2. CLASSIFICATION OF DISTRIBUTED ROUTING
MODELS
The Saint-Venant equations have various
simplified forms, each defining a one
dimensional distributed routing model.
Variations of Eqs. in conservation and
nonconservation forms, neglecting lateral
inflow, wind shear, and eddy losses, are used
to define various one-dimensional distributed
routing models as shown inTable 1.
3. The momentum equation consists of terms for the
physical processes that govern the flow momentum.
These terms are: the local acceleration term, which
describes the change in momentum due to the
change in velocity over time; the convective
acceleration term, which describes the change in
momentum due to change in velocity along the
channel; the pressure force term, proportional to the
change in the water depth along the channel; the
gravity force term, proportional to the bed slope S0;
and the friction force term, proportional to the friction
slope Sf.
5. The local and convective acceleration terms represent the
effect of inertial forces on the flow.
When the water level or flow rate is changed at a particular
point in a channel carrying a subcritical flow, the effects of
these changes propagate back upstream.
These backwater effects can be incorporated into distributed
routing methods through the local acceleration, convective
acceleration, and pressure terms.
Lumped routing methods may not perform well in
simulating the flow conditions when backwater effects are
significant and the river slope is mild, because these
methods have no hydraulic mechanisms to describe
upstream propagation of changes in flow momentum.
6. As shown inTable 1, alternative distributed flow routing models
are produced by using the full continuity equation while
eliminating some terms of the momentum equation.
The simplest distributed model is the kinematic wave model,
which neglects the local acceleration, convective acceleration,
and pressure terms in the momentum equation; that is, it
assumes S0= Sf and the friction and gravity forces balance each
other.
The diffusion wave model neglects the local and convective
acceleration terms but incorporates the pressure term.
The dynamic wave model considers all the acceleration and
pressure terms in the momentum equation.
The momentum equation can also be written in forms that take
into account whether the flow is steady or unsteady, and uniform
or nonuniform, as shown in Eqs. (1).
7. In the continuity equation, dAldt = 0 for a
steady flow, and the lateral inflow q is zero for
a uniform flow.
Conservation form:
Non conservation form:
(1a)
(1b)
8. WAVE MOTION
Kinematic waves govern flow when inertial and
pressure forces are not important.
Dynamic waves govern flow when these forces are
important, such as in the movement of a large flood
wave in a wide river.
In a kinematic wave, the gravity and friction forces
are balanced, so the flow does not accelerate
appreciably.
Fig. 1 illustrates the difference between kinematic
and dynamic wave motion within a differential
element from the viewpoint of a stationary observer
on the river bank.
9. Fig 1 Kinematic and dynamic waves in a short
reach of channel as seen by a stationary
observer.
10. For a kinematic wave, the energy grade line is
parallel to the channel bottom and the flow is
steady and uniform (S0 = Sf) within the
differential length, while for a dynamic wave
the energy grade line and water surface
elevation are not parallel to the bed, even
within a differential element.
11. Kinematic Wave Celerity
A wave is a variation in a flow, such as a change in flow rate or
water surface elevation, and the wave celerity is the velocity with
which this variation travels along the channel.
The celerity depends on the type of wave being considered and
may be quite different from the water velocity.
For a kinematic wave the acceleration and pressure terms in the
momentum equation are negligible, so the wave motion is
described principally by the equation of continuity.
The name kinematic is thus applicable, as kinematics refers to the
study of motion exclusive of the influence of mass and force; in
dynamics these quantities are included.
The kinematic wave model is defined by the following equations.
Continuity:
Momentum:
(1)
(2)
12. The momentum equation can also be
expressed in the form
For example, Manning's equation written
with S0 = Sf and R = AIP is
which can be solved for A as
(5)
(4)
(3)
13. Equation (1) contains two dependent variables, A
and Q, but A can be eliminated by differentiating
(3):
And substituting for dAldt in (1) to give
Kinematic waves result from changes in Q.
An increment in flow, dQ, can be written as
(7)
(6)
14. Dividing through by dx and rearranging
produces:
Equations (7) and (9) are identical if
And
(9)
(8)
(11)
(10)
15. Differentiating Eq. (3) and rearranging gives
and by comparing (11) and (12), it can be seen
that
or
(12)
(14)
16. where Ck is the kinematic wave celerity.
This implies that an observer moving at a
velocity dxldt =Ck with the flow would see the
flow rate increasing at a rate of dQIdx = q.
If q =0, the observer would see a constant
discharge.
Eqs. (10) and (14) are the characteristic
equations for a kinematic wave, two ordinary
differential equations that are
mathematically equivalent.
17. The kinematic wave celerity can also be
expressed in terms of the depth y as
Where dA=Bdy,
Both kinematic and dynamic wave motion are
present in natural flood waves.
In many cases the channel slope dominates in
the momentum equation (1); therefore, most of
a flood wave moves as a kinematic wave.
18. Lighthill and Whitham (1955) proved that the
velocity of the main part of a natural flood
wave approximates that of a kinematic wave.
If the other momentum terms [dVldt,
V(dV/dx), and (1/g)dy/dx] are not negligible,
then a dynamic wave front exists which can
propagate both upstream and downstream
from the main body of the flood wave, as
shown in Fig. 2.
20. As previously shown, if a wave is kinematic (Sf = S0) the kinematic
wave celerity varies with dQ/dA.
For Manning's equation, wave celerity increases as Q increases.
As a result, the kinematic wave theoretically should advance
downstream with its rising limb getting steeper.
However, the wave does not get longer, or attenuate, so it does
not subside, and the flood peak stays at the same maximum
depth.
As the wave becomes steeper the other momentum equation
terms become more important and introduce dispersion and
attenuation.
The celerity of a flood wave departs from the kinematic wave
celerity because the discharge is not a function of depth alone,
and, at the wave crest, Q and y do not remain constant.
21. Lighthill andWhitham (1955) illustrated that the
profile of a wave front can be determined by
combining the Chezy equation:
with the momentum equation (1b) to produce:
in which C is the Chezy coefficient and R is the
hydraulic radius.
22. Dynamic wave celerity
The dynamic wave celerity can be found by
developing the characteristic equations for
the Saint-Venant equations.
Beginning with the nonconservation form of
the Saint-Venant equations (Table 1), it may
be shown that the corresponding
characteristic equations are (Henderson,
1966):
In which cd is the dynamic wave celerity,
given for rectangular channel by
23. where y is the depth of flow. For a channel of arbitrary
cross section, cd = .
This celerity cd measures the velocity of a dynamic
wave with respect to still water.
As shown in Fig. 2, in moving water there are two
dynamic waves, one proceeding upstream with
velocity V- cd and the other proceeding downstream
with velocity V + cd.
For the upstream wave to move up the channel
requires V< cd, or, equivalently, that the flow be
subcritical, sinceV = gy is the critical velocity of a
rectangular, open channel flow.
24. Example 1.
A rectangular channel is 200 feet (60.96m)
wide, has bed slope 1 percent and Manning
roughness 0.035. Calculate the water velocity
V, the kinematic and dynamic wave celerities
Ck and cd and the velocity of propagation of
dynamic waves V ± Cd at a point in the channel
where the flow rate is 5000 cfs (141.58m3/s).
Solution. Manning's equation with R=y, S0 =
Sf, and channel width B is written
25. which is solved for y as
(0.88m)
Hence the water velocity is
(2.64m/s)
26. The kinematic celerity ck is given by:
The dynamic celerity cd is
27. In interpreting these results with Fig. 2, it can be seen that a
flood wave traveling at the kinematic wave celerity (14.4
ft/s) will move down the channel faster than the water
velocity (8.65 ft/s), while the dynamic waves move
upstream (-1.0 ft/s) and downstream (18.3 ft/s) at the same
time.
In the event that the approximation S0= Sf is not valid, the
various velocities and celerities can be determined using the
full momentum equation to describe Sf as in Eq. (17).
28. ANALYTICAL SOLUTION OF THE
KINEMATIC WAVE
The solution of the kinematic wave equations specifies the
distribution of the flow as a function of distance x along the
channel and time t.
The solution may be obtained numerically by using finite
difference approximations to Eq. (7), or analytically by
solving simultaneously the characteristic equations (10) and
(14).
In this section the analytical method is presented for the
special case when lateral inflow is negligible;
The solution for Q(x, t) requires knowledge of the initial
conditionQ(x,0), or the value of the flow along the channel
at the beginning of the calculations, and the boundary
conditionQ(O, t), the inflow hydrograph at the upstream end
of the channel.
29. The objective is to determine the outflow
hydrograph at the downstream end of the channel,
Q(L, t), as a function of the inflow hydrograph, any
lateral flow occurring along the sides of the channel,
and the dynamics of flow in the channel as
expressed by the kinematic wave equations.
If the lateral flow is neglected, (10) reduces to dQIdx
= 0, or Q =a constant.
Thus, if the flow rate is known at a point in time and
space, this flow value can be propagated along the
channel at the kinematic wave celerity, as given by
30. The solution can be visualized on an x-t plane, as shown in Fig.3
(b), where distance is plotted on the horizontal axis, and time on
the vertical axis.
Each point in the x-t plane has a value of Q associated with it,
which is the flow rate at that location along the channel, at that
point in time.
These values of Q may be thought of as being plotted on an axis
coming out of the page perpendicular to the x-t plane.
In particular, the inflow hydrograph Q(O, t) is shown in Fig. 3(a)
folded down to the left, and the outflow hydrograph Q(L, i) is
shown in Fig 1(c) folded down to the right of the x-t plane.
These two hydrographs are connected by the characteristic lines
shown in part (b) of the figure.The equations for these lines are
found by solving (1):
31. Fig 3 Kinematic wave routing of a flow hydrograph through a channel reach of length L using
propagation of the flow along characteristic lines in the x-t plane. If flow rate were plotted
on a third axis, perpendicular to the x-t plane {b), then the inflow hydrograph (a) is the
variation of flow through time at x = 0 folded down to the left of the x-t plane; the outflow
hydrograph (c) is the variation of flow rate through time at x = L and is folded down to the
right of the x-t plane in the figure. The dashed lines indicate the propogation of specific flow
rates along characteristic lines in the x-t plane.
32. so the time at which a discharge Q entering a channel of
length L at time to will appear at the outlet is
The slope of the characteristic line is Ck = dQIdA for the
particular value of flow rate being considered.
The lines shown in Fig. 1(b) are straight because q = 0, and Q
is constant along them.
If q ≠ 0, Q andCk vary along the characteristic lines, which
then become curved.
33. Example 2
A 200-foot-wide rectangular channel is
15,000 feet long, has a bed slope of 1 percent,
and a Manning's roughness factor of 0.035.
The inflow hydrograph to the channel is given
in columns 1 and 2 ofTable 1. Calculate the
outflow hydrograph by analytical solution of
the kinematic wave equations.
34. solution
The kinematic wave celerity for a given value
of the flow rate is calculated in the same
manner as shown in Example 1, where it was
shown that for this channel, Ck = 14.4 ft/s for
Q = 5000 cfs.The corresponding values for the
other flow
35. Table 1: Routing of a flow hydrograph by
analytical solution of the kinematic wave
36. rates on the inflow hydrograph are shown in column 3 of
Table 1.
The travel time through a reach of length L is L/ck so for L =
15,000 ft and c* = 14.4 ft/s, the travel time is 15,000/14.4 =
1042 s/60s/min = 17.4 min, as shown in column 4 of the
table.
The time when this discharge on the rising limb of the
hydrograph will arrive at the outlet of the channel is, by Eq.
(3) t = to + L/ck=48 + 17.4 = 65.4 min, as shown in column 5.
The inflow and outflow hydrographs for this example are
plotted in Fig. 3.
It can be seen that the kinematic wave is a wave of
translation without attenuation; the maximum discharge of
6000 cfs is undiminished by passage through the channel.
