Bailard and Inman (1981, 1984) formula

A. Bayram et al. / Coastal Engineering 44 (2001) 7999

97

A.4. Bailard and Inman (1981, 1984) formula

A.5. Van Rijn (1984, 1993) formula

Bailard and Inman (1981) extended the formula

Van Rjin (1984) presented comprehensive formu-

introduced by Bagnold to oscillatory flow in combi-

las for calculating the bed load and suspended load,

nation with a steady current over a plane sloping

and only a short description of the method is given in

bottom. The instantaneous bed load ( q0b,BI) and sus-

the following. For the bed load he adapted the

pended load ( q0s,BI) transport rate vectors are ex-

approach of Bagnold assuming that sediment particles

pressed as,

jumping under the influence of hydrodynamic fluid

"

#

forces and gravity forces dominate the motion of the

02 0 tanb 03

0:5f wqeb

bed load particles. The saltation (jumps) character-

istics were determined by solving the equation of

motion for an individual sediment particle. The bed

"

5 #

load can be defined as the product between the

3

0:5f wqes 0 0 es

particle concentration (cb; a reference concentration

for the bed load different from the reference concen-

s

tration for suspended load ca), the particle velocity

in which tanb is the local bottom slope, tanc a

(ub), and the layer thickness (db; taken to be equal to

dynamic friction factor, Ut0 the instantaneous velocity

the reference level a) according to,

vector near the bed (wave and current) and ib is a unit

qb;VR cbubdb

vector in the direction of the bed slope. Averaging

over a wave period, the total transport rate and

where,

direction are obtained containing both the wave- and

cb

T

current-related contributions. Assuming that a weak

0:18

longshore current prevails, neglecting effects of the

D

c0

slope term on the total transport rate for near-normal

!1=3

incident waves, the local time-averaged longshore

s 1g

D d50

sediment transport rate is (Bailard, 1984),

v2

dv

eb

d3

qt;BI 0:5qfwu3

s0b;wc sb;cr

v

0

qs q g tanc 2

T

sb;cr

es

dvu

0:5qfwu4

0

3

qs q gws

in which c0 ( = 0.65) is the maximum bed load

concentration, D * the dimensionless grain diameter,

where eb and es are efficiency factors, and:

T the excess bed shear stress parameter, and s0b,wc is

the effective bed shear stress for waves and current

V

dv

combined (calculated according to Van Rijn's own

u0

method, not discussed here). Substituting the above

expressions into the bed load transport formula

hjUt0j3i

u

together with some other relationships not given

3

u0

yields,

sffiffiffiffiffiffiffiffiffi "

#1:5

s0b;wc s0b;wc sb;cr

The following coefficient values are typically used

qb;VR 0:25cqsd50D0:3

in calculations: eb = 0.1, es = 0.02, tanc = 0.63. Thus,

q

sb;cr

the efficiency factors are assumed to be constant,

although work has indicated that eb and es are related

where,

rffiffiffiffiffi

to the bed shear stress and the particle diameter. It

Hs

should also be noted that the formula is derived for

c1

h

plane bed conditions.