The Manning formula or Manning's equation is an
empirical formula
In chemistry, the empirical formula of a chemical compound is the simplest whole number ratio of atoms present in a compound. A simple example of this concept is that the empirical formula of sulfur monoxide, or SO, is simply SO, as is the empir ...
estimating the average velocity of a liquid in an
open channel flow (flowing in a conduit that does not completely enclose the liquid). However, this equation is also used for calculation of flow variables in case of
flow in partially full conduits, as they also possess a free surface like that of open channel flow. All flow in so-called open channels is driven by
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
.
It was first presented by the French engineer in 1867, and later re-developed by the Irish engineer
Robert Manning in 1890.
Thus, the formula is also known in Europe as the Gauckler–Manning formula or Gauckler–Manning–Strickler formula (after
Albert Strickler).
The Gauckler–Manning formula is used to estimate the average velocity of water flowing in an open channel in locations where it is not practical to construct a
weir
A weir or low-head dam is a barrier across the width of a river that alters the flow characteristics of water and usually results in a change in the height of the water level. Weirs are also used to control the flow of water for outlets of l ...
or
flume to measure flow with greater accuracy. Manning's equation is also commonly used as part of a numerical step method, such as the
standard step method, for delineating the free surface profile of water flowing in an open channel.
[ Chow (1959) pp. 262-267]
Formulation
The Gauckler–Manning formula states:
:
where:
* is the cross-sectional average velocity (dimension of
L/
T; units of ft/s or m/s);
* is the Gauckler–Manning coefficient. Units of are often omitted, however is not dimensionless, having dimension of T/L
1/3 and units of s/m
1/3.
* is the hydraulic radius (L; ft, m);
* is the
stream slope or
hydraulic gradient, the linear
hydraulic head loss loss (dimension of L/L, units of m/m or ft/ft); it is the same as the
channel bed slope when the water depth is constant. ().
* is a conversion factor between
SI and
English units
English units were the units of measurement used in England up to 1826 (when they were replaced by Imperial units), which evolved as a combination of the Anglo-Saxons, Anglo-Saxon and Ancient Roman units of measurement, Roman systems of units. V ...
. It can be left off, as long as you make sure to note and correct the units in the term. If you leave in the traditional SI units, is just the dimensional analysis to convert to English. for SI units, and for English units. (Note: (1 m)
1/3/s = (3.2808399 ft)
1/3/s = 1.4859 ft/s)
Note: the Strickler coefficient is the reciprocal of Manning coefficient: 1/, having dimension of L
1/3/T and units of m
1/3/s; it varies from 20 m
1/3/s (rough stone and rough surface) to 80 m
1/3/s (smooth concrete and cast iron).
The
discharge formula, , can be used to rewrite Gauckler–Manning's equation by substitution for . Solving for then allows an estimate of the
volumetric flow rate
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes \do ...
(discharge) without knowing the limiting or actual flow velocity.
The formula can be obtained by use of
dimensional analysis. In the 2000s this formula was derived theoretically using the phenomenological theory of
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
.
Hydraulic radius
The hydraulic radius is one of the properties of a channel that controls water discharge. It also determines how much work the channel can do, for example, in moving sediment. All else equal, a river with a larger hydraulic radius will have a higher flow velocity, and also a larger cross sectional area through which that faster water can travel. This means the greater the hydraulic radius, the larger volume of water the channel can carry.
Based on the 'constant
shear stress
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
at the boundary' assumption,
hydraulic radius is defined as the ratio of the channel's cross-sectional area of the flow to its
wetted perimeter
Wetting is the ability of a liquid to displace gas to maintain contact with a solid surface science, surface, resulting from intermolecular interactions when the two are brought together. These interactions occur in the presence of either a ga ...
(the portion of the cross-section's perimeter that is "wet"):
:
where:
* is the hydraulic radius (
L);
* is the cross sectional area of flow (L
2);
* is the
wetted perimeter
Wetting is the ability of a liquid to displace gas to maintain contact with a solid surface science, surface, resulting from intermolecular interactions when the two are brought together. These interactions occur in the presence of either a ga ...
(L).
For channels of a given width, the hydraulic radius is greater for deeper channels. In wide rectangular channels, the hydraulic radius is approximated by the flow depth.
The hydraulic radius is ''not'' half the
hydraulic diameter
The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channe ...
as the name may suggest, but one quarter in the case of a full pipe. It is a function of the shape of the pipe, channel, or river in which the water is flowing.
Hydraulic radius is also important in determining a channel's efficiency (its ability to move water and
sediment
Sediment is a solid material that is transported to a new location where it is deposited. It occurs naturally and, through the processes of weathering and erosion, is broken down and subsequently sediment transport, transported by the action of ...
), and is one of the properties used by water engineers to assess the
channel's capacity.
Gauckler–Manning coefficient
The Gauckler–Manning coefficient, often denoted as , is an empirically derived coefficient, which is dependent on many factors, including surface roughness and
sinuosity. When field inspection is not possible, the best method to determine is to use photographs of river channels where has been determined using Gauckler–Manning's formula.
The friction coefficients across weirs and orifices are less subjective than along a natural (earthen, stone or vegetated) channel reach. Cross sectional area, as well as , will likely vary along a natural channel. Accordingly, more error is expected in estimating the average velocity by assuming a Manning's , than by direct sampling (i.e., with a
current flowmeter), or measuring it across weirs, flumes or
orifices.
In natural streams, values vary greatly along its reach, and will even vary in a given reach of channel with different
stages of flow. Most research shows that will decrease with stage, at least up to bank-full. Overbank values for a given reach will vary greatly depending on the time of year and the velocity of flow. Summer vegetation will typically have a significantly higher value due to leaves and seasonal vegetation. Research has shown, however, that values are lower for individual shrubs with leaves than for the shrubs without leaves.
This is due to the ability of the plant's leaves to streamline and flex as the flow passes them thus lowering the resistance to flow. High velocity flows will cause some vegetation (such as grasses and forbs) to lay flat, where a lower velocity of flow through the same vegetation will not.
In open channels, the
Darcy–Weisbach equation
In fluid dynamics, the Darcy–Weisbach equation is an Empirical research, empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressibl ...
is valid using the hydraulic diameter as equivalent pipe diameter.
It is the only best and sound method to estimate the energy loss in human made open channels. For various reasons (mainly historical reasons), empirical resistance coefficients (e.g. Chézy, Gauckler–Manning–Strickler) were and are still used. The
Chézy coefficient Chézy may refer to:
; People
* Antoine de Chézy (1718–1798), French hydraulic engineer
* Antoine-Léonard de Chézy (1773–1832), French orientalist
* Helmina von Chézy (1783–1856), German journalist, poet and playwright
; Communes in Fran ...
was introduced in 1768 while the Gauckler–Manning coefficient was first developed in 1865, well before the classical pipe flow resistance experiments in the 1920–1930s. Historically both the Chézy and the Gauckler–Manning coefficients were expected to be constant and functions of the roughness only. But it is now well recognised that these coefficients are only constant for a range of flow rates. Most friction coefficients (except perhaps the Darcy–Weisbach friction factor) are estimated ''100% empirically'' and they apply only to fully rough turbulent water flows under steady flow conditions.
One of the most important applications of the Manning equation is its use in sewer design. Sewers are often constructed as circular pipes. It has long been accepted that the value of varies with the flow depth in partially filled circular pipes.
A complete set of explicit equations that can be used to calculate the depth of flow and other unknown variables when applying the Manning equation to circular pipes is available.
These equations account for the variation of with the depth of flow in accordance with the curves presented by Camp.
Authors of flow formulas
*
Albert Brahms (1692–1758)
*
Antoine de Chézy
Antoine de Chézy (September 1, 1718 – October 5, 1798), also called Antoine Chézy, was a French physicist and hydraulics engineer who contributed greatly to the study of fluid mechanics and designed a canal for the Paris water supply. He i ...
(1718–1798)
*
Henry Darcy (1803–1858)
*
Julius Ludwig Weisbach (1806-1871)
* (1826–1905)
*
Robert Manning (1816–1897)
*Wilhelm Rudolf Kutter (1818–1888)
*
Henri Bazin (1843–1917)
*
Ludwig Prandtl (1875–1953)
*
Paul Richard Heinrich Blasius (1883–1970)
*
Albert Strickler (1887–1963)
*Cyril Frank Colebrook (1910–1997)
See also
*
Chézy formula
*
Darcy–Weisbach equation
In fluid dynamics, the Darcy–Weisbach equation is an Empirical research, empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressibl ...
*
Hydraulics
Hydraulics () is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counterpart of pneumatics, which concer ...
Notes and references
Further reading
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External links
*
Hydraulic Radius Design Equations Formulas CalculatorHistory of the Manning Formula*