Backwater calculations in open channel. (4) Standard step method depth calculated from distance

In steady gradually-varied open channel flows, the longitudinal free-surface profile is called the backwater profile. It may be predicted using the differential form of the energy equation, called 'backwater equation'. The backwater equation is often rewritten in a finite difference form. In the backwater equation, Sf is the friction slope or longitudinal slope of the total head line (T.H.L.).
Learn more on Flow resistance in open channels - Basic considerations { • Flow resistance in ope... }
A number of integration methods of the backwater equation exist. A most common is the standard step method. Herein, the standard step method 'distance computed from depth' is discussed (Henderson 1966, pp. 126-130; Chanson 2004a, pp. 107-108). This method is simple, extremely reliable and very stable. It is strongly recommended to practising engineers who are not necessarily hydraulic experts.
In the standard step method 'distance computed from depth' (Henderson 1966, pp. 126-130; Chanson 2004a, pp. 107-108), the engineer nominates a series of water depths, from which values of the distance are to be calculated.
In this video, we consider the following application. A spillway is controlled by a broad-crest, followed by a 1V:5H smooth chute, being 100 m wide. For a discharge of 110 m3/s, calculate the backwater profile down the smooth spillway chute. The chute is made of smooth concrete.
The concepts of backwater profiles and backwater calculations are essential to the hydraulic engineering of open channel flows and its practical applications, including flood modelling. These are discussed in the relevant UA-cam video movies in the same Playlist at:
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Fluid mechanics and hydraulics in Hubert Chanson UA-cam channel { / @hubert_chanson }
Fundamentals of open channel hydraulics [Playlist] { • Fundamentals of open c... }
Advanced hydraulics of open channel flows [Playlist] { • Advanced hydraulics of... }
Environmental hydraulics of open channel flows [Playlist] { • Environmental hydrauli... }
Applied hydraulics [Playlist] { • Applied hydrodynamics }
Backwater in open channels { • Backwater in open chan... }
Longitudinal free-surface profiles in gradually-varied open channel flows { • Longitudinal free-surf... }
Backwater profiles in gradually-varied open channel flows { • Backwater profiles in ... }
Flow resistance in open channels (1) Basic considerations { • Flow resistance in ope... }
Backwater calculations in open channel. (1) Presentation
Backwater calculations in open channel. (2) Standard step methods
Backwater calculations in open channel. (3) Standard step method distance calculated from depth
Backwater calculations in open channel. (4) Standard step method depth calculated from distance
Acknowledgements
Ya-Hui CHOU
André CHANSON
References
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. (2004a). "The Hydraulics of Open Channel Flow: An Introduction." Butterworth-Heinemann, 2nd edition, Oxford, UK, 630 pages (ISBN 978 0 7506 5978 9).
CHANSON, H. (2004b). "Environmental Hydraulics of Open Channel Flows." Elsevier-Butterworth-Heinemann, Oxford, UK, 483 pages (ISBN 978 0 7506 6165 2).