Geoscience Reference
In-Depth Information
A
j
R
2
/
3
j
K
j
=
/
n
j
(5.14)
where
z
b
,
j
is the bed elevation at station
j
,
y
j
is the distance between stations
j
and
j
1, and
n
j
is the Manning roughness coefficient in panel
j
.
The composite cross-sectional area of flow is defined as the sum of all panel subareas
and is the true area. The composite velocity is defined as the total discharge divided
by the cross-sectional area, conserving continuity. The composite hydraulic radius is
conveyance-weighted as
+
R
j
K
j
M
M
R
=
K
j
(5.15)
j
=
1
j
=
1
where
M
is the number of the wetted panels.
Because the alpha method ignores the effect of vertical walls, it is not adequate in
situations where vertical sidewalls or steep bank slopes exist.
Division of hydraulic radius
Einstein (1950) proposed a more adequate method for determining the composite
hydraulic properties for the cross-section with rough vertical sidewalls or steep bank
slopes, based on the division of hydraulic radius. This method assumes equal velocity
in all panels, and calculates all hydraulic variables in the normal way, except for the
composite Manning roughness coefficient.
The total shear stress
τ
in the cross-section can be computed as
M
χτ
=
1
χ
τ
(5.16)
j
j
j
=
χ
=
j
=
1
χ
j
, and
where
χ
is the total wetted perimeter, i.e.,
τ
j
is the shear stress in
panel
j
.
Einstein's method determines
τ
=
γ
RS
f
(5.17)
τ
j
=
γ
R
j
S
f
(5.18)
Applying the equal velocity assumption and the Manning equation in the entire
cross-section and each panel yields
S
1
/
2
S
1
/
2
3
/
2
,
R
j
3
/
2
R
=
(
nU
/
f
)
=
(
n
j
U
/
f
)
(5.19)
