Several hydrodynamic theories have been employed for deriving downstream hydraulic geometry relations of width, depth, velocity, and slope in terms of flow discharge. Five theories, the Smith theory, the Julien-Wargadalam (JW) theory, the Parker theory, the Griffiths theory, and the Ackers theory, are discussed in this chapter. These theories employ different forms of the continuity equation, friction equation, and transport equations. The Smith hydrodynamic theory also uses a morphological relation, whereas the JW theory uses an angle between transversal and downstream shear stress components, and the Parker theory uses a depth function.

Empirical equations of downstream hydraulic geometry, entailing width, depth, velocity, and bed slope, can be derived using the scaling theory. The theory employs the momentum equation, a flow resistance formula, and continuity equation for gradually varied open channel flow. The scaling equations are expressed as power functions of water discharge and bed sediment size, and are applicable to alluvial, ice, and bedrock channels. These equations are valid for any value of water discharge as opposed to just mean or bank-full values that are used in empirical equations. This chapter discusses the use of scaling theory for the derivation of downstream hydraulic geometry. The scaling theory-based hydraulic geometry equations are also compared with those derived using the regime theory, threshold theory, and stability index theory, and the equations are found to be consistent.

This chapter derives the at-a-station as well as the downstream hydraulic geometry using the tractive force theory. Here the threshold discharge does not necessarily correspond to the bankfull discharge for at-a-station hydraulic geometry. At a given section, there can be a threshold discharge where the channel is flowing partially. .

Hydraulic geometry is a quantitative description of the variation of river characteristics with variation in discharge and sediment load. It is impacted by climate, geology, and human interference. Hydraulic geometry relations have been expressed in power form and have been derived using a multitude of hypotheses. These relations play a fundamental role in the design of alluvial canals, river training works, and watershed management. The objective of this chapter is to introduce preliminary concepts that are deemed important for understanding different aspects of hydraulic geometry.