E7 Hydraulics of Rivers and Canals

Topic

River and canal hydraulics describe open-channel flow—that is, flow with a free surface in direct contact with the atmosphere, driven by gravity.

The Saint-Venant equations—previously mentioned in the context of floods and hydrological modeling—are the fundamental equations for this type of channel flow; they combine a continuity equation (expressing the conservation of water mass) with a momentum equation in one dimension along the channel. Together, they describe flood wave propagation, the filling and emptying of floodplains, and the flow morphology within the channel itself.

The Froude number—defined as the ratio of the flow velocity to the propagation speed of a small-amplitude surface wave within that same flow—determines the flow regime: when the ratio is less than one, the flow is subcritical, and downstream conditions control its behavior; when the ratio equals one, the flow is critical; and when the ratio exceeds one, the flow is supercritical, meaning upstream conditions control it. Most flow observed in natural rivers is, in fact, subcritical.

Manning's formula relates the mean flow velocity to the channel's geometry, slope, and roughness: this mean velocity increases with the channel's hydraulic radius raised to the two-thirds power and with the square root of the channel slope, while decreasing inversely with a roughness coefficient specific to each channel—a coefficient that is higher the rougher the channel bed and walls are.

Hydraulic geometry, in turn, describes how a river's width, depth, and velocity scale with discharge—both at a single point in the channel over time and along the entire length of the channel from one end to the other. The Leopold and Maddock relationships show that each of these three variables increases according to its own power of the discharge, and that the sum of these three exponents is always equal to one; this means that the entire increase in discharge is distributed—with nothing left over or missing—among the widening, deepening, and acceleration of the river.

Furthermore, meanders are the most common channel pattern in low-gradient rivers transporting a mixed sediment load, and the wavelength of these meanders scales with the channel width, measuring approximately eleven times that width. The expected behavior of this entire system includes a flood wave that propagates downstream at a velocity greater than the mean velocity of the flow itself. Thus, from the free-surface flow with which this document begins to the flood wave—faster than the flow—with which it concludes, and encompassing the Froude number, Manning’s formula, and hydraulic geometry, the hydraulics of rivers and channels dictate that the very form of a river—its width, depth, velocity, and meanders—is determined by the discharge it carries.

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