Geoscience Reference
In-Depth Information
of practical interest generally requires an approximate numerical solution algorithm. The first attempts
at an explicit finite difference solution of the St. Venant equations date back to Stoker (1957). There are
now well-established finite difference schemes such as the four-point implicit method described by Fread
(1973; see Box 5.6), that has been used even under the extreme conditions of routing the flood wave due
to a dam break (e.g. Fread, 1985). Routing for rainfall-runoff modelling is generally not so extreme.
The other requirements to apply such a model are information about the geometry of the channel, a
specification of the initial velocities and depths of flow at the start of a simulation and boundary conditions
at both the upstream and downstream boundaries of a channel reach. Geometry, roughness coefficients,
and initial and boundary conditions are only ever known imprecisely and some simplifying assumptions
are usually necessary.
In respect of the channel geometry, it has usually been assumed that the shape of the channel can be
interpolated between surveyed cross-sectional profiles measured at different distances along the channel.
At low flows, this will not give a good representation of the effects of the pool/riffle geometry of the
channel; at high overbank flows, it may not take proper account of the effects of embankments, field
boundaries and other obstructions to the flow. These will also affect the appropriate values of the effective
roughness parameter that must reflect all the causes of momentum loss in a reach of channel. Thus,
the effective values might be different from that inferred from a measured velocity profile at any single
point in the channel and may also require a parametric description of how the roughness coefficient
changes with depth of flow, especially for overbank flow under flood conditions (see, for example,
Knight
et al.
, 2010).
The boundary conditions will have an important effect on the solution. The St. Venant equations re-
quire boundary conditions to be specified for both upstream and downstream boundaries (in contrast to
the kinematic wave approximation discussed below). In fact, since there are two unknowns in the solu-
tion, velocity and depth for each cross-section, two upstream boundary conditions and two downstream
boundary conditions are required for every simulated reach. Junctions between reaches require some
special conditions to ensure consistency in the solutions for the upstream and downstream reaches. It is,
in fact, rare for the boundary conditions to be specified directly in terms of velocity and depth at each
boundary: they are not generally available. Water surface elevation or stream stage is more generally
available, at least at the gauging sites that often mark the boundary points for a solution in a larger
river. A measurement of stage can be used with an appropriate rating curve and cross-sectional survey
to get approximate estimates of discharge and cross-sectional area, from which a mean velocity can be
derived. The rating curve may be measured, may be a theoretical rating of a gauging structure or may be
derived by assuming that there is a locally uniform flow at the boundary. In the last case, the relationship
between velocity and stage can be described by one of the uniform flow equations such as the Manning
or Darcy-Weisbach equations, given knowledge of the appropriate roughness coefficient.
A little bit of care is necessary here however. Use of such a uniform flow rating curve implies that the
water surface is always parallel to the bed. The fully dynamic equations, however, imply that the water
surface should be steeper than the bed slope on the rising limb of a hydrograph and less steep on the
falling limb, resulting in a hysteretic or looped rating curve (see, for example, Dottori
et al.
2009). The
uniform flow assumption can, then, only provide an approximate boundary condition for the solution.
Again, it is important to assess what assumptions are being made in the description and solution for each
process. It is also the case that, even where a measured rating curve is available, the observations on
which it is based should be checked as they can show nonstationarities over time, can show a wide scatter
at some stations and may not extend to the highest flood flows of interest (for good logistical and health
and safety reasons!).
It must be remembered that this description of channel flow is a one-dimensional description, with
the solution variables being average velocities and depths of flow. This type of description will not be
as accurate during flood conditions, when local roughness coefficients, velocities and depths of flow
may vary dramatically in the cross-section. Recent developments have seen two-dimensional models
