Geoscience Reference
In-Depth Information
A number of studies have examined the theoretical limits of acceptability of approximations
to the full St. Venant equations for different surface flows. A study by Daluz Vieira (1983), based
on extensive numerical analysis, provided ranges of validity for different approximations to the
St. Venant equations, including the kinematic wave equation, in terms of two dimensionless
numbers (see Figure B5.6.1): a Froude number, Equation (B5.6.13), and a kinematic wave
number, Equation (B5.6.14).
C
tan
g
0
.
5
F
o
=
(B5.6.13)
g
3
Lsin
C
4
q
2
0
.
333
=
(B5.6.14)
These studies suggest where the simplified models give a reasonable theoretical agreement
with the full St. Venant equations. However, uncertainty in effective parameter values and
boundary conditions may mean that the simplified models are useful under a wider range
of conditions. The flood routing example of Figure 5.9 is a good illustration of this. In a real
application, the shear stress and lateral inflow terms are not usually well known. Both terms
may vary in both space and time as the river stage or depth of overland flow changes during an
event. For river flow, in particular, there may be important changes in the effective shear stress
if the flow exceeds the bankfull stage and starts to spill onto a flood plain (see, for example,
the work of Knight
et al.
, 1994, 2010).
Box 5.7 Derivation of the Kinematic Wave Equation
The kinematic wave equation arises from the combination of a mass balance or continuity
equation, expressed in terms of storage and flows, and a functional relationship between
storage and flow that may be nonlinear but is single-valued, that is to say that there is a single
value of flow at a point corresponding to any value of storage at that point. Consider a one-
dimensional downslope overland flow on a slope of constant width. Let
x
be distance along
the slope,
h
the depth of flow (which acts as the storage variable), and
q
the mean downslope
velocity at any
x
(which is the flow variable). The mass balance equation can then be expressed
as the partial differential equation
∂h
∂t
=−
∂q
∂x
+
r
(B5.7.1)
where
r
is a rate of addition or loss of water per unit length and width of slope at point
x
, and
t
is time.
The functional relation between
h
and
v
may be of many forms but a common assumption
is the power law (see, for example, the kinematic approximation for surface flows in Box 5.6):
bh
a
q
=
(B5.7.2)
Thus, assuming
a
and
b
are constant and combining these two equations, to yield a single
equation in
h
∂h
∂t
=−
abh
(
a
−1
)
∂h
∂x
+
r
(B5.7.3)
