Backwater calculations in open channel 3 Standard step method distance calculated from depth

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In steady gradually-varied open channel flows, the backwater profile may be predicted using the backwater equation, i.e. the differential form of the energy equation. For numerical integration, the backwater equation is rewritten in a finite difference form. • Learn more on Backwater in open channels { • Backwater in open channels } • Several integration methods of the backwater equation exist. A common one is the standard step method 'depth calculated from distance'. In this method, i.e. standard step method 'depth calculated from distance' (Henderson 1966, pp. 136-140; Chanson 2004a, pp. 284-286), the engineer selects a series of longitudinal locations, where the water depth is calculated by trial-and-error (i.e. iteratively) or graphically. At each integration step, the water depth di+1 is selected iteratively until the above equation is fulfilled for the selection locations xi and xi+1 (Chanson 2004a, pp. 285-286). • The following application presents an application of a smooth spillway chute controlled by a broad-crest. The chute slope is 1V:5H (11.3) and the chute is 100 m wide. For a discharge of 110 m3/s, let us calculate the backwater profile down the smooth spillway chute. The chute is made of smooth concrete. • Finally the step method 'depth calculated from distance’ (used in several softwares) assumes a mild slope with subcritical flow or steep slope with supercritical flow. It becomes indeterminate for critical flow conditions! • 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 Youtube video movies in the same Playlist at: • { / @hubert_chanson } • Fluid mechanics and hydraulics in Hubert Chanson Youtube channel { / @hubert_chanson } • Fundamentals of open channel hydraulics [Playlist] { • Fundamentals of open channel hydraulics } • Advanced hydraulics of open channel flows [Playlist] { • Advanced hydraulics of open channel f... } • Environmental hydraulics of open channel flows [Playlist] { • Environmental hydraulics of open chan... } • Applied hydraulics [Playlist] { • Applied hydrodynamics } • Backwater in open channels { • Backwater in open channels } • Longitudinal free-surface profiles in gradually-varied open channel flows { • Longitudinal free-surface profiles in... } • Backwater profiles in gradually-varied open channel flows { • Backwater profiles in gradually-varie... } • Flow resistance in open channels (1) Basic considerations { • Flow resistance in open channels (1) ... } • 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).

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