Geoscience Reference
In-Depth Information
Define the total resistance and the bed shear stress as
gn
2
U
2
R
1
/
3
=
ρ
c
f
U
2
τ
=
ρ
(10.15)
s
gn
b
U
2
R
1
/
3
=
ρ
c
fb
U
2
τ
=
ρ
(10.16)
b
s
where
c
f
and
n
are the friction factor and Manning coefficient corresponding to the
total roughness,
c
fb
and
n
b
are the friction factor and Manning coefficient correspond-
ing to the bed roughness, and
R
s
is the hydraulic radius of the bed with vegetation.
The hydraulic radius
R
s
has been defined differently in the literature. Many models
simply set
R
s
as the flow depth
h
, while Barfield
et al
. (1979) considered the effect of
vegetation on the flow “eddy size” and suggested the following relation:
hl
n
2
h
R
s
=
(10.17)
+
l
n
where
l
n
is the lateral spacing of vegetation elements.
Substituting Eqs. (10.7), (10.15), and (10.16) into Eq. (10.14) yields
1
n
2
n
b
+
C
d
N
a
A
v
R
1
/
3
=
(10.18)
s
2
g
(
1
−
c
v
)
For the channel with densely distributed vegetation, the drag of vegetation becomes
the major contributor to the total resistance, and thus the term of
n
b
in Eq. (10.18)
can be eliminated.
Eq. (10.18) does not include the effect of side banks, so it is used only for a wide
channel or in a depth-averaged 2-D model. When the effect of side banks needs to
be considered,
R
s
in Eq. (10.18) is set as
A
/
P
w
, or the following equation derived by
Petryk and Bosmajian (1975) may be used:
2
g
C
d
A
vi
A
χ
4
/
3
1
n
2
n
b
+
=
(10.19)
A
is the wetted perimeter, and
A
vi
is
the total frontal area of vegetation elements blocking the flow per unit channel length.
Note that 1
χ
where
A
is the cross-sectional area of flow,
1 and
C
d
represents an average drag coefficient of all vegetation
elements in Eq. (10.19). If the heterogeneity of vegetation elements is considered, a
variable
C
di
should be used for each element.
Eq. (10.19) is applicable to emergent, rigid vegetation distributed relatively uni-
formly in the lateral direction.
−
c
v
≈
