Geoscience Reference
In-Depth Information
2-D numerical models
If the vertical (or lateral) variations of flow and sediment quantities in a water body are
sufficiently small or can be determined analytically, their variations in the horizontal
plane (or longitudinal section) can be approximately described by a depth-averaged
(or width-averaged) 2-D model. Presented in this chapter are the governing equations,
boundary conditions, and numerical solutions of the depth-averaged and width-
averaged 2-D models of flow and sediment transport in open channels, as well as
the enhancement of the depth-averaged 2-D model to account for the effects of helical
flow on fluvial processes in curved and meandering channels.
6.1 DEPTH-AVERAGED 2-D SIMULATION OF FLOW
IN NEARLY STRAIGHT CHANNELS
6.1.1 Governing equations
For shallow water flows with low sediment concentration, the depth-averaged 2-D
hydrodynamic equations are Eqs. (2.79), (2.82), and (2.83). In the case of nearly
straight channels, the dispersion momentum transports due to the vertical non-
uniformity of flow velocity are combined with the turbulent stresses, so these equations
are rewritten as
∂
h
+
∂(
hU
x
)
+
∂(
hU
y
)
=
0
(6.1)
∂
t
∂
x
∂
y
hU
x
)
∂
∂(
hU
x
)
+
∂(
+
∂(
hU
y
U
x
)
gh
∂
z
s
1
ρ
∂(
hT
xx
)
1
ρ
∂(
hT
xy
)
=−
x
+
+
∂
t
x
∂
y
∂
∂
x
∂
y
1
ρ
(τ
+
−
τ
)
+
f
c
hU
y
(6.2)
sx
bx
hU
y
)
∂
+
∂(
∂(
hU
y
)
+
∂(
hU
x
U
y
)
gh
∂
z
s
1
ρ
∂(
hT
yx
)
1
ρ
∂(
hT
yy
)
=−
+
+
∂
t
∂
x
y
∂
y
∂
x
∂
y
1
ρ
(τ
+
−
τ
)
−
f
c
hU
x
(6.3)
sy
by
where
x
and
y
are the horizontal Cartesian coordinates (not necessarily along the
