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Backwater in open channels

In gradually-varied open channel flows, the free-surface profile is also called 'backwater profile'. It may be predicted using the differential form of the energy equation, called 'backwater equation', first proposed by J.B. Bélanger in 1828.
The backwater calculations are developed assuming a non-uniform-equilibrium flow, a steady flow motion, that the flow is gradually varied, and that, at a given section, the flow resistance is the same as for an uniform flow for the same depth and discharge, regardless of trends of the depth. The backwater equation has two singularities: critical flow conditions and uniform equilibrium flow. Thus the backwater calculations should only be conducted if it is known beforehand that critical flow conditions do not take place anywhere along the channel. Similarly, it should not be applied to rapidly-varied flows, e.g. hydraulic jumps, gates.
The backwater profile is calculated by using simultaneously the continuity equation (i.e. Q = V×A) and the backwater equation. The backwater equation is integrated from a location of known water depth and velocity. The calculations are non-linear usually because of the flow resistance term.
Finally, a video movie shows the establishment of a backwater profile. A downstream gate is partially closed, obstructing a supercritical flow. A hydraulic jump develops upstream of the gate and propagates upstream, until it becomes a stationary hydraulic jump.
Hubert Chanson.

Additional notes
YouTube channel >>
Hubert Chanson
Hubert Chanson
BELANGER, J.B. (1828). "Essai sur la Solution Numérique de quelques Problèmes Relatifs au Mouvement Permanent des Eaux Courantes." ('Essay on the Numerical Solution of Some Problems relative to Steady Flow of Water.') Carilian-Goeury, Paris, France, 38 pages & 5 tables (in French).
HENDERSON, F.M. (1966). "Open Channel Flow." MacMillan Company, New York, USA.
MONTES, J.S. (1998). "Hydraulics of Open Channel Flow." ASCE Press, New-York, USA, 697 pages.
CHANSON, H. (2004). "The Hydraulics of Open Channel Flow: An Introduction." Butterworth-Heinemann, 2nd edition, Oxford, UK, 630 pages (ISBN 978 0 7506 5978 9).

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