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

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.