POWER November 2020 - 24

HYDROPOWER
Understanding Open-Channel
Flow Equations for Hydro
Applications
Chezy and Manning developed equations that are used to determine the average
volumetric flowrate in open channels. This article explains a laboratory method that
was developed and tested to further identify and quantify the parameters that make
up the roughness coefficients of those equations. This method uses a hydraulic
flume, and makes use of the technique of dimensional homogeneity and a new
exponential form of an equation for instrument calibration.
Lee H. Sheldon, PE
A
ccurately measuring average velocities
in channels or culverts with
surfaces open to the atmosphere
has been a challenge for centuries. The
larger the flow cross-sectional area the
greater the inaccuracy or uncertainty of
measurement.
Open-channel flow is governed by
the Froude relation, the ratio of inertial
forces to gravitational forces. Thus, it
was recognized early in the history of
hydraulics that the formula for such average
velocity would need to be a balance
between gravity, causing the flow, and
channel roughness, seeking to retard
the flow. It was also recognized that any
such formula would have to be for uniform
flow, that is, for steady state flow,
such that the water depth relative to the
bottom of the waterway is a constant,
or d(y)/dx = 0.
It is noted that in pipe or pressurized
flow the word uniform has a different
meaning. In that application, it means
that the velocity profile has a constant
velocity over the entire cross-section. On
the other hand, open-channel hydraulics
has no word for constant velocity over
a cross-section. In this article, " normal "
means the first of these two definitions,
that is, steady state and constant depth.
All units in this article are engineering
units as commonly used in the U.S.
Equations Developed by Chezy and
Manning
The first recognized and most lasting
" resistance " formula for steady state,
open-channel flow is credited to Antoine
Chezy. He was tasked with determining
the cross-section and calculating the
discharge for the Paris water supply, and
24
increasing its flowrate. He did so in 1768
by comparing flow conditions between
two water courses, the Courpalet Canal
and the Seine River. His resulting formula
was published in his report on the
Canal de l'Yvette as:
Vavg = C x R1/2 x S1/2
where Vavg is the average velocity in feet
per second; C is Chezy's factor of flow
resistance in feet1/2/sec; R is the hydraulic
radius (the cross-sectional area divided
by the wetted perimeter) in feet; and
tems. In the U.S. system, it has units of
second/feet1/3. If using metric units, the
1.486 is replaced by 1.0 and its units are
second/meter1/3.
Manning's equation has been the
most successful of all open-channel
empirical equations, based on the resistance
to flow and derived from observation.
In fact, it is no exaggeration to say it
is the cornerstone of today's science of
hydraulic engineering.
However, in the classic sense, both
Chezy's and Manning's equations have
several similar shortcomings. First, they
Chezy's and Manning's equations have
several similar shortcomings. First, they
do not have dimensional homogeneity.
S is the slope, which is dimensionless.
However, Chezy's work received little attention
until many years after his death.
In 1889, an Irishman named Robert
Manning, who was Chief Engineer of Ireland's
Office of Public Works, presented
a paper entitled " On the Flow of Water
in Open Channels and Pipes. " Although
his main interest appears to have been
hydrology, he derived an average " resistance "
formula for open channels from
all the different resistance formulas published
up to that time. In today's format,
this equation, which we'll call Equation 1
for future reference, is:
Vavg = (1.486/n) x R2/3 x S1/2
where n is Manning's roughness coefficient,
which is the same numerically in
either U.S. or metric dimensional syswww.powermag.com
do
not have dimensional homogeneity,
that is, the units on the left side are not
the same as the units on the right side.
Such equations are usually derived by experimentation
or observation and quickly
lose accuracy if extrapolated beyond
their range of observation. It is known
that Manning's equation loses accuracy
with very steep or shallow slopes. Secondly,
to achieve dimensional homogeneity,
their constants or coefficients are
not pure numbers, but are artificially assigned
units.
Furthermore, Manning's equation
suggests the average velocity is more
sensitive to the hydraulic radius than
to the slope. This really is an incompatibility,
because the very nature of openchannel
flow is a function of the slope
component of gravity. The shape of the
water passage, as calculated by the hyPOWER
| November 2020
http://www.powermag.com
POWER November 2020

Table of Contents for the Digital Edition of POWER November 2020
