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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, 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

A B S T R A C TA 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

Sheet flow

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, and the acceleration coefficient, ac expressing the

wave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determined

empirically from both experimental and baseline kmodel 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 sheet flow 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. 17031714.]. 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. 2008 Elsevier B.V. All rights reserved.

1. Introduction

Many researchers have studied turbulent boundary layers and

bottom friction through laboratory experiments and numerical

models. The experimental studies have contributed significantly

towards understanding of turbulent behavior of sinusoidal oscillatory

boundary layersover smoothand 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

profiles, 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 by Hino

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

specific 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., Fredse 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. Schffer 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

Coastal Engineering 55 (2008) 11021112

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).

0378-3839/$ see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.coastaleng.2008.04.007

Contents lists available at ScienceDirect

Coastal Engineering

j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / c o a s t a l e n g
mailto:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.coastaleng.2008.04.007http://www.sciencedirect.com/science/journal/03783839http://www.sciencedirect.com/science/journal/03783839http://dx.doi.org/10.1016/j.coastaleng.2008.04.007mailto:[email protected]:[email protected]:[email protected]:[email protected]


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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) and Nielsen

(2006) applied a modified version of the formula proposed by Nielsen

(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. Fredse 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

turbulent flow condition. However, Sleath (1987) and Dick and Sleath

(1991) observed that the phase difference and shear stress were

depended on the cross-stream distance from the bed, z for the mobile

roughness bed. It is envisaged that the phase difference calculated at

base of sheet flow layer may be very close to 90, while the phase

difference just above undisturbed level may only 1020 and the

phase 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) 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 efficient 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 boundarylayer under saw-tooth wave over rough bedhavenot 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) kmodel proposed by

Menter (1994) was also employed to and the experimental data was

used for model verification. Moreover, a quantitative comparisonbetween 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 theeffect of wave skew-nesson thebottom shear stressunder

saw-toothwaves. The proposed acconstant was determinedempirically

from both experimental and the BSL k 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 sheet flow sediment transport rate data under skew waves by

Watanabeand Sato (2004). Moreover, theaccelerationeffect 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 a flow-measuring unit. The saw-tooth wave

profi

le used is as presented by Schffer and Svendsen (1986) bysmoothing the sharp crest and trough parts. The definition sketch for

saw-tooth wave after smoothing is shown in Fig. 1. Here, Umax is the

velocity at wave crest, T is wave period, tp is time interval measured

from the zero-up cross point to wave crest in the time variation of free

stream velocity, tis time and is the wave skew-ness parameter. The

smaller indicate more wave skew-ness, while the sinusoidal wave

(without skew-ness) would have =0.50.

The oscillatory flow 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 digitalanalog (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 to Sleath (1987) at least 50 wave cycles areneeded to

successfully compute statistical quantities for turbulent condition. A

schematic diagram of the experimental set-up is shown in Fig. 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 cmand 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. Thetriangular roughnesshaving a heightof

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 in Fig. 3. Moreover, it was confirmed

that the velocity measurement at the center of the roughness and at

the flaking off region around the roughness has shown a similar flow

distribution as shown in Jonsson 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

pressuredrag on the roughness elements. Viscositybecomesirrelevant

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

1103Suntoyo et al. / Coastal Engineering 55 (2008) 11021112


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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

flowis logarithmic.Therefore theusuallog-law canbe used to estimate

thetimevariation of bottom shearstress(t) over roughbed as shownby previous studies e.g., Jonsson and Carlsen (1976), Hino et al. (1983),

Jensen et al. (1989), Fredse and Deigaard (1992) and Fredse 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 stresscomputed from theusuallog-lawand 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 maybe under estimatedby as much as

20% up to 60% in accelerating flow and overestimated by as much as

20% up to 80% in decelerating flow, respectively, for unsteady flow 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 unsteady flow, as given by Soulsby 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, v is

the kinematics viscosity, am/ks is the roughness parameter, ks,

Nikuradse's equivalent roughness defined as ks= 30zo in which zo is

the roughness height, am= Umax/, the orbital amplitude offluid just

above the boundary layer, where, Umax, the velocity at wave crest, ,

the angular frequency, T, wave period, S(=Uo/(zh)), 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 () were used; 0.453, 0.400

and 0.320. Moreover, the maximum flow velocity at free stream, Umaxranges from 0.72 to 1.45 m/s. The sediment median diameters are

d50=0.20 mm and d50= 0.74 mm and the wave periods are T=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 1

q

Ap

Ax 1q

As

Az1

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

Au

At AU

At 1q

As

Az2

For turbulent flow,

s

q vAu

AzPuVvV 3

The Reynolds stress qPuVvV may be expressed as qPuVvVqvt Au=Az , where t is the eddy viscosity.

And Eq. (3) became,

s

qv vt Au

Az4

For practical computations, turbulent flows are commonly computed

by the NavierStokes 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 modelsare 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 energy k

and its dissipation . 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) kmodel 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) am/ks Re S ks/zh

SK1 4.0 398 0.145 0.314 168.9 6.96 105 25.3 0.15

SK2 4.0 399 0.147 0.363 169.3 6.89 105 25.4 0.15

SK3 4.0 400 0.147 0.406 169.8 6.93 105 25.5 0.15

SK4 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) 11021112


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The baseline (BSL) model is one of the two-equation turbulence

modelsproposed by Menter (1994). The basic ideaof the BSLkmodel

is to retainthe robustand accurate formulation of theWilcox kmodel

in the near wall region, and to take advantage of the free stream

independence of the kmodel in the outer part of boundary layer. It

means that this model is designed to give results similar to those of the

original kmodel of Wilcox, but without its strong dependency on

arbitrary free stream ofvalues. Therefore, the BSL kmodel gives

results similar to the kmodel ofWilcox (1988) in the inner part ofboundary layer but changes gradually to the kmodel of Jones and

Launder (1972)towards to the outer boundary layer and the free stream

velocity. In orderto be able to perform the computations within one set

of equations, theJonesLaunder model wasfirst transformed into the k

formulation. The blending between the two regions is done by a

blending function F1 changing gradually from one to zero in the desired

region. The governing equations of the transport equation for turbulent

kinetic energy k and the dissipation of the turbulent kinetic energy

from the BSL model as mentioned before are,

Ak

At AAz

v vtrkx Ak

Az

& ' vt Au

Az

2bTxk 5

AxAt

AAz

v vtrx AxAz

& ' g AuAz

2bx2 2 1 F1 rx2 1xAkAz

AxAz

6

From k and , the eddy viscosity can be calculated as

vt kx

7

where, the values of the model constants are given as k=0.5,

=0.09, =0.5, =0.553 and = 0.075 respectively, and F1 is a

blending function, given as:

F1 arg 41 8

where,

arg1 min maxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k

0:09xz

r;

500v

z2x

!;

4rx2k

CDkxz2

" #9

here, z is the distance to the next surface and CDk is the positive

portion of the cross-diffusion term of Eq. (6) defined as

CDkx max 2rx21

x

Ak

Az

Ax

Az; 1020

10

Thus, Eqs. (2), (5) and (6) were solved simultaneously after nor-

malizing by using the free stream velocity, U, angular frequency,

kinematics viscosity, and zh.

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, (at z =zh, u/z=k/

z=/z =0). The kmodel 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 by Saffman (1970), given as follow,

xw UTSR=v 11

where w is the surface boundary condition of the specific dissipation

at the wall in which the turbulent kinetic energy k reduces to 0,

UT Fffiffiffiffiffiffiffiffiffiffiffiffijs0j=qp is friction velocity and the parameter SR is related to

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

SR 50kS

2for ks b25 and SR

100

ksfor ks z25 12

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 energy k and the dissipation

of the turbulent kinetic energy 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 CrankNicolson type implicit finite-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. The first grid point was placed at a distance

ofz1= (r1) zh/(rn1), where ris the ratio between two consecutive

grid spaces and n is total number of grid points. The value of r was

selected such thatz1 shouldbe sufficiently small in order to maintain

fine resolution near the wall. In this study, the value of z1 is givenequal to 0.0042 cm from thewall which correspond to z+= zU/v =0.01.

It may be noted that in kmodel 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 for k and . In the kmodel, 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, k and at every time instant during

a wave cycle. Second stage of convergence was based on themaximum

wall shear stress in a wave cycle. The convergence limit was set to

1106 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 BSL k

model for the cases SK2 and SK4presented in Figs. 4 and 5, respectively.

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



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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. Theexperimental dataand theturbulence model show that

the velocity overshoot is much influenced 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 (=0.363). As expected this difference is not visible for

symmetric case (Case SK4) (=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 BSLkmodel could predict the mean velocity very well in the

wholewave cycle of asymmetriccase. Moreover, it predictedthe velocityovershooting satisfactorily (Fig. 4). For symmetrccase(Case SK4) as well

the model prediction is excellent. A similar result was obtained by Sana

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

steady flow by Nezu (1977),

uV 1:052ffiffiffikp 13

where k is the turbulent kinetic energy obtained in the turbulence

model.

Comparison made on the basis of approximation to calculate the

fluctuating velocity by Nezu (1977) may notbe applicable in thewhole

range of cross-stream dimension since it is based on theassumption 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 BSL k model can

predict very well the turbulent intensity across the depth almost all at

phases, but, near the wall underestimatesat phases A, C, D and E (Case

SK2)andat phasesA, C,D, E and H (Case SK3)as shown in Figs. 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,

u Uj

lnz

z0

14

where, u is the flow velocity in the boundary layer, is the von

Karman's constant (=0.4), z is the cross-stream distance from

theoretical bed level (z = y +z) (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

Fig. 6. Turbulent intensity comparison between BSL kmodel prediction and experimental data for Case SK2.

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



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rough turbulent flow is obtained by extrapolation of the logarithmic

velocity distribution above the bed to the value of z = zo where u

vanishes. The temporal variations ofz and zo are 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 ofz and zo are then averaged to get zo=0.05 cm for

all cases and z =0.015 cm, z =0.012 cm, z =0.023 cm and

z =0.011 cm, for Case SK1, Case SK2, Case SK3 and Case SK4,

respectively. The bottom roughness, ks can be obtained by applying

the Nikuradse's equivalent roughness in which zo= ks/30. By plotting u

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, o can then be obtained. The

obtained value ofz and zo as 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. 8 shows the time-variation of bottom shear stress under saw-

tooth waves with the variation in the wave skew-ness parameter . 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 bottomshearstress is causedby 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 for =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 modified by the phase

difference is proposed by Tanaka and Samad (2006), as follows:

so t ur

1

2qfwU t jU t j 15

Here o(t), the instantaneous bottom shear stress, t, time, , the

angular frequency, U(t) is the time history of free stream velocity, is

phase difference between bottom shear stress and free stream velocity

and fw is 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, ac for sawtooth wave.

Fig. 7. Turbulent intensity comparison between BSL kmodel prediction and experimental data for Case SK3.



7/11

Nielsen (2002) proposed a method for the instantaneous wave

friction velocity, U(t) incorporating the acceleration effect, as follows:

U t ffiffiffiffiffifw2

rcosuU t sin u

r

AU

At

& '16

so t

qU t j

U t j

17

This method is based on the assumption that the steady flow

componentis weak (e.g. in a strongundertow, in a surf zone, etc.). This

method is termed as Method 2 here. It seems reasonable to derive the

(t) from u(t) by means of a simple transfer function based on the

knowledge from simple harmonic boundary layer flows as has been

done by Nielsen (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 givenin Eq. (18). Both velocity and acceleration terms

are 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 calculation

method a new acceleration coefficient, ac is used expressing the wave

skew-ness effect on the bottom shear stress under saw-tooth waves,

that is determined empirically from both experimental and BSL k

modelresults. The instantaneous friction velocity, can be expressed as:

U t ffiffiffiffiffiffiffiffiffiffifw=2

qU tu

r

acr

AU t At

& '18

Here, the value of acceleration coefficient ac is obtained from the

average value of ac(t) calculated from experimental result as well as

the BSL k model results of bottom shear stress using following

relationship:

ac t U t

ffiffiffiffiffiffiffiffiffiffifw=2

pU t u

r

ffiffiffiffiffiffiffifw=2

pr

AU t At

19

Fig. 9 shows an example of the temporal variation of the accel-

eration coefficient ac(t) for =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 of

the wave skew-ness parameter, are plotted in Fig. 10. Hereafter, an

equation based on regression line to estimate the acceleration

coefficient acas a function of is proposed as:

ac 036 ln a 0:249 20

The increase in the wave skew-ness (or decreasing the value of)

brings about an increase in the value of acceleration coefficient, ac. For

the symmetric wave where =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 BSLkmodel, calculation methods and experimental

results of bottom shear stress, for Case SK1.

Fig.10. Acceleration coefficient acas function of.



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6.2.3. Wave friction factor and phase difference

The wave friction coefficient proposed by Tanaka and Thu (1994)

was used in all the calculation methods in the present study as

follows:

fw exp 7:53 8:07 amzo

0:100( )21

us 42:4C0:153 1 0:00279C0:357

1 0:127C0:563 degree 22

for smooth : C 0:111j fw2 Re

; for rough : C 1j

ffiffiffiffifw2

qamz0

23

u 2aus degree 24

Where, s is phase difference between free stream velocity and

bottom shear stress proposed by Tanaka and Thu (1994) based on

sinusoidal wave study and Cdefined by Eq. (23).

Fig. 11 shows the phase difference obtained from measured data

under saw-tooth waves, as well as from theory proposed by Tanaka

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

=0.500 in Eq. (24) yields the same result as Eq. (22). As seenin Fig.11

the phase difference at crest, trough and average between crest and

trough for Case SK4with=0.500 is about 19.1, this value agreeswell

with the result obtained from Eq. (22) as well as Eq. (24) for =0.500.

The increase in the wave skew-ness or decreasing causes the

average value of phase difference in experimental results to gradually

decrease as shown in Fig. 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 15 show a comparison among the BSL k

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 in Fig. 12.

While, it overestimated the same in the crest phase for Case SK2 and

SK3 as shown in Figs. 13 and 14, and in the trough phase for Case SK4

as shown in Fig. 15.

Fig. 13. Comparison among the BSLkmodel, calculation methods and experimental






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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 BSLkmodel results showed close agreement

with theexperimental data andMethod 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 by Watanabe and Sato (2004).

At first, the instantaneous sheet flow sediment transport rate q(t) is

expressed as a function of the Shields number (t) as given below:

U t q t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqs=q 1 gd350q

A sign s t f gjs t j0:5 js t j scrf g 25

Here, (t) is the instantaneous dimensionless sediment transport

rate, s is density of the sediment, gis gravitational acceleration, d50 is

median diameter of sediment, A is a coefficient, (t) is the Shields

parameter defined by ((t)/(((s/)1)gd50)) in which (t) is the

instantaneous bottom shear stress calculated from both Method 1 and

Method 3. Whilecris the critical Shields number for the initiation of

sediment movement (Tanaka and To, 1995).

s

cr 0:055 1

exp 0:09S0:58

0:09S0:72 26Where, S is dimensionless particle size defined as:

S ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqs=q 1 gd350

q4v

27

The net sediment transport rate, qnet, which is averaged over one-

period is expressed in the following expression according to Eq. (25).

U AF qnetffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqs=q 1 gd350

q 28

F 1

TZT

0 sign s

t f gjs

t j0:5

js

t j s

crf gdt 29Here, is the dimensionless net sediment transport rate, F is a

function of Shields parameter and qnet is 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 |(t)|Ncr and

during the phase |(t)|bcr 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 transport ks=2.5 d50 as shown

by Swart (1974), Nielsen (2002) and Nielsen 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 by Watanabe and

Sato (2004). The bottom shear stress calculated from Method 1 was

substituted into Eq. (29) and the result is shown in Fig. 16 by open

symbols. As expected that Method 1 yieldsa netsedimenttransport rate

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





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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 between F and the dimensionless netsediment transport rate () 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 of Ffrom

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 sheet flow

sediment transport rate from Watanabeand Sato (2004). Fig. 17 showsa comparison between the experimental data and calculations based

on Method 3 for the net sediment transport rates, qnet and maximum

velocity, Umax for the wave period T=3 s and the median diameter of

sediment particle d50=0.20 mm along with the wave skew-ness

parameter (). 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 in Fig. 18along with the net sediment transport. In this figure the values of

Umax, Tand d50 are fixed and only has been changed. As obvious fora

wave profile without skew-ness (=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 for T=5 s and d50=0.20 mm in Fig. 19.

Recently, Nielsen (2006) applied an extension of the domain filter

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 betweencalculated and experimental data of the net sediment transport was

found using =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 21 show 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 from Watanabe

and Sato (2004) and the net sediment transport calculated by the present model.

Fig.19. Comparison of experimental andcalculation result of thenet sedimenttransport

rates in variation of maximum velocity Umax and the wave skew-ness for

d50=0.20 mm and T=5 s.

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

Fig. 17. The relation between the net sediment transport rates and Umax in variation of

for T=3 s and d50=0.20 mm.



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sediment transport calculated by Nielsen'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 (ranges from 9.6 to 16.5). The model

performance is indicated by the coefficient 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 kturbulence model. The mean velocity distributions under saw-tooth

waves show different characteristics from those under sinusoidal

waves. The velocity overshooting is much influenced 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 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 factor ac, which is determined empirically from experimentaldata and the BSLkmodel 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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