Geoscience Reference
In-Depth Information
Summing Eqs. (2.108) and (2.110) leads to the 1-D mass balance equation of
total load:
AC
t
β
AUC
t
p
m
)
∂
A
b
∂
+
∂
∂
+
∂(
)
(
1
−
=
0
(2.111)
t
t
∂
x
t
where
C
t
is the total-load concentration averaged over the cross-section, defined as
C
t
A
s
UC
AU
β
β
=
(
Q
b
+
)/(
)
; and
t
is a correction factor for total load, related to
s
AC
t
AC
)
U
β
/ β
/ β
and
U
b
by
=
/(
+
Q
b
/
U
b
)
=
1
/
[
r
s
+
(
1
−
r
s
/
U
b
]
, which is similar
t
s
s
to Eq. (2.92).
2.4.4 Effects of sediment transport and bed change
on flow
Recall that the aforementioned 1-D, 2-D, and 3-D hydrodynamic equations ignore
the effects of flow density and bed change by assuming that the sediment con-
centration is low and that the bed varies much more slowly than the flow. This
assumption is not valid for high shear flows with strong sediment transport. In
addition, the flow density varies with salinity, temperature, and other factors.
In general, the 3-D hydrodynamic equations with a variable flow density
are
Eqs. (2.25) and (2.26), which are rewritten in the Cartesian coordinate system shown
in Fig. 2.6 as
ρ
∂ρ
∂
+
∂(ρ
u
x
)
+
∂(ρ
u
y
)
+
∂(ρ
u
z
)
=
0
(2.112)
t
∂
x
∂
y
∂
z
u
x
)
∂(ρ
u
x
)
+
∂(ρ
+
∂(ρ
u
y
u
x
)
+
∂(ρ
u
z
u
x
)
−
∂
p
x
+
∂τ
+
∂τ
+
∂τ
xx
xy
xz
=
F
x
∂
t
∂
x
∂
y
∂
z
∂
∂
x
∂
y
∂
z
(2.113)
u
y
)
+
∂(ρ
∂(ρ
u
y
)
+
∂(ρ
u
x
u
y
)
+
∂(ρ
u
z
u
y
)
y
+
∂τ
x
+
∂τ
y
+
∂τ
F
y
−
∂
p
yx
yy
yz
=
∂
∂
∂
∂
∂
∂
∂
∂
t
x
y
z
z
(2.114)
u
z
)
∂(ρ
u
z
)
+
∂(ρ
u
x
u
z
)
+
∂(ρ
u
y
u
z
)
+
∂(ρ
−
∂
p
z
+
∂τ
+
∂τ
+
∂τ
zy
zx
zz
=
F
z
∂
t
∂
x
∂
y
∂
z
∂
∂
x
∂
y
∂
z
(2.115)
Like Eq. (2.63), the
z
-momentum equation (2.115) can be simplified to the hydro-
static pressure equation (2.64) for gradually varied (shallow water) flows. When
the flow density is variable in the vertical direction, Eq. (2.64) has the following
solution:
z
s
z
ρ
p
=
p
a
+
gdz
(2.116)
