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266
Proceedings of the 7th International Conference on Asian and Pacific Coasts
(APAC 2013) Bali, Indonesia, September 24-26, 2013
A NEW MODEL FOR SEDIMENT TRANSPORT RATE UNDER
OSCILLATORY SHEET FLOW CONDITIONS
X. Chen1 and X. Yu
2
ABSTRACT: A general and simple unsteady type empirical formula for non-cohesive sediment movement under
oscillatory sheet flow conditions is obtained based on integration of assumed concentration and velocity profiles. The
formula includes basic erosion depth and free stream velocity as parameter but is in accordance with the traditional
formula expressed as 3/2 power of Shields parameter. An important factor including erosion depth and oscillatory
boundary layer thickness also appears in the formula due to integration of the non-uniform velocity profile. The erosion
depth formula applied is advanced with a special consideration to the effects of acceleration, phase-lag and sediment
size. The concentration profile obeys traditional exponential law based on mass conservation, and the velocity profile in
a significant part over the initial bed is similar to that in a laminar flow. Data used for validation are measured from
oscillating water tunnels covering quite a wide range of flow and sediment conditions. Several widely used empirical
formulas, steady and unsteady types, are used to compare with the present formula for both instantaneous and net
sediment transport rate. Results show that present formula is valid for both instantaneous and net sediment transport rate.
The process of the instantaneous rate is shown to vary synchronously with free stream velocity, and the decreasing rate
caused by increasing sediment size is also correctly predicted. The direction and magnitude of net transport rate are
correctly predicted, while existing instantaneous type formulas are failed in direction for fine sediment under pure
velocity-skewed flows. Net sediment transport rate is shown to be affected by not only the acceleration, sediment sizes,
phase-lag, but also the difference of boundary layer thickness between onshore stage and offshore stage.
Keywords: Sediment transport rate; oscillatory sheet flow; empirical formula; erosion depth.
1 State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing, 100084, CHINA. 2 State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing,
100084, CHINA.
INTRODUCTION
Both velocity skewness and acceleration skewness
are usually observed in nearshore oscillatory flows
induced by waves. When water depth decreases, the
wave crest is sharpened and the wave trough is flattened
to produce positive velocity skewness. When wave
enters into the surf zone, the wave front is steepened and
the wave rear is gentled, then positive acceleration
skewness occurs. The sediment transport becomes much
complex under the oscillatory sheet flows induced by
waves with velocity skewness and acceleration skewness
not only because the transport rate becomes significantly
larger but also because the mechanisms of sediment
transport is less clear. Accurate prediction of sediment
transport rates is essential for environmental and
morphological studies in coastal and estuarine areas. For
these reasons, many empirical formulas have been
proposed for sediment transport rate under oscillatory
sheet flow conditions. The accuracy of those formulas,
however, is still not enough and further studies are still
necessary for a better description of the sediment
transport in the nearshore areas.
The empirical models about net sediment transport
under wave generated sheet flows could be related to the
velocity or shear stress exerted on the bed by the fluid
over it. Quasi-steady models without considering phase-
lag between concentration and flow velocity include
those of Bailard (1981), Soulsby (1997), Ribberink
(1998), Nielsen and Callaghan (2003), Nielsen (2006),
Gonzalez-Rodriguez and Madsen (2007). Semi-unsteady
models include those of Dibajnia and Watanabe (1992),
Dohmen-Janssen et al. (2002), Ahmed and Sato (2003),
Watanabe and Sato (2004), Silva et al. (2006), all of
which account for the phase-lag between the
concentration and flow velocity in some way.
Acceleration is also known to play an important role in
the sediment transport process, and many quasi-steady or
semi-unsteady models considered its effects (Nielsen and
Callaghan, 2003; Nielsen, 2006; Silva et al., 2006;
Gonzalez-Rodriguez and Madsen, 2007). In spite of
many models proposed, it is still lack a model that can
well predict both the instantaneous and net sediment
transport rates, because many factors as sediment size
X. Chen and X. Yu
267
and phase-lag are hard to be reflected in the
instantaneous transport rate.
The present study aims to develop an empirical
model to predict both the instantaneous and net sediment
transport rates. Special consideration will be given to the
effects of acceleration, phase-lag, sediment size, and
oscillatory boundary layer thickness. Based on the
temporal and spatial description of both velocities and
concentrations above the immobile bed during the wave
period, a full unsteady empirical model on the sediment
transport under oscillatory sheet flow will be integrated.
The formula will be applied to pure velocity-skewed
flow and pure acceleration-skewed flow to isolate the
effects of velocity skewness and acceleration skewness
to sediment transport. In the following text, the empirical
model is introduced in Section 2, while the predictions
and discussions are shown in Section 3, about both
instantaneous and net sediment transport rate under a
wide range of sheet flow and sediment conditions as
velocity amplitudes, wave periods, wave shapes and
sediment sizes. Finally, the conclusions are drawn in
Section 4.
NEW MODEL
An instantaneous sediment transport rate relationship
may be obtained by integrating the concentration profile
and velocity profile. The profile of the free stream
velocity in general is shown in Fig. 1, where
U represents the velocity and T is the period of the
oscillatory flow; subscript c and t indicate the crest and
trough duration, respectively; subscript a and d denote
acceleration and deceleration duration, respectively.
Positive is the onshore direction, and negative the
offshore direction. Based on the mass conservation, the
sediment concentration profile can be well approximated
by the following exponential law as shown by Chen et al.
(2013),
exp 1, m
E
yS Sy t (1)
t
U
0
Tc Tt
Tac Tdc
Tat Tdt
Uc
Ut
T
Fig. 1 Free stream velocity duration in present study
where S is the volumetric concentration and subscript
m denotes the maximum value; y is the vertical
coordinate with its origin at the undisturbed bed;
2 T is the angular frequency; E is the erosion
depth given by
1 2E
m mt
tD
(2)
In Eq. (2), D is the median sediment diameter; t is the
time; is Shields parameter defined by
2 ,
2 1
wf U t
s gD (3)
where s is the sediment specific gravity; g is the
gravitational acceleration; wf is the wave friction factor.
It is assumed that wf varies linearly between its values
under wave crest cf and under wave trough tf evaluated
by Wilson’s (1989) formula:
0.4
,,
,
0.05464 1
c tc t
ac at
Uf
T s g (4)
Fig. 2 Minimum erosion depth
A New Model for Sediment Transport Rate under Oscillatory Sheet Flow Conditions
268
Fig. 3 Phase lag validation
By neglecting the velocity history before flow
reversal, the boundary layer then develops from the time
of flow reversal to the flow peak, similar to that induced
by a quarter of the sinusoidal flow (Liu and Sato, 2006;
Silva et al., 2006; Gonzalez-Rodriguez and Madsen,
2007). Parameters 1 and 2 in Eq. (2) denote the
minimum value and variation range of the erosion depth,
satisfying 1 2 3 so that 1.5max 3( / ) mE D .
Since O’Donoghue and Wright’s (2004a) data can be
well fitted by 2.5min 0.5( / ) mE D at the minimum
erosion depth as shown in Fig. 2, we take 1 0.5 m
when 6m . For large m , i.e., 6m , as shown in
the experiment of Dick and Sleath (1992), and also in
that of O’Donoghue and Wright (2004a), very small
variation of the erosion depth can be observed because
the suspended sediment is too hard to fall down. Thus,
we can take 1 3 . In Eq. (2), is the phase lag. As
seen in Fig. 3, O’Donoghue and Wright’s (2004a) data
leads to /S w , representing the time ratio
between sediment falling down and the wave period
introduced by Dohmen-Janssen (1999) and Dohmen-
Janssen et al. (2002). Here w is the sediment falling
velocity given by van Rijn (1993); 2S E is the sheet
flow layer thickness obtained by letting in Eq. (1)
mS =0.6 and S = 0.08 as the top of sheet flow layer
following Dohmen-Janssen (1999).
The velocity profile may be approximately expressed
as,
,1 exp 4.6
,
E
B
y t y
t
U
U (5)
0.82
0.09B
N N
A
k k (6)
where B is the wave boundary layer thickness given by
Fredsoe and Deiggard (1992); A varies linearly between
its values under wave crest and under wave trough
expressed by , , ,2 /c t c t ac atA U T ,respectively; Nk is the
roughness height related to given by Wilson (1990).
It can be seen , 0.99 ,B EU t U t . Eq. (5) is
close to the laminar flow solution, but without a 45º
phase lead between the immobile bed surface and the
outer free stream velocity. None phase lead may not be
bad because the phase lead in practice is actually not so
large. It is reported to be 15º-20º in Dick and Sleath
(1991, 1992) and Zala-Flores and Sleath (1998), and an
average of 21.5º in O'Donoghue and Wright (2004b).
Using Eq. (5) and (1), we obtain an expression for the
instantaneous sediment transport rate
, ,/
4.6
4.6E
m E
B E
y t y tS U
S U dy (7)
with E is given by Eq. (2), U given by Eq.(5) and B
given by Eq.(6). /B E is computed by integrating the
non-uniform velocity profile. It is noted that m ES U .
Net sediment transport rate could be obtained by
averaging Eq. (7) within a period. The non-dimensional
form is
2
3 /
4.6
4.6 11
m E
B E
S U
D s gDs gD (8)
Considering S = 0.08 at Ey , in present model
instantaneous bedload sediment transport rate can be
easily derived as
9.2 /
, ,
4.60.135 1
4.6 4.6
E
E
E B
b
m EE B
B E B E
y t y t
S u
S U dy
e (9)
If Eq. (3) and a linear relation /E D (Asano,
1992; Zala-Flores and Sleath, 1998) are used, Eq. (8)
becomes 1.5 when /B E is nearly a constant.
This is in accordance with the classical formula of
Meyer-Peter and Müller (1948): 1.5
8 cr , and
similar to the formulas of Nielsen (1992), Soulsby
(1997) and Ribberink (1998), where subscript cr
denotes critical value.
RESULTS AND DISCUSSIONS
We now compare the present model with
experiments and existing empirical models including: (1)
the instantaneous formula of Ribberink (1998) without
acceleration, and Nielsen (2006) accounting for velocity
acceleration in :
X. Chen and X. Yu
269
0.5 cos sin /wU t f U t U t (10)
and, (2) the net transport formula of Silva et al. (2006)
accounting for acceleration, which is based on Dibajinia
and Watanabe (1992) considering phase-lag between
concentration and flow velocity
3 *3 3 *3
0.55
2
c c c t t t t c
c c t t
n
U T UT
U T UT
(11)
where and n are empirical constants; denotes the
sediment carried up and taken away by present half
cycle; * denotes the sediment carried up by present half
cycle but taken away by next half cycle. In the following
tables and figures, experiment data is shorted as ‘Exp.’
and present model result is shorted as ‘Pres.’; empirical
formulas are shorted as ‘R98’, ‘S06’, and ‘N06’.
Instantaneous transport rate is discussed first in this part
under different wave shapes, sediment sizes and periods,
all of which are aimed to explain their contributions to
net sediment transport rate.
Instantaneous Transport Rate
Experimental and predicted net sediment transport
rate are also shown in Table 1, in which Case FA5010 to
CA7515 are pure velocity-skewed flows measured by
O’Donoghue and Wright (2004a, b) and Li et al. (2008)
under the free velocity
0 0cos( ) cos2( )1.2 0.3U t t t t (12)
Case S556015f to S706015c are pure acceleration-
skewed flows measured by vander A et al. (2010a) under
16
0 1 /1.3 2 1 sini
iU a i t i (13)
where 0 0U , and max max min/( )U U U .
Fig. 4 shows the predicted instantaneous bedload
sediment transport rate at different D and T by different
models for pure velocity-skewed flows. The response of
sediment movement to flow is very well as the transport
rate variation almost following the free stream velocity
in all cases. Predicted phase shift in Nielsen (2006) is
Table 1 Predictions of net sediment transport rate
caused by in Eq. (10). The bedload transport rate
process was predicted very well by the present empirical
formula by comparison with all experiments; while it
was not enough in fine sediment cases for the other two
formulas. Predicted transport rate of long period
(T=7.5s) is slightly smaller than of short period (T=5s)
due to decreasing wave friction factor. It should be
realized that the onshore or offshore transport rate
decreases with the increase of sediment size under the
same flow conditions, while only present model predicts
this tendency owing to decreasing erosion caused by
increased sediment size in Eq. (2). Phase-lag effects are
increased with increasing U, decreasing T and D
(Dibajnia and Watanabe, 1992; Dohmen-Janssen et al,
2002; O'Donoghue and Wright 2004a, b; vander A et al.,
2010a). The phase-lag effect of fine sediment is more
significant than the medium and coarse sediment. Fine
sediment is easy to be picked up to a relatively large
height above the bed in the suspension layer and hard to
fall down, which leads to erosion depth at offshore stage
as large as the onshore stage for the fine sediment case.
In addition, boundary layer thickness at onshore stage
with large Uc and kN should be larger than that at
offshore stage, so that periodic averaged near bed
velocity is negative in O’Donoghue and Wright (2004b).
These make experimental transport rate in the sheet flow
layer near offshore flow peak (t/T=0.6~0.8) very close to
that near onshore flow peak (t/T=0.15~0.3) though the
free stream velocity is much smaller in Fig. 4 (1)-(2).
Only present model could predict offshore transport rate
correctly with 1.51 m in Eq. (2) denoting sediment
amount could not fall down, and relatively small
offshore wave boundary layer thickness in Eq. (7)
denoting relatively larger near bed velocity. Negative net
sediment transport rates for fine sediment could be
predicted by present model as seen in Table 1. While for
Test case (mm2/s)
Exp. Pres. R98 N06
FA5010 -128.3 -117.4 78.1 52.0
LA612 -61.0 -99.6 67.5 48.9
FA7515 -88.3 -58.8 56.6 45.5
MA5010 52.6 64.6 54.2 68.2
MA7515 35.9 47.6 47.4 58.4
CA5010 44.1 77.4 60.1 85.3
CA7515 33.8 44.3 51.7 71.8
S556015f 24.2 23.3 0.0 12.6
S706015f 100.2 61.7 0.0 36.2
S556015m 4.4 14.3 0.0 15.7
S706015m 41.2 42.5 0.0 47.9
S556015c 4.2 9.9 0.0 19.6
S706015c 30.6 28.1 0.0 63.2
A New Model for Sediment Transport Rate under Oscillatory Sheet Flow Conditions
270
the coarse sediment case with relative high settling
velocity, sediment is hard to be picked up, and
suspended sediment entrained at high velocity would fall
back to the bed quickly as the flow velocity decreases.
The maximum E near onshore flow peak
(t/T=0.15~0.3) is significantly greater than the maximum
E near offshore flow peak (t/T=0.6~0.8), which leads
to a much stronger onshore rate than offshore rate in the
sheet flow layer. Phase-lag effects of medium sediment
case are between the fine and coarse sizes, and the ratio
of maximum onshore rate to maximum offshore rate
should be between them. In the experiment, positive net
transport rate is caused by summation of large onshore
rate and small offshore rate; all formulas can predict the
process as seen in Table 1 and Fig. 4.
Fig. 5 shows the predicted instantaneous sediment
transport rate of different D and β for pure acceleration-
skewed flows under the same T=6s. Present predicted
instantaneous transport rate follows the free stream
velocity very well, and the larger onshore transport rate
than offshore transport leads to positive net transport rate,
which is also seen in two-phase modeling of Hsu and
Hanes (2004). With the same D in present model, larger
β corresponds to larger onshore transport rate, smaller
offshore transport rate, and thus larger positive net
transport rate. Such positive net transport rates, and its
increasing tendency with increasing β, are well in
accordance with experiment in Table 1. One reason
would be the relatively larger onshore near bed velocity
represented by relatively smaller wave boundary layer
thickness in Eq. (7). Furthermore, observed onshore
shear stress is larger than offshore shear stress under
sawtooth waves as in Suntoyo et al. (2008). As
mentioned by Nielsen (1992) and vander A et al. (2010b),
larger peak bed shear stress is generated in the crest half
period Tac, because the boundary layer has less time to
develop before Uc comes. Accordingly, in the trough half
period Tat the peak bed shear stress is smaller because
the boundary layer has much more time to develop
before Ut comes. This effect is included in present model
by defining different friction factors for the wave crest
and trough in Eq. (4). This is another important reason
for the well estimation of all sediment sizes’ net
transport rate by present model. When D increases,
present predicted transport rate decrease because
decreased erosion depth in Eq. (2) with the decreased
shear stress, and less sediment is entrained to be carried
out. The Ribberink (1998) cannot predict the different
between onshore and offshore duration because the
formula does not include the acceleration skewness
effect. Nielsen (2006) result shows obviously difference
between onshore and offshore durations, and the
difference is larger when β increases. But it increases
when the sediment size increases due to increasing wave
friction factor, the prediction cannot agree well with
experiments for all sediment sizes. Over all in Fig. 5,
instantaneous rate of Nielsen (2006) fits present model
for medium sediment case better than other size cases,
while Ribberink (1998) fits present model for coarse
sediment case better than other size cases.
Net Sediment Transport Rate
With much more data, special discussion about net
sediment transport rate is conducted in the section.
Collected net sediment transport data include half
periodic sinusoidal flows (Sawamoto and Yamashita,
1986); pure velocity-skewed flows: 1st order cnoidal
flows (Dibajinia, 1991; Ahmed and Sato, 2003), 2nd
order Stokes flows (Ribberink and Al-Salem, 1994,
1995; O’Donoghue and Wright, 2004a, b; Li et al.,
2008); pure acceleration-skewed flows: sawtooth flows
(Watanabe and Sato, 2004; vander A et al., 2010a).
Prediction of net sediment transport rates by different
models are shown in Fig. 6, and the relevant errors are
shown in Table 2, where PD, P2 and P5 denote the
prediction percentage of correct direction, between twice
error (dashed line), and between fifth error (dashed-
dotted line); root-mean-square error is computed by 1 22[ ( ]) /rms PRED EXPE n , and logarithm error is
computed by 1 22
2{ }[log ( / )] /L PRED EXPE n . If
the direction is not correctly predicted, /PRED EXP is
given 0.01 / EXP to avoid negative in logarithm
function. Negative transport rate could not be predicted
without phase-lag effect in Ribberink (1998) and Nielsen
(2006) formulas, their rmsE and LE are larger than the
Silva et al. (2006) formula which includes phase-lag and
acceleration effects. With comprehensive consideration
about the effect of acceleration skewness, wave
boundary layer thickness and phase-lag between
concentration and flow velocity, net sediment transport
rates is predicted very well by present formula. Though
present net transport rate is an integration of Eq. (7), its
rmsE and LE are even less than that of Silva et al.
(2006) based on half period prediction.
Tabel 2 Errors for net sediment transport rate
PD(%) P2(%) P5(%) LE rmsE (mm2/s)
R98 42.9 32.8 42.2 8.6 66.8
N06 86.4 49.4 85.7 4.4 62.4
S06 93.5 59.7 89.0 2.8 42.7
Pres. 97.4 60.4 90.3 2.1 37.0
CONCLUSIONS
An empirical formula for sediment transport under
oscillatory flow conditions is derived by integration of
given concentration and velocity profiles in the present
X. Chen and X. Yu
271
study. An advanced erosion depth formula with special
consideration to acceleration, phase-lag and sediment
size is used in the exponential law of concentration
profile based on mass conservation. The velocity profile
is similar to that in a laminar flow but without a 45º
phase ahead of the free stream velocity near the initial
bed. The transport rate formula obtained includes basic
erosion depth and free stream velocity which could be
related to the traditional 3/2 power law in terms of
Shields parameter. In addition, the formula includes an
important factor of erosion depth and oscillatory
boundary layer thickness due to integration of non-
uniform velocity profile.
Sediment transport in pure velocity-skewed flows
and pure acceleration-skewed flows are well predicted,
and the effects of velocity skewness and acceleration
skewness are isolated. The tendency of transport rate
decreasing with the increase of sediment size under the
same flow conditions is well predicted by the present
model, because decreasing erosion depth is correctly
performed. Large offshore transport rate for fine
sediment in pure velocity-skewed flows could also be
predicted correctly by present model with well expressed
minimum erosion depth caused by phase-lag, and
relatively larger offshore near bed velocity represented
by relatively small offshore wave boundary layer
thickness. For pure acceleration-skewed flows with
β>0.5, positive net sediment transport rate and its
increasing tendency with increasing β are both correctly
predicted by present model. One reason is the expressed
larger onshore near bed velocity represented by
relatively smaller onshore wave boundary layer
thickness. The other is because larger peak bed shear
stress in crest half period than trough half period with
less time developed boundary layer is well expressed by
present friction factor definition.
(1) FA5010: D=0.13mm, T=5.0s (3) MA5010: D=0.27mm, T=5.0s (5) CA5010: D=0.46mm, T=5.0s
(2) FA7515: D=0.13mm, T=7.5s (4) MA7515: D=0.27mm, T=7.5s (6) CA7515: D=0.46mm, T=7.5s
Fig. 4 Predicted sediment transport rate for O’Donoghue and Wright (2010) experiments
A New Model for Sediment Transport Rate under Oscillatory Sheet Flow Conditions
272
(1) S556015f: D=0.15mm, β=0.58 (3) S556015m: D=0.27mm, β=0.58 (5) S556015c: D=0.46mm, β=0.58
(2) S706015f: D=0.15mm, β=0.70 (4) S706015m: D=0.27mm, β=0.70 (6) S706015c: D=0.46mm, β=0.71
Fig. 5 Predicted sediment transport rate for vander A et al. (2010a) experiments
(1) Ribberink (1998) (2) Nielsen (2006)
X. Chen and X. Yu
273
(3) Silva et al. (2006) (4) Present model
Fig. 6 Net sediment transport rate prediction
In summary, the present formula is validated widely
by data for both instantaneous and net sediment transport
rates. The magnitude and direction of sediment transport
under a wide range of flow and sediment conditions
could be well described by the present model. The
effects of acceleration, sediment sizes, wave boundary
layer thickness and phase-lag between concentration and
flow velocity are all considered. Both instantaneous and
net sediment transport rates can be predicted very well.
ACKNOWLEDGEMENTS
The study is supported by China Postdoctoral
Science Foundation under grant No. 2012M520291 and
State Key Laboratory of Hydro-science and Engineering,
Tsinghua University under grant No. 2011-KY-1.
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