Geoscience Reference
In-Depth Information
Table 5.2 Values of the parameter constants in the friction slope
S
f
as given by Equations (5.39) or (5.43)
Regime
Laminar
Turbulent
Parameter
Without rain
With rain
(Gauckler-Manning)
(Chezy)
g
3
ν
g
ν
(3
+
cS
f
P
e
)
C
r
n
−
1
C
r
a
2
2
2
/
3
1
/
2
b
1
1
1
/
2
1
/
2
Note:
P
is the rainfall intensity and values of the constants
c
,
d
and
e
are given following
Equation (5.33).
5.4
GENERAL CONSIDERATIONS AND SOME FEATURES
OF FREE SURFACE FLOW
The solution of the shallow water equations (5.13) and (5.22) (or (5.24) and (5.25)) is not
easy, and most flow problems encountered in natural situations have to be analyzed by
numerical methods. The availability of digital computation technology in recent decades
has greatly facilitated this, and rapid advances have been made in this field (see Liggett
and Cunge, 1975; Cunge
et al.
, 1980; Tan, 1992; Montes, 1998). Nevertheless for a better
understanding of their structure and the physical implications, it is useful to consider
simpler forms of these equations; these are valid in certain special situations, for which
solutions may be more easily obtainable, or for which important features of the flow can
be deduced by inspection.
5.4.1
Complete system of the shallow water equations: small disturbances
As can be seen in the second term and in the term containing
S
f
, Equation (5.22), describing
the conservation of momentum, is a nonlinear partial differential equation. However, if the
flow is a small departure from an initially uniform steady state, it is possible to linearize the
shallow water equations, which greatly facilitates the solution. More importantly, however,
not only is the solution easier, but it also brings out clearly some of the general physical fea-
tures, regarding the coexistence of different wave types, which are inherent in the nonlinear
system as well.
Consider for this purpose a small departure from uniform steady flow, by the substitution
of
V
=
V
0
+
V
p
and
h
=
h
0
+
h
p
, in which the subscript 0 refers to uniform steady condi-
tions and the subscript p refers to a small perturbation or disturbance. This is illustrated in
Figure 5.6 for the water depth
h
. Thus Equations (5.13) and (5.22) become, after retention
of the first-order terms,
V
0
∂
h
p
∂
x
+
h
0
∂
V
p
∂
x
+
∂
h
p
∂
t
−
i
=
0
and
(5.44)
∂
V
p
∂
t
+
V
0
∂
V
p
x
+
g
∂
h
p
x
+
g
(
S
f
−
S
0
)
=
0
∂
∂
