Geoscience Reference
In-Depth Information
1-D and 2-D model equations from 3-D model equations via section-, depth-, and
width-integrating (averaging) approaches is introduced in this section.
Before deriving the spatially-integrated models, let us introduce the 3-D flow and
sediment transport equations and the associated boundary conditions at the water
surface, channel bottom, and banks. In the Cartesian coordinate system shown in
Fig. 2.6, the 3-D continuity and momentum equations (2.42) and (2.43) of flow with
low sediment concentration are rewritten as
∂
u
x
∂
+
∂
u
y
∂
+
∂
u
z
∂
=
0
(2.60)
x
y
z
u
x
)
∂
∂
u
x
∂
+
∂(
+
∂(
u
y
u
x
)
+
∂(
u
z
u
x
)
1
ρ
1
ρ
∂
p
1
ρ
∂τ
xx
=
F
x
−
x
+
t
x
∂
y
∂
z
∂
∂
x
1
ρ
∂τ
xy
∂
1
ρ
∂τ
xz
+
+
(2.61)
y
∂
z
u
y
)
∂
+
∂(
∂
u
y
∂
t
+
∂(
u
x
u
y
)
∂
+
∂(
u
z
u
y
)
∂
∂τ
yx
∂
1
ρ
1
ρ
∂
p
1
ρ
=
F
y
−
y
+
∂
x
y
z
x
1
ρ
∂τ
1
ρ
∂τ
yy
yz
+
+
(2.62)
∂
y
∂
z
u
z
)
∂
∂
u
z
∂
+
∂(
u
x
u
z
)
+
∂(
u
y
u
z
)
+
∂(
1
ρ
1
ρ
∂
p
1
ρ
∂τ
zx
=
F
z
−
z
+
t
∂
x
∂
y
z
∂
∂
x
∂τ
1
ρ
1
ρ
∂τ
zy
zz
+
y
+
(2.63)
∂
∂
z
where
x
(
is the vertical
coordinate above a datum;
u
x
,
u
y
, and
u
z
are the components of mean velocity in the
x
-,
y
- and
z
-directions;
=
x
1
)
and
y
(
=
x
2
)
are the horizontal coordinates;
z
(
=
x
3
)
zz
are the stresses (including both molecular
and turbulent effects); and
F
x
,
F
y
, and
F
z
are the components of the resultant external
force in the
x
-,
y
- and
z
-directions. As gravity is assumed to be the only external force,
F
x
τ
xx
,
τ
xy
,
...
, and
τ
g
.
Note that the bar “-”, denoting time-averaged quantities, is omitted in Eqs. (2.60)-
(2.63) for simplicity.
For gradually varied (shallow water) flows, the inertia and diffusion effects in the
vertical momentum equation (2.63) are usually neglected, yielding the hydrostatic
pressure equation:
=
F
y
=
0 and
F
z
=−
ρ
∂
p
=−
ρ
g
(2.64)
∂
z
Under the assumption of constant
ρ
along the depth, Eq. (2.64) has an analytic
solution:
