Hydraulic Theory
Kinematic Flood Routing


From the MIDUSS Version 2
Reference Manual  Chapter 8
(c) Copyright Alan A. Smith Inc. 
Flood routing methods
can be classified as hydraulic  in which both continuity and dynamic
equations are used  or hydrologic, which generally uses the
continuity equation alone. MIDUSS uses a method based on a
kinematic wave equation and therefore
falls into the second category. For the type of conduits used in
storm drainage systems, kinematic routing
yields results of very acceptable accuracy.
The continuity equation is simply a statement
that the difference between inflow and outflow must equal the rate of
change of storage in the reach being considered. Equation [8.20] is a
general continuity equation in which lateral inflow is ignored.
If the flow in the channel can be assumed to be
quasiuniform then discharge is uniquely defined by the depth or
stage, i.e.
so that
Now separating variables in [8.20] yields:
Equation [8.22] is
seen to be a wave equation in which a function
Q(x)
is propagated with a celerity
c
which is given by
c =
dQ/dA. The variation of
Q
with respect to both time
t
and space
x
can be best described using a spacetime
coordinate system which defines changes in discharge
Q
in an elementary reach
Dx
and over a timestep
Dt.
Figure 8.4  An element
of a spacetime coordinate system.
Figure 8.4 shows such a situation. For this
element of the spacetime system equation [8.22] can be expressed in
terms of finite differences as follows.
This equation is applied around a 'nucleus' of
the spacetime element which is offcentre and defined by the
weighting factors a
and b which are
applied to the x and
t dimensions respectively as
shown in Figure 8.4.
The finite difference quotients of equation
[8.23] are expanded around the 'nucleus' point of Figure 8.4 in terms
of the values of Q at
points 5, 6, 7 & 8. These in turn can be expressed as weighted
averages of the values at points 1, 2, 3 & 4 and the weighting
coefficients a and
b.
Thus:
and
where
Substituting equations [8.24] in [8.23] yields:
The quantity (c.Dt/Dx)
is a dimensionless time ratio which is equivalent to the Courant
criterion for numerical stability. Denoting this by
t
and substituting for Q5
etc. results in equation [8.26].
Now the process of flood routing usually involves
a 'marching' solution in which the initial conditions are known at
time t and it is
required to predict conditions at time (t+Dt).
An upstream boundary condition is provided by the timehistory of the
inflow hydrograph at x=0.
In the case of kinematic wave routing it
is possible to advance the solution for all values of time
t so that the solution
advances over the whole time domain for each reach of channel.
In either case, the solution for the element of
Figure 8.4 involves an estimate of Q4
in terms of the other three known values. Collecting terms and
casting Q4 as the
dependant variable yields:
where
Equation [8.27] is a generalized form of the
Muskingum flood routing method but for the special case of
b =0.5
this reduces to the more familiar form shown below.
Before equation [8.27] can be applied two
additional pieces of information are required:
(i) What
values of a
and b
should be used to best represent the attenuation for a specific
hydraulic condition?
(ii) What conditions must apply for the
computation to be numerically stable?
Evaluation of
the Weighting Coefficients
Convergence is the condition in which the
solution of a finitedifference equation for a finite grid size
approximates the true solution of the partial differential equation
which it represents. It can be shown (Biesenthal
(1975) and Smith (1980)) for the noncentred
scheme of Figure 8.4 that as the coefficients
a and
b
depart from a value of 0.5, truncation errors of the order of
O(Dx)
and O(Dt)
increase respectively and independently. This property is fundamental
to the use and apparent success of kinematic
routing methods in modelling attenuation. It should be emphasized,
however, that this attenuation results from truncation error and is a
property of the numerical finite difference scheme and not of the
physical system. The trick is to find a way to make the numerical
truncation error a close approximation to the attenuation which the
flood wave will experience.
For a noncentred
finitedifference scheme the error may be included in the partial
differential equation as
e in
equation [8.29].
If only first order terms are included in the
error term this becomes:
or
Equation [8.31] is in the form of a diffusion
equation where D
is the coefficient of diffusion. In order to relate
D to the physical
characteristics of the channel an alternate diffusion equation can be
developed using the continuity equation [8.20] with a simplified form
of momentum equation in which the convective and temporal accelerative
terms are assumed to be negligible, i.e.
where h
= water surface elevation
Sf
= friction gradient
This can be developed (Smith(1980) see References
) to yield the diffusion equation of [8.33].
where K
= channel conveyance defined by
Sf
= Q^{2}/K^{2}
Now from the definition of the conveyance K the
total derivative is obtained as:
Comparison of the terms in equations [8.31] and
[8.33] provides a means of evaluating the diffusion coefficient
D and thus the
weighting coefficients
a and
b in
terms of the hydraulic characteristics of the channel. Thus:
where the friction head loss over the reach being
considered is given by
.
Equation [8.36] can be simplified by assuming an
initial value of
b = 0.5
so that:
If equation [8.37] yields a value for
a which
is less than zero, MIDUSS sets
a = 0.0
and solves for
b from [8.36]. This value will generally be in the
range 0.5
£
b
£
1.0. The generalized Muskingum coefficients of equation
[8.27] can then be evaluated and a solution obtained for
Q4.
In MIDUSS, evaluation of the diffusion
coefficient differs depending on whether the conduit is a pipe or
channel. The Manning equation [8.12] can be differentiated to obtain
an expression for
dQ/dy as follows:
Substituting in the 2nd part of [8.35] yields an
expression for D which can
be evaluated in terms of the channel crosssection parameters and the
channel gradient, thus:
For pipes, an alternative procedure is used in
which a fitted polynomial represents the ratio
Q/(dQ/dyr)
as a function of the proportional discharge
Qr which is the ratio
of actual discharge to full pipe capacity. With very acceptable
accuracy this can be represented as follows:
from which the diffusion coefficient
D can be found using equation
[8.35].
Criteria for
Numerical Stability in Flood Routing
Once values have been determined for the
weighting coefficients
a and
b it
is possible to carry out a check on the numerical stability of the
process. This involves the calculation of limiting values for the
grid dimensions
Dx
and
Dt.
Biesenthal (1975)
(see References ) obtained the following condition for numerical
stability.
In general, this means that the nucleus of Figure
8.4 must lie above and to the right of a diagonal through the centre
of the spacetime element which has a slope of
(1/c).
Figure 8.5  Stability
and numerical error characteristics of a spacetime element.
Figure 8.5 shows a
typical case in which the time step
dt
is approximately half of the Courant time step given by
Dtc
=
Dx/c.
The heavy dashed lines parallel to the main diagonal form a family of
lines each of which comprises the locus of points for which the
nucleus will generate a numerical error
e
of a specific value. It is significant that the same numerical
attenuation can be produced by a set of (
a,
b
) coordinates and MIDUSS makes use of this feature.
The shaded area of
Figure 8.5 indicates locations of the nucleus which are numerically
unstable. Because the values of
a
and
b
are constrained to provide a required numerical error, the criterion
for stability given by equation [8.38] must be satisfied by
manipulating either the routing timestep
dt
or the reach length
Dx.
The greater the coarseness of the spacetime grid the greater the
chance of numerical instability. Thus we need to determine upper
limits for both
Dx
and the routing timestep
dt
Equation [8.35] shows a relationship between the
diffusion coefficient D
and the weighting coefficients
a and
b. If
we assume initially that
b
= 0.5 this reduces to:
Similarly if
b
= 0.5 the inequality [8.41]
becomes:
Taking the 1st and 2nd parts of [8.43] and
substituting for
a from
[8.42] we obtain:
or
To get an upper bound for the routing timestep
we use the 2nd and 3rd parts of [8.43] and obtain:
or
MIDUSS uses the limiting criteria of [8.44] and
[8.45] to divide either the reach length
L or the hydrograph
timestep ?t into
submultiples which satisfy the conditions for stability.
In the case of very long reaches the time step
remains unchanged but routing is carried out over two or more
subreaches. The final outflow hydrograph
is the only one presented to the user.
In the case of very short reaches only a single
reach length is used but the routing timestep is set at
d
t =
Dt/n
(n = 2,3,4...). Only flow values at intervals of
Dt
are presented in the final outflow hydrograph.
