Hydraulic Theory
Channel Design


From the MIDUSS Version 2
Reference Manual  Chapter 8
(c) Copyright Alan A. Smith Inc. 
This section
summarizes the methods used to analyze the channel for uniform and
critical flow depth. As with the Pipe command, each reach of
channel is assumed to be prismatic, that is, of constant crosssection
and slope. As long as the channel flow has a free surface, the flow
in each reach is assumed to be quasiuniform, neglecting the variation
of flow with time. For this condition the friction slope
Sf
and the water surface are assumed to be parallel to the bed slope
S0.
The resistance is assumed to be represented by the Manning equation
[8.12].
MIDUSS lets you define
the crosssectional shape of the channel either as a trapezoidal shape
as shown in Figure 8.2 or as an arbitrary crosssection defined by the
coordinates of up to 50 points.
Figure 8.2 ‑ Definition
of a trapezoidal section.
Figure 8.3 shows a
crosssection defined by 9 points. For the purpose of illustration,
the remainder of this section assumes that the crosssection is
trapezoidal in shape. MIDUSS uses a simple routine to process the
coordinates of a more complex section with a specified water surface
elevation to yield the same crosssectional properties.
Figure 8.3  An arbitrary
crosssection using 9 points.
Normal Depth in
Channels
Figure 8.2 shows a
crosssection of arbitrary trapezoidal shape, of total depth
d
with a flow depth
y.
Using the Manning equation the normal discharge
Q
for any given depth y is given as follows.
where
Q
= normal discharge (c.m/s or
c.ft/s)
M
= 1.0 for metric units
1.49 for imperial or
US customary units (3.28 ft/m)^(1/3)
n
= Manning's roughness coefficient
A
= crosssectional area
R
= hydraulic radius = Area/Wetted perimeter
S0
= bed slope (m/m or ft/ft)
Evaluation of the
crosssection properties depends on whether a simple trapezoidal
crosssection or a more complex crosssection is defined. For a
general trapezoidal shape the following equations are used.
where
B
= base width
T
= top width
P
= wetted perimeter
y
= depth of flow.
GL
= slope of the left bank (GL
horiz : 1 vert)
GR
= slope of the right bank (GR
horiz : 1 vert)
and other terms are as
previously defined.
The maximum carrying
capacity Qfull for flow with a free
surface is found from equation [8.12] setting
y = d.
If the peak discharge
Q
is less than
Qfull
the depth of uniform flow is found by an interval halving technique.
Convergence is assumed when
Dy/y
<
0.000001.
The hydraulic gradient
is then computed by equation [8.14].
Critical Depth
in Channels
The calculation of
critical depth in a channel assumes that a free surface exists.
MIDUSS does not check to see if the critical depth is less than the
specified total depth
d.
If the basewidth is finite but the side slopes
are vertical the crosssection is rectangular and the critical depth
can be calculated explicitly by equation [8.15].
If the basewidth is
zero and at least one of the side slopes are finite the crosssection
is triangular and again an explicit solution for
ycr can be found
from equation [8.16].
For the case of a general trapezoidal
crosssection an iterative solution is required to solve the critical
flow criterion of equation [8.8]. This involves the application of
the NewtonRaphson method (equation [8.6])
in which the function and its derivative are defined by equations
[8.17] and [8.18] respectively.
Convergence is assumed when
Dy/y
< 0.0001. More information on the hydraulics of open
channels can be found in many standard texts (See Reference )
