Geoscience Reference
In-Depth Information
5.2.2.6 Auxiliary treatments for unsteady flow calculation
Representation of friction slope
The friction slope in Eq. (5.2) may be represented in various ways, such as arithmetic
mean, harmonic mean, geometric mean, and conveyance mean (French, 1985), as
expressed in Eqs. (5.42)-(5.45) for a steady flow model. For an unsteady flow model,
the arithmetic mean friction slope is introduced in Eq. (5.56), and the conveyance
mean friction slope is given as
2
R
Q
n
+
1
i
Q
n
+
1
i
R
Q
i
+
1
+
(
Q
i
]
θ
[
ψ
+
(
1
−
ψ
)
]+
(
1
−
θ)
[
ψ
1
−
ψ
)
R
R
+
1
S
f
=
θ
[
ψ
R
K
n
+
1
i
K
n
+
1
i
−
θ)
[
ψ
R
K
i
+
1
+
(
K
i
]
+
(
−
ψ
R
)
]+
(
−
ψ
R
)
1
1
1
+
1
(5.120)
The harmonic and geometric mean friction slopes are left to interested readers.
Small flow depth
Computational difficulties arise when the flow depth becomes small. As the flow depth
approaches zero, the conveyance and flow discharge go to zero, and thus the friction
slope becomes indeterminate. This was explained well by Cunge
et al
. (1980). Meselhe
and Holly (1993) showed that the characteristic curves are vertical and do not intersect
when the flow depth is zero; consequently, a solution does not exist.
Cunge
et al
. (1980) proposed and Meselhe and Holly (1993) developed further
an approach for handling the dry-bed problem. The basic idea is to switch the
weighting for friction slope from central (
ψ
=
0.5, highest accuracy) to upstream
R
(0
0.5). In a diffusive wave model, Langendoen (1996) related the weighting
factor to the flow depth as
≤
ψ
R
≤
0.5,
ah
b
ψ
R
=
min
(
)
(5.121)
where the coefficient
a
0.35.
In the author's experience, Eq. (5.121) may fail and
≈
0.7 and the exponent
b
≈
ψ
R
=
0 is occasionally necessary
to ensure stable solutions when using the dynamic wave model. One of the best choices
is to try several values of
ψ
R
and find the value closest to 0.5 that allows a stable
solution for a specific case.
Storage effect of still water zones
Still waters, or very slow flows, exist in sudden expansions, appendix channels, ponds,
or small lakes that are connected to the main stream. These still water zones do not
have significant momentum exchange with the main stream, but their storages may
affect the main flow. To consider the storage effect, the continuity equation (5.1) is
substituted by
∂(
A
+
A
0
)
+
∂
Q
∂
=
q
l
(5.122)
∂
t
x
