SHEETFLOW SEDIMENT TRANSPORT UNDER ASYMMETRIC WAVES AND STRONGCURRENTS
Le Phuong Dong1 and Shinji Sato1
Experiments have been conducted to investigate the sheetflow sediment transport of uniform sand under asymmetricoscillatory flows in combination with relatively strong opposite currents. Two kinds of nearshore waves were
performed, namely, velocity asymmetric waves and acceleration asymmetric waves. Image analysis technique isutilized to study major influences of wave shapes and current through observing the instantaneous sheetflow layer
thickness. Maximum sheetflow layer thickness was formulated and incorporated to an enhanced Watanabe and Sato’s
formulation. The new conceptual model is examined its validity for a wide range of experimental conditions
Keywords: sheetflow; oscillatory flow; sediment transport; net transport rates formula; asymmetric waves; currents

INTRODUCTION

Interaction of nearshore waves and strong currents is often observed at the river mouths wheremean current velocity of outflow uc is often comparable to the maximum near bottom orbital velocity
uw. For instance, at Ba Lat river mouth, the largest tributary of Red River Basin, Vietnam, measured
data show that the ratio between uc/uw is often larger than 0.5 (Pruszak et al. 2005). This phenomenon
thus effects to the sediment transport and topography changes in these areas. Under such conditions,
sheetflow transport regime is predominant as the Shields parameter is often large enough to wash out
sand ripples and transport sediment in a thin layer of a few mm to cm with high sand concentration
along the bed. Since observing and measuring the sheetflow sediment transport in the field is somehow
difficult because it occurs very much closed to the sea bottom, indoor oscillatory flow tunnel
measurement is an effective and economical approach to study various physical aspects of sheetflow
sand movement. Due to the limitation of experimental facilities, few studies have been conducted withwaves superimposed currents. Dohmen-Janssen (1999) is among the first study that measured thesediment transport under sinusoidal waves combined with relatively strong currents (Fig.1). However,in the surf zone (river mouth regions), asymmetric waves are dominant. Both field measurement (Elgar
and Guza 1985) and wave flume data (Sato et al. 1992) show that velocity asymmetric waves with sharp
crest and flat trough (Fig 2a) are developed just around the breaker line, whereas the acceleration
asymmetric waves with saw-tooth shapes are formed near the shore. As a consequence, the existence ofdifferent wave shapes may lead to different sediment transport behaviors. Up to now, various papers
have been published on such topic, however, there have been a small number of studies investigating
about sediment transport under such kind of waves and steady currents with relatively small velocity
against waves. Dibajnia and Watanabe (1992) carried out a series of oscillatory experiments with andwithout steady current to study the effect of wave nonlinearity in sediment transport. However, thesuperimposed current is restricted as the current to wave amplitude ratio is smaller than 0.3 (Fig.2). In
the experiments performed with acceleration asymmetric oscillation, Watanabe and Sato (2004) added
opposite steady currents to illustrate the effects of undertow but the ratio uc/uw is also smaller than 0.3(Fig 2). Still the existing researches on net transport rate models under collinear asymmetric wavesstrong currents produce the undecided conclusion due to the lack of concrete evidences. This study
attempts to deliver new experiments performed with different asymmetric oscillatory flow conditions in
combination with relatively strong superimposed current generated in the opposite direction with waves(focus of this study is highlighted by red solid symbols, Fig.1). It aims to further examine the role ofstrong opposite current on the net transport rates under oscillatory flow conditions. On the basis ofimage analysis technique, instantaneous sheetflow layer thickness was measured and its maximum
values are formulated. Incorporating the new expression for maximum sheetflow layer thickness, a newmodel based on Watanabe and Sato (2004)’s conception has been proposed and its applicability for awide range of experimental conditions is verified.

1

Department of Civil Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan;

1

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COASTAL ENGINEERING 2010

Figure 1. Existing studies and present experiments on sheetflow sediment transport under waves andcurrents (positive sign means currents were generated in the same direction with waves and vice versa).

Figure 2. A typical bottom velocity profile of : a) Velocity asymmetric waves. b) Acceleration asymmetricwaves

EXPERIMENTS

A set of experimental data has been obtained using the Oscillatory Flow Tunnel (OFT) at the
University of Tokyo. The OFT consists of a loop-shaped closed conduit and a hydraulically driven
piston. Negative or positive current to oscillatory flow can be generated through a circulation systemthat controlled by a hydraulic pump. The rectangular horizontal tunnel test section is 5.7 m in length,
height of 23.3 cm and width of 7.6 cm. The central 4.3 m of the test section with mild slopes at bothends is filled with 4.0 cm deep uniform sand (d50 = 0.2 mm). Asymmetric oscillations with the period
T= 3 s and the wave amplitude uw varied from 0.6 to 1.3 m/s (Fig.1) have been generated with different

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COASTAL ENGINEERING 2010

wave shapes based on the third order cnoidal wave theory. The waves shape is characterized by theasymmetric velocity index Rv:

Rv 

umax
umax  umin

(1)

and the dimensionless acceleration asymmetry index i:

i  1 

Tpi

(2)

Ti

where u is the flow velocity; subscript (i = c or t ) denotes for crest or trough with Ti being the halfwave period (s); Tpi is the time from the flow reversal to the maximum velocity at each half cycle(Fig.2). Noted that the definition of βi in Eq.(2) satisfies βc +βt =1. In total, two kinds of both velocity
asymmetry waves with Rv = 0.6 and 0.8 and acceleration asymmetry waves with c = 0.55 and 0.68
were performed. Relatively strong superimposed currents are generated in opposite direction withwaves at velocity of 0.3 and 0.5 m/s which results in the current to wave amplitude ratio between -1 and-0.3 as shown in figure 1. Table 1 listed experimental setup performed in this study. In total, 18
experiments were performed and no bed forms were generated for all cases that assure the sheetflow
regime. Experimental processes were recorded with a high speed video camera (400 frames/s). Net
transport rates were measured for 17 cases, while case 18 were performed to examine the calibration
relationship between brightness and sand concentration. The net transport rates were estimated on thebasis of sand mass difference in two halves of the tunnel after an experimental duration Δt as follows(Dibajnia et al. 2001):

qmeas 

M on  M off(3)2b s twhere qmeas is measured net sediment transport; ∆Mon, ∆Moff is the difference in the sand mass on theonshore and offshore side before and after experiment, respectively; b is the width of test section; ρs =2.65 g/cm3 is the sediment density.
Table 1. Experimental conditions in the present study.
Case
01
02
03
04
05
06
07
08
09
10
11
12
13
14
15
16
17
18

d50
(mm)
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2

T
(s)
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.6

uw
(m/s)
1.00
1.20
0.70
1.00
0.60
1.30
1.00
1.30
0.60
1.30
0.60
0.58
0.83
1.00
0.50
0.71
0.93
1.27

Rv

𝛃c

0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.5

0.55
0.55
0.55
0.55
0.55
0.55
0.68
0.68
0.68
0.68
0.68
0.5
0.5
0.5
0.5
0.5
0.5
0.5

uc
(m/s)
-0.31
-0.31
-0.30
-0.50
-0.50
-0.50
-0.51
-0.52
-0.50
-0.50
-0.50
-0.50
-0.50
-0.50
-0.50
-0.50
-0.50
0.00

Δt (s)
45.00
34.50
37.54
34.50
34.75
30.80
36.83
39.17
35.08
34.56
31.15
31.12
32.10
31.31
33.18
34.30
34.31
-

qmeas
(cm2/s)-0.566-0.480-0.232-1.664-0.818-1.814-0.687-0.231-0.647-0.177-0.844-0.833-1.562-1.445-0.431-0.446-0.986-

In order to study the sand motions processes, image analysis technique was utilized. The back sidewall of test section was painted in black, so sand particle is the only source of light reflection. A larger
brightness value may correspond to a higher sand concentration and vice versa. It is expected that thechanges of sand concentration may change the spatial distribution of brightness. At the unmovable bed,
the brightness reaches the maximum value where the maximum sand concentration is located. In orderto establish a relationship between sand concentration and brightness, the illumination conditions were
kept stable for all experiments. It can be achieved by adjusting beforehand the angles of spot lights and4

COASTAL ENGINEERING 2010

the aperture of camera’s curtain. Figure 3 illustrates the relationship between time average brightness
and the sand concentration for experimental case 18. This experiment was performed under the same
condition with case 1-1 in Horikawa et.al (1982) in which sand concentration was spatially measured.
In the present study, sheetflow layer thickness is of interest since it is experimentally observed that 8090% of sand is transported in this layer (Dohmen-Janssen et al. 2001). Many researchers (e.g, DohmenJanssen et al. 2001, O'Donoghue and Wright 2004a) defined such thickness as the distance between
undisturbed beds to the elevation where sand concentration equals to 8vol% (approximate 200g/l). The
temporal sheetflow layer thickness in present study is estimated in analogy with previous researches but
was correlated with brightness value. Since the sheetflow layer thickness can be measured through acertain ratio between suspended sand concentration and the maximum sand concentration at stationary
bed, it is also expected that the thickness of sheet layer can also be measured by considering the relative
relationship of temporal distribution of brightness above the bottom with the maximum value atunmovable bed. From figure 3, it is found that at the elevation where sand concentration equal to 200
g/l the brightness value reduces to 95. This brightness value corresponds to approximate 55% of
maximum brightness value (bmax = 175). Sheetflow layer thickness is measured here as the distance
between undisturbed bed where brightness is relatively maximum and the level where brightness
relatively reduces to 55% maximum brightness value.
However, it may be argued that it is somehow difficult to keep the experimental conditions stable
for all the tests. In order to further examine this, experimental condition 18 was repeated 3 times but thebrightness was varied by changing the aperture of camera in each test. Figure 4 shows the smoothed out
temporal sheetflow layer thickness for experimental case 18 under different illumination conditions(bmax). As can be seen from the figure, even for large different between maximum brightness values,
measurement of maximum sheetflow layer thickness are acceptable as the errors are as small as 1mm.
Therefore, it is concluded that a small fluctuation of illumination condition may not affect so much to
the experimental results.

Figure 3. Relationship between measured time average sand concentration and brightness for case 18

Figure 4. Temporal sheetflow layer thickness estimated under different illumination conditions for case 18

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COASTAL ENGINEERING 2010
MEASURED SHEETFLOW LAYER THICKNESS

Figures 5 to 7 show some examples of instantaneous sheetflow layer thickness that is measured in
the present study. Here we considered the top of sheetflow layer thickness which is visually determinedby investigating several peaks of instantaneous sheetflow layer thickness. In some cases, for example infigure 6a, it is observed that there is a maximum sheetflow layer thickness corresponding to a shortduration peak occuring at around flow reversal. We assumed that it was caused by rather highturbulence due to flow deceleration and may not reflect the effect of flow intensity. These peaks
therefore were removed. Figure 5 shows the influence of wave nonlinearity to sheetflow layer thicknessfor experimental case 13 and case 05 of which their maximum velocities are similar but wave
nonlinearities are different (Rv = 0.5 for case 05 and Rv = 0.6 for case 13, respectively). It seems that the
maximum sheetflow layer thickness increases with increasing asymmetric velocity index Rv. It isbecause for the higher wave nonlinearity, during the crest period, a larger amount of sand are entrained
and remained above the sheetflow layer due to higher onshore velocity. Such amount of sand cancontribute to an increase of sheetflow layer thickness when the offshore velocity occurs. Therefore it
could be said that the sheetflow layer thickness is proportional with wave nonlinearity index Rv.
Figure 6 illustrates the influence of steady current to the development of sheetflow layer thickness.
It is found that presence of strong steady current which is results in larger ratio uc/uw also increase thesheetflow layer thickness. For example, the sheetflow layer thickness for experimental case 04 is about
50% higher than that of case 01. The maximum sheetflow layer thickness is frequently related to theflow intensity (through Shield parameter) (Wilson 1989, Dohmen-Janssen 1999). It is noted that thedifference of flow intensity (~u2) in these two cases is about 30% and it cannot accurately reflect thechange of sheetflow layer thickness. However, this phenomenon can be explained by the concept of“pick up time” (Dohmen-Janssen 1999) which comes from the fact that flow requires sometime to erode
the bed and entrain the sediment to the maximum possible level. For the same flow condition, if Tc or Tt
is shorter compared to such “pick up time”, the sediment load entrained to flow is limited by theavailable time to erode the sand bed. The appearance of currents in opposite direction with waves could
enlarge Tt and hence enlarge the pick-up time for sheetflow layer thickness completely developed. Inorder to take into account the changes of times length due to appearance of steady current, a parameter
kTt is introduced as follows: T 
kTt   t 
T  t ,w 

0.4(4)

where Tt is the half wave period of collinear waves and currents and Tt,w is the half wave period for purewaves only. kTt increases with increasing the ratio uc/uw and without current, it returns unity. Therefore
we assumed it can represent for appearance of steady currents.
The influence of time length to the development of sheetflow layer thickness can be more clearlyseen in figure 7 where it demonstrates the instantaneous sheetflow layer thickness for different
acceleration asymmetric waves. The flow loads in these two experiments are the same but sheetflow
layer thickness is observed larger for the case with smaller βt (higher forward leaning degree). This isbecause for smaller βt, the acceleration time length Tpt (see Fig.2) is also longer and flow load has
more time to develop the thickness of sheetflow layer. This is identical with what observed by
Watanabe and Sato (2004) as the sheetflow layer thickness increases with increasing βc (or decreasingβt). The sheetflow layer thickness hence is expected that it is inversely proportional with βt.

6COASTAL ENGINEERING 2010

Figure 5. Instantaneous sheetflow layer thickness and particle for: a) case 13 and b) case 05

Figure 6. Instantaneous sheetflow layer thickness and particle for: a) case 01 and b) case 04

Figure 7. Instantaneous sheetflow layer thickness and particle for: a) case 04 and b) case 07

Figure 8 shows a new relationship of maximum sheetflow layer thickness as a linear function of
maximum Shield parameter Ψmax, waves shapes indices βt, Rv as well as kTt. A fairly good agreementwith the experiments shows that the maximum sheetflow layer thickness can be expressed as follows:s 24 Rv kTt max
d50
2t

(5)

where kTt is computed by Eq.(4); Ψmax is the maximum Shields parameter computed with maximum
velocity umax = uw,min + uc; the friction factor fw is calculated by method of Swart (1974) using ut as thevelocity amplitude and T = 2Tt as the oscillatory period. In sheetflow condition, the roughness height is

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COASTAL ENGINEERING 2010

often assumed as of the order of the sheetflow layer thickness (e.g, Wilson 1989, Nielsen 1992, Van
Rijn 1993, Ribberink 1998). This might be caused by the intensive energy dissipation in the sheetflow
layer, due to the grain-grain and grain-fluid interactions. Here, we considered the roughness height ks
defined by Wilson (1989):

ks  5 s d50(6)where Ψs is the Shields parameter estimated with ks = d50 and the equivalent sinusoidal velocity
amplitude ut (Wilson et al. 1995). Note that Eq.(5) is derived for the case that currents aresuperimposed in opposite direction with waves. For the conditions when current is in direction withwaves, we can apply the Eq.(5) but the subscript t (trough) should be changed to c (crest).

Figure 8. Measured non-dimensional sheetflow layer thickness and prediction by equation (5)NET SAND TRANSPORT FORMULA OF WATANABE AND SATO (2004)

In 1992, Dibajina and Watanabe (DW) proposed a conceptual model based on the assumption thatthe net transport rates should take into account the exchange process of suspension height during one
half cycle to the next half cycle. If the sand entrained and suspended by the positive (negative) velocity
reach high beyond a critical height, a part of it still remained in suspension when flow reverses and canbe transported into the negative (positive) direction. Since then, several subsequence models relied on
DW’s conception have been proposed of which Watanabe and Sato (2004) is among the latest version.
This version improved the formula of Dibajnia et.al (2001) by taking into account the influence ofacceleration asymmetric waves and expressed in the form as follows:

s 

qs
u ' T (  2  c t' )  u 't Tt (t  2 t c' ) 0.0019 c c c(7)
w0 d50T sgd50

where s is the immersed specific gravity of sediment; d50 is the sediment grain size; other parameters are
determined by Eqs.(8) to (12).

ui  2  urms i ; u 'i  ui / 2(1  i )(8)

i  q 'i min(1, cr /  'i ) ;  'i  q 'i max(0,1  cr /  'i )(9)qi  iTi sg / d50 ; q 'i  qi / [2(1  i )](10)

i  (i  cr )d50 / ( w0T j ) ;  'i  ( 'i  cr ) / 2i

(11)

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COASTAL ENGINEERING 2010

i  ui2 / (2sgd50 )   i / f w(12)Here ui, qi, ωi, θi are the equivalent velocity amplitude, representative suspension intensity, relativesuspension height and mobility number, respectively. Other parameters with prime marks (’) are
modified to take into account the influence of acceleration asymmetric waves. Ψi is the Shield
parameter, fw is the friction factor, θcr is the critical mobility number for the onset of general movement(θcr = Ψcr/fw ≈ 5; Ψcr = 0.05).
The sediment exchange between both half-cycles is controlled by the parameter 'i, once its criticalvalue, cr = 1.2 is exceeded. Thus, the net transport rate for each half-cycle is influenced by amount ofsand that is entrained (through q’i) and amount of sand that is exchanged to next half-cycle (through'i).
Fig. 9 shows the comparison of net transport rate estimated by WS04 model and with measured
data for the cases with weak current ( uc / uw ≤ 0.3) and for the cases with strong currents (uc / uw ≥ 0.3).
The comparison comprises 260 up to date oscillatory flow measurement found in literature together
with experiments in present study. To evaluate the model, we relied on the root mean square error(RMSE) between prediction and measurement (Ahmed and Sato 2003):

RMSE 

2
1 N
qcomp  qmeasN n 1

1 N 2
 qmeasN n 1(13)

where qcomp(meas) is computed (measured) net transport rates; N is the number of experiments.
Computational results reveal that the accuracy of WS04 is quite good if waves are dominants(RMSE = 0.53). However, considering the experiments performed with strong currents in this study, it
failed to predict not only the magnitude but also the direction of net transport rate (even if magnitude
problem seems to be more significant). Hence more detailed study have been made as follows in order
to figure out reason of WS04’s failure as well as to derive a new formula by modifying this formula set

Figure 9. Measured and predicted transport rates by WS04 model for: a) cases with weak current and b)
cases with strong currents. Three oblique lines demonstrate: the perfect agreement between prediction andmeasurement (center line), and where prediction equal to 50% and 200% of experimental measurement (twoother lines), respectively

9COASTAL ENGINEERING 2010
A NEW SAND TRANSPORT RATE FORMULA FOR WAVES AND CURRENTS
Suspension height and settling velocity in Dibajnia and Watanabe conception

Figure 10 shows the variation with uw,max of the four net transport rate components c, t, ’c, ’t
computed from c, t, ’c, and ’t using WS04 model for experimental conditions: uc = -0.5 m/s, c =0.55, Rv = 0.5:

c (t )  0.0019

 'c (t )  0.0019

u 'c (t ) Tc (t ) c (t )T sgd50

2 t ( c )u 't ( c ) Tt ( c )  'c (t )T sgd 50

(14)

(15)

It is clearly seen that sediment is carried to offshore direction mainly by contribution of t (foruw,max < 1.5 m/s, ’c = 0). Meanwhile, c and ’t represent for portions of sand deposited in onshore
direction but 't is of two to three times larger than c (Fig.10). The reason is that flow load in crest
cycle is smaller than that in trough cycle due to appearance of the strong opposite current. Hence, theamount of sand entrained to suspension under the action of positive velocity, q’c, is also much smaller
than the component, q’t. Therefore, if neglecting c component, it seems that the failure of WS04
model mainly comes from an underestimation of the sum (t − ’t) or t / ’t . Referring to the definitionof t and ’t, we may express as:t 
 't 

u 't Tt q 't cr( )T sgd50  'tu 'c Tc q 't

T sgd50

(1 

cr
)
 't

(16)

(17)

For a designed wave combined current profile, the ratio u’tTt / u’cTc is known, hence it is notablethat an underestimation of t /’t is resulted from an overestimation of ’t or that of representativesuspension height, 't in DW type model because of:

t

1
1

 cr 
 't  't / cr 1  't  't
 'i 

 'i
w0Ti

ui2
 'i 
2 sg 2  i

(18)

(19)

(20)

Representative suspension height Δ’i is derived based on assumption that the flow kinetic energy(Ek = ρVui2/2) is completely transferred to the required potential energy (Ep = (ρs−ρ)VΔi) to raise up thesand particles through the strong but confined eddies existing inside the water body. Derivation of DW
concept for Δ'i can be understood as it represents for the height at gravity centre of suspended sandalong vertical position. Such gravity centre of suspended sand particle should be correlated with the
sheetflow layer thickness. This is sound in the sense that almost sands are contained and transported
within the sheetflow layer (Dohmen-Janssen, 1999). However, as shown in figure 11, sheetflow layer
thickness measured in present study is about 4-6 times larger than predicted by Eq.(20). This is because
sheetflow layer thickness could be limited by the turbulent kinetic energy dissipation due to high sandconcentration inside sheetflow layer (Da Silva et al. 2006). So far, it is somehow difficult toquantitatively estimate the turbulent part. Therefore, it is expected that a more appropriate description

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COASTAL ENGINEERING 2010

for representative suspension height which is matched with observation could help improve theperformance of WS04 model.
Another parameter that controls ω’i is the settling velocity which is assumed equal to free settling
velocity at clear water. In sheetflow condition, this seems unacceptable since such settling velocity
gradually decreases due to high sand concentration inside the sheetflow layer. Under suchcircumstances, the hindered settling velocity in high sand concentrations proposed by Richardson andZaki (1954) is widely accepted:

whs  (1  Cv )n w0(21)

Here, whs is the settling velocity due to hindered settling and Cv is the volumetric sedimentconcentration. w0 is the free settling velocity of sediment particle at still water. The exponent n = 2.3 4.6 depends on sand Reynolds number (Fredsøe and Deigaard 1992, p200). For very fine to medium
sands, van Rijn (1984) suggested a value n = 4.
Throughout the sheetflow structure, measurements from Dohmen-Janssen (1999) and O’Donoghue
and Wright (2004a) show that there is a location where sand concentration is more or less constantthrough wave cycles. This elevation is described as the top of pick-up layer which corresponds to theboundary between the pick-up layer where the concentrations is in anti-phase with the main flow andthe upper sheetflow layer where the concentrations tends to be in phase with the main flow.
Experiments with different sand sizes reveal that the concentration value at that position lies around
600g/l. In addition, sand concentrations in sheetflow are approximately the same for different sands andonly the thickness of sheetflow layer varies (Dohmen-Janssen et al. 2001). Meanwhile, the top of pickup layer at which concentration is more or less constant can be seen as a fixed point about which sandconcentration profile pivot during the flow cycles (O'Donoghue and Wright 2004a). This means thecentre of sheetflow layer thickness might be located around this point. Therefore, it is simply
considered that the sand settling velocity at this elevation can be represented for the mean settling
velocity inside the sheetflow layer. Using Eq.(21) with n = 4, for the fine and medium sand size, if thesuspended sediment concentration reaches 600g/l, the hindered settling velocity could reduce to 36% ofsettling velocity of sand particle at clear water . The mean settling velocity inside the sheetflow layerthen can be expressed as:

whs  0.36w0(22)

The aim to briefly determine hindered settling velocity is that it may help to figure out the criticalvalue of cr which will be described in detail in next section.

Figure 10. Contribution of net transport components computed by WS04 model and measured data forexperimental condition: uc = -0.5 m/s, 𝛃c = 0.55

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COASTAL ENGINEERING 2010

Figure 11. Measured non-dimensional sheetflow layer thickness and prediction by Eq.(20)

New net transport rate formula

The expression of sheetflow layer thickness was introduced in a new net sediment transport
formulae which is rearranged to appear in similar form with WS04 model as follows:

s 

qs
u ' T (  2  c t' )  u 't Tt (t  2 t c' ) 0.3 c c c
w0 d50T sgd50(23)i  q 'i
 'i  cr 
 'i  0





  

i  q 'i  cr  1   i  1  cr  
 '     'i    'icr 
 i
 ' 
 ' j   i q 'i i cr


 'i


(24)

i  Rv kTi   i   cr  ;  'i  i / 2i

(25)

i 

i d50w0Ti

qi  iTi;  'i  'i d50
2i w0Tiqi
sgq 'i 
2(1  i )
d50

 T 
kTi   i T  i,w 

(26)

(27)

0.4

(28)

12

COASTAL ENGINEERING 2010 c  1  4 uc
 c
ut

if

ut
 0.25uc

u
if t  0.25
uc t  1& 
  4 uc
 t
ut

if

uc
 0.25ut

u
if c  0.25
ut

(29)

Here, i is the Shields parameter; fw is the friction factor computed with velocity ui and oscillation
period 2Ti following the friction law developed by Swart (1974) with the relative roughness heightcomputed by Eq.(6); cr = 0.05 is critical Shields parameter for onset of the initial movement; kTi is anintensive parameter which being considered that the presence of superimposed currents may enlarge or
reduce pick up time; Rv is asymmetric velocity index; i is the velocity leaning index; i is proposed by
Dibajnia et al. (2001) after considering whether kinetic energy of this half cycle is high enough to carry
the remaining sand of previous cycle. The critical cr is determined by considering that the exchange
process will occur whenever the falling time of suspended sand within the representative suspension
height or sheetflow layer thickness longer than a representative half-cycle period 2βiTi: 'i
12 iTi whs(30) 'i  24 'i d50(31)

24 'i d50
 1  67 'i  1  cr  0.015
0.36w0 *2iTi(32)

Comparison between measurements and computations

Figure 12 shows the net transport rates computed by present formula with 260 experimental data
being taken into account. The overall score for predictions with strong current cases (Fig.12b) isapparently much better. However, for that of weak current cases (Fig.12a), it is attributed that the newmodel does not show the well performance in comparison with WS04 model. Table 2 compares the root
mean square error between non-dimensional net transport rates calculated by current formula and WS04
model. Nevertheless, the evaluation is further divided into fine sand (d50 < 0.2mm) and coarser sand(d50 ≥ 0.2mm). The results precisely display that in circumstance of coarse sand (d50 ≥ 0.2 mm), newmodel not only helps improve the accuracy for predicting net transport rates under second order Stokes- cnoidal waves combined with weak currents, but also significantly enhances the prediction capacity
for collinear waves and strong currents cases. The large errors merely occur for finer sand (d50 <0.2mm). Dohmen-Janssen et al. (2002) and O’Donoghue and Wright (2004b) have found that under thesame flow conditions, sheetflow layer thickness for very fine sand (d50 = 0.13 - 0.15 mm)
approximately doubled that of coarser sand (d50 0.2 mm). There is no available appropriate
explanation and it might indicate the different transport for very fine sand, i.e., plug flow may easily
occur with fine sand (Sleath 1999). This is not accounted in the new model and therefore may lead to its
larger error. In order to take into consideration such unknown behavior for fine sand, it is assumed thatthe sheetflow layer thickness for fine sand cases (d50≤0.15mm) is two times larger than that of forcoarser sand as observed by prior researches. It is though that Δ’i is in linear relationship with θ’i(shown in Eq.31), modification is made for mobility number θi, (θ’i ) in Eq.(25) as follows:i ( 'i ) ( d 0.15mm )  2i ( 'i )( d 0.2mm )
50

(33)

50

For sand size: 0.15 < d50 < 0.2 mm, θi, (θ’i ) can be estimated by interpolating between mobility
numbers for (d50≤0.15mm) and that of (d50 0.2 mm) ; i.e:i ( 'i )  1   d50  0.15  0.2  0.15 i ( 'i )( d 0.2mm)
50

(34)

Figure 13 demonstrates the net transport rates estimated by modified formula. Undoubtedly theoverall score for fine sand under weak currents cases is drastically improved although a larger error is
obviously seen for a small number of experiments (N=6) under strong current cases. Until now, theCOASTAL ENGINEERING 201013

reason has not clearly found yet due to scarce experiment. Hence, further research is highly required
and added experiments should be conducted so as to examine the sand transport behavior for very finesand.
Table 2. The prediction error (RMSE) of models

WS04
New model
New model with modified 𝜃i (𝜃’i)uc/uw < 0.3
finecoarse
N=35
N=191
0.760
0.460
1.073
0.389
0.621
0.389

uc/uw ≥ 0.3
finecoarse
N=6
N=28
0.173
0.743
0.213
0.304
0.304
0.720

Figure 12.Comparison between computations by new formula and 260 up to date experimental cases

Figure 13.Comparison between computations by new formula with modified θi(θ’i) and 260 up to dateexperimental cases

14

COASTAL ENGINEERING 2010

CONCLUSIONS

Comprehensive experiments were conducted to investigate the behavior of sheetflow sedimenttransport under asymmetric waves with strong opposite currents, which have not been explored by any
prior researches. Major aspects of sediment transport under asymmetric waves and currents have beenstudied through introducing a new technique to measure the sheetflow layer thickness on basis of image
analysis. Net transport rates measurement in present study revealed that Watanabe and Sato(2004)’s
formula fails to deliver an accurate prediction. The reason pointed out due to the overestimation ofrepresentative suspension height in their formula. Maximum sheetflow layer thicknesses are formulated
and incorporated in a new enhanced DW type net transport rate model. The new model has been
examined with high caution. In addition, its validity is verified over a wide range of experimental data.
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