Abstract
Despite decades of effort, stable hydraulic geometry for an open channel water flow has hardly been established because of too many unknown variables for too few rational relationships. This article derives the most efficient channel cross section using calculus of variations for the given flow area at the minimum wetting perimeter length, which is equivalent to the principle of least action. Analysis indicates that water can most efficiently flow in a semi-ellipse section channel with minimum friction and erosion. Anisotropy in channel erodibility was found to be necessary in the natural stable channel characterization because gravitation force and channel bank consolidation cannot be ignored in earth surface material. This channel cross section, based on the principle of least action, may be regarded as the theoretical stable hydraulic section for erodible bed, which was comparable to the observed river cross-sections during high flow periods.
Highlights
A standard characterization of the river channel section, known as regime equation, was established early in the1950s using power functions of discharge Q, as follows:
This study explores theoretical stable hydraulic section based on the principle of least action using the variational calculus approach
Field observation of the stable hydraulic section is extremely difficult because the natural channel is never free from secondary flow due to meandering channel geometry and unsteady flow due to hydrological forcing
Summary
Stable hydraulic geometry for an open channel water flow has hardly been established because of too many unknown variables for too few rational relationships. This article derives the most efficient channel cross section using calculus of variations for the given flow area at the minimum wetting perimeter length, which is equivalent to the principle of least action. There are too many unknowns (e.g. water depth, river width, channel slope, roughness, and river cross-sectional geometry) and too few equations (e.g. mass conservation, friction law, and sediment transport equation) even for a steady uniform flow Additional relationships such as Maximum Flow Efficiency (MFE)[13], Maximum Sediment Transport Capacity (MSTC)[14], and Minimum Stream Power (MSP)[15], have been proposed since the 1950s to solve the problem. It is important that neither empirical flow-resistance relationship (e.g. Manning formula) nor sediment transport formulas (e.g. DuBoys formula) were used to derive the optimum cross section function
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