Geoscience Reference
In-Depth Information
bed sediment configurations:
z
=
z
b
+
δ
=
ω
s
∂
c
E
b
=−
ε
s
c
b
∗
(2.75)
∂
z
where
E
b
is the entrainment flux of sediment at the interface. Correspondingly, the
deposition flux at the interface is defined as
D
b
=
ω
s
c
b
, in which
c
b
is the suspended-
load concentration at the interface between the suspended-load and bed-load zones.
Note that
E
b
and
D
b
are defined per unit area of horizontal plane rather than bed
surface; the bed surface may be curved, whereas
E
b
and
D
b
are along the vertical
direction.
Eqs. (2.74) and (2.75) are often called “concentration” and “gradient” boundary
conditions, respectively. Eq. (2.74) is applicable for equilibrium sediment transport at
the interface, while Eq. (2.75) is applicable for both equilibrium and non-equilibrium
sediment transports. In particular, for equilibrium transport,
D
b
=
E
b
and Eq. (2.75)
becomes Eq. (2.74). Therefore, Eq. (2.75) is more general than Eq. (2.74). More
discussions about the near-bed suspended-load boundary condition are given in
Sections 2.5.2 and 7.3.1.
2.4.1 Depth-averaged 2-D model equations
Depth-averaged hydrodynamic equations
The depth-averaged quantity
of a three-dimensional variable
φ
is defined by
z
s
z
b
φ
1
h
=
dz
(2.76)
Integrating the continuity equation (2.60) over the flow depth yields
z
s
z
s
z
s
∂
u
y
∂
∂
u
x
∂
∂
u
z
∂
x
dz
+
y
dz
+
z
dz
=
0
(2.77)
z
b
z
b
z
b
which is reformulated using the Leibniz integral rule as
z
s
z
s
∂
∂
u
hx
∂
z
s
u
bx
∂
z
b
x
+
∂
u
hy
∂
z
s
u
by
∂
z
b
∂
u
x
dz
−
x
+
u
y
dz
−
+
x
∂
∂
∂
y
∂
y
y
z
b
z
b
+
u
hz
−
u
bz
=
0
(2.78)
Substituting boundary conditions (2.68) and (2.71) into Eq. (2.78) leads to the
depth-integrated 2-D continuity equation:
∂
h
+
∂(
hU
x
)
+
∂(
hU
y
)
∂
=
0
(2.79)
∂
t
∂
x
y
where
U
x
and
U
y
are the depth-averaged quantities of local velocities
u
x
and
u
y
, defined
