Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport
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
Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport
Suntoyo a , b , ⁎ b , Hitoshi Tanaka , Ahmad Sana c
b Department of Ocean Engineering, Faculty of Marine Technology, Institut Teknologi Sepuluh Nopember (ITS), Surabaya 60111, Indonesia Department of Civil Engineering, Tohoku University, 6-6-06 Aoba, Sendai 980-8579, Japan c Department of Civil and Architectural Engineering, Sultan Qaboos University, P.O. Box 33, AL-KHOD 123, Oman
ARTICLE INFO
ABSTRACT
A large number of studies have been done dealing with sinusoidal wave boundary layers in the past. Received 14 August 2007
Article history:
However, ocean waves often have a strong asymmetric shape especially in shallow water, and net of Received in revised form 30 March 2008
Accepted 4 April 2008 sediment movement occurs. It is envisaged that bottom shear stress and sediment transport behaviors
Available online 21 May 2008 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
Keywords: laboratory and numerical experiments. A new calculation method for bottom shear stress based on velocity Turbulent boundary layers
and acceleration terms, theoretical phase difference, φ and the acceleration coefficient, a c expressing the Sheet flow
wave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determined Sediment transport
empirically from both experimental and baseline k–ω model results. The new calculation has shown better Skew waves
agreement with the experimental data along a wave cycle for all saw-tooth wave cases compared by other Saw-tooth waves
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. 1703–1714.]. Moreover, the characteristics of the net sediment transport were also examined and a good agreement between the proposed method and experimental data has been found.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction and Sana and Shuy (2002) have compared the direct numerical simulation (DNS) data for sinusoidal oscillatory boundary layer on
Many researchers have studied turbulent boundary layers and smooth bed with various two-equation turbulence models and, a bottom friction through laboratory experiments and numerical
quantitative comparison has been made to choose the best model for models. The experimental studies have contributed significantly
specific purpose. However, these models were not applied to predict towards understanding of turbulent behavior of sinusoidal oscillatory
the turbulent properties for asymmetric waves over rough beds. boundary layers over smooth and rough bed (e.g., Jonsson and Carlsen,
Many studies on wave boundary layer and bottom friction asso- 1976; Tanaka et al., 1983; Sleath, 1987, Jensen et al., 1989 ). These
ciated with sediment movement induced by sinusoidal wave motion studies explained how the turbulence is generated in the near-bed
have been done (e.g., Fredsøe and Deigaard, 1992 ). These studies have region either through the shear layer instability or turbulence bursting
shown that the net sediment transport over a complete wave cycle is phenomenon. Such studies included measurement of the velocity
zero. In reality, however ocean waves often have a strongly non-linear profiles, bottom shear stress and some included turbulence intensity.
shape with respect to horizontal axes. Therefore it is envisaged that An extensive series of measurements and analysis for the smooth bed
turbulent structure, bottom shear stress and sediment transport be- boundary layer under sinusoidal waves has been presented by Hino
haviors are different from those in sinusoidal waves due to the effect et al. (1983) . Jensen et al. (1989) carried out a detailed experimental
of acceleration caused by the skew-ness of the wave. study on turbulent oscillatory boundary layers over smooth as well as
Tanaka (1988) estimated the bottom shear stress under non-linear rough bed under sinusoidal waves. Moreover, Sana and Tanaka (2000)
wave by modified stream function theory and proposed formula to predict bed load transport except near the surf zone in which the acceleration effect plays an important role. Schäffer and Svendsen
⁎ Corresponding author. Department of Civil Engineering, Tohoku University, 6-6-06 Aoba, Sendai 980-8579, Japan.
(1986) presented the saw-tooth wave as a wave profile expressing E-mail addresses: [email protected] , [email protected] (Suntoyo),
wave-breaking situation. Moreover, Nielsen (1992) proposed a bottom [email protected] (H. Tanaka), [email protected] (A. Sana).
shear stress formula incorporating both velocity and acceleration 0378-3839/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi: 10.1016/j.coastaleng.2008.04.007
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
terms for calculating sediment transport rate based on the King's
2. Experimental study
(1991) saw-tooth wave experiments with the phase difference of 45°. Recently, Nielsen (2002) , Nielsen and Callaghan (2003) and Nielsen
2.1. Turbulent boundary layer experiments
(2006) applied a modified version of the formula proposed by Nielsen (1992) and applied it to predict sediment transport rate with various
Turbulent boundary layer flow experiments under saw-tooth experimental data. They have shown that the phase difference
waves were carried out in an oscillating tunnel using air as the between free stream velocity and bottom shear stress used to evaluate
working fluid. The experimental system consists of the oscillatory the sediment transport is from 40° up to 51°. Whereas, many
flow generation unit and a flow-measuring unit. The saw-tooth wave researchers e.g. Fredsøe and Deigaard (1992) , Jonsson and Carlsen
profile used is as presented by Schäffer and Svendsen (1986) by (1976) , Tanaka and Thu (1994) have shown that the phase difference
smoothing the sharp crest and trough parts. The definition sketch for for laminar flow is 45° and drops from 45° to about 10° in the
saw-tooth wave after smoothing is shown in Fig. 1 . Here, U max is the turbulent flow condition. However, Sleath (1987) and Dick and Sleath
velocity at wave crest, T is wave period, t p is time interval measured (1991) observed that the phase difference and shear stress were
from the zero-up cross point to wave crest in the time variation of free depended on the cross-stream distance from the bed, z for the mobile
stream velocity, t is time and α is the wave skew-ness parameter. The roughness bed. It is envisaged that the phase difference calculated at
smaller α indicate more wave skew-ness, while the sinusoidal wave base of sheet flow layer may be very close to 90°, while the phase
(without skew-ness) would have α = 0.50. difference just above undisturbed level may only 10–20° and the
The oscillatory flow generation unit comprises of signal control phase difference about 51° as the best fit value obtained by Nielsen
and processing components and piston mechanism. The piston (2006) may be occurred at some depth below the undisturbed level.
displacement signal is fed into the instrument through a PC. Input More recently, Gonzalez-Rodriguez and Madsen (2007) presented
digital signal is then converted to corresponding analog data through
a digital–analog (DA) converter. A servomotor, connected through a asymmetric and skewed waves. The model used a time-varying
a simple conceptual model to compute bottom shear stress under
servomotor driver, is driven by the analog signal. The piston mecha- friction factor and a time-varying phase difference assumed to be the
nism has been mounted on a screw bar, which is connected to the linear interpolation in time between the values calculated at the crest
servomotor. The feed-back on piston displacement, from one instant and trough. However, this model does not parameterize the fluid
to the next, has been obtained through a potentiometer that com- acceleration effect or the horizontal pressure gradients acting on the
pared the position of the piston at every instant to the input signal, sediment particle. Moreover, this model under predicted most of
and subsequently adjusted the servomotor driver for position at the Watanabe and Sato's (2004) experimental data induced by skew
next instant. The measured flow velocity record was collected by waves or acceleration-asymmetric waves.
means of an A/D converter at 10 millisecond intervals, and the mean Hsu and Hanes (2004) examined in detail the effects of wave
velocity profile variation was obtained by averaging over 50 wave profile on sediment transport using a two-phase model. They have
cycles. According to Sleath (1987) at least 50 wave cycles are needed to shown that the sheet flow response to flow forcing typical of
successfully compute statistical quantities for turbulent condition. A asymmetric and skewed waves indicates a net sediment transport in
schematic diagram of the experimental set-up is shown in Fig. 2 . the direction of wave propagation. However, for a predictive near-
The flow-measuring unit comprises of a wind tunnel and one shore morphological model, a more efficient approach to calculate the
component Laser Doppler Velocimeter (LDV) for flow measurement. bottom shear stress is needed for practical applications. Moreover,
Velocity measurements were carried out at 20 points in the vertical investigation of a more reliable calculation method to estimate the
direction at the central part of the wind tunnel. The wind tunnel has a time-variation of bottom shear stress and that of turbulent boundary
length of 5 m and the height and width of the cross-section are 20 cm layer under saw-tooth wave over rough bed have not been done as yet.
and 10 cm, respectively ( Fig. 2 ). These dimensions of the cross-section Bottom shear stress estimation is the most important step, which is
of wind tunnel were selected in order to minimize the effect of required as an input to the practical sediment transport models.
sidewalls on flow velocity. The triangular roughness having a height of Therefore, the estimation of bottom shear stress from a sinusoidal
5 mm (a roughness height, H r = 5 mm) and 10 mm width was pasted wave is of limited value in connection with the sediment transport
over the bottom surface of the wind tunnel at a spacing of 12 mm estimation unless the acceleration effect is incorporated therein.
along the wind tunnel, as shown in Fig. 3 . Moreover, it was confirmed In the present study, the characteristics of turbulent boundary layers
that the velocity measurement at the center of the roughness and at under saw-tooth waves are investigated experimentally and numeri-
the flaking off region around the roughness has shown a similar flow cally. Laboratory experiments were conducted in an oscillating tunnel
distribution as shown in Jonsson and Carlsen (1976) . over rough bed with air as the working fluid and smoke particles as
These roughness elements protrude out of the viscous sub-layer at tracers. The velocity distributions were measured by means of Laser
high Reynolds numbers. This causes a wake behind each roughness Doppler Velocimeter (LDV). The baseline (BSL) k–ω model proposed by
element, and the shear stress is transmitted to the bottom by the Menter (1994) was also employed to and the experimental data was
pressure drag on the roughness elements. Viscosity becomes irrelevant 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, a c 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 a c constant was determined empirically
from both experimental and the BSL k–ω model results. The new calculation method of bottom shear stress under saw-tooth wave was
further applied to calculate sediment transport rate induced by skew or saw-tooth waves. Sediment transport rate was formulated by using the existing 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.
Fig. 1. Definition sketch for saw-tooth wave.
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
Table 1 Experimental conditions for saw-tooth waves
Case
T (s)
U max (cm/s)
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
Fig. 2. Schematic diagram of experimental set-up. integration with respect to time. These experiments consist of 33 cases. Three values of the wave skew-ness (α) were used; 0.453, 0.400
for determining either the velocity distribution or the overall drag on and 0.320. Moreover, the maximum flow velocity at free stream, U max the surface. And the velocity distribution near a rough bed for steady
ranges from 0.72 to 1.45 m/s. The sediment median diameters are flow is logarithmic. Therefore the usual log-law can be used to estimate
d 50 = 0.20 mm and d 50 = 0.74 mm and the wave periods are T = 3.0 s and the time variation of bottom shear stress τ ο (t) over rough bed as shown
T = 5.0 s.
by previous studies e.g., Jonsson and Carlsen (1976) , Hino et al. (1983) , Jensen et al. (1989) , Fredsøe and Deigaard (1992) and Fredsøe et al.
3. Turbulence model
(1999) . Moreover, some previous studies (e.g., Jonsson and Carlsen, 1976; Hino et al., 1983; Sana et al., 2006 ) also have shown that the
For the 1-D incompressible unsteady flow, the equation of motion values of bottom shear stress computed from the usual log-law and the
within the boundary layer can be expressed as momentum integral methods gave a quite similar, especially by virtue
ð1Þ 20% up to 60% in accelerating flow and overestimated by as much as
A A A of the phase difference in crest and trough values of the shear stress. s u 1 p 1
q Nevertheless, this usual log-law may be under estimated by as much as A t x þ q A z
At the axis of symmetry or outside boundary layer u = U, therefore 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
ð2Þ the bed shear stress for unsteady flow, as given by Soulsby and Dyer
A t¼ A tþ q A z
For turbulent flow,
Experiments have been carried out for four cases under saw-tooth
ð3Þ Table 1 . The maximum velocity was kept almost 400 cm/s for all the cases. The Reynolds number magnitude defined for each case has
waves. The experimental conditions of present study are given in
q ¼v A z
u VvV
q P uVvV may be expressed as q P uVvV¼ sufficed to locate these cases in the rough turbulent regime. Here, v is
The Reynolds stress
q v t ð A u=Az Þ, where ν t is the eddy viscosity. the kinematics viscosity, a m /k s is the roughness parameter, k s ,
And Eq. (3) became,
Nikuradse's equivalent roughness defined as k s = 30z o in which zo is the roughness height, a =U max
/σ, the orbital amplitude of fluid just
q ð vþv t Þ A z
above the boundary layer, where, U max , the velocity at wave crest, σ,
the angular frequency, T, wave period, S (=U o /(σz h )), the reciprocal of
the Strouhal number, z h , the distance from the wall to the axis of For practical computations, turbulent flows are commonly computed symmetry of the measurement section.
by the Navier–Stokes equation in averaged form. However, the averaging process gives rise to the new unknown term representing the transport of mean momentum and heat flux by fluctuating quantities. In order to determine these quantities, turbulence models are required. Two-equation turbulence models are complete turbu- lence models that fall in the class of eddy viscosity models (models which are based on a turbulent eddy viscosity are called as eddy viscosity models). Two transport equations are derived describing transport of two scalars, for example the turbulent kinetic energy k and its dissipation ε. 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–ω model was used to evaluate the turbulent properties to compare with the ex-
Fig. 3. Definition sketch for roughness.
perimental data.
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
The baseline (BSL) model is one of the two-equation turbulence UT ¼ F p ffiffiffiffiffiffiffiffiffiffiffiffi js 0 j=q is friction velocity and the parameter S R is related to models proposed by Menter (1994)
. The basic idea of the BSL k–ω model + the grain-roughness Reynolds number, k s =k s (U ⁎/v), is to retain the robust and accurate formulation of the Wilcox k–ω model
in the near wall region, and to take advantage of the free stream
s b25 and S independence of the k–ε model in the outer part of boundary layer. It R ¼ for k þ z 25
for k þ
ð12Þ means that this model is designed to give results similar to those of the
original k–ω model of Wilcox, but without its strong dependency on The instantaneous bottom shear stress can be determined using arbitrary free stream of ω values. Therefore, the BSL k–ω model gives
Eq. (4), in which the eddy viscosity was obtained by solving the results similar to the k–ω model of Wilcox (1988) in the inner part of
transport equation for turbulent kinetic energy k and the dissipation boundary layer but changes gradually to the k–ε model of Jones and
of the turbulent kinetic energy ω in Eq. (7). While, the instantaneous Launder (1972) towards to the outer boundary layer and the free stream
value of u(z,t) and v t can be obtained numerically from Eqs. (1)–(7) velocity. In order to be able to perform the computations within one set
with the proper boundary conditions.
of equations, the Jones–Launder model was first transformed into the k– ω formulation. The blending between the two regions is done by a
3.2. Numerical method
blending function F 1 changing gradually from one to zero in the desired region. The governing equations of the transport equation for turbulent
A Crank–Nicolson type implicit finite-difference scheme was used kinetic energy k and the dissipation of the turbulent kinetic energy ω
to solve the dimensionless non-linear governing equations. In order to from the BSL model as mentioned before are,
achieve better accuracy near the wall, the grid spacing was allowed to increase exponentially in the cross-stream direction to get fine
A u 2 resolution near the wall. The first grid point was placed at a distance
A t ¼ A z ð vþv t r kx Þ A z þv t A /(r z n bTxk ð5Þ of Δz 1 = (r − 1) z h − 1), where r is the ratio between two consecutive grid spaces and n is total number of grid points. The value of r was
2 selected such that Δz 1 A should be sufficiently small in order to maintain x A A x A u
A t ¼ A z ð vþv t r x Þ þg fine resolution near the wall. In this study, the value of Δz 1 is given
bx þ21 ð F 1 Þr x 2 x A z A z
equal to 0.0042 cm from the wall which correspond to z + = zU⁎/v = 0.01. It may be noted that in k–ε model where wall function method is used
From k and ω, the eddy viscosity can be calculated as to describe roughness the first grid point should be lie in the logarithmic region and corresponding boundary conditions should be
vk t ¼
applied for k and ε. In the k–ω model, as explained before the effect of x
roughness can be simply incorporated using Eq. (11). In space 100 and where, the values of the model constants are given as σ kω = 0.5,
in time 7200 steps per wave cycle were used. The convergence was
achieved through two stages; the first stage of convergence was based blending function, given as:
β ⁎ = 0.09, σ ω = 0.5, γ = 0.553 and β = 0.075 respectively, and F 1 is a
on the dimensionless values of u, k and ω at every time instant during
a wave cycle. Second stage of convergence was based on the maximum
F 1 ¼ arg 4 1 ð8Þ
wall shear stress in a wave cycle. The convergence limit was set to
1 × 10 −6 for both the stages.
where, "
4. Mean velocity distributions
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arg
500v
4r
Mean velocity profiles in a rough turbulent boundary layer under saw-tooth waves at selected phases were compared with the BSL k–ω here, z is the distance to the next surface and CD k ω is the positive
1 ¼ min max
0:09xz z 2 x
CD z 2 kx ð9Þ
model for the cases SK2 and SK4 presented in Figs. 4 and 5 , respectively. portion of the cross-diffusion term of Eq. (6) defined as
1 A k CD A x kx ¼ max 2r x 2 x 20 A z A z ; 10
Thus, Eqs. (2), (5) and (6) were solved simultaneously after nor- malizing by using the free stream velocity, U, angular frequency, σ
kinematics viscosity, ν and z h .
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=z h , ∂u/∂z = ∂k/
∂z = ∂ω/∂z=0). The k–ω model provides a natural way to incorporate the effects of surface roughness through the surface boundary condition.
The effect of roughness was introduced through the wall boundary condition of Wilcox (1988) , in which this equation was originally recognized by Saffman (1970) , given as follow,
x w ¼ UTS R =v
where ω w is the surface boundary condition of the specific dissipation ω at the wall in which the turbulent kinetic energy k reduces to 0,
Fig. 4. Mean velocity distribution for Case SK2 with α = 0.363.
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
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) ,
p ffiffiffi
uV¼ 1:052 k
ð13Þ 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–ω 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
Fig. 5. Mean velocity distribution for Case SK4 with α = 0.500. 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.
The solid line showed the turbulence model prediction while open and closed circles showed the experimental data for mean velocity profile 6. Bottom shear stress
distribution. The experimental data and the turbulence model show that the velocity overshoot is much influenced by the effect of acceleration 6.1. Experimental Results
and the velocity magnitude. The difference of the acceleration between the crest and trough phases is significant. The velocity overshooting is Bottom shear stress is estimated by using the logarithmic velocity
higher in the crest phase than the trough as shown at phase B and F for
distribution given in Eq. (14), as follows,
Case SK2 (α = 0.363). As expected this difference is not visible for
U⁎
symmetric case (Case SK4) (α = 0.500). Moreover, the asymmetry of the
ð14Þ flow velocity can be observed in phase A and E. Due to the higher 0
u¼ j ln z
acceleration at phase A the velocity overshooting is more distinguished where, u is the flow velocity in the boundary layer, κ is the von in the wall vicinity.
Karman's constant (= 0.4), z is the cross-stream distance from The BSL k–ω model could predict the mean velocity very well in the
theoretical bed level (z = y + Δz) ( Fig. 3 ). For a smooth bottom z o = 0, whole wave cycle of asymmetric case. Moreover, it predicted the velocity
but for rough bottom, the elevation of theoretical bed level is not a overshooting satisfactorily ( Fig. 4 ). For symmetrc case (Case SK4) as well
single value above the actual bed surface. The value of z o for the fully
Fig. 6. Turbulent intensity comparison between BSL k–ω model prediction and experimental data for Case SK2.
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
Fig. 7. Turbulent intensity comparison between BSL k–ω model prediction and experimental data for Case SK3.
rough turbulent flow is obtained by extrapolation of the logarithmic ing with acceleration effect. The increase in wave skew-ness causes an velocity distribution above the bed to the value of z = z o where u
increase the asymmetry of bottom shear stress. The wave without vanishes. The temporal variations of Δz and z o are obtained from the
skew-ness shows a symmetric shape, as seen in Case SK4 for α = 0.500 extrapolation results of the logarithmic velocity distribution on the
( Fig. 8 ).
fitting a straight line of the logarithmic distribution through a set of velocity profile data at the selected phases angle for each case. These
6.2. Calculation methods of bottom shear stress obtained values of Δz and z o are then averaged to get z o = 0.05 cm for all cases and Δz = 0.015 cm, Δz = 0.012 cm, Δz = 0.023 cm and
6.2.1. Existing methods
Δ z = 0.011 cm, for Case SK1, Case SK2, Case SK3 and Case SK4, There are two existing calculation methods of bottom shear stress respectively. The bottom roughness, k s can be obtained by applying
for non-linear wave boundary layers. The maximum bottom shear the Nikuradse's equivalent roughness in which z o =k s /30. By plotting u
stress within a basic harmonic wave-cycle modified by the phase against ln(z/z 0 ), a straight line is drawn through the experimental
difference is proposed by Tanaka and Samad (2006) , as follows: data, the value of friction velocity, U⁎ can be obtained from the slope
of this line and bottom shear stress, τ o can then be obtained. The
ð15Þ accuracy for application of logarithmic law in a wide range of velocity
2 f w Ut ð ÞjU t ð Þj
obtained value of Δz and z o as the above mentioned has a sufficient
Here τ o (t), the instantaneous bottom shear stress, t, time, σ, the profiles near the bottom. Suzuki et al. (2002) have given the details of
angular frequency, U(t) is the time history of free stream velocity, φ is this method and found good accuracy.
phase difference between bottom shear stress and free stream velocity Fig. 8 shows the time-variation of bottom shear stress under saw-
and f w is the wave friction factor. This method is referred as Method 1 tooth waves with the variation in the wave skew-ness parameter α. It
in the present study.
can be seen that the bottom shear stress under saw-tooth waves has an asymmetric shape during crest and trough phases. The asymmetry of bottom shear stress is caused by wave skew-ness effect correspond-
Fig. 8. The time-variation of bottom shear stress under saw-tooth waves. Fig. 9. Calculation example of acceleration coefficient, a c for sawtooth wave.
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
the BSL k–ω model results of bottom shear stress using following relationship:
U⁎t ðÞ
p f ffiffiffiffiffiffiffiffiffiffi
w =2 Utþ u
c ðÞ¼ r p ffiffiffiffiffiffiffi f w =2 A Ut
A ðÞ t
Fig. 9 shows an example of the temporal variation of the accel- eration coefficient a c (t) for α = 0.300 based on the numerical com- putations. The results of averaged value of acceleration coefficient a c from both experimental and numerical model results as function of the wave skew-ness parameter, α are plotted in Fig. 10 . Hereafter, an equation based on regression line to estimate the acceleration
coefficient a c as a function of α is proposed as:
Fig. 10. Acceleration coefficient a c as function of α.
ð20Þ Nielsen (2002) proposed a method for the instantaneous wave friction velocity, U⁎(t) incorporating the acceleration effect, as follows:
a c ¼ 036 ln a ðÞ
The increase in the wave skew-ness (or decreasing the value of α) r ffiffiffiffiffi
brings about an increase in the value of acceleration coefficient, a c . For
f sin u U⁎ t A ðÞ¼ w cos uU t U the symmetric wave where α = 0.500, the value of a is equal to zero. In
others words the acceleration term is not significant for calculating the bottom shear stress under symmetric wave. Therefore, for
s o ð Þ ¼ qU⁎ t t ð ÞjU⁎ t ð Þj
sinusoidal wave Method 3 yields the same result as Method 1. 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 τ ο (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, a c 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–ω model results. The instantaneous friction velocity, can be expressed as:
q ffiffiffiffiffiffiffiffiffiffi u U⁎ t A
ðÞ¼ f a w Ut =2 Utþ c r ðÞ þ r A t
Here, the value of acceleration coefficient a c is obtained from the
average value of a c (t) calculated from experimental result as well as
Fig. 12. Comparison among the BSL k–ω model, calculation methods and experimental Fig. 11. Phase difference between the bottom shear stress and the free stream velocity.
results of bottom shear stress, for Case SK1.
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
α = 0.500 in Eq. (24) yields the same result as Eq. (22). As seen in Fig. 11 the phase difference at crest, trough and average between crest and trough for Case SK4 with α = 0.500 is about 19.1°, this value agrees well with the result obtained from Eq. (22) as well as Eq. (24) for α = 0.500. The increase in the wave skew-ness or decreasing α causes 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–ω model, three calculation methods and experimental results of bottom shear stress under saw-tooth waves, for Case SK1, Case SK2, Case SK3 and Case 4, respectively.
Method 3 has shown the best agreement with the experimental results along a wave cycle for all saw-tooth wave cases. Method 2 slightly underestimated the bottom shear stress during acceleration phase for the higher wave skew-ness (Case SK1) as shown 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 BSL k–ω model, calculation methods and experimental results of bottom shear stress, for Case SK2.
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:
a 0:100 f ) w ¼ exp
7:53 þ 8:07 m z
u s ¼ 42:4C 0:153
for smooth : C ¼ j f w ; for rough : C ¼ q ffiffiffiffi
2 Re
u ¼ 2au s ð degree Þ
Where, φ 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 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
Fig. 14. Comparison among the BSL k–ω model, calculation methods and experimental effect under saw-tooth waves was included using Eq. (24). A value of
results of bottom shear stress, for Case SK3.
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
Here, Φ(t) is the instantaneous dimensionless sediment transport rate, ρ s is density of the sediment, g is gravitational acceleration, d 50 is median diameter of sediment, A is a coefficient, τ⁎(t) is the Shields parameter defined by (τ ο (t) / (((ρ s /ρ) − 1)gd 50 )) in which τ ο (t) is the instantaneous bottom shear stress calculated from both Method 1 and Method 3. While τ⁎ cr is the critical Shields number for the initiation of sediment movement ( Tanaka and To, 1995 ).
s ⁎ cr ¼ 0:055 1
ð26Þ Where, S ⁎ is dimensionless particle size defined as:
exp 0:09S 0:58
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð q
s =q
1 Þgd 50 3
ð27Þ The net sediment transport rate, q net , which is averaged over one-
4v
period is expressed in the following expression according to Eq. (25). U
q net
¼ AF ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s ⁎ cr gdt ð29Þ Here, Φ is the dimensionless net sediment transport rate, F is a
sign s
⁎t
0 f ðÞ gjs⁎ t ð Þj f js⁎ t ð Þj
function of Shields parameter and q net 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 |τ⁎(t)| N τ cr ⁎ and during the phase |τ⁎(t)| b τ cr ⁎ 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 (k s ) may be defined to be proportional to a characteristic grain size for evaluating the
friction factor. For sheet-flow sediment transport k s = 2.5 d 50 as shown by Swart (1974) , Nielsen (2002) and Nielsen and Callaghan (2003) .
Fig. 15. Comparison among the BSL k–ω model, calculation methods and experimental Therefore, in the present study the same relationship is used to
results of bottom shear stress, for Case SK4. 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
As expected, Method 1 yielded a symmetric value of the bottom Sato (2004) . The bottom shear stress calculated from Method 1 was shear stress at the crest and trough part for all the cases of saw-tooth
substituted into Eq. (29) and the result is shown in Fig. 16 by open waves. Moreover, the BSL k–ω model results showed close agreement
symbols. As expected that Method 1 yields a net sediment transport rate with the experimental data and Method 3 results. Therefore, Method 3
can be considered as a reliable calculation method of bottom shear stress under saw-tooth waves for all cases.
It can be concluded that the proposed method (Method 3) for calculating the instantaneous bottom shear stress under saw-tooth waves has a sufficient accuracy.
7. Application to the net sediment transport induced by skew waves
7.1. Sediment transport rate formulation The proposed calculation method of bottom shear stress is further
applied to formulate the sheet-flow sediment transport rate under skew wave using the experimental data 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 τ⁎(t) as given below:
U qt ðÞ
ðÞ¼ 0:5 t
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ¼ A sign s⁎ t f ðÞ gjs⁎ t ð Þj f js⁎ t ð Þj
s ⁎ cr g ð25Þ
ð q s =q
1 Þgd 50 Fig. 16. Formulation of sediment transport rate under skew waves.
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
Fig. 17. The relation between the net sediment transport rates and U max in variation of α Fig. 19. Comparison of experimental and calculation result of the net sediment transport for T = 3 s and d 50 = 0.20 mm.
rates in variation of maximum velocity U max and the wave skew-ness α for
d 50 = 0.20 mm and T = 5 s.
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
bottom shear stress and consequently yields a higher net sediment for calculating the net sediment transport rate under skew waves.
transport rate ( Fig. 17 ).
Furthermore, the relation between F and the dimensionless net Onshore and offshore sediment transport rate is shown in Fig. 18 sediment transport rate (Φ) obtained by the proposed method
along with the net sediment transport. In this figure the values of (Method 3) is shown in Fig. 16 by closed symbols. Since of the
U max , T and d 50 are fixed and only α has been changed. As obvious for a acceleration effect has been included in this calculation method (Eq.
wave profile without skew-ness (α = 0.500) the amount of onshore (18)), which causes the bottom shear stress at crest differ from that at
sediment transport is equal to that in offshore direction, therefore the trough, and therefore yields a net positive or negative value of F from
net sediment transport rate is zero. The difference between the Eq. (29). A linear regression curve is also shown in with the value of
onshore and the offshore sediment transport becomes more promi-
A = 11 (Eq. (28)). 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 d 50 = 0.20 mm in Fig. 19 . The characteristics of the net sediment transport induced by skew
7.2. Net sediment transport by skew waves
Recently, Nielsen (2006) applied an extension of the domain filter waves are studied using the present calculation method for bottom
method developed by Nielsen (1992) to evaluate the effect of shear stress (Method 3) and the experimental data for the sheet flow
acceleration skew-ness on the net sediment transport based on the sediment transport rate from Watanabe and Sato (2004) . Fig. 17 shows
data of Watanabe and Sato (2004) . A good agreement between
a comparison between the experimental data and calculations based calculated and experimental data of the net sediment transport was on Method 3 for the net sediment transport rates, q net and maximum
found using φ = 51°, a value much different from the usual notion that velocity, U max for the wave period T = 3 s and the median diameter of
the phase difference is of the order of 10 o for rough turbulent wave sediment particle d 50 = 0.20 mm along with the wave skew-ness
boundary layers.
parameter (α). It is clear that an increase in the wave skew-ness and Figs. 20 and 21 show the correlation of the net sediment transport the maximum velocity produces an increase in the net sediment
experimental data from Watanabe and Sato (2004) and the net 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
Fig. 20. Correlation of the net sediment transport experimental data from Watanabe Fig. 18. Change in amount of sediment transport rate according to an increasing α.
and Sato (2004) and the net sediment transport calculated by the present model.
1112
Suntoyo et al. / Coastal Engineering 55 (2008) 1102–1112
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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