Geoscience Reference
In-Depth Information
2.4.3 Section-averaged 1-D model equations
The cross-section-averaged quantity
of a three-dimensional variable
φ
is defined by
A
φ
z
s
b
2
b
1
φ
1
A
1
A
=
dA
=
dydz
(2.100)
z
b
where
A
is the flow area in the cross-section, as shown in Fig. 2.7.
Integrating the 3-D continuity equation (2.60) over the cross-section leads to
z
s
b
2
z
s
b
2
z
s
b
2
∂
∂
u
y
∂
∂
u
x
∂
u
z
∂
x
dydz
+
y
dydz
+
z
dydz
=
0
(2.101)
z
b
b
1
z
b
b
1
z
b
b
1
which is reformulated to the following 1-D continuity equation by applying the Leibniz
rule, the non-slip condition (2.68) at the channel bed and banks, and the kinematic
condition (2.71) at the water surface:
AU
∂
A
∂
+
∂(
)
=
0
(2.102)
t
∂
x
where
U
is the flow velocity averaged over the cross-section, defined by Eq. (2.100).
Integrating the 3-D momentum equation (2.66) over the cross-section yields the 1-D
momentum equation:
AU
AU
2
(
T
xx
+
D
xx
∂(
)
+
∂(
)
gA
∂
˜
z
s
1
ρ
∂
[
A
)
]
1
ρ
(
=−
x
+
+
B
ˆ
τ
−
χ
ˆ
τ
)
(2.103)
sx
bx
∂
t
∂
x
∂
∂
x
where
T
xx
is the normal stress averaged over the cross-section,
D
xx
is the dispersion
momentum transport,
B
is the channel width at the water surface,
χ
is the wetted
perimeter,
ˆ
τ
sx
is the wind driving force per unit horizontal area at the water surface,
ˆ
τ
bx
is the shear force per unit area of bed and bank surfaces.
The turbulent stress term in Eq. (2.103) is usually ignored, because it is much weaker
than the convection term. The dispersion term is often combined with the convection
term by introducing a correction factor. In inland rivers, the wind driving force usually
is negligible. Therefore, the resulting 1-D momentum equation is
and
AU
AU
2
+
∂(β
∂(
)
)
gA
∂
˜
z
s
1
ρ
χ
ˆ
τ
=−
x
−
(2.104)
bx
∂
t
∂
x
∂
β
=
A
u
2
dA
AU
2
β
where
is the correction factor for momentum, defined as
/(
)
, with
u
being the streamwise flow velocity in the 3-D model.
