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Credits, sources etc. - see also references at end
This has so far been abstracted (and slightly modified) from : On the Shoulders of Giants-part II; by
Francis E. Griggs Jr., Prof. of Civ. Engrg., Merrimack Coll., North Andover, MA 01845, and
Dir. of Quality Assurance, Clough Harbor Associates, Albany, NY 12205.
published in the ASCE Journal of Professional Issues in Engineering Education and
Practice, Vol. 122, No. 1, January, 1996. ASCE, ISSN 0733-9380/96/0001-0017-0025
The education of our
future water resources engineers is incomplete without an appreciation of the
men and women who created the profession of civil engineering in the late
eighteenth and nineteenth centuries. This module traces our understanding of
open channel flow and the development of the Manning equation. It begins in the
time of Aristotle and Archimedes and ends in the late nineteenth century with
the publication of Manning's work and equation.
When people asked Isaac Newton how he
had accomplished so much, he answered, "If I have seen further
it is by standing on the shoulders of giants." If Newton
acknowledged his debt to those who preceded him, we - teachers,
practitioners and students alike - should recognize our debt to
those early civil engineers who paved the way for us.
Who are these giants and why is it
important that students have knowledge of the history of our profession?
The reasons are many, but the primary reason is that lessons, which
are applicable today, can be learned by studying the people and
projects that transformed engineering from an art to a science over
the eighteenth and nineteenth centuries. Students, once exposed to
the past, will have a far greater appreciation of the contributions
made by these people and be able to place their profession in a
better historical and professional context.
Note from Francis
Briggs: I believe that professors, given the time and opportunity, would
be interested in incorporating history segments into many of their
engineering courses. The main obstacle is that many professors do
not have time to read enough background material to feel
comfortable with lecturing on historical matters. Having taken up
the study of the history of civil engineering, I can appreciate the
time it takes to become comfortable in this new, and yet old, area
of study. I have been so impressed, however, with the past and the
founders of my profession, that I am, like the Blues Brothers,
"on a mission from God." That mission is to make it as
easy as possible for any professor to include history in his or her
courses. This module that can be used in the teaching of fluid
mechanics. The module includes photographs of the engineers who
made it all possible, as it is important for the students to see,
and remember, the images of those upon whose shoulders they are
standing.
The key sources for
this paper were the two books by Hunter Rouse (1957, 1976) on the
history of hydraulics. The references given at the end of the
article, however, are the source of most of the information.
Preliminary
Investigations
The study of mechanics,
including fluid mechanics, had its start, as did most scientific matters, with
the Greeks, particularly Aristotle and Archimedes. Aristotle (384-322 B.C.)
believed that the motion exhibited by all elements, including water, was
related to their tendency to return to their natural level. He discussed the
movement of bodies through air and water, noting that "the medium causes
a difference because it impedes the moving thing. Most of all it is moving in
the opposite direction, but in a secondary degree even if it is at rest. . .
." He was, therefore, thinking about what we would now call drag, if the
body moved through a medium (air, water) or fluid resistance if the fluid
flowed over a stationary stream bed. Archimedes (287-212 B.C.) was much more
involved with fluid behavior, building the Archimedes' screw and formulating
the fundamental rules of hydrostatics: "Any solid lighter than a fluid
will, if placed in the fluid, be so far immersed that the weight of the solid
will be equal to the weight of the water displaced. . . . A solid heavier than
a fluid will, if placed in it, descend to the bottom of the fluid, and the
solid will, when weighed in the fluid, be lighter than its true weight by the
weight of the fluid displaced." Neither thinker appears to have
considered the flow of water through a channel or aqueduct.
The Romans were
perhaps the greatest aqueduct builders of all time, having built them
throughout their empire. Many of these aqueducts still remain a millennium and
a half later. Frontinus, one of the greatest aqueduct builders, was aware that
water only flowed downhill and that the greater the slope, for a given size
channel, the greater the amount of water that could be transmitted. He wrote,
for instance, that "the several aqueducts reach the city at different
elevations. Whence it comes that some deliver water on higher grounds while
others cannot elevate themselves to the higher summits." There is no
evidence that the Romans built their aqueducts at specific slopes to achieve
specific amounts of flow. The size of the channel appears to have come more
from structural necessity than hydraulic need. The slopes appear to be based
primarily on topography. Clemens Herschel (1899), translated Frontinus' works and was convinced that while the Romans may have
had a qualitative sense of the linkage between slope, area of conduit, and
discharge, it was very poorly defined and not, in his opinion, a factor in
their design of the aqueducts. Frontinus was aware of channel resistance.
"Let us not forget in this connection," he said, "that every
stream of water whenever it comes from a higher point and flows into a
delivery tank through a short length of pipe, not only comes up to its
measure, but yields, moreover, a surplus; but whenever it comes from a low
point, that is under a less head, and is conducted a tolerably long distance,
it will actually shrink in measure by the resistance of its own conduit."
In other words, long, flat aqueducts yield a lessened amount of flow or
discharge.
Leonardo da Vinci (1452-1519) also
studied the flow of water in open channels and developed a nine
chapter treatise on the subject. While he made many errors, as seen
from twentieth century knowledge, his observations on open channel
flow were remarkably accurate:
The water of straight rivers is the
swifter the farthest away it is from the walls, because of
resistances. . . .
Water has a higher speed on the surface than at the
bottom. This happens because water on the surface borders on air which
is of little resistance, because [it is] lighter than water, and
water at the bottom is touching the earth which is of higher resistance,
because [it is] heavier than water and not moving. From this
follows that the part which is more distant from the bottom has
less resistance than that below.
In other words, Leonardo had
correctly understood bed resistance and the impact it had on the
velocity distribution over the depth of flow. He also had a grasp
of what is now called the continuity of flow: "A river in each
part of its length in an equal time gives passage to an equal
quantity of water, whatever the width, depth, the slope, the
roughness, the tortuosity. . . ."
It
is clear that he knew that the flow of the river was a function of
the product of area and velocity, such that, as the area increased,
the velocity would drop, and vice versa. This fundamental equation
that
is known as the continuity equation.
Leonardo's work on fluid flow, however, like much of his work, was
not generally known by his peers and only became available to us
recently.
Mariotte - 1686
Edmé Mariotte
(1620-1684) was the next major investigator to experiment with the
flow of water in open channels and published his work in "A
Treatise of Hydrostaticks, Wherein the Motion of Water and Other
Fluids, Is Considered." His work (Mariotte 1978) was
originally published in 1686, shortly after his death, by his
friend De la Hire, himself one of the preeminent French scientists
of the time. Open channel flow was covered in "Discourse
III-Of the Equilibrium of Fluids by Their Impulse" and
"Discourse IV-Of the Measure of Water Running in an Aqueduct,
or in a River."
In
Discourse III he wrote that "you must consider, that the water
of a river does not go equally fast at its surface, and in its
other parts; for the water near the bottom is very much retarded by
meeting with stones, weeds, and other inequalities." He had
used wax floats and measured the time it took for the floats to
move a certain distance down the river. He described his experiment
as follows:
In Discourse IV he continued his wax
float idea to determine the amount of flow in an aqueduct.
We must place upon water a ball of wax, loaded within with something
heavy, insomuch that but very little of the wax lies above the
surface of the water, for fear of the wind; and after having
measur'd a length of 15 or 20 feet in the aqueduct, we shall know
by a half second pendulum in how much time the ball of wax, carried
by the current of the water, will run that distance. Afterwards we
must multiply the breadth of the aqueduct by the height of the
water, and the product by the space which the wax shall have run
thro'. The last product which is solid, will give all of the water
which shall have passed during the time of observation thro' one
section of the aqueduct. To make this observation with exactness,
it is necessary that the bed of the aqueduct should have the same inclination
as the superficies of the water that passes in it. And moreover we
suppose, that the water runs equally fast, on the top, and on the
sides.
The
latter statement would imply that Mariotte thought that the
velocity was the same throughout the cross section of flow. That he
did not make this oversight can be seen in a passage that followed
the one just quoted.
Mariotte assumed an average velocity of flow of
two-thirds of the surface velocity. He did not give a functional relationship
between depth and velocity, but the example he used indicates that
he did not view the relationship as linear.
By the end of the seventeenth century
it was known that the discharge of an open channel was a function
of the slope, the shape of the channel, and the roughness of the
channel bed. The functional relationship, however, was not known.
Thinkers like Mariotte could, however, determine the discharge of existing
channels by use of crude, surface velocity measuring devices and
the continuity equation.
Design of channels
With this study of
the flow of water in open channels up to the seventeenth century we
will now move into the study of how channels can be designed to
pass a desired amount of flow. We will show who determined the
relationship between the slope, roughness, and shape of a channel,
and the velocity of flow. The study will trace the development of
one of our most-used formulas, the Manning formula. This formula
(often called the Chézy-Manning formula), has its origins in the 1770s
when Antoine Chézy was assigned the task of increasing the water
supply of Paris.
Chézy -
1776
Chézy (1718-1798) graduated with honors from the Ecole des Ponts et Chaussées and
worked closely with Perronet, the first director of the school. He
was active in a wide range of civil engineering work, including the
construction of bridges and streets in Paris. He and Perronet were
given the task of determining how much flow could be brought from
the Yvette River into the city. They wanted to be able to predict
the flow of water based on analytical methods rather than on
experience and full-size tests. Chézy built model channels on
which he ran tests to determine the factors that influence flow in
an open channel. He ran his tests on a wooden flume approximately 200
m in length, made up of a channel 1.3 m wide and 0.52 m deep. At
about the same time that Chézy was conducting his experiments,
Pierre Louis DuBuat was also conducting a series of tests that
perhaps had a longer life and greater effect than those of Chézy,
even though Chézy's name survives and DuBuat's is known only to
historians of hydraulics.
Chézy began his study
with the following comments:
After having designed a channel, and having
well adjusted and regulated its slope, it is very interesting to
know if this channel will be sufficient for the water which is to
flow in it. For this, it is necessary to know the speed at which
the water can flow in the channel, which one assumes to have a
uniform slope. . . .
. . . Whatever that
initial velocity may be, it diminishes or augments rapidly enough
to reduce to a uniform and constant velocity which is due to the
slope of the channel and to gravity, of which, the effect is
restrained by the resistance of friction against the channel
boundaries. . . .
The velocity due to gravity, which acts continuously .
. . is only uniform when it no longer accelerates, and gravity does not
cease to accelerate except when its action upon water is equal to
the resistance occasioned by the boundary of the channel; but the
resistance is as the square of the velocity because of the number
and force of the particles colliding in a given time; it is also as
the part of the perimeter of the section of the flow which touches
the boundaries of the channel. . . .
Upon calling the velocity , and that
part of the perimeter , the resistance of friction will therefore
be as . On the other hand, the effect of gravity is as the area of
the section of the current, and as the slope of the channel or as
the height which it descends for each toise [1.949 meters] length.
Calling, therefore, the area of the section A, and the slope of the
channel H, the effect of gravity will be as AH.
Chézy then
concluded that the ratio of the driving force and the resisting
force of two streams (one with uppercase values and one with
lowercase values) would be thus:
Where, in his
analysis, the uppercase values were from a known stream and the
lowercase values from a channel he was designing. Since the ratios
are equal to each other they must be equal to a constant. Chézy
later equated the left ratio to a constant and expressed his
equation in the form in which it is usually seen.
where R = a/p
is the hydraulic radius, and S = h.
He knew that C was
not a universal constant but one that would have to be determined
on a case-by-case basis. However, based on his experiments he
determined that C ranged between 31 and 44 when using metric units.
The Chézy formula was usually expressed as:
or
in current terminology
It is clear that the
average velocity of flow varied with the square root of the
hydraulic radius and the square root of the slope of the channel.
Storrow -
1835
Charles
Storrow (1809-1904), an American civil engineer, born and educated in
Montreal, studied in France at
the Ecole Des Ponts et Chaussées in the early 1830s. He was
exposed to the work of the French physicists, mathematicians, and
engineers who were trying to combine theory and experiment into a
comprehensive approach to the determination of the laws governing
the flow of water in open channels. He came back to the United
States in 1832 to work on the railroad connecting Lowell, Mass. and
Boston. In 1835 he received a serious injury, which required a
period of recuperation. It was during this period that he wrote his
famous A Treatise on Water Works for Conveying and Distributing Supplies
of Water (Storrow 1835). In his chapter on the motion of water in
open channels he discussed only the work of de Prony, Eytelwein,
and Bélanger. In his introduction, however, he reported on the
works of Couplet, Chézy, Mariotte, Bossut, DuBuat, Girard, de
Prony, Bélanger, and Eytelwein, thus making known to the engineers
of America the best that the continent had to offer in the area of
flow of water in open channels. This book became the bible of
American engineers working in the area of hydraulics. In his book
he converted the de Prony and Eytelwein constants into equivalent
English units and put his calculations into a tabular format, which
made the work even more important to American engineers. Thus, while
Storrow contributed nothing new to the study of flow in open
channels, he did make known to American engineers what had been
developed in Europe.
Weisbach - 1845
In 1845, Julius Weisbach (1806-1871) wrote a
book which, for the first time, made the teaching of fluid mechanics an
integral part of engineering mechanics. He believed in the importance of
a non-dimensional coefficient in his open channel flow equation. His
equation was that
or
where
R = A/D
where z was a coefficient that varied with
V alone. Weisbach is
also credited with being the first man to write an accurate equation for
the flow through a weir.
Saint
Venant - 1851
The great Saint
Venant (1797-1886) in 1851 also made an approximation that
in metric units.
It was
clear to these investigators, with the exception of Saint Venant,
that based on theory and experimental data, the velocity was
approximately a function of the square root of the product of R and
S, times a constant.
Manning -
1851
Starting
in 1851, Robert Manning combined the results of many experimenters
into a single equation that best matched observed velocities of
flow in open channels. Manning was an engineer in the Irish Office
of Public Works during the years of the great famine. In 1848, as a
new district engineer, he read the second edition of Traité
d'Hydraulique by d'Aubisson des Voissons and from that time on had
a great interest in hydraulics. His first paper was given in 1851
to the Institution of Civil Engineers of Ireland and dealt with
hydraulics and hydrology. A later paper on the design of the water
system of Belfast, Ireland, won the Telford Medal from the
Institution of Civil Engineers, London in 1866.
D'Aubisson's
equation was
in English units and was
sometimes published as
Basing his
calculations on a form of the de Prony-Eytelwein formula, Manning
simplified d'Aubisson's equation (for computational purposes) to
where D = slope of the channel in ft/mile and
R is in feet.
He found
that this simplified result did not always match the observed data
very well. It was, however, of the same form as Chézy's formula of
1776.
Darcy and Bazin
- 1857 and 1865
Henri P. G. Darcy
(1803-1858) and Henri E. Bazin (1829-1917) performed a comprehensive set
of experiments on pipe flow in 1857, 1858, and 1859. Darcy, a
native of Dijon, France, was the designer and builder of the water
system for his hometown. This system later became a model for
several municipal water systems. In 1857, Darcy, working primarily
with pipe flow, suggested that
where D
= pipe diameter.
Darcy
died before the end of his studies, but his assistant Bazin carried
on the experiments to their conclusion. In 1865 Bazin proposed an
equation similar to Darcy's but this time for open channel flow.
where R the
hydraulic radius replaced D the pipe diameter.
Bazin wrote that
the "two coefficients are not, it is true, completely
independent of the slope; but they vary within limits far narrower
than those of the Prony binomial formula." He did not have a
specific S term in his coefficient of the V2 term as his "two
coefficients vary in an inverse sense as one modifies the slope of
the canal. . . . . There is thus established a sort of compensation
by which the formulas obtained for several slopes, although
different from the first, give within ordinary limits of
application almost identical values, and they can hence be replaced
without inconvenience by a single formula with mean
coefficients."
Gauckler - 1868
In 1868, Philipe Gaspard Gauckler (1826-1905), after
reviewing the data of Humphreys and Abbot and others, found that a
single equation could not model the flow in streams of significantly
differing slopes. He therefore developed two equations, one for slopes
of less than 0.0007 and one for slopes greater than 0.0007. They were
that
for
slopes < 0.0007 and
for slopes > 0.0007
The latter equation
has, as we will see, been credited to Manning but was in actual
fact determined, in part, by Gauckler 30 years previously.
Gauckler's C2 was not the same as Manning's, so even though the
power on R was the same, the velocities would be different.
Ganguillet and
Kutter - 1869
In 1869, the first
attempt to link an equation to the roughness of the channel came
from Emile Oscar Ganguillet (1818-1894) and Wilhelm Ruldoph Kutter
(1818-1888), two Swiss engineers. Ganguillet was the chief engineer
of the Bern Department of Public Works, and Kutter a member of his
staff. Their study was prompted by the Mississippi River report by
Humphreys and Abbot. Applying the American equation to the steep mountain
streams of Switzerland they found that it did not predict the
velocities or discharges of streams of this topography. They
concluded that the American equation was only good for gentle
slopes, and they developed a complex equation that for the first
time included a bed-roughness term. It was as follows:
Where n is the
Kutter roughness factor that runs between 0.01 and 0.035 for the
usual channel surfaces. For the usual ranges of S, n, and R, the
value of Kutter's C, to be used with Chézy's formula, ranged from
22 to 220. For values of S > 0.0005, the term m/S was frequently
neglected, thus simplifying the equation somewhat.
Fteley and
Stearns - 1883
In the late 1870s
Alphonse Fteley (1837-1903) and Frederic Stearns (1851-1919) were
engineers on the Boston water supply. They were assigned to design
and build the Sudbury Aqueduct in Massachusetts. As a part of that
design they ran many tests on various conduits and described them
in a Norman Award-winning article in the Transactions ASCE in 1883.
Both were later to become presidents of ASCE. Their equation
was
It can be seen that
they arrived at a power of R somewhat higher than earlier
investigators but less than that at which Manning was to arrive.
Manning -
1889
In 1889 at the age of 72, almost a professional lifetime
later, Manning (1816-1897)revisited the topic.
Manning went back and
restudied the previous data to determine whether or not a new
formula, based on experiment, could be determined. He made his
findings known in a classic paper to the Institution of Civil
Engineers, Ireland in 1889, with a supplement issued in 1895
(Manning 1889, 1895). In his 1889 paper he reviewed the equations
of DuBuat, Eytelwein, Weisbach, Saint Venant, Neville, Darcy and
Bazin, and Ganguillet and Kutter. He then computed the velocities
given by each formula for a given slope and for various values of
R. He chose to use the mean of the velocities given by all the formulas
as an approximation of the truth. He then curve-fit a line to data
formed by plotting R and V for each formula. He found several
equations that would fit the data, but he started with the
following one:
in metric units
This equation, he
admitted, "was purely empirical," and he tried another
approach, which yielded
in metric
units.
He arrived at this
formula by holding the power of his slope term at 1/2 and
determining, using the same approach, the best power of R and the
constant C. He knew that other formulas would also approximate the
data, but he liked the form of the equation as it was similar to
other, previously given formulas. Yet the 4/7 power on R concerned
him, so he went back to Bazin's data and determined which power of
R might fit the data better. Using a large number of Bazin's tests
he found that values for the power of R ran between 0.6351 and
0.6778 for one battery of tests and 0.6176 and 0.6733 for another
battery. Another set of tests, however, had a value range between 0.7635
and 0.8395.
He concluded, even
given the spread of data, that the formula was "sufficiently
accurate to take the value of the exponent at 0.666 or 2/3. . .
." He then compared his formula in the form
to 170 experiments
that had been conducted by Bazin, Kutter, Revy, Fteley and Stearns,
and Humphreys and Abbot. Based on this analysis he concluded that
"[if] these observations be correct, then the author's formula
gives better results than either Bazin or Kutter."
He knew this
equation did not fit all of the data available, especially the
Mississippi River tests by Humphreys and Abbot. He spent a great
deal of time discussing the Humphreys data and came to the
following conclusions:
He
therefore rejected Humphreys and Abbot's data as inaccurate due to
the sensitivity of measuring the very flat slopes of the
Mississippi, and with this data removed from the mix he found that
his formula was within 7% of the observed velocities in all but 17
of the remaining 160 tests. He had, of course, determined values of
C for various channel bottoms as a part of his analysis.
By the end of 1889,
Manning had published the results of his work, which effectively
modified Chézy's equation by having the power of the hydraulic
radius at 2/3 instead of the 1/2 used by Chézy and some
others. He did not like his own equation for two reasons. The first
was that it was difficult in those days to determine the cube root
of a number and then square it to arrive at a number to the 2/3
power. (If he didn't like the 2/3 power, one wonders what he
thought of his earlier 4/7 power.) In addition, the equation was
dimensionally incorrect, and so to obtain dimensional correctness
he developed the following equation:
where m =
"height of a column of mercury which balances the atmosphere,"
and C was a dimensionless number "which varies with the nature
of the surface."I
In a supplement to his famous paper (Manning 1895),
he wrote about his simpler formula that "it is worthy of
remark that the value of the reciprocal of C corresponds closely
with that of n, as determined by Ganguillet and Kutter; both
C and
n being constant for the same channel."
In some late
nineteenth century textbooks the Manning formula was written as
in metric units. The equation in
English units was first published by Bovey (1901) as
Note that the
1.486 is a unit conversion factor from the metric to the English
system and is the cube root of 3.28, the number of feet in a meter.
The equation was changed in this manner so that the value of n is
independent of the system of units chosen.
Values of Manning's
n for the
usual channel bottoms range from 0.012 to 0.035. He never mentioned
Kutter's n, so the n in the equation was based on knowledge of
similar streams and their flow characteristics.
Bazin
- 1897
It was after the
publication of the results of Ganguillet and Kutter that Bazin
changed the form of his equation to
where
B = a roughness factor. In
the English system of units the equation became
where
m ranged between 0.5 and 1.75
The Bazin and Kutter
equations were the commonly accepted equations up until early in
the twentieth century. The Bazin equation was used primarily in
France and England, and the Kutter Equation was used in most of the
rest of the world
Summary
For many years
Manning's equation was given in both forms, but as time went by with the
results being about the same, the more complex, dimensionally correct
equation was dropped and the more familiar equation adopted as Manning's
equation. For his dimensionally correct formula, Manning gave a range of
values of C in three different systems of units: dimensionless, feet,
and metric.
The simpler formula
slowly became accepted by engineers around the world and was
incorporated into English language textbooks as early as 1899. In the
United States, Horace King's Handbook of Hydraulics, published in 1918
(Second Edition 1929), firmly set the Manning formula as the accepted
equation to use in calculating the velocity of flow in an open channel.
His reasons for doing so were as follows:
Second edition
comment: In the first edition of this volume the author advocated the
adoption of the Manning formula for open channels. After eleven years,
many engineers have recognized the advantages of this simple formula and
its use has become quite extensive. In the present edition its adoption
as a general formula applicable to both open channels and pipes is
recommended.
With this
recommendation the Manning equation, not the one that Manning
liked, but the simplified equation, became the standard used in the
United States. Over the 120 years between Chézy and Manning, the
only thing that changed was the power on R from 1/2 to 2/3, and the
modification of Chézy's C into a constant using Kutter's n. Some
of the best engineers, mathematicians, and physicists of France,
England, Ireland, Germany, and the United States had, however, a
hand in shaping this apparently small change in the equation.
Today theoreticians
favor a logarithmic-type formula as it is more in keeping with what
we now know of the nature of the boundary layer between the bed of
a channel and the moving fluid. Yet even though we know much more
about turbulent flow than we did in the latter part of the
nineteenth century, Manning's formula is still used by practicing
engineers.
As
we move into the study of open channel flow we should remember the
work of da Vinci, Mariotte, Chézy, DuBuat, Eytelwein, Darcy, Bazin,
de Prony, Girard, Humphreys, Kutter, Manning, and many others who
developed, over a long period of time, the formula fundamental to
our understanding of open channel flow.
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