand its dissipation ε. The Reynolds stress tensor is then computedusing an assumption, which relates the Reynolds stress tensor to thevelocity gradients and an eddy viscosity. While in one-equationturbulence models (incomplete turbulence model), the transportequation is solved for a turbulent quantity (i.e. the turbulent kineticenergy, k) and a second turbulent quantity is obtained from algebraicexpression. In the present paper the base line (BSL) k–ω model wasused to evaluate the turbulent properties to compare with the ex-perimental data.

Fig. 4. Mean velocity distribution for Case SK2 with α=0.363.

1105Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

The baseline (BSL) model is one of the two-equation turbulencemodels proposed byMenter (1994). The basic idea of the BSL k–ωmodelis to retain the robust and accurate formulation of theWilcox k–ωmodelin the near wall region, and to take advantage of the free streamindependence of the k–ε model in the outer part of boundary layer. Itmeans that this model is designed to give results similar to those of theoriginal k–ω model of Wilcox, but without its strong dependency onarbitrary free stream of ω values. Therefore, the BSL k–ω model givesresults similar to the k–ω model of Wilcox (1988) in the inner part ofboundary layer but changes gradually to the k–ε model of Jones andLaunder (1972) towards to the outer boundary layer and the free streamvelocity. In order to be able to perform the computations within one setof equations, the Jones–Laundermodel wasfirst transformed into the k–ω formulation. The blending between the two regions is done by ablending function F1 changing gradually from one to zero in the desiredregion. The governing equations of the transport equation for turbulentkinetic energy k and the dissipation of the turbulent kinetic energy ωfrom the BSL model as mentioned before are,

AkAt

¼ A

Azvþ vtrkxð ÞAk

Az

� �þ vt

AuAz

� �2

�bTxk ð5Þ

AxAt

¼ A

Azvþ vtrxð ÞAx

Az

� �þ g

AuAz

� �2

�bx2 þ 2 1� F1ð Þrx2 1xAkAz

AxAz

ð6Þ

From k and ω, the eddy viscosity can be calculated as

vt ¼ kx

ð7Þ

where, the values of the model constants are given as σkω=0.5,β⁎=0.09, σω=0.5, γ=0.553 and β=0.075 respectively, and F1 is ablending function, given as:

F1 ¼ arg 41

� � ð8Þ

where,

arg1 ¼ min max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik

0:09xz

r;500vz2x

!;4rx2kCDkxz2

" #ð9Þ

here, z is the distance to the next surface and CDkω is the positiveportion of the cross-diffusion term of Eq. (6) defined as

CDkx ¼ max 2rx21xAkAz

AxAz

;10�20� �

ð10Þ

Thus, Eqs. (2), (5) and (6) were solved simultaneously after nor-malizing by using the free stream velocity, U, angular frequency, σkinematics viscosity, ν and zh.

3.1. Boundary conditions

Non slip boundary conditions were used for velocity and turbulentkinetic energy on the wall (u=k=0) and at the axis of symmetry ofthe oscillating tunnel, the gradients of velocity, turbulent kinetic energyand specific dissipation rate were equated to zero, (at z=zh, ∂u/∂z=∂k/∂z=∂ω/∂z=0). The k–ωmodel provides a natural way to incorporate theeffects of surface roughness through the surface boundary condition.The effect of roughness was introduced through the wall boundarycondition of Wilcox (1988), in which this equation was originallyrecognized by Saffman (1970), given as follow,

xw ¼ UTSR=v ð11Þ

where ωw is the surface boundary condition of the specific dissipationω at the wall in which the turbulent kinetic energy k reduces to 0,

UT ¼ Fffiffiffiffiffiffiffiffiffiffiffiffijs0j=q

pis friction velocity and the parameter SR is related to

the grain-roughness Reynolds number, ks+=ks(U⁎/v),

SR ¼ 50kþS

� �2

for kþs b25 and SR ¼ 100kþs

for kþs z25 ð12Þ

The instantaneous bottom shear stress can be determined usingEq. (4), in which the eddy viscosity was obtained by solving thetransport equation for turbulent kinetic energy k and the dissipationof the turbulent kinetic energy ω in Eq. (7). While, the instantaneousvalue of u(z,t) and vt can be obtained numerically from Eqs. (1)–(7)with the proper boundary conditions.

3.2. Numerical method

A Crank–Nicolson type implicit finite-difference scheme was usedto solve the dimensionless non-linear governing equations. In order toachieve better accuracy near the wall, the grid spacing was allowed toincrease exponentially in the cross-stream direction to get fineresolution near the wall. The first grid point was placed at a distanceof Δz1=(r−1) zh/(rn−1), where r is the ratio between two consecutivegrid spaces and n is total number of grid points. The value of r wasselected such thatΔz1 should be sufficiently small in order tomaintainfine resolution near the wall. In this study, the value of Δz1 is givenequal to 0.0042 cm from thewall which correspond to z+=zU⁎/v=0.01.It may be noted that in k–εmodel where wall function method is usedto describe roughness the first grid point should be lie in thelogarithmic region and corresponding boundary conditions should beapplied for k and ε. In the k–ωmodel, as explained before the effect ofroughness can be simply incorporated using Eq. (11). In space 100 andin time 7200 steps per wave cycle were used. The convergence wasachieved through two stages; the first stage of convergence was basedon the dimensionless values of u, k and ω at every time instant duringawave cycle. Second stage of convergencewas based on themaximumwall shear stress in a wave cycle. The convergence limit was set to1×10−6 for both the stages.

4. Mean velocity distributions

Mean velocity profiles in a rough turbulent boundary layer undersaw-tooth waves at selected phases were compared with the BSL k–ωmodel for the cases SK2 and SK4 presented in Figs. 4 and 5, respectively.

Fig. 5. Mean velocity distribution for Case SK4 with α=0.500.

1106 Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

The solid line showed the turbulence model prediction while open andclosed circles showed the experimental data for mean velocity profiledistribution. The experimental data and the turbulencemodel show thatthe velocity overshoot is much influenced by the effect of accelerationand the velocity magnitude. The difference of the acceleration betweenthe crest and trough phases is significant. The velocity overshooting ishigher in the crest phase than the trough as shown at phase B and F forCase SK2 (α=0.363). As expected this difference is not visible forsymmetric case (Case SK4) (α=0.500). Moreover, the asymmetry of theflow velocity can be observed in phase A and E. Due to the higheracceleration at phase A the velocity overshooting is more distinguishedin the wall vicinity.

The BSL k–ωmodel could predict the mean velocity very well in thewholewave cycle of asymmetric case.Moreover, it predicted thevelocityovershooting satisfactorily (Fig. 4). For symmetrc case (Case SK4) aswell

Fig. 6. Turbulent intensity comparison between BSL k–ω m

the model prediction is excellent. A similar result was obtained by Sanaand Shuy (2002) using DNS data for model verification.

5. Prediction of turbulence intensity

The fluctuating velocity in x-direction u' can be approximatedusing Eq. (13) that is a relationship derived from experimental data forsteady flow by Nezu (1977),

uV¼ 1:052ffiffiffik

pð13Þ

where k is the turbulent kinetic energy obtained in the turbulencemodel.

Comparison made on the basis of approximation to calculate thefluctuating velocity by Nezu (1977)may not be applicable in thewholerange of cross-stream dimension since it is based on the assumption ofisotropic turbulence. This assumption may be valid far from the wall,where the flow is practically isotropic, whereas the flow in the regionnear the wall is essentially non-isotropic. The BSL k–ω model canpredict very well the turbulent intensity across the depth almost all atphases, but, near the wall underestimates at phases A, C, D and E (CaseSK2) and at phases A, C, D, E and H (Case SK3) as shown in Figs. 6 and 7,respectively. However, the model qualitatively reproduces theturbulence generation and mixing-processes very well.

6. Bottom shear stress

6.1. Experimental Results

Bottom shear stress is estimated by using the logarithmic velocitydistribution given in Eq. (14), as follows,

u ¼ U⁎j

lnzz0

� �ð14Þ

where, u is the flow velocity in the boundary layer, κ is the vonKarman's constant (=0.4), z is the cross-stream distance fromtheoretical bed level (z=y+Δz) (Fig. 3). For a smooth bottom zo=0,but for rough bottom, the elevation of theoretical bed level is not asingle value above the actual bed surface. The value of zo for the fully

odel prediction and experimental data for Case SK2.

Fig. 7. Turbulent intensity comparison between BSL k–ω model prediction and experimental data for Case SK3.

1107Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

rough turbulent flow is obtained by extrapolation of the logarithmicvelocity distribution above the bed to the value of z=zo where uvanishes. The temporal variations of Δz and zo are obtained from theextrapolation results of the logarithmic velocity distribution on thefitting a straight line of the logarithmic distribution through a set ofvelocity profile data at the selected phases angle for each case. Theseobtained values of Δz and zo are then averaged to get zo=0.05 cm forall cases and Δz=0.015 cm, Δz=0.012 cm, Δz=0.023 cm andΔz=0.011 cm, for Case SK1, Case SK2, Case SK3 and Case SK4,respectively. The bottom roughness, ks can be obtained by applyingthe Nikuradse's equivalent roughness in which zo=ks/30. By plotting uagainst ln(z/z0), a straight line is drawn through the experimentaldata, the value of friction velocity, U⁎ can be obtained from the slopeof this line and bottom shear stress, τo can then be obtained. Theobtained value of Δz and zo as the above mentioned has a sufficientaccuracy for application of logarithmic law in a wide range of velocityprofiles near the bottom. Suzuki et al. (2002) have given the details ofthis method and found good accuracy.

Fig. 8 shows the time-variation of bottom shear stress under saw-tooth waves with the variation in the wave skew-ness parameter α. Itcan be seen that the bottom shear stress under saw-tooth waves hasan asymmetric shape during crest and trough phases. The asymmetryof bottom shear stress is caused bywave skew-ness effect correspond-

Fig. 8. The time-variation of bottom shear stress under saw-tooth waves.

ing with acceleration effect. The increase inwave skew-ness causes anincrease the asymmetry of bottom shear stress. The wave withoutskew-ness shows a symmetric shape, as seen in Case SK4 for α=0.500(Fig. 8).

6.2. Calculation methods of bottom shear stress

6.2.1. Existing methodsThere are two existing calculation methods of bottom shear stress

for non-linear wave boundary layers. The maximum bottom shearstress within a basic harmonic wave-cycle modified by the phasedifference is proposed by Tanaka and Samad (2006), as follows:

so t � ur

¼ 1

2qfwU tð ÞjU tð Þj ð15Þ

Here τo(t), the instantaneous bottom shear stress, t, time, σ, theangular frequency, U(t) is the time history of free stream velocity, φ isphase difference between bottom shear stress and free streamvelocityand fw is the wave friction factor. This method is referred as Method 1in the present study.

Fig. 9. Calculation example of acceleration coefficient, ac for sawtooth wave.

Fig. 10. Acceleration coefficient ac as function of α.

1108 Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

Nielsen (2002) proposed a method for the instantaneous wavefriction velocity, U⁎(t) incorporating the acceleration effect, as follows:

U⁎ tð Þ ¼ffiffiffiffiffifw2

rcosuU tð Þ þ sin u

rAUAt

� �ð16Þ

so tð Þ ¼ qU⁎ tð ÞjU⁎ tð Þj ð17Þ

This method is based on the assumption that the steady flowcomponent is weak (e.g. in a strong undertow, in a surf zone, etc.). Thismethod is termed as Method 2 here. It seems reasonable to derive theτο(t) from u(t) by means of a simple transfer function based on theknowledge from simple harmonic boundary layer flows as has beendone by Nielsen (1992).

6.2.2. Proposed methodThe new calculation method of bottom shear stress under saw-

tooth waves (Method 3) is based on incorporating velocity andacceleration terms provided through the instantaneous wave frictionvelocity, U⁎(t) as given in Eq. (18). Both velocity and acceleration termsare adopted from the calculation method proposed by Nielsen (1992,2002) (Eq. (16)). The phase difference was determined from anempirical formula for practical purposes. In the new calculationmethod a new acceleration coefficient, ac is used expressing the waveskew-ness effect on the bottom shear stress under saw-tooth waves,that is determined empirically from both experimental and BSL k–ωmodel results. The instantaneous friction velocity, can be expressed as:

U⁎ tð Þ ¼ffiffiffiffiffiffiffiffiffiffifw=2

qU t þ u

r

þ ac

rAU tð ÞAt

� �ð18Þ

Here, the value of acceleration coefficient ac is obtained from theaverage value of ac(t) calculated from experimental result as well as

Fig. 11. Phase difference between the bottom shear stress and the free stream velocity.

the BSL k–ω model results of bottom shear stress using followingrelationship:

ac tð Þ ¼ U ⁎ tð Þ �ffiffiffiffiffiffiffiffiffiffifw=2

pU t þ u

r

� �ffiffiffiffiffiffiffifw=2

pr

AU tð ÞAt

ð19Þ

Fig. 9 shows an example of the temporal variation of the accel-eration coefficient ac(t) for α=0.300 based on the numerical com-putations. The results of averaged value of acceleration coefficient acfrom both experimental and numerical model results as function ofthe wave skew-ness parameter, α are plotted in Fig. 10. Hereafter, anequation based on regression line to estimate the accelerationcoefficient ac as a function of α is proposed as:

ac ¼ �036 ln að Þ � 0:249 ð20Þ

The increase in the wave skew-ness (or decreasing the value of α)brings about an increase in the value of acceleration coefficient, ac. Forthe symmetric wavewhere α=0.500, the value of ac is equal to zero. Inothers words the acceleration term is not significant for calculatingthe bottom shear stress under symmetric wave. Therefore, forsinusoidal wave Method 3 yields the same result as Method 1.

Fig. 12. Comparison among the BSL k–ω model, calculation methods and experimentalresults of bottom shear stress, for Case SK1.

Fig. 13. Comparison among the BSL k–ω model, calculation methods and experimentalresults of bottom shear stress, for Case SK2.

Fig. 14. Comparison among the BSL k–ω model, calculation methods and experimentalresults of bottom shear stress, for Case SK3.

1109Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

6.2.3. Wave friction factor and phase differenceThe wave friction coefficient proposed by Tanaka and Thu (1994)

was used in all the calculation methods in the present study asfollows:

fw ¼ exp �7:53þ 8:07amzo

� ��0:100( )

ð21Þ

us ¼ 42:4C0:153 1þ 0:00279C�0:357

1þ 0:127C0:563 degreeð Þ ð22Þ

for smooth : C ¼ 0:111

j fw2 Re

; for rough : C ¼ 1

jffiffiffiffifw2

qamz0

ð23Þ

u ¼ 2aus degreeð Þ ð24Þ

Where, φs is phase difference between free stream velocity andbottom shear stress proposed by Tanaka and Thu (1994) based onsinusoidal wave study and C defined by Eq. (23).

Fig. 11 shows the phase difference obtained from measured dataunder saw-tooth waves, as well as from theory proposed by Tanakaand Thu (1994) in Eq. (22) for sinusoidal wave. The wave skew-nesseffect under saw-tooth waves was included using Eq. (24). A value of

α=0.500 in Eq. (24) yields the same result as Eq. (22). As seen in Fig. 11the phase difference at crest, trough and average between crest andtrough for Case SK4with α=0.500 is about 19.1°, this value agrees wellwith the result obtained from Eq. (22) as well as Eq. (24) for α=0.500.The increase in the wave skew-ness or decreasing α causes theaverage value of phase difference in experimental results to graduallydecrease as shown in Fig. 11.

6.3. Comparison for bottom shear stress

In the previous section it has been shown that the bottom shearstress under saw-tooth waves has an asymmetric shape in both wavecrest and trough phases. The increase in wave skew-ness causes anincrease in the asymmetry of bottom shear stress under saw-toothwaves. Figs. 12, 13, 14 and 15 show a comparison among the BSL k–ωmodel, three calculation methods and experimental results of bottomshear stress under saw-tooth waves, for Case SK1, Case SK2, Case SK3and Case 4, respectively.

Method 3 has shown the best agreement with the experimentalresults along a wave cycle for all saw-tooth wave cases. Method 2slightly underestimated the bottom shear stress during accelerationphase for the higher wave skew-ness (Case SK1) as shown in Fig. 12.While, it overestimated the same in the crest phase for Case SK2 andSK3 as shown in Figs. 13 and 14, and in the trough phase for Case SK4as shown in Fig. 15.

Fig. 16. Formulation of sediment transport rate under skew waves.

Fig. 15. Comparison among the BSL k–ω model, calculation methods and experimentalresults of bottom shear stress, for Case SK4.

1110 Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

As expected, Method 1 yielded a symmetric value of the bottomshear stress at the crest and trough part for all the cases of saw-toothwaves. Moreover, the BSL k–ω model results showed close agreementwith the experimental data andMethod 3 results. Therefore, Method 3can be considered as a reliable calculation method of bottom shearstress under saw-tooth waves for all cases.

It can be concluded that the proposed method (Method 3) forcalculating the instantaneous bottom shear stress under saw-toothwaves has a sufficient accuracy.

7. Application to the net sediment transport inducedby skewwaves

7.1. Sediment transport rate formulation

The proposed calculation method of bottom shear stress is furtherapplied to formulate the sheet-flow sediment transport rate underskewwave using the experimental data byWatanabe and Sato (2004).At first, the instantaneous sheet flow sediment transport rate q(t) isexpressed as a function of the Shields number τ⁎(t) as given below:

U tð Þ ¼ q tð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqs=q� 1ð Þgd350

q ¼ A sign s⁎ tð Þf gjs⁎ tð Þj0:5 js⁎ tð Þj � s⁎crf g ð25Þ

Here, Φ(t) is the instantaneous dimensionless sediment transportrate, ρs is density of the sediment, g is gravitational acceleration, d50 ismedian diameter of sediment, A is a coefficient, τ⁎(t) is the Shieldsparameter defined by (το(t) / (((ρs/ρ)−1)gd50)) in which το(t) is theinstantaneous bottom shear stress calculated from both Method 1 andMethod 3. While τ⁎cr is the critical Shields number for the initiation ofsediment movement (Tanaka and To, 1995).

s⁎cr ¼ 0:055 1� exp 0:09S0:58⁎

� �� �þ 0:09S�0:72⁎ ð26Þ

Where, S⁎ is dimensionless particle size defined as:

S⁎ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqs=q� 1ð Þgd350

q4v

ð27Þ

The net sediment transport rate, qnet, which is averaged over one-period is expressed in the following expression according to Eq. (25).

U ¼ AF ¼ qnetffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqs=q� 1ð Þgd350

q ð28Þ

F ¼ 1T

Z T

0sign s⁎ tð Þf gjs⁎ tð Þj0:5 js⁎ tð Þj � s⁎crf gdt ð29Þ

Here, Φ is the dimensionless net sediment transport rate, F is afunction of Shields parameter and qnet is the net sediment transportrate in volume per unit time and width. Moreover, the integration ofEq. (29) is assumed to be done only in the phase |τ⁎(t)|Nτcr⁎ andduring the phase |τ⁎(t)|bτcr⁎ the function of integration is assumed tobe 0.

Sheet-flow condition occurs when the tractive force exceeds acertain limit, sand ripples disappear, replaced by a thin moving layerof sand in high concentration. Many researchers have shown that thecharacteristic of Nikuradse's roughness equivalent (ks) may be definedto be proportional to a characteristic grain size for evaluating thefriction factor. For sheet-flow sediment transport ks=2.5 d50 as shownby Swart (1974), Nielsen (2002) and Nielsen and Callaghan (2003).Therefore, in the present study the same relationship is used toformulate the sheet-flow sediment transport rate under skew wave.

First of all, the wave velocity profile, U(t) which was obtained fromthe time variation of acceleration of first order cnoidal wave theory byintegrating with respect to time as in the experiment by Watanabe andSato (2004). The bottom shear stress calculated from Method 1 wassubstituted into Eq. (29) and the result is shown in Fig. 16 by opensymbols. As expected thatMethod 1 yields a net sediment transport rate

Fig.19. Comparison of experimental and calculation result of the net sediment transportrates in variation of maximum velocity Umax and the wave skew-ness α ford50=0.20 mm and T=5 s.

Fig. 17. The relation between the net sediment transport rates and Umax in variation of αfor T=3 s and d50=0.20 mm.

1111Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

to be zero, because the integral value of F for a complete wave cycle iszero. In other word, it can be concluded that (Method 1) is not suitablefor calculating the net sediment transport rate under skew waves.

Furthermore, the relation between F and the dimensionless netsediment transport rate (Φ) obtained by the proposed method(Method 3) is shown in Fig. 16 by closed symbols. Since of theacceleration effect has been included in this calculation method (Eq.(18)), which causes the bottom shear stress at crest differ from that attrough, and therefore yields a net positive or negative value of F fromEq. (29). A linear regression curve is also shown in with the value ofA=11 (Eq. (28)).

7.2. Net sediment transport by skew waves

The characteristics of the net sediment transport induced by skewwaves are studied using the present calculation method for bottomshear stress (Method 3) and the experimental data for the sheet flowsediment transport rate fromWatanabe and Sato (2004). Fig. 17 showsa comparison between the experimental data and calculations basedon Method 3 for the net sediment transport rates, qnet and maximumvelocity, Umax for the wave period T=3 s and the median diameter ofsediment particle d50=0.20 mm along with the wave skew-nessparameter (α). It is clear that an increase in the wave skew-ness andthe maximum velocity produces an increase in the net sedimenttransport rate depicted in both experimental data and calculationresults. The proposed method shows very good agreement with thedata with minor differences. However, the present model has alimitation that does not simulate the sediment suspension. Asmentioned previously higher wave skew-ness produces a higher

Fig. 18. Change in amount of sediment transport rate according to an increasing α.

bottom shear stress and consequently yields a higher net sedimenttransport rate (Fig. 17).

Onshore and offshore sediment transport rate is shown in Fig. 18along with the net sediment transport. In this figure the values ofUmax, T and d50 are fixed and only α has been changed. As obvious for awave profile without skew-ness (α=0.500) the amount of onshoresediment transport is equal to that in offshore direction, therefore thenet sediment transport rate is zero. The difference between theonshore and the offshore sediment transport becomes more promi-nent due to an increase in the wave skew-ness and thus causing in asignificant increase the net sediment transport.

A similar comparison is made for another of experimentalcondition for T=5 s and d50=0.20 mm in Fig. 19.

Recently, Nielsen (2006) applied an extension of the domain filtermethod developed by Nielsen (1992) to evaluate the effect ofacceleration skew-ness on the net sediment transport based on thedata of Watanabe and Sato (2004). A good agreement betweencalculated and experimental data of the net sediment transport wasfound using φ=51°, a value much different from the usual notion thatthe phase difference is of the order of 10o for rough turbulent waveboundary layers.

Figs. 20 and 21 show the correlation of the net sediment transportexperimental data from Watanabe and Sato (2004) and the net

Fig. 20. Correlation of the net sediment transport experimental data from Watanabeand Sato (2004) and the net sediment transport calculated by the present model.

Fig. 21. Correlation of the net sediment transport experimental data fromWatanabe andSato (2004) and the net sediment transport calculated by Nielsen's model (2006).

1112 Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112

sediment transport calculated by Nielsen's model (2006) and by thepresent model, respectively. The present method shows a slightlybetter correlation than Nielsen's model (2006) with a reasonablevalue of the phase difference (φ ranges from 9.6° to 16.5°). The modelperformance is indicated by the coefficient of determination. Thepresent model shows the coefficient of determination (R2=0.655),which higher than that for Nielsen's model as (R2=0.557). Althoughthe present model is marginally better than the Nielsen's model(2006), the present model used a more realistic value of the phasedifference obtained from well-established formula.

8. Conclusions

The characteristics of the turbulent boundary layer under saw-tooth waves were studied using experiments and the BSL k–ωturbulence model. The mean velocity distributions under saw-toothwaves show different characteristics from those under sinusoidalwaves. The velocity overshooting is much influenced by the effect ofacceleration and the velocity magnitude. The velocity overshootinghas different appearance in the crest and trough phases caused by thedifference of acceleration. The BSL k–ω model shows a goodagreement with all the experimental data for saw-tooth waveboundary layer by virtue of velocity and turbulence kinetics energy(T.K.E). The model prediction far from the bed is generally good, whilenear the bed some discrepancies were found for all the cases.

A new calculation method for calculating bottom shear stressunder saw-tooth waves has been proposed based on velocity andacceleration termswhere the effect of wave skew-ness is incorporatedusing a factor ac, which is determined empirically from experimentaldata and the BSL k–ω model results. The new method has shown thebest agreement with the experimental data along a wave cycle for allsaw-tooth wave cases in comparison with the existing calculationmethods.

The new calculation method of bottom shear stress (Method 3)was applied to the net sediment transport experimental data undersheet flow condition by Watanabe and Sato (2004) and a goodagreement was found.

The inclusion of the acceleration effect in the calculation of bottomshear stress has significantly improved the net sediment transportcalculation under skew waves. It is envisaged that the new calculationmethod may be used to calculate the net sediment transport rateunder rapid acceleration in surf zone in practical applications, thusimproving the accuracy of morphological models in real situations.

Acknowledgments

The first author is grateful for the support provided by JapanSociety for the Promotion of Science (JSPS), Tohoku University, Japanand Institut Teknologi Sepuluh Nopember (ITS), Surabaya, Indonesiafor completing this study. This research was partially supported byGrant-in-Aid for Scientific Research from JSPS (No. 18006393).

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