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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