86
A. Bayram et al. / Coastal Engineering 44 (2001) 7999
performed. A Shields curve was employed to deter-
being finer) and d90 = 0.24 mm (diameter correspond-
mine the criterion for the initiation of motion based on
ing 90% being finer) as typical values (Birkemeier et
h, which was included in the formulas that have this
al., 1985).
feature.
The roughness height was estimated in the follow-
ing manner:
4. Evaluation of the formulas
Measured hydrodynamics were employed as much
Flat bed: r is set equal to 2.5d50, where d50 is
as possible in the formulas to reduce the uncertainties
the median grain size (Nielsen, 1992)
in the transport calculations. The rms wave height and
Rippled bed: r is calculated from the ripple
peak spectral wave period was used as the character-
height and length and the Shields parameter
istic input parameters to quantify the random wave
according to Nielsen (1992)
field. Values at intermediate locations where no meas-
Sheet flow: r is calculated from the Shields
urements were made were obtained by linear interpo-
parameter and d90 according to Van Rijn
lation (note that this is the cause for the discontinuities
(1984), where d90 is the grain size that 10%
in the derivative of the calculated transport rate dis-
of the sediment exceeds by weight.
tributions). It was assumed that the incident wave
angle was small, implying that the angle between the
waves and the longshore current was approximately
on r using the formula proposed by Swart (1976),
90. In the VR formula the undertow velocity is
which is based on an implicit relationship given by
needed if the resultant shear stress is calculated (i.e.,
Jonsson (1966), assuming rough turbulent flow,
the shear stress resulting from the cross-shore and
0:194
longshore currents combined). No undertow measure-
r
r
ments were available and the model of Dally and
lnfw 5:98 5:2
for :
< 0:63
ab
ab
Brown (1995) was employed to calculate this velocity.
r
The influence of the shear stress from the undertow
fw 0:3
! 0:63
for :
ab
was typically small compared to that of the longshore
1
current and waves. In a few cases extrapolation of the
current from the most shoreward or seaward measure-
ment point was needed. On the shoreward side the
where ab is the amplitude of the horizontal near-bed
current was assumed to decrease linearly to become
water particle excursion. For the purpose of compar-
zero at the shoreline, whereas at the seaward end the
ing the predictive capabilities of the formulas, coef-
current was taken to be proportional to the ratio of
ficient values proposed by the developers and
breaking waves (i.e., assuming that most of the
coworkers were employed without any particular
current was wave-generated in this region).
tuning of the coefficients. Predicted transport rates
The roughness height (r), which determines the
with the formulas were converted to mass flux per unit
friction factors for waves and current, is a decisive
width before comparison with the measurements.
parameter that may markedly influence the sediment
transport rate, especially the bed load transport. Here,
4.1. Comparisons with DUCK85 data
the calculation of the roughness height was divided
into three different cases depending on the bottom
Although simulations were carried out for all runs
conditions, namely flat bed, rippled bed, and sheet
listed in Table 2, only four of the runs were selected
flow. The division between these cases was made
for detailed discussion here. However, the overall
based on the Shields parameter (h), where h < 0.05
conclusions given are based on the results from all
implied flat bed, 0.05 < h < 1.0 rippled bed, and h > 1.0
simulations. Fig. 4 shows predicted cross-shore dis-
sheet flow (Van Rijn, 1993). An iterative approach
tributions of the longshore sediment transport rate
was needed because the bottom conditions are not
with the different formulas for the experimental run
known a priori when the roughness calculation is
at 0957 on September 5, 1985 (denoted 859050957
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