Manning formula

The Manning formula, known also as the Gauckler-Manning formula, or Gauckler-Manning-Strickler formula in Europe, is an empirical formula for open channel flow, or free-surface flow driven by gravity. It was first presented by the French engineer Philippe Gauckler and later re-developed by the Irish engineer Robert Manning in 1890. For more than a hundred years, this formula lacked a theoretical derivation. Recently this formula was derived theoretically using the phenomenological theory of turbulence.

The Gauckler-Manning formula states:

$V={\frac {k}{n}}R_{h}^{\frac {2}{3}}\cdot S^{\frac {1}{2}}$

where:

$V$ is the cross-sectional average velocity (ft/s, m/s)

$k$ is a conversion constant equal to 1.486 for U.S. customary units or 1.0 for SI units

$n$ is the Gauckler-Manning coefficient (independent of units)

$R_{h}$ is the hydraulic radius (ft, m)

$S$ is the slope of the water surface or the linear hydraulic head loss (ft/ft, m/m) ($S=h_{f}/L$)

The discharge formula, $Q=A\,V$, can be used to manipulate 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 Gauckler-Manning Formula is used to estimate flow in open channel situations where it is not practical to construct a weir or flume to measure flow with greater accuracy. Error rates of +/- 30% or more are common using the Gauckler-Manning Formula while error rates within +/- 10% are possible with properly constructed weirs or flumes.

Hydraulic radius

The hydraulic radius is a measure of a river-channel flow efficiency. Water speed along the channel depends on its cross-sectional shape (among other factors), and the hydraulic radius is a characterisation of the channel that intends to capture such efficiency. It is defined as the ratio of the channel's cross-sectional area to its perimeter:

$R_{h}={\frac {A}{P}}$

where:

$R_{h}$ is the hydraulic radius (m)

$A$ is the cross sectional area of flow (m²)

$P$ is wetted perimeter (m)

The greater the hydraulic radius, the greater the efficiency of the channel and the less likely the river is to flood. The highest values occur when channels are deep, narrow, and semi-circular in shape.

The hydraulic radius is not half the hydraulic diameter as the name may suggest. It is a function of the shape of the pipe, channel, or river in which the water is flowing. In wide rectangular channels, the hydraulic radius is approximated by the flow depth. The measure of a river's channel efficiency (its ability to move water and sediment) is used by water engineers to assess the likelihood of flooding. The hydraulic radius of a channel is defined as the ratio of its cross-sectional area to its wetted perimeter (the part of the cross-section – bed and bank – that is in contact with the water).

Gauckler-Manning coefficient

The Gauckler-Manning coefficient, often denoted as $n$, is an empirically derived coefficient, which is dependent on many factors, including river-bottom roughness and sinuosity. Often the best method is to use photographs of river channels where $n$ has been determined using Gauckler-Manning's formula.

Values vary greatly in natural stream channels 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. 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 is valid using the hydraulic diameter as equivalent pipe diameter. It is the only sound method to estimate the energy loss in man-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.

References

• Walkowiak,D (Ed.) Open Channel Flow Measurement Handbook, 6th Ed. Teledyne ISCO; 2006 ISBN 0962275735.

See also

• Hydraulics
• Chézy formula
• Darcy–Weisbach equation

External links

• History of the Manning Formula
• Manning formula calculator for several channel shapes
• Manning n values associated with photos