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

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Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves

and 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, Indonesia b_{Department of Civil Engineering, Tohoku University, 6-6-06 Aoba, Sendai 980-8579, Japan}

c

Department of Civil and Architectural Engineering, Sultan Qaboos University, P.O. Box 33, AL-KHOD 123, Oman

A B S T R A C T A R T I C L E I N F O

Article history: Received 14 August 2007

Received in revised form 30 March 2008 Accepted 4 April 2008

Available online 21 May 2008

Keywords:

Turbulent boundary layers Sheetﬂow

Sediment transport Skew waves Saw-tooth waves

A large number of studies 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 of sediment movement occurs. It is envisaged that bottom shear stress and sediment transport behaviors influenced by the effect of asymmetry are different from those in sinusoidal waves. Characteristics of the turbulent boundary layer under breaking waves (saw-tooth) are investigated and described through both laboratory and numerical experiments. A new calculation method for bottom shear stress based on velocity and acceleration terms, theoretical phase difference,φand the acceleration coefficient,acexpressing the wave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determined empirically from both experimental and baselinek–ωmodel results. The new calculation has shown better agreement with the experimental data along a wave cycle for all saw-tooth wave cases compared by other existing methods. It was further applied into sediment transport rate calculation induced by skew waves. Sediment transport rate was formulated by using the existing sheetflow sediment transport rate data under skew waves by Watanabe and Sato [Watanabe, A. and Sato, S., 2004. A sheet-flow transport rate formula for asymmetric, forward-leaning waves and currents. Proc. of 29th ICCE, ASCE, pp. 1703–1714.]. Moreover, the characteristics of the net sediment transport were also examined and a good agreement between the proposed method and experimental data has been found.

1. Introduction

Many researchers have studied turbulent boundary layers and bottom friction through laboratory experiments and numerical models. The experimental studies have contributed signi_ﬁcantly towards understanding of turbulent behavior of sinusoidal oscillatory boundary layers over smooth and rough bed (e.g.,Jonsson and Carlsen, 1976; Tanaka et al., 1983; Sleath, 1987, Jensen et al., 1989). These studies explained how the turbulence is generated in the near-bed region either through the shear layer instability or turbulence bursting phenomenon. Such studies included measurement of the velocity pro_ﬁles, bottom shear stress and some included turbulence intensity. An extensive series of measurements and analysis for the smooth bed boundary layer under sinusoidal waves has been presented byHino et al. (1983).Jensen et al. (1989)carried out a detailed experimental study on turbulent oscillatory boundary layers over smooth as well as rough bed under sinusoidal waves. Moreover,Sana and Tanaka (2000)

and Sana and Shuy (2002) have compared the direct numerical simulation (DNS) data for sinusoidal oscillatory boundary layer on smooth bed with various two-equation turbulence models and, a quantitative comparison has been made to choose the best model for speci_ﬁc purpose. However, these models were not applied to predict the 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 motion have been done (e.g.,Fredsøe and Deigaard, 1992). These studies have shown that the net sediment transport over a complete wave cycle is zero. In reality, however ocean waves often have a strongly non-linear shape with respect to horizontal axes. Therefore it is envisaged that turbulent structure, bottom shear stress and sediment transport be-haviors are different from those in sinusoidal waves due to the effect of acceleration caused by the skew-ness of the wave.

Tanaka (1988)estimated the bottom shear stress under non-linear wave by modiﬁed stream function theory and proposed formula to predict bed load transport except near the surf zone in which the acceleration effect plays an important role.Schäffer and Svendsen (1986)presented the saw-tooth wave as a wave proﬁle expressing wave-breaking situation. Moreover,Nielsen (1992)proposed a bottom shear stress formula incorporating both velocity and acceleration ⁎ Corresponding author. Department of Civil Engineering, Tohoku University, 6-6-06

Aoba, Sendai 980-8579, Japan.

E-mail addresses:[email protected],[email protected](Suntoyo),

[email protected](H. Tanaka),[email protected](A. Sana).

Contents lists available atScienceDirect

Coastal Engineering

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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)andNielsen (2006)applied a modified version of the formula proposed byNielsen (1992)and applied it to predict sediment transport rate with various experimental data. They have shown that the phase difference between free stream velocity and bottom shear stress used to evaluate the sediment transport is from 40° up to 51°. Whereas, many researchers e.g.Fredsøe and Deigaard (1992), Jonsson and Carlsen (1976),Tanaka and Thu (1994)have shown that the phase difference for laminar flow is 45° and drops from 45° to about 10° in the turbulentflow condition. However,Sleath (1987)andDick and Sleath (1991) observed that the phase difference and shear stress were depended on the cross-stream distance from the bed,zfor the mobile roughness bed. It is envisaged that the phase difference calculated at base of sheetflow layer may be very close to 90°, while the phase difference just above undisturbed level may only 10–20° and the phase difference about 51° as the bestfit value obtained byNielsen (2006)may be occurred at some depth below the undisturbed level.

More recently,Gonzalez-Rodriguez and Madsen (2007)presented a simple conceptual model to compute bottom shear stress under asymmetric and skewed waves. The model used a time-varying friction factor and a time-varying phase difference assumed to be the linear interpolation in time between the values calculated at the crest and trough. However, this model does not parameterize the_ﬂuid acceleration effect or the horizontal pressure gradients acting on the sediment particle. Moreover, this model under predicted most of Watanabe and Sato's (2004) experimental data induced by skew waves or acceleration-asymmetric waves.

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

In the present study, the characteristics of turbulent boundary layers under saw-tooth waves are investigated experimentally and numeri-cally. Laboratory experiments were conducted in an oscillating tunnel over rough bed with air as the working_fluid and smoke particles as tracers. The velocity distributions were measured by means of Laser Doppler Velocimeter (LDV). The baseline (BSL)k–ωmodel proposed by Menter (1994)was also employed to and the experimental data was used for model verification. Moreover, a quantitative comparison between turbulence model and experimental data was made. A new calculation method for bottom shear stress is proposed incorporating both velocity and acceleration terms. In this method a new acceleration coefficient,acand a phase difference empirical formula were proposed to express the effect of wave skew-ness on the bottom shear stress under saw-tooth waves. The proposedacconstant was determined empirically from both experimental and the BSL k–ω model results. The new calculation method of bottom shear stress under saw-tooth wave was further applied to calculate sediment transport rate induced by skew or saw-tooth waves. Sediment transport rate was formulated by using the existing sheetflow sediment transport rate data under skew waves by Watanabe and Sato (2004). Moreover, the acceleration effect on both the bottom shear stress and sediment transport under skew waves were examined.

2. Experimental study

2.1. Turbulent boundary layer experiments

Turbulent boundary layer flow experiments under saw-tooth waves were carried out in an oscillating tunnel using air as the working fluid. The experimental system consists of the oscillatory flow generation unit and aflow-measuring unit. The saw-tooth wave pro_file used is as presented by Schäffer and Svendsen (1986) by smoothing the sharp crest and trough parts. The definition sketch for saw-tooth wave after smoothing is shown inFig. 1. Here,Umaxis the velocity at wave crest,Tis wave period,tpis time interval measured from the zero-up cross point to wave crest in the time variation of free stream velocity,tis time andαis the wave skew-ness parameter. The smallerα_{indicate more wave skew-ness, while the sinusoidal wave} (without skew-ness) would haveα= 0.50.

The oscillatoryflow generation unit comprises of signal control and processing components and piston mechanism. The piston displacement signal is fed into the instrument through a PC. Input digital signal is then converted to corresponding analog data through a digital–analog (DA) converter. A servomotor, connected through a servomotor driver, is driven by the analog signal. The piston mecha-nism has been mounted on a screw bar, which is connected to the servomotor. The feed-back on piston displacement, from one instant to 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 the next instant. The measured flow velocity record was collected by means of an A/D converter at 10 millisecond intervals, and the mean velocity profile variation was obtained by averaging over 50 wave cycles. According toSleath (1987)at least 50 wave cycles are needed to successfully compute statistical quantities for turbulent condition. A schematic diagram of the experimental set-up is shown inFig. 2.

The flow-measuring unit comprises of a wind tunnel and one component Laser Doppler Velocimeter (LDV) for_flow measurement. Velocity measurements were carried out at 20 points in the vertical direction at the central part of the wind tunnel. The wind tunnel has a length of 5 m and the height and width of the cross-section are 20 cm and 10 cm, respectively (Fig. 2). These dimensions of the cross-section of wind tunnel were selected in order to minimize the effect of sidewalls onflow velocity. The triangular roughness having a height of 5 mm (a roughness height,Hr= 5 mm) and 10 mm width was pasted over the bottom surface of the wind tunnel at a spacing of 12 mm along the wind tunnel, as shown inFig. 3. Moreover, it was confirmed that the velocity measurement at the center of the roughness and at theflaking off region around the roughness has shown a similarflow distribution as shown inJonsson and Carlsen (1976).

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

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for determining either the velocity distribution or the overall drag on the surface. And the velocity distribution near a rough bed for steady flow is logarithmic. Therefore the usual log-law can be used to estimate the time variation of bottom shear stressτο(t) over rough bed as shown by previous studies e.g.,Jonsson and Carlsen (1976),Hino et al. (1983), Jensen et al. (1989),Fredsøe and Deigaard (1992)andFredsøe 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 the values of bottom shear stress computed from the usual log-law and the momentum integral methods gave a quite similar, especially by virtue of the phase difference in crest and trough values of the shear stress. Nevertheless, this usual log-law may be under estimated by as much as 20% up to 60% in acceleratingflow and overestimated by as much as 20% up to 80% in deceleratingflow, respectively, for unsteadyflow as shown by Soulsby and Dyer (1981). The usual log-law should be modified by incorporating velocity and acceleration terms to estimate the bed shear stress for unsteadyflow, as given bySoulsby and Dyer (1981).

Experiments have been carried out for four cases under saw-tooth waves. The experimental conditions of present study are given in Table 1. The maximum velocity was kept almost 400 cm/s for all the cases. The Reynolds number magnitude defined for each case has sufficed to locate these cases in the rough turbulent regime. Here,vis the kinematics viscosity, am/ks is the roughness parameter, ks, Nikuradse's equivalent roughness defined asks= 30zoin which zo is the roughness height,am=Umax/σ, the orbital amplitude offluid just above the boundary layer, where,Umax, the velocity at wave crest,σ, the angular frequency,T, wave period,S(=Uo/(σzh)), the reciprocal of the Strouhal number,zh, the distance from the wall to the axis of symmetry of the measurement section.

2.2. Sediment transport experiment

The experimental data from Watanabe and Sato (2004) for oscillatory sheet_flow sediment transport under skew waves motion were used in the present study. Theflow velocity wave profile was the acceleration asymmetric or skew wave profile obtained from the time variations of acceleration of _first-order cnoidal wave theory by integration with respect to time. These experiments consist of 33 cases. Three values of the wave skew-ness (α) were used; 0.453, 0.400 and 0.320. Moreover, the maximum_flow velocity at free stream,Umax ranges from 0.72 to 1.45 m/s. The sediment median diameters are

d50= 0.20 mm andd50= 0.74 mm and the wave periods areT= 3.0 s and

T= 5.0 s.

3. Turbulence model

For the 1-D incompressible unsteady_ﬂow, the equation of motion within the boundary layer can be expressed as

Au

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

A_u

For practical computations, turbulentflows are commonly computed by the Navier–Stokes equation in averaged form. However, the averaging process gives rise to the new unknown term representing the transport of mean momentum and heat flux by fluctuating quantities. In order to determine these quantities, turbulence models are required. Two-equation turbulence models are complete turbu-lence models that fall in the class of eddy viscosity models (models which are based on a turbulent eddy viscosity are called as eddy viscosity models). Two transport equations are derived describing transport of two scalars, for example the turbulent kinetic energyk

and its dissipationε. The Reynolds stress tensor is then computed using an assumption, which relates the Reynolds stress tensor to the velocity gradients and an eddy viscosity. While in one-equation turbulence models (incomplete turbulence model), the transport equation is solved for a turbulent quantity (i.e. the turbulent kinetic energy,k) and a second turbulent quantity is obtained from algebraic expression. In the present paper the base line (BSL)k–ωmodel was used to evaluate the turbulent properties to compare with the ex-perimental data.

Fig. 3.Deﬁnition sketch for roughness.

Table 1

Experimental conditions for saw-tooth waves

Case T(s) Umax(cm/s) v(cm2_/s) _α _am/ks _Re _S _ks/zh

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The baseline (BSL) model is one of the two-equation turbulence models proposed byMenter (1994). The basic idea of the BSLk–ωmodel is to retain the robust and accurate formulation of the Wilcoxk_–ωmodel in the near wall region, and to take advantage of the free stream independence of thek–εmodel in the outer part of boundary layer. It means that this model is designed to give results similar to those of the originalk–ωmodel of Wilcox, but without its strong dependency on arbitrary free stream ofωvalues. Therefore, the BSLk–ωmodel gives results similar to thek–ωmodel ofWilcox (1988)in the inner part of boundary layer but changes gradually to thek–εmodel ofJones and Launder (1972)towards to the outer boundary layer and the free stream velocity. In order to be able to perform the computations within one set of equations, the Jones–Launder model wasﬁrst transformed into thek–

ωformulation. The blending between the two regions is done by a blending functionF1changing gradually from one to zero in the desired region. The governing equations of the transport equation for turbulent kinetic energykand the dissipation of the turbulent kinetic energyω

from the BSL model as mentioned before are,

A_k

Fromkandω, the eddy viscosity can be calculated as

vt¼k

x 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 a

here,zis the distance to the next surface andCDkωis the positive portion of the cross-diffusion term of Eq. (6) deﬁned as

CDkx¼max 2rx2

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

kinematics viscosity,νandzh.

3.1. Boundary conditions

Non slip boundary conditions were used for velocity and turbulent kinetic energy on the wall (u=k= 0) and at the axis of symmetry of the oscillating tunnel, the gradients of velocity, turbulent kinetic energy and speciﬁc dissipation rate were equated to zero, (atz=zh,∂u/∂z=∂k/ ∂_z₌∂_ω_/∂_z_{= 0). The}_k_–_ω_{model provides a natural way to incorporate the} effects of surface roughness through the surface boundary condition. The effect of roughness was introduced through the wall boundary condition of Wilcox (1988), in which this equation was originally recognized bySaffman (1970), given as follow,

xw¼UTSR=v ð11Þ

whereωwis the surface boundary condition of the speciﬁc dissipation

ωat the wall in which the turbulent kinetic energykreduces to 0,

UT¼F ffiffiffiffiffiffiffiffiffiffiffiffi_js0j=q

p

is friction velocity and the parameterSRis related to the grain-roughness Reynolds number,ks+=ks(U⁎/v),

The instantaneous bottom shear stress can be determined using Eq. (4), in which the eddy viscosity was obtained by solving the transport equation for turbulent kinetic energykand the dissipation of the turbulent kinetic energyωin Eq. (7). While, the instantaneous value ofu(z,t) andvtcan be obtained numerically from Eqs. (1)–(7) with the proper boundary conditions.

3.2. Numerical method

A Crank_–Nicolson type implicitfinite-difference scheme was used to solve the dimensionless non-linear governing equations. In order to achieve better accuracy near the wall, the grid spacing was allowed to increase exponentially in the cross-stream direction to get _fine resolution near the wall. Thefirst grid point was placed at a distance ofΔz1= (r−1)zh/(rn−1), whereris the ratio between two consecutive grid spaces andnis total number of grid points. The value ofrwas selected such thatΔz1should be sufficiently small in order to maintain fine resolution near the wall. In this study, the value ofΔz1is given equal to 0.0042 cm from the wall which correspond toz+=zU⁎/v= 0.01. It may be noted that ink–εmodel where wall function method is used to describe roughness the first grid point should be lie in the logarithmic region and corresponding boundary conditions should be applied forkandε. In thek_–ωmodel, as explained before the effect of roughness can be simply incorporated using Eq. (11). In space 100 and in time 7200 steps per wave cycle were used. The convergence was achieved through two stages; thefirst stage of convergence was based on the dimensionless values ofu,kandωat every time instant during a wave cycle. Second stage of convergence was based on the maximum wall shear stress in a wave cycle. The convergence limit was set to 1 × 10−6_{for both the stages.}

4. Mean velocity distributions

Mean velocity proﬁles in a rough turbulent boundary layer under saw-tooth waves at selected phases were compared with the BSLk–ω

model for the cases SK2 and SK4 presented inFigs. 4 and 5, respectively.

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The solid line showed the turbulence model prediction while open and closed circles showed the experimental data for mean velocity profile distribution. The experimental data and the turbulence model show that the velocity overshoot is much in_fluenced by the effect of acceleration and the velocity magnitude. The difference of the acceleration between the crest and trough phases is significant. The velocity overshooting is higher in the crest phase than the trough as shown at phase B and F for Case SK2 (α= 0.363). As expected this difference is not visible for symmetric case (Case SK4) (α= 0.500). Moreover, the asymmetry of the flow velocity can be observed in phase A and E. Due to the higher acceleration at phase A the velocity overshooting is more distinguished in the wall vicinity.

The BSLk–ωmodel could predict the mean velocity very well in the whole wave cycle of asymmetric case. Moreover, it predicted the velocity overshooting satisfactorily (Fig. 4). For symmetrc case (Case SK4) as well

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

5. Prediction of turbulence intensity

The ﬂuctuating velocity in x-direction u' can be approximated using Eq. (13) that is a relationship derived from experimental data for steadyﬂow byNezu (1977),

u_V¼1:052 ffiffiffi

k

p

ð13_Þ

wherekis the turbulent kinetic energy obtained in the turbulence model.

Comparison made on the basis of approximation to calculate the fluctuating velocity byNezu (1977)may not be applicable in the whole range of cross-stream dimension since it is based on the assumption of isotropic turbulence. This assumption may be valid far from the wall, where the_flow is practically isotropic, whereas the_flow in the region near the wall is essentially non-isotropic. The BSLk–ωmodel can predict very well the turbulent intensity across the depth almost all at phases, but, near the wall underestimates at phases A, C, D and E (Case SK2) and at phases A, C, D, E and H (Case SK3) as shown inFigs. 6 and 7, respectively. However, the model qualitatively reproduces the turbulence generation and mixing-processes very well.

6. Bottom shear stress

6.1. Experimental Results

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

u¼U_j⁎ ln z

z0

ð14Þ

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

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rough turbulentflow is obtained by extrapolation of the logarithmic velocity distribution above the bed to the value ofz=zo where u vanishes. The temporal variations ofΔzandzoare obtained from the extrapolation results of the logarithmic velocity distribution on the fitting a straight line of the logarithmic distribution through a set of velocity profile data at the selected phases angle for each case. These obtained values ofΔzandzoare then averaged to getzo= 0.05 cm for all cases and Δ_z= 0.015 cm, Δ_z= 0.012 cm, Δ_z= 0.023 cm and Δz= 0.011 cm, for Case SK1, Case SK2, Case SK3 and Case SK4, respectively. The bottom roughness,kscan be obtained by applying the Nikuradse's equivalent roughness in whichzo=ks/30. By plottingu against ln(z/z0), a straight line is drawn through the experimental data, the value of friction velocity,U⁎can be obtained from the slope of this line and bottom shear stress,τocan then be obtained. The obtained value ofΔzandzoas the above mentioned has a sufficient accuracy for application of logarithmic law in a wide range of velocity profiles near the bottom.Suzuki et al. (2002)have given the details of this method and found good accuracy.

Fig. 8shows the time-variation of bottom shear stress under saw-tooth waves with the variation in the wave skew-ness parameterα. It can be seen that the bottom shear stress under saw-tooth waves has an asymmetric shape during crest and trough phases. The asymmetry of bottom shear stress is caused by wave skew-ness effect

correspond-ing with acceleration effect. The increase in wave skew-ness causes an increase the asymmetry of bottom shear stress. The wave without skew-ness shows a symmetric shape, as seen in Case SK4 forα= 0.500 (Fig. 8).

6.2. Calculation methods of bottom shear stress

6.2.1. Existing methods

There are two existing calculation methods of bottom shear stress for non-linear wave boundary layers. The maximum bottom shear stress within a basic harmonic wave-cycle modi_ﬁed by the phase difference is proposed byTanaka and Samad (2006), as follows:

so t u r

¼1₂qfwU tð ÞjU tð Þj ð15Þ

Hereτo(t), the instantaneous bottom shear stress,t, time,σ, the angular frequency,U(t) is the time history of free stream velocity,φis phase difference between bottom shear stress and free stream velocity andfwis the wave friction factor. This method is referred as Method 1 in the present study.

Fig. 8.The time-variation of bottom shear stress under saw-tooth waves. Fig. 9.Calculation example of acceleration coefﬁcient,acfor sawtooth wave.

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

U⁎_{ð Þ ¼}t

ffiffiffiffiffi

fw 2

r

cosuU t_{ð Þ þ} sinu

r A_U

A_t

ð16_Þ

soð Þ ¼t qU⁎ð Þjt U⁎ð Þjt ð17Þ

This method is based on the assumption that the steadyﬂow component is weak (e.g. in a strong undertow, in a surf zone, etc.). This method is termed as Method 2 here. It seems reasonable to derive the

το(t) fromu(t) by means of a simple transfer function based on the knowledge from simple harmonic boundary layerﬂows as has been done byNielsen (1992).

6.2.2. Proposed method

The new calculation method of bottom shear stress under saw-tooth waves (Method 3) is based on incorporating velocity and acceleration terms provided through the instantaneous wave friction velocity,U⁎(t) as given in Eq. (18). Both velocity and acceleration terms are adopted from the calculation method proposed byNielsen (1992, 2002) (Eq. (16)). The phase difference was determined from an empirical formula for practical purposes. In the new calculation method a new acceleration coefﬁcient,acis used expressing the wave skew-ness effect on the bottom shear stress under saw-tooth waves, that is determined empirically from both experimental and BSLk–ω

model results. The instantaneous friction velocity, can be expressed as:

U⁎ð Þ ¼t

ffiffiffiffiffiffiffiffiffiffi

fw=2

q

U tþu_rþa_rcAU t_Að Þ_t

ð18Þ

Here, the value of acceleration coefﬁcientacis obtained from the average value ofac(t) calculated from experimental result as well as

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

acð Þ ¼t

U⁎ð Þt ffiffiffiffiffiffiffiffiffiffi

fw=2

p

U tþu r

ffiffiffiffiffiffiffi

fw=2 p

r AU tð Þ

At

ð19_Þ

Fig. 9shows an example of the temporal variation of the accel-eration coefficientac(t) for α= 0.300 based on the numerical com-putations. The results of averaged value of acceleration coef_ficientac from both experimental and numerical model results as function of the wave skew-ness parameter,αare plotted inFig. 10. Hereafter, an equation based on regression line to estimate the acceleration coefficientacas a function ofαis proposed as:

ac¼ 036 lnð Þa 0:249 ð20Þ

The increase in the wave skew-ness (or decreasing the value ofα) brings about an increase in the value of acceleration coef_ﬁcient,ac. For the symmetric wave whereα= 0.500, the value ofacis equal to zero. In others words the acceleration term is not signiﬁcant for calculating the bottom shear stress under symmetric wave. Therefore, for sinusoidal wave Method 3 yields the same result as Method 1.

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

Fig. 12.Comparison among the BSLk–ωmodel, calculation methods and experimental results of bottom shear stress, for Case SK1.

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6.2.3. Wave friction factor and phase difference

The wave friction coefﬁcient proposed byTanaka and Thu (1994) was used in all the calculation methods in the present study as follows:

fw¼exp 7:53þ8:07

am

zo

0:100

( )

ð21_Þ

us¼42:4C0:153

1_þ0:00279C 0:357

1_þ0:127C0:563 ðdegreeÞ ð22Þ

for smooth :C¼0:111 jfw2Re

; for rough :C¼ 1 j

ffiffiffiffi

fw

2

q

am z0

ð23Þ

u¼2ausðdegreeÞ ð24Þ

Where,φsis phase difference between free stream velocity and bottom shear stress proposed byTanaka and Thu (1994) based on sinusoidal wave study andCdeﬁned by Eq. (23).

Fig. 11shows the phase difference obtained from measured data under saw-tooth waves, as well as from theory proposed byTanaka and Thu (1994)in Eq. (22) for sinusoidal wave. The wave skew-ness effect 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 inFig. 11 the phase difference at crest, trough and average between crest and trough for Case SK4 withα= 0.500 is about 19.1°, this value agrees well with the result obtained from Eq. (22) as well as Eq. (24) forα= 0.500. The increase in the wave skew-ness or decreasing α causes the average value of phase difference in experimental results to gradually decrease as shown inFig. 11.

6.3. Comparison for bottom shear stress

In the previous section it has been shown that the bottom shear stress under saw-tooth waves has an asymmetric shape in both wave crest and trough phases. The increase in wave skew-ness causes an increase in the asymmetry of bottom shear stress under saw-tooth waves.Figs. 12, 13, 14 and 15show a comparison among the BSLk–ω

model, three calculation methods and experimental results of bottom shear stress under saw-tooth waves, for Case SK1, Case SK2, Case SK3 and Case 4, respectively.

Method 3 has shown the best agreement with the experimental results along a wave cycle for all saw-tooth wave cases. Method 2 slightly underestimated the bottom shear stress during acceleration phase for the higher wave skew-ness (Case SK1) as shown inFig. 12. While, it overestimated the same in the crest phase for Case SK2 and SK3 as shown inFigs. 13 and 14, and in the trough phase for Case SK4 as shown inFig. 15.

Fig. 13.Comparison among the BSLk–ωmodel, calculation methods and experimental results of bottom shear stress, for Case SK2.

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As expected, Method 1 yielded a symmetric value of the bottom shear stress at the crest and trough part for all the cases of saw-tooth waves. Moreover, the BSLk–ωmodel results showed close agreement with the experimental data and Method 3 results. Therefore, Method 3 can be considered as a reliable calculation method of bottom shear stress under saw-tooth waves for all cases.

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

7. Application to the net sediment transport induced by skew waves

7.1. Sediment transport rate formulation

The proposed calculation method of bottom shear stress is further applied to formulate the sheet-flow sediment transport rate under skew wave using the experimental data byWatanabe and Sato (2004). Atfirst, the instantaneous sheetflow sediment transport rateq(t) is expressed as a function of the Shields numberτ⁎(t) as given below:

U_{ð Þ ¼}t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq tð Þ

qs=q 1

ð Þgd3 50

q ¼Asignfs⁎ð Þtgjs⁎ð Þjt

0:5

js⁎_{ð Þj}t s⁎cr

f g ð25_Þ

Here,Φ(t) is the instantaneous dimensionless sediment transport rate,ρsis density of the sediment,gis gravitational acceleration,d50is median diameter of sediment,Ais a coefﬁcient,τ⁎(t) is the Shields parameter deﬁned by (το(t) / (((ρs/ρ)−1)gd50)) in whichτο(t) is the instantaneous bottom shear stress calculated from both Method 1 and Method 3. Whileτ⁎cris the critical Shields number for the initiation of sediment movement (Tanaka and To, 1995).

s⁎

cr¼0:055 1 exp 0:09S

0:58

⁎

þ0:09S 0:72

⁎ ð26Þ

Where,S⁎is dimensionless particle size deﬁned as:

S⁎¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qs=q 1

ð Þgd3 50

q

4v ð27Þ

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

U¼AF¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqnet

qs=q 1

ð Þgd3 50

q ð28Þ

F_¼1 T

ZT

0

sign_fs⁎_{ð Þ}t _gjs⁎_{ð Þj}t 0:5

js⁎_{ð Þj}t s⁎cr

f gdt _ð29_Þ

Here,Φis the dimensionless net sediment transport rate,Fis a function of Shields parameter andqnetis the net sediment transport rate in volume per unit time and width. Moreover, the integration of Eq. (29) is assumed to be done only in the phase |τ⁎(t)|Nτcr⁎ and during the phase |τ⁎(t)|bτcr⁎the function of integration is assumed to be 0.

Sheet-flow condition occurs when the tractive force exceeds a certain limit, sand ripples disappear, replaced by a thin moving layer of sand in high concentration. Many researchers have shown that the characteristic of Nikuradse's roughness equivalent (ks) may be defined to be proportional to a characteristic grain size for evaluating the friction factor. For sheet-_flow sediment transportks= 2.5d50as shown bySwart (1974),Nielsen (2002)andNielsen and Callaghan (2003). Therefore, in the present study the same relationship is used to formulate the sheet-_flow sediment transport rate under skew wave.

First of all, the wave velocity proﬁle,U(t) which was obtained from the time variation of acceleration ofﬁrst order cnoidal wave theory by integrating with respect to time as in the experiment byWatanabe and Sato (2004). The bottom shear stress calculated from Method 1 was substituted into Eq. (29) and the result is shown inFig. 16by open symbols. As expected that Method 1 yields a net sediment transport rate

Fig. 16.Formulation of sediment transport rate under skew waves. Fig. 15.Comparison among the BSLk–ωmodel, calculation methods and experimental

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

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

A =11 (Eq. (28)).

7.2. Net sediment transport by skew waves

The characteristics of the net sediment transport induced by skew waves are studied using the present calculation method for bottom shear stress (Method 3) and the experimental data for the sheetﬂow sediment transport rate fromWatanabe and Sato (2004).Fig. 17shows a comparison between the experimental data and calculations based on Method 3 for the net sediment transport rates,qnetand maximum velocity,Umaxfor the wave periodT= 3 s and the median diameter of sediment particle d50= 0.20 mm along with the wave skew-ness parameter (α). It is clear that an increase in the wave skew-ness and the maximum velocity produces an increase in the net sediment transport rate depicted in both experimental data and calculation results. The proposed method shows very good agreement with the data with minor differences. However, the present model has a limitation that does not simulate the sediment suspension. As mentioned previously higher wave skew-ness produces a higher

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

Onshore and offshore sediment transport rate is shown inFig. 18 along with the net sediment transport. In thisﬁgure the values of

Umax,Tandd50arefixed and onlyαhas been changed. As obvious for a wave pro_file without skew-ness (α= 0.500) the amount of onshore sediment transport is equal to that in offshore direction, therefore the net sediment transport rate is zero. The difference between the onshore and the offshore sediment transport becomes more promi-nent due to an increase in the wave skew-ness and thus causing in a significant increase the net sediment transport.

A similar comparison is made for another of experimental condition forT= 5 s andd50= 0.20 mm inFig. 19.

Recently,Nielsen (2006)applied an extension of the domainﬁlter method developed by Nielsen (1992) to evaluate the effect of acceleration skew-ness on the net sediment transport based on the data of Watanabe and Sato (2004). A good agreement between calculated and experimental data of the net sediment transport was found usingφ= 51°, a value much different from the usual notion that the phase difference is of the order of 10o_{for rough turbulent wave} boundary layers.

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

Fig. 20.Correlation of the net sediment transport experimental data fromWatanabe and Sato (2004)and the net sediment transport calculated by the present model. Fig. 19.Comparison of experimental and calculation result of the net sediment transport rates in variation of maximum velocity Umax and the wave skew-ness α for d50= 0.20 mm andT= 5 s.

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sediment transport calculated byNielsen's model (2006)and by the present model, respectively. The present method shows a slightly better correlation than Nielsen's model (2006) with a reasonable value of the phase difference (φranges from 9.6° to 16.5°). The model performance is indicated by the coef_ﬁcient of determination. The present model shows the coefﬁcient of determination (R2_{= 0.655),} which higher than that for Nielsen's model as (R2_{= 0.557). Although} the present model is marginally better than the Nielsen's model (2006), the present model used a more realistic value of the phase difference 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–ω

turbulence model. The mean velocity distributions under saw-tooth waves show different characteristics from those under sinusoidal waves. The velocity overshooting is much in_ﬂuenced by the effect of acceleration and the velocity magnitude. The velocity overshooting has different appearance in the crest and trough phases caused by the difference of acceleration. The BSL k–ω model shows a good agreement with all the experimental data for saw-tooth wave boundary layer by virtue of velocity and turbulence kinetics energy (T.K.E). The model prediction far from the bed is generally good, while near the bed some discrepancies were found for all the cases.

A new calculation method for calculating bottom shear stress under saw-tooth waves has been proposed based on velocity and acceleration terms where the effect of wave skew-ness is incorporated using a factorac, which is determined empirically from experimental data and the BSLk–ωmodel results. The new method has shown the best agreement with the experimental data along a wave cycle for all saw-tooth wave cases in comparison with the existing calculation methods.

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

The inclusion of the acceleration effect in the calculation of bottom shear stress has signiﬁcantly improved the net sediment transport calculation under skew waves. It is envisaged that the new calculation method may be used to calculate the net sediment transport rate under rapid acceleration in surf zone in practical applications, thus improving the accuracy of morphological models in real situations.

Acknowledgments

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

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