Geoscience Reference
In-Depth Information
where
S
fb
is the bed friction slope,
F
d
is the drag force defined in Eq. (10.7), and
K
is
the conveyance.
When the vegetation concentration
c
v
is small, Eqs. (10.51) and (10.52) can be
simplified as Eqs. (5.1) and (5.2) by eliminating 1
c
v
.
If the entire cross-section is covered by nearly uniformly distributed vegetation, the
conveyance
K
can be determined by
−
A
5
/
3
n
K
=
(10.54)
2
/
3
χ
where the Manning
n
accounts for the effects of both channel bed friction and
vegetation drag, and is determined by one of the relations described in Section 10.1.2.
If the cross-section is partially covered by vegetation or the vegetation density varies
along the cross-section, the flow velocity significantly varies in the vegetated and non-
vegetated zones or even in different vegetated zones. Thus, the cross-section needs to
be divided into a suitable number of subsections, either vegetated or non-vegetated.
The conveyance in each subsection is determined by
A
5
/
3
j
K
j
=
(10.55)
2
/
3
n
j
χ
j
where
K
j
,
A
j
,
j
, and
n
j
are the conveyance, flow area, wetted perimeter, and Manning
roughness coefficient of subsection
j
, respectively. The total conveyance
K
can be
obtained by summing the conveyances of all subsections as
χ
K
=
K
j
(10.56)
j
The flow velocity in each subsection is determined using the Manning equation:
K
j
S
1
/
2
f
A
j
U
j
=
(10.57)
Eqs. (10.51) and (10.52) are iteratively solved together with Eqs. (10.53) and
(10.55)-(10.57) and a relation between the Manning roughness coefficient and flow
conditions introduced in Section 10.1.2. This approach is similar to but more
complicated than that used for compound channels in Section 5.1.1.4.
10.2.2 Numerical solutions
The 1-D, 2-D, and 3-D governing equations can be solved using the numerical methods
described in Chapters 5-7. For example, the 1-D equations (10.51) and (10.52)
can be solved using the Preissmann scheme and the Thomas algorithm described in
