1

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

by

SUNTOYO*

Department of Ocean Engineering, Faculty of Marine Technology, Institut Teknologi Sepuluh Nopember (ITS), Surabaya 60111, Indonesia

* Corresponding author: Department of Civil Engineering, Tohoku University 6-6-06 Aoba, Sendai 980-8579, Japan, e-mail: suntoyo@kasen1.civil.tohoku.ac.jp

Hitoshi TANAKA Department of Civil Engineering, Tohoku University

6-6-06 Aoba, Sendai 980-8579, Japan, e-mail: tanaka@tsunami2.civil.tohoku.ac.jp

Ahmad SANA Department of Civil and Architectural Engineering, Sultan Qaboos University P.O. Box 33, AL-KHOD 123, Sultanate of Oman, e-mail: sana@squ.edu.om

Abstract

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 k- model 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 (2004). 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.

Keywords: Turbulent boundary layers, sheet flow, sediment transport, skew waves, saw-tooth waves.

2

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 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 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 associated 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 behaviors 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 terms for calculating sediment transport rate

based on the Kings (1991) saw-tooth wave experiments with the phase difference of 45o. 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

3

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 40o up to 51o. 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 45o and drops from 45o to about 10o 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 90o, while the phase

difference just above undisturbed level may only 10 20o and the phase difference about 51o 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 Satos (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 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 numerically. Laboratory experiments were conducted in an oscillating

tunnel over rough bed with air as the working fluid and smoke particles as tracers. The velocity

4

distributions were measured by means of Laser Doppler Velocimeter (LDV). The baseline (BSL) k-

model 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, ac and 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 proposed ac constant was determined empirically 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

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 a flow-measuring unit. The saw-tooth wave profile used is as presented by Schffer and

Svendsen (1986) by smoothing 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,

t is 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 digital-analog (DA) converter. A

servomotor, connected through a servomotor driver, is driven by the analog signal. The piston mechanism

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 compared the

position of the piston at every instant to the input signal, and subsequently adjusted the servomotor driver

5

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 are needed 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 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 on flow 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

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

pressure drag on the roughness elements. Viscosity becomes irrelevant 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 stress (t) over rough bed as shown by 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), and 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

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

6

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,

Nikuradses equivalent roughness defined as ks=30zo in which zo is the roughness height, am=Umax/, the

orbital amplitude of fluid 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. The flow 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, Umax ranges 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=3.0s and T=5.0s.

3. Turbulence Model

For the 1-D incompressible unsteady flow, the equation of motion within the boundary layer can be

expressed as

zx

pt

u

+

=

11

(1)

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

zt

Ut

u

+

=

1

(2)

For turbulent flow,

'v'uz

u

=

(3)

7

The Reynolds stress ''vu may be expressed as ( )zuvvu t = /'' , where t is the eddy viscosity. And equation (3) became,

( )z

ut

+=

(4)

For practical computations, turbulent flows 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 turbulence

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) k- model was

used to evaluate the turbulent properties to compare with the experimental data.

The baseline (BSL) model is one of the two-equation turbulence models proposed by Menter

(1994). The basic idea of the BSL k- model is to retain the robust and accurate formulation of the

Wilcox k- model in the near wall region, and to take advantage of the free stream independence of the k-

model in the outer part of boundary layer. It means that this model is designed to give results similar to

those of the original k- model of Wilcox, but without its strong dependency on arbitrary free stream of

values. Therefore, the BSL k- model gives results similar to the k- model of Wilcox (1988) in the inner

part of boundary layer but changes gradually to the k- model of Jones-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 was first 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,

8

( ) kz

uv

z

kvv

zt

ktkt *

2

+

+

=

(5)

( ) ( )zz

kFz

u

zvv

zt t

+

+

+

=

112 212

2 (6)

From k and , the eddy viscosity can be calculated as

kt = (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:

( )411 arg=F (8) where,

= 2

221

4;

500;

09.0maxminarg

zCDk

z

v

z

k

k

(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

=20

2 10,12max

zz

kCDk

(10)

Thus, Eqs. (2), (5) and (6) were solved simultaneously after normalizing 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 k- 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 by Saffman (1970), given as follow,

/* Rw SU= (11)

9

where w is the surface boundary condition of the specific dissipation at the wall in which the turbulent

kinetic energy k reduces to 0, /* oU = is friction velocity and the parameter SR is related to the

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

250

=

+s

Rk

S for ks+< 25 and

+=

s

Rk

S 100for ks+ 25 (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 instantaneous value 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 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 of z1 = (r-1) zh/(rn-1), where r is the ratio between two consecutive grid

spaces and n is total number of grid points. The value of r was selected such that z1 should be

sufficiently small in order to maintain fine resolution near the wall. In this study, the value of z1 is

given equal to 0.0042 cm from the wall which correspond to z+ = zU*/v = 0.01. It may be noted that in k-

model 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 k-

model, as explained before the effect of roughness can be simply incorporated using Eq. (9). In space 100

and in time 7200 steps per wave cycle were used. The convergence was achieved through two stages; the

first stage of convergence was based on the dimensionless values of u, k and at 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 x 10-6 for both the stages.

10

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 SK4 presented in Figs. 4 and 5, respectively.

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 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 BSL k- model 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 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 equation (13) that is a relationship

derived from experimental data for steady flow by Nezu (1977),

ku 052.1'= (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 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 BSL k-

model 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 in Figs. 6 and 7, respectively. However, the model qualitatively reproduces the turbulence

generation and mixing-processes very well.

11

6. Bottom Shear Stress

6.1. Experimental Results

Bottom shear stress is estimated by using the logarithmic velocity distribution given in equation (14),

=

0ln*

z

zUu

(14)

where, u is the flow velocity in the boundary layer, is the von Karmans 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 a single value above the actual bed surface. The value

of zo for the fully 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 of z 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 of z 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 Nikuradses 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 of z 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 bottom shear stress is caused by

wave skew-ness effect corresponding 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

12

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:

( ) ( )tUtUft wo

21

=

(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.

Nielsen (2002) proposed a method for the instantaneous wave friction velocity, U*(t)

incorporating the acceleration effect, as follows:

( ) ( ) ( )

+=t

tUtU

ftU w

sincos2

* (16)

( ) ( ) ( )tUtUto ** = (17) This method is based on the assumption that the steady flow 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

(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 given in 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 an empirical

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- model results. The instantaneous friction

velocity, can be expressed as:

( ) ( )

+

+=

t

tUatUftU cw

2/* (18)

13

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:

( )( )

( )t

tUf

tUftUta

w

w

c

+

=

2

2* (19)

Fig. 9 shows an example of the temporal variation of the acceleration coefficient ac(t) for =

0.300 based on the numerical computations. The results of averaged value of acceleration coefficient ac

from 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

ac as a function of is proposed as:

( ) 249.0ln36.0 = ca (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 of ac is 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.

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:

+=

100.007.853.7exp

o

mw

z

af (21)

563.0

357.0153.0

127.0100279.014.42

CCCs

+

+=

(degree) (22)

for smooth:

ew R

fC2

111.0

= ; for rough:

02

1

z

afC mw= (23)

s 2= (degree) (24)

14

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 C defined by equation (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 seen in Fig. 14 the phase difference at crest, trough and average between crest

and trough for Case SK4 with = 0.500 is about 19.1o, this value agrees well 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.

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 BSL k- model 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.

15

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:

( )( ) 3501/

)(gd

tqt

s

=

( ){ } ( ) ( ){ }crtttsignA **** 5.0 = (25)

Here, (t) is the instantaneous dimensionless sediment transport rate, s is density of the sediment, g is

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. While *cr is the critical Shields number for the initiation

of sediment movement (Tanaka and To (1995)).

( ){ } 72.0*58.0** 09.009.0exp1055.0 += SScr (26) Where, S* is dimensionless particle size defined as:

( )

4

1/ 350*

gdS s

= (27)

The net sediment transport rate, qnet, which is averaged over one-period is expressed in the following

expression according to equation (25).

( ) 3501/ gdq

FAs

net

==

(28)

( ){ } ( ) ( ){ } dttttsignT

F crT ****1 5.0

0 = (29)

Here, 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 equation

(29) is assumed to be done only in the phase |*(t)|> *cr and during the phase |*(t)|< *cr the function of integration is assumed to be 0.

16

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 Nikuradses 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 of first 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 yields a net sediment

transport rate to be zero, because the integral value of F for 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 net sediment 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 F from Eq.

(29). A linear regression curve is also shown in with the value of A = 11 (Eq. 28).

7.3. 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 Watanabe and Sato (2004). Fig. 17 shows a 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 = 3s 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

17

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. (18) along with the net sediment

transport. In this figure the values of Umax, T and d50 are fixed and only has been changed. As obvious

for a 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 prominent 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 = 5s 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 skewness 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 using = 51o, 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 sediment transport calculated by Nielsens model (2006) and by

the present model, respectively. The present method shows a slightly better correlation than Nielsens

model (2006) with a reasonable value of the phase difference ( ranges from 9.6 o to 16.5 o). 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 Nielsens model as (R2=0.557). Although the present

model is marginally better than the Nielsens 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- turbulence 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

18

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 skewness is incorporated

using a factor ac, which is determined empirically from experimental data and the BSL k- model 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).

References

Dick, J.E. and Sleath, J.F.A. 1991. Velocities and concentrations in oscillatory flow over beds of sediment.

Journal of Fluids Mechanics, 233, 165-196.

Fredse, J. and Deigaard, R., 1992. Mechanics of coastal sediment transport. Advanced Series on Ocean

Engineering, vol. 3. World Scientific Publication.

19

Fredse, J., Andersen, K. H.. and Sumer, B. M. 1999. Wave plus current over a ripple-covered bed.

Coastal Engineering, 38, 177-221.

Hino, M., Kashiwayanag, M., Nakayama, A. and Nara, T., 1983. Experiments on the turbulence statistics

and the structure of a reciprocating oscillatory flow. Journal of Fluid Mechanics 131, 363-400.

Hsu, T. J. and Hanes, D. M., 2004. Effects of wave shape on sheet flow sediment transport. Journal of

Geophysical Research 109, (C05025), doi:10.1029 /2003JC002075.

Jensen, B. L., Sumer, B. M. and Fredse, J., 1989. Turbulent oscillatory boundary layers at high Reynolds

numbers. Journal of Fluid Mechanics 206, 265-297.

Jones, W. P. and Launder, B. E., 1972. The prediction of laminarization with a two-equation model of

turbulence. Int. Journal of Heat Mass Transfer 15, 301-314.

Jonsson, I. G. and Carlsen, N.A., 1976. Experimental and theoretical investigations in an oscillatory

turbulent boundary layer. Journal of Hyd. Research, 14 (1), 45-60.

King, D.B., 1991. Studies in oscillatory flow bed load sediment transport. PhD Thesis, University of

California, San Diego, USA.

Menter, F. R., 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA

Journal, 32 (8), 1598-1605.

Nezu, I., 1977. Turbulent structure in open channel flow. Ph.D Dissertation, Kyoto University, Japan.

Nielsen, P., 1992. Coastal bottom boundary layers and sediment transport. Advanced Series on Ocean

Engineering, vol. 4. World Scientific Publication.

Nielsen, P., 2002. Shear stress and sediment transport calculations for swash zone modeling. Coastal

Engineering 45, 53-60.

Nielsen, P. and Callaghan, D. P., 2003. Shear stress and sediment transport calculations for sheet flow

under waves. Coastal Engineering 47, 347-354.

Nielsen, P., 2006. Sheet flow sediment transport under waves with acceleration skewness and boundary

layer streaming. Coastal Engineering 53, 749-758.

Saffman, P. G., 1970. Dependence on Reynolds Number of High-Order Moments of Velocity Derivatives

in Isotropic Turbulence. Physics Fluids 13, 21922193.

Sana, A. and Tanaka, H., 2000. Review of k- model to analyze oscillatory boundary layers. Journal of

Hydraulic Engineering 126(9), 701-710.

20

Sana, A. and Shuy, E. B., 2002. Two-equation turbulence models for smooth oscillatory boundary layers.

Journal of Waterway, Port, Coastal and Ocean Engineering 128 (1), 38-45.

Sana, A., Tanaka, H., Yamaji, H. and Kawamura, I., 2006. Hydrodynamic behavior of asymmetric

oscillatory boundary layers at low Reynolds Numbers. Journal of Hydraulic Engineering 132 (10),

1086-1096.

Schffer, A. H. and Svendsen, I. A., 1986. Boundary layer flow under skew waves. Inst. Hydrodynamics

and Hydraulic Engineering, Tech. Univ. Denmark, Prog. Report 64, 13 33.

Sleath, J.F. A., 1987. Turbulent oscillatory flow over rough beds. Journal of Fluid Mechanics 182, 369-

409.

Soulsby, R. L. and Dyer, K.R., 1981. The form of the near-bed velocity profile in a tidally accelerating

flow. Journal of Geophysical Research 86 (C9), 8067-8074.

Suzuki, T., Tanaka, H. and Yamaji, H., 2002. Investigation of rough bottom boundary layer under

irregular waves. Annual Journal of Hydraulic Engineering 46, 869-874.(in Japanese)

Swart, D.H., 1974. Offshore Sediment Transport and Equilibrium Beach Profile. Delft Hydraulics

Laboratory Publication, No. 131.

Tanaka, H., Chian, C. S. and Shuto, N., 1983. Experiments on an oscillatory flow accompanied with a

unidirectional motion. Coastal Engineering in Japan 26, 19-37.

Tanaka, H., 1988. Bed load transport due to non-linear wave motion. Proceedings of 21st International

Conference on Coastal Engineering, ASCE, Malaga, Spain, pp. 1803-1817.

Tanaka, H. and Thu, A., 1994. Full-range equation of friction coefficient and phase difference in a wave-

current boundary layer. Coastal Engineering 22, 237-254.

Tanaka, H. and To, D.V., 1995. Initial motion of sediment under waves and wave-current combined

motions. Coastal Engineering 25, 153-163.

Tanaka, H. and Samad, M.A., 2006. Prediction of instantaneous bottom shear stress for turbulent plane

bed condition under irregular wave. Journal of Hydraulic Research, 44 (1), 94-106.

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. Wilcox, D.C., 1988. Reassessment of the scale-determining equation for advanced turbulent models.

AIAA Journal 26 (11), 1299-1310.

21

Figure Captions

Fig. 1. Definition sketch for saw-tooth wave

Fig. 2. Schematic diagram of experimental set-up

Fig.3 Definition sketch for roughness

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

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

Fig. 6. Turbulent intensity comparison between BSL k- model prediction and experimental data for Case

SK2

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

SK3 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. 10. Acceleration coefficient ac as function of

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

Fig. 12. Comparison among the BSL k- model, calculation methods and experimental results of bottom

shear stress, for Case SK1

Fig. 13. Comparison among the BSL k- model, calculation methods and experimental results of bottom

shear stress, for Case SK2

Fig. 14. Comparison among the BSL k- model, calculation methods and experimental results of bottom

shear stress, for Case SK3

Fig. 15. Comparison among the BSL k- model, calculation methods and experimental results of bottom

shear stress, for Case SK4 Fig. 16. Formulation of sediment transport rate under skew waves

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

d50=0.20mm

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

Fig. 19. Comparison of experimental and calculation result of the net sediment transport rates in variation

of maximum velocity Umax and the wave skewness for d50 = 0.20 mm and T = 5s.

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. 21. Correlation of the net sediment transport experimental data from Watanabe and Sato (2004) and the net sediment transport calculated by Nielsens model (2006)

22

tp T/2

T

Umaxt

U

Fig. 1. Definition sketch for saw-tooth wave

Fig. 2. Schematic diagram of experimental set-up

Measuring Section

90 90 5.0 m

A

A

20

10

Section A-A (Dim. in cm)

Wind Tunnel

Input Signal through a PC

DA Converter

Signal Controller

Servo Motor

Piston Servo Motor Driver to Wind

Tunnel

Potentiometer

(Unit:cm)

23

Fig.3 Definition sketch for roughness

0 2 4Time (s)

-400-200

0200400

U(cm

/s)

A

B CD E

FG

HSK 2

-1 -0.5 0 0.5 1u/U

510-3

510-2

510-1

5100

z/z

h

A B CDEFG H

BSLk- Model Exp.

-1 -0.5 0 0.5 1u/U

510-3

510-2

510-1

5100

z/z

h

Hr

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

z0

y z

z=0 y=0

u

u

z

10mm 12mm

5.0mm

24

0 2 4Time (s)

-400-200

0200400

U(cm

/s)

AB

CD

E

FG

HSK 4

-1 -0.5 0 0.5 1u/U

510-3

510-2

510-1

5100

z/z

h

Hr

-1 -0.5 0 0.5 1u/U

510-3

510-2

510-1

5100

z/z

hA B CD

EFG H

BSLk- Model Exp.

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

25

0 20 40 60u'(cm/s)

H

Hr

Hr

BSL k- Model Exp.(SK2)

10-3

10-2

10-1

100

z/z

h

A B C D

0 20 40 60u'(cm/s)

10-3

10-2

10-1

100

z/z

h

E

0 20 40 60u'(cm/s)

F

0 20 40 60u'(cm/s)

G

Fig. 6. Turbulent intensity comparison between BSL k- model prediction and experimental data for Case SK2

26

0 20 40 60u'(cm/s)

H

Hr

Hr

BSL k- Model Exp.(SK3)

10-3

10-2

10-1

100

z/z

h

A B C D

0 20 40 60u'(cm/s)

10-3

10-2

10-1

100

z/z

h

E

0 20 40 60u'(cm/s)

F

0 20 40 60u'(cm/s)

G

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

0 2 4Time (s)

-2000

0

2000

o(t)

/

(cm2 /s

2 )

Case SK1 (=0.314) Case SK2 (=0.363) Case SK3 (=0.406) Case SK4 (=0.500)

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

27

-400-200

0200400

U(t)

(c

m/s

)Umax = 400 cm/s

= 0.300

0 1 2 3Time (s)

-0.2

0

0.2

0.4

ac

average = 0.194

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

0.2 0.4Wave skewness parameter ()

0

0.1

0.2

0.3

ac

ac = -0.36ln()-0.249

BSL k- Model Exp.

Fig. 10. Acceleration coefficient ac as function of

28

100 101 102 103

am/ks

0

10

20

30

40

(d

egre

e)

Eq.(24), =0.500 Eq.(22) Eq.(24), =0.406 Eq.(24), =0.363 Eq.(24), =0.314

Crest (Exp.) Trough (Exp.) Average (Exp.)

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

29

-400-200

0200400

U(t)

(cm

/s)

Umax = 398 cm/s

= 0.314

0 1 2 3 4Time (s)

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK1

BSL k- Model Exp.

-1500

0

1500 o

(t)/

(cm

2 /s2 ) Case SK1

Method 1 Exp.

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK1

Method 2 Exp.

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK1

Method 3 Exp.

Fig. 12. Comparison among the BSL k- model, calculation methods and experimental results of bottom shear stress, for Case SK1

30

-400-200

0200400

U(t)

(cm

/s)

Umax = 399 cm/s

= 0.363

0 1 2 3 4Time (s)

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK2

BSL k- Model Exp.

-1500

0

1500 o

(t)/

(cm

2 /s2 ) Case SK2

Method 1 Exp.

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK2

Method 2 Exp.

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK2

Method 3 Exp.

Fig. 13. Comparison among the BSL k- model, calculation methods and experimental results of bottom shear stress, for Case SK2

31

-400-200

0200400

U(t)

(cm

/s)

Umax = 400 cm/s

= 0.406

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK3

Method 3 Exp.

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK3

Method 2 Exp.

-1500

0

1500 o

(t)/

(cm

2 /s2 ) Case SK3

Method 1 Exp.

0 1 2 3 4Time (s)

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK3

BSL k- Model Exp.

Fig. 14. Comparison among the BSL k- model, calculation methods and experimental results of bottom shear stress, for Case SK3

32

-400-200

0200400

U(t)

(cm

/s)

Umax = 400 cm/s

= 0.500

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK4

Method 3 Exp.

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK4

Method 2 Exp.

-1500

0

1500 o

(t)/

(cm

2 /s2 ) Case SK4

Method 1 Exp.

0 1 2 3 4Time (s)

-1500

0

1500

o(t)

/

(cm2 /s

2 ) Case SK4

BSL k- Model Exp.

Fig. 15. Comparison among the BSL k- model, calculation methods and experimental results of bottom shear stress, for Case SK4

33

0 0.5 1 1.5F

0

5

10

15

Method1 3 = 11 F d50 T

0.20mm, 3s

0.74mm, 3s

0.20mm, 5s

0 0.5 1 1.5F

0

5

10

15

Method1 3

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

0 50 100 150Umax(cm/s)

0

0.5

1

1.5

2

q ne

t(cm

2 /s)

=0.320 =0.400 =0.453

exp. cal.

T=3sd50=0.20mm

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

34

0.10.20.30.40.5

-4

-2

0

2

4

q net(c

m2 /s

)

qon

qnetqoff

Umax = 108 cm/s

T = 3sd50 = 0.20 mm

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

0 50 100 150Umax(cm/s)

0

0.5

1

1.5

2

q ne

t(cm

2 /s)

=0.320 =0.400 =0.453

exp. cal.

T=5sd50=0.20mm

Fig. 19. Comparison of experimental and calculation result of the net sediment transport rates in variation of maximum velocity Umax and the wave skewness for d50 = 0.20 mm and T = 5s.

35

0 1 2 3Calculated qnet (cm2/s)

0

1

2

Expe

rimen

tal

q ne

t

(cm2 /

s) Present modelR2=0.655

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.

0 1 2 3Calculated qnet (cm2/s)

0

1

2

Expe

rimen

tal

q ne

t

(cm2 /

s) Nielsen's model (2006)R2=0.557

Fig. 21. Correlation of the net sediment transport experimental data from Watanabe and Sato (2004) and the net sediment transport calculated by Nielsens model (2006)

36

Tables

Table 1 Experimental conditions for saw-tooth waves

Table 1 Experimental conditions for saw-tooth waves

Case T (s) Umax

(cm/s) (cm2/s) am/ks Re S ks/zh

SK1 4.0 398 0.145 0.314 168.9 6.96x105 25.3 0.15

SK2 4.0 399 0.147 0.363 169.3 6.89 x105 25.4 0.15

SK3 4.0 400 0.147 0.406 169.8 6.93 x105 25.5 0.15

SK4 4.0 400 0.151 0.500 169.8 6.75x105 25.5 0.15