The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. 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.
It was first presented by the French engineer Philippe Gaspard Gauckler[fr] in 1867,[1] and later re-developed by the Irish engineer Robert Manning in 1890.[2] 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 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.[3]
Formulation
The Gauckler–Manning formula states:
where:
V is the cross-sectional average velocity (L/T; ft/s, m/s);
n is the Gauckler–Manning coefficient. Units of n are often omitted, however n is not dimensionless, having units of: (T/[L1/3]; s/[m1/3]).
k is a conversion factor between SI and English units. It can be left off, as long as you make sure to note and correct the units in the n term. If you leave n in the traditional SI units, k is just the dimensional analysis to convert to English. k = 1 for SI units, and k = 1.49 for English units. (Note: (1 m)1/3/s = (3.2808399ft)1/3/s = 1.4859ft/s)
NOTE: Ks strickler = 1/n manning. The coefficient Ks strickler varies from 20 (rough stone and rough surface) to 80 m1/3/s (smooth concrete and cast iron).
The discharge formula, Q = AV, can be used to rewrite Gauckler–Manning's equation by substitution for V. Solving for Q then allows an estimate of the volumetric flow rate (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.[4][5]
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 at the boundary' assumption,[6] hydraulic radius is defined as the ratio of the channel's cross-sectional area of the flow to its wetted perimeter (the portion of the cross-section's perimeter that is "wet"):
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 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), 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 n, 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 n is to use photographs of river channels where n has been determined using Gauckler–Manning's formula.
The friction coefficients across weirs and orifices are less subjective than n along a natural (earthen, stone or vegetated) channel reach. Cross sectional area, as well as n, will likely vary along a natural channel. Accordingly, more error is expected in estimating the average velocity by assuming a Manning's n, than by direct sampling (i.e., with a current flowmeter), or measuring it across weirs, flumes or orifices.
In natural streams, n 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 n will decrease with stage, at least up to bank-full. Overbank n 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 n value due to leaves and seasonal vegetation. Research has shown, however, that n values are lower for individual shrubs with leaves than for the shrubs without leaves.[7] 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.[8]
In open channels, the Darcy–Weisbach equation 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 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 n varies with the flow depth in partially filled circular pipes.[9] 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.[10] These equations account for the variation of n with the depth of flow in accordance with the curves presented by Camp.
↑ Gauckler, Ph. (1867), Etudes Théoriques et Pratiques sur l'Ecoulement et le Mouvement des Eaux, vol.Tome 64, Paris, France: Comptes Rendues de l'Académie des Sciences, pp.818–822
↑ Manning, R. (1891). "On the flow of water in open channels and pipes". Transactions of the Institution of Civil Engineers of Ireland. 20: 161–207.
↑ Freeman, Gary E.; Copeland, Ronald R.; Rahmeyer, William; Derrick, David L. (1998). Field Determination of Manning'snValue for Shrubs and Woody Vegetation. pp.48–53. doi:10.1061/40382(1998)7. ISBN978-0-7844-0382-2.{{cite book}}: |journal= ignored (help)
↑ Camp, T. R. (1946). "Design of Sewers to Facilitate Flow". Sewage Works Journal. 18 (1): 3–16. JSTOR25030187. PMID21011592.
↑ Akgiray, Ömer (2005). "Explicit solutions of the Manning equation for partially filled circular pipes". Canadian Journal of Civil Engineering. 32 (3): 490–499. doi:10.1139/l05-001. ISSN0315-1468.
In fluid dynamics, the Darcy–Weisbach equation is an 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 incompressible fluid. The equation is named after Henry Darcy and Julius Weisbach. Currently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the Moody diagram or Colebrook equation.
Robert Manning was an Irish hydraulic engineer best known for creation of the Manning formula.
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In fluid mechanics and hydraulics, open-channel flow is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways but differ in one important respect: open-channel flow has a free surface, whereas pipe flow does not, resulting in flow dominated by gravity but not hydraulic pressure.
The Fanning friction factor, named after John Thomas Fanning, is a dimensionless number used as a local parameter in continuum mechanics calculations. It is defined as the ratio between the local shear stress and the local flow kinetic energy density:
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The term friction loss has a number of different meanings, depending on its context.
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In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the Darcy friction factor, a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open-channel flow.
In fluid mechanics, pipe flow is a type of liquid flow within a closed conduit, such as a pipe or tube. The other type of flow within a conduit is open channel flow. These two types of flow are similar in many ways, but differ in one important aspect. Pipe flow does not have a free surface which is found in open-channel flow. Pipe flow, being confined within closed conduit, does not exert direct atmospheric pressure, but does exert hydraulic pressure on the conduit.
In engineering, the Moody chart or Moody diagram is a graph in non-dimensional form that relates the Darcy–Weisbach friction factor fD, Reynolds number Re, and surface roughness for fully developed flow in a circular pipe. It can be used to predict pressure drop or flow rate down such a pipe.
The Chézy formula is an semi-empirical resistance equation which estimates mean flow velocity in open channel conduits. The relationship was realized and developed in 1768 by French physicist and engineer Antoine de Chézy (1718–1798) while designing Paris's water canal system. Chézy discovered a similarity parameter that could be used for estimating flow characteristics in one channel based on the measurements of another. The Chézy formula relates the flow of water through an open channel with the channel's dimensions and slope. The Chézy equation is a pioneering formula in the field of fluid mechanics and was expanded and modified by Irish Engineer Robert Manning in 1889. Manning's modifications to the Chézy formula allowed the entire similarity parameter to be calculated by channel characteristics rather than by experimental measurements. Today, the Chézy and Manning equations continue to accurately estimate open channel fluid flow and are standard formulas in all fields that relate to fluid mechanics and hydraulics, including physics, mechanical engineering and civil engineering.
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In fluid mechanics, the Reynolds number is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers, flows tend to be turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow. These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation.
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Albert Strickler was a Swiss mechnical engineer recognized for contributions to our understanding of hydraulic roughness in open channel and pipe flow. Strickler proposed that hydraulic roughness could be characterized as a function of measurable surface roughness and described the concept of relative roughness, the ratio of hydraulic radius to surface roughness. He applied these concepts to the development of a dimensionally homogeneous form of the Manning formula.
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