Geoscience Reference
In-Depth Information
Again, the St. Venant equations are nonlinear partial differential equations (hyperbolic, in this
case) that do not have analytical solutions except for some very special cases and approximate
numerical solutions are necessary. As noted in the main text, the first attempt at formulating
a numerical solution was by Stoker (1957). This used explicit time stepping which generally
requires very short time steps to achieve adequate accuracy. Most solution algorithms used
today are based on implicit time stepping (see the explanation in Box 5.3), such as the four-
point implicit method described by Fread (1973).
B5.6.1 Summary of Model Assumptions
We can summarise the assumptions made in developing this form of the St. Venant equations
as follows:
A1 The flow can be adequately represented by the average flow velocity and average flow
depth at any cross-section.
A2 The amplitude of the flood wave is small relative to its wavelength so that pressure in
the water column at any cross-section is approximately hydrostatic (pressure is directly
proportional to depth below the water surface).
A3 The water is incompressible and of constant temperature and density.
A4 The friction slope may be estimated approximately using one of the uniform flow equa-
tions (such as the Manning or Darcy-Weisbach equations) with actual flow velocities and
depths.
B5.6.2 Simplifications of the St. Venant Equations
The St. Venant equations are based on hydraulic principles but they are clearly an approxi-
mation to the fully three-dimensional flow processes in any stream channel. There are various
further approximations to the St. Venant equations that are produced by assuming that one or
more of the terms in the momentum equation, Equation B5.6.1, can be neglected. The two
main approximate solutions are the diffusion wave approximation:
∂Agh
∂x
=
gA
(
S
o
−
S
f
)
(B5.6.7)
and the kinematic wave approximation:
gA
(
S
o
−
S
f
)
=
0
(B5.6.8)
so that
S
o
=
S
f
(B5.6.9)
which reflects the assumption for the kinematic wave equation that the water surface is always
parallel to the bed. This is also, of course, the assumption made in equations describing a
uniform flow so, assuming again that the Darcy-Weisbach uniform flow equation is a good
approximation for the transient flow case,
2
g
f
S
o
R
h
0
.
5
v
=
(B5.6.10)
For a channel that is wide relative to its depth or an overland flow on a relatively smooth
slope, then
R
h
≈
h
and this equation has the form of a power law relationship between velocity
and storage:
bh
0
.
5
v
=
(B5.6.11)
