Geoscience Reference
In-Depth Information
longitudinal and transverse directions); and
τ
bx
and
τ
by
are the bed shear stresses,
determined by
c
f
m
b
U
x
U
x
+
c
f
m
b
U
y
U
x
+
τ
bx
=
ρ
U
y
,
τ
by
=
ρ
U
y
(6.4)
h
1
/
3
, with
n
being the Manning roughness coefficient of the channel
bed; and
m
b
is the bed slope coefficient defined in Eq. (2.82). For a movable bed
with sediment grains and bed forms, the Manning
n
can be evaluated using one of
the empirical formulas introduced in Section 3.3.3; however,
n
is in general treated
as a calibrated parameter because of its complexity, as discussed in Section 5.1.1.3.
In addition, one may set the bed slope coefficient
m
b
as 1 and lump its effect into the
Manning
n
.
Note that unlike the 1-D model, the depth-averaged 2-D model can simulate the
effects of large-scale roughness structures, such as channel contraction, expansion,
and curvature, on the flow field, using fine meshes. In addition, the depth-averaged
2-D model accounts for the effect of channel banks through boundary conditions
and considers the effect of horizontal turbulent diffusion through the eddy viscosity.
Therefore, the values of Manning
n
in the 1-D and 2-D models are not exactly the
same.
In Eqs. (6.2) and (6.3),
gn
2
where
c
f
=
/
τ
sx
and
τ
sy
represent the forces acting on the water surface,
usually caused by wind driving:
τ
sx
=
ρ
a
c
fa
U
wind
,
x
U
wind
,
x
+
τ
sy
=
ρ
a
c
fa
U
wind
,
y
U
wind
,
x
+
U
wind
,
y
,
U
wind
,
y
(6.5)
where
U
wind
,
x
and
U
wind
,
y
are the
x
- and
y
-components of wind velocity,
ρ
a
is the air
density, and
c
fa
is the friction coefficient at the water surface.
The last terms in Eqs. (6.2) and (6.3) represent the Coriolis force due to the rotation
of the earth. The Coriolis coefficient
f
c
is determined by
f
c
=
2
sin
ϕ
(6.6)
where
is the rotation velocity of the earth in radians per second, and
ϕ
is the latitude
degree of the water body of interest.
The Coriolis and wind driving forces are important in large water bodies, such as
coastal waters, estuaries, and large lakes, but they are usually negligible in inland
rivers.
The stresses
T
ij
, which include both viscous and turbulent effects, are
determined using the Boussinesq assumption:
(
i
,
j
=
x
,
y
)
)
∂
U
x
∂
2
3
ρ
T
xx
=
2
ρ(ν
+
ν
−
k
t
x
∂
U
x
∂
+
∂
U
y
∂
T
xy
=
T
yx
=
ρ(ν
+
ν
)
t
y
x
