Minimum energy as the general form of critical flow and maximum flow efficiency and for explaining variations in river channel pattern

Abstract

[1] Although the Bélanger-Böss theorem of critical flow has been widely applied in open channel hydraulics, it was derived from the laws governing ideal frictionless flow. This study explores a more general expression of this theorem and examines its applicability to flow with friction and sediment transport. It demonstrates that the theorem can be more generally presented as the principle of minimum energy (PME), with maximum efficiency of energy use and minimum friction or minimum energy dissipation as its equivalents. Critical flow depth under frictionless conditions, the best hydraulic section where friction is introduced, and the most efficient alluvial channel geometry where both friction and sediment transport apply are all shown to be the products of PME. Because PME in liquids characterizes the stationary state of motion in solid materials, flow tends to rapidly expend excess energy when more than minimally demanded energy is available. This leads to the formation of relatively stable but dynamic energy-consuming meandering and braided channel planforms and explains the existence of various extremal hypotheses.

Altered Ecologies: Fire, climate and human influence on terrestrial landscapes: Terra Australis 32

“…Despite remarkably frequent bankfull flow, they transport almost no bedload (Nanson 2006) and therefore exhibit low width/depth ratios close to the optimum for transporting water alone (Chow 1959;Huang et al 2002;Huang et al 2004). That these channels frequently accommodate bankfull flow and rarely flood overbank prompted Nanson (2006) to suggest that they may be self-forming systems adjusted to contemporary flow conditions.…”

“…While Huang et al (2004) suggested that, despite such narrow geometries of ~2 being theoretically optimal for sediment-free rectangular channels in equilibrium, insufficient bank strength in natural streams greatly limits their occurrence. Nanson et al (2010) have shown that peatland channels can indeed provide such optimal conditions.…”

Least action principle, equilibrium states, iterative adjustment and the stability of alluvial channels

“…For an example of a quantum system, electrons achieve a 'stationary' state when, even though the electrons are not stationary, the system's total energy reaches a minimum above that of minimum potential energy. As shown by Huang et al (2004a), the stationary equilibrium state in an open channel (E = E min ) is where the available energy is just able to transport the fluid and sediment load. Unlike a pendulum, but as in quantum mechanics, in this situation a river is not stationary or unable to flow; however, the concept is the same as for solid mechanics, hence the acquired terminology.…”

“…As a result, part of this minimum amount of potential energy must be converted into kinetic energy, leading the principle of minimum potential energy to be more specifically expressed as the principle of minimum (total) (specific) energy (PmE). As demonstrated by Huang et al (2004a), PmE applies to all kinds of open channel flow: ideal frictionless or frictional flow in nonadjustable channels or flow in adjustable alluvial channels that transport sediment. In solid mechanics, minimum energy occurs when the material remains stationary.…”

“…For flow in an open channel, Huang and Nanson (2000, 2001 and Huang et al ( , 2004aHuang et al ( , 2004b have examined the applicability of the LAP and found that, due to the action of gravity, the LAP can be written as the principle of minimum potential energy. For transporting water discharge and sediment load, however, flow needs to have a certain amount of kinetic energy to achieve equilibrium.…”