Geoscience Reference
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and it moves with the free surface, i.e.,
dx
dt
u hx , dy
dt
u hy , dz
dt
=
=
=
u hz
(2.70)
where u hx , u hy , and u hz are the components of flow velocity at the water surface in
the x -, y - and z -directions. Thus, differentiating Eq. (2.69) with respect to t leads to
the following kinematic condition of free surface:
z s
u hx
z s
u hy
z s
+
x +
=
u hz
(2.71)
t
y
The three-dimensional sediment transport equation (2.44) closed using Eq. (2.46)
is rewritten as
c
+ ∂(
u x c
)
+ ∂(
u y c
)
+ ∂(
u z c
)
∂(ω
s c
)
=
s
c
+
s
c
ε
ε
t
x
y
z
z
x
x
y
y
ε
+
s
c
(2.72)
z
z
In general, Eq. (2.72) is approximately applicable to all sediment loads (if fine
enough) in the entire water column. However, because bed load and suspended load
behave differently, the water column is often divided into two zones: a bed-load zone
from the bed elevation z b to z b + δ
and a suspended-load zone from z b + δ
to z s ,
as shown in Fig. 2.6. Here,
is the thickness of the bed-load zone, which is usually
assumed to be about twice the sediment diameter (Einstein, 1950) or half the bed-form
height.
The net vertical sediment flux across the water surface should be zero and, thus, the
suspended-load boundary condition at the water surface is
δ
ε
s c
s
c
z + ω
z s =
0
(2.73)
z
=
There are usually two approaches to specify the suspended-load boundary condition
at the interface between the suspended-load and bed-load zones. One approach is to
assume the near-bed suspended-load concentration to be at equilibrium:
c
|
z b + δ =
c b
(2.74)
z
=
where c b
is the equilibrium (capacity) sediment concentration at the interface.
The other approach is to assume that the near-bed sediment entrainment flux is at
the capacity of flow picking up sediment under the considered flow conditions and
 
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